230 results
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152. On a Realization of Prime Tangles and Knots
- Author
-
Quach Thi Cam Van
- Subjects
Algebra ,General Mathematics ,Realization (systems) ,Prime (order theory) ,Mathematics - Abstract
The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles.
- Published
- 1983
153. Multipliers for Amalgams and the Algebra S0(G)
- Author
-
Maria L. Torres De Squire
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Dual group ,Function (mathematics) ,01 natural sciences ,Algebra ,0103 physical sciences ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Abelian group ,Algebra over a field ,Haar measure ,Mathematics - Abstract
Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by
- Published
- 1987
154. On Difference Operators and their Factorization
- Author
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Rodney Nillsen and Patrick J. Browne
- Subjects
General Mathematics ,010102 general mathematics ,Incomplete LU factorization ,Operator theory ,01 natural sciences ,Congruence of squares ,Algebra ,Factorization ,Factorization of polynomials ,0103 physical sciences ,010307 mathematical physics ,Dixon's factorization method ,0101 mathematics ,Mathematics - Abstract
Throughout this paper A will be used to denote a given set and g a permutation of it. We shall assume that there is a subset C ⊆ A so that1Here Z denotes the set of integers. For x ∈ A it now follows that there is an unique α(x) ∈ Z so that2and then alsoIn general we shall be concerned with solving the following equation for u3where pi, n ≤ i ≤ r, and v are given real valued functions on A and pnpr does not vanish on A. For B ⊆ A, F(B) will denote the set of all real valued functions defined on B.
- Published
- 1983
155. Algebras of Analytic Operators associated with a Periodic Flow on a Von Neumann Algebra
- Author
-
Baruch Solel
- Subjects
Jordan algebra ,General Mathematics ,010102 general mathematics ,Tomita–Takesaki theory ,Operator theory ,01 natural sciences ,C*-algebra ,Algebra ,Von Neumann's theorem ,symbols.namesake ,Von Neumann algebra ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Abelian von Neumann algebra ,Affiliated operator ,Mathematics - Abstract
Let M be a σ-finite von Neumann algebra and {σt}t∊T be a σ-weakly continuous representation of the unit circle, T, as *-automorphisms of M. Let H∞(σ) be the set of all x ∊ M such thatThe structure of H∞(σ) was studied by several authors (see [2-13]).The main object of this paper is to study the σ-weakly closed subalgebras of M that contain H∞(σ). In [12] this was done for the special case where H∞(σ) is a nonselfadjoint crossed product.Let Mn, for n ∊ Z, be the set of all x ∊ M such that
- Published
- 1985
156. The Transfer of the Krull Dimension and the Gabriel Dimension to Subidealizers
- Author
-
Mark L. Teply and Günter Krause
- Subjects
Algebra ,Krull's principal ideal theorem ,Pure mathematics ,Transfer (group theory) ,Dimension (vector space) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Krull dimension ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let M be a right ideal of the ring T with identity. A unital subring R of T which contains M as a two-sided ideal is called a subidealizer ; the largest such subring is the idealizer I (M) of M in T. M is said to be generative if TM = T. In this case M is idempotent, and it follows from the dual basis lemma that T is finitely generated projective as a right R-module (see [7, Lemma 2.1]); we make frequent use of these two facts in this paper.
- Published
- 1977
157. Criteria for Total Projectivity
- Author
-
Paul Hill
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.
- Published
- 1981
158. The Decomposition of the Module of n-th Order Differentials in Arbitrary Characteristic
- Author
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Klaus G. Fischer
- Subjects
Algebra ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Decomposition (computer science) ,Order (group theory) ,Mathematics - Abstract
Throughout this paper, it is assumed that A is the complete, equicharacteristic, local ring of an algebraic curve at a one-branch singularity whose residue field is algebraically closed and contained in A. Hence, the domain A is dominated by only one valuation ring in its quotient held F, and if t is a uniformizing parameter, then the integral closure of A in F, denoted by Ā, is [[t]].
- Published
- 1978
159. On the Structure of Certain Nest Algebra Modules
- Author
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G. J. Knowles
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Structure (category theory) ,010307 mathematical physics ,Nest algebra ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let be a nest algebra of operators on some Hilbert space H. Weakly closed -modules were first studied by J. Erdos and S. Power in [4]. It became apparent that many interesting classes of non self-adjoint operator algebras arise as just such a module. This paper undertakes a systematic investigation of the correspondence which arises between such modules and order homomorphisms from Lat into itself. This perspective provides a basis to answer some open questions arising from [4]. In particular, the questions concerning unique “determination” and characterization of maximal and minimal elements under this correspondence, are resolved. This is then used to establish when the determining homomorphism is unique.
- Published
- 1987
160. 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
161. 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
162. A Characterization of the Hyperhomology Groups of the Tensor Product
- Author
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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
163. 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
164. 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
165. 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
166. 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
167. 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
168. 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
169. Systems Of Linear Congruences
- Author
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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
170. 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
171. 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
172. 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
173. 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
174. 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
175. 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
176. 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
177. 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
178. Topics in Direct Differential Geometry
- Author
-
Ralph Park
- Subjects
Algebra ,Differential geometry ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In the theory of curves, one often makes differentiability assumptions in order that analytic methods can be used. Then one tries to weaken these assumptions as much as possible. The theory of curves which is presented here uses geometric methods, such as central projection, rather than analysis. In this way, no analytic assumptions are needed and a purely geometric theory results. Since this theory is not so well known as the analytic one, I have tried to make the treatment as self-contained as possible. It is hoped that this paper will form a quick introduction for a reader who has had no previous acquaintance with the subject.We assume that our curves satisfy a condition, which we call direct differentiability. Roughly this condition is that, at each point of the curve, all the osculating spaces exist.
- Published
- 1972
179. On Certain Onto Maps
- Author
-
Isaac Namioka
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.
- Published
- 1962
180. Spectral Theory for a Class of Nonnormal Operators II
- Author
-
Harry Gonshor
- Subjects
Algebra ,Discrete mathematics ,Class (set theory) ,Spectral theory ,General Mathematics ,Notation ,Unitary state ,Equivalence (measure theory) ,Mathematics - Abstract
In a previous paper (2) we have developed a spectral theory and a unitary equivalence theory for a certain class of non-normal operators. We dealt primarily with operators which were called J 2 operators. At present we are interested in studying the uniformly closed rings generated by such operators. The notation will be the same as in (2).
- Published
- 1958
181. Defining Families for Integral Domains of Real Finite Character
- Author
-
William Heinzer and Jack Ohm
- Subjects
Algebra ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Finite character ,Mathematics - Abstract
Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).
- Published
- 1972
182. Hermitian Varieties in a Finite Projective Space PG(N, q2)
- Author
-
Raj Chandra Bose and I. M. Chakravarti
- Subjects
Hermitian symmetric space ,Pure mathematics ,Quadric ,Projective unitary group ,General Mathematics ,010102 general mathematics ,Galois theory ,01 natural sciences ,Hermitian matrix ,Prime (order theory) ,Algebra ,0103 physical sciences ,Projective space ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.
- Published
- 1966
183. On Nilpotent Products of Cyclic Groups
- Author
-
Ruth Rebekka Struik
- Subjects
Algebra ,Nilpotent ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Cyclic group ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.
- Published
- 1960
184. Induced Representations and Invariants
- Author
-
G. de B. Robinson
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.
- Published
- 1950
185. Linear Transformations on Grassmann Spaces
- Author
-
Roy Westwick
- Subjects
Algebra ,Linear map ,General Mathematics ,Mathematics - Abstract
1. Let U denote an n-dimensional vector space over a field F and let Gnr denote the set of non-zero decomposable r-vectors of the Grassmann product space ΛrU. Let T be a linear transformation of ΛrU into itself which maps Gnr into itself. If F is algebraically closed, or if T is non-singular, then the structure of T is known. In this paper we show that if T is singular, then the image of ΛrU has a very special form with dimension equal to the larger of the integers r + 1 and n – r + 1. We give an example to show that this can occur.
- Published
- 1969
186. The Extent of the Sequence Space Associated with a Basis
- Author
-
William H. Ruckle
- Subjects
Algebra ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Sequence space ,Mathematics - Abstract
The associated sequence space S of a sequence of vectors {xn} in a Banach space consists of all scalar sequences (sn) for which converges. My primary motivation in writing this paper was to present a new proof to a recent theorem of N. I. and V. I. Gurarii concerning limits of extent on S when {xn} is a basis of a uniformly convex or a uniformly smooth Banach space [5], This theorem is stated as Theorem 2.4. Several interesting consequences of this theorem were noted by N. I. Gurarii in [3] and [4].
- Published
- 1972
187. Some Theorems On Matrices With Real Quaternion Elements
- Author
-
N. A. Wiegmann
- Subjects
Hurwitz quaternion ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Hypercomplex analysis ,Divisor (algebraic geometry) ,01 natural sciences ,Unitary state ,Matrix similarity ,Algebra ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Quaternion matrix ,010307 mathematical physics ,Isomorphism ,0101 mathematics ,Quaternion ,Mathematics - Abstract
Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.
- Published
- 1955
188. The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra
- Author
-
Ole A. Nielsen
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,Algebra ,Set (abstract data type) ,symbols.namesake ,Von Neumann algebra ,0103 physical sciences ,symbols ,Direct integral ,010307 mathematical physics ,0101 mathematics ,Abelian von Neumann algebra ,Affiliated operator ,Mathematics - Abstract
The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.
- Published
- 1971
189. An Inequality Concerning Analytic Functions with a Positive Real Part
- Author
-
Thomas H. MacGregor
- Subjects
Algebra ,Discrete mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Log sum inequality ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Cauchy–Schwarz inequality ,Mathematics ,media_common ,Analytic function - Abstract
This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for |z| < 1 and has values lying in a strip of width δ. This result is known and was first proved by Tammi (1).Our second theorem is a generalization of this. Namely, ifis analytic for |z| < 1 and satisfies Re{zmf(m>(z)} ≧ A andthenconverges.Another application of our fundamental inequality is the following. Let be analytic for |z| < 1 and satisfy Re p(z) ≧ 0 and set and .
- Published
- 1969
190. On the Derivation Algebras of Lie Algebras
- Author
-
Shigeaki Tôgô
- Subjects
General Mathematics ,010102 general mathematics ,Non-associative algebra ,Killing form ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,0103 physical sciences ,010307 mathematical physics ,Nest algebra ,0101 mathematics ,Generalized Kac–Moody algebra ,Mathematics - Abstract
LetLbe a Lie algebra over a field of characteristic 0 and letD(L)be the derivation algebra ofL, that is, the Lie algebra of all derivations ofL. Then it is natural to ask the following questions: What is the structure ofD(L)?What are the relations of the structures ofD(L)andL? It is the main purpose of this paper to present some results onD(L)as the answers to these questions in simple cases.Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebraLis not completely determined by the structure ofD(L). However, there is still some intimate connection between the structure ofD(L)and that ofL.
- Published
- 1961
191. On Additive Operators
- Author
-
N. A. Friedman and A. E. Tong
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4])
- Published
- 1971
192. Variety Invariants for Modular Lattices
- Author
-
Rudolf Wille
- Subjects
Algebra ,business.industry ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Variety (universal algebra) ,Modular design ,business ,Mathematics - Abstract
A variety (primitive class) is a class of abstract algebras which is closed under the formation of subalgebras, homomorphic images, and products. For a given variety we shall call a function μ*, which assigns to each algebra a natural number or ∞, denoted by μ*(A), a variety invariant if for every natural number n the class of all with μ*(A) ≦ n is again a variety. In this paper, a general method of finding variety invariants for the variety of all modular lattices will be developed. This method will be based on the concept of a quotient tree of a modular lattice. As examples of variety invariants we shall define, using the general result, the primitive length and the primitive width of modular lattices.
- Published
- 1969
193. A Regular Singular Functional
- Author
-
A. D. Martin
- Subjects
Algebra ,General Mathematics ,Mathematics - Abstract
1. Introduction. In a joint paper with Leighton (2), the author considered quadratic functionals of the type1.1 (0 < a < b)in which x = 0 is a singular point of the functional which is otherwise regular on [0, b]. The hypothesis on a regular functional includes the assumption that r is continuous and positive on a closed interval [0, b].
- Published
- 1956
194. On D. E. Littlewood's Algebra of S-Functions
- Author
-
D. G. Duncan
- Subjects
Filtered algebra ,Algebra ,Jordan algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Division algebra ,Cellular algebra ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Mathematics - Abstract
Several papers have been written on the “new” multiplication of S-functions since Littlewood [3, p. 206] first suggested the problem. M. Zia-ud-Din [13] calculated the case {m} ⊗ {n} for mn ≤ 12, making use of the tables of the characters of the symmetric group of degree mn. Later Thrall [10,pp. 378-382] developed explicit formulae for the cases {m} ⊗ {2},{m} ⊗ {3}, {2} ⊗ {m} (where m is any integer).
- Published
- 1952
195. On the Number of Prolongations of a Finite Rank Valuation
- Author
-
Michael J. Wright
- Subjects
Algebra ,Valuation (logic) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Rank (computer programming) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
A (non-archimedean) valuation v on a field K is said to be henselian if it has a unique prolongation to a valuation on Ka, the algebraic closure of K. A henselization (Kh, vh) of a valuated field (K, v) is a smallest separable extension of K containing a henselian prolongation vh of v. (Kh, vh) is unique up to K-isomorphism, and (Kh, vh) = (K, v) if and only if v is henselian. In this paper we confine ourselves to valuations of finite rank.If v is a non-henselian rank one valuation on K, and if [Ks:K] = ∞, Ks being the separable closure of K, then it is known that v has infinitely many prolongations to Ka [1, (27.11)]. We shall see that this is no longer true if the rank of v is greater than one.
- Published
- 1971
196. Normal and Canonical Representations in Free Products of Lattices†
- Author
-
H. Lakser
- Subjects
Algebra ,Free product ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In solving the word problem for free lattices, Whitman [4] showed that free lattices admit canonical representations, that is, of all polynomials over the generating set representing an element of the lattice, the polynomial of shortest length is unique up to commutativity and associativity. These well-defined shortest polynomials have proved very important in analyzing the internal structure of free lattices in detail; see, e.g., [5].Sorkin [3] proved that the free product of chains also admits canonical representations; these were exploited by Rolf [2]. In the above-mentioned paper, Sorkin also suggested that the free product of two copies of 22 does not admit canonical representations.
- Published
- 1970
197. Unique Factorization Theorems for Subalgebras of the Incidence Algebra
- Author
-
K. L. Yocom
- Subjects
Algebra ,Incidence algebra ,General Mathematics ,Factorization of polynomials ,Unique factorization domain ,Mathematics - Abstract
H. Scheid [4] has found necessary and sufficient conditions on a partially ordered set S(≦) which is a direct sum of a countable number of trees for a certain subalgebra G(+, *) of the incidence algebra F(+, *) to be an integral domain. In this paper we prove that under similar conditions on S, G(+, *) is actually a unique factorization domain or, failing this, that there is a subalgebra H(+, *) of F(+, *) which is a unique factorization domain and contains G. Similar results are then obtained as corollaries in the regular convolution rings of Narkiewicz.
- Published
- 1972
198. Some Configurations in Finite Projective Spaces and Partially Balanced Incomplete Block Designs
- Author
-
D. K. Ray-Chaudhuri
- Subjects
Algebra ,General Mathematics ,Incomplete block ,Projective test ,Mathematics ,Block design - Abstract
Using the methods developed in (2 and 3), in this paper we study some properties of the configuration of generators and points of a cone in an w-dimensional finite projective space. The configuration of secants and external points of a quadric in a finite plane of even characteristic is also studied. I t is shown that these configurations lead to several series of partially balanced incomplete block (PBIB) designs. PBIB designs are defined in Bose and Shimamoto (1). A PBIB design with m associate classes is an arrangement of v treatments in b blocks such that.
- Published
- 1965
199. Elementary Factorization in π-Regular Rings
- Author
-
Arthur Steger
- Subjects
Principal ideal ring ,Ring (mathematics) ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Unique factorization domain ,01 natural sciences ,Set (abstract data type) ,Algebra ,Factorization ,Principal ideal ,0103 physical sciences ,Partition (number theory) ,010307 mathematical physics ,0101 mathematics ,Commutative property ,Mathematics - Abstract
This paper extends the results of A. L. Foster (1) on elementary factorization in Boolean-like rings to commutative π-regular rings. After proving some preliminary lemmas we proceed to the partition of the set of non-units of a π-regular ring into irreducible and composite elements. Finally, we prove a number of theorems concerning factorization rings, weakly unique factorization rings, principal ideal rings, etc. The principal result is that a π-regular ring is a weakly unique factorization ring if and only if it is a principal ideal ring.
- Published
- 1966
200. A Note on Induced Modules
- Author
-
Charles W. Curtis
- Subjects
Algebra ,General Mathematics ,Mathematics - Abstract
In this paper, A denotes a ring with an identity element 1, and B a subring of A containing 1 such that B satisfies the left and right minimum conditions, and A is a finitely generated left and right B-module. The identity element 1 is required to act as the identity operator on all modules which we shall consider. For any left B-module V, there is a standard construction of a left A -module which is, roughly speaking, the smallest A -module containing V.
- Published
- 1961
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