Intuitive dyadic calculus: The basics

This article is divided into two parts. In the first part we present a general theory of the dyadic lattices. In the second part we show several applications of this theory to harmonic analysis: a decomposition of an arbitrary measurable function in terms of its local mean oscillations, and a pointwise bound of Calderón–Zygmund operators by sparse operators.     

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.     

Integral spinor norms in dyadic local fields III

In this paper we give an alternative computation of integral spinor norms over dyadic local fields by using the Jordan decomposition of W-type. In particular, we emphasize the striking similarity between the theory over dyadic local fields and that over the local fields of characteristic 2.     

Integral spinor norms in dyadic local fields Ⅲ

In this paper we give an alternative computation of integral spinor norms over dyadic local fields by using the Jordan decomposition of W-type.In particular,we emphasize the striking similarity between the theory over dyadic local fields and that over the local fields of characteristic 2.     

Ghost length, cone length and complete level of DG modules

In this paper, we introduce a new numerical invariant complete level for a DG module over a local chain DG algebra and give a characterization of it in terms of ghost length. We also study some of its upper bounds. The cone length of a DG module is an invariaut closely related with the invariant level. We discover some important results on it.     

On Characterization of Nonuniform Tight Wavelet Frames on Local Fields

In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic.     

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.     

Self-Conformal Multifractal Measures

A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure.     

Dyadic Diaphony of Digital Nets Over ℤ<Subscript>2</Subscript>

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ₂. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.The first author is supported by the Australian Research Council under its Center of Excellence Program.The second author is supported by the Austrian Research Foundation (FWF), Project S 8305 and Project P17022-N12.     

Embedding Toeplitz systems in triangular maps: The last but one problem of the Sharkovsky classification program

We give an example of a triangular map of the unit square containing a minimal Li–Yorke chaotic set and such that, in the whole system, there are no DC3-pairs. This solves the last but one problem of the Sharkovsky program of classification of triangular maps. We use completely new methods, in fact we show that every zero-dimensional almost 1–1 extension of the dyadic odometer can be realized as the unique nonperiodic minimal set in a triangular map of type 2^∞. In case of a regular Toeplitz system we can additionally arrange that all invariant measures are supported by minimal sets.     

Dyadic Diaphony of Digital Nets Over ?2

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ₂. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the L₂{\cal L}_2 discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.     

Integral spinor norm groups over dyadic local fields

The spinor norms of integral rotations of an arbitrary quadratic lattice over an arbitrary dyadic local field are determined. The results are given in terms of BONGs, short for “bases of norm generators”. This approach provides a new way to describe lattices over dyadic local fields.     

CONSTRUCTIONS OF THE INVARIANT SETS AND INVARIANT MEASURES

In this paper, we will discuss the constructiOn problems about the invariant sets and invariant measures of continues maps~ which map complexes into themselves, using simplical approximation and Markov cbeirs. In [7], the author defined a matrix by using r-normal subdivi of the w,dimensional unit cube, considered it a Markov matrix, and constructed the invariantset and invafiant measure, In fact, the matrix he defined is not Markov matrix generally. So wewill give [7] and amendment in the last pert of this paper. We also construct an invariant set thatis the chain-recurrent set of the map by means of a non-negative matrix which only depends on themap. At hst, we will prove the higher dimension?Banach variation formuls that can simplify thetransition matrix.     

Optimal use of components of a dyadic wavelet representation

In practice, one may encounter linear transforms whose result is important for a certain set of vector components only. In particular this relates to vector components restored from its dyadic wavelet representation. In this paper we study optimization of the index set of the dyadic wavelet representation which can be used to restore required original vector components.     

Invariant properties of representations under cleft extensions

The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.     

On the existence of H-surfaces into Riemannian manifolds

This paper considers the existence of a local minimizer of a conformally invariant functional defined on a space of maps of a closed Riemann surface into a compact Riemannian manifold . The functional is defined for a given tensor on of type (1,2) and we call its extremal an -surface. In fact, we prove that there exists a local minimizer of the functional in a given homotopy class under certain conditions on , and the minimum of the Dirichlet integral of maps of the homotopy class. Received January 21, 1994 / Received in revised form October 24, 1995 / Accepted March 15, 1996     

Multifractal and Multi-Multifractal of One-Dimensional Expanding Markov Maps

In measure theory, one is interested in local behaviours, for example in local dimensions, local entropies or local Lyapunov exponents. It has been relevant to study dynamical systems where one can develop further the study of multifractal and multi-multifractal, particularly when there exist strange attractors or repellers. Multifractal and multi-multifractal refer to a notion of size, which emphasizes the local variations of different values coming from the theory of dynamical systems and generated by the dimension theory of invariant measures. This paper gives some part of the literature in this field. Many results are already known, but the large deviations approach allows us to reprove these results and to obtain quite easily results concerning extremal points and extremal measures.     

Pointwise convergence to the initial data for nonlocal dyadic diffusions

We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in ?⁺. First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator. As a consequence we obtain the pointwise convergence to the initial data.