303 results
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2. An Existence Result for Stepanoff Almost-Periodic Differential Equations
- Author
-
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
3. Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts
- Author
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Tuoc Phan
- Subjects
General Mathematics ,Lorentz transformation ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,symbols.namesake ,Bounded function ,symbols ,Vector field ,Maximal function ,0101 mathematics ,Divergence (statistics) ,Mathematics - Abstract
This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.
- Published
- 2019
4. 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
5. Isomorphisms of Twisted Hilbert Loop Algebras
- Author
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Timothée Marquis and Karl-Hermann Neeb
- Subjects
17B65, 17B70, 17B22, 17B10 ,General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics - Rings and Algebras ,01 natural sciences ,Combinatorics ,Loop (topology) ,symbols.namesake ,Isomorphism theorem ,Rings and Algebras (math.RA) ,Affine root system ,Product (mathematics) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Isomorphism ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Representation Theory ,Mathematics - Abstract
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections
- Published
- 2017
6. 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
7. 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
8. 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
9. Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues
- Author
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Marc Chamberland
- Subjects
Polynomial ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,Jacobian conjecture ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Jacobian matrix and determinant ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Constant (mathematics) ,Eigenvalues and eigenvectors ,Mathematics ,Variable (mathematics) - Abstract
Recent papers have shown that C1 maps whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or F is a polynomial. Specifically, F = (u, v) must take the formfor some constants a, b, c, d, e, f , α, β and a C1 function ϕ in one variable. If, in addition, the function ϕ is not affine, thenThis paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are ±1/2 and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.
- Published
- 2003
10. Rigidity of Hamiltonian Actions
- Author
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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
11. Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum
- Author
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Kenneth S. Williams and Alfred J. van der Poorten
- Subjects
Discrete mathematics ,symbols.namesake ,Quadratic equation ,General Mathematics ,Dedekind sum ,symbols ,Binary quadratic form ,Dedekind eta function ,Isotropic quadratic form ,Mathematics - Abstract
Habib Muzaffar of Carleton University has pointed out to the authors that in their paper [A] only the resultfollows from the prime ideal theorem with remainder for ideal classes, and not the stronger resultstated in Lemma 5.2. This necessitates changes in Sections 5 and 6 of [A]. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of [A], each and every occurrence of o(s − 1) should be replaced by o(1).Sections 5 and 6 of [A] have been rewritten to incorporate the abovementioned correction and are given below. They should replace the original Sections 5 and 6 of [A].
- Published
- 2001
12. A Mountain Pass to the Jacobian Conjecture
- Author
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Gary Meisters and Marc Chamberland
- Subjects
geography ,geography.geographical_feature_category ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Jacobian conjecture ,01 natural sciences ,Algebra ,symbols.namesake ,Jacobian matrix and determinant ,symbols ,Mountain pass ,0101 mathematics ,Mathematics - Abstract
This paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem 1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of are uniformly bounded away from zero for x ∊ Rn, where is a class C1 map, then F is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class C1 maps F of Rn into itself is that all the eigenvalues of F′(x) are bounded away from zero on Rn. This is stated as Conjecture 2.1. If true, it would imply (via Reduction-of-Degree) injectivity of polynomial mapssatisfying the hypothesis, det F′(x) ≡ 1, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.
- Published
- 1998
13. Completion versus removal of redundancy by perturbation
- Author
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Ole Christensen and Marzieh Hasannasab
- Subjects
Completeness ,Sequence ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Hilbert space ,Perturbation (astronomy) ,Riesz bases ,010103 numerical & computational mathematics ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Combinatorics ,Frames ,symbols.namesake ,Redundancy ,Redundancy (information theory) ,FOS: Mathematics ,symbols ,42C40 ,0101 mathematics ,Mathematics - Abstract
A sequence $\left \{g_k\right \}_{k=1}^{\infty }$ in a Hilbert space ${\cal H}$ has the expansion property if each $f\in \overline {\text {span}} \left \{g_k\right \}_{k=1}^{\infty }$ has a representation $f=\sum _{k=1}^{\infty } c_k g_k$ for some scalar coefficients $c_k.$ In this paper, we analyze the question whether there exist small norm-perturbations of $\left \{g_k\right \}_{k=1}^{\infty }$ which allow to represent all $f\in {\cal H};$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\left \{g_k\right \}_{k=1}^{\infty }$ such that $g_k\to 0$ as $k\to \infty ,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.
- Published
- 2021
14. 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
15. On Homogeneous Images of Compact Ordered Spaces
- Author
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Jacek Nikiel and E. D. Tymchatyn
- Subjects
Discrete mathematics ,Pure mathematics ,Continuum (topology) ,General Mathematics ,First-countable space ,010102 general mathematics ,Hausdorff space ,Mathematics::General Topology ,Disjoint sets ,01 natural sciences ,Jordan curve theorem ,symbols.namesake ,Metrization theorem ,0103 physical sciences ,Homogeneous space ,symbols ,010307 mathematical physics ,0101 mathematics ,Indecomposable module ,Mathematics - Abstract
We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.
- Published
- 1993
16. Brill-Noether generality of binary curves
- Author
-
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
17. Embedding of Dirichlet type spaces into tent spaces and Volterra operators
- Author
-
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
18. Dyson’s rank, overpartitions, and universal mock theta functions
- Author
-
Helen W. J. Zhang
- Subjects
Ramanujan theta function ,Combinatorics ,symbols.namesake ,General Mathematics ,Modulo ,010102 general mathematics ,0103 physical sciences ,symbols ,Rank (graph theory) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average: $$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$ Based on Appell–Lerch sum properties and universal mock theta functions, we obtain the stronger version of the results of Bringmann and Lovejoy.
- Published
- 2020
19. Large values of Dirichlet L-functions at zeros of a class of L-functions
- Author
-
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
20. Generalized -Einstein Real Hypersurfaces in and
- Author
-
Yaning Wang
- Subjects
symbols.namesake ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,symbols ,0102 computer and information sciences ,0101 mathematics ,Einstein ,01 natural sciences ,Mathematical physics ,Mathematics - Abstract
In this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.
- Published
- 2020
21. Stability of Almost Periodic Nicholson’s Blowflies Model Involving Patch Structure and Mortality Terms
- Author
-
Si Fu, Lihong Huang, Chuangxia Huang, and Xin Long
- Subjects
Lyapunov function ,Class (set theory) ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Exponential stability ,symbols ,Applied mathematics ,0101 mathematics ,Differential inequalities ,Mathematics - Abstract
Taking into account the effects of patch structure and nonlinear density-dependent mortality terms, we explore a class of almost periodic Nicholson’s blowflies model in this paper. Employing the Lyapunov function method and differential inequality technique, some novel assertions are developed to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recently published literatures. Particularly, an example and its numerical simulations are arranged to support the proposed approach.
- Published
- 2019
22. 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
23. One-Level Density of Low-lying Zeros of Quadratic and Quartic Hecke -functions
- Author
-
Peng Gao and Liangyi Zhao
- Subjects
Field (physics) ,General Mathematics ,Gaussian ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Quadratic equation ,Quartic function ,symbols ,Point (geometry) ,0101 mathematics ,Lying ,Mathematical physics ,Mathematics - Abstract
In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.
- Published
- 2019
24. Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains
- Author
-
Jean Lagacé
- Subjects
General Mathematics ,Dimension (graph theory) ,0211 other engineering and technologies ,02 engineering and technology ,35P20, 11H06, 52C07 ,01 natural sciences ,Dirichlet distribution ,Mathematics - Spectral Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Neumann boundary condition ,Number Theory (math.NT) ,0101 mathematics ,Remainder ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,021103 operations research ,Mathematics - Number Theory ,010102 general mathematics ,Mathematical analysis ,Torus ,Mathematics::Spectral Theory ,symbols ,Cube ,Laplace operator ,Analysis of PDEs (math.AP) - Abstract
This paper is concerned with the maximisation of the k'th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension d as k goes to infinity. We show that in any dimension maximisers exist for any given k, but that any sequence of maximisers degenerates as k goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the k'th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions., Comment: 20 pages
- Published
- 2019
25. 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
26. Ricci Solitons on Almost Co-Kähler Manifolds
- Author
-
Yaning Wang
- Subjects
General Mathematics ,010102 general mathematics ,Lie group ,010103 numerical & computational mathematics ,01 natural sciences ,Manifold ,Ricci soliton ,symbols.namesake ,Dimension (vector space) ,Minkowski space ,symbols ,0101 mathematics ,Einstein ,Mathematics ,Mathematical physics - Abstract
In this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$-Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.
- Published
- 2018
27. A Special Case of Completion Invariance for thec2Invariant of a Graph
- Author
-
Karen Yeats
- Subjects
Combinatorics ,symbols.namesake ,Computer Science::Information Retrieval ,General Mathematics ,symbols ,Feynman diagram ,Invariant (physics) ,Special case ,Graph property ,Graph ,Mathematics - Abstract
Thec2invariant is an arithmetic graph invariant defined by Schnetz. It is useful for understanding Feynman periods. Brown and Schnetz conjectured that thec2invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of thec2invariant in the case where we are over the field with 2 elements and the completed graph has an odd number of vertices. The methods involve enumerating certain edge bipartitions of graphs; two different constructions are needed.
- Published
- 2018
28. The Oscillatory Hyper-Hilbert Transform Associated with Plane Curves
- Author
-
Junfeng Li and Haixia Yu
- Subjects
symbols.namesake ,Plane curve ,General Mathematics ,Bounded function ,010102 general mathematics ,Mathematical analysis ,symbols ,010103 numerical & computational mathematics ,Hilbert transform ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, the bounded properties of oscillatory hyper-Hilbert transformalong certain plane curves γ(t),are studied. For general curves, these operators are bounded in L2() if β ≥ 3α. Their boundedness in Lp() is also obtained, whenever β > 3α and .
- Published
- 2018
29. 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
30. 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
31. On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis
- Author
-
Michael Th. Rassias and Helmut Maier
- Subjects
Combinatorics ,symbols.namesake ,Riemann hypothesis ,Critical line ,General Mathematics ,symbols ,Expression (computer science) ,Infimum and supremum ,Dirichlet distribution ,Riemann zeta function ,Mathematics - Abstract
A crucial role in the Nyman–Beurling–Báez-Duarte approach to the Riemann Hypothesis is played by the distancewhere the infimum is over all Dirichlet polynomialsof length N. In this paper we investigate under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.
- Published
- 2018
32. Area Integral Means of Analytic Functions in the Unit Disk
- Author
-
Chunjie Wang, Kehe Zhu, and Xiaohui Cui
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Measure (physics) ,Hardy space ,01 natural sciences ,Unit disk ,010101 applied mathematics ,Range (mathematics) ,symbols.namesake ,Logarithmically convex function ,Bergman space ,symbols ,0101 mathematics ,Mathematics ,Analytic function - Abstract
For an analytic function ऒ on the unit disk , we show that the L2 integral mean of ऒ on c < |z| < r with respect to the weighted area measure (1 − |z|2)αd A(z) is a logarithmically convex function of r on (c, 1), where −3 ≤ ∞ ≤ 0 and c ∈ [0, 1). Moreover, the range [−3, 0] for ∞ is best possible. When c = 0, our arguments here also simplify the proof for several results we obtained in earlier papers.
- Published
- 2018
33. Euler-type Relative Equilibria and their Stability in Spaces of Constant Curvature
- Author
-
Juan Manuel Sánchez-Cerritos and Ernesto Pérez-Chavela
- Subjects
Geodesic ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Space (mathematics) ,01 natural sciences ,Stability (probability) ,Measure (mathematics) ,010305 fluids & plasmas ,Constant curvature ,symbols.namesake ,0103 physical sciences ,Euler's formula ,symbols ,Algebraic curve ,0101 mathematics ,Curved space ,Mathematics - Abstract
We consider three point positivemasses moving onS2andH2. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of S2, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classiûcation of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibriumis spectrally stable or unstable. OnH2, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.
- Published
- 2018
34. Periodic Steady-state Solutions of a Liquid Film Model via a Classical Method
- Author
-
Ahmad Alhasanat and Chunhua Ou
- Subjects
Surface (mathematics) ,Steady state ,General Mathematics ,Uniform convergence ,Mathematical analysis ,Reynolds number ,Physics::Fluid Dynamics ,symbols.namesake ,Liquid film ,Scheme (mathematics) ,symbols ,Asymptotic formula ,Uniqueness ,Mathematics - Abstract
In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness as well as the asymptotic formula of the periodic solutions.
- Published
- 2018
35. Hilbert Transformation and Representation of the ax + b Group
- Author
-
Tao Qian, Pei Dang, and Hua Liu
- Subjects
Pure mathematics ,Mathematics - Complex Variables ,Group (mathematics) ,Semigroup ,General Mathematics ,Operator (physics) ,symbols.namesake ,Transformation (function) ,Unit circle ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Hilbert transform ,Complex Variables (math.CV) ,Symmetry (geometry) ,Commutative property ,Mathematics - Abstract
In this paper we study the Hilbert transformations over L2() and L2() fromthe viewpoint of symmetry. For a linear operator over L2() commutative with the ax + b group, we show that the operator is of the form λI+ηH, where I and H are the identity operator and Hilbert transformation, respectively, and λ, η are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the ax + b group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the ax + b group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle L2().
- Published
- 2018
36. QpSpaces and Dirichlet Type Spaces
- Author
-
Stamatis Pouliasis, Guanlong Bao, and Nihat Gökhan Göğüş
- Subjects
Invariant function ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,06 humanities and the arts ,Type (model theory) ,0603 philosophy, ethics and religion ,Space (mathematics) ,01 natural sciences ,Unit disk ,Dirichlet distribution ,symbols.namesake ,Intersection ,060302 philosophy ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, we show that the Möbius invariant function space Qpcan be generated by variant Dirichlet type spaces 𝒟μ,pinduced by finite positive Borel measures μ on the open unit disk. A criterion for the equality between the space 𝒟μ,pand the usual Dirichlet type space 𝒟pis given. We obtain a sufficient condition to construct different 𝒟μ,pspaces and provide examples. We establish decomposition theorems for 𝒟μ,pspaces and prove that the non-Hilbert space Qpis equal to the intersection of Hilbert spaces 𝒟μ,p. As an application of the relation between Qpand 𝒟μ,pspaces, we also obtain that there exist different 𝒟μ,pspaces; this is a trick to prove the existence without constructing examples.
- Published
- 2017
37. Anisotropic Hardy-Lorentz Spaces with Variable Exponents
- Author
-
Jorge J. Betancor, Lourdes Rodríguez-Mesa, and Víctor Almeida
- Subjects
Mathematics::Functional Analysis ,Physics::General Physics ,Variable exponent ,Mathematics::Complex Variables ,General Mathematics ,Lorentz transformation ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,Physics::Classical Physics ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Atomic decomposition ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Maximal function ,0101 mathematics ,Anisotropy ,Mathematical physics ,Mathematics ,Variable (mathematics) - Abstract
In this paper we introduceHardy-Lorentz spaces with variable exponents associated with dilations in ℝn. We establishmaximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.
- Published
- 2017
38. Rational Function Operators from Poisson Integrals
- Author
-
Laiyi Zhu and Xu Xu
- Subjects
Order of integration (calculus) ,Algebra ,symbols.namesake ,General Mathematics ,Singular integral operators of convolution type ,symbols ,Rational function ,Operator theory ,Poisson distribution ,Mathematics - Abstract
In this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.
- Published
- 2017
39. Springer’s Weyl Group Representation via Localization
- Author
-
James B. Carrell and Kiumars Kaveh
- Subjects
Pure mathematics ,Weyl group ,14M15, 14F43, 55N91 ,Group (mathematics) ,General Mathematics ,Flag (linear algebra) ,010102 general mathematics ,Group Theory (math.GR) ,Type (model theory) ,01 natural sciences ,Cohomology ,Mathematics - Algebraic Geometry ,symbols.namesake ,Algebraic group ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,symbols ,Equivariant cohomology ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G$ denote a reductive algebraic group over $\mathbb{C}$ and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer variety $\mathcal{B}_x$ is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing the Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable property that the Weyl group $W$ of $G$ admits a representation on the cohomology of $\mathcal{B}_x$ even though $W$ rarely acts on $\mathcal{B}_x$ itself. Well-known constructions of this action due to Springer et al use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when $x$ is what we call parabolic-surjective. The idea is to use localization to construct an action of $W$ on the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic subtorus of $G$. This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type $A$ and, more generally, all nilpotents for which it is known that $W$ acts on $H_S^*(\mathcal{B}_x)$ for some torus $S$. Our result is deduced from a general theorem describing when a group action on the cohomology of the fixed point set of a torus action on a space lifts to the full cohomology algebra of the space., 6 pages, title changed and made shorter, the presentation of the paper totally revised, final version to appear in the Canadian Mathematical Bulletin
- Published
- 2017
40. Weak Factorizations of the Hardy Space H1(ℝn) in Terms of Multilinear Riesz Transforms
- Author
-
Ji Li and Brett D. Wick
- Subjects
Mathematics::Functional Analysis ,Multilinear map ,Pure mathematics ,Constructive proof ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,42B35, 42B20, 42B35 ,Hardy space ,Characterization (mathematics) ,01 natural sciences ,Dual (category theory) ,010101 applied mathematics ,symbols.namesake ,Riesz transform ,Factorization ,Mathematics - Classical Analysis and ODEs ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms., Comment: improved some typos
- Published
- 2017
41. The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators
- Author
-
Djordjije Vujadinovic and David Kalaj
- Subjects
Unit sphere ,Newtonian potential ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Operator (computer programming) ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Boundary data ,symbols ,0101 mathematics ,Poisson's equation ,Bessel function ,Möbius transformation ,Mathematics - Abstract
In this paper we determine the L1 ⟶ L1 and L∞ ⟶ L∞ norms of an integral operator N related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.
- Published
- 2017
42. The Classical N-body Problem in the Context of Curved Space
- Author
-
Florin Diacu
- Subjects
Differential equation ,General Mathematics ,n-body problem ,010102 general mathematics ,Mathematical analysis ,Equations of motion ,Newton's laws of motion ,Dynamical Systems (math.DS) ,01 natural sciences ,010101 applied mathematics ,Constant curvature ,symbols.namesake ,FOS: Mathematics ,Gaussian curvature ,symbols ,Mathematics - Dynamical Systems ,0101 mathematics ,Constant (mathematics) ,Curved space ,Mathematics - Abstract
We provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature, k, for all k real. In previous studies, the equations of motion made sense only for k nonzero. The system derived here does more than just include the Euclidean case in the limit when k tends to 0: it recovers the classical equations for k=0. This new expression of the laws of motion allows the study of the N-body problem in the context of constant curvature spaces and thus offers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals., Comment: 19 pages, 1 figure
- Published
- 2017
43. Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
- Author
-
Arash Ghaani Farashahi
- Subjects
Pure mathematics ,Trace (linear algebra) ,Dual space ,General Mathematics ,010102 general mathematics ,Hilbert space ,Operator theory ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Compact group ,Homogeneous space ,symbols ,Coset ,0101 mathematics ,Mathematics - Abstract
This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. LetGbe a compact group and letHbe a closed subgroup ofG. LetG/Hbe the left coset space ofHinGand letμbe the normalized G-invariant measure onG-Hassociated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert spaceL2(G/H,μ).
- Published
- 2017
44. Cokernels of Homomorphisms from Burnside Rings to Inverse Limits
- Author
-
Masaharu Morimoto
- Subjects
Normal subgroup ,Pure mathematics ,Finite group ,General Mathematics ,010102 general mathematics ,Burnside ring ,Cartesian product ,01 natural sciences ,symbols.namesake ,Cokernel ,0103 physical sciences ,symbols ,Order (group theory) ,Homomorphism ,010307 mathematical physics ,Inverse limit ,0101 mathematics ,Mathematics - Abstract
Let G be a finite group and let A(G) denote the Burnside ring of G. Then an inverse limit L(G) of the groups A(H) for proper subgroups H of G and a homomorphism res from A(G) to L(G) are obtained in a natural way. Let Q(G) denote the cokernel of res. For a prime p, let N(p) be the minimal normal subgroup of G such that the order of G/N(p) is a power of p, possibly 1. In this paper we prove that Q(G) is isomorphic to the cartesian product of the groups Q(G/N(p)), where p ranges over the primes dividing the order of G.
- Published
- 2017
45. The Thickness of the Cartesian Product of Two Graphs
- Author
-
Yichao Chen and Xuluo Yin
- Subjects
General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Cartesian product ,01 natural sciences ,Complete bipartite graph ,Upper and lower bounds ,Graph ,Planar graph ,Combinatorics ,symbols.namesake ,Planar ,010201 computation theory & mathematics ,symbols ,Bipartite graph ,0101 mathematics ,Mathematics - Abstract
The thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t that contains no proper subgraph of thickness t. In this paper, upper and lower bounds are obtained for the thickness, t(G ⎕ H), of the Cartesian product of two graphs G and H, in terms of the thickness t(G) and t(H). Furthermore, the thickness of the Cartesian product of two planar graphs and of a t-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph K4k,4k, the thickness of the Cartesian product of two complete bipartite graphs Kn,n and Kn,n is also given for n≠4k + 1.
- Published
- 2016
46. A Note on 3-choosability of Planar Graphs Related to Montanssier’s Conjecture
- Author
-
Haihui Zhang
- Subjects
Conjecture ,General Mathematics ,Existential quantification ,010102 general mathematics ,Process (computing) ,0102 computer and information sciences ,01 natural sciences ,Planar graph ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Graph (abstract data type) ,0101 mathematics ,Mathematics - Abstract
For a given list assignment L = {L(v) : v ∈ V(G)}, a graph G = (V, E) is L-colorable if there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V. If G is L-colorable for every list assignment L having |L(v)| ≥ k for all v ∈ V, then G is said to be k-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.
- Published
- 2016
47. Discrete Space-time and Lorentz Transformations
- Author
-
Christian Pommerenke and Gerd Jensen
- Subjects
Pure mathematics ,Spinor ,010308 nuclear & particles physics ,Gaussian integer ,General Mathematics ,Discrete space ,Lorentz transformation ,010102 general mathematics ,Context (language use) ,Mathematical proof ,01 natural sciences ,General Relativity and Quantum Cosmology ,symbols.namesake ,Tensor (intrinsic definition) ,0103 physical sciences ,symbols ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
Alfred Schild established conditions where Lorentz transformations map world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild’s number-theoretic arguments, the subject is also interesting when isolated from its physical background. Schild’s paper is not easy to understand. Therefore, we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers
- Published
- 2016
48. Constrained Approximation with Jacobi Weights
- Author
-
Igor A. Shevchuk, Kirill A. Kopotun, and Dany Leviatan
- Subjects
General Mathematics ,Jacobi method for complex Hermitian matrices ,010102 general mathematics ,Mathematical analysis ,Jacobi method ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Jacobi eigenvalue algorithm ,Jacobi rotation ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we prove that for ℓ = 1 or 2 the rate of best ℓ- monotone polynomial approximation in the Lp norm (1 ≤ p ≤) weighted by the Jacobi weight with , is bounded by an appropriate (ℓ + 1)-st modulus of smoothness with the same weight, and that this rate cannot be bounded by the (ℓ + 2)-nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.
- Published
- 2016
49. Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems
- Author
-
Cristian Carcamo and Claudio Vidal
- Subjects
Lyapunov function ,symbols.namesake ,Planar ,General Mathematics ,Mathematical analysis ,Degenerate energy levels ,symbols ,Linear approximation ,Hamiltonian (quantum mechanics) ,Cubic function ,Phase diagram ,Mathematics ,Hamiltonian system - Abstract
In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters. In particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated with mechanical Hamiltonian systems and Hamiltonian systems defined by cubic polynomials. Finally, we point out important diòerences with the continuous case.
- Published
- 2015
50. Lyapunov Stability and Attraction Under Equivariant Maps
- Author
-
Carlos J. Braga Barros, Victor H. L. Rocha, and Josiney A. Souza
- Subjects
Lyapunov stability ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Lyapunov exponent ,01 natural sciences ,Attraction ,010101 applied mathematics ,symbols.namesake ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Equivariant map ,Lyapunov equation ,0101 mathematics ,Mathematics - Abstract
LetMandNbe admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume thatis a semigroup acting on bothMandN. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors, and Lyapunov stable sets (all concepts deûned for the action of the semigroup) under equivariant maps and semiconjugations fromMtoN.
- Published
- 2015
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