Geometry Dynamics Research Articles

PENGGUNAAN TEKNOLOGI 3 DIMENSI SEBAGAI METODE PEMBELAJARAN GEOMETRI PADA ANAK USIA 5 - 6 TAHUN

Geometry is an important field for life and need to be teaching as early as possible so that children’s understanding ability about space and shape can get better. Old learning geometry methode only using paper and pencil while now as techonologies growing up so fast, teacher and school start using technology as a media learning for geometry. By using technology, such as; virtual reality, dynamics geometry software, etc, children can more develop and explore their geometric ability and get to know more about geometry. Plus, children around age 5- years old usually learning something through concrete so that’s why we need techonology as a media to make geometric learning more realistic so that children can develop their cognitive ability, especially in geometry field. This is qualitative study and using literature study as a collecting data methode. So, the conclusion is that by using technology as a geometry learning media is really help for children aged 5-6 years old to get to more understand about geometry.

Annular Dynamics and Leaflet Geometry in Patch Reconstruction of the Posterior Mitral Leaflet After Adding a Flexible Annuloplasty Ring.

Patch reconstruction of the posterior mitral leaflet using small intestinal submucosa extracellular matrix has been successfully performed in a porcine study. The patch reconstruction, however, resulted in non-physiological systolic widening of the mitral annulus, suggesting the need for an annuloplasty ring. The objective was to characterize the impact on annular dynamics and leaflet geometry of adding a flexible annuloplasty ring to the posterior mitral leaflet patch reconstruction. Measurements were performed in an acute 80-kg porcine model, with seven pigs acting as their own controls. The posterior mitral leaflet was reconstructed with a 2-ply small intestinal submucosa extracellular matrix patch (CorMatrix®). Additionally, a Simulus® Flexible Annuloplasty Ring (Medtronic Inc., Minneapolis, MN, USA) was inserted. Mitral annular dynamics were evaluated using sonomicrometry, and leaflet geometry was described using echocardiography. The annuloplasty ring reduced mitral annular dimensions and restricted cyclic changes in mitral annular area (126 ± 19 vs. 30 ± 13mm2, p < 0.001), septal-lateral and commisure-commisure distances. Ring annuloplasty prevented systolic widening in the mitral annulus after posterior mitral leaflet reconstruction. The annular saddle shape and leaflet coaptation length (8.7 ± 2.3 vs. 9.7 ± 1.3mm, p = 0.221) were comparable before and after ring insertion. The flexible annuloplasty ring resulted in a downsized annulus with restriction of cyclic annular changes in the reconstructed mitral valve. Ring insertion preserved the annular saddle shape and coaptation length. The ring annuloplasty counteracted the non-physiological annular dynamics, and this may improve durability of the posterior mitral leaflet patch reconstruction.

Hopfield Neural Network Flow: A Geometric Viewpoint.

We provide gradient flow interpretations for the continuous-time continuous-state Hopfield neural network (HNN). The ordinary and stochastic differential equations associated with the HNN were introduced in the literature as analog optimizers and were reported to exhibit good performance in numerical experiments. In this work, we point out that the deterministic HNN can be transcribed into Amari's natural gradient descent, and thereby uncover the explicit relation between the underlying Riemannian metric and the activation functions. By exploiting an equivalence between the natural gradient descent and the mirror descent, we show how the choice of activation function governs the geometry of the HNN dynamics. For the stochastic HNN, we show that the so-called "diffusion machine," while not a gradient flow itself, induces a gradient flow when lifted in the space of probability measures. We characterize this infinite-dimensional flow as the gradient descent of certain free energy with respect to a Wasserstein metric that depends on the geodesic distance on the ground manifold. Furthermore, we demonstrate how this gradient flow interpretation can be used for fast computation via recently developed proximal algorithms.

Quantitating Interfraction Target Dynamics During Concurrent Chemoradiation for Glioblastoma: A Prospective Serial Imaging Study

Magnetic resonance image (MRI) guided radiation therapy has the potential to improve outcomes for glioblastoma by adapting to tumor changes during radiation therapy. This study quantifies interfraction dynamics (tumor size, position, and geometry) based on sequential magnetic resonance imaging scans obtained during standard 6-week chemoradiation. Sixty-one patients were prospectively imaged with gadolinium-enhanced T1 (T1c) and T2/FLAIR axial sequences at planning (Fx0), fraction 10 (Fx10), fraction 20 (Fx20), and 1 month after the final fraction of chemoradiation therapy (P1M). Gross tumor volumes (GTVs) and clinical target volumes (CTVs) were contoured at all time points. Target dynamics were quantified by absolute volume (V), volume relative to Fx0 (Vrel), and the migration distance (dmigrate; the linear displacement of the GTV or CTV relative to Fx0). Temporal changes were assessed using a linear mixed-effects model. Median volumes at Fx0, Fx10, Fx20, and P1M for the GTV were 18.4 cm3 (range, 1.1-110.5 cm3), 14.7 cm3 (range, 0.9-115.1 cm3), 13.7 cm3 (range, 0.6-174.2 cm3), and 13.0 cm3 (range, 0.9-76.3 cm3), respectively, with corresponding median Vrel of 0.88 at Fx10, 0.77 at Fx20, and 0.71 at P1M relative to Fx0 (P < .001 for all). The GTV (CTV) migration distances were greater than 5 mm in 46% (54%) of patients at Fx10, 50% (58%) of patients at Fx20, and 52% (57%) of patients at P1M. Dynamic tumor morphologic changes were observed, with 40% of patients exhibiting a decreased GTV (Vrel ≤1) with a dmigrate >5 mm during chemoradiation therapy. Clinically meaningful tumor dynamics were observed during chemoradiation therapy for glioblastoma, supporting evaluation of daily MRI guided radiation therapy and treatment plan adaptation.

Using Dynamical Geometry Softwares in the study of Plane Geometry: potentialities and limitations

Background: Some inconsistencies that appeared when teaching Geometry - using Dynamical Geometry Softwares - to Mathematics undergraduate students, inspired this work . Objectives: To stress the potentialities and, specially, the limitations of Dynamical Geometry Softwares in order of using it correctly by teachers, pre-service teachers and students for learning and teaching situations and investigations in Geometry. Design: Critical analysis study of situations, with examples produced using Dynamical Geometry Softwares, in order to reveal some inconsistencies with respect to the theory. In particular, Geogebra was the software used. Setting and participants: Some of the examples presented here, elaborated in GeoGebra, were briefly discussed in classes of pre-service teachers of mathematics aiming at an awareness of the inconsistencies that may appear using a Dynamical Geometry Software. The authors are the unique participants of the elaboration of those examples. Data collection and analysis: There was not data collection, but only elaboration of examples in order to provide some arguments for future discussions. Results: Examples production shows some limitations of Dynamical Geometry Softwares and that those limitations are insurmountable due to epistemological reasons. Conclusions: Awareness of Dynamical Geometry Software limitations is fundamental for its correct use. Those limitations do not invalidate the software potential. On the contrary, being conscious of both potentialities and limitations of a hardware is a necessary condition to a fruitful use of it.

Inclusion complexes of tricyclic drugs and β-cyclodextrin: Inherent chirality and dynamic behaviour.

Amitriptyline (AMT) and cyclobenzaprine (CBZ) are tricyclic drugs used as antidepressant and muscle relaxant, respectively. They show inherently chirality, i.e. they are chiral due to the lack of any symmetry element. As they are used as racemic mixture, diastereomeric inclusion complexes are formed via encapsulation in homochiral βCD. In this work we show that a suitable combination of NMR methods easily provides details on the chiral recognition, geometry of complexation, rotational dynamics and spatial proximity of selected atom pairs. In particular, we show that 13C NMR can be used to unambiguously assess chiral recognition, demonstrating a higher performance over 1H NMR. The mole fraction of the bound drug and the association constant can be worked out through diffusion experiments, whereas the combination of non-selective, selective and bi-selective relaxation spectra gave insights into the rotational motion of the complexed drug and the spatial proximity of selected proton pairs. The toolkit here proposed provides a thorough characterization of CD/drug inclusion complexes from a physicochemical point of view. This can constructively complement the conventional pharmacological and pharmacokinetic experiments, and can shed light on the understanding of CD/drug formulations.

Fluctuations of the magnetization for Ising models on Erdős–Rényi random graphs—the regimes of small p and the critical temperature * * The research of the first two authors was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: dynamics–geometry–structure. The third author has been supported by the German Research Foundation (DFG) via

We continue our analysis of Ising models on the (directed) Erdős–Rényi random graph. This graph is constructed on N vertices and every edge has probability p to be present. These models were introduced and first studied by Bovier and Gayrard (1993 J. Stat. Phys. 72 643–64) and further analyzed by the authors in a previous note, in which we consider the case of p = p(N) satisfying p 3 N 2 → ∞ and β < 1. In the current note we prove a quenched central limit theorem for the magnetization for p satisfying pN → ∞ in the high-temperature regime β < 1. We also show a non-standard central limit theorem for p 4 N 3 → ∞ at the critical temperature β = 1. For p 4 N 3 → 0 we obtain a Gaussian limiting distribution for the magnetization. Finally, in the critical regime p 4 N 3 → c the limiting distribution for the magnetization contains a Gaussian component as well as a -term. Hence, at β = 1 we observe a phase transition in p for the fluctuations of the magnetization.

An approximation strategy to compute accurate initial density matrices for repeated self-consistent field calculations at different geometries

ABSTRACT Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimisations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, little is known on how to efficiently exploit the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix, expanded in a set of localised basis functions, within the self-consistent field iterations for parametrised Hartree–Fock problems where the nuclear coordinates are changed along with a few user-specified collective variables, such as the molecule's normal modes. Our approach is based on an offline-stage where the Hartree–Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value. The method allows nonlinear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold, by mapping such a manifold onto the tangent space. Numerical tests on different amino acids show promising initial results.

Efficient Treatment of Large Active Spaces through Multi-GPU Parallel Implementation of Direct Configuration Interaction.

We have extended our graphical processing unit (GPU)-accelerated direct configuration interaction program to multiple devices, reducing iteration times for configuration spaces of 165 million determinants to only 3 s using NVIDIA P100 GPUs. Similar improvements in the one- and two-particle reduced density matrix formation allow for fast analytical energy gradients and electronic properties. Our parallel algorithm enables the calculation of arbitrarily large configuration spaces (limited only by available system memory), with iteration times of 13 min for an active space of 18 electrons in 18 orbitals (2.4 billion determinants) using six consumer grade NVIDIA 1080Ti GPUs. These advances enable routine molecular dynamics simulations, geometry optimizations, and absorption spectrum calculations for molecules with large configuration spaces, a task that has heretofore required massive computational effort. In this work, we demonstrate the utility of our program by generating the absorption spectrum for diphenyl acetylene at the floating occupation molecular orbital complete active space configuration interaction level of theory. Several active spaces were investigated to assess the dependence of spectral features on orbital space dimension.

Exploring the Isoptics of Fermat Curves in the Affine Plane Using DGS and CAS

Isoptic curves of plane curves are a live domain of study, mostly for closed, smooth, strictly convex curves. A technology-rich environment allows for a two-fold development: dynamical geometry systems enable us to perform experiments and to derive conjectures, and computer algebra systems (CAS) are the appropriate environments for an algebraic approach for determining isoptics, with its Grobner bases packages, amongst others. Closed Fermat curves in the affine plane present a specific problem: the variables of the involved polynomials represent the coordinates of points on the Fermat curve, which avoid the dense set of points with two rational coordinates (excepted 4 “trivial” points). Therefore, automated computation has to be performed over the field $$\mathbb {R}$$ of the real numbers, with which the Grobner bases packages do not work. Instead, other packages implemented in CASs have to be used. First, we present a study of the orthoptics of closed Fermat curves of even order. Then we proceed to an algebraic study of these curves using a CAS. The generalization to angles other than $$90^\circ $$ is performed afterwards, with an algebraic approach using support functions, then using numerical methods.

Diffuse to fuse EEG spectra – Intrinsic geometry of sleep dynamics for classification

We propose a novel algorithm for sleep dynamics visualization and automatic annotation by applying diffusion geometry based sensor fusion algorithm to fuse spectral information from two electroencephalograms (EEG). The diffusion geometry approach helps organize the nonlinear dynamical structure hidden in the EEG signal. The visualization is achieved by the nonlinear dimension reduction capability of the chosen diffusion geometry algorithms. For the automatic annotation purpose, the support vector machine is trained to predict the sleep stage. The prediction performance is validated on a publicly available benchmark database, Physionet Sleep-EDF [extended] SC* (SC = Sleep Cassette) and ST* (ST = Sleep Telemetry), with the leave-one-subject-out cross validation. When we have a single EEG channel (Fpz-Cz), the overall accuracy, macro F1 and Cohen's kappa achieve 82.72%, 75.91% and 76.1% respectively in Sleep-EDF SC* and 78.63%, 73.58% and 69.48% in Sleep-EDF ST*. This performance is compatible with the state-of-the-art results. When we have two EEG channels (Fpz-Cz and Pz-Oz), the overall accuracy, macro F1 and Cohen's kappa achieve 84.44%, 78.25% and 78.36% respectively in Sleep-EDF SC* and 79.05%, 74.73% and 70.31% in Sleep-EDF ST*. The results suggest the potential of the proposed algorithm in practical applications.

Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Escape from a potential well can occur in different physical systems, such as capsize of ships, resonance transitions in celestial mechanics, and dynamic snap-through of arches and shells, as well as molecular reconfigurations in chemical reactions. The criteria and routes of escape in one-degree of freedom systems have been well studied theoretically with reasonable agreement with experiment. The trajectory can only transit from the hilltop of the one-dimensional potential energy surface. The situation becomes more complicated when the system has higher degrees of freedom since the system state has multiple routes to escape through an equilibrium of saddle-type, specifically, an index-1 saddle. This paper summarizes the geometry of escape across a saddle in some widely known physical systems with two degrees of freedom and establishes the criteria of escape providing both a methodology and results under the conceptual framework known as tube dynamics. These problems are classified into two categories based on whether the saddle projection and focus projection in the symplectic eigenspace are coupled or uncoupled when damping and/or gyroscopic effects are considered. For simplicity, only the linearized system around the saddle points is analyzed, but the results generalize to the nonlinear system. We define a transition region, Th, as the region of initial conditions of a given initial energy h which transit from one side of a saddle to the other. We find that in conservative systems, the boundary of the transition region, ∂Th, is a cylinder, while in dissipative systems, ∂Th is an ellipsoid.

Geometrical features of a Jewish monument: study with a DGS

We study a couple of geometrical elements of a monumental synagogue (Dohany Synagogue in Budapest). Then we describe how undergraduate students from the Jewish orthodox population have performed a technology-based study, using a Dynamical Geometry System, notions, and tools of analytic geometry. These students come without a classical background in mathematics, and such activities relying on artefacts form their everyday life help to attract them to mathematics learning.

An automated study of isoptic curves of an astroid

Learning mathematics in a technology rich environment enables to revive classical topics which have been removed from the curriculum a long time ago. Theoretical issues and their applications can be studied within an experimental process, using automated proofs. We present a technology based study of isoptic curves of an astroid, which is a non smooth and non convex curve. We take advantage of both algebraic symbolic features, such as resultants and Gröbner bases packages implemented in a Computer Algebra System (CAS), together with a Dynamical Geometry System (DGS). The usage of algebraic methods is made possible by the translation of the trigonometric parametrization into rational expressions, and then into polynomial equations. The dynamics provided by the DGS enable the discovery of special features of the isoptics, in particular the fact that the isoptic has singular points and, according to the viewing angle, the isoptic may be inscribed in the astroid or not.

Mutual Correlation in the Shock Wave Geometry

We probe the shock wave geometry with the mutual correlation in a spherically symmetric Reissner-Nordström AdS black hole on the basis of the gauge/gravity duality. In the static background, we find that the regions living on the boundary of the AdS black holes are correlated provided the considered regions on the boundary are large enough. We also investigate the effect of the charge on the mutual correlation and find that the bigger the value of the charge is, the smaller the value of the mutual correlation will be. As a small perturbation is added at the AdS boundary, the horizon shifts and a dynamical shock wave geometry form after long time enough. In this dynamic background, we find that the greater the shift of the horizon is, the smaller the mutual correlation will be. Especially for the case that the shift is large enough, the mutual correlation vanishes, which implies that the considered regions on the boundary are uncorrelated. The effect of the charge on the mutual correlation in this dynamic background is found to be the same as that in the static background.

Tuning STEM: Tailoring the Incident Probe, Scattering Dynamics and Detector Geometry for Maximum Specimen Information

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An application-based characterization of dynamical distance geometry problems

The dynamical Distance Geometry Problem (dynDGP) is the problem of finding a realization in a Euclidean space of a weighted undirected graph G representing an animation by relative distances, so that the distances between realized vertices are as close as possible to the edge weights. In the dynDGP, the vertex set of the graph G is the set product of V , representing certain objects, and T , representing time as a sequence of discrete steps. We suppose moreover that distance information is given together with the priority of every distance value. The dynDGP is a special class of the DGP where the dynamics of the problem comes to play an important role. In this work, we propose an application-based characterization of dynDGP instances, where the main criteria are the presence or absence of a skeletal structure, and the rigidity of such a skeletal structure. Examples of considered applications include: multi-robot coordination, crowd simulations, and human motion retargeting.

Automated Study of a Regular Trifolium

We present a study of a classical plane real curve in a technology-rich environment. The interplay between implicit and parametric presentations is enhanced. Cooperation of a computer algebra system and of a dynamical geometry system enables to discover properties that may be hidden in hand-made computations. Moreover, technology does not act only as a solver, but the outputs reveal a need to expand the framework from geometry over the reals to some usage of complex variables. The paper is a case study of an undergraduate approach, that is, its main audience is educators at the university level.

Experimental and dynamics simulation studies of the molecular modeling and reactivity of the Yaojie oil shale kerogen

In this study, the chemical structural parameters of a kerogen from Yaojie oil shale were identified and investigated by using experimental data obtained from solid-state 13C nuclear magnetic resonance spectroscopy, X-ray photoelectron spectroscopy, X-ray diffraction, flash pyrolysis–gas chromatography–mass spectrometry, and Fourier transform infrared spectroscopy. Twelve constitutional isomers of the Yaojie 2D models with different grids and their corresponding 3D models were constructed by the combination of molecular simulation methods and multiple testing techniques. Finally, a reasonable 3D model of Yaojie kerogen was determined by a series of anneal dynamics simulations and geometry optimization calculations. From the results obtained by Mulliken population analysis and electron density analysis, the π–π conjugation effect among the aromatic ring structures is significantly impacted by the heteroatoms, leading to the accumulation of electrons on heteroatoms. Meanwhile, sulfur, oxygen, and nitrogen in the heterocyclic structures exhibited higher electronic charge density and considerably more reactivity compared with those of other atoms. Moreover, the molecular orbitals of HOMO and LUMO were calculated to examine the properties of chemical reaction of kerogen. The HOMO–LUMO energy gap indicated that the Yaojie kerogen exhibited higher reactivities compared to Type Ⅰ kerogen.

Geometric generalised Lagrangian-mean theories

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian-mean (GLM) theory of Andrews &amp; McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) This finite-amplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin’s circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold’s metric. The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the ‘glm’ definition proposed by Soward &amp; Roberts (J. Fluid Mech., vol. 661, 2010, pp. 45–72). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean velocity and Lagrangian-mean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating–stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating–stratified fluids.