7 results
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2. Macphail’s theorem revisited
- Author
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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
3. Algebras whose units satisfy a $$*$$-Laurent polynomial identity
- Author
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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
4. Some remarks on small values of $$\tau (n)$$
- Author
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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
5. When does the canonical module of a module have finite injective dimension?
- Author
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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
6. Integral geometry of pairs of planes
- Author
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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
7. Cheeger–Gromoll splitting theorem for the Bakry–Emery Ricci tensor
- Author
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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
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