Fast convolutions meet Montgomery

Arithmetic in large ring and field extensions is an important problem of symbolic computation, and it consists essentially of the combination of one multiplication and one division in the underlying ring. Methods are known for replacing one division by two short multiplications in the underlying ring, which can be performed essentially by using convolutions.

However, while using school-book multiplication, modular multiplication may be grouped into operations (where denotes the number of operations of one multiplication in the underlying ring), the short multiplication problem is an important obstruction to convolution. It raises the costs in that case to . In this paper we give a method for understanding and bypassing this problem, thus reducing the costs of ring arithmetic to roughly when also using fast convolutions. The algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper.

     

Hilbert transforms and maximal functions along variable flat curves

We study certain Hilbert transforms and maximal functions along variable flat curves in the plane. We obtain their boundedness by considering the oscillatory singular integrals which arise from an application of a partial Fourier transform.

     

The Berezin transform and radial operators

We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial symbol.

     

Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domains

This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with $\alpha\in(0,1)$. We prove the existence and uniqueness of tempered pullback random attractors for the equations in $L^{2}(\mathbf{R}^{3})$. In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in $L^{2}(\mathbf{R}^{3})$ by the tail-estimates of solutions.     

Wavelet bases in

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of . These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces . Moreover, -free vector wavelets are constructed and analysed. The relationship between and are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains . As an application, we obtain wavelet multilevel preconditioners in and .

     

A counterexample to a weak-type estimate for potential spaces and tangential approach regions

We show that for every potential space , there exists an approach region for which the associated maximal function is of weak-type, but the boundedness for the completed region is false, which is in contrast with the nontangential case.

     

Powers and roots of Toeplitz operators

We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in which commute with Toeplitz operators whose symbols are of the form .

     

The flat model structure on complexes of sheaves

Let be the category of chain complexes of -modules on a topological space (where is a sheaf of rings on ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on . As a corollary, we have a general framework for doing homological algebra in the category of -modules. I.e., we have a natural way to define the functors and in .

     

Uniqueness of left invariant symplectic structures on the affine Lie group

We show the uniqueness of left invariant symplectic structures on the affine Lie group under the adjoint action of , by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with , and also by giving an unexpected identity on it which relates two left invariant symplectic structures. As an application of this result, we classify maximum rank left invariant Poisson structures on the simple Lie groups and . This result is a generalization of Stolin's classification of constant solutions of the classical Yang-Baxter equation for and .

     

The zero set of semi-invariants for extended Dynkin quivers

We show that the set of common zeros of all semi-invariants vanishing at 0 on the variety of all representations with dimension vector of an extended Dynkin quiver under the group is a complete intersection if is ``big enough'. In case does not contain an open -orbit, which is the case not considered so far, the number of irreducible components of grows with , except if is an oriented cycle.

     

Completeness in the set of wavelets

We study the completeness properties of the set of wavelets in . It is well-known that this set is not closed in the unit ball of . However, if one considers the metric inherited as a subspace (in the Fourier transform side) of , we do obtain a complete metric space.

     

On the location of the essential spectrum of Schrödinger operators

We give estimates on the bottom of the essential spectrum of Schrödinger operators in .

     

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

Two estimates for curves in the plane

We obtain a Fourier transform estimate and an convolution estimate for certain measures on a class of convex curves in the plane.

     

Embeddings of some classical Banach spaces into modulation spaces

We give sufficient conditions for a tempered distribution to belong to certain modulation spaces by showing embeddings of some Besov-Triebel-Lizorkin spaces into modulation spaces. As a consequence we have a new proof that the Hölder-Lipschitz space embeds into the modulation space when d$">. This embedding plays an important role in interpreting recent modulation space approaches to pseudodifferential operators.

     

Linear independence of time-frequency translates

The refinement equation plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates of , it is natural to ask if there exist similar dependencies among the time-frequency translates of . In other words, what is the effect of replacing the group representation of induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection , the set of all functions such that is independent is an open, dense subset of . It is conjectured that this set is all of .

     

Notes on a -group generated by the Lévy Laplacian

In this paper we shall give some results on a -group generated by the Lévy Laplacian and operators approximating that group in the space of continuous linear operators defined on a certain locally convex space in

     

Segal-Bargmann空间上带无界符号的Toeplitz算子

In this paper,we construct a function φ in L2(Cn,d Vα) which is unbounded on any neighborhood of each point in Cnsuch that Tφ is a trace class operator on the SegalBargmann space H2(Cn,d Vα).In addition,we also characterize the Schatten p-class Toeplitz operators with positive measure symbols on H2(Cn,d Vα).     

Algebraic gamma monomials and double coverings of cyclotomic fields

We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex , to compute , where is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension , we define a double covering of to be an extension of degree , such that is Galois. We demonstrate that each class gives rise to a double covering of , by . When lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of is abelian and hence . The may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.