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

On the role of the coefficients in the strong convergence of a general type Mann iterative scheme

Let H be a Hilbert space. Let $(W_{n})_{n\in\mathbb{N}}$ be a suitable family of mappings. Let S be a nonexpansive mapping and D be a strongly monotone operator. We study the convergence of the general scheme $x_{n+1}=W_{n}(\alpha_{n}Sx_{n}+(1-\alpha_{n})(I-\mu_{n}D)x_{n})$ in dependence on the coefficients $(\alpha_{n})_{n\in\mathbb{N}}$ , $(\mu_{n})_{n\in\mathbb{N}}$

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

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 $\mathrm{PG}(1,q^5)$

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

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

Un Galateo para Oriente. Introducción a los Advertimentos e avisos dos co0stumes e catangues de Japao (1581)

Un dels escrits clau de la accomodatio jesuítica en la missió de Japó van ser els Advertimentos eavisos dos costumes e catangues de Japão, ideats pel visitador de l’Índia, Xina i Japó AlessandroValignano, i redactats presumiblement en 1581 entre Funai, capital de Bungo, i Usuki. Desprésde ser traduït a l’italià en 1946, el tractat es va catalogar com una creació del gènere cerimonial icom un dels manifestos de l’obertura jesuïta a l’univers japonès i de la «gran trobada» entre Orienti Occident. Aquest estudi pretén fer un abordatge de tipus panoràmic dels Advertimentosbasats en tres punts: adaptació, emulació i familiarització; al costat d’una interpretació en la quals’aproxima l’obra de Valignano amb els escrits descriptius d’autors occidentals des de l’antiguitatclàssica. Entre les analogies registrades, obres com el Galateo de Giovanni della Casa vaninfluir en el comportament social al llarg dels segles, impulsant la cortesia, les bones maneres i lacivilitat, nocions que Valignano va englobar en la seva obra per a Japó.One of the famous written works on accomodatio by Jesuits during their mission in Japan was Advertimentos e avisos dos costumes e catangues de Japão, a book compiled by the Visitor of India, Alessandro Valignano, presumably written in 1581 between Funai, the capital of the province Bungo, and Usuki. After its translation into Italian in 1946, the work was ascribed to the ceremonial genre. It was also upheld as a record of the Jesuits’ opening to the Japanese universe and the “great encounter” between the East and the West.This study presents a panoramic approach to the Advertimentos based on three key points: adaptation, emulation, and familiarization; combined with an interpretation in which Valignano’s work contrasts with descriptive writings by Western authors from Classic Antiquity onwards. Among other analogies, works such as Galateo: The Rules of Polite Behavior by Giovanni della Casa influenced the social behavior over the course of centuries, moving towards politeness, good manners, and civility in the upper class; concepts which Valignano encompassed in his work for Japan.Resumen Uno de los escritos clave de la accomodatio jesuítica en la misión de Japón fueron los Advertimentos e avisos dos costumes e catangues de Japão, ideados por el Visitador de la India, Alessandro Valignano, y redactados, presumiblemente, en 1581 entre Funai, capital de Bungo, y Usuki. Tras ser traducido al italiano en 1946, el tratado se catalogó como una creación del género ceremonial y como uno de los manifiestos de la apertura jesuita al universo japonés, del gran «encuentro» entre Oriente y Occidente. Este ensayo pretende ser una visión panorámica de los Advertimentos, en la que se destacan tres puntos principales: adaptación, emulación y familiarización. A esta se suman las interpretaciones realizadas por Valignano en sus escritos descriptivos procedentes de autores occidentales desde la Antigüedad clásica. Entre las analogías registradas, se encuentran obras como el Galateo, de Giovanni della Casa, que influyó en el comportamiento social a lo largo de los siglos, impulsando la cortesía, las buenas maneras y la civilidad en las clases altas, nociones que Valignano englobó en su obra para Japón