Problems and results on additive properties of general sequences. II

1986 ◽  
Vol 48 (1-2) ◽  
pp. 201-211 ◽  
Author(s):  
P. Erdős ◽  
A. Sárközy
Keyword(s):  
General Sequences

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Frequent Closed Partial Orders Mining in Sequences

2013 ◽  
Vol 846-847 ◽  
pp. 1304-1307
Author(s):  
Ye Wang ◽  
Yan Jia ◽  
Lu Min Zhang
Keyword(s):  
Sequence Data ◽  
Real Data ◽  
Partial Orders ◽  
Hard Problem ◽  
Important Data ◽  
Data Set ◽  
Pruning Algorithm ◽  
Equal Chance ◽  
Np Hard Problem ◽  
General Sequences

Mining partial orders from sequence data is an important data mining task with broad applications. As partial orders mining is a NP-hard problem, many efficient pruning algorithm have been proposed. In this paper, we improve a classical algorithm of discovering frequent closed partial orders from string. For general sequences, we consider items appearing together having equal chance to calculate the detecting matrix used for pruning. Experimental evaluations from a real data set show that our algorithm can effectively mine FCPO from sequences.

Polynomial and power series ring extensions from sequences

2020 ◽  
pp. 2250048
Author(s):  
Gyu Whan Chang ◽  
Phan Thanh Toan
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Trivial Case ◽  
Addition Formula ◽  
Power Series Ring ◽  
Series Ring ◽  
Ring Extensions ◽  
General Sequences ◽  
Divisibility Properties ◽  
Positive Integers

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

scholarly journals On additive properties of general sequences

2005 ◽  
Vol 71 (3) ◽  
pp. 479-485 ◽  
Author(s):  
Min Tang ◽  
Yong-Gao Chen
Keyword(s):  
Infinite Sequence ◽  
Number Of Solutions ◽  
Quantitative Form ◽  
General Sequences ◽  
Positive Integers

Let A = {a1, a2,…} (a1 < a2 < …) be an infinite sequence of positive integers. Let A(n) be the number of elements of A not exceeding n, and denote by R2(n) the number of solutions of ai + aj = n, i ≤ j. In 1986, Erdős, Sárközy and Sós proved that if (n − A(n))/log n → ∞(n → ∞), then . In this paper, we generalise this theorem and give its quantitative form. For example, one of our conclusions implies that if limsup(n − A(n))/log n = ∞, then for infinitely many positive integers N.

scholarly journals Retrograde diagenesis, a widespread process on a regional scale

Clay Minerals ◽  
2005 ◽  
Vol 40 (1) ◽  
pp. 93-104 ◽  
Author(s):  
F. Nieto ◽  
M. Pilar Mata ◽  
B. Bauluz ◽  
G. Giorgetti ◽  
P. Árkai ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Regional Metamorphism ◽  
Regional Scale ◽  
Tectonic Stress ◽  
Textural Features ◽  
X Ray ◽  
Transmission Electron ◽  
Regional Basis ◽  
General Sequences ◽  
Basic Rocks

AbstractPelitic and basic rocks occurring within prograde sequences in Portugal, Spain and Hungary have been studied by X-ray powder diffraction (XRD), scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The minerals formed in typical prograde reactions define the general sequences, but smectite, chlorite-smectite (corrensite) and/or berthierine were found to have replaced chlorite, whereas kaolinite and mixed-layer illite-smectite replaced illite-muscovite. Alteration occurred under conditions normally associated with diagenesis, subsequent to regional metamorphism, and we therefore refer to such processes with the term “retrograde diagenesis”. In the cases studied and in other cited examples, reactions occurred on a regional basis via pervasive fluids under open-system conditions inferred to be related to tectonic stress. The observed alterations could generally not be inferred from XRD data, although the presence of pure smectite in sediments other than bentonite is suggestive of retrograde relations, especially where other minerals are consistent with a higher grade of diagenesis. Retrograde diagenesis is readily observed through imaging of textures by TEM, however. Textural features show that retrograde reactions are more common than generally assumed, and that care should be used in interpreting geological events where appropriate textural relations are not seen.

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