Abstract
Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part ...and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is ±1/3, while the index of the line field is ±1. Moreover, for an elliptic quadratic point whose cubic form is semi-homogeneous, we can use Loewner’s conjecture to show that the index is at most 1. From the above local results, we can conclude some global results: A generic compact elliptic surface has at least 6 quadratic points, a compact elliptic surface with semi-homogeneous cubic forms has at least 2 quadratic points and the number of quadratic points in a hyperbolic disc is odd. By studying the behavior of the cubic form in a neighborhood of the parabolic curve, we also obtain a relation between the indices of the quadratic points of a generic surface with non-empty elliptic and hyperbolic regions.
•This paper introduces the Distributed-Delay Connectome Tensor Neural Mass Model.•The DD-NMM Toolbox allows modeling networks with arbitrary connectivities and realistic transmission delays.•Compared ...to more traditional methods, an order-of-magnitude reduction in the computation time is possible.•Feasible explorations of extensive networks are possible to investigate physiological phenomena such as the splitting alpha peak of the EEG spectrum.•Our software can be used stand-alone or integrated into other platforms such as The Virtual Brain (TVB).
This paper introduces methods and a novel toolbox that efficiently integrates high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations (RDEs) of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. To date, simplistic assumptions prevail about delays in the CT, often assumed to be Dirac-delta functions. In reality, delays are distributed due to heterogeneous conduction velocities of the axons connecting neural masses. These distributed-delay CTs are challenging to model. Our approach implements these models by leveraging several innovations. Semi-analytical integration of RDEs is done with the Local Linearization (LL) scheme for each neural mass, ensuring dynamical fidelity to the original continuous-time nonlinear dynamic. This semi-analytic LL integration is highly computationally-efficient. In addition, a tensor representation of the CT facilitates parallel computation. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism. This ease of implementation includes models with distributed-delay CTs. Consequently, our algorithm scales linearly with the number of neural masses and the number of equations they are represented with, contrasting with more traditional methods that scale quadratically at best. To illustrate the toolbox's usefulness, we simulate a single Zetterberg-Jansen and Rit (ZJR) cortical column, a single thalmo-cortical unit, and a toy example comprising 1000 interconnected ZJR columns. These simulations demonstrate the consequences of modifying the CT, especially by introducing distributed delays. The examples illustrate the complexity of explaining EEG oscillations, e.g., split alpha peaks, since they only appear for distinct neural masses. We provide an open-source Script for the toolbox.
This article describes the study and digital implementation of a system onboard a TMS 3208F28335 ® DSP for vector control of the bearing motor speed with four poles split winding with 250 W of power. ...Smart techniques: ANFIS and Neural Networks were investigated and computationally implemented to evaluate the bearing motor performance under the following conditions: operating as an estimator of uncertain parameters and as a speed controller. Therefore, the MATLAB program and its toolbox were used for the simulations and the parameter adjustments involving the structure ANFIS (Adaptive-Network-Based Fuzzy Inference System) and simulations with the Neural Network. The simulated results showed a good performance for the two techniques applied differently: the estimator and a speed controller using both a model of the induction motor operating as a bearing motor. The experimental part for velocity vector control uses three control loops: current, radial position, and speed, where the configurations of the peripherals, that is, the interfaces or drivers for driving the bearing motor.
Originality/value--Unlike in developed countries, the Brazilian stock market, young and not very representative of the economy, was not able to anticipate changes in the banks' rating. This study ...anticipates information to investors who aid in the decision to buy, hold or sell securities, and signals that the financial system is more susceptible to macroeconomic shocks in unstable economies.
The centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses. In this article, we obtain the canonical equations of these ...ellipses. Moreover, we complement the previous results about incenters obtained by Romaskevich. Also the geometric locus defined by barycenters of the triangles of an elliptic billiard are ellipses as established by Schwartz and Tabachnikov and explicit equations of these ellipses are also given. In addition, we obtain that the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse and the canonical equation is obtained.
Darboux curves on surfaces I GARCIA, Ronaldo; LANGEVIN, Rémi; WALCZAK, Paweł
Journal of the Mathematical Society of Japan,
1/2017, Letnik:
69, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Mobius group and share many properties with geodesics. Here, we characterize ...these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable when the surface is a special canal.
In H. Brézis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73–97. Brézis and Friedman prove that certain nonlinear parabolic ...equations, with the
δ
-measure as initial data, have no solution. However in J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186–196. Colombeau and Langlais prove that these equations have a unique solution even if the
δ
-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais’ result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371–399..
•We create lifespan Riemannian qEEG norms for cross-spectral tensors.•EEG from 1564 subjects provided by 9 countries, 12 devices, and 14 studies were used.•We demonstrate qEEG “batch effects”, ...providing harmonization methods to remove them.•Multinational harmonized z-scores increase diagnostic accuracy of brain dysfunction.•Data and software are available for norm and individual z-scores calculation.
This paper extends frequency domain quantitative electroencephalography (qEEG) methods pursuing higher sensitivity to detect Brain Developmental Disorders. Prior qEEG work lacked integration of cross-spectral information omitting important functional connectivity descriptors. Lack of geographical diversity precluded accounting for site-specific variance, increasing qEEG nuisance variance. We ameliorate these weaknesses. (i) Create lifespan Riemannian multinational qEEG norms for cross-spectral tensors. These norms result from the HarMNqEEG project fostered by the Global Brain Consortium. We calculate the norms with data from 9 countries, 12 devices, and 14 studies, including 1564 subjects. Instead of raw data, only anonymized metadata and EEG cross-spectral tensors were shared. After visual and automatic quality control, developmental equations for the mean and standard deviation of qEEG traditional and Riemannian DPs were calculated using additive mixed-effects models. We demonstrate qEEG “batch effects” and provide methods to calculate harmonized z-scores. (ii) We also show that harmonized Riemannian norms produce z-scores with increased diagnostic accuracy predicting brain dysfunction produced by malnutrition in the first year of life and detecting COVID induced brain dysfunction. (iii) We offer open code and data to calculate different individual z-scores from the HarMNqEEG dataset. These results contribute to developing bias-free, low-cost neuroimaging technologies applicable in various health settings.
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