Standard bases over Euclidean domains

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.

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Biorthogonal Systems in lp-Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-071-6 ◽

1969 ◽

Vol 21 ◽

pp. 625-638 ◽

Cited By ~ 1

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).

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Complete Decomposability in the Exterior Algebra of a Free Module

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-003-3 ◽

1980 ◽

Vol 32 (1) ◽

pp. 27-33 ◽

Cited By ~ 2

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.

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