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.     

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.     

λ-Convergence of multiple Walsh–Paley series and sets of uniqueness

Mathematical Notes - λ-convergent multiple Walsh–Paley series on a multidimensional dyadic group are studied. It is proved that, for all λ > 1, any arbitrary finite union of...     

Electromagnetic Excitation of a Spherical Medium by an Arbitrary Dipole and Related Inverse Problems

A piecewise homogeneous spherical medium is excited by an external or internal electric dipole with arbitrary location and polarization. The dyadic Green's function of the medium is determined analytically. Then, the vector electric fields and far‐field patterns are obtained. Low‐frequency approximations of the far‐field patterns are subsequently derived, which encode the dipole's locations coordinates and polarization components in the different orders of the associated expansions. This fact enables the establishment of far‐field inverse scattering algorithms referring to the electromagnetic interior or exterior excitation of a small sphere by an arbitrary dipole. Inverse medium and inverse source problems are considered concerning, respectively, the determination of the scatterer's material parameters and the dipole's characteristics. The developed inverse algorithms determine exactly the unknown parameters of problems fulfilling the low‐frequency assumption, which is indeed the case in most relevant applications, like, e.g., in biomedical imaging.     

Dedekind–MacNeille Completion of n-ordered Sets

A completion of an n-ordered set is defined, by analogy with the case of posets (2-ordered sets), as a pair , where Q is a complete n-lattice and is an n-order embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-ordered set. The completion reduces to the Dedekind–MacNeille completion in the dyadic case, the case of posets. A characterization theorem is provided, analogous to the well-known dyadic one, for the case of joined n-ordered sets. The condition of joinedness is trivial in the dyadic case and, therefore, this characterization theorem generalizes the uniqueness theorem for the Dedekind–MacNeille completion of an arbitrary poset.      

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.     

The mean value of the product of class numbers of paired quadratic fields III

This is the final part in a series of three papers. In this part, we evaluate the previously unevaluated local densities at dyadic places that appear in the density theorem stated in the first part. For this purpose we introduce an invariant, the level, attached to a pair of ramified quadratic extensions of a dyadic local field. This invariant measures how close the fields are in their arithmetic properties and its evaluation may be of interest independent of its application here.     

具有重结点的B-样条曲面G1的连续条件

本文得到了关于两个双三次内部重结点B-样条曲面片G1连续的充分必要条件和在公共边界线上控制向量的本征条件.这些条件直接由两个B-样条曲面的控制向量表示.文[10]证明了使用内部单结点的双三次B-样条曲面来构造G1光滑曲面,局部格式不存在.使用本文的这些条件就可以构造出具有局部各式的G1光滑造型.     

On the structure invariants of proper rational matrices with prescribed finite poles

Abstract

The algebraic structure of matrices defined over arbitrary fields whose elements are rational functions with no poles at infinity and prescribed finite poles is studied. Under certain very general conditions, they are shown to be matrices over an Euclidean domain that can be classified according to the corresponding invariant factors. The relationship between these invariants and the local Wiener–Hopf factorization indices will be clarified. This result can be seen as an extension of the classical theorem on pole placement by Rosenbrock in control theory.     

Counting codes over rings

We extend the definition of free codes to codes over local rings and arbitrary Frobenius rings. The number of free codes over finite Frobenius rings is determined by calculating the number for local rings and applying the Chinese Remainder Theorem. A formula for the number of codes of arbitrary type over a finite chain ring is given and this is applied to determine the number of linear codes over a finite principal ideal ring.     

Criteria for monotonicity properties of Musielak-Orlicz spaces equipped with the Amemiya norm

Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity, and uniform monotonicity of Musielak-Orlicz spaces over any σ-finite and complete measure space, endowed with the Amemiya norm are given. The fact that the spaces are considered over arbitrary σ-finite measure space is essential because, as it is shown in Example 3, the Musielak-Orlicz spaces need not be strictly monotone even if their restrictions to the nonatomic part and the purely atomic part are strictly monotone.     

具有重结点的B-样条曲面G1的连续条件

本文得到了关于两个双三次内部重结点B-样条曲面片G¹连续的充分必要条件和在公共边界线上控制向量的本征条件.这些条件直接由两个B-样条曲面的控制向量表示.文[10]证明了使用内部单结点的双三次B-样条曲面来构造G¹光滑曲面,局部格式不存在.使用本文的这些条件就可以构造出具有局部各式的G¹光滑造型.     

Scalar restriction for formal groups

The aim of this work is to generalize the results on the scalar restriction functor for formal groups stated in the paper “Explicit classification of formal groups over local fields” by the first two authors. These results are extended to formal groups over an arbitrary commutative ring, which we assume to be an integral domain or a ring of characteristic zero if necessary. Bibliography: 6 titles. Deceased. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 59–70.     

Local decoding and testing of polynomials over grids

We study the local decodability and (tolerant) local testability of low‐degree n‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}ⁿ. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals.     

Nonlinear approximation schemes associated with nonseparable wavelet bi-frames

In the present paper, we study nonlinear approximation properties of multivariate wavelet bi-frames. For a certain range of parameters, the approximation classes associated with best N-term approximation are determined to be Besov spaces and thresholding the wavelet bi-frame expansion realizes the approximation rate. Our findings extend results about dyadic wavelets to more general scalings. Finally, we verify that the required linear independence assumption is satisfied for particular families of nondyadic wavelet bi-frames in arbitrary dimensions.     

Jacquet functors and unrefined minimal K-types

The notion of an unrefined minimal K-type is extended to an arbitrary reductive group over a non archimedean local field. This allows one to define the depth of a representation. The relationship between unrefined minimal K-types and the functors of Jacquet is determined. Analogues of fundamental results of Borel are proved for representations of depth zero.     

On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.     

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.