In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.

Markerless 3D pose estimation has become an indispensable tool for kinematic studies of laboratory animals. Most current methods recover 3D pose by multi-view triangulation of deep network-based 2D pose estimates. However, triangulation requires multiple, synchronized cameras and elaborate calibration protocols that hinder its widespread adoption in laboratory studies.Here, we describe LiftPose3D, a deep network-based method that overcomes these barriers by reconstructing 3D poses from a single 2D camera view. We illustrate LiftPose3D's versatility by applying it to multiple experimental systems using flies, mice, rats, and macaque monkeys and in circumstances where 3D triangulation is impractical or impossible. Our framework achieves accurate lifting for stereotyped and non-stereotyped behaviors from different camera angles. Thus, LiftPose3D permits high-quality 3D pose estimation in the absence of complex camera arrays, tedious calibration procedures, and despite occluded body parts in freely behaving animals.

Quantum Hall effect (QHE) is the basis of modern resistance metrology. In Quantum Hall Array Resistance Standards (QHARS), several individual QHE elements, each one having the same QHE resistance (typically half of the von Klitzing constant), are arranged in networks that realize resistance values close to decadic values (such as 1 kΩ or 100 kΩ), of direct interest for dissemination. The same decadic value can be approximated with different grades of precision, and even for the same approximation several networks of QHE elements can be conceived. The paper investigates the design of QHARS networks by giving methods to find a proper approximation of the resistance of interest, and to design the corresponding network with a small number of elements. Results for several decadic cases are given.

This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number of colored compositions for any coloration. The interesting consequences arising from this relationship also give an immediate and simple criterion to determine whether a sequence of integers counts the number of some colored compositions. Applications to Catalan and Fibonacci numbers naturally emerge, allowing to clearly answer to some open questions. Moreover, the definition of colored compositions with the "black tie" provides straightforward combinatorial proofs to a new identity involving multinomial coefficients and to a new closed formula for the Invert operator. Finally, colored compositions with the "black tie" give rise to a new combinatorial interpretation for the convolution operator, and to a new and easy method to count the number of parts of colored compositions.

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Markerless 3D pose estimation has become an indispensable tool for kinematic studies of laboratory animals. Most current methods recover 3D pose by multi-view triangulation of deep network-based 2D pose estimates. However, triangulation requires multiple, synchronised cameras per keypoint and elaborate calibration protocols that hinder its widespread adoption in laboratory studies. Here, we describe LiftPose3D, a deep network-based method that overcomes these barriers by reconstructing 3D poses from a single 2D camera view. We illustrate LiftPose3D’s versatility by applying it to multiple experimental systems using flies, mice, and macaque monkeys and in circumstances where 3D triangulation is impractical or impossible. Thus, LiftPose3D permits high-quality 3D pose estimation in the absence of complex camera arrays, tedious calibration procedures, and despite occluded keypoints in freely behaving animals.

In this paper the properties of Rédei rational functions are used to derive rational approximations for square roots and both Newton and Padé approximations are given as particular cases. Moreover, Rédei 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.

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.