Geometric Explanation Research Articles

On the integration of transitive Lie algebroids

We revisit the problem of integrating Lie algebroids A\Rightarrow M to Lie groupoids G\rightrightarrows M , for the special case that the Lie algebroid A is transitive . We obtain a geometric explanation of the Crainic–Fernandes obstructions for this situation, and an explicit construction of the integration whenever these obstructions vanish. We also indicate an extension of this approach to regular Lie algebroids.

Determination of electrons location using mirror effect phenomena in scanning electron microscope

In sense of the phenomena of mirror effects, the behavior of electrons inside the chamber of scanning electron microscope (SEM) investigated. Indeed, a simplified geometrical explanation for the behavior of incident electrons introduced. The presented description is mainly concerns with simple trigonometric functions. However, the synthesis of these functions provide a tool which can be used to trace electron as it leaves the column diaphragm until it reaches the detectors. Accordingly, the position of landing electrons throughout its travel being determinable in terms of the sample potential the operation variables. Results have shown that introduced approach could commendably use to simulate behavior of electrons inside the chamber of SEM.

On the Convergence of Stochastic Process Convergence Proofs

Convergence of a stochastic process is an intrinsic property quite relevant for its successful practical for example for the function optimization problem. Lyapunov functions are widely used as tools to prove convergence of optimization procedures. However, identifying a Lyapunov function for a specific stochastic process is a difficult and creative task. This work aims to provide a geometric explanation to convergence results and to state and identify conditions for the convergence of not exclusively optimization methods but any stochastic process. Basically, we relate the expected directions set of a stochastic process with the half-space of a conservative vector field, concepts defined along the text. After some reasonable conditions, it is possible to assure convergence when the expected direction resembles enough to some vector field. We translate two existent and useful convergence results into convergence of processes that resemble to particular conservative vector fields. This geometric point of view could make it easier to identify Lyapunov functions for new stochastic processes which we would like to prove its convergence.

Geometric Isomerism and DFT Theoretical Explanation of the Unexpected Formation of N, N-disubstituted Formamidines from 2-amino-3-cyano-4.6- diarylpyridines

Introduction:Unexpected series of N,N-disubstituted formamidines were obtained upon the reaction of 2-amino-3-cyano-4.6-diarylpyridines with triflouroroacetic anhydride (TFAA) in dimethyl formamide (DMF).Methods:The syn/anti ratio of N,N-disubstituted formamidines was calculated by applying the 1H-NMR additivy increment rule. The mechanism of formamidine formation is suggested using the experimental results, 1H NMR and 13C NMR spectroscopy and successfully computed with DFT calculations using the B3LYP functional and 6-311G (p,d) basis set.Results:The reaction mechanism classified as unexpected nucleophilic attack reaction of the solvent (DMF) on the carbonyl group leaving a positively charged intermediate which was attacked with nitrogen atom of the amine form N,N-disubstituted formamidines. Bond lengths and Mulliken atomic charges were calculated to support the reliability of the predicted reaction mechanism. The activation energy of each species appears in the reaction pathway was calculated.Conclusion:HOMO-LUMO analysis showed that the reaction pathway is preferred to proceed through the suggested mechanism. It is clear that DMF is not the appropriate solvent in the case of preparation of triflouroacetamide.

Switchbacks Explained: Super-Parker Fields—The Other Side of the Sub-Parker Spiral

Abstract We provide a simple geometric explanation for the source of switchbacks and associated large and one-sided transverse flows in the solar wind observed by the Parker Solar Probe (PSP). The more radial, sub-Parker spiral structure of the heliospheric magnetic field observed previously by Ulysses, ACE, and STEREO is created within rarefaction regions where footpoint motion from the source of fast into slow wind at the Sun creates a magnetic fieldline connection across solar wind speed shear. Conversely, when footpoints move from the source of slow wind into faster wind, a super-Parker spiral field structure is formed: below the Alfvén critical point, one-sided transverse field-aligned flows develop; above the Alfvén critical point, the field structure contracts between adjacent solar wind flows, and the radial field component decreases in magnitude with distance from the Sun, eventually reversing into a switchback. The sub-Parker and super-Parker spirals behave functionally as opposites. Observations from PSP confirm the paucity of switchbacks within rarefaction regions and immediately outside these rarefaction regions, we observe numerous switchbacks in the magnetic field that are directly associated with abrupt transients in solar wind speed. The magnetic field strength, the radial component of the magnetic field, the speed gradients, radial Alfvén speed, and the ratio of the sound speed to the radial Alfvén speed all conform to predictions based on the sub-Parker and super-Parker spirals within rarefaction regions and solar wind speed enhancements (spikes or jets), respectively. Critically, the predictions associated with the super-Parker spiral naturally explain the observations of switchbacks being associated with unexpectedly large and one-sided tangential flows.

Fano Varieties of K3-Type and IHS Manifolds

AbstractWe construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure, and we link some of them to the projective families of irreducible holomorphic symplectic manifolds.

Univoque graphs for non-integer base expansions

Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. We give a precise description of the set of sequences generated by these graphs. This provides a geometric explanation of many former abstract results in this domain. Our results are illustrated by many examples.

On the geometry and environment of repeating FRBs

ABSTRACT We propose a geometrical explanation for periodically and non-periodically repeating fast radio bursts (FRBs) under neutron star (NS)–companion systems. We suggest a constant critical binary separation, rc, within which the interaction between the NS and companion can trigger FRBs. For an elliptic orbit with the minimum and maximum binary separations, rmin and rmax, a periodically repeating FRB with an active period could be reproduced if rmin < rc < rmax. However, if rmax < rc, the modulation of orbital motion will not work due to persistent interaction, and this kind of repeating FRBs should be non-periodic. We test relevant NS–companion binary scenarios on the basis of FRB 180916.J0158+65 and FRB 121102 under this geometrical frame. It is found that the pulsar–asteroid belt impact model is more suitable to explain these two FRBs since this model is compatible with different companions (e.g. massive stars and black holes). At last, we point out that FRB 121102-like samples are potential objects that can reveal the evolution of star-forming region.

A noncommutative geometric LR rule

The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given by Vakil and later generalized by Coskun. In this paper we give a noncommutative version of the geometric LR rule. As a consequence, we establish a geometric explanation for the positivity of noncommutative LR coefficients in certain cases.

Gravitational decoupling in cosmology

Abstract Whereas the nature of dark components in the Universe remains unknown, alternative models of gravity have been developed to offer a geometric explanation to the origin of such components. In this work we use the Minimal Geometric Deformation approach to study extensions of the theory of General Relativity in a cosmological context. This is possible since such approach allows the decoupling of gravitational sources, and the Einstein field equations can be analytically solved with the presence of a new gravitational sector once a known GR solution is considered. In particular, we implement such approach in Friedmann–Robertson–Walker and Kantowski–Sachs universes. We demonstrate that the gravitational decoupling leads to modifications of well known cosmological solutions. For instance, we show that an effective spatial curvature in the Friedmann–Robertson–Walker metric, as well as several kind of matter components in the Kantowski–Sachs case, are obtained. Thus, we found that it is possible to obtain spatial curvature and new matter terms from geometry, which in cosmology they could be useful in addressing problems such as the spatial flatness of the Universe, dark matter and dark energy.

The Geometrical Demonstration of the Order of Resonant Saddle Points in $${\mathbb {C}}^2$$

Here we investigate the Minkowski box dimension of complex integral curves of the vector fields near resonant saddles in $${\mathbb {C}}^2$$. The results provide the geometrical explanation of the order of the saddle points and a quantitative description for the non-integrability via monodromy.

Equivalent permeability of shale rocks: Simple and accurate empirical coupling of organic and inorganic matter

Coupling the organic/inorganic permeabilities in shale is fundamental. Simple treatments, including the arithmetic/geometric average methods, are problematic and can lead to large errors. More complicated methods can generate reasonable results but are much more expensive. In this work, a simple and accurate empirical model for the equivalent permeability of shale considering permeability contrast between organic and inorganic matter and TOC is established. Geometric explanation of the parameters in the model is also provided. Afterwards, model validation against numerical data demonstrates the accuracy of the model in all realistic scenarios. Moreover, model comparison with arithmetic/geometric average methods shows the superiority of the model in accurate calculation while keeping the form simple. Finally, the model is applied to liquid flow in shale, and the results are analyzed. The model is highly recommended for theoretical studies in shale gas/oil that need coupling of the organic and inorganic permeabilities, including single-phase and multi-phase flows.

Texture evolution of commercially pure copper processed by multiple compressions in a channel die

Texture evolution of copper, subjected to multiple compressions in a channel die was studied and interpreted using a simple pure geometric explanation. In this geometric interpretation, deformation in each pass of multiple compressions in a channel die was considered to be analogous to that of flat rolling. After each pass of the compressions, the sample is rotated around two mutually perpendicular directions. The locations of the stable texture components of the rolling were determined with respect to the new sample frame after each pass. In this approach, if the previous stable texture component locates in the stable position in the current pressing pass, this position is considered as the stable texture component position; otherwise, these components rotate into more stable ones. In addition, texture evolution for even passes were predicted via the modified Taylor model which correlate with that estimated by the proposed pure geometric texture evolution explanation and show good agreement with experiments.

Why 3D fragmentation usually leads to cuboids: a simple geometric explanation

Why 3D fragmentation usually leads to cuboids: a simple geometric explanation

Double Additive Margin Softmax Loss for Face Recognition

Learning large-margin face features whose intra-class variance is small and inter-class diversity is one of important challenges in feature learning applying Deep Convolutional Neural Networks (DCNNs) for face recognition. Recently, an appealing line of research is to incorporate an angular margin in the original softmax loss functions for obtaining discriminative deep features during the training of DCNNs. In this paper we propose a novel loss function, termed as double additive margin Softmax loss (DAM-Softmax). The presented loss has a clearer geometrical explanation and can obtain highly discriminative features for face recognition. Extensive experimental evaluation of several recent state-of-the-art softmax loss functions are conducted on the relevant face recognition benchmarks, CASIA-Webface, LFW, CALFW, CPLFW, and CFP-FP. We show that the proposed loss function consistently outperforms the state-of-the-art.

Porous structure of cylindrical packed beds with aspect ratios between 1 and 2

The paper provides a comprehensive summary of the analysis of the porous structure of cylindrical packed beds with aspect ratios between 1 and 2. For the first time a proper geometrical explanation is given for the shape of the curve for the overall porosity as a function of the aspect ratio. Errors found in the published equations for the lateral coordinates of the spheres in packed beds with aspect ratios between 1.866 and 2.0 are corrected. A geometrical explanation for the presence and the behaviour of the inflection point (“porosity surge wave”) observed in the curves for the radial variation in the porosity is also given for the first time. A systematic analysis of the axial variation in the porosity is also presented for the first time and based on the geometry analytical equations are derived for the minimum and maximum axial porosity as a function of aspect ratio. The results can serve as a benchmark for experimental and numerical studies on the porous structure of small aspect ratio cylindrical packed beds.

Underlying structure in the dynamics of chase and escape interactions

Chase and escape behaviors are important skills in many sports. Previous studies have described the behaviors of the attacker (escaper) and defender (chaser) by focusing on their positional relationship and have presented several key parameters that affect the outcome (successful attack or defense). However, it remains unclear how each individual agent moves, and how the outcome is determined in this type of interaction. To address these questions, we constructed a chase and escape task in a virtual space that allowed us to manipulate agents’ kinematic parameters. We identified the basic strategies of each agent and their robustness to changes in their parameters. Moreover, we identified the determinants of the outcome and a geometrical explanation of their importance. Our results revealed the underlying structure of a simplified human chase and escape interaction and provided the insight that, although each agent apparently moves freely, their strategies in two-agent interactions are in fact rather constrained.

Hierarchical frequency clusters in adaptive networks of phase oscillators.

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency multiclusters by investigating the effect of the time scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multiclusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.

Asymmetrical voltage response in resonant neurons shaped by nonlinearities.

The conventional impedance profile of a neuron can identify the presence of resonance and other properties of the neuronal response to oscillatory inputs, such as nonlinear response amplifications, but it cannot distinguish other nonlinear properties such as asymmetries in the shape of the voltage response envelope. Experimental observations have shown that the response of neurons to oscillatory inputs preferentially enhances either the upper or lower part of the voltage envelope in different frequency bands. These asymmetric voltage responses arise in a neuron model when it is submitted to high enough amplitude oscillatory currents of variable frequencies. We show how the nonlinearities associated to different ionic currents or present in the model as captured by its voltage equation lead to asymmetrical response and how high amplitude oscillatory currents emphasize this response. We propose a geometrical explanation for the phenomenon where asymmetries result not only from nonlinearities in their activation curves but also from nonlinearites captured by the nullclines in the phase-plane diagram and from the system's time-scale separation. In addition, we identify an unexpected frequency-dependent pattern which develops in the gating variables of these currents and is a product of strong nonlinearities in the system as we show by controlling such behavior by manipulating the activation curve parameters. The results reported in this paper shed light on the ionic mechanisms by which brain embedded neurons process oscillatory information.

Identities on factorial Grothendieck polynomials

Gustafson and Milne proved an identity which can be used to express a Schur function sμ(x1,x2,…,xn) with μ=(μ1,μ2,…,μk) in terms of the Schur function sλ(x1,x2,…,xn), where λ=(λ1,λ2,…,λk) is a partition such that λi=μi+n−k for 1≤i≤k. On the other hand, Fehér, Némethi and Rimányi found an identity which relates sμ(x1,x2,…,xn) to the Schur function sλ(x1,x2,…,xℓ), where λ=(λ1,λ2,…,λℓ) is a partition obtained from μ by removing some of the largest parts of μ. Fehér, Némethi and Rimányi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Fehér-Némethi-Rimányi identity.