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2. Remark on a paper by Aczél and Ostrowski
- Author
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Wolfgang Walter
- Subjects
Pure mathematics ,General Mathematics ,Of the form ,Function (mathematics) ,Mathematics - Abstract
The objective of the paper Aczél and Ostrowski (1973) is to show in an elementary way that any real-valued function f, defined on (0,1) and satisfying he inquality , where n ≧ 3 is fixed, is necessary of the form .
- Published
- 1976
3. LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS
- Author
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Saminathan Ponnusamy, Navneet Lal Sharma, and Karl-Joachim Wirths
- Subjects
010101 applied mathematics ,Combinatorics ,Logarithm ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Convex function ,01 natural sciences ,Upper and lower bounds ,Unit disk ,Mathematics ,Univalent function - Abstract
Let${\mathcal{S}}$be the family of analytic and univalent functions$f$in the unit disk$\mathbb{D}$with the normalization$f(0)=f^{\prime }(0)-1=0$, and let$\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$denote the logarithmic coefficients of$f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families${\mathcal{F}}(c)$and${\mathcal{G}}(c)$of functions$f\in {\mathcal{S}}$defined by$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)for some$c\in (0,3]$and$c\in (0,1]$, respectively. We obtain the sharp upper bound for$|\unicode[STIX]{x1D6FE}_{n}|$when$n=1,2,3$and$f$belongs to the classes${\mathcal{F}}(c)$and${\mathcal{G}}(c)$, respectively. The paper concludes with the following two conjectures:∙If$f\in {\mathcal{F}}(-1/2)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$for$n\geq 1$, and$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$where$\text{Li}_{2}(x)$denotes the dilogarithm function.∙If$f\in {\mathcal{G}}(c)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$for$n\geq 1$.
- Published
- 2019
4. THE LATTICE OF VARIETIES OF STRICT LEFT RESTRICTION SEMIGROUPS
- Author
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Peter R. Jones
- Subjects
010101 applied mathematics ,Pure mathematics ,Unary operation ,General Mathematics ,Lattice (order) ,010102 general mathematics ,0101 mathematics ,Identity element ,01 natural sciences ,Mathematics - Abstract
Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.
- Published
- 2018
5. NILPOTENCY IN UNCOUNTABLE GROUPS
- Author
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Marco Trombetti, Francesco de Giovanni, De Giovanni, Francesco, and Trombetti, Marco
- Subjects
Pure mathematics ,nilpotent group ,uncountable group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics (all) ,Uncountable set ,010307 mathematical physics ,0101 mathematics ,soluble group ,01 natural sciences ,Mathematics - Abstract
The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.
- Published
- 2016
6. 2-ARC-TRANSITIVE REGULAR COVERS OF HAVING THE COVERING TRANSFORMATION GROUP
- Author
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Shaofei Du and Wenqin Xu
- Subjects
Transitive relation ,Matching (graph theory) ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Complete bipartite graph ,Prime (order theory) ,Combinatorics ,Arc (geometry) ,010201 computation theory & mathematics ,Covering graph ,2-transitive group ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
This paper contributes to the regular covers of a complete bipartite graph minus a matching, denoted $K_{n,n}-nK_{2}$, whose fiber-preserving automorphism group acts 2-arc-transitively. All such covers, when the covering transformation group $K$ is either cyclic or $\mathbb{Z}_{p}^{2}$ with $p$ a prime, have been determined in Xu and Du [‘2-arc-transitive cyclic covers of $K_{n,n}-nK_{2}$’, J. Algebraic Combin.39 (2014), 883–902] and Xu et al. [‘2-arc-transitive regular covers of $K_{n,n}-nK_{2}$ with the covering transformation group $\mathbb{Z}_{p}^{2}$’, Ars. Math. Contemp.10 (2016), 269–280]. Finally, this paper gives a classification of all such covers for $K\cong \mathbb{Z}_{p}^{3}$ with $p$ a prime.
- Published
- 2016
7. DUALITY FOR QUASIPOLYTOPES
- Author
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Anna B. Romanowska and Anna Mućka
- Subjects
Pure mathematics ,Fenchel's duality theorem ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Metric Geometry ,Strong duality ,Duality (optimization) ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.
- Published
- 2016
8. ON POWER MOMENTS OF THE HECKE MULTIPLICATIVE FUNCTIONS
- Author
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Makoto Minamide and Kalyan Chakraborty
- Subjects
Moment (mathematics) ,Pure mathematics ,Conjecture ,General Mathematics ,Multiplicative function ,Ergodicity ,Order (ring theory) ,Quantum ,Hecke operator ,Mathematics ,Power (physics) - Abstract
In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called ‘Hecke multiplicative’ functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general$k$th power and provide an analogous estimate for the$k$th power moment of the Hecke multiplicative functions. This may be of general interest.
- Published
- 2015
9. CLASSIFICATION OF UNIVALENT HARMONIC MAPPINGS ON THE UNIT DISK WITH HALF-INTEGER COEFFICIENTS
- Author
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Jinjing Qiao and Saminathan Ponnusamy
- Subjects
General Mathematics ,Mathematical analysis ,Half-integer ,Harmonic (mathematics) ,Unit disk ,Mathematics - Abstract
Let ${\mathcal{S}}$ denote the set of all univalent analytic functions $f$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ on the unit disk $|z|. In 1946, Friedman [‘Two theorems on Schlicht functions’, Duke Math. J.13 (1946), 171–177] found that the set ${\mathcal{S}}_{\mathbb{Z}}$ of those functions in ${\mathcal{S}}$ which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa [‘Univalent functions with half-integer coefficients’, Comput. Methods Funct. Theory13(1) (2013), 133–151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set ${\mathcal{S}}_{\mathbb{Z}}$. In this paper, we determine the class of all normalized sense-preserving univalent harmonic mappings $f$ on the unit disk with half-integer coefficients for the analytic and co-analytic parts of $f$. It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.
- Published
- 2014
10. THE VECTOR-VALUED TENT SPACES AND
- Author
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Mikko Kemppainen
- Subjects
Stochastic integration ,Pure mathematics ,Atomic decomposition ,Argument ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Duality (optimization) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Tent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to $H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases $p=1$ and $p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case $p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.
- Published
- 2014
11. SYMMETRIC GRAPHS WITH 2-ARC TRANSITIVE QUOTIENTS
- Author
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Guangjun Xu and Sanming Zhou
- Subjects
Transitive relation ,05C25, 05E18 ,General Mathematics ,Symmetric graph ,Automorphism ,Quotient graph ,Block design ,Combinatorics ,Ordered pair ,FOS: Mathematics ,Mathematics - Combinatorics ,Partition (number theory) ,Combinatorics (math.CO) ,Quotient ,Mathematics - Abstract
A graph $\Ga$ is $G$-symmetric if $\Ga$ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Ga$, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\Ga)$, namely when $V(\Ga)$ admits a nontrivial $G$-invariant partition $\BB$, the quotient graph $\Ga_{\BB}$ of $\Ga$ with respect to $\BB$ is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for $\Ga_{\BB}$ to be $(G, 2)$-arc transitive (regardless of whether $\Ga$ is $(G, 2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of $\BB$ and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v, k$ and two other parameters with respect to $(\Ga, \BB)$ together with a certain 2-point transitive block design induced by $(\Ga, \BB)$. We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for $\Ga_{\BB}$ to be $(G, 2)$-arc transitive., To appear in Journal of the Australian Mathematical Society. (The previous title of this paper was "Finite symmetric graphs with two-arc transitive quotients III")
- Published
- 2014
12. A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY
- Author
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Ganna Kudryavtseva
- Subjects
Algebra ,General Mathematics ,Stone duality ,Object (computer science) ,Noncommutative geometry ,Mathematics - Abstract
The aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.
- Published
- 2013
13. FELL BUNDLES AND IMPRIMITIVITY THEOREMS: MANSFIELD’S AND FELL’S THEOREMS
- Author
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John Quigg, Dana P. Williams, Paul S. Muhly, and Steven Kaliszewski
- Subjects
Surjective function ,Pure mathematics ,geography ,Transformation (function) ,geography.geographical_feature_category ,Series (mathematics) ,General Mathematics ,Fell ,Mathematics - Abstract
S. KALISZEWSKI, PAUL S. MUHLY, JOHN QUIGG, AND DANA P. WILLIAMSAbstract. In the third and latest paper in this series, we recover the imprim-itivity theorems of Mansfield and Fell using our technique of Fell bundles overgroupoids. Also, we apply the Rieffel Surjection of the first paper in the seriesto relate our version of Mansfield’s theorem to that of an Huef and Raeburn,and to give an automatic amenability result for certain transformation Fellbundles.
- Published
- 2013
14. THE ESSENTIAL SPECTRUM OF A PERTURBED OPERATOR ARISING IN TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS
- Author
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M. Faierman and Reinhard Mennicken
- Subjects
Algebra ,Pure mathematics ,General Mathematics ,Operator (physics) ,Essential spectrum ,Magnetohydrodynamics ,Mathematics - Abstract
Descloux and Geymonat considered a model problem in two-dimensional magnetohydrodynamics and conjectured that the essential spectrum has an explicitly given band structure. This conjecture was recently proved by Faierman, Mennicken, and Möller by reducing the problem to that for a 2×2 block operator matrix. In a subsequent paper Faierman and Mennicken investigated the essential spectrum for the problem arising from a particular type of perturbation of precisely one of the operator entries in the matrix representation cited above of the original problem considered by Descloux and Geymonat. In this paper we extend the results of that work by investigating the essential spectrum for the problem arising from particular types of perturbations of all but one of the aforementioned operators. It remains an open question whether one can perturb the exceptional operator in such a way as to leave the essential spectrum unchanged.
- Published
- 2010
15. SMALL-SCALE EQUIDISTRIBUTION OF RANDOM WAVES GENERATED BY AN UNFAIR COIN FLIP
- Author
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Miriam J. Leonhardt and Melissa Tacy
- Subjects
Coin flipping ,Scale (ratio) ,General Mathematics ,Fair coin ,Wavelength scale ,Mathematical analysis ,Random waves ,Mathematics - Abstract
In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.
- Published
- 2021
16. NONUNIFORM MULTIRESOLUTION ANALYSIS
- Author
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S. Pitchai Murugan and G. P. Youvaraj
- Subjects
General Mathematics ,Multiresolution analysis ,Algorithm ,Mathematics - Abstract
Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.
- Published
- 2021
17. A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS
- Author
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Lingling Huang and Chao Ma
- Subjects
Pure mathematics ,General Mathematics ,Product (mathematics) ,Quotient ,Mathematics - Abstract
This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.
- Published
- 2021
18. ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS
- Author
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Gerard O'Reilly, Craig Miller, Nik Ruskuc, Martyn Quick, University of St Andrews. Pure Mathematics, and University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra
- Subjects
20M10 ,Semigroup ,Residual finiteness ,General Mathematics ,T-NDAS ,Separability ,Finiteness condition ,Group Theory (math.GR) ,Congruence ,Commutative semigroup ,Schutzenberger group ,FOS: Mathematics ,Congruence (manifolds) ,QA Mathematics ,Schützenberger group ,QA ,Mathematics - Group Theory ,Mathematical economics ,Mathematics - Abstract
Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Sch\"utzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup $S$, these three separability conditions coincide and are equivalent to every $\mathcal{H}$-class of $S$ being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal{H}$-classes, we investigate whether it has one of these properties if and only if all its Sch\"utzenberger groups have the property., Comment: 27 pages
- Published
- 2021
19. On systems of diagonal forms
- Author
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Michael P. Knapp
- Subjects
Combinatorics ,Discrete mathematics ,Polynomial ,Degree (graph theory) ,Exponential growth ,Integer ,General Mathematics ,Diagonal ,Zero (complex analysis) ,Field (mathematics) ,Prime (order theory) ,Mathematics - Abstract
In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field Qp provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in Qp. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.
- Published
- 2007
20. SOME -HARDY AND -RELLICH TYPE INEQUALITIES WITH REMAINDER TERMS
- Author
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Yongyang Jin and Shoufeng Shen
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Type (model theory) ,Remainder ,01 natural sciences ,Mathematics ,media_common - Abstract
In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.
- Published
- 2021
21. The generalized condition numbers of bounded linear operators in Banach spaces
- Author
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Yimin Wei, Yifeng Xue, and Guoliang Chen
- Subjects
Unbounded operator ,Approximation property ,Computer Science::Information Retrieval ,General Mathematics ,Spectrum (functional analysis) ,Mathematical analysis ,Finite-rank operator ,C0-semigroup ,Compact operator ,Strictly singular operator ,Bounded operator ,Mathematics - Abstract
For any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.
- Published
- 2004
22. A new approach to thek(GV)-problem
- Author
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Thomas Michael Keller
- Subjects
Combinatorics ,Discrete mathematics ,Finite group ,Semidirect product ,Conjugacy class ,Character (mathematics) ,Group (mathematics) ,General Mathematics ,Line (geometry) ,Order (group theory) ,Mathematics - Abstract
This paper is concerned with the well-known and long-standingk(G V)-problem: If the finite groupGacts faithfully and irreducibly on the finiteGF(p)-moduleVandpdoes not divide the order ofG, is the numberk(GV) of conjugacy classes of the semidirect productGVbounded above by the order ofV? Over the past two decades, through the work of numerous people, by using deep character theoretic arguments this question has been answered in the affirmative except for ρ = 5 for which it is still open. In this paper we suggest a new approach to thek(G V)-problem which is independent of most of the previous work on the problem and which is mainly group theoretical. To demonstrate the potential of the new line of attack we use it to solve thek(G V)-problem for solvableGand large ρ.
- Published
- 2003
23. GRADIENT FLOWS OF HIGHER ORDER YANG–MILLS–HIGGS FUNCTIONALS
- Author
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Pan Zhang
- Subjects
General Mathematics ,010102 general mathematics ,Yang–Mills existence and mass gap ,Riemannian manifold ,01 natural sciences ,010101 applied mathematics ,Higgs field ,Higgs boson ,Order operator ,Gravitational singularity ,0101 mathematics ,Balanced flow ,Mathematics ,Gauge fixing ,Mathematical physics - Abstract
In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the$L^2$-bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgsk-functional with Higgs self-interaction, we show that, provided$\dim (M), for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local$L^2$-derivative estimates, energy estimates and blow-up analysis.
- Published
- 2021
24. GENERATORS OF FINITE FIELDS WITH PRESCRIBED TRACES
- Author
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Lucas Reis and Sávio Ribas
- Subjects
010101 applied mathematics ,Finite field ,Mathematics - Number Theory ,Distribution (number theory) ,Field extension ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,Special class ,01 natural sciences ,Mathematics - Abstract
This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions., Comment: This new version contains many corrections, and some minor results were removed. ** To appear in J. Aust. Math. Soc
- Published
- 2021
25. Landau-Lifshitz equation of ferromagnetism with external magnetic field
- Author
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Peter Y. H. Pang, Feng Zhou, and J. Xiao
- Subjects
Magnetism ,Principal curvature ,General Mathematics ,Second fundamental form ,Mathematical analysis ,Uniform boundedness ,Initial value problem ,Uniqueness ,Landau–Lifshitz–Gilbert equation ,Magnetic field ,Mathematics ,Mathematical physics - Abstract
In this article, we prove the existence and uniqueness of solution for the Cauchy problem of the Landau- Lifshitz equation of ferromagnetism with external magnetic field. We also show that the solution is globally regular with the exception of at most finitely many blow-up points. An energy identity at blow-up points is presented. In this paper we discuss the Landau-Lifshitz equation (10) which models the fer- romagnetic spin chain on surfaces with the Gilbert damping term in the presence of an external magnetic field h.x; t/. This equation plays an important rin the understanding of non-equilibrium magnetism. Let and be closed oriented Riemannian surfaces with metric tensors D ./1;2 and g D.g ij /1i; j2 respectively. Let be isometrically embedded in 3 and denote its second fundamental form by A. We shall assume that the principal curvatures of and their derivatives are uniformly bounded. Throughout this paper, C will denote constants which depend only on and . The notation '' for the dot product of vectors will be suppressed. For u0 V ! and t > 0, the Landau-Lifshitz system is given by
- Published
- 2002
26. Automorphisms of Cayley graphs of metacyclic groups of prime-power order
- Author
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Hyo-Seob Sim and Cai Heng Li
- Subjects
Combinatorics ,Vertex-transitive graph ,Cayley's theorem ,Cayley graph ,Cayley table ,Group (mathematics) ,General Mathematics ,Graph theory ,Automorphism ,Prime (order theory) ,Mathematics - Abstract
This paper inverstigates the automorphism groups of Cayley graphs of metracyclicp-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclicp-group for odd primep. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2pof a nonabelian metacyclicp-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.
- Published
- 2001
27. CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS
- Author
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Le Huy Tien and Manseob Lee
- Subjects
010101 applied mathematics ,Pure mathematics ,Class (set theory) ,Property (philosophy) ,Chain (algebraic topology) ,General Mathematics ,010102 general mathematics ,Vector field ,Homoclinic orbit ,0101 mathematics ,Riemannian manifold ,01 natural sciences ,Mathematics - Abstract
Let M be a closed n-dimensional smooth Riemannian manifold, and let X be a $C^1$-vector field of $M.$ Let $\gamma $ be a hyperbolic closed orbit of $X.$ In this paper, we show that X has the $C^1$-stably shadowing property on the chain component $C_X(\gamma )$ if and only if $C_X(\gamma )$ is the hyperbolic homoclinic class.
- Published
- 2021
28. NEW CONSTRUCTIONS OF SELF-COMPLEMENTARY CAYLEY GRAPHS
- Author
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Shu Jiao Song, Guang Rao, and Cai Heng Li
- Subjects
Discrete mathematics ,Cayley graph ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnicket al.[‘On orbital partitions and exceptionality of primitive permutation groups’,Trans. Amer. Math. Soc.356(2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.
- Published
- 2021
29. INFINITARY COMMUTATIVITY AND ABELIANIZATION IN FUNDAMENTAL GROUPS
- Author
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Jeremy Brazas and Patrick Gillespie
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Commutative property ,Mathematics - Abstract
Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$ -operations at a point $x\in X$ . Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.
- Published
- 2020
30. CRITICAL BINOMIAL IDEALS OF NORTHCOTT TYPE
- Author
-
D. Llena, Pedro A. García-Sánchez, and Ignacio Ojeda
- Subjects
Monomial ,Pure mathematics ,Binomial (polynomial) ,Semigroup ,General Mathematics ,010102 general mathematics ,Complete intersection ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Numerical semigroup ,Genus (mathematics) ,Affine space ,0101 mathematics ,Mathematics - Abstract
In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.
- Published
- 2020
31. ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS
- Author
-
Peng Gao and Liangyi Zhao
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,Corollary ,Quadratic equation ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Class number ,Lying ,The Imaginary ,Mathematics - Abstract
In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke $L$-functions of imaginary quadratic number fields of class number one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1710.04909
- Published
- 2020
32. UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES
- Author
-
Arindam Roy and Nicolas Robles
- Subjects
Residue (complex analysis) ,General Mathematics ,Central object ,Liouville function ,010102 general mathematics ,Expected value ,01 natural sciences ,Riemann zeta function ,010101 applied mathematics ,Constant factor ,Combinatorics ,symbols.namesake ,Prime factor ,Elementary proof ,symbols ,0101 mathematics ,Mathematics - Abstract
In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.
- Published
- 2020
33. ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE
- Author
-
José Felipe Voloch, Geertrui Van de Voorde, and John Sheekey
- Subjects
Trace (semiology) ,010201 computation theory & mathematics ,business.industry ,General Mathematics ,Product (mathematics) ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,Process engineering ,business ,01 natural sciences ,Mathematics - Abstract
This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $xy$, where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.
- Published
- 2020
34. ON THE REGULARITY OF SETS IN RIEMANNIAN MANIFOLDS
- Author
-
A. Barani and A. Sepahvand
- Subjects
Pure mathematics ,021103 operations research ,Closed set ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Context (language use) ,02 engineering and technology ,Subderivative ,Riemannian manifold ,01 natural sciences ,Set (abstract data type) ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.
- Published
- 2020
35. BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS
- Author
-
Hong Chuong Doan
- Subjects
Pure mathematics ,Smoothness (probability theory) ,Semigroup ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Upper and lower bounds ,Manifold ,010101 applied mathematics ,Operator (computer programming) ,Integer ,Maximal function ,0101 mathematics ,Heat kernel ,Mathematics - Abstract
Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$. Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1 where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.
- Published
- 2020
36. SPHERICALIZATION AND FLATTENING PRESERVE UNIFORM DOMAINS IN NONLOCALLY COMPACT METRIC SPACES
- Author
-
Yaxiang Li, Saminathan Ponnusamy, and Qingshan Zhou
- Subjects
Pure mathematics ,Metric space ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Flattening ,Mathematics - Abstract
The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.
- Published
- 2020
37. ON LINEAR RELATIONS FOR DIRICHLET SERIES FORMED BY RECURSIVE SEQUENCES OF SECOND ORDER
- Author
-
Carsten Elsner and Niclas Technau
- Subjects
Pure mathematics ,Fibonacci number ,General Mathematics ,Recurrence formula ,010102 general mathematics ,Elliptic function ,01 natural sciences ,symbols.namesake ,Lucas number ,symbols ,Order (group theory) ,Linear independence ,0101 mathematics ,Dirichlet series ,Mathematics - Abstract
Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by $$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n}^{s}}}.\end{eqnarray}$$ As a consequence of Nesterenko’s proof of the algebraic independence of the three Ramanujan functions $R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$ and $P(\unicode[STIX]{x1D70C})$ for any algebraic number $\unicode[STIX]{x1D70C}$ with $0, the algebraic independence or dependence of various sets of these numbers is already known for positive even integers $s$. In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer $m$ the solutions of $$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode[STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701}_{L}^{\ast }(2s))=0\end{eqnarray}$$ with $t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$$(1\leq s\leq m)$ form a $\mathbb{Q}$-vector space of dimension $m$. This proves a conjecture from the Ph.D. thesis of Stein, who, in 2012, was inspired by the relation $-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_{L}^{\ast }(2)=0$. All the results are also true for zeta functions in $2s$, where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second-order recurrence formula.
- Published
- 2020
38. DERIVATION RELATION FOR FINITE MULTIPLE ZETA VALUES IN
- Author
-
Hideki Murahara and Tomokazu Onozuka
- Subjects
010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Derivation relation ,Mathematics - Abstract
Ihara et al. proved the derivation relation for multiple zeta values. The first-named author obtained its counterpart for finite multiple zeta values in ${\mathcal{A}}$. In this paper, we present its generalization in $\widehat{{\mathcal{A}}}$.
- Published
- 2020
39. PERFECT TRIANGLES ON THE CURVE
- Author
-
Shahrina Ismail
- Subjects
Combinatorics ,Elliptic curve ,Infinite set ,Number theory ,Median ,biology ,General Mathematics ,Open problem ,biology.animal ,Homogeneous space ,Heron ,Mathematics - Abstract
A Heron triangle is a triangle that has three rational sides $(a,b,c)$ and a rational area, whereas a perfect triangle is a Heron triangle that has three rational medians $(k,l,m)$. Finding a perfect triangle was stated as an open problem by Richard Guy [Unsolved Problems in Number Theory (Springer, New York, 1981)]. Heron triangles with two rational medians are parametrized by the eight curves $C_{1},\ldots ,C_{8}$ mentioned in Buchholz and Rathbun [‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly 104(2) (1997), 106–115; ‘Heron triangles and elliptic curves’, Bull. Aust. Math.Soc. 58 (1998), 411–421] and Bácskái et al. [Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, 2003]. In this paper, we reveal results on the curve $C_{4}$ which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to prove the nonexistence of a perfect triangle arising from this curve.
- Published
- 2019
40. SOBOLEV’S INEQUALITY FOR MUSIELAK–ORLICZ–MORREY SPACES OVER METRIC MEASURE SPACES
- Author
-
Takao Ohno and Tetsu Shimomura
- Subjects
Pure mathematics ,Inequality ,Generalization ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Poincaré inequality ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,Corollary ,Metric (mathematics) ,symbols ,0101 mathematics ,media_common ,Variable (mathematics) ,Mathematics - Abstract
Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.
- Published
- 2019
41. FINITE SOLVABLE GROUPS WITH DISTINCT MONOMIAL CHARACTER DEGREES
- Author
-
Guohua Qian and Yong Yang
- Subjects
Monomial ,Pure mathematics ,Nonlinear system ,Character (mathematics) ,Solvable group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we classify the finite solvable groups in which distinct nonlinear monomial characters have distinct degrees.
- Published
- 2019
42. A RESULT OF PALEY AND WIENER ON DAMEK–RICCI SPACES
- Author
-
Mithun Bhowmik
- Subjects
Pure mathematics ,Paley–Wiener theorem ,General Mathematics ,010102 general mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,0101 mathematics ,Mathematics - Abstract
A classical result due to Paley and Wiener characterizes the existence of a nonzero function in $L^{2}(\mathbb{R})$, supported on a half-line, in terms of the decay of its Fourier transform. In this paper, we prove an analogue of this result for Damek–Ricci spaces.
- Published
- 2019
43. TH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE
- Author
-
Huaiqing Zuo, Stephen S.-T. Yau, and Naveed Hussain
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Isolated singularity ,01 natural sciences ,Moduli ,Hypersurface ,0103 physical sciences ,Lie algebra ,Maximal ideal ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$, that is, $L(V)=\text{Der}(A(V),A(V))$. It is known that $L(V)$ is finite dimensional and its dimension $\unicode[STIX]{x1D706}(V)$ is called the Yau number. We introduce a new series of Lie algebras, that is, $k$th Yau algebras $L^{k}(V)$, $k\geq 0$, which are a generalization of the Yau algebra. The algebra $L^{k}(V)$ is defined to be the Lie algebra of derivations of the $k$th moduli algebra $A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$, that is, $L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$, where $m$ is the maximal ideal of ${\mathcal{O}}_{n}$. The $k$th Yau number is the dimension of $L^{k}(V)$, which we denote by $\unicode[STIX]{x1D706}^{k}(V)$. In particular, $L^{0}(V)$ is exactly the Yau algebra, that is, $L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$. These numbers $\unicode[STIX]{x1D706}^{k}(V)$ are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that $\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$ We prove this conjecture for a large class of singularities.
- Published
- 2019
44. WEIGHTED WEAK TYPE ENDPOINT ESTIMATES FOR THE COMPOSITIONS OF CALDERÓN–ZYGMUND OPERATORS
- Author
-
Guoen Hu
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Commutator (electric) ,Bilinear interpolation ,Weak type ,01 natural sciences ,law.invention ,Composite operator ,law ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Symbol (formal) ,Mathematics - Abstract
Let $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$, $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$ while the composite operator $T_{1,b}T_{2}$ satisfies $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}^{2}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$
- Published
- 2019
45. COMPLETE WEINGARTEN HYPERSURFACES SATISFYING AN OKUMURA TYPE INEQUALITY
- Author
-
Henrique F. de Lima and Eudes L. de Lima
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Type inequality ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Mathematics::Algebraic Geometry ,Hypersurface ,Operator (computer programming) ,Maximum principle ,Product (mathematics) ,Tensor (intrinsic definition) ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we deal with complete linear Weingarten hypersurfaces immersed into Riemannian space forms. Assuming an Okumura type inequality on the total umbilicity tensor of such hypersurfaces, we prove that either the hypersurface is totally umbilical or it holds an estimate for the norm of the total umbilicity tensor, which is also shown be sharp in the sense that the product of space forms realize them. Our approach is based on a version of the Omori–Yau maximum principle for a suitable Cheng–Yau type operator.
- Published
- 2019
46. A NOTE ON THE INTERSECTIONS OF THE BESICOVITCH SETS AND ERDŐS–RÉNYI SETS
- Author
-
Min Wu and Jinjun Li
- Subjects
Combinatorics ,General Mathematics ,Hausdorff dimension ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
For $x\in (0,1]$ and a positive integer $n,$ let $S_{\!n}(x)$ denote the summation of the first $n$ digits in the dyadic expansion of $x$ and let $r_{n}(x)$ denote the run-length function. In this paper, we obtain the Hausdorff dimensions of the following sets: $$\begin{eqnarray}\bigg\{x\in (0,1]:\liminf _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FC},\limsup _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FD},\lim _{n\rightarrow \infty }\frac{r_{n}(x)}{\log _{2}n}=\unicode[STIX]{x1D6FE}\bigg\},\end{eqnarray}$$ where $0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$, $0\leq \unicode[STIX]{x1D6FE}\leq +\infty$.
- Published
- 2019
47. CHARACTERIZATIONS OF BMO AND LIPSCHITZ SPACES IN TERMS OF WEIGHTS AND THEIR APPLICATIONS
- Author
-
Jiang Zhou, Zhidong Teng, and Dinghuai Wang
- Subjects
Pure mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,0101 mathematics ,Lp space ,Lipschitz continuity ,01 natural sciences ,Mathematics - Abstract
Let $0 with $1/p-1/q=\unicode[STIX]{x1D6FC}/n$, $\unicode[STIX]{x1D714}\in A_{p,q}$, $\unicode[STIX]{x1D708}\in A_{\infty }$ and let $f$ be a locally integrable function. In this paper, it is proved that $f$ is in bounded mean oscillation $\mathit{BMO}$ space if and only if $$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q} where $\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$ and $f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$. We also show that $f$ belongs to Lipschitz space $Lip_{\unicode[STIX]{x1D6FC}}$ if and only if $$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q} As applications, we characterize these spaces by the boundedness of commutators of some operators on weighted Lebesgue spaces.
- Published
- 2019
48. ON SOME NEW MOCK THETA FUNCTIONS
- Author
-
Nancy S. S. Gu and Li-Jun Hao
- Subjects
Computer Science::Robotics ,Ramanujan theta function ,Algebra ,symbols.namesake ,General Mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,symbols ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Mathematics - Abstract
In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.
- Published
- 2018
49. INFINITELY MANY SOLUTIONS FOR NONLOCAL SYSTEMS INVOLVING FRACTIONAL LAPLACIAN UNDER NONCOMPACT SETTINGS
- Author
-
S. M. Sbai, M. Khiddi, and S. Benmouloud
- Subjects
Pure mathematics ,Class (set theory) ,Critical point (set theory) ,General Mathematics ,Fractional Laplacian ,Mathematics - Abstract
In this paper, we study a class of Brezis–Nirenberg problems for nonlocal systems, involving the fractional Laplacian $(-\unicode[STIX]{x1D6E5})^{s}$ operator, for $0 , posed on settings in which Sobolev trace embedding is noncompact. We prove the existence of infinitely many solutions in large dimension, namely when $N>6s$ , by employing critical point theory and concentration estimates.
- Published
- 2018
50. CLASSICAL PROPERTIES OF COMPOSITION OPERATORS ON HARDY–ORLICZ SPACES ON PLANAR DOMAINS
- Author
-
Michał Rzeczkowski
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,Disjoint sets ,Characterization (mathematics) ,Composition (combinatorics) ,01 natural sciences ,010101 applied mathematics ,Planar ,Compact space ,0101 mathematics ,Mathematics - Abstract
In this paper we study composition operators on Hardy–Orlicz spaces on multiply connected domains whose boundaries consist of finitely many disjoint analytic Jordan curves. We obtain a characterization of order-bounded composition operators. We also investigate weak compactness and the Dunford–Pettis property of these operators.
- Published
- 2018
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