Rings, Ideals and Standard Bases

2002 ◽  
pp. 1-88 ◽  
Author(s):  
Gert-Martin Greuel ◽  
Gerhard Pfister
Keyword(s):  
Standard Bases

International Standard Bases and Safety Classification

2014 ◽  
pp. 31-60
Author(s):  
Michael A. Yastrebenetsky ◽  
Grygoriy Gromov
Keyword(s):  
International Atomic Energy Agency ◽  
Atomic Energy ◽  
International Electrotechnical Commission ◽  
Energy Agency ◽  
International Standard ◽  
Standard Bases ◽  
Technical Requirements ◽  
General Safety

The main standard bases for NPP I&C systems are documents of the International Atomic Energy Agency (IAEA) and International Electrotechnical Commission (IEC). Standards are interconnected through the following: IAEA develops general safety principles for NPP I&C systems, and IEC develops technical requirements that use and specify safety principles. Structures of the bases are considered. Classifications of I&C systems and their components are given on the basis of their safety impact. According to the IAEA classification, all systems are divided into safety important and non-safety important. According to IEC, functions to be performed by I&C systems shall be assigned to categories according to their importance to safety. The importance to safety of a function shall be identified by means of the consequences in the event of its failure, when it is required to be performed, and by the consequences in the event of a spurious actuation. All functions are divided into categories A, B, C.

Biorthogonal Systems in lp-Spaces

1969 ◽  
Vol 21 ◽  
pp. 625-638 ◽  
Author(s):  
R. Keown ◽  
C. Conatser
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Natural Generalization ◽  
Standard Basis ◽  
Biorthogonal Systems ◽  
Lp Spaces ◽  
Standard Bases

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).

Complete Decomposability in the Exterior Algebra of a Free Module

1980 ◽  
Vol 32 (1) ◽  
pp. 27-33 ◽  
Author(s):  
M. Boratynski ◽  
E. D. Davis ◽  
A. V. Geramita
Keyword(s):  
Commutative Ring ◽  
Present Note ◽  
Free Module ◽  
Exterior Algebra ◽  
Standard Bases ◽  
Classical Criterion ◽  
Generally True

Recall the classical criterion for the complete decomposability of exterior vectors: the completely decomposable vectors in ∧pRn, R a field, are precisely the “Plücker vectors,” i.e. those whose coordinates (relative to the standard bases for ∧pRn) satisfy the Plücker equations. For R an arbitrary commutative ring, completely decomposable exterior vectors are still Plücker vectors, but the converse is not generally true. Rings for which the converse is true (for all 1 ≤ p ≤ n) are called Towber rings. Noetherian Towber rings are regular and, in fact, are characterized by the property that every Plücker vector in ∧2R4 is completely decomposable. (See [10] for these two results as well as for the above mentioned facts.) The present note develops a new characterization of Towber rings, combining it with results of Kleiner [9] and Estes-Matijevic [5] in (1) below.

scholarly journals Standard bases over Euclidean domains

2021 ◽  
Vol 102 ◽  
pp. 21-36
Author(s):  
Christian Eder ◽  
Gerhard Pfister ◽  
Adrian Popescu
Keyword(s):  
Standard Bases ◽  
Euclidean Domains

Alternative algorithms for computing generic μ∗-sequences and local Euler obstructions of isolated hypersurface singularities

2019 ◽  
Vol 18 (08) ◽  
pp. 1950156
Author(s):  
Katsusuke Nabeshima ◽  
Shinichi Tajima
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Rational Functions ◽  
Isolated Singularity ◽  
Euler Obstruction ◽  
Standard Bases ◽  
Hypersurface Singularities ◽  
Previous Algorithm ◽  
Computation Speed

A new algorithm is introduced for computing [Formula: see text]-sequences of isolated hypersurface singularities. It is shown that the new algorithm results in better performance, compared to our previous algorithm that utilizes parametric local cohomology systems, in computation speed. Furthermore, it can be used to compute local Euler obstruction of a hypersurface with an isolated singularity. The key idea of the new algorithm is computing standard bases in a local ring over a field of rational functions.

scholarly journals Reduction Mod p of Standard Bases

2005 ◽  
Vol 33 (5) ◽  
pp. 1635-1661 ◽  
Author(s):  
Matthias Aschenbrenner
Keyword(s):  
Standard Bases ◽  
Mod P

Pipeline Leakage Recognition Based on the Projection Singular Value Features and Support Vector Machine

2010 ◽  
Author(s):  
Mingda Wang ◽  
Laibin Zhang ◽  
Wei Liang ◽  
Jinqiu Hu
Keyword(s):  
Support Vector Machine ◽  
Feature Extraction ◽  
Field Experiments ◽  
Small Sample ◽  
Singular Value ◽  
Support Vector ◽  
Leakage Detection ◽  
Feature Extraction Method ◽  
Standard Bases ◽  
Value Decomposition

Identification of negative pressure waveform is the key of pipeline leakage detection. The feature extraction and the choice of the classifier are two main contents to solve the recognition problem. In this paper, a new feature extraction method based on the Projection Singular Value is presented. First of all, the two orthogonal singular value decomposition matrixes of the typical leakage waveform are extracted as the standard bases. Then the projection singular value features of the other pressure wave matrixes are extracted by being projected to the two standard bases. As the pipeline leakage is a small probability event, it is difficult to obtain the leakage samples. A multi-classification Support Vector Machine, which has the advantage of small sample learning, is constructed to classify these features in this paper. The field experiments indicate that the leakage detection based on this feature extraction and recognition model has a higher accuracy of leakage recognition.

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