Showing papers in "Canadian Journal of Mathematics in 2003"
TL;DR: In this article, the shortest-time problem on a Riemannian space with an external force was studied, and it was shown that the problem can be converted to a shortest path problem on the Randers space.
Abstract: In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.
129 citations
TL;DR: The infinitely divisible distributions of random variables with Laplace transforms are characterized for various in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Levy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transform.
Abstract: The infinitely divisible distributions on of random variables with Laplace transforms respectively are characterized for various in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Levy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet -function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.
111 citations
TL;DR: In this article, the authors considered the model subspaces K = H 2 ⊖ �H 2 of the Hardy space H 2 generated by an inner function in the upper half plane.
Abstract: This paper is a continuation of (6). We consider the model subspaces K� = H 2 ⊖ �H 2 of the Hardy space H 2 generated by an inner functionin the upper half plane. Our main object is the class of admissible majorants for K�, denoted by Admand consisting of all functions ! defined on R such that there exists an f 6 0, f ∈ Ksatisfying | f(x)| ≤ !(x) almost everywhere on R. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any Kgenerated by a meromorphic inner function. In contrast with (6), we consider the generating functionssuch that the unit vector �(x) winds up fast as x grows from −∞ to ∞. In particular, we consider � = B where B is a Blaschke product with "horizontal" zeros, i.e., almost uniformly distributed in a strip parallel to and separated from R. It is shown, among other things, that for any such B, any even ! decreasing on (0, ∞) with a finite logarithmic integral is in Adm B (unlike the "vertical" case treated in (6)), thus generalizing (with a new proof) a classical result related to Adm exp(iz), � > 0. Some oscillating !'s in Adm B are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm exp(iz), � > 0, and to de Branges' space H(E).
68 citations
TL;DR: In this paper, the authors give a constant for the convergence time of a multiplicative function f in terms of the location of the absolute value of y that minimizes the sum above.
Abstract: For given multiplicative function f, with | f(n)| ≤ 1 for all n, we are interested in how fast its mean value (1/x) P nx f(n) converges. Halshowed that this depends on the minimum M (over y ∈ R) of P px 1 − Re( f(p)p i y ) � /p, and subsequent authors gave the upper bound ≪ (1 + M)e M. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Hallemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of y that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k-th powers mod p.
61 citations
TL;DR: In this article, the authors extend the concept of Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admis- sible) operators that are similar to self-adjoint operators.
Abstract: We extend the concept of Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admis- sible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V 1 b HV for some self-adjoint operator b H; (ii) the operators H and b H are resolvent comparable, i.e., the difference of the resolvents of H and b H is a trace class operator (for non-real values of the spectral parameter); (iii) tr(V R RV ) = 0 whenever R is bounded and the commutator V R RV is a trace class operator. The spectral shift function �(�,H,A) associated with the pair of resolvent comparable admissible operators (H,A) is introduced then by the equality �(�,H,A) = �(�, b H, b A) where �(�, b
57 citations
TL;DR: In this article, the authors present a model with independent mutation events at the sites, and crossover events between them, which admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkaged equilibrium, derived by means of M\"obius inversion.
Abstract: It is well known that rather general mutation-recombination models can besolved algorithmically (though not in closed form) by means of Haldanelinearization. The price to be paid is that one has to work with a multipletensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, withindependent mutation events at the sites, and crossover events between them. Itadmits a closed solution of the corresponding differential equation on thebasis of the original state space, and also closed expressions for the linkagedisequilibria, derived by means of M\"obius inversion. As an extra benefit, theapproach can be extended to a model with selection of additive type acrosssites. We also derive a necessary and sufficient criterion for the mean fitnessto be a Lyapunov function and determine the asymptotic behaviour of thesolutions.
51 citations
TL;DR: In this article, the authors studied the singular integral operator defined on all test functions, where is a bounded function, is an integrable function on the unit sphere satisfying certain cancellation conditions.
Abstract: We study the singular integral operator defined on all test functions , where is a bounded function, is an integrable function on the unit sphere satisfying certain cancellation conditions. We prove that, for , extends to a bounded operator from the Sobolev space to the Lebesgue space with being a distribution in the Hardy space where . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for on the Hardy spaces, as well as the boundedness for the truncated maximal operator .
41 citations
TL;DR: In this article, the authors provide an explicit thick and thin decomposition for oriented hyperbolic manifolds M of dimension 5, which is based on the identification of the isometry group of the universal space by the matrix group PSL(2, H) of quaternionic 2 × 2-matrices with Dieudonndeterminant.
Abstract: We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds M of dimension 5. The result implies improved universal lower bounds for the volume vol5(M) and, for M compact, new estimates relating the injectivity radius and the diameter of M with vol5(M). The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group PSL(2, H) of quaternionic 2 × 2-matrices with Dieudonndeterminant � equal to 1 and isolation properties of PSL(2, H).
40 citations
TL;DR: In this article, two new variations on the Fels and Olver algorithm are presented, which under some conditions on the group action, sim- plify a moving frame construction, leading to a better understanding of invariant differential forms on the jet bundles.
Abstract: The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, sim- plify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.
40 citations
TL;DR: In this paper, a complete, explicit and self-contained geometric construction of abelian varieties associated to Gauss' and Appell-Lauricella hypergeometric series is presented.
Abstract: In this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, Wustholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.
37 citations
TL;DR: The generalized Riesz factorization (GRF) as mentioned in this paper is a factorization of inner functions in Hardy spaces that coincides with the GRF when ''z'' = z. In this paper we prove several results about the generalized GRF and apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on Hardy spaces.
Abstract: Every classical inner function ' in the unit disk gives rise to a certain factorization of func- tions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when '(z) = z. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.
TL;DR: In this article, the authors define a total ordering of the pure braid groups which is invariant under multiplication on both sides, and remark that the same type of ordering can be applied to the fundamental groups of complex hyperplane arrangements.
Abstract: We define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic. By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.
TL;DR: In this article, the authors describe new construction principles for infinite-dimensional Lie groups, and show that the weak direct product of an arbitrary family of Lie groups can be made a Lie group, modelled on the locally convex direct sum L i2I L(Gi).
Abstract: We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space (X, �,µ) and (possibly infinite-dimensional) Lie group G, we construct a Lie group L1(X, G), which is a Fr´ echet-Lie group if G is so. We also show that the weak direct product Q � i2I Gi of an arbitrary family (Gi)i2I of Lie groups can be made a Lie group, modelled on the locally convex direct sum L i2I L(Gi).
TL;DR: In this paper, upper and lower Gaussian estimates for the diffusion kernel of a divergence and non-divergence form elliptic operator in the Lipschitz domain were given.
Abstract: We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain.
TL;DR: In this paper, an explicit skein theoretical algorithm was developed to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3manifolds.
Abstract: We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology of U(1)- representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra N = Z/2 ⋉ V � R 2 on the other side. We find that both TQFT's are SL(2, R)- equivariant functors and, as such, are isomorphic. The SL(2, R)-action in the Hennings construction comes from the natural action on N and in the case of the Frohman-Nicas theory from the Hard-Lefschetz decomposition of the U(1)-moduli spaces given that they are naturally Kahler. The irreducible components of this TQFT, corresponding to simple representations of SL(2, Z) and Sp(2g, Z), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg-Witten theories, Casson type theories for homology circlesla Donaldson, higher rank gauge theories following Frohman and Nicas, and the Z/pZ reductions of Reshetikhin-Turaev theories over the cyclotomic integers Z(ζp). We also conjecture that the Hennings TQFT for quantum-sl2 is the product of the Reshetikhin-Turaev TQFT and such a homological TQFT. 1
TL;DR: In this article, the ideal structures of Cuntz algebras are determined by quasi-free actions of abelian groups and another proof of Kishimoto's result on the simplicity of such crossed products is given.
Abstract: We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and K-groups of our algebras.
TL;DR: In this paper, the moduli spaces M n of stable SL(2, C)-bundles on a Hopf surface H, from the point of view of symplectic geometry, were studied.
Abstract: A Hopf surface is the quotient of the complex surface C2 \ {0} by an infinite cyclic group of dilations of C 2 . In this paper, we study the moduli spaces M n of stable SL(2, C)-bundles on a Hopf surface H, from the point of view of symplectic geometry. An important point is that the surface H is an elliptic fibration, which implies that a vector bundle on H can be considered as a family of vector bundles over an elliptic curve. We define a map G: M n → P 2n+1 that associates to every bundle on H a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on M n . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kit is an elliptic fibration that does not admit a section.
TL;DR: In this article, Riemann-Roch et al. introduced the concept of a diviseurs equivariant to define le determinant and le degre d a tel faisceau.
Abstract: Le cadre du travail presente dans cette these est celui de la theorie equivariante des courbes, c'est-a-dire l'etude des courbes munies d'une action d'un groupe G, qu'on considere toujours fini. Le resultat essentiel est un theoreme de Riemann-Roch a valeurs dans l'anneau des caracteres du groupe considere, et qui releve le theoreme classique. Il est obtenu pour des G-faisceaux de rang quelconque grâce a l'introduction d'un groupe de diviseurs a coefficients equivariants qui permet en particulier de definir le determinant et le degre d'un tel faisceau. On applique ce theoreme au calcul de structures galoisiennes d'origine geometrique.
TL;DR: In this paper, a topology on Z, which gives a nice proof that the set of prime integers is infinite, is char- acterised and examined, and it is found to be homeomorphic to Q, with a compact completion homeo- morphic to the Cantor set.
Abstract: A topology on Z, which gives a nice proof that the set of prime integers is infinite, is char- acterised and examined. It is found to be homeomorphic to Q, with a compact completion homeo- morphic to the Cantor set. It has a natural place in a family of topologies on Z, which includes the p-adics, and one in which the set of rational primes P is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and k-free numbers.
TL;DR: In this paper, Cohen-Wu gave an integral homotopy decomposition of a double-suspension in a co-H space, where the double suspension is replaced by a cocommutative co-h space.
Abstract: James gave an integral homotopy decomposition of ��X, Hilton-Milnor one for (�X∨�Y), and Cohen-Wu gave p-local decompositions ofX if X is a suspension. All are natu- ral. Usingidempotentsandtelescopes weshowthattheJamesandHilton-Milnordecompositions have analogueswhen the suspensionsare replaced by coassociative co-H spaces, and the Cohen-Wudecom- position has an analogue when the (double) suspension is replaced by a coassociative, cocommutative co-H space.
TL;DR: For a Sturm-Liouville equation whose leading coefficient function changes sign, the authors of as mentioned in this paper established inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions, and derived the discontinuities of each eigenvalue under a different indexing scheme.
Abstract: For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prufer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.
TL;DR: Schottky-Burnside automorphic functions have been used for extremal properties of polynomials with respect to the maximum norm, the behavior of zeros and recurrence coefficients.
Abstract: It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szego polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szego polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.
TL;DR: In this paper, it was shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defined functions of the analytic variety Vt in the open set in Cn.
Abstract: Let V be an analytic variety in some open set in Cn For a real analytic curve with (0) = 0 and d ≥ 1, define Vt = t d (V − (t)) It was shown in a previous paper that the currents of integration over Vt converge to a limit current whose support T,dV is an algebraic variety as t tends to zero Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the Vt As a corollary, it is shown that T,dV is either inhomogeneous or coincides with T,�V for allin some neighborhood of d As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragm´¨ of conditions Corresponding results for limit varieties T�,�W of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case
TL;DR: In this paper, the authors present conditions under which the homomorphism induced by the inclusion of a dominant morphism is injective, i.e., injective if and only if
Abstract: Let f : E → B be a dominant morphism, where E and B are smooth irreducible complex quasi-projective varieties, and let Fb be the general fiber of f. We present conditions under which the homomorphism�1(Fb)→�1(E) induced by the inclusion is injective.
TL;DR: In this paper, the authors characterized those C 2 Hn such that every matrix in the convex hull of U(C) can be written as the average of two matrices inU(C).
Abstract: LetHn be the real linear space of n n complex Hermitian matrices. The unitary (similarity) orbitU(C) of C 2 Hn is the collection of all matrices unitarily similar to C. We characterize those C 2 Hn such that every matrix in the convex hull ofU(C) can be written as the average of two matrices inU(C). The result is used to study spectral properties of submatrices of matrices inU(C), the convexity of images ofU(C) under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.
TL;DR: In this paper, the µ-asymptotic cycle was extended to the one-dimensional case, and sufficient conditions for the existence of such a quantifier were given, together with some applications.
Abstract: Given a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure �, Sullivan has defined an element of Hp(Mn, R). This generalized the notion of a µ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure µ. In this one-dimensional case there was a natural 1-1 corre- spondencebetween transversalinvariantmeasuresandinvariantmeasuresµwhenonehada smooth flow without stationary points. For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.
TL;DR: In this article, the isomorphism classes of nongraded Hamiltonian Lie algebras are determined by employing a "sandwich" method and by studying some features of these Lie algesbras.
Abstract: Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a "sandwich" method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are deter- mined.
TL;DR: In this article, the uniform boundedness of a particular zonal kernel in the L 1 norm on the unit sphere of R 2n was shown for the projectors πll', when l, l'belong to a proper angular sector in N × N.
Abstract: In this paper we estimate the (Lp − L2)-norm of the complex harmonic projectors πll', 1 ≤ p ≤ 2, uniformly with respect to the indexes l, l ' . We provide sharp estimates both for the projectors πll', when l, l ' belong to a proper angular sector in N × N, and for the projectors πl0 and π0l. The proof is based on an extension of a complex interpolation argument by C. Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the L 1 norm on the unit sphere of R 2n .
TL;DR: In this article, it was shown that the restriction is a bijection of level zero discrete series (Proposition 3.2) and that it is invariant under (Theorem 4.1).
Abstract: Let be a -adic local field and let be the unit group of a central simple -algebra of reduced degree . Let denote the set of irreducible discrete series representations of . The “Abstract Matching Theorem” asserts the existence of a bijection, the “Jacquet-Langlands” map, which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map 𝒥ℒ, but only for “level zero” representations. We prove that the restriction is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra and is invariant under (Theorem 4.1).