Showing papers in "Canadian Mathematical Bulletin in 1996"
TL;DR: In this paper, the spectral flow is defined as the dimension of the nonnegative eigenspace at the end of a path of a self-adjoint Fredholm operator on a separable Hilbert space.
Abstract: We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt } is a path of such operators, we can associate to {Bt } an integer, sf({Bt }), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {Bt } the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0. We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.
229 citations
TL;DR: In this paper, the convex set of all positive linear maps from the matrix algebra Mn (ℂ) into itself is characterized and a join homomorphism from the complete lattice of all faces of the complete set of faces of all subspaces of ℂn is constructed.
Abstract: Let denote the convex set of all positive linear maps from the matrix algebra Mn (ℂ) into itself. We construct a join homomorphism from the complete lattice of all faces of into the complete lattice of all join homomorphisms between the lattice of all subspaces of ℂn . We also characterize all maximal faces of .
44 citations
TL;DR: In this article, the authors considered the delay differential equation where α(t and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer.
Abstract: Consider the delay differential equation where α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.
36 citations
TL;DR: In this article, a countably compact topological subsemigroup of which a group is not a group was constructed under MA-countable, hence a counterexample for the Wallace problem.
Abstract: We construct, under MAcountable, a countably compact topological subsemigroup of which is not a group, hence a counterexample for the Wallace problem. We also show that there is no p-compact counterexample for the Wallace problem, answering a question of D. Grant. Finally, we show that—in some sense—our counterexample for the Wallace problem constructed under MAcountable cannot be done in ZFC.
32 citations
TL;DR: In this article, it was shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic.
Abstract: It is shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic. As a corollary, we deduce vanishing and finiteness theorems for Riemannian foliations without assuming the harmonicity of the basic mean curvature.
32 citations
TL;DR: In this article, it was shown that for any positive integer m, there exists at most one pair of integers with 0 ≤ x ≤ y ≤ m such that (x, y, m) is a solution to the Markoff equation: x 2 + y 2 + m 2 = 3xym.
Abstract: In 1913 Frobenius conjectured that for any positive integer m, there exists at most one pair of integers (x, y) with 0 ≤ x ≤ y ≤ m such that (x, y, m) is a solution to the Markoff equation: x 2 + y 2 + m 2 = 3xym. We show this is true if either m, 3m — 2 or 3m + 2 is prime, twice a prime or four times a prime.
29 citations
TL;DR: In this paper, it was shown that the uniqueness of standard diagonals in non-self-adjoint AF C*-algebras fails for any non-Self-Adjoint AF operator algebra.
Abstract: Abstract AF C*-algebras contain natural AF masas which, here, we call standard diagonals. Standard diagonals are unique, in the sense that two standard diagonals in an AF C*-algebra are conjugate by an approximately inner automorphism. We show that this uniqueness fails for non-selfadjoint AF operator algebras. Precisely, we construct two standard diagonals in a particular non-selfadjoint AF operator algebra which are not conjugate by an approximately inner automorphism of the non-selfadjoint algebra.
17 citations
TL;DR: In this paper, a new proof of the Doob-Meyer decomposition of a supermartingale into martingale and decreasing parts was given, and the proof is elementary in the sense that nothing more sophisticated than Doob's inequality is used.
Abstract: A new proof is given of the Doob-Meyer decomposition of a supermartingale into martingale and decreasing parts. Although not the most concise proof, the proof is elementary in the sense that nothing more sophisticated than Doob's inequality is used. If the supermartingale is bounded and the jump times are totally inaccessible, then it is shown that discrete time approximations converge to the decreasing part in L2. The general case is handled by reduction to the above special case.
17 citations
TL;DR: In this article, the authors characterize, by means of congruence identities, all varieties having a weak difference term and all neutral varieties, and characterize all varieties with a difference term even in the particular case of locally finite varieties.
Abstract: Abstract We characterize, by means of congruence identities, all varieties having a weak difference term, and all neutral varieties. Our characterization of varieties with a difference term is new even in the particular case of locally finite varieties.
15 citations
TL;DR: In this paper, it was shown that the ring of invariants in three variables is always Cohen-Macaulay when |G| is divisible by the characteristic of at least 4.
Abstract: Let be a representation of the finite group G over the field . If the order |G| of G is relatively prime to the characteristic of or n = 1 or 2, then it is known that the ring of invariants is Cohen-Macaulay. There are examples to show that need not be Cohen-Macaulay when |G| is divisible by the characteristic of . In all such examples is at least 4. In this note we fill the gap between these results and show that rings of invariants in three variables are always Cohen-Macaulay.
15 citations
TL;DR: In this article, the authors give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its value on the integers from 1 to 2n −1.
Abstract: Abstract We give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its values on the integers from 1 to 2n —1.
TL;DR: In this article, the dynamics of the shift map on a specific class of shift invariant spaces of nonnegative integer sequences exactly models the maps Tu for u € (0,4) associated with the generalized Renyi maps on [0,1].
Abstract: Abstract This paper continues our investigation of backward continued fractions, associated with the generalized Renyi maps on [0,1). We first show that the dynamics of the shift map on a specific class of shift invariant spaces of nonnegative integer sequences exactly models the maps Tu for u € (0,4). In the second part we construct a new family of explicit invariant measures for certain values of the parameter u.
TL;DR: A long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base such as b, is described.
Abstract: Complex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.
TL;DR: In this article, it was shown that for each critically sharp inequality of the form where is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞], there exists a continuous function with 0 ≤ Ψ ≤ Φ but with not being attained even if the supremum in (1) is attained.
Abstract: Abstract We study the question of nonexistence of extremal functions for perturbations of some sharp inequalities such as those of Moser-Trudinger (1971) and Chang- Marshall (1985). We shall show that for each critically sharp (in a sense that will be precisely defined) inequality of the form where is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞), we have a continuous Ψ on [0, ∞) with 0 ≤ Ψ ≤ Φ, but with not being attained even if the supremum in (1) is attained. We then apply our results to the Moser-Trudinger and Chang-Marshall inequalities. Our result is to be contrasted with the fact shown by Matheson and Pruss (1994) that if Ψ(t) = o(Φ(t) as t —> ∞ then the supremum in (2) is attained. In the present paper, we also give a converse to that fact.
TL;DR: In this article, a non-zero idempotent element can be represented as a sum of two nilpotent elements, where the latter is the sum of a ring and the former is a ring.
Abstract: Abstract In this paper we study in which rings a non-zero idempotent element can be presented as a sum of two nilpotent elements.
TL;DR: In this article, the pure infiniteness of C * -crossed products by endomorphisms and automorphisms was studied, and it was shown that any simple C *-crossed product A ×αZ by an automorphism α is purely infinite.
Abstract: Abstract We study the pure infiniteness of C* -crossed products by endomorphisms and automorphisms. Let A be a purely infinité simple unital C*-algebra. At first we show that a crossed product A × p N by a corner endomorphism p is purely infinite if it is simple. From this observation we prove that any simple C*-crossed products A ×αZ by an automorphism α is purely infinite. Combining this with the result in [Je] on pure infiniteness of crossed products by finite groups, one sees that if α is an outer action by a countable abelian group G then the simple C*-algebra A ×α G is purely infinite.
TL;DR: In this paper, it was shown that certain integral conditions, known to be sufficient for normality, are in fact sufficient for strong normality of a function from the hyperbolic disk of the complex plane to Riemann sphere.
Abstract: Abstract Loosely speaking, a function (meromorphic or harmonic) from the hyperbolic disk of the complex plane to the Riemann sphere is normal if its dilatation is bounded. We call a function strongly normal if its dilatation vanishes at the boundary. A sequential property of this class of functions is proved. Certain integral conditions, known to be sufficient for normality, are shown to be in fact sufficient for strong normality.
TL;DR: In this paper, it was shown that if the radical of R is countably generated as a left ideal then R is quasi-Frobenius, and the same conclusion can be drawn if r(A ∩ B) = r( A) + r(B) for all left ideals A and B of R.
Abstract: Abstract Let R be a left and right perfect right self-injective ring. It is shown that if the radical of R is countably generated as a left ideal then R is quasi-Frobenius. It is also shown that the same conclusion can be drawn if r(A ∩ B) = r(A) + r(B) for all left ideals A and B of R.
TL;DR: The notion of quasi-duality was introduced by Kraemer as discussed by the authors, who showed that each left linearly compact ring has a quasi duality if and only if it has a power series ring.
Abstract: AS a generalization of Morita duality, Kraemer introduced the notion of quasi-duality and showed that each left linearly compact ring has a quasi-duality. Let R be an associative ring with identity and R[[x]] the power series ring. We prove that (1) R[[x]] has a quasi-duality if and only if R has a quasi-duality; (2) R[[x]] is left linearly compact if and only if R is left linearly compact and left noetherian; and (3) R[[x]] has a Morita duality if and only if R is left noetherian and has a Morita duality induced by a bimodule RUS such that S is right noetherian.
TL;DR: In this paper, it was shown that any nonsingular self-generator Σ-CS module is a direct sum of uniserial Noetherian quasi-injective submodules.
Abstract: It is shown that if M is a nonsingular CS-module with an indecomposable decomposition M = ⊕ i∊I Mi , then the family {Mi | i € I} is locally semi-T"- nilpotent. This fact is used to prove that any nonsingular self-generator Σ-CS module is a direct sum of uniserial Noetherian quasi-injective submodules. As an application, we provide a new proof of Goodearl's characterization of non-singular rings over which all nonsingular right modules are projective.
TL;DR: In this article, the Abbena-Thurston manifold (M,g) is a critical point of the functional where Q is the Ricci operator and R is the scalar curvature.
Abstract: Abstract The Abbena-Thurston manifold (M,g) is a critical point of the functional where Q is the Ricci operator and R is the scalar curvature, and then the index of I(g) and also the index of — I(g) are positive at (M,g).
TL;DR: In this paper, it was shown that the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant coefficients.
Abstract: As it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant —n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.
Abstract: Abstract Nonnegative product linearization of the Askey-Wilson polynomials is shown for a wide range of parameters. As a corollary we obtain Rahman's result on the continuous q-Jacobi polynomials with α ≥ β > — 1 and α + β + 1 ≥ 0.
TL;DR: In this article, the authors determine all pairs (R 1, R 2) of irreducible root systems such that R 1 −> R 2 and R 2 − > R 2.
Abstract: If R 1 and R 2 are root systems and there is a linear map which maps R 1 ∪{0} onto R 2∪{0} we write R 1 —> R 2. We determine all pairs (R 1, R 2) of irreducible root systems such that R 1 —> R 2.
TL;DR: In this article, it was shown that every Turaev-Viro invariant for 3-manifolds is a sum of three new invariants and discuss their properties.
Abstract: We show that every Turaev-Viro invariant for 3-manifolds is a sum of three new invariants and discuss their properties. We also find a solution of a conjecture of L. H. Kauffman and S. Lins. Tables of the invariants for closed orientable 3-manifolds of complexity ≤ 3 are presented at the end of the paper.
TL;DR: In this article, it was shown that the relation [CK] ⊂ int(ΠK) holds for K in a dense open set of convex bodies, in the Hausdorff metric.
Abstract: Abstract Let d ≥ 2, and K ⊂ ℝd be a convex body with 0 ∈ int K. We consider the intersection body IK, the cross-section body CK and the projection body ΠK of K, which satisfy IK ⊂ CK ⊂ ΠK. We prove that [bd(IK)] ∩ [bd(CK)] ≠ (a joint observation with R. J. Gardner), while for d ≥ 3 the relation [CK] ⊂ int(ΠK) holds for K in a dense open set of convex bodies, in the Hausdorff metric. If IK = c ˙ CK for some constant c > 0, then K is centred, and if both IK and CK are centred balls, then K is a centred ball. If the chordal symmetral and the difference body of K are constant multiples of each other, then K is centred; if both are centred balls, then K is a centred ball. For d ≥ 3 we determine the minimal number of facets, and estimate the minimal number of vertices, of a convex d-polytope P having no plane shadow boundary with respect to parallel illumination (this property is related to the inclusion [CP] ⊂ int(ΠP)).
TL;DR: In this paper, the derived length of an A-group G is bounded by the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists.
Abstract: Abstract An A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.
TL;DR: A geometric invariant is associated to the parabolic moduli space on a marked surface and is related to the symplectic structure of the moduli spaces as mentioned in this paper, where the invariant can be expressed as
Abstract: A geometric invariant is associated to the parabolic moduli space on a marked surface and is related to the symplectic structure of the moduli space.
TL;DR: In this paper, the Lindelöf number of Y with Z of X is compared with the corresponding Lindeløf number for a dense subset of X. The authors show that if X has a topological property that is closed hereditary, closed under countable unions and closed under continuous images, then X has isproperty if and only if Y has.
Abstract: Abstract Let X and Y be Tychonov spaces and suppose there exists a continuous linear bijection from Cp(X)to CP(Y). In this paper we develop a method that enables us to compare the Lindelöf number of Y with the Lindelöf number of some dense subset Z of X. As a corollary we get that if for perfect spaces X and Y, CP(X) and Cp(Y)are linearly homeomorphic, then the Lindelöf numbers of Jf and Fare equal. Another result in this paper is the following. Let X and Y be any two linearly ordered perfect Tychonov spaces such that Cp(X)and Cp(Y)are linearly homeomorphic. Let be a topological property that is closed hereditary, closed under taking countable unions and closed under taking continuous images. Then X has isproperty if and only if Y has. As examples of such properties we consider certain cardinal functions.