arXiv subjects
Publications and source records attributed to Pedro J. Freitas.
It is well known that Charles Hermite kept an intense correspondence with many of the word's leading mathematicians of his time. This paper focuses on Hermite's letters to Francisco Gomes Teixeira, a Portuguese mathematician, who exchanged letters with Hermite for more than twenty years.
We prove that the golden angle (an angle that divides the circle in the golden ratio) is not constructible using straightedge and compass.
There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed. The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the $n$-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a joint mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all.
If $\mathscr{J}$ is a finite-dimensional nilpotent algebra over a finite field $\Bbbk$, the algebra group $P = 1+\mathscr{J}$ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If $\mathscr{J}$ is endowed with an involution $\widehatς$, then $\widehatς$ naturally defines a group automorphism of $P = 1+\mathscr{J}$, and we may consider the fixed point subgroup $C_{P}(\widehatς) = \{x\in P : \widehatς(x) = x^{-1}\}$. Assuming that $\Bbbk$ has odd characteristic $p$, we use the standard supercharacter theory for $P$ to construct a supercharacter theory for $C_{P}(\widehatς)$. In particular, we obtain a supercharacter theory for the Sylow $p$-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by André and Neto for the special case of the symplectic and orthogonal groups.
In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.
The sum-product conjecture of Erd\H os and Szemerédi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+δ}$ for every $δ<1$. In this paper we consider the set of all spectral radii of $n\times n$ matrices with entries in $A$, and find lower bounds for the cardinality of this set. In the case $n=2$, this cardinality is necessarily larger than $\max\{|A+A|,|A\cdot A|\}$.
In recent papers, R. Bhatia, T. Jain and P. Grover obtained formulas for directional derivatives, of all orders, of the determinant, the permanent, the $m$-th compound map and the $m$-th induced power map. In this paper we generalize these results for immanants and for other symmetric powers of a matrix.
In this paper we present the exact value for the norm of directional derivatives, of all orders, for symmetric tensor powers of operators on finite dimensional vector spaces. Using this result we obtain an upper bound for the norm of all directional derivatives of immanants. This work is inspired in results by R. Bhatia, J. Dias da Silva, P. Grover and T. Jain.
In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some positive integer $k$. We show that if the system has a nontrivial solution then there exists a nontrivial solution $\x=(x_1,...,x_n)\trans$ such that $\frac{|x_j|}{|x_i|}\le k^{n-1}$ for each $i,j$ satisfying $x_ix_j\ne 0$. This inequality is sharp. We also prove a conjecture of A. Tyszka related to our results.