On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2

For an integer $k\geq 2$, let $\{F^{(k)}_{n} \}_{n\geq 0}$ be the $k$--generalized Fibonacci sequence which starts with $0, \ldots, 0, 1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $c$ having at least two presentations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$

Entanglement, Non-linear Dynamics, and the Heisenberg Limit

We show that the quantum Fisher information provides a sufficient condition to recognize multi-particle entanglement in a $N$ qubit state. The same criterion gives a necessary and sufficient condition for sub shot-noise phase sensitivity in the estimation of a collective rotation angle $\theta$. The analysis therefore singles out the class of entangled states which are {\it useful} to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the non-linear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.Comment: Phys. Rev. Lett. 102, 100401 (2009

BEC in Nonextensive Statistical Mechanics

We discuss the Bose-Einstein condensation (BEC) for an ideal gas of bosons in the framework of Tsallis's nonextensive statistical mechanics. We study the corrections to the standard BEC formulas due to a weak nonextensivity of the system. In particular, we consider three cases in the D-dimensional space: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. The results show that small deviations from the extensive Bose statistics produce remarkably large changes in the BEC transition temperature.Comment: LaTex, 7 pages, no figures, to be published in Mod. Phys. Lett. B; corrected a typo in Eq. (2

On a nonintegrality conjecture

It is conjectured that the sum $$S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k}$$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in {\mathbb Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools are some deep results on the distribution of primes in short intervals

On Petersson's partition limit formula

For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $\chi=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $\lambda>0$ such that $\chi(\lambda)=\pm 1$. Petersson proved a beautiful limit formula for the ratio of $p_+(n)$ to $p_-(n)$ as $n\to\infty$ expressed in terms of important invariants of the real quadratic field $\mathbb{Q}(\sqrt{p})$. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz-Ces\`aro theorem. In this paper we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman-Erd\H{o}s

The Naples Systematic Series – Second part: Irregular waves, seakeeping in head sea

Abstract The main aim of this study is to characterize the dynamic behavior of the Naples Systematic Series (NSS) in irregular head sea. A further aim of the study is to provide data to detect the influence of hull form on the sea-keeping performances in the planing and semi-planing speed range. The NSS derives from a parent hull that has shown to behave well in rough seas, characterized by high deadrise angles of the bottom at the bow, to reduce acceleration. All the models of NSS were tested in three different sea states and in Fr range from 0.515 to 1.197. The relatively high Froude numbers associated with the forms of the series make inappropriate the statistical analysis usually carried out to describe the behavior of the displacement ships. To overcome these unsuitableness, Cartwright Lounguet Higgins, extreme value and normal distribution fittings have been furnished for heave and pitch maxima and minima; gamma and extreme value have been furnished to represent acceleration in the centre of gravity and bow. Finally, a case study is presented to show a useful procedure for designer evaluation

LA SCUOLA OFFICINA MECCANICA PRESSO IL VILLAGGIO MONTE DEGLI ULIVI A RIESI. RICOSTRUZIONE DI UN PROCESSO TRA ANALISI COMPOSITIVE E GRAFICO-GEOMETRICHE

Il presente contributo trae origine da una pi\uf9 ampia ricerca che ha consentito di approfondire la comprensione dei principi progettuali e geometrico-compositivi dell\u2019opera in oggetto; avvalendosi anche di un rilievo scientifico integrato che, attraverso nuove tecniche digitali applicate, ha permesso di condurre una specifica analisi grafico-geometrica sull\u2019impianto architettonico1. L\u2019edificio della scuola officina meccanica, inserito in un pi\uf9 ampio e articolato complesso edilizio2, \ue8 composto da una singolare configurazione in pianta ai diversi livelli, oggi non pi\uf9 chiaramente leggibile a causa dei numerosi interventi di trasformazione subiti. I disegni di restituzione grafica, realizzati sulla base dei rilievi architettonici, costituiscono un imprescindibile documento di consultazione che documenta l\u2019attuale distribuzione dei locali che hanno alterato l\u2019idea progettuale. Con l\u2019intento di acquisire nuovi elementi di conoscenza propedeutici a futuri interventi di restauro conservativo e a una pi\uf9 consapevole fruizione del bene, lo studio descrive ed esamina il processo progettuale dell\u2019opera, ricercando i principi ordinatori, compositivi e geometrici che ne hanno determinato la particolare struttura. Il testo \ue8 articolato in due parti: la prima sintetizza le questioni generali del progetto, dalla ideazione alla realizzazione del Villaggio Monte degli Ulivi e dell\u2019edificio della ex scuola officina meccanica; la seconda, sulla scorta dei risultati di uno studio sulle funzioni grafiche digitali applicate alla geometria, indaga la natura geometrica dei profili conici che regolano il progetto e la realizzazione dell\u2019edificio.This contribution originates from a wider research which allowed to analyse in depth both the design and geometric-compositional principles of the Scuola Officina Meccanica's building - located within the wide and structured building complex in Villaggio \u201cMonte degli Ulivi\u201d. A scientific integrated survey was also used that, thanks to newly-applied digital techniques, allowed to carry out a specific graphic-geometric analysis of the architectural layout. The study is developed in two parts. The first summarizes the general matters of the project, from the design to the realisation of both Villaggio \u201cMonte degli Ulivi\u201d and the building of the former Scuola Officina Meccanica. The second part, on the basis of the results from digital graphic functions applied to the geometry, investigates the geometric nature of the conic profiles regulating both the project and the realisation of the building

The zero-multiplicity of Berstel type sequences

A ternary linear recurrence $(u_n)_{n\ge 0}$ is of Berstel type if it satisfies the recurrence relation $u_{n+3}=2u_{n+2}-4u_{n+1}+4u_{n}$ for all $n\ge 0$. In this paper, we investigate the zero-multiplicity of such sequences. We prove that, except for nonzero multiples of shifts of the Berstel sequence with initial values $0,0,1$, which has zero-multiplicity 6, and nonzero multiples of shifts of the sequence with initial values $0,1,4$, which has zero-multiplicity 3, all other sequences have zero multiplicity at most 2

On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

In this paper, we show that if (X_n, Y_n) is the nth solution of the Pell equation X^2−dY^2= ±1 for some non-square d, then given any int eger c, the equation c = X_n−2^m has at most 2 integer solutions (n, m)withn ≥ 0andm ≥ 0, except for the only pair (c, d)=(−1, 2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences