Some Characterizations of Left Weakly Regular Ordered Semigroups

In this paper, the notion of left weakly regular ordered semigroups is introduced. Furthermore, left weakly regular ordered semigroups are characterized by the properties of their left ideals, right ideals and （generalized） bi-ideals, and also by the properties of their fuzzy left ideals, fuzzy right ideals and fuzzy （generalized） bi-ideals.     

Principal Quasi-Baerness of Rings of Skew Generalized Power Series

Let R be a ring and （S, 〈） be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,（R） has a generalized join in S,（R）. Several known results follow as consequences of our results.     

形式三角矩阵的PP和半遗传环(英文)

In this paper the sufficient and necessary conditions are given for a formal triangular matrix ring to be right PP, generalized right PP, or semihereditary, respectively.     

Measures not charging polar sets and Schrödinger equations in L p

We study the SchrSdinger equation （q -￡）u ＋μu = f, where ￡ is the generator of a Borel right process and μ is a signed measure on the state space. We prove the existence and uniqueness results in Lp, 1 ≤p 〈∞ . Since we consider measures μcharging no polar set, we have to use new tools： the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation （subordination） operators （in the sense of G Mokobodzki） induced by the regular strongly supermedian kernels. We extend the results on the SchrSdinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on Lp. If the measure μ is positive then the perturbed process solves the martingale problem for ￡- μ and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having μ as Revuz measure.     

Zip模(英文)

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.     

Extensions of zip rings

A ring R is called right zip provided that if the right annihilator r_R(X) of a subset X of R is zero, r_R(Y)=0 for a finite subset Y⊆X. Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; Characterize a ring R such that Mat_n(R) is right zip; When does R being a right zip ring imply R[G] being right zip when G is a finite group? In this note, we continue the study of the extensions of noncommutative zip rings based on Faith's questions.     

On zip and weak zip rings of skew generalized power series

In this paper we show under certain conditions that the skew generalized power series R[[S, w]] is a right zip (weak zip) ring if and only if R is a right zip (weak zip) ring.     

Counting the number of isotone selfmappings of crowns

Letn>1. The number of all strictly increasing selfmappings of a 2n-element crown is . The number of all order-preserving selfmappings of a 2n-element crown is

Tests for the convergence of continued fractions,based on the fundamental system of inequalities

We have proved that if the partial numerators of the continued fraction f(c)=1/1+c₂/l+c₃/l+... are all nonzero and for at least some number n?1 satisfy the inequalities $$p_n \left| {1 + c_n + c_{n + 1} } \right| \ge p_{n - 2} p_n \left| {c_n } \right| + \left| {c_{n + 1} } \right|(n \ge 1,p_{ - 1} = p_0 = c_1 = 0,p_n \ge 0),$$ then f(c) converges in the wide sense if and only if at least one of the series $$\begin{array}{l} \sum\nolimits_{n = 1}^\infty {\left| {c_3 c_5 \ldots c_{2n - 1} /(c_2 c_4 \ldots c_{2n} )} \right|} , \\ \sum\nolimits_{n = 1}^\infty {\left| {c_2 c_4 \ldots c_{2n} /(c_3 c_5 \ldots c_{2n + 1} )} \right|} \\ \end{array}$$      

On some results of Hua in short intervals

In this note, we prove some results of Hua in short intervals. For example, each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k}}, \hfill \\ {\left| {{p_j} - \sqrt {N/5} } \right| \leqslant U,\left| {p - {{\left( {N/5} \right)}^{\tfrac{1}{k}}}} \right|\leqslant UN - \tfrac{1}{2} + \tfrac{1}{k},j = 1,2,3,4,} \hfill \\ \end{array} } \right. $

where \( U = N\tfrac{1}{2} - \eta + \varepsilon \) with \( \eta = \frac{2}{{\kappa \left( {K + 1} \right)\left( {{K^2} + 2} \right)}} \) and \( K = {2^{k - 1}},k\geqslant 3. \)

     

A note on zip rings

A ring R is called right zip if the right annihilator r _R (S) of a subset S of R is zero, r _R (X)=0 for a finite subset X of S. In this note we will prove that for any u.p.-monoid M a right uniform ring R is right zip if and only if the monoid ring R[M] is right zip.     

An explicit formula for |FM(1+1+n)|

We show that the number of elements in FM(1+1+n), the modular lattice freely generated by two single elements and an n-element chain, is 1 $$\frac{1}{{6\sqrt 2 }}\sum\limits_{k = 0}^{n + 1} {\left[ {2\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {2k} \\ {k - 2} \\ \end{array} } \right)} \right]} \left( {\lambda _1^{n - k + 2} - \lambda _2^{n - k + 2} } \right) - 2$$ , where \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) .     

The formula for the number of order-preserving selfmappings of a fence

LetY be a fence of sizem andr=?m?1/2?. The numberb(m) of order-preserving selfmappings ofY is equal toA _r-B_r-C_r-D_r, where, ifm is odd, $$\begin{gathered} A_r = 2(r + 1)\sum\limits_{s = 0}^r {\left( {\begin{array}{*{20}c} {r + s} \\ {2s} \\ \end{array} } \right)} 4^s , B_r = 2r\sum\limits_{s = 1}^r {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ {s - 1} \\ \end{array} } \right),} \hfill \\ C_r = 4r\sum\limits_{s = 0}^{r - 1} {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right), D_r = \sum\limits_{s = 0}^{r - 1} {(2s + 1)} \left( {\begin{array}{*{20}c} {r + s - 1} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right)} \hfill \\ \end{gathered} $$ . Ifm is even, a similar formula forb(m) is true. The key trick in the proof is a one-to-one correspondence between order-preserving selfmappings ofY and pairs consisted of a partition ofY and a strictly increasing mapping of a subfence ofY toY.     

General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Padé approximants. Examples include the multivariate forms of the exponential and the -exponential functions

and

as well as the Appell function

and the multivariate form of the partial theta function

     

I sottogruppi del gruppo modulare con coefficienti del corpo di Jacobi-Eisenstein e un teorema sui gruppi finiti

Sommario Introduzione — § 1 – 1. L'indice μ(n) dei sottogruppi Г_μ(n) del gruppo Γ di sostituzioni lineari unimodulari con coefficienti del campo diJacobi-Eisenstein — 2. Il poliedro fondamentale del sottogruppo Г_μ(1−ε) — § 2 – 3. I campi fondamentali dei gruppi Г_μ(n) — 4. Impossibilità di limitare con un numero finito di piani e sfere di riflessione i poliedri fondamentali dei gruppi Г_μ(n), conn intero razionale pari, diverso da 2 — § 3 – 5. Relazioni fondamentali fra le sostituzioni generatrici del gruppo di sostituzioni lineari con coefficienti del corpo K_ε con determinante ±1 — § 4 – 6. Sulla indipendenza delle sostituzioniS,T,U, generatrici del gruppo finito G_2μ(n) e sulle loro relazioni caratteristiche nel gruppo G_2μ(n) — § 5 – 7. Dimostrazione del teorema fondamentale sui gruppi G_2μ(n). Lemmi preliminari — 8, Dimostrazione del teorema fondamentale nel caso di moduli primi con 2(1−ε) — § 6 – 9. Il teorema fondamentale per i modulim(1−ε), 3m, 2m, 2m(1−ε), 6m conm primo con 6 – 10. Immagine geometrica dei gruppi G_2μ(1−ε) — § 7 – 11. Il gruppo delle sostituzioni unimodulari , [c/1+4ma]=+1, e il caso eccezionale dei moduli 4m – 12. Il gruppo delle sostituzioni unimodulari [c/1+3m(1−ε)a]₃=+1 e il caso eccezionale dei moduli 3(1−ε)m.     

STABILIZED FEM FOR CONVECTION-DIFFUSION PROBLEMS ON LAYER-ADAPTED MESHES

The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.     

Almost-Perfect Rings and Modules

Following [1 Amini , A. , Ershad , M. , Sharif , H. ( 2008 ). Rings over which flat covers of finitely generated modules are projective . Comm. Algebra 36 : 2862 – 2871 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP --In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Projections,Birkhoff Orthogonality and Angles in Normed Spaces

Let X be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle Aq（x, y） between two vectors x and y in X, such that x is Birkhoff orthogonal to y if and only if Aq（x,y）=π/2. Some other properties of this angle are also discussed.     

PREDUAL SPACES FOR Q SPACES

To find the predual spaces Pα（R^n） of Qα（R^n） is an important motivation in the study of Q spaces. In this article, wavelet methods are used to solve this problem in a constructive way. First, an wavelet tent atomic characterization of Pα（Rn） is given, then its usual atomic characterization and Poisson extension characterization are given. Finally, the continuity on Pα of Calderon-Zygmund operators is studied, and the result can be also applied to give the Morrey characterization of Pα（Rn）.