Helical axis analysis to quantify humeral kinematics during shoulder rotation.

© 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.Information pertaining to the helical axis during humeral kinematics during shoulder rotation may be of benefit to better understand conditions such as shoulder instability. The aim of this study is to quantify the behavior of humeral rotations using helical axis (HA) parameters in three different conditions. A total of 19 people without shoulder symptoms participated in the experiment. Shoulder kinematics was measured with an optoelectric motion capture system. The subjects performed three different full range rotations of the shoulder. The shoulder movements were analyzed with the HA technique. Four parameters were extracted from the HA of the shoulder during three different full-range rotations: range of movement (RoM), mean angle (MA), axis dispersion (MDD), and distance of their center from the shoulder (D). No significant differences were observed in the RoM for each condition between left and right side. The MA of the axis was significantly lower on the right side compared to the left in each of the three conditions. The MDD was also lower for the right side compared to the left side in each of the three conditions.The four parameters proposed for the analysis of shoulder kinematics showed to be promising indicators of shoulder instability.Peer reviewe

Linear divisibility sequences and Salem numbers

We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1

Periodic representations and rational approximations of square roots

In this paper the properties of R\'edei rational functions are used to derive rational approximations for square roots and both Newton and Pad\'e approximations are given as particular cases. As a consequence, such approximations can be derived directly by power matrices. Moreover, R\'edei rational functions are introduced as convergents of particular periodic continued fractions and are applied for approximating square roots in the field of p-adic numbers and to study periodic representations. Using the results over the real numbers, we show how to construct periodic continued fractions and approximations of square roots which are simultaneously valid in the real and in the p-adic field

Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials

In this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences related to SCPs

Groups and monoids of Pythagorean triples connected to conics

We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and $3 \times 3$ matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations

Polynomial sequences on quadratic curves

In this paper we generalize the study of Matiyasevich on integer points over conics, introducing the more general concept of radical points. With this generalization we are able to solve in positive integers some Diophantine equations, relating these solutions by means of particular linear recurrence sequences. We point out interesting relationships between these sequences and known sequences in OEIS. We finally show connections between these sequences and Chebyshev and Morgan-Voyce polynomials, finding new identities

The Biharmonic mean

We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we call the biharmonic mean. The biharmonic mean allows to introduce the biharmonic numbers, providing a new characterization for primes. Moreover, we highlight some interesting divisibility properties and we characterize the semi--prime biharmonic numbers showing their relationship with linear recurrent sequences that solve certain Diophantine equations

Sodium hydroxide pretreatment as an effective approach to reduce the dye/holes recombination reaction in P-Type DSCs

We report the synthesis of a novel squaraine dye (VG21-C12) and investigate its behavior as p-type sensitizer for p-type Dye-Sensitized Solar Cells. The results are compared with O4-C12, a well-known sensitizer for p-DSC, and sodium hydroxide pretreatment is described as an effective approach to reduce the dye/holes recombination. Various variable investigation such as dipping time, dye loading, photocurrent, and resulting cell efficiency are also reported. Electrochemical impedance spectroscopy (EIS) was utilized for investigating charge transport properties of the different photoelectrodes and the recombination phenomena that occur at the (un)modified electrode/electrolyte interface