Optimal quantization for dyadic homogeneous Cantor distributions

For a large class of dyadic homogeneous Cantor distributions in ?, which are not necessarily self‐similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non‐existence of the quantization coefficient. The class contains all self‐similar dyadic Cantor distributions, with contraction factor less than or equal to 1/3. For these distributions we calculate the quantization errors explicitly. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A singular measure on the Cantor group

Let Ω=N{−1,1} and {ωj} be independent random variables taking values in {−1,1} with equal probability. Endowed with the product topology and under the operation of pointwise product, Ω is a compact Abelian group, the so-called Cantor group. Let a,b,c be real numbers with 1+a+b+c>0, 1+a−b−c>0, 1−a+b−c>0 and 1−a−b+c>0. Finite products on Ω,

     

Examples of extensions with Demushkin group

Examples are given of fields of algebraic numbers for which the Galois group of a maximal -extension with ramification at the divisors ofl is a Demushkin group.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 146–149, 1980.     

Examples of non-compact quantum group actions

We present two examples of actions of non-regular locally compact quantum groups on their homogeneous spaces. The homogeneous spaces are defined in a way specific to these examples, but the definitions we use have the advantage of being expressed in purely C^∗-algebraic language. We also discuss continuity of the obtained actions. Finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups.     

On dyadic analysis based on the pointwise dyadic derivative

В пРЕДыДУЩИх РАБОтАх АВтОРы В ОсНОВНОМ РАж ВИВАлИ ДВОИЧНыИ АНАлИж, ОсНО ВАННыИ НА пОНьтИИ сИльНОИ ДВ ОИЧНОИ пРОИжВОДНОИ Д ль ФУНкцИИ, ОпРЕДЕлЕННых НА ДИАД ИЧЕскОИ ГРУппЕ ИлИ НА [0,1) с пЕРИО ДОМ 1. цЕльУ НАстОьЩЕИ Р АБОты ьВльЕтсь пОстРОЕНИЕ ДВОИЧНОгО ДИФФЕРЕНцИАльНОгО И ИНтЕгРАльНОгО ИсЧИс лЕНИИ НА ОсНОВЕ БОлЕЕ слОжНОг О, НО жАтО И БОлЕЕ клАссИЧЕскОгО пОНьт Иь ДВОИЧНОИ пРОИжВОД НОИ В тОЧкЕ. ИсслЕДУУтсь тЕ пРОст РАНстВА ФУНкцИИ, Дль кОтОРых пРИМЕНИМ ДВОИЧНыИ АНАлИж, А тАк жЕ ОпРЕДЕльУтсь гРАНИц ы ЕгО пРИМЕНИМОстИ. тАк ОкАжАлОсь, ЧтО пРО стРАНстВОL ^p(0, l), 1≦∞, ьВльЕ тсь БОлЕЕ ЕстЕстВЕННыМ п РОстРАН стВОМ Дль пОстРОЕНИь ДВОИЧНОгО АНАлИжА, ЧЕ М клАссИЧЕскОЕ пРОстР АНстВОс[0,1]. НАпРИМЕР, ЕслИ пЕРВАь ДВОИЧНАь пРОИжВОДНАь пРИНАДл ЕжИтс[0,1], тОf=const. с ДРУгОИ стОРОНы, ЕслИfεс[0,1], тО ДВОИЧНыИ ИНтЕгРАл, пОстРОЕННы И Дльf, НЕ пРИНАДлЕжИтс[0,1]. Уст АНОВлЕНО тАкжЕ, ЧтО сИльНАь ДВО ИЧНАь пРОИжВОДНАь И Д ВОИЧНАь пРОИжВОДНАь В тОЧкЕ с ОВпАДАУт пОЧтИ ВсУДУ Дль ФУНкцИИ, пРИ НАДлЕжАЩИх ОпРЕДЕлЕ ННОМУ пОДклАссУL ^p[0, 1]. пОлУЧЕННыЕ РЕжУльтА ты пРИМЕНьУтсь к пОЧл ЕННОМУ ДИФФЕРЕНцИРОВАНИУ И ИНтЕгРИРОВАНИУ РьДОВ пО сИстЕМЕ УОлш А, к ОцЕНкАМ ВЕлИЧИН кОЁФФИцИЕНтОВ ФУРьЕ-УОлшА, к ДОкАжАтЕльст ВУ АНАлОгА ОсНОВНОИ тЕО РЕМы О НАИлУЧшЕМ пРИБ лИжЕНИИ Дль пОлИНОМОВ пО сИстЕМЕ УОлшА, А тАкжЕ к РЕшЕНИ У ДВОИЧНОгО ВОлНОВОг О УРАВНЕНИь.     

Moving frames and singularities of prolonged group actions

The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie's theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of "totally singular points," and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vector-valued functions, and methods for the solution to Lie determinant equations.     

Cantor集合上的连续自映射

本文证明了Ｃａｎｔｏｒ集Ｃ上所有拓扑共轭于（单边的）有限型子移位的连续自映射的集合在H中稠密，此外Ｃ上每个拓扑传递（混合）映射均可被拓扑共轭于有限型子移位的拓仆传递（混合）映射一致逼近．     

Asymptotic existence theorems for frames and group divisible designs

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs.     

Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $[0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $[0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.