212 results
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2. Third Hankel determinants for two classes of analytic functions with real coefficients
- Author
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Paweł Zaprawa and Young Jae Sim
- Subjects
010101 applied mathematics ,Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematics ,Analytic function - Abstract
In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.
- Published
- 2021
3. The Kobayashi–Royden metric on punctured spheres
- Author
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Junqing Qian and Gunhee Cho
- Subjects
Rational number ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Exponential function ,Bell polynomials ,010101 applied mathematics ,Metric (mathematics) ,Backslash ,SPHERES ,0101 mathematics ,Asymptotic expansion ,Mathematics - Abstract
This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ ℙ 1 ∖ { 0 , 1 , ∞ } {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of ℂ ℙ 1 ∖ { a 1 , … , a n } {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} , as well, and the metric on ℂ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 i } {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.
- Published
- 2020
4. Generalized Riesz potentials of functions in Morrey spaces L (1,ϕ;κ)(G) over non-doubling measure spaces
- Author
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Yoshihiro Sawano, Masaki Shigematsu, and Tetsu Shimomura
- Subjects
010104 statistics & probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,01 natural sciences ,Measure (mathematics) ,Mathematics - Abstract
This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L ( 1 , φ ; κ ) ( G ) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X ) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X} . An example in the end of this article convincingly explains why the modification parameter τ must be introduced.
- Published
- 2019
5. On commutator Krylov transitive and commutator weakly transitive Abelian p-groups
- Author
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Andrey R. Chekhlov and Peter V. Danchev
- Subjects
010101 applied mathematics ,Pure mathematics ,Transitive relation ,law ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Commutator (electric) ,0101 mathematics ,Abelian group ,01 natural sciences ,Mathematics ,law.invention - Abstract
We define the concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive Abelian p-groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive p-groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a commutator weakly transitive Abelian p-group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of p-primary Abelian groups.
- Published
- 2019
6. On the bounded approximation property on subspaces of ℓ p when 0 < p < 1 and related issues
- Author
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Félix Cabello Sánchez, Jesús M. F. Castillo, and Yolanda Moreno
- Subjects
Pure mathematics ,Approximation property ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Linear subspace ,Mathematics - Abstract
This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 < p ≤ 1 {0 , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.
- Published
- 2019
7. Exceptional sets for sums of almost equal prime cubes
- Author
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Mengdi Wang
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Waring–Goldbach problem ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Prime (order theory) ,Mathematics - Abstract
In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.
- Published
- 2019
8. Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II
- Author
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Sungmun Cho
- Subjects
Pure mathematics ,Residue (complex analysis) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Hermitian matrix ,Mathematics - Abstract
This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions E / F {E/F} , where F is a finite unramified field extension of ℚ 2 {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice ( L , H ) {(L,H)} , when a base field is unramified over ℚ {\mathbb{Q}} at a prime ( 2 ) {(2)} .
- Published
- 2018
9. Homogeneous Finsler spaces and the flag-wise positively curved condition
- Author
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Ming Xu and Shaoqiang Deng
- Subjects
Mathematics - Differential Geometry ,22E46, 53C30 ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Flag (linear algebra) ,010102 general mathematics ,Lie group ,Space (mathematics) ,01 natural sciences ,Differential Geometry (math.DG) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,Compact Lie algebra ,Tangent space ,Mathematics::Differential Geometry ,0101 mathematics ,Invariant (mathematics) ,010306 general physics ,Hopf conjecture ,Mathematics - Abstract
In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank $2$ non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space $S^3\times \mathbb{R}$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on $S^2\times S^3$ and $S^6\times S^7$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on $S^3\times S^3$, shedding some light on the long standing general Hopf conjecture., 23 pages. The newest version has strengthened the main results in the paper, and provides more examples. We add a short survey on the most recent progress inspired by this paper in the introduction section
- Published
- 2018
10. Space-time L 2 estimates, regularity and almost global existence for elastic waves
- Author
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Kunio Hidano and Dongbing Zha
- Subjects
010101 applied mathematics ,Applied Mathematics ,General Mathematics ,Space time ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .
- Published
- 2018
11. Rational homology and homotopy of high-dimensional string links
- Author
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Paul Arnaud Songhafouo Tsopméné and Victor Turchin
- Subjects
Homotopy group ,Pure mathematics ,Conjecture ,Hochschild homology ,Direct sum ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Codimension ,Homology (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Isotopy ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of π 0 {\pi_{0}} .
- Published
- 2018
12. Fourier transforms of powers of well-behaved 2D real analytic functions
- Author
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Michael Greenblatt
- Subjects
Class (set theory) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Newton polygon ,Function (mathematics) ,01 natural sciences ,Subclass ,symbols.namesake ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,010307 mathematical physics ,42B20 ,0101 mathematics ,Analytic function ,Mathematics - Abstract
This paper is a companion paper to [G4], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [G4] are stated in a rather general form. In this paper, we expand on the results of [G4] and show that there is a class of "well-behaved" functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way., 13 pages
- Published
- 2017
13. A generalization of a graph theory Mertens’ theorem: Galois covering case
- Author
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Iwao Sato, Seiken Saito, and Takehiro Hasegawa
- Subjects
Mertens conjecture ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Fundamental theorem of Galois theory ,010102 general mathematics ,Galois group ,0102 computer and information sciences ,01 natural sciences ,Embedding problem ,Combinatorics ,Differential Galois theory ,symbols.namesake ,010201 computation theory & mathematics ,Mertens function ,Mertens' theorems ,symbols ,Galois extension ,0101 mathematics ,Mathematics - Abstract
In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.
- Published
- 2017
14. Period relations for cusp forms of GSp4
- Author
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Harald Grobner and Ronnie Sebastian
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Cusp (anatomy) ,010307 mathematical physics ,0101 mathematics ,Geodesy ,01 natural sciences ,Period (music) ,Mathematics - Abstract
Let F be a totally real number field and let π be a cuspidal automorphic representation of GSp 4 ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} , which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period p ( π ) {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods p ( π ) {p(\pi)} and all of their twists p ( π ⊗ ξ ) {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that ( 𝔤 , K ) {(\mathfrak{g},K)} -cohomological cusp forms of GSp 4 ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for GSp 4 ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.
- Published
- 2017
15. Group actions and geometric combinatorics in 𝔽qd$\mathbb{F}_{q}^{d}$
- Author
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Michael Rudnev, Jonathan Pakianathan, Alex Iosevich, Derrick Hart, and Michael Bennett
- Subjects
Discrete mathematics ,Group action ,010201 computation theory & mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,Geometric combinatorics ,01 natural sciences ,Mathematics - Abstract
In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists s 0 ( d ) < d ${s_{0}(d) such that if E ⊂ 𝔽 q d ${E\subset{\mathbb{F}}_{q}^{d}}$ , d ≥ 2 ${d\geq 2}$ , with | E | ≥ C q s 0 ${|E|\geq Cq^{s_{0}}}$ , then | T d d ( E ) | ≥ C ′ q ( d + 1 2 ) ${|T^{d}_{d}(E)|\geq C^{\prime}q^{d+1\choose 2}}$ , where T k d ( E ) ${T^{d}_{k}(E)}$ denotes the set of congruence classes of k-dimensional simplices determined by k + 1 ${k+1}$ -tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for T k d ( E ) ${T^{d}_{k}(E)}$ with 2 ≤ k ≤ d - 1 ${2\leq k\leq d-1}$ . A non-trivial result for T 2 2 ( E ) ${T^{2}_{2}(E)}$ in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 4 3 ${\frac{4}{3}}$ , previously established in [4] for the q ≡ 3 mod 4 ${q\equiv 3\textup{ mod }4}$ case to the case q ≡ 1 mod 4 ${q\equiv 1\textup{ mod }4}$ , and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in 𝔽 q d ${{\mathbb{F}}_{q}^{d}}$ , where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.
- Published
- 2016
16. An estimate on the heat kernel of Schrödinger operators with non-negative potentials on nilpotent Lie groups and its applications
- Author
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Jianfeng Dong, Jizheng Huang, and Yu Liu
- Subjects
Algebra ,Nilpotent ,symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Lie group ,Heat kernel ,Schrödinger's cat ,Mathematics - Abstract
In this paper we investigate the heat kernel of the Schrödinger operator L = - Δ G + W $L=-\Delta _G+W$ on the nilpotent Lie group G, where Δ G is the sub-Laplacian on G and the non-negative potential W belongs to the reverse Hölder class B q 1 . The main aim of the paper is to give a pointwise estimate for the heat kernel of Schrödinger operators with non-negative potentials on the nilpotent Lie group G. As its applications, we obtain the Lp estimates for parabolic Schrödinger operators with certain non-negative potentials.
- Published
- 2013
17. Nielsen numbers of maps of aspherical figure-eight type polyhedra
- Author
-
Seung Won Kim and Peter Yi
- Subjects
Pure mathematics ,Polyhedron ,Applied Mathematics ,General Mathematics ,Type (model theory) ,Mathematics - Abstract
Let X be an aspherical polyhedron of the homotopy type of the figure-eight and let f : X → X be a self-map. The Wagner algorithm [Trans. Amer. Math. Soc. 351 (1999), 41–62] provides computations for the Nielsen number of self-maps of X satisfying the remnant condition. If f is without remnant, then using the concept of mutant by Jiang [Math. Ann. 311 (1998), 467–479] we may assume that f #(b) is an initial segment of f #(a), where f # is the induced endomorphism of π1(X) and a, b are generators of π1(X). Let f #(b) = U and f #(a) = Un R, where n is the maximal such positive integer. If R is not an initial segment of U, we say that f is of Type Y. In this paper, we prove that if f is of Type Y, then f can be mutated either to a map that has remnant or to an exceptional form for which we can calculate the Nielsen number directly. Not all self-maps of X are of Type Y. However, making use of the results in this paper, an algorithm is presented by Kim [J. Pure Appl. Algebra 216 (2012), 1652–1666] that does compute the Nielsen number for all self-maps of X.
- Published
- 2013
18. Algebraic supergroups of Cartan type
- Author
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Fabio Gavarini
- Subjects
Pure mathematics ,algebraic supergroups ,General Mathematics ,Lie superalgebras of Cartan type ,Lie superalgebra ,Type (model theory) ,Mathematics - Algebraic Geometry ,Simple (abstract algebra) ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Algebraic number ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics ,Applied Mathematics ,Mathematics::Rings and Algebras ,Tangent ,Mathematics - Rings and Algebras ,Settore MAT/02 - Algebra ,Rings and Algebras (math.RA) ,14M30, 14A22 (Primary), 17B10, 17B20, (Secondary) ,Settore MAT/03 - Geometria ,Affine transformation ,Supergroup ,Mathematics - Representation Theory - Abstract
I present a construction of connected affine algebraic supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups G_V that I introduced. In particular, the supergroup constructed in this way associated with g := W(n) and its standard representation is described somewhat more in detail. In addition, *** an "Erratum" is added here *** after the main text to fix a mistake which was kindly pointed out to the author by prof. Masuoka after the paper was published: this "Erratum" is accepted for publication in "Forum Mathematicum", it appears here in its final form (but prior to proofreading). In it, I also explain more in detail the *Existence Theorem* for algebraic supergroups of Cartan type which comes out of the main result in the original paper., Comment: Main file: La-TeX file, 47 pages, already published (see below). Erratum: La-TeX file, 6 pages, to appear (see below). For the main file, the original publication is available at www.degruyter.com (cf. the journal reference here below)
- Published
- 2012
19. Clifford–Wolf translations of Finsler spaces
- Author
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Ming Xu and Shaoqiang Deng
- Subjects
Pure mathematics ,Killing vector field ,Homogeneous ,Applied Mathematics ,General Mathematics ,Isometry ,Mathematics::Differential Geometry ,Characterization (mathematics) ,Special case ,Constant (mathematics) ,Translation (geometry) ,Mathematics - Abstract
In this paper, we study Clifford–Wolf translations of Finsler spaces. We give a characterization of those Clifford–Wolf translations generated by Killing vector fields. In particular, we show that there is a natural interrelation between the local one-parameter groups of Clifford–Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for a Killing vector field to have a constant length, in which case the local one-parameter group of isometries generated by the Killing field consist of Clifford–Wolf translations. Finally, we construct explicit examples to explain some of the results of this paper.
- Published
- 2012
20. Primitive ideals in quantum Schubert cells: Dimension of the strata
- Author
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Stéphane Launois, Karel L Casteels, and Jason P. Bell
- Subjects
Pure mathematics ,Weyl group ,Applied Mathematics ,General Mathematics ,Subalgebra ,Torus ,Representation theory ,Stratification (mathematics) ,Bruhat order ,symbols.namesake ,Mathematics::Quantum Algebra ,Lie algebra ,symbols ,Mathematics::Representation Theory ,Quantum ,Mathematics - Abstract
The aim of this paper is to study the representation theory of quantum Schubert cells. Let 𝔤 $\mathfrak {g}$ be a simple complex Lie algebra. To each element w of the Weyl group W of 𝔤 $\mathfrak {g}$ , De Concini, Kac and Procesi have attached a subalgebra U q [ w ] $U_q[w]$ of the quantised enveloping algebra U q ( 𝔤 ) $U_q(\mathfrak {g})$ . Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of U q [ w ] $U_q[w]$ . More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux–Cauchon and Yakimov, that the primitive spectrum of U q [ w ] $U_q[w]$ admits a stratification indexed by those elements v ∈ W $v \in W$ with v ≤ w $v \le w$ in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair v ≤ w $v \le w$ .
- Published
- 2012
21. Semi-classical limits of the first eigenfunction and concentration on the recurrent sets of a dynamical system
- Author
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Ivan Kupka and David Holcman
- Subjects
Singular perturbation ,Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,FOS: Physical sciences ,G.1.8, analysis on manifold, nonself adjoint operator, concentration phenomena ,Mathematical Physics (math-ph) ,Eigenfunction ,Riemannian manifold ,Elliptic operator ,Order operator ,Limit (mathematics) ,Dynamical system (definition) ,Mathematical Physics ,Mathematics - Abstract
Dear Reader, please find the third and last part of a series of papers on the singular perturbation of the first eigenfunction associated to a non self-adjoint second order elliptic operators. This series started in 1999 and we presented the early results in 2000 at Columbia University. We published two notes in CRAS in 2001 and 2005 summarizing our results. The present paper contains the proofs of the announced theorems and many open questions. We tried to publish these results in the the top tier of mathematical journals (Annals, Acta, Duke...) but our results were not deemed sufficiently interesting for them and probably not trendy enough. Some of you may like this work, so here it is. Best Regards, Ivan and David. We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold, when the diffusion constant $\epsilon$ goes to zero. If the drift of the diffusion is given by a Morse-Smale vector field $b$, the limits of the eigenfunctions concentrate on the recurrent set of $b$. A blow-up analysis enables us to find the main properties of the limit measures on a recurrent set., Comment: around 70 pages. Can't be read in one shot
- Published
- 2011
22. Hankel determinants for starlike and convex functions associated with sigmoid functions
- Author
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Amina Riaz, Mohsan Raza, and Derek K. Thomas
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Sigmoid function ,Convex function ,Mathematics - Abstract
This paper is concerned with Hankel determinants for starlike and convex functions related to modified sigmoid functions. Sharp bounds are given for second and third Hankel determinants.
- Published
- 2021
23. Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type
- Author
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Jiecheng Chen and Shaoyong He
- Subjects
Pure mathematics ,Homogeneous ,Applied Mathematics ,General Mathematics ,Type (model theory) ,Lipschitz continuity ,Mathematics - Abstract
The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of general product singular integral operators introduced by Han, Li and Lin [Y. Han, J. Li and C.-C. Lin, Criterion of the L 2 L^{2} boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 2016, 3, 845–907] on the multiparameter Lipschitz spaces of homogeneous type M ~ = M 1 × ⋯ × M n {\tilde{M}=M_{1}\times\cdots\times M_{n}} . Each factor space M i {M_{i}} , 1 ≤ i ≤ n {1\leq i\leq n} , is a space of homogeneous type in the sense of Coifman and Weiss. These operators generalize those studied by Journé on the Euclidean space and include operators studied by Nagel and Stein on Carnot–Carathéodory spaces. The main tool used here is the discrete Littlewood–Paley–Stein theory and almost orthogonality together with a density argument for the product Lipschitz spaces in the weak sense.
- Published
- 2021
24. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
- Author
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Qian Li and Zhi-Yi Wu
- Subjects
symbols.namesake ,Fourier transform ,Planar ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,symbols ,Spectral measure ,Numerical digit ,Mathematics - Abstract
We consider the self-similar measure μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} generated by an expanding real matrix M = ( ρ - 1 0 0 ρ - 1 ) ∈ M 2 ( ℝ ) {M=\begin{pmatrix}\rho^{-1}&0\\ 0&\rho^{-1}\end{pmatrix}\in M_{2}({\mathbb{R}})} and a digit set 𝒟 = { ( 0 0 ) , ( a b ) , ( c d ) , ( a + c b + d ) } ⊆ ℤ 2 . {{\mathcal{D}}=\Biggl{\{}\begin{pmatrix}0\\ 0\end{pmatrix},\begin{pmatrix}a\\ b\end{pmatrix},\begin{pmatrix}c\\ d\end{pmatrix},\begin{pmatrix}a+c\\ b+d\end{pmatrix}\Biggr{\}}\subseteq{\mathbb{Z}}^{2}}. In this paper, we study the spectral and non-spectral problems of μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} . In this case that ( a b ) {(\begin{smallmatrix}a\\ b\end{smallmatrix})} and ( c d ) {(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two independent vectors, we prove that if ρ - 1 ∈ ℤ {\rho^{-1}\in{\mathbb{Z}}} , then μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} is a spectral measure if and only if ρ - 1 ∈ 2 ℤ {\rho^{-1}\in 2{\mathbb{Z}}} . For the case that ( a b ) {(\begin{smallmatrix}a\\ b\end{smallmatrix})} and ( c d ) {(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two dependent vectors, we first give the sufficient and necessary condition for L 2 ( μ M , 𝒟 ) {L^{2}(\mu_{M,{\mathcal{D}}})} to contain an infinite orthogonal set of exponential functions. Based on this result, we can give the exact cardinality of orthogonal exponential functions in L 2 ( μ M , 𝒟 ) {L^{2}(\mu_{M,{\mathcal{D}}})} when L 2 ( μ M , 𝒟 ) {L^{2}(\mu_{M,{\mathcal{D}}})} does not admit any infinite orthogonal set of exponential functions by classifying the values of ρ.
- Published
- 2021
25. A note on special cubic fourfolds of small discriminants
- Author
-
Hoang Le Truong and Hoang Ngoc Yen
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Vector bundle ,Mathematics - Abstract
In this paper, our purpose is to give a characterization of the generic special cubic fourfold which contains a smooth rational surface of degree 9 not homologous to a complete intersection. As corollaries, we will give an explicit construction of families of smooth surfaces in generic special cubic fourfolds X ∈ 𝒞 δ {X\in\mathcal{C}_{\delta}} for 6 < δ ≤ 30 {6 and δ ≡ 0 ( mod 6 ) {\delta\equiv 0~{}(\bmod~{}6)} . This applies in particular to give an explicit construction of two different liaison class of smooth surfaces in all such special cubic fourfolds with the prescribed invariants.
- Published
- 2021
26. Extrapolation to product Herz spaces and some applications
- Author
-
Mingquan Wei
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Mathematics::Operator Algebras ,law ,Applied Mathematics ,General Mathematics ,Product (mathematics) ,Mathematics::Classical Analysis and ODEs ,Extrapolation ,Commutator (electric) ,Mathematics ,law.invention - Abstract
This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.
- Published
- 2021
27. The exact number of orthogonal exponentials on the spatial Sierpinski gasket
- Author
-
Qi Wang
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Sierpinski triangle ,Mathematics ,Exponential function - Abstract
Let μ M , D {\mu_{M,D}} be a self-affine measure associated with an expanding real matrix M = diag [ ρ 1 , ρ 2 , ρ 3 ] {M=\operatorname{diag}[\rho_{1},\rho_{2},\rho_{3}]} and the digit set D = { 0 , e 1 , e 2 , e 3 } {D=\{0,e_{1},e_{2},e_{3}\}} in the space ℝ 3 {\mathbb{R}^{3}} , where | ρ 1 | , | ρ 2 | , | ρ 3 | ∈ ( 1 , ∞ ) {\lvert\rho_{1}\rvert,\lvert\rho_{2}\rvert,\lvert\rho_{3}\rvert\in(1,\infty)} and e 1 , e 2 , e 3 {e_{1},e_{2},e_{3}} is the standard basis of unit column vectors in ℝ 3 {\mathbb{R}^{3}} . In this paper, we mainly consider the case ρ 1 ∈ { p q : p ∈ 2 ℤ , q ∈ 2 ℤ - 1 } , ρ 2 , ρ 3 ∈ { p q : p , q ∈ 2 ℤ - 1 } . \rho_{1}\in\Bigl{\{}\frac{p}{q}:p\in 2\mathbb{Z},\,q\in 2\mathbb{Z}-1\Bigr{\}}% ,\quad\rho_{2},\rho_{3}\in\Bigl{\{}\frac{p}{q}:p,q\in 2\mathbb{Z}-1\Bigr{\}}. We prove that if ρ 2 = ρ 3 {\rho_{2}=\rho_{3}} , then there exist at most 4 mutually orthogonal exponential functions in the Hilbert space L 2 ( μ M , D ) {L^{2}(\mu_{M,D})} , where the number 4 is the best upper bound. If ρ 2 = - ρ 3 {\rho_{2}=-\rho_{3}} , then there exist at most 8 mutually orthogonal exponential functions in L 2 ( μ M , D ) {L^{2}(\mu_{M,D})} , where the number 8 is the best upper bound. If | ρ 3 | ≠ | ρ 2 | {\lvert\rho_{3}\rvert\neq\lvert\rho_{2}\rvert} , then there are any number of orthogonal exponentials in L 2 ( μ M , D ) {L^{2}(\mu_{M,D})} . This gives the exact number of orthogonal exponentials on the spatial Sierpinski gasket in the above case.
- Published
- 2021
28. Simply transitive NIL-affine actions of solvable Lie groups
- Author
-
Jonas Deré and Marcos Origlia
- Subjects
Mathematics - Differential Geometry ,Solvable Lie algebra ,Transitive relation ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lie group ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Characterization (mathematics) ,01 natural sciences ,Mathematics - Geometric Topology ,Nilpotent ,Morphism ,Differential Geometry (math.DG) ,0103 physical sciences ,Simply connected space ,Lie algebra ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $\rho: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $\varphi: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$., Comment: 22 pages, 8 tables. Comments are welcome
- Published
- 2021
29. Solvability for nonlocal boundary value problems with generalized 𝑝-Laplacian on an unbounded domain
- Author
-
Marek Matyjasik and Katarzyna Szymańska-Dȩbowska
- Subjects
010101 applied mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Domain (ring theory) ,Mathematical analysis ,Nonlocal boundary ,Half line ,0101 mathematics ,01 natural sciences ,Laplace operator ,Value (mathematics) ,Mathematics - Abstract
This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results.
- Published
- 2021
30. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
- Author
-
Guozhen Lu, Jiao Chen, and Liang Huang
- Subjects
010101 applied mathematics ,Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Pseudo-differential operator ,Mathematics ,Flag (geometry) - Abstract
In this paper, we establish the endpoint estimate ( 0 < p ≤ 1 {0 ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B S 1 , 0 0 {BS^{0}_{1,0}} . The study of this operator is motivated from the L p {L^{p}} ( 1 < p < ∞ {1 ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L ∞ L^{\infty} and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the L p {L^{p}} ( 1 < p < ∞ {1 ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, L p L^{p} -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., h p 1 × h p 2 × h p 3 → L p {h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}} with 1 p 1 + 1 p 2 + 1 p 3 = 1 p {\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}} with 0 < p < ∞ {0 (see Theorem 1.6 and Theorem 1.7).
- Published
- 2021
31. Incidences between Euclidean spaces over finite fields
- Author
-
Semin Yoo
- Subjects
Pure mathematics ,Finite field ,Spectral graph theory ,Applied Mathematics ,General Mathematics ,Euclidean geometry ,Mathematics - Abstract
Let 𝔽 q {\mathbb{F}_{q}} be the finite field of order q, where q is an odd prime power. Then a k-dimensional quadratic subspace ( W , Q ) {(W,Q)} of ( 𝔽 q n , x 1 2 + x 2 2 + ⋯ + x n 2 ) {(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})} is called dot 𝐤 {\operatorname{dot}_{\mathbf{k}}} -subspace if Q is isometrically isomorphic to x 1 2 + x 2 2 + ⋯ + x k 2 {x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}} . In this paper, we obtain bounds for the number of incidences I ( 𝒦 , ℋ ) {I(\mathcal{K},\mathcal{H})} between a collection 𝒦 {\mathcal{K}} of dot k {\operatorname{dot}_{k}} -subspaces and a collection ℋ {\mathcal{H}} of dot h {\operatorname{dot}_{h}} -subspaces when h ≥ 4 k - 4 {h\geq 4k-4} , which is given by | I ( 𝒦 , ℋ ) - | 𝒦 | | ℋ | q k ( n - h ) | ≲ q k ( 2 h - n - 2 k + 4 ) + h ( n - h - 1 ) - 2 2 | 𝒦 | | ℋ | . \Bigl{\lvert}I(\mathcal{K},\mathcal{H})-\frac{\lvert\mathcal{K}\rvert\lvert% \mathcal{H}\rvert}{q^{k(n-h)}}\Bigr{\rvert}\lesssim q^{\frac{k(2h-n-2k+4)+h(n-% h-1)-2}{2}}\sqrt{\lvert\mathcal{K}\rvert\lvert\mathcal{H}\rvert}. In particular, we improve the error term in [N. D. Phuong, P. V. Thang and L. A. Vinh, Incidences between planes over finite fields, Proc. Amer. Math. Soc. 147 2019, 5, 2185–2196] obtained by Phuong, Thang and Vinh for general collections of affine subspaces in the presence of our additional conditions.
- Published
- 2021
32. The cohomology groups of the outer Whitehead automorphism group of a free product
- Author
-
John Meier and Ethan Berkove
- Subjects
Combinatorics ,Pure mathematics ,Kernel (algebra) ,Free product ,Applied Mathematics ,General Mathematics ,Spectral sequence ,Equivariant map ,Outer automorphism group ,Automorphism ,Cohomology ,Quotient ,Mathematics - Abstract
ji 2 Aut() induced by gj i (gk )= g k if k 6= i g g j k if k = i (where gk 2 Gk and g g j k is shorthand for conjugating gk by gj )i s anelementary Whitehead automorphism. The Whitehead automorphism group, Wh(), is the subgroup of Aut() generated by the g j i . If none of the Gi are infinite cyclic, then Wh() is the kernel of the map Aut() Out(G1 ◊···◊ Gn). In particular, if the Gi are finite, then Wh() is a finite-index subgroup of Aut(). In this paper we compute the cohomology groups of OWh(), the quotient of Wh() in the outer-automorphism group, with field coecients. Our approach is to analyze the equivariant spectral sequence associated to the action of OWh() on a contractible, simplicial complex introduced by McCullough and Miller [10]. Throughout the paper, cohomology groups will be assumed to have field coecients unless otherwise indicated, and () n denotes the n-fold product ◊ ···◊ | {z } n copies
- Published
- 2010
33. Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras
- Author
-
Sergei Silvestrov and Abdenacer Makhlouf
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Non-associative algebra ,Formal group ,Witt algebra ,Universal enveloping algebra ,Affine Lie algebra ,Lie conformal algebra ,Algebra ,Mathematics::Category Theory ,Algebra representation ,Generalized Kac–Moody algebra ,Mathematics - Abstract
The aim of this paper is to extend to Hom-algebra structures the theory of 1-parameter formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis and Richardson. In this paper, formal deformations of Hom-associative and Hom-Lie algebras are studied. The first groups of a deformation cohomology are constructed and several examples of deformations are given. We also provide families of Hom-Lie algebras deforming Lie algebra sl(2)(K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl(2)(K).
- Published
- 2010
34. A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas
- Author
-
Xiangqian Guo, Yan-an Cai, Hongjia Chen, Yao Ma, and Mianmian Zhu
- Subjects
Pure mathematics ,Class (set theory) ,Tensor product ,Quantum group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Abstract
In this paper, we construct a class of new modules for the quantum group U q ( s l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ( s l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ( s l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ( s l 2 ) U_{q}(\mathfrak{sl}_{2}) .
- Published
- 2021
35. Weighted boundedness of multilinear Calderón commutators
- Author
-
Yanping Chen and Xueting Han
- Subjects
010101 applied mathematics ,Multilinear map ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Lipschitz continuity ,01 natural sciences ,Mathematics - Abstract
The main result of this paper is to give that if b ∈ Lip ( ℝ n ) {b\in\operatorname{Lip}(\mathbb{R}^{n})} , h j ∈ BMO ( ℝ n ) {h_{j}\in\operatorname{BMO}(\mathbb{R}^{n})} , j = 1 , … , k {j=1,\ldots,k} , k ∈ ℤ + {k\in\mathbb{Z}^{+}} and w ∈ A p {w\in A_{p}} , 1 < p < ∞ {1 , then the multilinear Calderón commutators T Ω , b , h → {T_{\Omega,b,\vec{h}}} with variable kernels are bounded on L p ( w ) {L^{p}(w)} . In addition, the authors extend the above result to the Morrey space.
- Published
- 2021
36. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Author
-
Robert Stegliński
- Subjects
Lyapunov function ,Partial differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,Applied mathematics ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.
- Published
- 2021
37. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Author
-
Silas L. Carvalho and Alexander Condori
- Subjects
Correlation dimension ,Pure mathematics ,010201 computation theory & mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Shift systems ,0101 mathematics ,Invariant (mathematics) ,01 natural sciences ,Fractal dimension ,Mathematics - Abstract
In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each q > 0 {q>0} , zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system ( X , T ) {(X,T)} (where X = M ℤ {X=M^{\mathbb{Z}}} is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q > 1 {q>1} , infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s ∈ ( 0 , 1 ) {s\in(0,1)} and each q > 1 {q>1} , zero lower s-generalized and infinite upper q-generalized dimensions.
- Published
- 2021
38. Natural pseudo-distances between closed curves
- Author
-
Patrizio Frosini, Pietro Donatini, P. Donatini, and P. Frosini
- Subjects
Set (abstract data type) ,Pure mathematics ,MEASURING FUNCTION ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,NATURAL PSEUDO-DISTANCE ,PERSISTENT TOPOLOGY ,Infimum and supremum ,Natural (archaeology) ,Mathematics - Abstract
Let us consider two closed curves $mathcal{M}$, $mathcal{N}$ of class $C^1$ and two functions $varphi:{mathcal{M}}rightarrow R$, $psi:mathcal{N}rightarrow R$ of class $C^1$, called measuring functions. The natural pseudo-distance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. The problem of finding the possible values for $d$ naturally arises. In this paper we prove that under appropriate hypotheses the natural pseudo-distance equals either $|c_1-c_2|$ or $frac{1}{2}|c_1-c_2|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This equality shows that the relations between the natural pseudo-distance and the critical values of the measuring functions previously obtained in higher dimensions can be made stronger in the particular case of closed curves. Moreover, the examples we give in this paper show that our result cannot be further improved, and therefore it completely solves the problem of determining the possible values for $d$ in the $1$-dimensional case.
- Published
- 2009
39. Spinor L-functions, theta correspondence, and Bessel coefficients
- Author
-
Ramin Takloo-Bighash and Philippe Michel
- Subjects
Pure mathematics ,Symplectic group ,Spinor ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Cuspidal representation ,Algebra ,symbols.namesake ,Spinor field ,symbols ,Bessel function ,Mathematics ,Dual pair ,Real number - Abstract
In this paper we prove two seemingly unrelated theorems. First we establish the entireness of the spinor L-functions of certain automorphic cuspidal representations of the similitude symplectic group of order four over the rational numbers. We also prove a theorem related to the existence of Bessel models for generic discrete series representations of the same group over the real numbers. The two results are linked by the method of proof; in both cases it is based on the pull-back of an appropriately chosen global Bessel functional via the theta correspondence for the dual pair (GO(2, 2),GSp(4)). The first main theorem is related to analytic properties of spinor L-functions. We prove the entireness of the spinor L-function for those generic automorphic cuspidal representation which satisfy a condition at the archimedean place (see below). Our study of the spinor L-function is based on an integral representation which works for generic representations. These integrals which were introduced by M. Novodvorsky in the Corvallis conference [26] serve as one of the few available integral representations for the Spinor L-function of GSp(4). Some of the details missing in Novodvorsky’s original paper have been reproduced in Daniel Bump’s survey article [4]. Further details have been supplied by [40]. Novodvorsky’s integral was first generalized by Ginzburg [10], and further generalized by Soudry [39], to orthogonal groups of arbitrary odd degree. In light of the results of [40], it is sufficient to study the integral of Novodvorsky at the archimedean place. Archimedean computations are often forbidding, and unless one expects major simplifications due to the nature of the parameters, the resulting integrals are often quite hard to manage. In our case of interest, the work of Moriyama [25] benefits from exactly such simplifications when he treats the case of cuspidal representations with archimedean components in the generic (limit of) discrete series. In this work, we concentrate on those archimedean representations for which direct computations have yielded very little. For this reason, our methods are a bit indirect, in fact somewhat more indirect than what at first seems necessary. Our method is based on the theta correspondence. First we observe in Lemma 2.2 that Novodvorsky’s integral is in fact a split Bessel functional. Then in 2.1 we pull the Bessel functional back via the theta correspondence for the dual reductive pair (GO(2, 2),GSp(4)), and prove that the resulting functional on GO(2, 2) is Eulerian. On the other hand
- Published
- 2007
40. On a Lévy process pinned at random time
- Author
-
Mohammed Louriki, Mohamed Erraoui, and Astrid Hilbert
- Subjects
010104 statistics & probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Statistical physics ,0101 mathematics ,01 natural sciences ,Lévy process ,Mathematics - Abstract
In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.
- Published
- 2020
41. The distinction problems for Sp4 and SO3,3
- Author
-
Hengfei Lu
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Epistemology - Abstract
This paper studies the Prasad conjecture for the special orthogonal group SO 3 , 3 \mathrm{SO}_{3,3} . Then we use the local theta correspondence between Sp 4 \mathrm{Sp}_{4} and O ( V ) \mathrm{O}(V) to study the Sp 4 \mathrm{Sp}_{4} -distinction problems over a quadratic field extension E / F E/F and dim V = 4 \dim V=4 or 6. Thus we can verify the Prasad conjecture for a square-integrable representation of Sp 4 ( E ) \mathrm{Sp}_{4}(E) .
- Published
- 2020
42. Coherent state transform for Landau levels on quasi-tori
- Author
-
Mohammed Ziyat
- Subjects
010101 applied mathematics ,Hermite polynomials ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Coherent states ,Torus ,Landau quantization ,0101 mathematics ,01 natural sciences ,Mathematical physics ,Mathematics - Abstract
The spectrum of the Laplacian operator on the positive theta line bundle over the quasi-torus reduces to eigenvalues π ℓ {\pi\ell} , ℓ = 0 , 1 , … {\ell=0,1,\ldots{}} , which are called Landau levels. This paper discusses the coherent state transform for each eigenspace associated with a Landau level. We construct a unitary transform valid for each eigenspace. A concrete form of the inverse formula for the proposed transform is also obtained.
- Published
- 2020
43. Free cyclic group actions on highly-connected 2n-manifolds
- Author
-
Jianqiang Yang and Yang Su
- Subjects
Class (set theory) ,Pure mathematics ,57R65, 57R19, 57S17, 57S25 ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Geometric Topology (math.GT) ,Cyclic group ,01 natural sciences ,Mathematics - Geometric Topology ,0103 physical sciences ,Prime factor ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,Topological conjugacy ,Mathematics - Abstract
In this paper we study smooth orientation-preserving free actions of the cyclic group $\mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $\sharp g (S^n \times S^n)\sharp \Sigma$, where $\Sigma$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classification up to smooth conjugation. When $n \ge 4$ we obtain a classification up to smooth conjugation when the prime factors of $m$ are larger than a constant $C(n)$., Comment: 18 pages
- Published
- 2020
44. A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)
- Author
-
Tapendu Rana
- Subjects
010101 applied mathematics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Lorentz transformation ,010102 general mathematics ,symbols ,0101 mathematics ,01 natural sciences ,Mathematical physics ,Mathematics - Abstract
In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for L p , 1 ( G ) {L^{p,1}(G)} ( 1 ≤ p < 2 {1\leq p ), with G = SL ( 2 , ℝ ) {G=\mathrm{SL}(2,\mathbb{R})} .
- Published
- 2020
45. From subcategories to the entire module categories
- Author
-
Rasool Hafezi
- Subjects
Pure mathematics ,Functor ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Quiver ,Triangular matrix ,01 natural sciences ,Representation theory ,Morphism ,Artin algebra ,Mathematics::Category Theory ,Category of modules ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Connection (algebraic framework) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander-Reiten quiver of the subcategories is obtained only by the Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $\Lambda$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $\Lambda$ and the stable Auslander algebra of $\Lambda$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $\Lambda$., Comment: Accepted for publication in Forum Mathematicum
- Published
- 2020
46. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
- Author
-
Shōta Inoue and Kenta Endo
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Distribution (number theory) ,Logarithm ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Riemann zeta function ,Riemann hypothesis ,symbols.namesake ,Critical line ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,0101 mathematics ,Complex plane ,Mathematics - Abstract
We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis., Comment: 15 pages
- Published
- 2020
47. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
- Author
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Félicien Comtat
- Subjects
Pure mathematics ,Uniform norm ,Character (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Automorphic form ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Recently, the problem of bounding the sup norms of L 2 {L^{2}} -normalized cuspidal automorphic newforms ϕ on GL 2 {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ ϕ ∥ ∞ ≪ N 1 4 + ϵ , \|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon}, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ ϕ ∥ ∞ ≪ N 1 2 + ϵ {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.
- Published
- 2020
48. Automorphic Schwarzian equations
- Author
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Abdellah Sebbar and Hicham Saber
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Plane (geometry) ,Differential equation ,11F03, 11F11, 34M05 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,0102 computer and information sciences ,01 natural sciences ,Representation theory ,symbols.namesake ,010201 computation theory & mathematics ,Eisenstein series ,FOS: Mathematics ,symbols ,Equivariant map ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
This paper concerns the study of the Schwartz differential equation { h , τ } = s E 4 ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . This also leads to the solutions to the Fuchsian differential equation y ′′ + s E 4 y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} .
- Published
- 2020
49. Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4
- Author
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Zhaohui Huo and Yueling Jia
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Bilinear interpolation ,010307 mathematical physics ,0101 mathematics ,Space (mathematics) ,01 natural sciences ,Well posedness ,Mathematics - Abstract
The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t u + ∂ x ( ∂ x x + ∂ y y ) u + u u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.
- Published
- 2020
50. When the image of a derivation on a uniformly complete 𝑓-algebra is contained in the radical
- Author
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Mohamed Ali Toumi
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Derivation ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Image (mathematics) ,Mathematics - Abstract
In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.
- Published
- 2020
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