Automorphisms of order three on numerical Godeaux surfaces

We prove that a numerical Godeaux surface cannot have an automorphism of order three

Caterina Sforza and Experimenti Translation into English and historical-linguistic analysis of some of her recipes

Caterina Sforza, Countess of Forlì from 1480 to 1500, was an intrepid and shrewd ruler, a mother, a lover, a woman with an eager interest in every field of knowledge, especially in science and experiments. Throughout her whole life, she devoted herself to experimenting and collecting hundreds of alchemical, cosmetic and medical recipes in a manuscript today known as “Experimenti”. This paper will include two chapters: in the first one, a biography of Caterina Sforza will be presented, together with a paragraph dedicated to her passion for botany and chemistry, which represent the foundation for her collection of recipes. Then, the reader will find a narration of the many changes of ownership and adventurous vicissitudes her manuscript experienced, thanks to which it is today possible to access Caterina’s remedies. The second chapter will contain the translation into present-day English of six recipes found in “Experimenti”. Each translation will be followed by the historical-linguistic analysis of some key words extracted from the original recipes: the analysis will base on the comparison between their Renaissance Italian translation and their current English meaning and spelling. An essential tool for this will be John Florio’s dictionary, first published in 1598 and considered the first extensive Italian-English dictionary. The aim of this paper is not only to make Caterina’s experiments accessible to an English public, but also to explore the development of the English language on different levels. Firstly, its evolution in time, from Early Modern English to today’s English. Secondly, its evolution in space, taking into consideration its immense spread all over the world through colonization (which, at the time Caterina Sforza and John Florio were alive, had only just begun), and the consequent many varieties of English existing now. Thirdly, its evolution with regard to the Italian language and culture

Semilinear wave equation on compact Lie groups

In this note, we study the semilinear wave equation with power nonlinearity $|u|^p$ on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any $p>1$, under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.Comment: arXiv admin note: substantial text overlap with arXiv:2005.1347

Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces

In this paper we consider the following Cauchy problem for the semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Delta u+\dfrac{\mu_1}{1+t} u_t+\dfrac{\mu_2^2}{(1+t)^2}u=|u|^p, \\ u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x), \end{cases}\tag{$\star$} \end{align} where $\mu_1, \mu_2^2$ are nonnegative constants and $p>1$. On the one hand we will prove a global (in time) existence result for \eqref{CP abstract} under suitable assumptions on the coefficients $\mu_1, \mu_2^2$ of the damping and the mass term and on the exponent $p$, assuming the smallness of data in exponentially weighted energy spaces. On the other hand a blow-up result for \eqref{CP abstract} is proved for values of $p$ below a certain threshold, provided that the data satisfy some integral sign conditions. Combining these results we find the critical exponent for \eqref{CP abstract} in all space dimensions under certain assumptions on $\mu_1$ and $\mu_2^2$. Moreover, since the global existence result is based on a contradiction argument, it will be shown firstly a local (in time) existence result

Quillen stratification for the Steenrod algebra

Let A be the mod 2 Steenrod algebra, and let Q denote the category of exterior sub-Hopf algebras of A, where the morphisms are given by inclusions. The restriction maps Ext_A (Z/2,Z/2) --> Ext_E (Z/2,Z/2), for E in Q, can be assembled into a map i:Ext_A (Z/2, Z/2) --> lim_Q Ext_E (Z/2,Z/2). There is an action of A on this inverse limit, and i factors through the invariants under this action, giving a map g:Ext_A (Z/2, Z/2) --> ( lim_Q Ext_E (Z/2,Z/2) )^A. We show that g is an F-isomorphism.Comment: 29 pages, published versio

On Chern ratios for surfaces with ample cotangent bundle

In this paper we study the problem of density in (1,3] for the Chern ratio of surfaces with ample cotangent bundle. In particular we prove density in (1,2) by constructing a family of complete intersection surfaces in a product of varieties with big cotangent bundle. We also analyse the case of complete intersections in a product of curves of genus at least 2.Comment: 10 pages, corrected typos, updated reference

Vanishing lines in generalized Adams spectral sequences are generic

We show that in a generalized Adams spectral sequence, the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property.Comment: 11 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper7.abs.htm