# Showing papers in "Canadian Mathematical Bulletin in 2017"

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TL;DR: In this paper, double commutativity and the reverse order law for the core inverse of a regular element are considered and new characterizations of the Moore-Penrose inverse are given by one-sided invertibilities in a ring.

Abstract: In this paper, double commutativity and the reverse order law for the core inverse are considered. Then new characterizations of the Moore–Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

54 citations

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TL;DR: In this article, the authors characterized the graphs having maximum and minimum degree Kirchhoff index among all n-vertex bicyclic graphs with exactly two cycles, where n is the number of cycles.

Abstract: Abstract Let $G$ be a connected graph with vertex set $V\left( G \right)$ .The degree Kirchhoff index of $G$ is defined as ${{S}^{\prime }}\left( G \right)\,=\,\sum{_{\left\{ u,v \right\}\,\subseteq \,V\left( G \right)}d\left( u \right)d\left( v \right)R\left( u,\,v \right)}$ , where $d\left( u \right)$ is the degree of vertex $u$ , and $R\left( u,\,v \right)$ denotes the resistance distance between vertices $u$ and $v$ . In this paper, we characterize the graphs having maximum and minimum degree Kirchhoff index among all $n$ -vertex bicyclic graphs with exactly two cycles.

28 citations

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TL;DR: In this article, the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials is discussed and connections between inner functions and optimal polynomial approximants to, where is a function in the space.

Abstract: We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to , where is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiued to produce inner functions.

26 citations

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TL;DR: In this paper, it was shown that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph-algebra, where the quantum symmetry of a graph coincides with the quantum symmetries of the graph -algebra.

Abstract: The study of graph -algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph -algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph -algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

25 citations

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TL;DR: In this article, the generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces have been established for generalized weighted matrices, where the latter is a special case of the former.

Abstract: In this paper, we establish estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

22 citations

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TL;DR: In this article, the endpoint regularity properties of the multisublinear fractional maximal operators were investigated, including the Hardy-Littlewood maximal operator, and some new bounds for the derivative of the one-dimensional fractional matrix maximal operators acting on the vector-valued function with all being -functions were obtained.

Abstract: In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all being -functions.

21 citations

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TL;DR: In this article, the authors give characterizations of the $(b,c)$ -inverse in terms of the direct sum decomposition, the annihilator, and the invertible elements.

Abstract: Abstract Let $R$ be a ring and $b,c\\in R$ . In this paper, we give some characterizations of the $(b,c)$ -inverse in terms of the direct sum decomposition, the annihilator, and the invertible elements. Moreover, elements with equal $(b,c)$ -idempotents related to their $(b,c)$ -inverses are characterized, and the reverse order rule for the $(b,c)$ -inverse is considered.

21 citations

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TL;DR: In this paper, the authors studied the warped structures of Finsler metrics and obtained the differential equation that characterizes FINsler warped product metrics with vanishing Douglas curvature.

Abstract: In this paper, we study the warped structures of Finsler metrics. We obtain the differential equation that characterizes Finsler warped product metrics with vanishing Douglas curvature. By solving this equation, we obtain all Finsler warped product Douglas metrics. Some new Douglas Finsler metrics of this type are produced by using known spherically symmetric Douglas metrics.

19 citations

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TL;DR: In this article, the co-annihilating ideal graph of a commutative ring with identity was studied and its properties were investigated, where the vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever.

Abstract: Let be a commutative ring with identity. The co-annihilating-ideal graph of , denoted by , is a graph whose vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever . In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

19 citations

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TL;DR: In this article, the strong Birkhoff-James orthogonality for inner product -modules and certain elements of Hilbert -modules has been characterized and a Pythagorean relation for bounded linear operators has been obtained.

Abstract: In this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert -modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , is the strong Birkhoff–James orthogonal to if and only if there exists a unit vector such that and . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product -modules.

18 citations

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TL;DR: In this paper, a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups is introduced, and a generalized abstract notion of the trace formula for the Hilbert space is presented.

Abstract: This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let be a compact group and let be a closed subgroup of . Let be the left coset space of in and let be the normalized -invariant measure on associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space .

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TL;DR: In this paper, it was shown that for any immersed cycle, the degree of the covering map is bounded by minus the Euler characteristic of the pullback to the original cycle, and that any finitely generated subgroup of a one-relator group has a Schur multiplier.

Abstract: We prove Wise's $W$
-cycles conjecture. Consider a compact graph $\Gamma '$
immersing into another graph $\Gamma $
. For any immersed cycle $\Lambda :{{S}^{1}}\to \Gamma $
, we consider the map $\Lambda '$
from the circular components $\mathbb{S}$
of the pullback to $\Gamma '$
. Unless $\Lambda '$
is reducible, the degree of the covering map $\mathbb{S}\to {{S}^{1}}$
is bounded above by minus the Euler characteristic of $\Gamma '$
. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

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TL;DR: In this article, a dictionary between models and simplicial complexes was constructed by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Abstract: In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

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TL;DR: In this article, a constructive proof of the weak factorization of the classical Hardy space in terms of multilinear Riesz transforms is given, and a new proof for the characterization of the dual of the Hardy space via commutators of the MRS transforms is obtained.

Abstract: This paper provides a constructive proof of the weak factorization of the classical Hardy space in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of (the dual of ) via commutators of the multilinear Riesz transforms.

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TL;DR: In this article, an interpolation formula for the values of p-adic Rankin-Selberg functions associated to non-ordinary modular forms is presented, and the interpolation is shown to be polynomial.

Abstract: We prove an interpolation formula for the values of certain p-adic Rankin–Selberg Lfunctions associated to non-ordinary modular forms.

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TL;DR: In this article, the authors established equivalent characterizations of Besov-type spaces, Triebel-Lizorkin type spaces, and BesovMorrey spaces via the sequence consisting of the difference between and the ball average.

Abstract: Let and In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence consisting of the difference between and the ball average These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and spaces with , which are of independent interest

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TL;DR: In this paper, the Jacobi coefficients are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space.

Abstract: The Jacobi coefficients c`j (; ) (1 j `, ; > 1) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the jacobi polynomials P(;) k (k 0; ; > 1) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented.

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TL;DR: In this article, it was shown that the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category of Gorenstein projective modules.

Abstract: For any ring , we show that, in the bounded derived category of left -modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category of Gorenstein projective (resp. injective) modules. As a consequence, we get that if is a left and right noetherian ring admitting a dualizing complex, then and are equivalent.

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TL;DR: For lattice polytopes, Gubeladze as discussed by the authors showed that there is no dioerence between - and -convex-normality (for ) and improved the bound to.

Abstract: In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of -convex-normal polytopes to show that integral polytopes all of whose edges are longer than have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no dioerence between - and -convex-normality (for ) and improve the bound to . In the second part we extend the definition to pairs of polytopes. Given two rational polytopes and , where the normal fan of is a refinement of the normal fan of , if every edge of is at least times as long as the corresponding face (edge or vertex) of , then .

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TL;DR: In this article, it was shown that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-mansifolds.

Abstract: It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group is a generalized torsion element.

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TL;DR: In this article, the authors define the connected number of a plane curve for a Galois cover, which is similar to the splitting number for a plane irreducible curve, and distinguish the embedded topology of Artal arrangements of degree b ≥ 4.

Abstract: The splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

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TL;DR: Numerical experiments show that the classical $\text{CSCS}$ iterative methods work slightly better than the Gauss–Seidel $(\text{GS})$ iteratives methods if the CSCS itself is convergent, and that there is always a constant $\alpha $ such that the shifted $\text {CSCs}$ iteration converges much faster than theGauss-Seidel iteration.

Abstract: It is known that every Toeplitz matrix enjoys a circulant and skew circulant splitting (denoted ) i.e., a circulant matrix and a skew circulant matrix. Based on the variant of such a splitting (also referred to as ), we first develop classical iterative methods and then introduce shifted iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical iterative methods work slightly better than the Gauss–Seidel iterative methods if the is convergent, and that there is always a constant such that the shifted iteration converges much faster than the Gauss–Seidel iteration, no matter whether the itself is convergent or not.

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TL;DR: In this paper, various properties of Seifert homology spheres were collected from the viewpoint of Dehn surgery along a seifert fiber and used in the study of concordance and homology cobordism.

Abstract: In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place which we hope to be useful in the study of concordance and homology cobordism.

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TL;DR: In this article, it was shown that if is even, there exists an infinite set of values of such that the distance between two distinct vertices is different from the distance of and.

Abstract: . A subset of the vertex set of a graph is called a resolving set of if for every pair of distinct vertices , of , there is such that the distance of and is different from the distance of and . The cardinality of a smallest resolving set is called the metric dimension of , denoted by . The circulant graph consists of the vertices and the edges , where , the indices are taken modulo . Grigorious, Manuel, Miller, Rajan, and Stephen proved that for , and they presented a conjecture saying that for , where . We disprove both statements. We show that if is even, there exists an infinite set of values of such that . We also prove that for , where and are even, , and .

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TL;DR: In this article, the authors generalize these results to commutative noetherian -rings and show that the local cohomology functor associated with does not increase injective dimension.

Abstract: Let be a commutative noetherian ring, let be an ideal, and let be an injective -module. A basic result in the structure theory of injective modules states that the -module consisting of -torsion elements is also an injective -module. Recently, de Jong proved a dual result: If is a flat -module, then the -adic completion of is also a flat -module. In this paper we generalize these facts to commutative noetherian -rings: let be a commutative non-positive -ring such that is a noetherian ring and for each -module is finitely generated. Given an ideal , we show that the local cohomology functor associated with does not increase injective dimension. Dually, the derived -adic completion functor does not increase flat dimension.

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TL;DR: In this paper, the authors determine the and norms of an integral operator related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

Abstract: In this paper we determine the and norms of an integral operator related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

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TL;DR: In this article, it was shown that the first Betti number of the duals of the free unitary quantum groups is one, and that all Betti numbers vanish for the dual of the quantum automorphism groups of full matrix algebras.

Abstract: We show that the first \({\ell }^2\)-Betti number of the duals of the free unitary quantum groups is one, and that all \({\ell }^2\)-Betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras.

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TL;DR: In this paper, it was shown that for any length function of a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a compact quantum metric space).

Abstract: Let $L$ be a length function on a group $G$, and let $M_L$ denote the operator of pointwise multiplication by $L$ on $\ell^2(G)$. Following Connes, $M_L$ can be used as a "Dirac" operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function of a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a "compact quantum metric space"). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

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TL;DR: In this paper, a sufficient and necessary condition for weighted translation operators on the Lebesgue space to be densely disjoint hypercyclic is given. And the characterization for the dual of a weighted translation is also obtained.

Abstract: Let $1\\le p<\\infty $ , and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space ${{\\ell }^{p}}(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.