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 $(\alpha_{n})_{n\in\mathbb{N}}$ , $(\mu_{n})_{n\in\mathbb{N}}$

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

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 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

Iban "sbagliato" e responsabilità delle banche nell'esecuzione dell'operazione di bonifico

Il commento analizza le decisioni del Collegio romano dell’A.B.F. che riconoscono, nell’ipotesi di errata indicazione da parte dell’ordinante dell’IBAN in un ordine di bonifico, la responsabilita` della banca del beneficiario (e non della banca dell’ordinante), per non aver verificato la discrasia tra il titolare del conto di pagamento identificato dall’IBAN inesatto e il nominativo del destinatario esplicitamente menzionato. In senso critico, si prospetta una esegesi dell’art. 24 d. legis. n. 11/2010, in materia di ‘‘identificativi unici inesatti’’, diversa rispetto a quella del Collegio, e se ne vagliano le ricadute sulla condotta richiesta ai prestatori di servizi di pagamento sul piano dell’esatta identificazione del destinatario del bonifico e sulla responsabilita` delle banche verso i clienti e gli utenti, non clienti, del sistema dei pagamenti