Showing papers in "Canadian Journal of Mathematics in 1969"
TL;DR: A special case of the theorem of Marcinkiewicz as discussed by the authors states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q, q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the
Abstract: A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q,q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper (5), as part of a more general theorem, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the functions in X. One of Calderόn's results implies that if X is a function space in the sense of Luxemburg (9), then X must be a rearrangement-invariant space.
199 citations
TL;DR: In this article, it was shown that Euclidean Lie algebras are not simple, and they were shown that E((A ij)) is not simple under the assumption that A ij does not fall in the list given in (4, Table).
Abstract: Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however. The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.
100 citations
TL;DR: In this paper, the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B is studied, and the question of when such a group is free is investigated.
Abstract: We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the form while Chang, Jennings, and Ree (4) took them in the form
80 citations
TL;DR: In this article, it was shown that if α and β are positive irrational numbers satisfying 1 then the sets [nα], [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers, if and only if β and α are irrational.
Abstract: The following result is well known (as usual, [x]denotes the integral part of x): (A) Let α and β be positive irrational numbers satisfying 1 Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem. The theorem has a converse, and the following holds: (B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.
79 citations
TL;DR: In this paper, it was shown that a ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilator terminates at some point, and this condition is inherited by subrings.
Abstract: Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring. Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly. Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings. Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.
70 citations
TL;DR: A great number of papers have been written on Stone lattices and a very satisfactory theory evolved as mentioned in this paper, and it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattice.
Abstract: Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).
57 citations
TL;DR: In this paper, the authors relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prufer domain, a Dedekind domain, or an almost-definite domain.
Abstract: Let D be an integrally closed domain with identity having quotient field L. If {Vα } is the set of valuation overrings of D and if A is an ideal of D, then A = ∪ α AV α is an ideal of D called the completion of A. If X is an indeterminate over D and f ∈ D[X], then we denote by Af the ideal of D generated by the coefficients of f. The Kronecker function ring DK of D is defined by DK = {f/g| f, g ∈ D[X], Af ⊆ Ag } (4, p. 558); and the domain D(X) is defined by D(X) = {f/g| f, g ∈ D[X], Ag = D} (5, p. 17). In this paper we wish to relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prufer domain, a Dedekind domain, or an almost Dedekind domain.
56 citations
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49 citations
TL;DR: In this article, a generalization of Herstein's result to semi-simple rings was proposed, where the degree of the lowest monomial of a polynomial is in the left-hand side of (1) ((1*)).
Abstract: Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that 1 then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition: (1)* Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).
47 citations
TL;DR: Theorem 2 below is a generalization of Arveson's theorem as mentioned in this paper for weakly closed algebra with an m.a.s.y operator on the complex Hilbert space.
Abstract: If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies . Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.) ARVESON's THEOREM. If is a weakly closed algebra which contains an m.a.s.a.y and if Lat , then is the algebra of all operators on . A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.
TL;DR: In this paper, the centralizer of every involution has a normal 2-complement, and we call such a group an I-group, i.e., a group whose centralizer has a 2 complementary centralizer.
Abstract: In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity,
we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition. As is well-known, the automorphism group G = PΓL(2, q) of H= PSL(2, q), q= pn , is of the form G = LF, where L = PGL(2, q), L ⨞ G, F is cyclic of order n, L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q) acts; cf. (3, Lemma 2.1) or (7, Lemma 3.3). It follows at once (4, Lemma 2.1; 8, Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.
TL;DR: In this article, a discussion of hyperbolic geometry on the open unit disc D, its metric and its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜
Abstract: A Blaschke product on the unit disc, where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the
condition C is satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product and for some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metric and its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜
TL;DR: In this article, it was shown that all automorphisms of a Chevalley group can be conjugated to any automorphism induced by a group of positive roots.
Abstract: In (8, § 3.2) Steinberg proved the following result. THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1’) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms. For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.
TL;DR: In this paper, it was shown that the question of whether a Fuchs 5-group is a direct sum of countable groups is open and the answer is negative even in the commutative case.
Abstract: The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property. Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G. Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.
TL;DR: In this paper, the problem of reconstructing cacti with a single cycle was solved by assuming both Kelly's Theorem and the result of Manvel in (10).
Abstract: Following the work of Kelly (8), Harary and Palmer (5), and Bondy (1) on the reconstruction of trees, and of Manvel (10) on the reconstruction of connected graphs with a single cycle, it was a natural step to attempt to solve the reconstruction problem for cacti. The solution of this problem, presented here, assumes both Kelly's Theorem and the result of Manvel in (10). Any definitions not given here can be found in (2). Let graph G have point set V = {v1 v2, …, vp} and line set X = {x1, x2, …, xq}. For each vi ∈ V, Gi = G – vi is the maximal subgraph of G which does not contain vi and is formed by deleting vi and all lines incident with it from G.
TL;DR: The concept of an n-abelian group is central to group theory and many generalizations of the concept have been considered and exploited as discussed by the authors, for example, in the case of n = p, a prime.
Abstract: The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if, (xy) n = xnyn for all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2. Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.
TL;DR: In this paper, the authors studied the question of automorphisms of integral group rings of finite groups with rational coefficients, and they proved that if two groups have isomorphic integral groups, then the groups are isomorphic.
Abstract: In this note we study the question of automorphisms of the integral group ring Z(G) of a finite group G. We prove that if G is nilpotent of class two, any automorphism of Z(G) is composed of an automorphism of G and an inner automorphism by a suitable unit of Q(G), the group algebra of G with rational coefficients. In § 3, we prove that if two finitely generated abelian groups have isomorphic integral group rings, then the groups are isomorphic. This is an extension of the classical result of Higman (2) for the case of finite abelian groups. In the last section we give a new proof of the fact that an isomorphism of integral group rings of finite groups preserves the lattice of normal subgroups. Other proofs are given in (1;4).
TL;DR: The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity to the notion of a multiplicative ideal.
Abstract: The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used. Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC. Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RP is the quotient ring formed using the complement of P.
TL;DR: In this paper, the structure of a relative Stone lattice is investigated in terms of prime ideals, topological representation, completeness, and the reduced triple associated with a Stone algebra L, as introduced in the first part of this paper.
Abstract: Using the triple associated with a Stone algebra L, as introduced in the first part of this paper (1), we will investigate certain problems concerning the structure of a Stone lattice. The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple. It is assumed that the reader is familiar with §§ 2–4 of (1). For the sake of convenience, we will write L= 〈C, D, ϕ〉 to indicate that 〈C, D, ϕ〉 is the triple associated with L, and whenever convenient we will write the elements of L as ordered pairs 〈x, a〉, as it is given in (1, § 4, the Construction Theorem).
TL;DR: In this article, the authors dealt with the number of zeros of a solution of the nth order linear differential equation 1.1 where the functions pj(z) (j = 0, 1, …, n-2) are assumed to be regular in a given domain D of the complex plane.
Abstract: In this paper we deal with the number of zeros of a solution of the nth order linear differential equation 1.1 where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.) The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function 1.2 where y 1(z) and y 2(z) are two linearly independent solutions of 1.3 is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.
TL;DR: In this paper, it was shown that the design of points and hyperplanes of a finite projective or affine space is isomorphic to that of a point and two distinct blocks, if and only if there are positive integers v, k, and y.
Abstract: A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following. Theorem 1. A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4 if and only if there are positive integers v, k, and y, with μ > 1 and (μ – l)(v — k) ≠ (k — μ)2 such that the following assumptions hold. (I) Every block is on k points, and every two intersecting blocks are on μ common points. (II) Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks. (III) Given two distinct points p and q, there is a block on p but not on q. (IV) There are v points, and v– 2 ≧ k > μ.
TL;DR: For a ring R with a 1 and n ≧ 2, the (n × n) matrix ring over R and its subring of lower triangular matrices are UA-rings as discussed by the authors.
Abstract: A semigroup (R, ⋅) is said to be a unique addition ring (UA-ring) if there exists a unique binary operation + making (R, ⋅, + ) into a ring. All our results can be presented in this semigroup theoretic setting. However, we prefer the following equivalent ring theoretic formulation: a ring R is a UA-ring if and only if any semigroup isomorphism α: (R, ⋅) ≅ (S, ⋅) with another ring S is always a ring isomorphism. UA-rings have been studied in (8; 4) and are also touched on in (1; 2; 6; 7). In this note we generalize Rickart's methods to much wider classes of rings. In particular, we show that, for a ring R with a 1 and n ≧ 2, the (n × n) matrix ring over R and its subring of lower triangular matrices are UA-rings.
TL;DR: In this paper, the second in a series of papers discussing linear groups of prime degree, the first being (8), was presented. But the authors only considered linear groups with degree 7.
Abstract: 1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.
TL;DR: In this article, the selection problem of a multifunction from X to Y is studied, where the objective is to determine which structures on X and Y and which definitions of measurability of F ensure that F will have a measurable selector.
Abstract: Let F: X → Y be a multifunction from X to Y. Then, given measure-theoretic or topological structures on X and Y, it is possible in various ways to define the measurability of F. The selection problem is to determine which structures on X and Y and which definitions of measurability of F ensure that F will have a measurable selector. This problem has been studied recently in papers by Castaing (2) and Kuratowski and Ryll-Nardzewski (6). In the latter paper, the problem is studied for its own interest. The former uses solutions of the problem to obtain general Filippov-type theorems. (See, for example, the corollaries following Theorems 2 and 3 of Castaing's paper.) For other material on Filippov's results see, among others, (3; 4; 5; 7; 9).