Quantum geometrical properties of topological materials.

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models.

Online Handwriting Recognition Method with a Non-Inertial Reference Frame Based on the Measurement of Linear Accelerations and Differential Geometry: An Alternative to Quaternions.

This work describes a mathematical model for handwriting devices without a specific reference surface (SRS). The research was carried out on two hypotheses: the first considers possible circular segments that could be made during execution for the reconstruction of the trace, and the second is the combination of lines and circles. The proposed system has no flat reference surface, since the sensor is inside the pencil that describes the trace, not on the surface as in tablets or cell phones. An inertial sensor was used for the measurements, in this case, a commercial Micro-Electro Mechanical sensor of linear acceleration. The tracking device is an IMU sensor and a processing card that allows inertial measurements of the pen during on-the-fly tracing. It is essential to highlight that the system has a non-inertial reference frame. Comparing the two proposed models shows that it is possible to construct shapes from curved lines and that the patterns obtained are similar to what is recognized; this method provides an alternative to quaternion calculus for poorly specified orientation problems.

Quantum mechanics of particles constrained to spiral curves with application to polyene chains.

CONTEXT: Due to advances in synthesizing lower-dimensional materials, there is the challenge of finding the wave equation that effectively describes quantum particles moving on 1D and 2D domains. Jensen and Koppe and Da Costa independently introduced a confining potential formalism showing that the effective constrained dynamics is subjected to a scalar geometry-induced potential; for the confinement to a curve, the potential depends on the curve's curvature function. METHOD: To characterize the π electrons in polyenes, we follow two approaches. First, we utilize a weakened Coulomb potential associated with a spiral curve. The solution to the Schrödinger equation with Dirichlet boundary conditions yields Bessel functions, and the spectrum is obtained analytically. We employ the particle-in-a-box model in the second approach, incorporating effective mass corrections. The π - π ∗ transitions of polyenes were calculated in good experimental agreement with both approaches, although with different wave functions.

A superposition free method for protein conformational ensemble analyses and local clustering based on a differential geometry representation of backbone.

Here a differential geometry (DG) representation of protein backbone is explored on the analyses of protein conformational ensembles. The protein backbone is described by curvature, κ, and torsion, τ, values per residue and we propose 1) a new dissimilarity and protein flexibility measurement and 2) a local conformational clustering method. The methods were applied to Ubiquitin and c-Myb-KIX protein conformational ensembles and results show that κ\τ metric space allows to properly judge protein flexibility by avoiding the superposition problem. The dmax measurement presents equally good or superior results when compared to RMSF, especially for the intrinsically unstructured protein. The clustering method is unique as it relates protein global to local dynamics by providing a global clustering solutions per residue. The methods proposed can be especially useful to the analyses of highly flexible proteins. The software written for the analyses presented here is available at https://github.com/AMarinhoSN/FleXgeo for academic usage only.

Swimming in Curved Surfaces and Gauss Curvature / Natación en superficies curvas y curvatura gaussiana / Natação em superfícies curvas e curvatura Gaussiana

Abstract The Cartesian-Newtonian paradigm of mechanics establishes that, within an inertial frame, a body either remains at rest or moves uniformly on a line, unless a force acts externally upon it. This crucial assertion breaks down when the classical concepts of space, time and measurement reveal to be inadequate. If, for example, the space is non-flat, an effective translation might occur from rest in the absence of external applied force. In this paper we examine mathematically the motion of a small object or lizard on an arbitrary curved surface. Particularly, we allow the lizard's shape to undergo a cyclic deformation due exclusively to internal forces, so that the total linear momentum is conserved. In addition to the fact that the deformation produces a swimming event, we prove -under fairly simplifying assumptions- that such a translation is somewhat directly proportional to the Gauss curvature of the surface at the point where the lizard lies.

Resumen El paradigma newtoniano de la mecánica establece que, en un sistema de referencia inercial, un cuerpo permanece en reposo o se mueve uniformemente en una recta, a menos que una fuerza externa actúe sobre él. Esta afirmación crucial falla cuando los conceptos clásicos de espacio, tiempo y medición son inadecuados. Si, por ejemplo, el espacio no es euclidiano, el cuerpo podría abandonar el reposo en ausencia de fuerza externa aplicada. En este artículo examinamos matemáticamente el movimiento de un pequeño objeto o lagartija en una superficie curva cualquiera. En particular, permitimos que la forma del lagarto sufra una deformación cíclica debida exclusivamente a fuerzas internas, de modo que la cantidad de movimiento lineal se conserva. Además del fenómeno de traslación o natación, probamos -bajo ciertas suposiciones simplificadoras- que dicha traslación es directamente proporcional a la curvatura gaussiana de la superficie en el punto donde yace la lagartija.

Resumo O paradigma Newtoniano da mecânica prevé que, em um referencial inercial (Galileano), um corpo está parado ou se movimentando em linha reta e com velocidade constante, a menos que uma força externa atue sobre ele. Essa declaração crucial falha quando os conceitos clássicos de espaço, tempo e medição são inadequados. Se, por exemplo, o espaço não é euclidiano, o corpo pode sair do repouso sem ser impelido por uma força externa. Neste artigo, examinamos matematicamente o movimento de um pequeno objeto (lagartixa) em qualquer superficie curva. Em particular, permitimos que a forma da lagartixa sofra uma deformação causada exclusivamente por forças internas, de modo que o momento linear seja conservado. Além do fenômeno da translação ou da natação, provamos -sob certos pressupostos simplificadores- que a translação efetiva é diretamente proporcional à curvatura Gaussiana da superfície.