101 results
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2. Strict Topologies for Vector-Valued Functions
- Author
-
Robert A. Fontenot
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Network topology ,01 natural sciences ,Vector-valued function ,Mathematics - Abstract
This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].
- Published
- 1974
3. On Open Extensions of Maps
- Author
-
Stan Franklin and J. K. Kohli
- Subjects
Discrete mathematics ,Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains. For example, in 1953 Whyburn showed that every map is the restriction of a compact map [17]. Similarly, Krolevec proved in 1967 that each locally perfect map can be extended to a perfect map [12] and in an as yet unpublished paper, Dickman obtained the same result for arbitrary maps [4]. In this paper we show that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. These results are then used to obtain analogues and improvements of recent theorems of Arhangel'skiï, Ĉoban, Hodel, and Proizvolov.
- Published
- 1970
4. A Theorem on Pure Submodules
- Author
-
George Kolettis
- Subjects
Discrete mathematics ,Algebra ,General Mathematics ,Mathematics - Abstract
In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.
- Published
- 1960
5. Identities of Non-Associative Algebras
- Author
-
J. Marshall Osborn
- Subjects
Algebra ,General Mathematics ,Associative property ,Mathematics - Abstract
In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.
- Published
- 1965
6. Generation of Local Integral Orthogonal Groups in Characteristic 2
- Author
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Barth Pollak
- Subjects
Combinatorics ,Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In two previous papers (see4;5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groupsS(L), Xh(L), andO(L) are introduced for a unimodular latticeLand the relationship betweenS(L) andO(L) studied. As in the previously cited papers, generation by symmetries means thatS(L) =O(L). The following result is obtained.
- Published
- 1968
7. A Characterization of the Algebra of Functions Vanishing at Infinity
- Author
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Robert E. Mullins
- Subjects
Circular points at infinity ,Filtered algebra ,Algebra ,General Mathematics ,media_common.quotation_subject ,Vanish at infinity ,Point at infinity ,Algebra over a field ,Characterization (mathematics) ,Infinity ,media_common ,Mathematics - Abstract
1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.
- Published
- 1969
8. The Kernel of the General-Sum Four-Person Game
- Author
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Bezalel Peleg
- Subjects
Algebra ,General Mathematics ,Kernel (statistics) ,Mathematics - Abstract
In this paper we apply various results and methods of previous papers on the kernel to four-person games.Section 2 contains the basic definitions needed. In §3 we prove that the kernel of the general-sum four-person game consists of a line segment (which may shrink to a point). A method for classifying games according to their kernels is suggested in §4 and is used there to characterize all four-person games whose kernel consists of a non-degenerate interval. In the last section, §5, we offer a bargaining procedure, based on principles established in (1), which leads to the kernel in the case of a non-degenerate interval.
- Published
- 1966
9. Rings with Finite Norm Property
- Author
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Kathleen B. Levitz and Joe L. Mott
- Subjects
Algebra ,General Mathematics ,Norm (mathematics) ,010102 general mathematics ,0103 physical sciences ,Matrix norm ,Ideal norm ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Field norm - Abstract
A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.
- Published
- 1972
10. Representations Subduced on an Ideal of a Lie Algebra
- Author
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B. Noonan
- Subjects
General Mathematics ,010102 general mathematics ,Universal enveloping algebra ,Lie superalgebra ,01 natural sciences ,Affine Lie algebra ,Super-Poincaré algebra ,Lie conformal algebra ,Graded Lie algebra ,Algebra ,0103 physical sciences ,Algebra representation ,Fundamental representation ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).
- Published
- 1962
11. Szegö Polynomials on a Compact Group with Ordered Dual
- Author
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I. I. Hirschman
- Subjects
Algebra ,Pure mathematics ,Compact group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Dual (category theory) ,Mathematics - Abstract
The Szegö polynomials are defined on T, the real numbers modulo 1. In this paper and in its sequel we give a generalization of Szegö polynomials in which T is replaced by an arbitrary locally compact abelian group θ on whose dual there has been distinguished a measurable order relation compatible with the group structure. The present paper is devoted to the case where θ is compact and therefore discrete. The general case will be taken up in the sequel mentioned above. It is desirable to proceed in this way because the case θ compact is much simpler and much more like the classical situation than is the general case, in which various measure-theoretic difficulties obtrude. Moreover, as it happens, it is possible to develop the theory in this way with relatively little repetition.
- Published
- 1966
12. On the Generality of the AP-Integral
- Author
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G. E. Cross
- Subjects
Algebra ,Generality ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.
- Published
- 1971
13. Finiteness of Semigroups of Operators in Universal Algebra
- Author
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Evelyn Nelson
- Subjects
Matrix unit ,General Mathematics ,010102 general mathematics ,Universal enveloping algebra ,Operator theory ,01 natural sciences ,Filtered algebra ,Algebra ,0103 physical sciences ,Universal algebra ,Special classes of semigroups ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper is a partial solution of problem 24 in (2) which suggests that the finiteness of the partially ordered semigroups generated by various combinations of operators on classes of universal algebras be investigated. The main result is that the semigroups generated by the following sets of operators (for definitions see §2) are finite: {H, S, P, Ps}, {C, H, S, P, PF} {C, H, S, PU, PF}.This paper is part of the author's Master's thesis written in the Department of Mathematics at McMaster University. The author is indebted to the referee for his helpful suggestions.
- Published
- 1967
14. An Elementary Proof of a Theorem About the Representation of Primes by Quadratic Forms
- Author
-
W. E. Briggs
- Subjects
General Mathematics ,010102 general mathematics ,Representation (systemics) ,Prime number ,01 natural sciences ,Algebra ,symbols.namesake ,Factorization ,Furstenberg's proof of the infinitude of primes ,0103 physical sciences ,Elementary proof ,symbols ,Dirichlet's theorem on arithmetic progressions ,010307 mathematical physics ,0101 mathematics ,Analytic proof ,Mathematics - Abstract
The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.
- Published
- 1954
15. On a Theorem of Beurling and Livingston
- Author
-
Felix E. Browder
- Subjects
Algebra ,Discrete mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).
- Published
- 1965
16. Some Remarks Concerning Categories and Subspaces
- Author
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J. R. Isbell
- Subjects
General Mathematics ,010102 general mathematics ,Structure (category theory) ,Bicategory ,01 natural sciences ,Linear subspace ,Algebra ,Development (topology) ,0103 physical sciences ,Point (geometry) ,010307 mathematical physics ,0101 mathematics ,Axiom ,Elaboration ,Mathematics - Abstract
This paper is primarily a brief elaboration on the axioms for a bicategory introduced in (3). From this point of view, the main aim is the development of the structure of certain systems of topological and uniform spaces, and the present paper merely points out some very general properties which follow from axioms so weak that they are satisfied by any system likely to be considered.
- Published
- 1957
17. Integral Extensions of Commutative Banach Algebras
- Author
-
John A. Lindberg
- Subjects
Discrete mathematics ,Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Commutative property ,Mathematics - Abstract
In this paper, we continue the study of integral extensions begun in [7]. Whereas in the previous paper, we dealt exclusively with the extension A[x]/(α(x)), α(x) a monic polynomial over A, we now deal with arbitrary integral extensions. Applications of the results presented herein will be made in subsequent papers.To simplify our presentation, we make the following conventions. By an algebra, we will always mean a commutative complex algebra with an identity element, usually denoted by e
- Published
- 1973
18. Semi-Prime Modules
- Author
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E. H. Feller and E. W. Swokowski
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Prime (order theory) ,Mathematics - Abstract
Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.
- Published
- 1966
19. Generators of Monothetic Groups
- Author
-
D. L. Armacost
- Subjects
Algebra ,General Mathematics ,Mathematics - Abstract
A topological group G is called monothetic if it contains a dense cyclic subgroup. An element x of G is called a generator of G if x generates a dense cyclic subgroup of G. We denote by E(G) the set of generators of G; the complement of E(G) in G, consisting of the “non-generators” of G, we write as N(G) Throughout this paper we consider only locally compact abelian (LCA) groups satisfying the T2 separation axiom (note that a monothetic group is automatically abelian). In [1] certain problems of measurability concerning the set E(G) are discussed. In this paper we shall consider some algebraic and topological properties of the sets E(G) and N(G)
- Published
- 1971
20. Systems of equations and generalized characters in groups
- Author
-
I. M. Isaacs
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,System of linear equations ,01 natural sciences ,Mathematics - Abstract
Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.
- Published
- 1970
21. Some Results on Quadrics in Finite Projective Geometry Based on Galois Fields
- Author
-
D. K. Ray-Chaudhuri
- Subjects
General Mathematics ,Fundamental theorem of Galois theory ,010102 general mathematics ,Galois group ,01 natural sciences ,Differential Galois theory ,Embedding problem ,Algebra ,symbols.namesake ,Galois geometry ,0103 physical sciences ,symbols ,Projective space ,010307 mathematical physics ,Projective differential geometry ,0101 mathematics ,Mathematics ,Projective geometry - Abstract
In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.
- Published
- 1962
22. Logarithmic Capacity of Sets and Double Trigonometric Series
- Author
-
V. L. Shapiro
- Subjects
Discrete mathematics ,Pythagorean trigonometric identity ,Logarithm ,General Mathematics ,010102 general mathematics ,Trigonometric integral ,Trigonometric polynomial ,01 natural sciences ,Trigonometric series ,Algebra ,symbols.namesake ,0103 physical sciences ,symbols ,Inverse trigonometric functions ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
It is the purpose of this paper to establish a closer connection between the logarithmic capacity of sets and double trigonometric series. In (9), closed sets of logarithmic capacity zero were established as sets of uniqueness for a particular class of double trigonometric series under circular (C, 1) summability. By slightly changing this class of series but still maintaining closed sets of logarithmic capacity zero as sets of uniqueness, it is shown in this paper that closed sets of positive logarithmic capacity form sets of multiplicity.
- Published
- 1954
23. A General Perron Integral, II
- Author
-
P. S. Bullen
- Subjects
Algebra ,General Mathematics ,Calculus ,Perron's formula ,Mathematics - Abstract
This paper continues work begun in a previous paper of the same title (7), which will be called I; results from I will be referred to as Theorem 1.4, Axiom 1.1 etc. The notation used in the present paper will, except where noted, be that of I, to which reference should be made for further details.In § 2, certain ideas presented in I are modified to give a neater and more general theory and then some new results of this theory are added. The remaining two sections develop some of the examples mentioned in I, § 5.
- Published
- 1967
24. On the Structure of Finite T0 + T5 Spaces
- Author
-
Shawpawn Kumar Das
- Subjects
Algebra ,General Mathematics ,Structure (category theory) ,Mathematics - Abstract
The object of this paper is to study some structural aspects of finite T0 + T4 and T0 + T5 spaces in order to establish certain recursion relations that can be used to obtain the number of (labelled as well as unlabelled) T0 + T5 topologies on a finite set.Here, as in [2], a topology 𝒥 is a T4(T5) space provided for any pair of disjoint closed sets A and B (separated sets A and B = A ∩ closure B = B ∩ closure A = 0) there exist disjoint open sets 0A and 0B 𝒥 such that A ⊆ 0A and B ⊆ 0B. An almost immediate consequence of these investigations is that the inherent simplicity of the connected T0 + T5 topologies ensures that they are reconstructable.
- Published
- 1973
25. On Continuous Images of Moore Spaces
- Author
-
G. M. Reed
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In [4-8], the author has obtained several counterexamples to conjectures involving chain conditions, normality conditions, completeness, and the existence of point countable bases in Moore spaces. Each of these examples was obtained by constructing, by various means, a Moore space based on another space X0. In this paper, the author unifies these construction techniques and states some of the relationships between the original spaces and the derived Moore spaces.
- Published
- 1974
26. Extrema of a Class of Functions on a Finite Set
- Author
-
Kenneth W. Lebensold
- Subjects
Maxima and minima ,Algebra ,Discrete mathematics ,Class (set theory) ,General Mathematics ,Finite set ,Mathematics - Abstract
In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.
- Published
- 1974
27. Improved Versions of Forms of Plessner's Theorem
- Author
-
Peter Colwell
- Subjects
Algebra ,General Mathematics ,Mathematics - Abstract
With the aid of a theorem about the Julia points of a function meromorphic in the unit disk, this paper strengthens a theorem of K. Meier. As a consequence a stronger form of Plessner's Theorem is seen to hold which contains a theorem of E. F. Collingwood. An additional consequence is a stronger form of Meier's analogue to Plessner's Theorem.
- Published
- 1974
28. Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras
- Author
-
C. Y. Lee
- Subjects
Algebra ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Universal enveloping algebra ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Representation Theory ,01 natural sciences ,Mathematics - Abstract
An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).
- Published
- 1974
29. Growth Conditions and Decomposable Operators
- Author
-
Mehdi Radjabalipour
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Throughout this paper T will denote a bounded linear operator which is defined on a Banach space and whose spectrum lies on a rectifiable Jordan curve J .The operators having some growth conditions on their resolvents have been the subject of discussion for a long time. Many sufficient conditions have been found to ensure that such operators have invariant subspaces [2 ; 3 ; 7 ; 8 ; 12 ; 13; 14; 21; 27; 28; 29], are S-operators [14], are quasidecomposable [9], are decomposable [4 ; 11], are spectral [7 ; 10 ; 15 ; 17], are similar to normal operators [16 ; 23 ; 25 ; 26], or are normal [15 ; 18 ; 22]. In this line we are going to show that many such operators are decomposable.
- Published
- 1974
30. Ranges of Products of Operators
- Author
-
Sandy Grabiner
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.
- Published
- 1974
31. Semi-Metrics on the Normal States of a W*-Algebra
- Author
-
David Promislow
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with some extensions of the Bures metric d defined on the set of normal states of a W*-algebra [2]. Each subgroup G of the automorphism group of leads naturally to a semi-metric dG. (See Definition 1.1 below.) When G is the identity group dG = d.
- Published
- 1973
32. The Dual of Frobenius' Reciprocity Theorem
- Author
-
G. de B. Robinson
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Reciprocity law ,01 natural sciences ,Algebra ,Eisenstein reciprocity ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Artin reciprocity law ,0101 mathematics ,Frobenius theorem (real division algebras) ,Mathematics - Abstract
In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.
- Published
- 1973
33. Sheets of Real Analytic Varieties
- Author
-
Andrew H. Wallace
- Subjects
Algebra ,General Mathematics ,Mathematics - Abstract
In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.
- Published
- 1960
34. Separability in an Algebra with Semi-Linear Homomorphism
- Author
-
David J. Winter
- Subjects
Algebra ,Symmetric algebra ,Filtered algebra ,General Mathematics ,Subalgebra ,Algebra representation ,Universal enveloping algebra ,Homomorphism ,Composition algebra ,Tensor algebra ,Mathematics - Abstract
The purpose of this paper is to outline a simple theory of separability for a non-associative algebra A with semi-linear homomorphism σ. Taking A to be a finite dimensional abelian Lie p-algebra L and σ to be the pth power operation in L, this separability is the separability of [2]. Taking A to be an algebraic field extension K over k and σ to be the Frobenius (pth power) homomorphism in K, this separability is the usual separability of K over k. The theory also applies to any unital non-associative algebra A over a field k and any unital homomorphism σ from A to A such that σ(ke) ⊂ ke, e being the identity element of A.
- Published
- 1972
35. A Characterization of the Hyperhomology Groups of the Tensor Product
- Author
-
Thomas W. Hungerford
- Subjects
Algebra ,Tensor product ,General Mathematics ,Characterization (materials science) ,Hyperhomology ,Mathematics - Abstract
If K and L are chain complexes of abelian groups (to which we restrict ourselves throughout this paper), then denotes the graded hyperhomology group of K and L, as defined in Car tan and Eilenberg (1) by means of free double complex resolutions of K and L. Hyperhomology groups have proved convenient in proving various versions of the Künneth theorem (see, for example, (4; 1; 2)).
- Published
- 1968
36. Extensions of Lie Algebras and the Third Cohomology Group
- Author
-
S. I. Goldberg
- Subjects
Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Mathematics::K-Theory and Homology ,General Mathematics ,Group cohomology ,Equivariant cohomology ,Étale cohomology ,Mathematics::Algebraic Topology ,Representation theory ,Cohomology ,Lie conformal algebra ,Mathematics - Abstract
Cohomology theories of various algebraic structures have been investigated by several authors. The most noteworthy are due to Hochschild, MacLane and Eckmann, Chevalley and Eilenberg, who developed the theory of cohomology groups of associative algebras, abstract groups, and Lie algebras respectively. In this paper we are concerned primarily with a characterization of the third cohomology group of a Lie algebra by its extension properties.
- Published
- 1953
37. Some Algebraic Structure in the Dual of a Compact Group
- Author
-
Richard Iltis
- Subjects
Abelian variety ,General Mathematics ,010102 general mathematics ,Dimension of an algebraic variety ,01 natural sciences ,Representation theory ,Algebra ,Compact group ,Unitary group ,Algebraic group ,0103 physical sciences ,010307 mathematical physics ,Compact quantum group ,0101 mathematics ,Group theory ,Mathematics - Abstract
Throughout this paper, G will denote a compact (Hausdorff) topological group with identity e. When G is abelian, there is no difficulty in relating the group multiplication in G to the multiplication in the dual of G since characters are homomorphisms with respect to pointwise multiplication and pointwise multiplication of characters yields another character. However, in the non-abelian case, there are two multiplications associated with the dual of G: (1) representations are homomorphisms with respect to composition multiplication, and (2) the tensor product of representations yields another representation.
- Published
- 1968
38. Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators
- Author
-
Kurt Kreith
- Subjects
Algebra ,Fourth order ,General Mathematics ,Essential spectrum ,Differential operator ,Mathematics - Abstract
In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).
- Published
- 1969
39. On Self-Adjoint Factorization of Operators
- Author
-
Heydar Radjavi
- Subjects
General Mathematics ,010102 general mathematics ,Spectral theorem ,Incomplete LU factorization ,Operator theory ,01 natural sciences ,Algebra ,Factorization ,Factorization of polynomials ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Self-adjoint operator ,Mathematics - Abstract
The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .
- Published
- 1969
40. On the Integral Part of a Linear form with Prime Variables
- Author
-
I. Danicic
- Subjects
Algebra ,General Mathematics ,Linear form ,Logarithmic integral function ,Prime (order theory) ,Mathematics - Abstract
The object of this paper is to prove the following:Theorem. Suppose that λ, μ are real non-zero numbers, not both negative, λ is irrational, and k is a positive integer. Then there exist infinitely many primes p and pairs of primes p1, p2 such thatIn particular [λp1 + μp2] represents infinitely many primes.Here [x] denotes the greatest integer not exceeding x.
- Published
- 1966
41. The Construction of Representations of Lie Algebras of Characteristic Zero
- Author
-
B. Noonan
- Subjects
Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,General Mathematics ,Non-associative algebra ,Fundamental representation ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Lie conformal algebra ,Mathematics - Abstract
In this paper a procedure is given whereby, from a representation of an ideal contained in the radical, explicit representations of a Lie algebra by matrices can be constructed in an algebraically closed field of characteristic zero. The construction is sufficiently general to permit one arbitrary eigenvalue to be assigned to the representation of each basis element of the radical not in the ideal. The theorem of Ado is proved as an application of the construction. While Ado's theorem has several proofs (1; 3; 5; 6), the present one has a value in its explicitness and in the fact that the degree of the representation can be given.
- Published
- 1962
42. Systems Of Linear Congruences
- Author
-
A. T. Butson and B. M. Stewart
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Congruence relation ,01 natural sciences ,Mathematics - Abstract
1. Introduction. On recent occasions papers have been presented concerned with the problem of solving a system of linear congruences. Apparently the authors were not aware that this problem was solved very neatly and completely a long time ago by H. J. S. Smith (5; 6).
- Published
- 1955
43. An Existence Theorem for Generalized Direct Products with Amalgamated Subgroups
- Author
-
C. Y. Tang
- Subjects
Algebra ,Mathematics::Logic ,Mathematics::Group Theory ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Existence theorem ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Generalized direct products with amalgamated subgroups were introduced by B. H. Neumann and Hanna Neumann in their joint paper (4). In general, we call a given collection of groups with specified subgroups amalgamated an amalgam of groups; if all groups are abelian we speak of an abelian amalgam. The group freely generated by the amalgam is called the abelian free sum of the amalgam provided it contains the amalgam isomorphically. The free abelian sum need not exist. Hence one of the problems is to find necessary and sufficient conditions for its existence.
- Published
- 1966
44. Residue Free Differentials and the Cartier Operator for Algebraic Function Fields of one Variable
- Author
-
Tetsuo Kodama
- Subjects
Algebra ,Residue (complex analysis) ,General Mathematics ,Invertible sheaf ,Algebraic function ,Mathematics - Abstract
Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:D(A) : the K-module of all differentials,G(A) : the K-module of all differentials of the first kind,R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).
- Published
- 1972
45. Characteristically Nilpotent Algebras
- Author
-
T. S. Ravisankar
- Subjects
Algebra ,Nilpotent ,Quantitative Biology::Neurons and Cognition ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Nilpotent group ,01 natural sciences ,Physics::History of Physics ,Mathematics - Abstract
Our aim in this paper is to extend (Theorem 1.7) to general algebras a classical result of Lie algebras due to Léger and Togo [6]. This extension requires, in turn, extension to general algebras of the concept of characteristically nilpotent algebras introduced by Dixmier and Lister [3] for Lie algebras. Based on this extended concept, we introduce in § 2 a new concept of radical (and semisimplicity) for general algebras and Lie triple systems. We study in some detail the consequences of the newly introduced concepts, furnishing necessary examples. With a stronger notion of characteristically nilpotent Mal'cev algebra arising out of these concepts, we obtain (Proposition 3.6) for such an algebra the parallel to the Leger-Tôgô result mentioned at the outset. In § 4, we deal with a further generalization of the concept of characteristic nilpotency leading to extension of very recent results of Chao [1] and Tôgô [12].
- Published
- 1971
46. Generalized Discrete Valuation Rings
- Author
-
H.-H. Brungs
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Discrete valuation ,01 natural sciences ,Mathematics - Abstract
Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).In the following, R will always denote a ring with unit element satisfying (W).
- Published
- 1969
47. Direct Products of Normed Linear Spaces
- Author
-
William B. Jones
- Subjects
Algebra ,Functional analysis ,General Mathematics ,Reflexive space ,Mathematics - Abstract
In this paper we shall study properties of a locally convex space (l.c.s.) which guarantee that it is a direct product of normed linear spaces or Banach spaces. The conditions will be given both as properties of the original space itself and as properties of the dual, and will take the form of a completeness condition and the existence of sub-basic sets of pseudo-norms with certain properties (a set of pseudo-norms is basic if the set of unit balls of its members is a base of neighbourhoods of 0.
- Published
- 1967
48. Singular Integrals on Ultraspherical Series
- Author
-
Charles F. Dunkl
- Subjects
Algebra ,Series (mathematics) ,General Mathematics ,Singular integral ,Mathematics - Abstract
One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G.
- Published
- 1972
49. A Note on the Mathieu Groups
- Author
-
Lowell J. Paige
- Subjects
Algebra ,General Mathematics ,Mathematics - Abstract
The principal result of this paper is the representation of the Mathieu group M23 as a group of 11 × 11 matrices over the Galois Field GF(2). This is a new representation of M23 and in §5 an indication of how the techniques of this result might be extended to the Mathieu group M11 is given.
- Published
- 1957
50. On Hereditary and Cohereditary Modules
- Author
-
M. S. Shrikhande
- Subjects
Algebra ,Section (category theory) ,Generalization ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
A recent paper by Goro Azumaya on M-projective and M-injective modules [1] suggests a generalization of the concept of hereditary rings to modules which is also capable of dualization. Section 2 is devoted to preliminaries on M-projective and M-infective modules.In section 3, we introduce the notion of hereditary and cohereditary modules. An R-module is called hereditary if every R-submodule of it is projective. Cohereditary modules are defined dually.
- Published
- 1973
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