185 results
Search Results
2. The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation
- Author
-
Ihyeok Seo and Yoonjung Lee
- Subjects
symbols.namesake ,General Mathematics ,Open problem ,symbols ,Initial value problem ,Beta (velocity) ,Lambda ,Nonlinear Schrödinger equation ,Energy (signal processing) ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.
- Published
- 2021
3. Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials
- Author
-
Yuanyang Yu and Zhipeng Yang
- Subjects
Combinatorics ,Nonlinear system ,Elliptic systems ,General Mathematics ,Operator (physics) ,Spectrum (functional analysis) ,Mathematics - Abstract
In this paper, we study the following nonlinear elliptic systems: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$ - Δ u 1 + V 1 ( x ) u 1 = ∂ u 1 F ( x , u ) x ∈ R N , - Δ u 2 + V 2 ( x ) u 2 = ∂ u 2 F ( x , u ) x ∈ R N , where $$u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2$$ u = ( u 1 , u 2 ) : R N → R 2 , F and $$V_i$$ V i are periodic in $$x_1,\ldots ,x_N$$ x 1 , … , x N and $$0\notin \sigma (-\,\Delta +V_i)$$ 0 ∉ σ ( - Δ + V i ) for $$i=1,2$$ i = 1 , 2 , where $$\sigma (-\,\Delta +V_i)$$ σ ( - Δ + V i ) stands for the spectrum of the Schrödinger operator $$-\,\Delta +V_i$$ - Δ + V i . Under some suitable assumptions on F and $$V_i$$ V i , we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).
- Published
- 2020
4. More about singular traces on simply generated operator ideals
- Author
-
Albrecht Pietsch
- Subjects
Large class ,Sequence ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.
- Published
- 2020
5. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
-
Tao Ma, Wei Liu, and Guixiang Hong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Variational inequality ,symbols ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.
- Published
- 2020
6. A spectral characterization of isomorphisms on $$C^\star $$-algebras
- Author
-
Rudi Brits, F. Schulz, and C. Touré
- Subjects
General Mathematics ,Star (game theory) ,010102 general mathematics ,Spectrum (functional analysis) ,Characterization (mathematics) ,01 natural sciences ,Surjective function ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Isomorphism ,0101 mathematics ,Algebra over a field ,Commutative property ,Banach *-algebra ,Mathematics - Abstract
Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).
- Published
- 2019
7. Generalization of the $${\varvec{lq}}$$lq-modular closure theorem and applications
- Author
-
El Hassane Fliouet
- Subjects
Discrete mathematics ,Modularity (networks) ,business.industry ,Generalization ,General Mathematics ,010102 general mathematics ,Separable extension ,Field (mathematics) ,Extension (predicate logic) ,Modular design ,01 natural sciences ,Integer ,Field extension ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,business ,Mathematics - Abstract
Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.
- Published
- 2018
8. Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε-function on the Fock–Bargmann–Hartogs domains
- Author
-
Enchao Bi and Huan Yang
- Subjects
Combinatorics ,E-function ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics ,Fock space - Abstract
In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.
- Published
- 2018
9. Some remarks on the Lehmer conjecture
- Author
-
José Antonio de la Peña
- Subjects
Polynomial ,Conjecture ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Coxeter group ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Tree (descriptive set theory) ,Mahler measure ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.
- Published
- 2018
10. The lattices of invariant subspaces of a class of operators on the Hardy space
- Author
-
Zeljko Cuckovic and Bhupendra Paudyal
- Subjects
Discrete mathematics ,Pure mathematics ,Volterra operator ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Hardy space ,Reflexive operator algebra ,01 natural sciences ,Linear subspace ,symbols.namesake ,Operator (computer programming) ,Lattice (order) ,FOS: Mathematics ,symbols ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**
- Published
- 2018
11. On the number of monic integer polynomials with given signature
- Author
-
Artūras Dubickas
- Subjects
010101 applied mathematics ,Combinatorics ,Real roots ,Integer ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Lambda ,Signature (topology) ,01 natural sciences ,Monic polynomial ,Mathematics - Abstract
In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertok, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.
- Published
- 2018
12. Sharp $$L_p$$ estimates for paraproducts on general measure spaces
- Author
-
Adam Osękowski
- Subjects
Pure mathematics ,Identification (information) ,General Mathematics ,Structure (category theory) ,Function method ,Measure (mathematics) ,Mathematics - Abstract
The paper contains the identification of the $$L_p$$ L p norms of paraproducts, defined on general measure spaces equipped with a dyadic-like structure. The proof exploits the Bellman function method.
- Published
- 2021
13. Macphail’s theorem revisited
- Author
-
Janiely Silva and Daniel Pellegrino
- Subjects
Combinatorics ,Mathematics::Functional Analysis ,Sequence ,Constructive proof ,Series (mathematics) ,General Mathematics ,Banach space ,Convergent series ,Mathematics - Abstract
In 1947, M.S. Macphail constructed a series in $$\ell _{1}$$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach space theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky–Rogers theorem asserts that in every infinite-dimensional Banach space E, there exists an unconditionally convergent series $$\sum x^{\left( j\right) }$$ such that $$\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty $$ for all $$\varepsilon >0$$ . Their proof is non-constructive and Macphail’s result for $$E=\ell _{1}$$ provides a constructive proof just for $$\varepsilon \ge 1$$ . In this note, we revisit Macphail’s paper and present two alternative constructions that work for all $$\varepsilon >0.$$
- Published
- 2021
14. Algebras whose units satisfy a $$*$$-Laurent polynomial identity
- Author
-
M. Ramezan-Nassab, Mai Hoang Bien, and M. Akbari-Sehat
- Subjects
Combinatorics ,Polynomial ,Identity (mathematics) ,Group (mathematics) ,General Mathematics ,Laurent polynomial ,Free algebra ,Torsion (algebra) ,Field (mathematics) ,Algebraic number ,Mathematics - Abstract
Let R be an algebraic algebra over an infinite field and $$*$$ be an involution on R. We show that if the units of R, $${\mathcal {U}}(R)$$ , satisfy a $$*$$ -Laurent polynomial identity, then R satisfies a polynomial identity. Also, let G be a torsion group and F a field. As a generalization of Hartley’s Conjecture, in Broche et al. (Arch Math 111:353–367, 2018), it is shown that if $${\mathcal {U}}(FG)$$ satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $$F[\alpha , \beta :\alpha ^2=\beta ^2=0]$$ , then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for $${\mathcal {U}}(FG)$$ to satisfy a Laurent polynomial identity.
- Published
- 2021
15. Some remarks on small values of $$\tau (n)$$
- Author
-
Anne Larsen and Kaya Lakein
- Subjects
Conjecture ,Series (mathematics) ,Mathematics::Number Theory ,General Mathematics ,Function (mathematics) ,Congruence relation ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Integer ,Lucas number ,Prime factor ,symbols ,Mathematics - Abstract
A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .
- Published
- 2021
16. When does the canonical module of a module have finite injective dimension?
- Author
-
V. H. Jorge Pérez and T. H. Freitas
- Subjects
Pure mathematics ,Ring (mathematics) ,Conjecture ,Mathematics::Commutative Algebra ,Dimension (vector space) ,General Mathematics ,ANÉIS E ÁLGEBRAS COMUTATIVOS ,Mathematics::Rings and Algebras ,Local ring ,Local cohomology ,Injective function ,Mathematics - Abstract
Foxby (Math Scand 2:175–186, 1971–1972) showed that a Cohen-Macaulay module over a Gorenstein local ring has finite projective dimension if and only if its canonical module has finite injective dimension. In this paper, we establish the result given by Foxby in a general setting. As a byproduct, some criteria to detect the Cohen-Macaulay property of a ring are provided in terms of intrinsic properties of certain local cohomology modules. Also, as an application, we show that any Cohen-Macaulay module that has a canonical module with finite injective dimension satisfies the Auslander–Reiten conjecture.
- Published
- 2021
17. Integral geometry of pairs of planes
- Author
-
Julià Cufí, Agustí Reventós, and Eduardo Gallego
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Differential Geometry (math.DG) ,Euclidean space ,General Mathematics ,FOS: Mathematics ,Convex set ,Mathematics::Metric Geometry ,52A15 (Primary), 53C65 (Secondary) ,Visual angle ,Invariant (mathematics) ,Integral geometry ,Mathematics - Abstract
We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke., 16 pages
- Published
- 2021
18. Cheeger–Gromoll splitting theorem for the Bakry–Emery Ricci tensor
- Author
-
Junhan Tang and Jia-Yong Wu
- Subjects
General Mathematics ,media_common.quotation_subject ,Zero (complex analysis) ,Riemannian manifold ,Type (model theory) ,Infinity ,Mathematics::Metric Geometry ,Splitting theorem ,Vector field ,Mathematics::Differential Geometry ,Ricci curvature ,Mathematics ,Mathematical physics ,media_common - Abstract
In this paper, we obtain a new Cheeger–Gromoll splitting theorem on a complete Riemannian manifold admitting a smooth vector field such that its Bakry–Emery Ricci tensor is non-negative and the vector field tends to zero at infinity. The result generalizes the classical Cheeger–Gromoll splitting theorem and the splitting type results of Lichnerowicz, Wei–Wylie, Fang–Li–Zhang, Wylie, Khuri–Woolgar–Wylie, Lim, and more.
- Published
- 2021
19. Riccati technique and oscillation of linear second-order difference equations
- Author
-
Michal Veselý and Petr Hasil
- Subjects
Class (set theory) ,Oscillation ,Differential equation ,General Mathematics ,Riccati equation ,Order (group theory) ,Applied mathematics ,Contrast (statistics) ,Linear equation ,Mathematics - Abstract
In this paper, we analyse oscillatory properties of a general class of linear difference equations. Applying the modified Riccati technique, we prove an oscillation criterion for the studied equations and we formulate its consequences. In contrast to many known criteria, in the presented results, there are not considered any auxiliary sequences. The results are based directly on the coefficients of the treated equations, i.e., the obtained results are easy to use. In addition, recently, we have proved a non-oscillatory counterpart of the presented criterion. The combination implies that the studied equations are conditionally oscillatory.
- Published
- 2021
20. Gradient estimates and Liouville type theorems for $$(p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0$$ on Riemannian manifolds
- Author
-
Mingfang Zhu and Bingqing Ma
- Subjects
Combinatorics ,Delta ,General Mathematics ,Type (model theory) ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we study gradient estimates of positive smooth solutions to the p-Laplace equation $$\begin{aligned} (p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0, \end{aligned}$$ which is related to the $$L^p$$ -log-Sobolev constant on Riemannian manifolds, where a is a nonzero constant. As applications, some Liouville type results are provided.
- Published
- 2021
21. Noninner automorphisms of order p for finite p-groups of restricted coclass
- Author
-
S. Mohsen Ghoraishi
- Subjects
Mathematics::Group Theory ,Pure mathematics ,Conjecture ,General Mathematics ,Order (group theory) ,Automorphism ,Upper and lower bounds ,Mathematics - Abstract
A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper, we give a lower bound for the coclass of finite nonabelian p-groups G having no noninner automorphism of order p leaving the Frattini subgoup $$\Phi (G)$$ elementwise fixed. As a consequence, the verification of the conjecture is reduced to the case of finite nonabelian p-groups G in which the coclass of G is greater than the minimum number of generators of G.
- Published
- 2021
22. Existence of left invariant Ricci flat metrics on nilpotent Lie groups
- Author
-
Yujian Xiang and Zaili Yan
- Subjects
Nilpotent Lie algebra ,Pure mathematics ,Nilpotent ,General Mathematics ,Metric (mathematics) ,Lie algebra ,Lie group ,Mathematics::Differential Geometry ,Extension (predicate logic) ,Abelian group ,Invariant (mathematics) ,Mathematics - Abstract
In this paper, we study the problem of the existence of left invariant Ricci flat metrics on nilpotent Lie groups. We mainly prove that any nilpotent Lie algebra obtained by a double extension of an Abelian Lie algebra admits at least one left invariant Ricci flat metric. As an application, we obtain certain new nilpotent Lie algebras which admit left invariant Ricci flat metrics.
- Published
- 2021
23. Tartar’s method for the Riesz–Thorin interpolation theorem
- Author
-
Yoichi Miyazaki
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,chemistry.chemical_compound ,chemistry ,General Mathematics ,Norm (mathematics) ,Lp space ,Thorin ,Interpolation ,Mathematics - Abstract
Tartar gave an alternative proof of the Riesz–Thorin interpolation theorem for operators of strong types (1, 1) and $$(\infty ,\infty )$$ . His method characterizes the $$L^{p}$$ norm in terms of the Lebesgue spaces $$L^{1}$$ and $$L^{\infty }$$ , and works not only for complex Lebesgue spaces but also for real Lebesgue spaces. The aim of this paper is to extend the proof for operators of strong types $$(p_{1},q_{1})$$ and $$(\infty ,\infty )$$ with $$1\le p_{1}\le q_{1}
- Published
- 2021
24. Multiplier completion of Banach algebras with application to quantum groups
- Author
-
Mehdi Nemati and Maryam Rajaei Rizi
- Subjects
Mathematics::Functional Analysis ,Quantum group ,General Mathematics ,Locally compact quantum group ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Cardinality ,Compact space ,Closure (mathematics) ,Norm (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Banach *-algebra ,Mathematics - Abstract
Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.
- Published
- 2021
25. A sufficient condition for random zero sets of Fock spaces
- Author
-
Pham Trong Tien and Xiang Fang
- Subjects
Sequence ,Zero set ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Function (mathematics) ,01 natural sciences ,Fock space ,Combinatorics ,0103 physical sciences ,Almost surely ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let $$(r_n)_{n=1}^\infty $$ be a non-decreasing sequence of radii in $$(0, \infty )$$ , and let $$(\theta _n)_{n=1}^\infty $$ be a sequence of independent random arguments uniformly distributed in $$[0, 2\pi )$$ . In this paper, we establish a new sufficient condition on the sequence $$(r_n)_{n=1}^\infty $$ under which $$(r_ne^{i\theta _n})_{n=1}^\infty $$ is almost surely a zero set for Fock spaces. The condition is in terms of the sum of two characteristics involving the counting function. The sharpness of this condition is discussed and examples are presented to illustrate it.
- Published
- 2021
26. A note on compactness theorems for the Bakry–Émery Ricci tensor and generalized quasi-Einstein tensors
- Author
-
Sanghun Lee
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,symbols.namesake ,Compact space ,0103 physical sciences ,symbols ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Einstein ,Ricci curvature ,Mathematical physics ,Mathematics - Abstract
In this paper, we extend compactness theorems of Cheeger, Gromov, Taylor, and Sprouse to the Bakry–Emery Ricci tensor and generalized quasi-Einstein tensors. Our results generalize previous results obtained by Yun and Wan.
- Published
- 2021
27. On the strong maximum principle for a fractional Laplacian
- Author
-
Nguyen Ngoc Trong, Bui Le Trong Thanh, and Do Duc Tan
- Subjects
General Mathematics ,010102 general mathematics ,Boundary (topology) ,Lipschitz continuity ,01 natural sciences ,Omega ,Combinatorics ,Maximum principle ,Dirichlet laplacian ,Bounded function ,0103 physical sciences ,Radon measure ,010307 mathematical physics ,0101 mathematics ,Fractional Laplacian ,Mathematics - Abstract
In this paper, we obtain a version of the strong maximum principle for the spectral Dirichlet Laplacian. Specifically, let $$d \in \{1,2,3,\ldots \}$$ , $$s \in (\frac{1}{2},1)$$ , and $$\Omega \subset \mathbb {R}^d$$ be open, bounded, connected with Lipschitz boundary. Suppose $$u \in L^1(\Omega )$$ satisfies $$u \ge 0$$ a.e. in $$\Omega $$ and $$(-\Delta )^s u$$ is a Radon measure on $$\Omega $$ . Then u has a quasi-continuous representative $${\tilde{u}}$$ . Let $$a \in L^1(\Omega )$$ be such that $$a \ge 0$$ a.e. in $$\Omega $$ . Then if $$\begin{aligned} (-\Delta )^s u + au \ge 0 \quad \text {a.e.} \text { in } \Omega \end{aligned}$$ and $${\tilde{u}} = 0$$ on a subset of positive $$H^s$$ -capacity of $$\Omega $$ , then $$u = 0$$ a.e. in $$\Omega $$ .
- Published
- 2021
28. Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$
- Author
-
Kun Tu
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Compact space ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Limit (mathematics) ,0101 mathematics ,01 natural sciences ,Measure (mathematics) ,Mathematics - Abstract
In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in $$\ell _1$$ coincide with those having the Banach-Saks property. Namely, we prove that the measure of the weak noncompactness based on the Eberlein double limit criterion is equal to the measure of the non-Banach-Saks property defined by the arithmetic separation of sequences.
- Published
- 2021
29. The arithmetic-geometric mean inequality of indefinite type
- Author
-
Mohammad Sal Moslehian, Kota Sugawara, and Takashi Sano
- Subjects
Pauli matrices ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Hilbert space ,Inequality of arithmetic and geometric means ,Type (model theory) ,01 natural sciences ,law.invention ,Combinatorics ,Matrix (mathematics) ,symbols.namesake ,Invertible matrix ,law ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying $$I \ge ^J A$$ and $${\mathrm{sp}}(A) \subseteq [1, \infty ),$$ the inequality $$\begin{aligned} \frac{I + A}{2} \le ^J \sqrt{A} \end{aligned}$$ holds, and the reverse does for A with $$I \ge ^J A$$ and $${\mathrm{sp}}(A) \subseteq [0, 1]$$ . We also prove that for J-positive invertible operators A, B acting on a Hilbert space of arbitrary dimension, the inequality $$\begin{aligned} \frac{A + B}{2} \ge ^J A \sharp ^J B \end{aligned}$$ holds, where $$A \sharp ^J B:= J \bigl ( (JA) \sharp (JB) \bigr )$$ . Several examples involving Pauli matrices are provided to illustrate the main results.
- Published
- 2021
30. On the invariants of inseparable field extensions
- Author
-
El Hassane Fliouet
- Subjects
Combinatorics ,Degree (graph theory) ,Field extension ,General Mathematics ,Field (mathematics) ,Extension (predicate logic) ,Finitely-generated abelian group ,Characterization (mathematics) ,Mathematics ,Separable space - Abstract
Let K be a finitely generated extension of a field k of characteristic $$p\not =0$$ . In 1947, Dieudonne initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$ , and which are called the distinguished subfields of K/k. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which K is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of K/k by means of new inseparability invariants.
- Published
- 2021
31. Continuous functionals for unbounded convergence in Banach lattices
- Author
-
Zili Chen, Zhangjun Wang, and Jinxi Chen
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Functional Analysis (math.FA) ,Dual (category theory) ,Mathematics - Functional Analysis ,Closed and exact differential forms ,0103 physical sciences ,Convergence (routing) ,FOS: Mathematics ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well., Comment: 9 pages
- Published
- 2021
32. Rigidity theorems for complete $$\lambda $$-hypersurfaces
- Author
-
Saul Ancari and Igor Miranda
- Subjects
Polynomial ,General Mathematics ,Second fundamental form ,010102 general mathematics ,Lambda ,Curvature ,01 natural sciences ,Combinatorics ,Mathematics::Algebraic Geometry ,Hypersurface ,Hyperplane ,Bounded function ,0103 physical sciences ,Classification theorem ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this article, we study hypersurfaces $$\Sigma \subset {\mathbb {R}}^{n+1}$$ with constant weighted mean curvature, also known as $$\lambda $$ -hypersurfaces. Recently, Wei-Peng proved a rigidity theorem for $$\lambda $$ -hypersurfaces that generalizes Le–Sesum’s classification theorem for self-shrinkers. More specifically, they showed that a complete $$\lambda $$ -hypersurface with polynomial volume growth, bounded norm of the second fundamental form, and that satisfies $$|A|^2H(H-\lambda )\le H^2/2$$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions, if the reverse inequality holds, then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete $$\lambda $$ -hypersurface with $$H\ge 0$$ must either be a hyperplane or a generalized cylinder.
- Published
- 2021
33. A sharp integral inequality for closed spacelike submanifolds immersed in the de Sitter space
- Author
-
Lucas S. Rocha, Fábio R. dos Santos, and Henrique F. de Lima
- Subjects
Mean curvature ,De Sitter space ,General Mathematics ,Second fundamental form ,010102 general mathematics ,Submanifold ,01 natural sciences ,Square (algebra) ,General Relativity and Quantum Cosmology ,Norm (mathematics) ,0103 physical sciences ,Vector field ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematical physics ,Mathematics ,Scalar curvature - Abstract
In this paper, we establish a sharp integral inequality for n-dimensional closed spacelike submanifolds with constant scalar curvature immersed with parallel normalized mean curvature vector field in the de Sitter space $$\mathbb S_p^{n+p}$$ of index p, and we use it to characterize totally umbilical round spheres $$\mathbb S^n(r)$$ , with $$r>1$$ , of $$\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}$$ . Our approach is based on a suitable lower estimate of the Cheng-Yau operator acting on the square norm of the traceless second fundamental form of such a spacelike submanifold.
- Published
- 2021
34. Weighted exponential inequality for differentially subordinate martingales
- Author
-
Michał Brzozowski
- Subjects
Generality ,Pure mathematics ,Integrable system ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Corollary ,Bounded function ,Jump ,0101 mathematics ,Differential (infinitesimal) ,Mathematics - Abstract
The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition$$A_{\infty }$$A∞. The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and càdlàg - we do not assume any path-continuity restrictions. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted$$L^p$$Lpestimates for differential subordinates of bounded martingales.
- Published
- 2021
35. Certain monomial ideals whose numbers of generators of powers descend
- Author
-
Reza Abdolmaleki and Shinya Kumashiro
- Subjects
Monomial ,Mathematics::Commutative Algebra ,General Mathematics ,Polynomial ring ,010102 general mathematics ,Monomial ideal ,Function (mathematics) ,Type (model theory) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,Integer ,0103 physical sciences ,FOS: Mathematics ,Irreducibility ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $��(I^2)\ge 9$ for the number of minimal generators of $I^2$ with $��(I)\geq 6$. Recently, Gasanova constructed monomial ideals such that $��(I)>��(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $��(I)>��(I^2)>\cdots >��(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $��(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends., 10 pages
- Published
- 2021
36. Tate–Hochschild cohomology for periodic algebras
- Author
-
Satoshi Usui
- Subjects
Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ring ,Cohomology ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
This paper is devoted to studying the Tate–Hochschild cohomology for periodic algebras. We will prove that the Tate–Hochschild cohomology ring of a periodic algebra can be written as the localization of the non-negative part of the Tate–Hochschild cohomology ring.
- Published
- 2021
37. On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions
- Author
-
Weinan Wang
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,Stability (probability) ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,Strong solutions ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we first prove the local existence of strong solutions to the 3D Boussinesq equations in a bounded domain with Navier boundary conditions. Then we show the global stability of strong large solutions under a suitable integral condition.
- Published
- 2021
38. On a question of f-exunits in $$\mathbb {Z}/n\mathbb {Z}$$
- Author
-
Bidisha Roy, Anand, and Jaitra Chattopadhyay
- Subjects
Combinatorics ,Polynomial ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Commutative ring ,0101 mathematics ,01 natural sciences ,Unit (ring theory) ,Mathematics - Abstract
In a commutative ring R with unity, a unit u is called exceptional if $$u-1$$ is also a unit. For $$R = {\mathbb {Z}}/n{\mathbb {Z}}$$ and for any $$f(X) \in {\mathbb {Z}}[X]$$ , an element $${\overline{u}} \in {\mathbb {Z}}/n{\mathbb {Z}}$$ is called an “f-exunit” if $$gcd(f(u),n) = 1$$ . Recently, we obtained the number of representations of a non-zero element of $${\mathbb {Z}}/n{\mathbb {Z}}$$ as a sum of two f-exunits for a particular infinite family of polynomials $$f(X) \in {\mathbb {Z}}[X]$$ . In this paper, we complete this problem by proving a similar formula for any non-constant polynomial $$f(X) \in {\mathbb {Z}}[X]$$ .
- Published
- 2021
39. Mixtures of classical and free independence
- Author
-
Janusz Wysoczanski and Roland Speicher
- Subjects
Pure mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematics - Operator Algebras ,16. Peace & justice ,Lambda ,01 natural sciences ,0103 physical sciences ,Homogeneous space ,FOS: Mathematics ,Independence (mathematical logic) ,010307 mathematical physics ,0101 mathematics ,Operator Algebras (math.OA) ,Quantum ,Random variable ,Cumulant ,Mathematics - Probability ,Mathematics - Abstract
We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered., Comment: We rewrote and shortened the earlier version. The third version contains mainly the results which are new compared to the paper of Mlotkowski
- Published
- 2016
40. Gaps for geometric genera
- Author
-
Flaminio Flamini, Ciro Ciliberto, Mikhail Zaidenberg, Dipartimento di Matematica (Roma Tre), Università degli Studi di Roma Tor Vergata [Roma], Dipartimento di Matematica, Universitá degli Studi di Roma 'Tor Vergata', Università degli Studi di Roma Tor Vergata [Roma]-Università degli Studi di Roma Tor Vergata [Roma], Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
- Subjects
Surface (mathematics) ,0209 industrial biotechnology ,Pure mathematics ,General Mathematics ,Geometric genus ,Dimension (graph theory) ,02 engineering and technology ,Interval (mathematics) ,01 natural sciences ,Upper and lower bounds ,Mathematics - Algebraic Geometry ,020901 industrial engineering & automation ,FOS: Mathematics ,projective hypersurface ,14N25, 14J70, 32J25, 32Q45 ,0101 mathematics ,GEOM ,Algebraic Geometry (math.AG) ,Projective variety ,Geometric Genera ,Mathematics ,010102 general mathematics ,Mathematical analysis ,Geometric Genera, Divisors, Singularities ,geometric genus ,14N25, 14J70, 14C20, 14J29, 32Q45 ,Divisors ,Gravitational singularity ,Settore MAT/03 - Geometria ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Singularities - Abstract
We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach., 9 pages, submitted preprint
- Published
- 2016
41. Sine functions on hypergroups
- Author
-
László Székelyhidi and Żywilla Fechner
- Subjects
Mathematics::Functional Analysis ,Polynomial ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Multiplicative function ,Mathematics::Classical Analysis and ODEs ,20N20, 43A62, 39B99 ,01 natural sciences ,Mathematics - Functional Analysis ,010101 applied mathematics ,Mathematics::Quantum Algebra ,Homomorphism ,Sine ,0101 mathematics ,Commutative property ,Mathematics - Abstract
In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.
- Published
- 2016
42. On the integrability of the wave propagator arising from the Liouville–von Neumann equation
- Author
-
Yoonjung Lee, Youngwoo Koh, and Ihyeok Seo
- Subjects
Density matrix ,Quantum particle ,General Mathematics ,010102 general mathematics ,Motion (geometry) ,Propagator ,Mathematics::Spectral Theory ,01 natural sciences ,Schrödinger equation ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Wave function ,Mathematics ,Von Neumann architecture ,Mathematical physics - Abstract
The Liouville–von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not as an equation for density functions. This setting leads to an extended form of the Schrodinger wave equation governing the motion of a quantum particle. In this paper, we obtain the integrability of the wave propagator arising from the Liouville–von Neumann equation in this setting.
- Published
- 2020
43. On sylowizers in finite groups proposed by Wolfgang Gaschütz
- Author
-
Xiang Li and Jia Zhang
- Subjects
Discrete mathematics ,Continuation ,Intersection ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Arch ,01 natural sciences ,Mathematics - Abstract
In this paper, we mainly investigate the conjugation of the sylowizer that was introduced by Gaschutz (Math Z 122(4):319–320, 1971) and study the p-supersolvability of finite groups by analyzing the intersection between $$O^{p}(G)$$ and sylowizers of p-subgroups. As a continuation of research (Lei and Li in Arch Math (Basel) 114:367–376, 2020), we also give some characterizations on p-nilpotent groups by using the permutability of a sylowizer of a p-subgroup.
- Published
- 2020
44. Spectrality of a class of planar self-affine measures with three-element digit sets
- Author
-
Yan Chen, Peng-Fei Zhang, and Xin-Han Dong
- Subjects
Class (set theory) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Orthogonal basis ,Combinatorics ,Integer matrix ,Planar ,Integer ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Element (category theory) ,Nuclear Experiment ,Mathematics - Abstract
Let $$\mu _{M, D}$$ be the self-affine measure generated by an expanding integer matrix $$M\in M_{2}(\mathbb {Z})$$ and an integer three-element digit set $$D=\{(0,0)^T, (\alpha ,\beta )^T,(\gamma ,\eta )^T\}$$ . In this paper, we show that if $$3\mid \det (M)$$ and $$3\not \mid \alpha \eta -\beta \gamma $$ , then $$L^2(\mu _{M,D})$$ has an orthogonal basis of exponential functions if and only if $$M^*\varvec{u}\in 3\mathbb {Z}^2$$ , where $$\varvec{u}=(\eta -2\beta ,\; 2\alpha -\gamma )^T$$ .
- Published
- 2020
45. Weighted/unweighted composition operators which are Ritt or unconditional Ritt operators
- Author
-
Mahesh Kumar
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Mathematics::Commutative Algebra ,Mathematics::Complex Variables ,Composition operator ,General Mathematics ,010102 general mathematics ,Banach space ,Holomorphic function ,Composition (combinatorics) ,01 natural sciences ,Operator (computer programming) ,0103 physical sciences ,Computer Science::Symbolic Computation ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we study when a composition operator or a weighted composition operator on a Banach space of holomorphic functions is a Ritt operator or an unconditional Ritt operator. It turns out that for composition operators or weighted composition operators on a Banach space of holomorphic functions, if a composition operator or a weighted composition operator is a Ritt operator, then it is also an unconditional Ritt operator.
- Published
- 2020
46. On finite factorized groups with permutable subgroups of factors
- Author
-
A. A. Trofimuk and Victor S. Monakhov
- Subjects
Pure mathematics ,General Mathematics ,Product (mathematics) ,010102 general mathematics ,0103 physical sciences ,Sylow theorems ,010307 mathematical physics ,Permutable prime ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Two subgroups A and B of a group G are called msp-permutable if the following statements hold: AB is a subgroup of G; the subgroups P and Q are mutually permutable, where P is an arbitrary Sylow p-subgroup of A and Q is an arbitrary Sylow q-subgroup of B, $${p\ne q}$$ . In the present paper, we investigate groups that are factorized by two msp-permutable subgroups. In particular, the supersolubility of the product of two supersoluble msp-permutable subgroups is proved.
- Published
- 2020
47. A counterexample to Zarrin’s conjecture on sizes of finite nonabelian simple groups in relation to involution sizes
- Author
-
Chimere Anabanti
- Subjects
Involution (mathematics) ,Finite group ,Conjecture ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Simple group ,0103 physical sciences ,Prime factor ,010307 mathematical physics ,Classification of finite simple groups ,0101 mathematics ,Mathematics ,Counterexample - Abstract
Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.
- Published
- 2018
48. The split common null point problem in Banach spaces
- Author
-
Wataru Takahashi
- Subjects
Discrete mathematics ,Pure mathematics ,Fréchet space ,General Mathematics ,Topological tensor product ,Eberlein–Šmulian theorem ,Banach space ,Interpolation space ,Birnbaum–Orlicz space ,Banach manifold ,Lp space ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
In this paper, we consider the split common null point problem in Banach spaces. Then using the metric resolvents of maximal monotone operators and the metric projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. The result of this paper seems to be the first one to study it outside Hilbert spaces.
- Published
- 2015
49. On the vanishing of self extensions over algebras
- Author
-
Ali Mahin Fallah
- Subjects
Noetherian ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Finitely-generated abelian group ,0101 mathematics ,01 natural sciences ,Commutative property ,Mathematics - Abstract
Recently, Araya, Celikbas, Sadeghi, and Takahashi proved a theorem about the vanishing of self extensions of finitely generated modules over commutative Noetherian rings. The aim of this paper is to obtain extensions of their result over algebras.
- Published
- 2020
50. An extension of the Cameron–Martin translation theorem via Fourier–Hermite functionals
- Author
-
Un Gi Lee and Jae Gil Choi
- Subjects
Pure mathematics ,Hermite polynomials ,General Mathematics ,Structure (category theory) ,Function (mathematics) ,Extension (predicate logic) ,Translation (geometry) ,symbols.namesake ,Abstract Wiener space ,Fourier transform ,Mathematics::Probability ,symbols ,Subspace topology ,Mathematics - Abstract
In this paper, we use the Fourier–Hermite functionals to extend the structure of the Cameron–Martin translation theorem on an abstract Wiener space $$(H,B,\nu )$$ . The directional function in our translation theorem may not be in the Cameron–Martin subspace H of B. We then proceed to obtain an explicit formula for our general translation theorem.
- Published
- 2020
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.