1,227 results
Search Results
52. 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
53. A Pólya–Vinogradov inequality for short character sums
- Author
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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
54. Further inequalities and properties of p-inner parallel bodies
- Author
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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
55. Surjective isometries of metric geometries
- Author
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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
56. Brill-Noether generality of binary curves
- Author
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Xiang He
- Subjects
Pure mathematics ,Generality ,Sequence ,Rank (linear algebra) ,General Mathematics ,Ramification (botany) ,010102 general mathematics ,Binary number ,Space (mathematics) ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,Dimension (vector space) ,0103 physical sciences ,FOS: Mathematics ,symbols ,010307 mathematical physics ,0101 mathematics ,Noether's theorem ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We show that the space of linear series of certain multi-degree (including the balanced ones) and rank $r$ on a general binary curve has the expected dimension if nonempty. This generalizes Theorem 24 of Caporaso's paper about binary curves from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$. In addition, we show that this space of linear series is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points lying on different components of the curve., Comment: Minor changes, to appear in Canadian Mathematical Bulletin
- Published
- 2020
57. Embedding of Dirichlet type spaces into tent spaces and Volterra operators
- Author
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Xiangling Zhu and Ruishen Qian
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,Embedding ,010307 mathematical physics ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Dirichlet distribution ,Mathematics - Abstract
In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.
- Published
- 2020
58. algebra structure on certain Banach algebra products
- Author
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Fatemeh Abtahi
- Subjects
010101 applied mathematics ,Pure mathematics ,Algebraic structure ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Banach *-algebra ,Mathematics - Abstract
Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $\theta \in \Delta (\mathcal B)$ . In this paper, we prove that $\mathcal A\times _{\theta }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A\times _{\theta }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.
- Published
- 2020
59. Complete boundedness of multiple operator integrals
- Author
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Clément Coine
- Subjects
Pure mathematics ,Multilinear map ,General Mathematics ,010102 general mathematics ,Compact operator ,01 natural sciences ,Mathematics - Functional Analysis ,Multiplier (Fourier analysis) ,Operator (computer programming) ,Tensor product ,Factorization ,Symbol (programming) ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we characterize the multiple operator integrals mappings which are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated to the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur multipliers., Comment: 13 pages
- Published
- 2020
60. Large values of Dirichlet L-functions at zeros of a class of L-functions
- Author
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Junxian Li
- Subjects
Pure mathematics ,symbols.namesake ,Class (set theory) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Dirichlet distribution ,Mathematics - Abstract
In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$ where $\chi $ is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of $GL(n)$ over ${\mathbb {Q}}$ .
- Published
- 2020
61. Khovanov–Rozansky homology for infinite multicolored braids
- Author
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Michael Willis
- Subjects
Large class ,Khovanov homology ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,Limiting ,Homology (mathematics) ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Tensor product ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,57M25, 57M27 ,0103 physical sciences ,FOS: Mathematics ,Braid ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Twist ,Mathematics - Abstract
We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids., 37 pages, 13 figures
- Published
- 2020
62. On the structure of Kac–Moody algebras
- Author
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Timothée Marquis and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Nipotent algebras ,Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Solvable algebras ,Mathematics - Rings and Algebras ,01 natural sciences ,Nilpotent ,Bracket (mathematics) ,Rings and Algebras (math.RA) ,Homogeneous ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Cartan matrix ,Kac-Moody algebras ,0101 mathematics ,Algebra over a field ,Element (category theory) ,Mathematics::Representation Theory ,17B67, 17B30 ,Mathematics - Abstract
Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y \in \mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$., 32 pages. Final version, to appear in Canadian Journal of Mathematics
- Published
- 2020
63. A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC
- Author
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Masaki Matsuno
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Graded ring ,Mathematics - Rings and Algebras ,Type (model theory) ,Automorphism ,01 natural sciences ,Elliptic curve ,Quadratic equation ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,Noncommutative algebraic geometry ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Twist ,Mathematics - Abstract
Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb{P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras which cannot be written as a twist of a Calabi-Yau AS-regular algebra by a graded algebra automorphism., Comment: 18 pages
- Published
- 2020
64. Growth of Fine Selmer Groups in Infinite Towers
- Author
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Debanjana Kundu
- Subjects
Class (set theory) ,Pure mathematics ,Conjecture ,Selmer group ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,0102 computer and information sciences ,Iwasawa theory ,01 natural sciences ,Arbitrarily large ,010201 computation theory & mathematics ,0101 mathematics ,Invariant (mathematics) ,Tower ,Mathematics - Abstract
In this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple $\mathbb{Z}_{p}$-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the $\unicode[STIX]{x1D707}=0$ conjecture for cyclotomic $\mathbb{Z}_{p}$-extensions. We show that in certain non-cyclotomic $\mathbb{Z}_{p}$-towers, the $\unicode[STIX]{x1D707}$-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified $p$-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-$p$-adic analytic.
- Published
- 2020
65. Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators
- Author
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José L. Torrea and Zhang Chao
- Subjects
Pure mathematics ,Sequence ,Series (mathematics) ,Semigroup ,General Mathematics ,010102 general mathematics ,Singular integral ,01 natural sciences ,Operator (computer programming) ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Differential (infinitesimal) ,Laplace operator ,Mathematics - Abstract
In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$, $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}, ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence.Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$, of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$.It is also shown that the local size of the maximal differential transform operators (with $V=0$) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$, we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.
- Published
- 2020
66. Generalized Beilinson Elements and Generalized Soulé Characters
- Author
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Kenji Sakugawa
- Subjects
Pure mathematics ,Polylogarithm ,Cyclotomic character ,Generalization ,General Mathematics ,010102 general mathematics ,Algebraic number field ,Cyclotomic field ,01 natural sciences ,Image (mathematics) ,Character (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The generalized Soulé character was introduced by H. Nakamura and Z. Wojtkowiak and is a generalization of Soulé’s cyclotomic character. In this paper, we prove that certain linear sums of generalized Soulé characters essentially coincide with the image of generalized Beilinson elements in K-groups under Soulé’s higher regulator maps. This result generalizes Huber–Wildeshaus’ theorem, which is a cyclotomic field case of our results, to an arbitrary number fields.
- Published
- 2020
67. Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers
- Author
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Mehdi Nemati and Maryam Rajaei Rizi
- Subjects
Pure mathematics ,Quantum group ,General Mathematics ,Locally compact quantum group ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Multiplier (Fourier analysis) ,Compact space ,Bounded function ,Homomorphism ,Ideal (ring theory) ,0101 mathematics ,Approximate identity ,Mathematics - Abstract
Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.
- Published
- 2020
68. Metrizability of Holonomy Invariant Projective Deformation of Sprays
- Author
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S. G. Elgendi and Zoltán Muzsnay
- Subjects
Invariant function ,Pure mathematics ,Geodesic ,General Mathematics ,010102 general mathematics ,Holonomy ,010103 numerical & computational mathematics ,Lambda ,01 natural sciences ,Principal curvature ,Metrization theorem ,0101 mathematics ,Projective test ,Invariant (mathematics) ,Mathematics - Abstract
In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$ , we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$ . We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$ , such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.
- Published
- 2020
69. Concordance, Crossing Changes, and Knots in Homology Spheres
- Author
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Christopher William Davis
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,Homotopy ,010102 general mathematics ,Geometric Topology (math.GT) ,Cobordism ,Homology (mathematics) ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology ,01 natural sciences ,Homology sphere ,Mathematics - Geometric Topology ,Knot (unit) ,Mathematics::K-Theory and Homology ,57M25 ,0103 physical sciences ,FOS: Mathematics ,Equivalence relation ,Slice knot ,0101 mathematics ,Unknot ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere., Comment: 10 pages, 1 figure. Changes from Version 1: Theorem 1.6 from version 1 was previously proven by the same technique by Austin-Rolfsen. The result of Theorem 1.9 frmo version 1 appears in a remark of Daemi
- Published
- 2019
70. The John–Nirenberg Inequality for the Regularized BLO Space on Non-homogeneous Metric Measure Spaces
- Author
-
Haibo Lin, Zhen Liu, and Chenyan Wang
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Non homogeneous ,Metric (mathematics) ,0101 mathematics ,Nirenberg and Matthaei experiment ,media_common ,Mathematics - Abstract
Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this paper, the authors establish the John-Nirenberg inequality for the regularized BLO space $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.
- Published
- 2019
71. Maximal Inequalities of Noncommutative Martingale Transforms
- Author
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Fedor Sukochev, Yong Jiao, and Dejian Zhou
- Subjects
Atomic decomposition ,Pure mathematics ,Mathematics::Operator Algebras ,General Mathematics ,Algebraic number ,Martingale (probability theory) ,Mathematical proof ,Noncommutative geometry ,Mathematics - Abstract
In this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decomposition and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities.
- Published
- 2019
72. Positive Definiteness on Products of Compact Two-point Homogeneous Spaces and Locally Compact Abelian Groups
- Author
-
C. P. Oliveira and V. A. Menegatto
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Positive-definite matrix ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Kernel (algebra) ,Fourier transform ,Positive definiteness ,Homogeneous ,symbols ,Point (geometry) ,TRANSFORMADA DE FOURIER ,Locally compact space ,0101 mathematics ,Abelian group ,Mathematics - Abstract
In this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group $G$ with dual group $\widehat{G}$, a compact two-point homogeneous space $\mathbb{H}$ with normalized geodesic distance $\unicode[STIX]{x1D6FF}$ and a profile function $\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$ satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel $((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$ is equivalent to the positive definiteness of the Fourier transformed kernels $(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$, $\unicode[STIX]{x1D6FE}\in \widehat{G}$, where $\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$, $u\in G$. We also provide some results on the strict positive definiteness of the kernel.
- Published
- 2019
73. Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces
- Author
-
Tetsu Shimomura and Takao Ohno
- Subjects
Pure mathematics ,Generalization ,Riesz potential ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,01 natural sciences ,Measure (mathematics) ,Sobolev space ,Corollary ,Bounded function ,Metric (mathematics) ,Maximal function ,0101 mathematics ,Mathematics - Abstract
Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials$I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$of order$\unicode[STIX]{x1D6FC}(\,\cdot \,)$with$f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.
- Published
- 2019
74. Maximal Operator for the Higher Order Calderón Commutator
- Author
-
Xudong Lai
- Subjects
42B20, 42B25 ,Mathematics::Functional Analysis ,Pure mathematics ,Multilinear map ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Commutator (electric) ,Space (mathematics) ,01 natural sciences ,law.invention ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs ,law ,Product (mathematics) ,Maximal operator ,Order (group theory) ,0101 mathematics ,Mathematics ,Weighted space - Abstract
In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calder\'on commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^p(\mathbb{R}^d,w)$ space, including some peculiar endpoint estimates of the higher dimensional Calder\'on commutator., Comment: 36 pages, Canadian Journal of Mathematics, to appear. arXiv admin note: text overlap with arXiv:1712.09020
- Published
- 2019
75. Orlicz Addition for Measures and an Optimization Problem for the -divergence
- Author
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Deping Ye and Shaoxiong Hou
- Subjects
Pure mathematics ,Optimization problem ,General Mathematics ,010102 general mathematics ,f-divergence ,Star (graph theory) ,01 natural sciences ,Dual (category theory) ,Interpretation (model theory) ,010101 applied mathematics ,Affine transformation ,0101 mathematics ,Isoperimetric inequality ,Divergence (statistics) ,Mathematics - Abstract
This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.
- Published
- 2019
76. Slice-torus Concordance Invariants and Whitehead Doubles of Links
- Author
-
Alberto Cavallo and Carlo Collari
- Subjects
Pure mathematics ,General Mathematics ,Concordance ,Computation ,010102 general mathematics ,Geometric Topology (math.GT) ,Torus ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Link concordance ,57M25, 57M27 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Link (knot theory) ,Mathematics::Symplectic Geometry ,Mathematics ,Slice genus - Abstract
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant., 31 pages, 19 figures, 4 tables. Improved exposition, typos fixed, slight improvement of Propositions 2.10 and 3.5, and added a comment on a result of A. Conway related to Theorem 1.4. Comments are welcome!
- Published
- 2019
77. Yamabe Solitons and Ricci Solitons on Almost co-Kähler Manifolds
- Author
-
Young Jin Suh and Uday Chand De
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,Object (computer science) ,01 natural sciences ,Mathematics - Abstract
The object of this paper is to study Yamabe solitons on almost co-Kähler manifolds as well as on $(k,\unicode[STIX]{x1D707})$-almost co-Kähler manifolds. We also study Ricci solitons on $(k,\unicode[STIX]{x1D707})$-almost co-Kähler manifolds.
- Published
- 2019
78. On Annelidan, Distributive, and Bézout Rings
- Author
-
Ryszard Mazurek and Greg Marks
- Subjects
Pure mathematics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Distributive lattice ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Annihilator ,Chain (algebraic topology) ,Distributive property ,Ideal (ring theory) ,0101 mathematics ,Symmetry (geometry) ,Mathematics - Abstract
A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.
- Published
- 2019
79. Rings whose Elements are the Sum of a Tripotent and an Element from the Jacobson Radical
- Author
-
Yiqiang Zhou, Tülay Yildirim, and M. Tamer Koşan
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,Idempotence ,Boolean ring ,Context (language use) ,Jacobson radical ,Nilpotent group ,Abelian group ,Element (category theory) ,Mathematics ,Group ring - Abstract
This paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$. These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.
- Published
- 2019
80. Calabi–Yau Quotients of Hyperkähler Four-folds
- Author
-
Alice Garbagnati, Chiara Camere, Giovanni Mongardi, Camere, Chiara, Garbagnati, Alice, and Mongardi, Giovanni
- Subjects
irreducible holomorphic symplectic manifold, Hyperkähler manifold, Calabi-Yau 4-fold, Borcea-Voisin construction, automorphism, quotient map, non symplectic involution ,automorphism ,Pure mathematics ,quotient map ,General Mathematics ,010102 general mathematics ,Hyperkähler manifold ,irreducible holomorphic symplectic manifold ,Calabi-Yau 4-fold ,Borcea-Voisin construction ,non symplectic involution ,Automorphism ,01 natural sciences ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Calabi–Yau manifold ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Quotient ,Mathematics - Abstract
The aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.
- Published
- 2019
81. Linear Conjugacy
- Author
-
Benjamin Steinberg
- Subjects
Linear representation ,Pure mathematics ,Mathematics::Commutative Algebra ,Group (mathematics) ,Semigroup ,General Mathematics ,20M30 ,010102 general mathematics ,Field (mathematics) ,Group Theory (math.GR) ,Mathematics - Rings and Algebras ,01 natural sciences ,Conjugacy class ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics ,Conjugate - Abstract
We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.
- Published
- 2019
82. Cyclicity in Dirichlet Spaces
- Author
-
I. Labghail and Y. Elmadani
- 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 $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$ -capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ ), $f$ vanishes on $E$ , and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ .
- Published
- 2019
83. On the Weak Order of Coxeter Groups
- Author
-
Matthew Dyer
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Coxeter group ,Group Theory (math.GR) ,010103 numerical & computational mathematics ,01 natural sciences ,Power set ,Bruhat order ,Complete lattice ,Lattice (order) ,FOS: Mathematics ,20F55 (Primary) 17B22(Secondary) ,Closure operator ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general)., 37 pages, submitted
- Published
- 2019
84. Commuting and Semi-commuting Monomial-type Toeplitz Operators on Some Weakly Pseudoconvex Domains
- Author
-
Xing Tang Dong, Ze-Hua Zhou, and Cao Jiang
- Subjects
Unit sphere ,Monomial ,Pure mathematics ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Commutator (electric) ,Type (model theory) ,01 natural sciences ,Toeplitz matrix ,law.invention ,law ,Bergman space ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Toeplitz operator - Abstract
In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.
- Published
- 2019
85. Embeddings of Müntz Spaces in
- Author
-
Ihab Al Alam and Pascal Lefèvre
- Subjects
Pure mathematics ,General Mathematics ,Muntz metal ,Mathematics - Abstract
In this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$.
- Published
- 2019
86. The Steklov Problem on Differential Forms
- Author
-
Mikhail Karpukhin
- Subjects
Pure mathematics ,Differential form ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Spectral properties ,01 natural sciences ,law.invention ,law ,0103 physical sciences ,Shape optimization ,010307 mathematical physics ,0101 mathematics ,Manifold (fluid mechanics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.
- Published
- 2019
87. Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes
- Author
-
Benjamin Steinberg and Samuel J. van Gool
- Subjects
Pure mathematics ,Semigroup ,General Mathematics ,010102 general mathematics ,Mathematical proof ,01 natural sciences ,Aperiodic graph ,0103 physical sciences ,Homomorphism ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Mathematics ,Merge (linguistics) ,Decomposition theorem - Abstract
This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_{1}$, $T_{2}$, which together generate $T$, and the subsemigroup generated by their setwise product $T_{1}T_{2}$. In this sense we decompose $T$ by merging the subsemigroups $T_{1}$ and $T_{2}$. More generally, our technique merges semigroup homomorphisms from free semigroups.
- Published
- 2019
88. A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth
- Author
-
Shu-Cheng Chang, Yingbo Han, Der-Chen Chang, and Jingzhi Tie
- Subjects
Pure mathematics ,Polynomial ,Conjecture ,Degree (graph theory) ,Volume growth ,General Mathematics ,Mean value ,Space (mathematics) ,Heat kernel ,Mathematics ,Sobolev inequality - Abstract
In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most$d$in a complete noncompact pseudohermitian$(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.
- Published
- 2019
89. Boundary Quotient -algebras of Products of Odometers
- Author
-
Dilian Yang and Hui Li
- Subjects
Product system ,Pure mathematics ,Semigroup ,If and only if ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Zappa–Szép product ,01 natural sciences ,Odometer ,Quotient ,Mathematics - Abstract
In this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.
- Published
- 2019
90. adic -functions for
- Author
-
Daniel Barrera Salazar and Chris Williams
- Subjects
Pure mathematics ,Distribution (number theory) ,General Mathematics ,010102 general mathematics ,Modular form ,Automorphic form ,Function (mathematics) ,Algebraic number field ,01 natural sciences ,0103 physical sciences ,Eigenform ,010307 mathematical physics ,Isomorphism ,Modular symbol ,0101 mathematics ,Mathematics - Abstract
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct$p$-adic$L$-functions for non-critical slope rational modular forms, the theory has been extended to construct$p$-adic$L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the$L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the$p$-adic$L$-function of the eigenform to be this distribution.
- Published
- 2019
91. Injectivity of the Connecting Homomorphisms in Inductive Limits of Elliott–Thomsen Algebras
- Author
-
Zhichao Liu
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Homomorphism ,010103 numerical & computational mathematics ,0101 mathematics ,Direct limit ,01 natural sciences ,Injective function ,Mathematics - Abstract
Let $A$ be the inductive limit of a sequence $$\begin{eqnarray}A_{1}\xrightarrow[{}]{\unicode[STIX]{x1D719}_{1,2}}A_{2}\xrightarrow[{}]{\unicode[STIX]{x1D719}_{2,3}}A_{3}\longrightarrow \cdots\end{eqnarray}$$ with $A_{n}=\bigoplus _{i=1}^{n_{i}}A_{[n,i]}$, where all the $A_{[n,i]}$ are Elliott–Thomsen algebras and $\unicode[STIX]{x1D719}_{n,n+1}$ are homomorphisms. In this paper, we will prove that $A$ can be written as another inductive limit $$\begin{eqnarray}B_{1}\xrightarrow[{}]{\unicode[STIX]{x1D713}_{1,2}}B_{2}\xrightarrow[{}]{\unicode[STIX]{x1D713}_{2,3}}B_{3}\longrightarrow \cdots\end{eqnarray}$$ with $B_{n}=\bigoplus _{i=1}^{n_{i}^{\prime }}B_{[n,i]^{\prime }}$, where all the $B_{[n,i]^{\prime }}$ are Elliott–Thomsen algebras and with the extra condition that all the $\unicode[STIX]{x1D713}_{n,n+1}$ are injective.
- Published
- 2019
92. Finsler Warped Product Metrics of Douglas Type
- Author
-
Xiaohuan Mo and Huaifu Liu
- Subjects
Pure mathematics ,Mathematics::Operator Algebras ,Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Optimization and Control ,Product metric ,Type (model theory) ,Curvature ,01 natural sciences ,Mathematics::K-Theory and Homology ,Product (mathematics) ,0103 physical sciences ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics - 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.
- Published
- 2019
93. On Knörrer Periodicity for Quadric Hypersurfaces in Skew Projective Spaces
- Author
-
Kenta Ueyama
- Subjects
Pure mathematics ,Quadric ,General Mathematics ,Skew ,Projective test ,Mathematics - Abstract
We study the structure of the stable category $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ of graded maximal Cohen–Macaulay module over $S/(f)$ where $S$ is a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree 1, and $f=x_{1}^{2}+\cdots +x_{n}^{2}$. If $S$ is commutative, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is well known by Knörrer’s periodicity theorem. In this paper, we prove that if $n\leqslant 5$, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to $\mathbb{P}^{1}$.
- Published
- 2018
94. Homogeneous Einstein Finsler Metrics on -dimensional Spheres
- Author
-
Xiaohuan Mo and Libing Huang
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Homogeneous ,Metric (mathematics) ,symbols ,SPHERES ,Sectional curvature ,0101 mathematics ,Einstein ,Mathematics - Abstract
In this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.
- Published
- 2018
95. Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds
- Author
-
Bin Shen
- Subjects
Pure mathematics ,General Mathematics ,Holomorphic function ,Vector bundle ,Vector field ,Mathematics::Differential Geometry ,Curvature ,Mathematics::Symplectic Geometry ,Hermitian matrix ,Mathematics - Abstract
In this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.
- Published
- 2018
96. The Quotient Problem for Entire Functions
- Author
-
Ji Guo
- Subjects
Pure mathematics ,General Mathematics ,Entire function ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Quotient ,Mathematics - Abstract
Let $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.
- Published
- 2018
97. On the Pointwise Bishop–Phelps–Bollobás Property for Operators
- Author
-
Sun Kwang Kim, Vladimir Kadets, Miguel Martín, Han Ju Lee, and Sheldon Dantas
- Subjects
Pointwise ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,Banach space ,Regular polygon ,46B04 (Primary), 46B07, 46B20 (Secondary) ,Space (mathematics) ,Compact operator ,01 natural sciences ,Mathematics - Functional Analysis ,Range (mathematics) ,Dimension (vector space) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollob\'as property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(\mu)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators., Comment: 19 pages, to appear in the Canadian J. Math. In this version, section 6 and the appendix of the previous version have been removed
- Published
- 2018
98. On the First Zassenhaus Conjecture and Direct Products
- Author
-
M.A. Serrano, Andreas Bächle, and Wolfgang Kimmerle
- Subjects
Ring (mathematics) ,Pure mathematics ,16S34, 16U60, 20C05 ,General Mathematics ,010102 general mathematics ,Sylow theorems ,Group Theory (math.GR) ,Mathematics - Rings and Algebras ,01 natural sciences ,Hall subgroup ,Mathematics::Group Theory ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Frobenius group ,Mathematics - Group Theory ,Direct product ,Group ring ,Mathematics - Abstract
In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products as well as the General Bovdi Problem (Gen-BP) which turns out to be a slightly weaker variant of (ZC1). Among others we prove that (Gen-BP) holds for Sylow tower groups, so in particular for the class of supersolvable groups. (ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group. We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G \times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group., 17 pages. Comments welcome!
- Published
- 2018
99. Infinitesimal Hilbertianity of Weighted Riemannian Manifolds
- Author
-
Danka Lučić and Enrico Pasqualetto
- Subjects
Mathematics - Differential Geometry ,Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,Infinitesimal ,010102 general mathematics ,Riemannian manifold ,01 natural sciences ,Sobolev space ,differentiaaligeometria ,symbols.namesake ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,symbols ,Mathematics::Metric Geometry ,53C23, 46E35, 58B20 ,010307 mathematical physics ,Finsler manifold ,Mathematics::Differential Geometry ,0101 mathematics ,monistot ,Carnot cycle ,funktionaalianalyysi ,Mathematics - Abstract
The main result of this paper is the following: anyweightedRiemannian manifold$(M,g,\unicode[STIX]{x1D707})$,i.e., a Riemannian manifold$(M,g)$endowed with a generic non-negative Radon measure$\unicode[STIX]{x1D707}$, isinfinitesimally Hilbertian, which means that its associated Sobolev space$W^{1,2}(M,g,\unicode[STIX]{x1D707})$is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold$(M,F,\unicode[STIX]{x1D707})$can be isometrically embedded into the space of all measurable sections of the tangent bundle of$M$that are$2$-integrable with respect to$\unicode[STIX]{x1D707}$.By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.
- Published
- 2020
100. Spherical Fundamental Lemma for Metaplectic Groups
- Author
-
Caihua Luo
- Subjects
Pure mathematics ,Metaplectic group ,Formalism (philosophy) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Fundamental lemma ,Topology ,01 natural sciences ,Mathematics - Abstract
In this paper, we prove the spherical fundamental lemma for metaplectic group Mp2n based on the formalism of endoscopy theory by J. Adams, D. Renard, and W.-W. Li.
- Published
- 2018
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