Showing papers in "The São Paulo Journal of Mathematical Sciences in 2021"

On power integral bases of certain pure number fields defined by $$x^{3^r} - m$$ x 3 r - m

Abstract: Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{3^r} -m$$ , with $$m e \pm 1$$ is a square-free rational integer and r is a positive integer. In this paper, we study the monogenity of K. We prove that if $$m ot \equiv \pm 1 \ \text { (mod }{9})$$ , then K is monogenic. We give also sufficient conditions on r and m for K to be not monogenic. Some illustrating examples are given too.

Submanifold geometry in symmetric spaces of noncompact type

Abstract: In this survey article we provide an introduction to submanifold geometry in symmetric spaces of noncompact type. We focus on the construction of examples and the classification problems of homogeneous and isoparametric hypersurfaces, polar and hyperpolar actions, and homogeneous CPC submanifolds.

On graded 1-absorbing prime ideals

Abstract: Let G be a group with identity e and R be a G-graded commutative ring with unity 1. In this article, we introduce and study the concept of graded 1-absorbing prime ideals which is a generalization of graded prime ideals. A proper graded ideal P of R is called graded 1-absorbing prime if for all nonunit elements $$x,y,z\in h(R)$$ such that $$xyz\in P$$ , then either $$xy\in P$$ or $$z\in P$$ .

Morse theory for the area functional

Abstract: In this article we will discuss a Morse-theoretic existence result of closed minimal hypersurfaces based on the notion of volume spectrum.

On Mazur rotations problem and its multidimensional versions

Abstract: The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach’s famous book in 1932, and known as the Mazur rotations problem Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraisse theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last 10 years The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces $$L_p$$ The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results Some new results are also included

On graded $$I_{e}$$ I e -primary submodules of graded modules over graded commutative rings

Abstract: Let G be a group with identity e Let R be a G-graded commutative ring with identity and M a graded R-module In this paper, we introduce the concept of graded $$J_{e}$$ -primary submodule as a generalization of a graded primary submodule for $$\ J=\oplus _{g\in G}J_{g}$$ a fixed graded ideal of R We give a number of results concerning of these classes of graded submodules and their homogeneous components A proper graded submodule C of M is said to be a graded $$J_{e}$$ -primary submodule of M if whenever $$r_{h}\in h(R)$$ and $$m_{\lambda }\in h(M)$$ with $$r_{h}m_{\lambda }\in C\backslash J_{e}C$$ , implies either $$m_{\lambda }\in C$$ or $$r_{h}\in Gr((C:_{R}M))$$

Common fixed points of Kannan, Chatterjea and Reich type pairs of self-maps in a complete metric space

Abstract: In this paper, we establish existence and uniqueness of common fixed points for Kannan, Reich and Chatterjea type pairs of self-maps in a complete metric space.

On the existence of pairs of primitive normal elements over finite fields

Abstract: We consider the extension field $${\mathbb {F}}_{q^n}$$ of degree n over $${\mathbb {F}}_{q}$$ , where $$q=p^k$$ for some prime p and positive integer k. In this paper, we provide a sufficient condition for the existence of a primitive normal element $$\alpha$$ of $${\mathbb {F}}_{q^n}$$ over $${\mathbb {F}}_{q}$$ , such that $$f(\alpha )$$ is also a primitive normal element of $${\mathbb {F}}_{q^n}$$ for any polynomial f(x) of degree $$>1$$ in $${\mathbb {F}}_{q^n}[x]$$ under some conditions, given in Sect. 3. More precisely we generalize the result, by Booker et al. given [Booker et al. in Math Comput 88:1903–1912, 2019. https://doi.org/10.1090/mcom/3390 , Theorem 1], for any degree polynomial and provide the existence of primitive normal pairs for quadratic polynomials. Moreover, for $$p=3$$ , we find 21 pairs (q, n) for which such primitive normal elements $$(\alpha , f(\alpha ))$$ may not exist for some quadratic polynomial f(x).

On a class of almost Kenmotsu manifolds admitting an Einstein like structure

Abstract: In the present paper, we introduce the notion of $$*$$ -gradient $$\rho$$ -Einstein soliton on a class of almost Kenmotsu manifolds. It is shown that if a $$(2n+1)$$ -dimensional $$(k,\mu )'$$ -almost Kenmotsu manifold M admits $$*$$ -gradient $$\rho$$ -Einstein soliton with Einstein potential f, then (1) the manifold M is locally isometric to $$\mathbb {H}^{n+1}(-4)$$ $$\times$$ $$\mathbb {R}^n$$ , (2) the manifold M is $$*$$ -Ricci flat and (3) the Einstein potential f is harmonic or satisfies a physical Poisson’s equation. Finally, an illustrative example is presented.

On the Hodge conjecture for quasi-smooth intersections in toric varieties

Abstract: We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ has Picard group $${\mathbb {Z}}$$ . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

A closed-form expression of a remarkable sequence of polynomials originating from a family of entire functions connecting the Bessel and Lambert functions

Abstract: In the paper, by virtue of the Faa di Bruno formula and some properties of the Bell polynomials of the second kind, the authors derive a closed-form expression and discuss some properties of a remarkable sequence of polynomials originating from a family of entire functions connecting the Bessel and Lambert functions.

Mathematical modeling applied to epidemics: an overview

Abstract: This work presents an overview of the evolution of mathematical modeling applied to the context of epidemics and the advances in modeling in epidemiological studies. In fact, mathematical treatments have contributed substantially in the epidemiology area since the formulation of the famous SIR (susceptible-infected-recovered) model, in the beginning of the 20th century. We presented the SIR deterministic model and we also showed a more realistic application of this model applying a stochastic approach in complex networks. Nowadays, computational tools, such as big data and complex networks, in addition to mathematical modeling and statistical analysis, have been shown to be essential to understand the developing of the disease and the scale of the emerging outbreak. These issues are fundamental concerns to guide public health policies. Lately, the current pandemic caused by the new coronavirus further enlightened the importance of mathematical modeling associated with computational and statistical tools. For this reason, we intend to bring basic knowledge of mathematical modeling applied to epidemiology to a broad audience. We show the progress of this field of knowledge over the years, as well as the technical part involving several numerical tools.

On monogenity of certain pure number fields defined by $$x^{20}-m$$ x 20 - m

Abstract: Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a complex root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{20}-m$$ , with $$m e \mp 1$$ a square free rational integer. In this paper, we study the monogenity of K. We prove that if $$m ot \equiv 1\ \text{(mod } {4})$$ and $$\overline{m} ot \in \{\overline{1}, \overline{7}, \overline{18}, \overline{24}\} \ \text{(mod } {25})$$ , then K is monogenic. But if $$m\equiv 1\ \text{(mod } {16})$$ or $$m\equiv 1\ \text{(mod } {25})$$ , then K is not monogenic. Some illustrating examples are given too.

Periodic string complexes over string algebras

Abstract: In this paper we develop combinatorial techniques for the case of string algebras with the aim to give a characterization of string complexes with infinite minimal projective resolution. These complexes will be called periodic string complexes. As a consequence of this characterization, we give two important applications. The first one, is a sufficient condition for a string algebra to have infinite global dimension. In the second one, we exhibit a class of indecomposable objects in the derived category for a special case of string algebras. Every construction, concept and consequence in this paper is followed by some illustrative examples.

Explicit soliton for the Laplacian co-flow on a solvmanifold

Abstract: We apply the general Ansatz proposed by Lauret (Rend Semin Mat Torino 74:55–93, 2016) for the Laplacian co-flow of invariant $$\mathrm {G}_2$$ -structures on a Lie group, finding an explicit soliton on a particular almost Abelian 7–manifold. Our methods and the example itself are different from those presented by Bagaglini and Fino (Ann Mat Pura Appl 197(6):1855–1873, 2018).

Recent results on the spectra of lens spaces

Abstract: In this paper we report on recent results by several authors, on the spectral theory of lens spaces and orbifolds and similar locally symmetric spaces of rank one. Most of these results are related to those obtained by Lauret et al. (IMRN 2016(4):1054–1089, 2016. https://doi.org/10.1093/imrn/rnv159 ), where the spectra of lens spaces were described in terms of the one-norm spectrum of a naturally associated congruence lattice. As a consequence, the first examples of Riemannian manifolds isospectral on p-forms for all p but not strongly isospectral were constructed. We also give a new elementary proof in the case of the spectrum on functions. In this proof, representation theory of compact Lie groups is avoided and replaced by the use of Molien’s formula and a manipulation of the one-norm generating function associated to a congruence lattice. In the last four sections we present several recent results, open problems and conjectures on the subject.

A retrospect of the research in nonassociative algebras in IME-USP

Abstract: The aim of this paper is to give an overview of the participation of our research group in the development of nonassociative algebras.

Geometric structures and configurations of flags in orbits of real forms

Abstract: This is an introduction and a survey on geometric structures modelled on closed orbits of real forms acting on spaces of flags. We focus on 3-manifolds and the flag space of all pairs of a point and a line containing it in $${\mathbb{P}}({\mathbb{C}}^3)$$ . It includes a description of general flag structures which are not necessarily flat and a combinatorial description of flat structures through configurations of flags in closed orbits of real forms. We also review volume and Chern–Simons invariants for those structures.

Representations of Compact Lie Groups of Low Cohomogeneity

Abstract: We survey different tools to classify representations of compact Lie groups according to their cohomogeneity and apply these methods to the case of irreducible representations of cohomogeneity 6, 7 and 8.

The Kempf–Ness Theorem and invariant theory for real reductive representations

Abstract: This paper does not contain any new result. We give new proofs of the Kempf–Ness Theorem and Hilbert–Mumford criterion for real reductive representations avoiding any algebraic results.

A nonlinear theory of generalized functions

Abstract: Generalized Functions are crucial in the development of theories modeling physical reality. Starting with the linear theory, we give a partial account of what is known, focussing on the non-linear theory and some of its more recent achievements.

On certain new differential equations for ratio of theta functions

Abstract: In this article, we deduce certain differential equations for the quotient of Ramanujan theta functions.

Generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations associated to the number operator

Abstract: By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.

Some remarks on blueprints and $${\pmb {{\mathbb {F}}}_1}$$ F 1 -schemes

Abstract: Over the past two decades several different approaches to defining a geometry over $${{\mathbb F}_1}$$ have been proposed. In this paper, relying on Toen and Vaquie’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category $${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$ of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoid M and a relation on the semiring $$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$ , is a monoid object in a certain symmetric monoidal category $${\mathsf B}$$ , which is shown to be complete, cocomplete, and closed. We prove that every $${\widetilde{{\mathsf B}}}$$ -scheme $$\Sigma $$ can be associated, through adjunctions, with both a classical scheme $$\Sigma _{\mathbb Z}$$ and a scheme $$\underline{\Sigma }$$ over $${{\mathbb F}_1}$$ in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhauser, Boston, 87–100, 2005), together with a natural transformation $$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$ . Furthermore, as an application, we show that the category of “ $${{\mathbb F}_1}$$ -schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of $${\widetilde{{\mathsf B}}}$$ -schemes to obtain a larger category, whose objects we call “ $${{\mathbb F}_1}$$ -schemes with relations”.

A Brauer-Clifford-long group for the category of dyslectic (S, H)-dimodule algebras

Abstract: In this article, we define the notion of Brauer-Clifford group for the category of dyslectic (S, H)-dimodules, where H is a commutative and cocommutative Hopf algebra and S is an H-commutative dimodule algebra This Brauer group turns out to be an example of the Brauer group of a braided monoidal category We also show that this Brauer group is anti-isomorphic to the Brauer group of the category of dyslectic Hopf Yetter-Drinfel’d $$(S^{op},H)$$ -modules

Caputo-Hadamard implicit fractional differential equations with delay

Abstract: This article deals with some existence results for some classes of Caputo-Hadamard implicit fractional differential equations with two boundary conditions and delay. The results are based on some fixed point theorems. We illustrate our results by some examples in the last section.

Generalized Ricci soliton and paracontact geometry

Abstract: In the present paper, we study generalized Ricci soliton in the framework of paracontact metric manifolds. First, we prove that if the metric of a paracontact metric manifold M with $$Q\varphi =\varphi Q$$ is a generalized Ricci soliton (g, X) and if $$X e 0$$ is pointwise collinear to $$\xi$$ , then M is K-paracontact and $$\eta$$ -Einstein. Next, we consider closed generalized Ricci soliton on K-paracontact manifold and prove that it is Einstein provided $$\beta (\lambda +2n\alpha ) e 1$$ . Next, we study K-paracontact metric as gradient generalized almost Ricci soliton and in this case we prove that (i) the scalar curvature r is constant and is equal to $$-2n(2n+1)$$ ; (ii) the squared norm of Ricci operator is constant and is equal to $$4n^2(2n+1)$$ , provided $$\alpha \beta e -1$$ .

Iterative method to find approximate solution of system of integral equations via generalized Meir–Keeler condensing operator

Abstract: We propose some generalization of Meir–Keeler fixed point theorem involving measure of noncompactness and prove the existence of solution of infinite systems of integral equations by using this new type fixed point theorem in Banach space. With the help of an example we illustrate our results, and find an approximate solution of the infinite systems integral equations using an iterative algorithm method.

The higher Riemann-Hilbert correspondence and principal 2-bundles

Abstract: In this note we show that the holonomies for higher local systems provided by the higher Riemann-Hilbert correspondence coincide with those associated to principal 2-bundles where both formalisms meet

On cubic torsors, biextensions and Severi-Brauer varieties over Abelian varieties

Abstract: We study homogeneous irreducible Severi–Brauer varieties over an Abelian variety A. Such objects were classified by Brion (Algebra Number Theory 7(10):2475–2510, 2013). Here we interpret that result within the context of cubic structures and biextensions for certain $$\mathbb {G}_m$$ -torsors over finite subgroups of A. Our results build on the theory of Breen (Fonctions theta et theoreme du cube, Springer, Berlin, 1983), and Moret-Bailly (Pinceaux de varietes abeliennes. Asterisque 129:266, 1985).