Showing papers in "Canadian Journal of Mathematics in 2015"
TL;DR: In this article, the authors studied a multimarginal optimal transportation problem in one dimension, and they showed that, given a minimizing transport plan, its symmetrization is induced by a cyclical map and that the symmetric optimal plan is unique.
Abstract: We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.
119 citations
TL;DR: In this article, the authors define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function and prove a realization formula and an Oka-Weil theorem for free analytic functions.
Abstract: We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.
107 citations
TL;DR: In this article, the authors consider random i.i.d. samples of continuous measures on bounded connected domains and prove an upper bound on the distance between the measure and the empirical measure of the sample.
Abstract: We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the -transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.
98 citations
TL;DR: In this article, the notions of bi-free independence and combinatorial-bi-free independent of two-faced families are equivalent using a diagrammatic view of bi non-crossing partitions.
Abstract: We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.
54 citations
TL;DR: In this paper, the relation between tropical and complex intersection theories was established, and local obstructions to approximating tropical curves in smooth tropical surfaces were provided based on the relationship between complex intersection theory and tropical curves.
Abstract: We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a non-singular fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.
30 citations
TL;DR: In this paper, a variational method for the Loewner equation in higher dimensions was developed based on the Pontryagin maximum principle from optimal control theory for several complex variables, and the conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball were obtained.
Abstract: In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and
Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary
conditions for the extremal mappings for a wide class of extremal problems over the set of normalized
biholomorphic mappings on the unit ball in .
21 citations
TL;DR: In this paper, it was shown that the spectral triple arising from an action of the reals on a -algebra with invariant trace satisfies the hypotheses of the non-unital local index formula.
Abstract: We prove two results about nonunital index theory left open in a previous paper. The
first is that the spectral triple arising from an action of the reals on a -algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow.
20 citations
TL;DR: In this paper, a category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to Brat- teli diagrams is constructed.
Abstract: A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Brat- teli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF alge- bras (and at the same time, of Glimm's classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.
17 citations
TL;DR: In this article, the authors investigated the number of complex zeros of polynomials p(z) with coefficients f 1; 1g inside or on the unit circle, denoted by N(p) and U(p), respectively, and showed that if n + 1 is a prime number, then for each integer k, 0 6 k 6 n 1, there exists a Littlewood polynomial with n = k and U = 0.
Abstract: We investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficientsf 1; 1g) inside or on the unit circlejzj = 1, denoted by N(p) and U (p), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for N(p), U (p) for polynomials p(z) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 6 k 6 n 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U (p) = 0. Furthermore, we describe some cases where the ratios N(p)=n and U (p)=n have limits as n!1 and find the corresponding limit values.
16 citations
TL;DR: In this article, the authors discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature and prove a priori bounds for the trace of the second fundamental form and extend the result to -dimensions.
Abstract: In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form and extend the result to -dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.
15 citations
TL;DR: In this paper, the authors established the non-tangential maximal function characterization of the Hardy space associated with for, and when and with, the associated Riesz transform is bounded from to the weighted classical Hardy space.
Abstract: Let be either in the Muckenhoupt class of weights or in the class of weights, and let be the degenerate elliptic operator on the Euclidean space , . In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with for , and when and with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .
TL;DR: In this paper, a generalization of a plane de Jonquimap to higher dimensional space P n with n 3 is presented. But the generalization is restricted to the case where n = 3.
Abstract: This paper is concerned with suitable generalizations of a plane de Jonquimap to higher dimensional space P n with n 3. For each given point of P n there is a subgroup of the entire Cre- mona group of dimension n consisting of such maps. We study both geometric and group-theoretical properties of this notion. In the case where n = 3 we describe an explicit set of generators of the group and give a homological characterization of a basic subgroup thereof.
TL;DR: In this article, it was shown that the continuously controlled coarse structure on a locally compact space X induced by somemetrizable compactiucation X is determined only by the topology of the remainder X ∖ X.
Abstract: Let TB be the category of totally bounded, locally compact metric spaces with the C0 coarse structures. We show that if X and Y are in TB then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, theHigson corona functor gives an equivalence of categories TB→ K, whereK is the category of compactmetrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space X induced by somemetrizable compactiucation X is determined only by the topology of the remainder X ∖ X.
TL;DR: In this paper, the behavior of limit sets, prolongations, prolongational limit sets and Lyapunov stable sets under equivariant maps and semiconjugations from $M$ to $N$ is studied.
Abstract: Abstract Let $M$ and $N$ be admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume that $\\mathcal{S}$ is a semigroup acting on both $M$ and $N$ . In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors, and Lyapunov stable sets (all concepts defined for the action of the semigroup $\\mathcal{S}$ ) under equivariant maps and semiconjugations from $M$ to $N$ .
TL;DR: In this paper, it was shown that if and are any two unitaries in a unital simple-algebra such that and commutes with and, then the subalgebra generated by and is is isomorphic to a quotient of some rotation algebra, provided that has a unique tracial state.
Abstract: We show that if and are any two unitaries in a unital –algebra such that and commutes with and , then the –subalgebra generated by and is isomorphic to a quotient of some rotation algebra , provided that has a unique tracial state. We also show that the Exel trace formula holds in any unital –algebra. Let be a real number. For any , we prove that there exists satisfying the following: if and are two unitaries in any unital simple –algebra with tracial rank zero such that for all tracial states of , then there exists a pair of unitaries and in such that
TL;DR: In this paper, the maximum number of edges in an -vertex graph that does not contain vertex-disjoint triangles was found, for all sufficiently large and each edge having a constant number of vertices.
Abstract: We find, for all sufficiently large and each , the maximum number of edges in an -vertex graph that does not contain vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96–102], which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi–Hajnal Theorem.
TL;DR: In this paper, the authors studied the hyperspace dynamics induced from generic continuous maps and from generic homeomorphisms of the Cantor space, with emphasis on the notions of Li-Y Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing, and recurrence.
Abstract: We study the hyperspace dynamics induced fromgeneric continuous maps and fromgeneric homeomorphisms of the Cantor space, with emphasis on the notions of Li– Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing, and recurrence.
TL;DR: In this paper, the authors investigated the Bochner-Schoenberg-Eberlein property and the algebra of Bochnerschoenberg and Eberlein functions on the spectrum.
Abstract: Abstract Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta $ of $B$ is a certain Banach algebra product $A\,{{\times }_{\theta }}\,B$ , which is a splitting extension of $B$ by $A$ . We investigate two topics for the algebra $A\,{{\times }_{\theta }}\,B$ in relation to the corresponding ones of $A$ and $B$ . The first one is the Bochner–Schoenberg–Eberlein property and the algebra of Bochner–Schoenberg–Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\,{{\times }_{\theta }}\,B$ .
TL;DR: In this article, the authors give explicit criteria for the induced action of a finite group G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g-2.
Abstract: Given a faithful action of a finite group G on an algebraic curve X of genus g > 1, we give explicit criteria for the induced action of G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g-2. This leads to a concise answer to the question when the action of G on the space H^0(X, \Omega_X^m) of global holomorphic polydifferentials of order m is faithful. If X is hyperelliptic, we furthermore provide an explicit basis of H^0(X, \Omega_X^m). Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of G on the first homology H_1(X, Z/mZ) if X is a Riemann surface.
TL;DR: In this article, the authors studied the first elementary divisor of the reduction of modulo a prime, and proved the existence of the density of the primes for which is fixed.
Abstract: Let be a generic Drinfeld module of rank . We study the first elementary divisor of the reduction of modulo a prime , as varies. In particular, we prove the existence of the density of the primes for which is fixed. For , we also study the second elementary divisor (the exponent) of the reduction of modulo and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.
TL;DR: For any set with positive lower asymptotic density, there are infinite sets such that they can be found in the union of and a translate of. as mentioned in this paper showed that for positive lower density, the same can also be said for positive Banach density.
Abstract: Erdős conjectured that for any set with positive lower asymptotic density, there are infinite sets such that . We verify Erdős’ conjecture in the case where has Banach density exceeding . As a consequence, we prove that, for with positive Banach density (a much weaker assumption than positive lower density), we can find infinite such that is contained in the union of and a translate of . Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.
TL;DR: In this paper, it was shown that the extended rotation algebra introduced by the authors in J. Reine Angew. Math. 665(2012), pp. 1−71, is always an algebra.
Abstract: Let be any irrational number. It is shown that the extended rotation algebra introduced by the authors in J. Reine Angew. Math. 665(2012), pp. 1–71, is always an algebra.
TL;DR: In this paper, Gompf introduced a -valued invariant of torsion combings on closed 3-manifolds, and showed that all torsions have the same -structure.
Abstract: Combings of compact, oriented, 3-dimensional manifolds are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying -structure. A combing is called torsion if this Euler class is a torsion element of . Gompf introduced a -valued invariant of torsion combings on closed 3-manifolds, and he showed that distinguishes all torsion combings with the same -structure. We give an alternative definition for and we express its variation as a linking number. We define a similar invariant of combings for manifolds bounded by . We relate to the -invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula , where is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.
TL;DR: In this paper, the authors elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine 3D space, and constant mean curvature one surfaces and flat surfaces in hyperbolic 3D spaces.
Abstract: We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.
TL;DR: In this article, the authors studied exponential trichotomy for infinite dimensional discrete time dy-namical systems and proved that finite-time exponential-trichotomy conditions allow us to derive exponential tricotomy for arbitrary times.
Abstract: In this article we study exponential trichotomy for infinite dimensional discrete time dy- namical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.
TL;DR: In this paper, one-parameter families of overconvergent Siegel-Hilbert modular forms are constructed for automorphic forms of non-cohomological weights.
Abstract: We construct one-parameter families of overconvergent Siegel–Hilbert modular forms. This result has applications to the construction of Galois representations for automorphic forms of non-cohomological weights.
TL;DR: In this paper, the Chern class relation for the transversal intersection of two nonsingular varieties was generalized to a relation for possibly singular varieties, under a splayedness assumption.
Abstract: We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern-Schwartz-MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a `strong splayedness' assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern-Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.
TL;DR: In this paper, a representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral -lattice.
Abstract: A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral -lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac–Moody Lie algebra of type are recovered by the new method.
TL;DR: In this paper, a tropical method for computing Gromov-witten type invariants of Fano manifolds of special type is presented, which applies to those Fano manifold types that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions.
Abstract: In this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type and some moduli space of rank two bundles on a genus two curve.
Abstract: We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over C, using equa- tions derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on N-graded Lie algebras of maximal class. As shown by A. Fialowski (see also (SZ, M2)) there are only three isomorphism types of N-graded Lie algebras L = ⊕ 1=1Li of maximal class generated by L1 and L2, L = hL1,L2i. Vergne described the structure of these algebras with the property L = hL1i. In this paper we study those generated by the first and q-th components where q > 2, L = hL1,Lqi. Under some technical condition, there can only be one isomorphism type of such algebras. For q = 3 we fully classify them. This gives a partial answer to a question posed by Millionshchikov.