Computing Characteristic Sets of Ordinary Radical Differential Ideals

2006 ◽  
Vol 13 (3) ◽  
pp. 515-527
Author(s):  
Brahim Sadik
Keyword(s):  
Upper Bounds ◽  
Algebraic Methods ◽  
Characteristic Set ◽  
Finitely Generated ◽  
Differential Ideal ◽  
Characteristic Sets

Abstract We give upper bounds for the order of the elements in a characteristic set of a regular differential ideal or a radical of a finitely generated differential ideal with respect to some specific orderings. We then show how to compute characteristic sets of these ideals using algebraic methods.

Numerical upper bounds on growth of automaton groups

2021 ◽  
pp. 1-33
Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi
Keyword(s):  
Group Theory ◽  
Upper Bounds ◽  
Geometric Invariant ◽  
Finitely Generated ◽  
Grigorchuk Group ◽  
Finitely Generated Group ◽  
Intermediate Growth ◽  
Mealy Automata ◽  
Optimal Bounds ◽  
Algorithmic Procedure

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

scholarly journals Multivariable dimension polynomials and new invariants of differential field extensions

2001 ◽  
Vol 27 (4) ◽  
pp. 201-214 ◽  
Author(s):  
Alexander B. Levin
Keyword(s):  
Field Extension ◽  
Differential Polynomials ◽  
Finitely Generated ◽  
Field Extensions ◽  
Differential Field ◽  
Differential Dimension ◽  
Characteristic Sets ◽  
Differential Dimension Polynomial ◽  
Dimension Polynomial

We introduce a special type of reduction in the ring of differential polynomials and develop the appropriate technique of characteristic sets that allows to generalize the classical Kolchin's theorem on differential dimension polynomial and find new differential birational invariants of a finitely generated differential field extension.

Artinianness and Attached Primes of Formal Local Cohomology Modules

2014 ◽  
Vol 21 (02) ◽  
pp. 307-316 ◽  
Author(s):  
Mohammad Hasan Bijan-Zadeh ◽  
Shahram Rezaei
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Upper Bounds ◽  
Finitely Generated ◽  
Lower And Upper Bounds ◽  
Local Cohomology Modules ◽  
Formal Local Cohomology ◽  
Cohomology Modules

Let 𝔞 be an ideal of a local ring (R, 𝔪) and M a finitely generated R-module. In this paper we study the Artinianness properties of formal local cohomology modules and we obtain the lower and upper bounds for Artinianness of formal local cohomology modules. Additionally, we determine the set [Formula: see text] and we show that the set of all non-isomorphic formal local cohomology modules [Formula: see text] is finite.

scholarly journals On the Turns of Digital Images

2015 ◽  
Vol 20 (2) ◽  
pp. 157-165
Author(s):  
P. G. Parfenov
Keyword(s):  
Nearest Neighbor ◽  
Hexagonal Lattice ◽  
Rectangular Lattice ◽  
Characteristic Set ◽  
Arbitrary Angle ◽  
Characteristic Sets ◽  
Predicted Values ◽  
Analytical Expressions ◽  
Eulerian Characteristic ◽  
Good Agreement

The images built on the basis of rectangular and hexagonal lattices are discussed in the article. For images on a rectangular lattice a formula is proposed, which gives approximate values of the components of a characteristic set of coefficients when turning at an arbitrary angle by the method of the nearest neighbor. The characteristic sets are presented in the form of diagrams, an experimental evaluation of errors is made. It was confirmed a good agreement with the predicted value component of characteristic sets and those which were obtained experimentally. For images built on the basis of a hexagonal lattice was offered a similar formula for the approximation of the components of the characteristic set for rotating at any angle, when this was applied to the modification of the nearest neighbor method for the preservation of coherence, as it was discovered its violation in some cases on a hexagonal lattice. On the basis of four-pixel fragments are built diagrams, which show a good agreement of predicted values and the obtained ones in the experiment. It was defined a system of three-pixel hexagonal fragments to which the theorem is proved on the Eulerian characteristic and were offered analytical expressions, which allow to avoid experimental detection of the characteristic sets of coefficients for all possible reference angles. Their use requires to produce only one such experiment.

scholarly journals Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal

2001 ◽  
Vol 31 (6) ◽  
pp. 631-649 ◽  
Author(s):  
Driss Bouziane ◽  
Abdelilah Kandri Rody ◽  
Hamid Maârouf
Keyword(s):  
Finitely Generated ◽  
Differential Ideal ◽  
Dimensional Decomposition

scholarly journals Lower and upper bounds for Cohen-Macaulay dimension

2005 ◽  
Vol 71 (2) ◽  
pp. 337-346
Author(s):  
J. Asadollahi ◽  
Sh. Salarain
Keyword(s):  
Noetherian Ring ◽  
Upper Bounds ◽  
Finitely Generated ◽  
Lower And Upper Bounds ◽  
Finitely Generated Module ◽  
Basic Properties ◽  
Homological Dimensions

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.

A Fuzzy Temporal System and its Characteristic Set Representation

2011 ◽  
Vol 217-218 ◽  
pp. 383-389
Author(s):  
Guang Peng Hu ◽  
Hai Yang Huang
Keyword(s):  
The Other ◽  
Temporal Relations ◽  
Characteristic Set ◽  
Time Intervals ◽  
Temporal System ◽  
Random Events ◽  
Characteristic Sets ◽  
Composition Table ◽  
Heisenberg's Uncertainty Principle ◽  
Better Than

The probability that the starting points or ending points of time intervals of two random events are equal is zero. Hence, in order to improve the efficiency of temporal reasoning, We introduced Heisenberg’s uncertainty principle to delete the fifteen basic temporal relations that the probabilities they happen were zero from INDU temporal system. The other ten basic temporal relations that the probabilities they happen were greater than zero were reserved. We used the characteristic sets to represent the ten basic temporal relations and got a fuzzy temporal system--UINDU. We gave the composition table of UINDU. It simplified INDU greatly. The experiments showed that the efficiency of UINDU was better than INDU.

scholarly journals Characteristic Sets for Learning k-Acceptable Languages

1970 ◽  
Vol 5 (1) ◽  
pp. 39-45
Author(s):  
Anuchit Jitpattanakul ◽  
Athasit Surarerks
Keyword(s):  
Formal Language ◽  
Learning Algorithm ◽  
Language Identification ◽  
Grammatical Inference ◽  
Challenging Problem ◽  
Characteristic Set ◽  
Learning Technique ◽  
Characteristic Sets ◽  
Theoretical Results

Learnability of languages is a challenging problem in the domain of formal language identification. It is known that the efficiency of a learning technique can be measured by the size of some good samples (representative or distinctive samples) formally called a characteristic set. Our research focuses on the characteristic set of k-acceptable languages. We proposed a Gold-style learning algorithm called KRPNI which applied the grammatical inference technique to identify a language and expressed it by a k-DFA. In this paper, we study the existence of such characteristic sets. Our theoretical results show that there exists a polynomial characteristic set for a k-acceptable language. It is found that the size of the characteristic set depends on the value of k, instead of the size of an alphabet.

Characteristic sets verses generalised characteristic sets for ordinary differential polynomial sets

2020 ◽  
Vol 10 (2) ◽  
pp. 146-157
Author(s):  
Farkhanda Afzal ◽  
Muhammad Ashiq
Keyword(s):  
Differential Equations ◽  
Differential Polynomial ◽  
Differential Polynomials ◽  
Characteristic Set ◽  
Algebraic Differential Equations ◽  
Standard Tool ◽  
Characteristic Sets

The concept of characteristic sets developed initially by Ritt and Wu has turned into a standard tool for the study of sets or systems of polynomial and algebraic differential equations. With the help of constructing characteristic sets, an arbitrary set or system of polynomials or differential polynomials can be triangularised. It means that it can be decomposed into a particular set or system of triangular forms. In this paper, a comparison of Ritt–Wu’s characteristic sets by Wang for the ordinary differential polynomial sets with the generalised characteristic sets of the ordinary differential polynomial sets by the authors has been presented.   Keywords: Characteristic set, differential polynomial, pseudo-division, admissible reduction.

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