Expositiones Mathematicae最新文献

The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a “meander”: a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.

We investigate holomorphic webs tangent to real-analytic Levi-flat hypersurfaces on compact complex surfaces. Under certain conditions, we prove that a holomorphic web tangent to a real-analytic Levi-flat hypersurface admits a multiple-valued meromorphic first integral. We also prove that the Levi foliation of a Levi-flat hypersurface induced by an irreducible real-analytic curve in the Grassmannian $G(n+1,n)$ extends to an algebraic web on the complex projective space.

In this paper, we investigate three general forms of multiple zeta(-star) values. We use these values to give three new sum formulas for multiple zeta(-star) values with height $\le 2$ and the evaluation of ${\zeta }^{\star }({\left\{1\right\}}^m,{\left\{2\right\}}^{n+1})$. We also give a new proof of the sum formula of multiple zeta values.

We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).

This paper aims to use Cartan’s original method in proving Theorems A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the sheaf is (locally) constant and the degree is positive. In the first case, we can further use Godement’s argument to show the topological dimension of a paracompact topological manifold is less than or equal to its real dimension.

In this paper, we construct a derivative-free multi-step iterative scheme based on Steffensen’s method. To avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, we freeze the divided differences used from the second step and use a weight function on already evaluated operators. Therefore, we define a family of multi-step methods with convergence order $2m$, where $m$ is the number of steps, free of derivatives, with several parameters and with dynamic behaviour, in some cases, similar to Steffensen’s method. In addition, we study how to increase the convergence order of the defined family by introducing memory in two different ways: using the usual divided differences and the Kurchatov divided differences. We perform some numerical experiments to see the behaviour of the proposed family and suggest different weight functions to visualize with dynamical planes in some cases the dynamical behaviour.

We prove that $\sum _{k=1}^{\infty }\left(\frac{n+k-1}{k-1}\right)\frac{k^{k-2}}{{(x+k)}^{n+k}}<\frac{1}{x^{n+1}}$holds for all integers $n\ge 0$ and real numbers $x>0$. This extends a result of Ramanujan, who submitted the inequality with $n=0$ as a problem to the “Journal of the Indian Mathematical Society”.

Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.

In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of $SO\left(8\right)$, is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.

In 1996, it was published the seminal work of Rochberg “Higher order estimates in complex interpolation theory” (Rochberg, 1996). Among many other things, the paper contains a new method to construct new Banach spaces having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each $z\in S$ the space formed by the arrays of the truncated sequence of the Taylor coefficients of the elements of the Calderón space. What was probably unforeseen is that the Rochberg constructions would lead to a deep theory connecting Interpolation theory, Homology, Operator Theory and the Geometry of Banach spaces. This work aims to synthetically present such connections, an up-to-date account of the theory and a list of significative open problems.