On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the $P(q,\ell)$ ($q$ odd prime power and $\ell>1$ odd) commutative semifields constructed by Bierbrauer in \cite{BierbrauerSub} are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth in \cite{BuHe2008}. Also, we show that they are strongly isotopic if and only if $q\equiv 1(mod\,4)$. Consequently, for each $q\equiv -1(mod\,4)$ there exist isotopic commutative presemifields of order $q^{2\ell}$ ($\ell>1$ odd) defining CCZ--inequivalent planar DO polynomials.Comment: References updated, pag. 5 corrected Multiplication of commutative LMPTB semifield

On symplectic semifield spreads of PG(5,q2), q odd

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ( 5 , q2), for q2> 2 .38odd, whose associated semifield has center containing Fq. Equivalently, we classify, up to isotopy, commutative semifields of order q6, for q2> 2 .38odd, with middle nucleus containing q2Fq2and center containing q Fq

Krasnoselskii-Mann method for non-self mappings

AbstractLet H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If $T:C\to H$ T : C → H is a non-self and non-expansive mapping, we can define a map $h:C\to\mathbb{R}$ h : C → R by $h(x):=\inf\{\lambda\geq 0:\lambda x+(1-\lambda)Tx\in C\}$ h ( x ) : = inf { λ ≥ 0 : λ x + ( 1 − λ ) T x ∈ C } . Then, for a fixed $x_{0}\in C$ x 0 ∈ C and for $\alpha_{0}:=\max\{1/2, h(x_{0})\}$ α 0 : = max { 1 / 2 , h ( x 0 ) } , we define the Krasnoselskii-Mann algorithm $x_{n+1}=\alpha _{n}x_{n}+(1-\alpha_{n})Tx_{n}$ x n + 1 = α n x n + ( 1 − α n ) T x n , where $\alpha_{n+1}=\max\{\alpha_{n},h(x_{n+1})\}$ α n + 1 = max { α n , h ( x n + 1 ) } . We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping

A Minimum problem for finite sets of real numbers with non-negative sum

Let $n$ and $r$ be two integers such that $0 < r \le n$; we denote by $\gamma(n,r)$ [$\eta(n,r)$] the minimum [maximum] number of the non-negative partial sums of a sum $\sum_{1=1}^n a_i \ge 0$, where $a_1, \cdots, a_n$ are $n$ real numbers arbitrarily chosen in such a way that $r$ of them are non-negative and the remaining $n-r$ are negative. Inspired by some interesting extremal combinatorial sum problems raised by Manickam, Mikl\"os and Singhi in 1987 \cite{ManMik87} and 1988 \cite{ManSin88} we study the following two problems: \noindent$(P1)$ {\it which are the values of $\gamma(n,r)$ and $\eta(n,r)$ for each $n$ and $r$, $0 < r \le n$?} \noindent$(P2)$ {\it if $q$ is an integer such that $\gamma(n,r) \le q \le \eta(n,r)$, can we find $n$ real numbers $a_1, \cdots, a_n$, such that $r$ of them are non-negative and the remaining $n-r$ are negative with $\sum_{1=1}^n a_i \ge 0$, such that the number of the non-negative sums formed from these numbers is exactly $q$?} \noindent We prove that the solution of the problem $(P1)$ is given by $\gamma(n,r) = 2^{n-1}$ and $\eta(n,r) = 2^n - 2^{n-r}$. We provide a partial result of the latter problem showing that the answer is affirmative for the weighted boolean maps. With respect to the problem $(P2)$ such maps (that we will introduce in the present paper) can be considered a generalization of the multisets $a_1, \cdots, a_n$ with $\sum_{1=1}^n a_i \ge 0$. More precisely we prove that for each $q$ such that $\gamma(n,r) \le q \le \eta(n,r)$ there exists a weighted boolean map having exactly $q$ positive boolean values.Comment: 15 page

Subspace code constructions

We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant dimension subspace codes. We also construct two new non-equivalent $(6, (q^3-1)(q^2+q+1), 4; 3)_q$ constant dimension subspace orbit-codes

A Carlitz type result for linearized polynomials

For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$

La transmisión del Renacimiento cultural europeo en China. Un itinerario por las cartas de Alessandro Valignano (1575-1606)

Alessandro Valignano is key to understanding the entry of the Jesuits Michele Ruggieri and Matteo Ricci into the Ming Empire. His insight and experience of the mission in Japan made him aware of the importance of mastering the Chinese language as a sine qua non for gaining access to the Middle Kingdom. The Visitor of the mission was the prime authority of the China Mission and the true founder of this enterprise. Nevertheless, relations between Valignano and China were either considered from a generic point of view or, conversely, in such detail that they had been reduced to a few specific circumstances. This article aims to locate and consider some unpublished European sources about the Father Visitor of the Jesuits, Alessandro Valignano, and his opinions and organization of the China Mission during the sixteenth and seventeenth centuries. The study concentrates on primary sources ranging from Valignano’s first contact with China to his grand plan of a Roman Embassy to China (1588-1603), while also examining his relationship with Matteo Ricci.Alessandro Valignano es la clave para comprender la entrada de los jesuitas Michele Ruggieri y Matteo Ricci en el Imperio Ming. Gracias a su intuición y a la experiencia adquirida en la misión de Japón, Valignano tomó consciencia de lo importante que era aprender chino como condición sine qua non para acceder al interior del territorio oriental. El Visitador de la India, como así se conocía a Alessandro Valignano, fue la primera autoridad de la misión de China y el verdadero fundador de esta empresa. Sin embargo, la relación de este con la tierra de Confucio fue considerada simplemente desde un punto de vista general o, al contrario, tan específicamente que dicha concomitancia se redujo a unas pocas anécdotas históricas.Este artículo tiene como objetivo localizar y considerar algunas fuentes europeas desconocidas hasta ahora sobre el padre Visitador de los jesuitas, así como sus opiniones y su organización de la misión de China durante los siglos XVI y XVII. Además, el estudio analiza las fuentes primarias, que van desde el primer contacto de Valignano con China hasta su gran objetivo de solicitar una embajada romana para el emperador chino (1588-1603), y su relación con Matteo Ricci

On the duality between p-Modulus and probability measures

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and probability measures with barycenter in $L^q(X,\mathfrak m)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)Comment: Minor corrections, typos fixe