1,227 results
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2. On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa
- Author
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Jarosław A. Wiśniewski
- Subjects
Surjective function ,Ample line bundle ,Pure mathematics ,Morphism ,Betti number ,General Mathematics ,Embedding ,Projective space ,Projective test ,Space (mathematics) ,Mathematics - Abstract
Let π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.
- Published
- 1992
3. Erratum to the Paper 'A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold'
- Author
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Wei Zhao
- Subjects
Pure mathematics ,Geodesic ,General Mathematics ,Geometry ,Finsler manifold ,Upper and lower bounds ,Mathematics - Abstract
We correct two clerical errors made in the paper ”A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold”.
- Published
- 2014
4. Quasi-uniform topological spaces, by M.G. Murdeshwar and S.A. Naimpally. Series A: Preprints of Research Papers. No.4, vol. 2. (Février 1966). Nordhoff. $2.00 (Stechert)
- Author
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J. Troué
- Subjects
Pure mathematics ,Series (mathematics) ,General Mathematics ,Topological space ,Mathematics - Published
- 1968
5. Corrigendum to the Paper 'Nilpotency of Derivations'
- Author
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Jiang Luh and L. O. Chung
- Subjects
Pure mathematics ,General Mathematics ,Mathematics - Published
- 1986
6. Correction to the Paper*
- Author
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Casper Goffman and G. M. Petersen
- Subjects
Matrix (mathematics) ,Pure mathematics ,General Mathematics ,Arithmetic ,Mathematics - Published
- 1962
7. Remark to my Paper: Introduction to Von Neumann Algebras and Continuous Geometry
- Author
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Israel Halperin
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Von Neumann algebra ,General Mathematics ,symbols ,Tomita–Takesaki theory ,Abelian von Neumann algebra ,Affiliated operator ,Continuous geometry ,Von Neumann architecture ,Mathematics - Published
- 1962
8. An Existence Result for Stepanoff Almost-Periodic Differential Equations
- Author
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S. Zaidman
- Subjects
Pure mathematics ,symbols.namesake ,Differential equation ,General Mathematics ,Ordinary differential equation ,Short paper ,Hilbert space ,symbols ,Sense (electronics) ,Mathematics - Abstract
In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.
- Published
- 1971
9. On Involutions of Quasi-Division Algebras
- Author
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Lowell Sweet
- Subjects
Involution (mathematics) ,Pure mathematics ,General Mathematics ,Short paper ,Structure (category theory) ,Order (ring theory) ,Multiplication ,Division (mathematics) ,Automorphism ,Associative property ,Mathematics - Abstract
All algebras are assumed to be finite dimensional and not necessarily associative. An involution of an algebra is an algebra automorphism of order two. A quasi-division algebra is any algebra in which the non-zero elements form a quasi-group under multiplication. The purpose of this short paper is to determine the structure of all involutions of quasi-division algebras and to give an application of this result.
- Published
- 1975
10. SNC Log Symplectic Structures on Fano Products
- Author
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Katsuhiko Okumura
- Subjects
Pure mathematics ,Mathematics::Algebraic Geometry ,General Mathematics ,Poisson manifold ,010102 general mathematics ,0103 physical sciences ,Projective space ,010307 mathematical physics ,Fano plane ,0101 mathematics ,01 natural sciences ,Symplectic geometry ,Mathematics - Abstract
This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.
- Published
- 2020
11. Non-cocompact Group Actions and -Semistability at Infinity
- Author
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Ross Geoghegan, Michael L. Mihalik, and Craig R. Guilbault
- Subjects
Class (set theory) ,Pure mathematics ,Property (philosophy) ,Group (mathematics) ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Infinity ,01 natural sciences ,Group action ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Counterexample ,media_common - Abstract
A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.
- Published
- 2019
12. Corrigendum to: A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups
- Author
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Roger R. Smith and Jan Cameron
- Subjects
Pure mathematics ,Crossed product ,Group (mathematics) ,Simple (abstract algebra) ,General Mathematics ,Unital ,Bimodule ,Mathematics - Abstract
This note corrects an error in our paper “A Galois correspondence for reduced crossed products of unital simple $\text{C}^{\ast }$-algebras by discrete groups”, http://dx.doi.org/10.4153/CJM-2018-014-6. The main results of the original paper are unchanged.
- Published
- 2019
13. On the Structure of the Schild Group in Relativity Theory
- Author
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Ch. Pommerenke and Gerd Jensen
- Subjects
Pure mathematics ,Group (mathematics) ,General Mathematics ,Lorentz transformation ,010102 general mathematics ,Integer lattice ,Structure (category theory) ,010103 numerical & computational mathematics ,Lattice of subgroups ,01 natural sciences ,symbols.namesake ,Theory of relativity ,Matrix group ,symbols ,0101 mathematics ,Group theory ,Mathematics - Abstract
Alfred Schild has established conditions that Lorentz transformationsmap world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in SL(z, ℂ). We consider the lattice of subgroups of the group originated in Schild’s paper and obtain generating sets for the full group and its subgroups.
- Published
- 2017
14. Tannakian Categories With Semigroup Actions
- Author
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Michael Wibmer and Alexey Ovchinnikov
- Subjects
Class (set theory) ,Pure mathematics ,Semigroup ,General Mathematics ,010102 general mathematics ,Braid group ,Tannakian category ,Group Theory (math.GR) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,010101 applied mathematics ,Linear differential equation ,Mathematics::Category Theory ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,Finite set ,Differential (mathematics) ,Axiom ,Mathematics - Abstract
Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications., Comment: minor revision
- Published
- 2017
15. Ghosts and Strong Ghosts in the Stable Category
- Author
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Jan Minac, Sunil K. Chebolu, and Jon F. Carlson
- Subjects
Subcategory ,Pure mathematics ,Finite group ,Functor ,Computer Science::Information Retrieval ,General Mathematics ,Sylow theorems ,Zero (complex analysis) ,Order (group theory) ,Field (mathematics) ,Topology ,Cohomology ,Mathematics - Abstract
Suppose that G is a finite group and k is a field of characteristic p > 0. A ghost map is a map in the stable category of finitely generated kG-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow p-subgroup of G is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.
- Published
- 2016
16. Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups
- Author
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Leonid V. Kovalev and Miroslav Bačák
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics::General Topology ,Metric Geometry (math.MG) ,Space (mathematics) ,Lipschitz continuity ,01 natural sciences ,Functional Analysis (math.FA) ,Hadamard space ,Mathematics - Functional Analysis ,symbols.namesake ,Metric space ,Cardinality ,Hausdorff distance ,Mathematics - Metric Geometry ,Hadamard transform ,0103 physical sciences ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
Let X(n), for n ∊ ℕ, be the set of all subsets of a metric space (X, d) of cardinality at most n. The set X(n) equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions r: X(n)→ X(n − 1) for n ≥ 2. It is known that such retractions do not exist if X is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if X is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when X is a Hadamard space. In this paper we answer the question in the positive.
- Published
- 2016
17. On Classes for Hyperbolic Riemann Surfaces
- Author
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Huaihui Chen and Rauno Aulaskari
- Subjects
Unit sphere ,symbols.namesake ,Pure mathematics ,Property (philosophy) ,General Mathematics ,Riemann surface ,symbols ,Holomorphic function ,Nesting (computing) ,Meromorphic function ,Mathematics - Abstract
The Qpspaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to theclasses of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) ofclasses on hyperbolic Riemann surfaces. The same property for Qp spaces was also established systematically and precisely in earlier work by the authors of this paper.
- Published
- 2016
18. Exact Morphism Category and Gorenstein-projective Representations
- Author
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Xiu-Hua Luo
- Subjects
Subcategory ,Pure mathematics ,Ideal (set theory) ,General Mathematics ,010102 general mathematics ,Quiver ,01 natural sciences ,Algebra ,Morphism ,Corollary ,If and only if ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Projective test ,Mathematics - Abstract
Let Q be a finite acyclic quiver, let J be an ideal of kQ generated by all arrows in Q, and let A be a finite-dimensional k-algebra. The category of all finite-dimensional representations of (Q, J2) over A is denoted by rep(Q, J2, A). In this paper, we introduce the category exa(Q, J2, A), which is a subcategory of rep (Q, J2, A) of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in rep(Q, J2, A), via the exact representations plus an extra condition. As a corollary, A is a self-injective algebra if and only if the Gorensteinprojective representations are exactly the exact representations of (Q, J2) over A.
- Published
- 2015
19. Spectral Properties of a Family of Minimal Tori of Revolution in the Five-dimensional Sphere
- Author
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Mikhail Karpukhin
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Spectral properties ,Torus ,Surface (topology) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Operator (computer programming) ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The normalized eigenvalues Ʌi(M, g) of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics g on a fixed surface M. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in 𝕊3 or 𝕊4. In this paper a family of extremal metrics induced by minimal immersions in 𝕊5 is investigated.
- Published
- 2015
20. A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of K–algebras
- Author
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Jeffrey C. Lagarias and Jason P. Bell
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,Algebraic extension ,Field (mathematics) ,Dynamical Systems (math.DS) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,16. Peace & justice ,Automorphism ,Mathematics - Algebraic Geometry ,Skolem–Mahler–Lech theorem ,Scheme (mathematics) ,FOS: Mathematics ,Affine space ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,Primary: 11D45. Secondary: 14R10. 11Y55, 11D88 ,Algebra over a field ,Algebraic Geometry (math.AG) ,Finite set ,Mathematics - Abstract
This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in {\rm Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic., Comment: 29 pages; to appear in the Canadian Journal of Mathematics
- Published
- 2015
21. Weighted Carleson Measure Spaces Associated with Different Homogeneities
- Author
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Xinfeng Wu
- Subjects
Carleson measure ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.
- Published
- 2014
22. Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ1-seminorm Topologies
- Author
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Murray Marshall, Sven Wagner, and Mehdi Ghasemi
- Subjects
Pure mathematics ,Representation theorem ,Semigroup ,General Mathematics ,Polynomial ring ,010102 general mathematics ,Closure (topology) ,Primary 43A35 Secondary 44A60, 13J25 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Semigroup with involution ,Integer ,Cone (topology) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Commutative property ,Mathematics - Abstract
In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares in the polynomial ring in the topology induced by the ℓ1-norm is equal to Pos([–1; 1]n), the cone consisting of all polynomials that are non-negative on the hypercube [–1,1]n. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted -seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theoremfrom2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer d ≥ 1, the closure of the cone of sums of 2d-powers in the topology induced by the -norm is equal to Pos([–1; 1]n).
- Published
- 2014
23. Existence of Taut Foliations on Seifert Fibered Homology 3-spheres
- Author
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Shanti Caillat-Gibert and Daniel Matignon
- Subjects
Pure mathematics ,General Mathematics ,Taut foliation ,General Topology (math.GN) ,Physics::Physics Education ,Fibered knot ,Geometric Topology (math.GT) ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mathematics - Geometric Topology ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematics - General Topology ,Mathematics - Abstract
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we prove that all but the 3-sphere and the Poincar\'e 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres., Comment: 34 pages, 1 figure
- Published
- 2014
24. The Ample Cone for a K3 Surface
- Author
-
Arthur Baragar
- Subjects
Surface (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Divisor (algebraic geometry) ,Algebraic number field ,01 natural sciences ,K3 surface ,Fractal ,Cone (topology) ,Hausdorff dimension ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we give several pictorial fractal representations of the ample or Kahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in P×P×P defined over a sufficiently large number field K, which have a line parallel to one of the axes, and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface’s group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296± .010. The ample cone or Kahler cone for a surface is a significant and often complicated geometric object. Though much is known about the ample cone, particularly for K3 surfaces, only a few non-trivial examples have been explicitly described. These include the ample cones with a finite number of sides (see [N1] for n = 3, and [N2, N3] for n ≥ 5; the case n = 4 is attributed to Vinberg in an unpublished work [N1]); the ample cone for a class of K3 surfaces with n = 3 [Ba3]; and the ample cones for several Kummer surfaces, which are K3 surfaces with n = 20 [V, K-K, Kon]. Though the complexity of the problem generically increases with n, the problem for K3 surfaces with maximal Picard number (n = 20) appear to be tractable because of the small size of the transcendental lattice. In this paper, we introduce accurate pictorial representations of the ample cone and the associated fractal for surfaces within a class of K3 surfaces with Picard number n = 4 (see Figures 1, 3, 4, and 5). As far as the author is aware, the associated fractal has not been studied in any great depth for any ample cone for which the fractal has a non-integer dimension, except the one in [Ba3]. The fractal in that case is Cantor-like (it is a subset of S) and rigorous bounds on its Hausdorff dimension are calculated in [Ba1]. The Hausdorff dimension of the fractal of this paper is estimated to be 1.296± .010. Our second main result is to relate the Hausdorff dimension of the fractal to the growth of the height of curves for an orbit of curves on a surface in this class. Precisely, let V be a surface within our class of K3 surfaces and let A = Aut(V/K) be its group of automorphisms over a sufficiently large number field K. Let D be an ample divisor on V and let C be a curve on V . Define NA(C)(t,D) = #{C′ ∈ A(C) : C′ ·D < t}. Here we have abused notation by letting C′ also represent the divisor class that contains C′. The intersection C′ ·D should be thought of as a logarithmic height of 2000 Mathematics Subject Classification. 14J28, 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05.
- Published
- 2011
25. Locally Indecomposable Galois Representations
- Author
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Eknath Ghate and Vinayak Vatsal
- Subjects
Pure mathematics ,Galois cohomology ,General Mathematics ,Fundamental theorem of Galois theory ,010102 general mathematics ,Galois group ,Galois module ,01 natural sciences ,Normal basis ,Embedding problem ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Galois extension ,0101 mathematics ,Indecomposable module ,Mathematics - Abstract
In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation. School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India. e-mail: eghate@math.tifr.res.in Department of Mathematics, University of British Columbia, Vancouver, BC e-mail: vatsal@math.ubc.ca Received by the editors August 5, 2008. Published electronically December 29, 2010. AMS subject classification: 11F80. 1
- Published
- 2011
26. On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces
- Author
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G. Loaiza and M. E. Puerta
- Subjects
Pure mathematics ,Multiplication operator ,General Mathematics ,Norm (mathematics) ,Mathematical analysis ,Interpolation space ,Maximal operator ,Finite-rank operator ,Compact operator ,Bitwise operation ,Coincidence ,Mathematics - Abstract
The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ℓp spaces. In a previous paper, an interpolation space, defined via the real method and using ℓp spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.
- Published
- 2010
27. A Generalization of Integrality
- Author
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Jim Coykendall and Tridib Dutta
- Subjects
Pure mathematics ,Property (philosophy) ,Generalization ,General Mathematics ,Calculus ,Daniell integral ,Mathematics - Abstract
In this paper, we explore a generalization of the notion of integrality. In particular, we study a near-integrality condition that is intermediate between the concepts of integral and almost integral. This property (referred to as the Ω-almost integral property) is a representative independent specialization of the standard notion of almost integrality. Some of the properties of this generalization are explored in this paper, and these properties are compared with the notion of pseudo-integrality introduced by Anderson, Houston, and Zafrullah. Additionally, it is shown that the Ω-almost integral property serves to characterize the survival/lying over pairs of Dobbs and Coykendall.
- Published
- 2010
28. On 6-Dimensional Nearly Kähler Manifolds
- Author
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Yoshiyuki Watanabe and Young Jin Suh
- Subjects
Pure mathematics ,Homogeneous ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dimension (graph theory) ,Simply connected space ,Kähler manifold ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.
- Published
- 2010
29. Commutativity via spectra of exponentials
- Author
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C. Touré, Rudi Brits, and F. Schulz
- Subjects
Pure mathematics ,General Mathematics ,Commutative property ,Spectral line ,Mathematics ,Exponential function - Abstract
Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$ . Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$ .
- Published
- 2021
30. On nonmonogenic number fields defined by
- Author
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Anuj Jakhar and Surender Kumar
- Subjects
Pure mathematics ,General Mathematics ,Algebraic number field ,Mathematics - Abstract
Let q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.
- Published
- 2021
31. Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions
- Author
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Bart Rodrigues
- Subjects
Surface (mathematics) ,Pure mathematics ,Intersection ,General Mathematics ,Open problem ,Geometry ,A fibers ,Mathematics - Abstract
In this paper we study ruled surfaces which appear as exceptional surface in a succession of blowing-ups. In particular we prove that the e-invariant of such a ruled exceptional surface E is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of E). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of e to the study of the poles of the well-known topological, Hodge and motivic zeta functions.
- Published
- 2007
32. On semidirectly closed pseudovarieties of finite semigroups and monoids
- Author
-
Jiří Kad’ourek
- Subjects
Pure mathematics ,General Mathematics ,Mathematics - Abstract
For every pseudovariety $\mathbf {V}$ of finite monoids, let $\mathbf {LV}$ denote the pseudovariety of all finite semigroups all of whose local submonoids belong to $\mathbf {V}$ . In this paper, it is shown that, for every nontrivial semidirectly closed pseudovariety $\mathbf {V}$ of finite monoids, the pseudovariety $\mathbf {LV}$ of finite semigroups is also semidirectly closed if, and only if, the given pseudovariety $\mathbf {V}$ is local in the sense of Tilson. This finding resolves a long-standing open problem posed in the second volume of the classic monograph by Eilenberg.
- Published
- 2021
33. The epsilon constant conjecture for higher dimensional unramified twists of (1)
- Author
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Werner Bley and Alessandro Cobbe
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,01 natural sciences ,Mathematics - Abstract
Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.
- Published
- 2021
34. On Density Conditions for Interpolation in the Ball
- Author
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Xavier Massaneda and Nicolas Marco
- Subjects
Unit sphere ,Pure mathematics ,Bergman space ,General Mathematics ,Entire function ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Ball (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics ,Interpolation - Abstract
In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of Cn, n > 1. We first give density conditions for a sequence to be interpolating for the class A−∞ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension 1, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for n = 1 coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.
- Published
- 2003
35. Rigidity of Hamiltonian Actions
- Author
-
Frédéric Rochon
- Subjects
Pure mathematics ,Geometric analysis ,Mathematical society ,General Mathematics ,010102 general mathematics ,Prove it ,Lie group ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Rigidity (electromagnetism) ,General theory ,symbols ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic geometry - Abstract
This paper studies the following question: Given an ω -symplectic action of a Lie group on a manifold M which coincides, as a smooth action, with a Hamiltonian ω-action, when is this action a Hamiltonian ω -action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ω -action, provided that M is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuffPolterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only. Received by the editors May 17, 2001. Supported by a postgraduate scholarship from theNatural Sciences and Engineering Research Council of Canada. AMS subject classification: 53D05, 37J25. c ©Canadian Mathematical Society 2003. 277
- Published
- 2003
36. On the triple correlations of fractional parts of
- Author
-
Aled Walker and Niclas Technau
- Subjects
Pure mathematics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
For fixed$\alpha \in [0,1]$, consider the set$S_{\alpha ,N}$of dilated squares$\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $modulo$1$. Rudnick and Sarnak conjectured that, for Lebesgue, almost all such$\alpha $the gap-distribution of$S_{\alpha ,N}$is consistent with the Poisson model (in the limit asNtends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of$S_{\alpha ,N}$in a random interval of length$L/N$, provided that$L> N^{1/4+\varepsilon }$. The threshold of$\tfrac {1}{4}$is substantially smaller than the threshold of$\tfrac {1}{2}$(which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations$(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $for a nonlacunary sequence$(a_n)_{n=1}^{\infty } $of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.
- Published
- 2021
37. Cancellation of two classes of dirichlet coefficients over Beatty sequences
- Author
-
Qiang Ma
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Dirichlet distribution ,Mathematics - Abstract
Let $\pi $ be an automorphic irreducible cuspidal representation of $\mathrm{GL}_{m}$ over $\mathbb {Q}$ . Denoted by $\lambda _{\pi }(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi )$ associated with $\pi $ . Let $\pi _{1}$ be an automorphic irreducible cuspidal representation of $\mathrm{SL}(2,\mathbb {Z})$ . Denoted by $\lambda _{\pi _{1}\times \pi _{1}}(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi _{1}\times \pi _{1})$ associated with $\pi _{1}\times \pi _{1}$ . In this paper, we study the cancellations of $\lambda _{\pi }(n)$ and $\lambda _{\pi _{1}\times \pi _{1}}(n)$ over Beatty sequences.
- Published
- 2021
38. Unitary equivalence of multiplication operators on the Bergman spaces of polygons
- Author
-
Hansong Huang and Dechao Zheng
- Subjects
Pure mathematics ,General Mathematics ,Multiplication ,Equivalence (measure theory) ,Unitary state ,Mathematics - Abstract
In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.
- Published
- 2021
39. On the Curves Associated to Certain Rings of Automorphic Forms
- Author
-
Kamal Khuri-Makdisi
- Subjects
Pure mathematics ,Quaternion algebra ,General Mathematics ,010102 general mathematics ,Automorphic form ,Complex multiplication ,Congruence relation ,01 natural sciences ,Algebra ,Elliptic curve ,0103 physical sciences ,Representation ring ,010307 mathematical physics ,Compactification (mathematics) ,0101 mathematics ,Hecke operator ,Mathematics - Abstract
In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.
- Published
- 2001
40. Non-existence of conformally flat real hypersurfaces in both the complex quadric and the complex hyperbolic quadric
- Author
-
Zeke Yao, Zejun Hu, and Bangchao Yin
- Subjects
Pure mathematics ,Quadric ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, by applying for a new approach of the so-called Tsinghua principle, we prove the nonexistence of locally conformally flat real hypersurfaces in both the m-dimensional complex quadric $Q^m$ and the complex hyperbolic quadric $Q^{m\ast }$ for $m\ge 3$ .
- Published
- 2021
41. Torsion in thin regions of Khovanov homology
- Author
-
Adam M. Lowrance, Alex Chandler, Radmila Sazdanovic, and Victor Summers
- Subjects
Khovanov homology ,Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Diagonal ,Geometric Topology (math.GT) ,Torus ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,57M25, 57M27 ,0103 physical sciences ,FOS: Mathematics ,Torsion (algebra) ,010307 mathematical physics ,0101 mathematics ,Link (knot theory) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb{Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki's conjecture that 3-braids have only $\mathbb{Z}_2$ torsion in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family., Comment: 20 pages, 11 figures. Section 4 has been simplified
- Published
- 2021
42. Rank conditions for finite group actions on 4-manifolds
- Author
-
Semra Pamuk and Ian Hambleton
- Subjects
Pure mathematics ,Finite group ,57M60, 57S17, 20J06 ,General Mathematics ,010102 general mathematics ,Geometric topology ,Geometric Topology (math.GT) ,Algebraic topology ,Rank (differential topology) ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Set (abstract data type) ,Mathematics - Geometric Topology ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Spectral sequence ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discrete singular set., 22 pages (v2). Accepted for publication in the Canadian Journal of Mathematics
- Published
- 2021
43. A weak Lefschetz result for Chow groups of complete intersections
- Author
-
Jenan Shtayat and James Lewis
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We introduce a weak Lefschetz-type result on Chow groups of complete intersections. As an application, we can reproduce some of the results in [P]. The purpose of this paper is not to reproduce all of [P] but rather illustrate why the aforementioned weak Lefschetz result is an interesting idea worth exploiting in itself. We hope the reader agrees.
- Published
- 2020
44. An estimate for the composition of rough singular integral operators
- Author
-
Guoen Hu and Xiangxing Tao
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Composition (combinatorics) ,Singular integral operators ,01 natural sciences ,Mathematics - Abstract
Let $\Omega $ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$ , $T_{\Omega }$ be the convolution singular integral operator with kernel $\frac {\Omega (x)}{|x|^d}$ . In this paper, we prove that if $\Omega \in L\log L(S^{d-1})$ , and U is an operator which is bounded on $L^2(\mathbb {R}^d)$ and satisfies the weak type endpoint estimate of $L(\log L)^{\beta }$ type, then the composition operator $UT_{\Omega }$ satisfies a weak type endpoint estimate of $L(\log L)^{\beta +1}$ type.
- Published
- 2020
45. A Pólya–Vinogradov inequality for short character sums
- Author
-
Matteo Bordignon
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Vinogradov ,02 engineering and technology ,021001 nanoscience & nanotechnology ,01 natural sciences ,Character (mathematics) ,0101 mathematics ,0210 nano-technology ,Mathematics ,media_common - Abstract
In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$ , with $0\le \gamma \le 1/3$ . We prove that $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$ with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.
- Published
- 2020
46. Stable Bi-Period Summation Formula and Transfer Factors
- Author
-
Yuval Z. Flicker
- Subjects
Discrete mathematics ,Pure mathematics ,Transfer (group theory) ,Conjugacy class ,Group (mathematics) ,General Mathematics ,Automorphic form ,Fundamental lemma ,Algebraic number field ,Reductive group ,Unit (ring theory) ,Mathematics - Abstract
This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ! F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals overthe group of F-adele points of G ,o f cusp forms on the group ofE-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to "periodic" automorphic forms, are related to analogous bi-period distributions associated to "periodic" au- tomorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the "fundamental lemma", which conjectures that the unit elements of the Hecke algebras on GandH havematching orbitalintegrals. Evenin stating this conjecture, oneneeds to intro- duce a "transfer factor". A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for SL(2). The geometric side of the trace formula for a test function f ! on the group of adele points of a reductive group G over a number field F ,i s as um of orbital integrals off ! parametrized by rational conjugacy classes, in G(F). It is obtained on integrating over the diagonal x = y the kernel Kf ! (x, y )o f ac onvolution operatorr(f ! ). Each such orbital integral can be expressed as an average of weighted sums of such orbital integrals over the stable conjugacy class, which is the set of rational points in the conjugacy class under the points of the group over the algebraic closure. Each such weighted sum is conjecturally related to a stable (a sum where all coefficients are equal to 1) such sum on an endoscopic group H of the group G.T his process of stabilization has been introduced by Langlands to establishliftingofautomorphicandadmissiblerepresentationsfromtheendoscopicgroups H to the original group G. The purpose of this paper is to develop an analogue in the context of the symmetric space G(E)/G(F), where E/F is a quadratic number field extension. Integrating the kernel Kf ! (x, y )o f the convolution operatorr(f ! ) for the test function f ! on the group of E- adele points of the group G over two independent variables x and y in the subgroup of F-adele points of G, we obtain a sum of bi-orbital integrals of f ! over rational bi-conjugacy classes. We introduce a notion of stable bi-conjugacy, and stabilize the geometric side of the bi-period summation formula. Thus we express the weighted sums in the stable bi
- Published
- 1999
47. Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
- Author
-
H. E. A. Campbell, R. J. Shank, David L. Wehlau, Anthony V. Geramita, and I.P. Hughes
- Subjects
Principal ideal ring ,Discrete mathematics ,Reduced ring ,Ring (mathematics) ,Pure mathematics ,General Mathematics ,Gorenstein ring ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Primitive ring ,Simple ring ,0101 mathematics ,Quotient ring ,Mathematics ,Group ring - Abstract
This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.
- Published
- 1999
48. Ward’s Solitons II: Exact Solutions
- Author
-
Christopher Kumar Anand
- Subjects
Surface (mathematics) ,Pure mathematics ,Function field of an algebraic variety ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Dimension of an algebraic variety ,Algebraic geometry ,01 natural sciences ,Algebraic cycle ,0103 physical sciences ,Real algebraic geometry ,010307 mathematical physics ,0101 mathematics ,Differential algebraic geometry ,Mathematics - Abstract
In a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.
- Published
- 1998
49. Further inequalities and properties of p-inner parallel bodies
- Author
-
Dongmeng Xi, Zhenbing Zeng, and Yingying Lou
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,media_common - Abstract
A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.
- Published
- 2020
50. Surjective isometries of metric geometries
- Author
-
D. Minda and A. F. Beardon
- Subjects
Surjective function ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Metric (mathematics) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Many authors define an isometry of a metric space to be a distance-preserving map of the space onto itself. In this note, we discuss spaces for which surjectivity is a consequence of the distance-preserving property rather than an initial assumption. These spaces include, for example, the three classical (Euclidean, spherical, and hyperbolic) geometries of constant curvature that are usually discussed independently of each other. In this partly expository paper, we explore basic ideas about the isometries of a metric space, and apply these to various familiar metric geometries.
- Published
- 2020
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