The Supremum of a Family of Additive Functions

Any system S in which an addition is defined for some, but not necessarily all, pairs of elements can be imbedded in a natural way in a commutative semi-group G, although different elements in S need not always determine different elements in G (see §2). Theorem 2.1 gives necessary and sufficient conditions in order that a functional p(x) on S can be represented as the su prémuni of some family of additive functionals on S, and one such set of conditions is in terms of possible extensions of p(x) to G. This generalizes the case with 5 a Boolean ring treated by Lorentz [4], Lorentz imbeds the Boolean ring in a vector space and this could be done for the general S; but we prefer to imbed S in a commutative semi-group and to give a proof (see § 1) generalizing the classical Hahn-Banach theorem to the case of an arbitrary commutative semigroup.

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The algebra of differential forms on a full matric bialgebra

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100071450 ◽

1993 ◽

Vol 114 (1) ◽

pp. 111-130 ◽

Cited By ~ 13

Author(s):

A. Sudbery

Keyword(s):

Vector Space ◽

Commutative Algebra ◽

Differential Algebra ◽

Differential Forms ◽

Sufficient Conditions ◽

Classical Case ◽

Algebra Of Functions ◽

The Matrix ◽

Constant Coefficients ◽

Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

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The Integral Extension of Isometries of Quadratic Forms Over Local Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1966-092-x ◽

1966 ◽

Vol 18 ◽

pp. 920-942 ◽

Cited By ~ 6

Author(s):

Allan Trojan

Keyword(s):

Vector Space ◽

Quadratic Forms ◽

Sufficient Conditions ◽

Local Fields ◽

The Other ◽

Quadratic Space ◽

Integral Extension ◽

Ring Of Integers ◽

Extension Of Isometries ◽

Necessary And Sufficient

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

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Regular, Commutative, Maximal Semigroups of Quotients

Canadian Mathematical Bulletin ◽

10.4153/cmb-1975-018-3 ◽

1975 ◽

Vol 18 (1) ◽

pp. 99-104 ◽

Cited By ~ 3

Author(s):

Jurgen Rompke

Keyword(s):

Commutative Semigroup ◽

Regular Ring ◽

Sufficient Conditions ◽

Commutative Rings ◽

Necessary And Sufficient Conditions ◽

Natural Question ◽

Commutative Case ◽

Ring Of Quotients ◽

Singular Ideal ◽

Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

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On commutative semigroup algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100060515 ◽

1983 ◽

Vol 93 (2) ◽

pp. 237-246 ◽

Cited By ~ 18

Author(s):

W. D. Munn

Keyword(s):

Commutative Semigroup ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Semigroup Algebras ◽

Necessary And Sufficient

This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).

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On the solvability of systems of linear equations on commutative semigroup

Demonstratio Mathematica ◽

10.1515/dema-2013-0270 ◽

2010 ◽

Vol 43 (4) ◽

Author(s):

Nguyen Thi Thu Huyen ◽

Nguyen Minh Tuan

Keyword(s):

Linear Operator ◽

Linear Space ◽

Commutative Semigroup ◽

Linear Equations ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Operator Equations ◽

Systems Of Linear Equations ◽

Necessary And Sufficient ◽

Linear Operator Equations

AbstractThis paper deals with the solvability of systems of linear operator equations in a linear space. Namely, the paper provides necessary and sufficient conditions for the operators under which certain kinds of systems of operator equations are solvable.

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The Hahn-Banach theorem for non-Archimedean-valued fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100027353 ◽

1952 ◽

Vol 48 (1) ◽

pp. 41-45 ◽

Cited By ~ 35

Author(s):

A. W. Ingleton

Keyword(s):

Banach Spaces ◽

Normed Linear Space ◽

General Theorem ◽

Sufficient Conditions ◽

Linear Operators ◽

Necessary Condition ◽

Linear Functionals ◽

Banach Theorem ◽

Necessary And Sufficient ◽

Special Case

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

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The quadratic form positive definite on a linear manifold

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100026293 ◽

1951 ◽

Vol 47 (1) ◽

pp. 1-6 ◽

Cited By ~ 6

Author(s):

S. N. Afriat

Keyword(s):

Quadratic Form ◽

Vector Space ◽

Linear Manifold ◽

Sufficient Conditions ◽

Positive Definite ◽

Necessary And Sufficient Conditions ◽

Real Vector Space ◽

Real Vector ◽

Necessary And Sufficient ◽

Real Quadratic

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

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Projectivity and flatness over the endomorphism ring of a finitely generated quasi-Poisson comodule

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500601 ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450060

Author(s):

T. Guédénon

Keyword(s):

Vector Space ◽

Endomorphism Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Module Algebra ◽

Finitely Generated ◽

Enveloping Algebra ◽

Grouplike Element ◽

Necessary And Sufficient ◽

Dual Ring

Let k be a field of characteristic 0, A a noncommutative Poisson k-algebra, U(A) the ordinary enveloping algebra of A, 𝒞 a quasi-Poisson A-coring that is projective as a left A-module, *𝒞 the left dual ring of 𝒞 (it is a right U(A)-module algebra) and Λ a right quasi-Poisson 𝒞-comodule that is finitely generated as a right U(A)#*𝒞-module. The vector space End 𝒫,𝒞(Λ) of right quasi-Poisson 𝒞-colinear maps from Λ to Λ is a ring. We give necessary and sufficient conditions for projectivity and flatness of a module over End 𝒫,𝒞(Λ). If 𝒞 contains a fixed quasi-Poisson grouplike element, we can replace Λ with A.

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On Statistics of Permutations Chosen From the Ewens Distribution

Combinatorics Probability Computing ◽

10.1017/s0963548314000376 ◽

2014 ◽

Vol 23 (6) ◽

pp. 889-913

Author(s):

TATJANA BAKSHAJEVA ◽

EUGENIJUS MANSTAVIČIUS

Keyword(s):

Probability Measure ◽

Sufficient Conditions ◽

Random Permutation ◽

Asymptotic Distributions ◽

Additive Functions ◽

Factorial Moments ◽

Permutation Matrices ◽

Poisson Limit ◽

Number Of Cycles ◽

Necessary And Sufficient

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

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The factorial moments of additive functions with rational argument

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700014403 ◽

2006 ◽

Vol 81 (3) ◽

pp. 425-440

Author(s):

J. Šiaulys ◽

G. Stepanauskas

Keyword(s):

Weak Convergence ◽

Limit Distribution ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Additive Functions ◽

Factorial Moments ◽

Rational Argument ◽

Necessary And Sufficient ◽

The Given

AbstractWe consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

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