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2. Geometric Syncronisation by Multi-pendulum and Electronic Models of Neurodynamics
- Author
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Resconi, Germano, Kozma, Robert, Kacprzyk, Janusz, Series editor, Madani, Kurosh, editor, Correia, António Dourado, editor, Rosa, Agostinho, editor, and Filipe, Joaquim, editor
- Published
- 2015
- Full Text
- View/download PDF
3. Proposal on a Riemannian Approach to Path Modeling in a Navigational Decision Support System
- Author
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Kopacz, Piotr and Mikulski, Jerzy, editor
- Published
- 2013
- Full Text
- View/download PDF
4. On Variability and Interdependence of Local Porosity and Local Tortuosity in Porous Materials: a Case Study for Sack Paper
- Author
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Matthias Neumann, Eduardo Machado Charry, Volker Schmidt, and Karin Zojer
- Subjects
Statistics and Probability ,Geodesic ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Copula (linguistics) ,Sinuosity ,01 natural sciences ,Tortuosity ,010104 statistics & probability ,Goodness of fit ,Gumbel distribution ,Joint probability distribution ,0101 mathematics ,Porosity ,Mathematics - Abstract
The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.
- Published
- 2020
5. Statistical Information Theory and Geometry for SAR Image Analysis : Invited Paper
- Author
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Alejandro C. Frery
- Subjects
Synthetic aperture radar ,Geodesic ,Computer science ,business.industry ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Image processing ,Disjoint sets ,Information theory ,Image (mathematics) ,Computer vision ,Information geometry ,Artificial intelligence ,Road map ,business - Abstract
This paper is intended as a road map for the literature on Synthetic Aperture Radar (SAR) image processing and analysis with Information-Theoretic and Information-Geometric Statistical tools. We do this by commenting upon seemingly disjoint problems, and showing that they can be tackled successfully within the same framework: tools that arise from using Information Theory (entropies and divergences) and Information Geometry (geodesic distances) enhanced with Statistics.
- Published
- 2019
6. The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 2. The weak field approximation in the 3-orthogonal gauges and Hamiltonian post-minkowskian gravity: the N-body problem and gravitational waves with asymptotic background 1This paper is one of three companion papers published in the same issue of Can. J. Phys
- Author
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David M. Alba and Luca Lusanna
- Subjects
Electromagnetic field ,Hamiltonian mechanics ,Physics ,Geodesic ,Gravitational wave ,n-body problem ,General Physics and Astronomy ,General Relativity and Quantum Cosmology ,symbols.namesake ,Linearization ,symbols ,Hamiltonian (quantum mechanics) ,Tetrad ,Mathematical physics - Abstract
In this second paper we define a post-minkowskian (PM) weak field approximation leading to a linearization of the Hamilton equations of Arnowitt–Deser–Misner (ADM) tetrad gravity in the York canonical basis in a family of nonharmonic 3-orthogonal Schwinger time gauges. The York time 3K (the relativistic inertial gauge variable, not existing in newtonian gravity, parametrizing the family, and connected to the freedom in clock synchronization, i.e., to the definition of the the shape of the instantaneous 3-spaces) is set equal to an arbitrary numerical function. The matter are considered point particles, with a Grassmann regularization of self-energies, and the electromagnetic field in the radiation gauge: an ultraviolet cutoff allows a consistent linearization, which is shown to be the lowest order of a hamiltonian PM expansion. We solve the constraints and the Hamilton equations for the tidal variables and we find PM gravitational waves with asymptotic background (and the correct quadrupole emission formula) propagating on dynamically determined non-euclidean 3-spaces. The conserved ADM energy and the Grassmann regularization of self-energies imply the correct energy balance. A generalized transverse–traceless gauge can be identified and the main tools for the detection of gravitational waves are reproduced in these nonharmonic gauges. In conclusion, we get a PM solution for the gravitational field and we identify a class of PM Einstein space–times, which will be studied in more detail in a third paper together with the PM equations of motion for the particles and their post-newtonian expansion (but in the absence of the electromagnetic field). Finally we make a discussion on the gauge problem in general relativity to understand which type of experimental observations may lead to a preferred choice for the inertial gauge variable 3K in PM space–times. In the third paper we will show that this choice is connected with the problem of dark matter.
- Published
- 2012
7. Further Comments on Some Comments on a Paper by Bradley Efron
- Author
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Dawid, A. P.
- Published
- 1977
8. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending
- Author
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Pengbo Bo and Wenping Wang
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Surface (mathematics) ,Developable surface ,Geodesic ,Computer science ,Mathematical analysis ,Bent molecular geometry ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Geometry ,Bending ,Computer Graphics and Computer-Aided Design ,Piecewise ,Envelope (mathematics) ,Representation (mathematics) ,ComputingMethodologies_COMPUTERGRAPHICS - Abstract
We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications. The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the surface. We manipulate the geodesic to provide intuitive shape control for modeling paper bending. Our method ensures a natural continuous isometric deformation from a piece of bent paper to its flat state without any stretching. Test examples show that the new scheme is fast, accurate, and easy to use, thus providing an effective approach to interactive paper bending. We also show how to handle non-convex piecewise smooth developable surfaces.
- Published
- 2007
9. Letter to the Editor Regarding Paper 'Automatic Computation of Left Ventricular Volume Changes over a Cardiac Cycle from Echocardiography Images by Nonlinear Dimensionality Reduction'
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Ahmad Shalbaf and Hamid Behnam
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Surface (mathematics) ,Geodesic ,Heart Ventricles ,Image registration ,Motion (geometry) ,Geometry ,Image processing ,Article ,Pattern Recognition, Automated ,Cardiovascular Physiological Phenomena ,Motion estimation ,Image Interpretation, Computer-Assisted ,Image Processing, Computer-Assisted ,Humans ,Radiology, Nuclear Medicine and imaging ,Letter to the Editor ,Mathematics ,Ultrasonography ,Radiological and Ultrasound Technology ,Nonlinear dimensionality reduction ,Reproducibility of Results ,Function (mathematics) ,Organ Size ,Computer Science Applications ,Transformation (function) ,Algebraic function ,Isomap ,Rotation (mathematics) ,Algorithm ,Algorithms - Abstract
Question: 1—Echocardiography images Images were acquired from end of diastole to end of systole at different phases (7 phases) within a cardiac cycle by a Vivid 3 GE Health echocardiography machine. To obtain worthy images at denoted phases, it would be dependent to operator (for the best clinical examination). Of course, having assumed images not only were acquired from A4C and A2C but also from short-axis views. Short axis views have not been used at your papers, so some radial, circumferential and sample rotatory data would be missed. Response: It is simply possible for us to record and analyze images of short-axis views. However, according to American Society of Echocardiography (ASE) [1, 2], in the modified Simpson’s rule which is a method for LV volume computation from 2-D echocardiography images, it suffices to acquire apical two and four chamber views (A4C, A2C) for left ventricular (LV) volume computation. Question: 2—Observed data yi′ s Echocardiography images paly role of observed data and they are symbolized by yi′ s. there is a sequence of observed data y1, y2,…,yN. N is the number of obtained images. In fact we have made a chain of displacement, rotation and pure strain or nonrigid transformation (deformation) by these observed data that is started from y1 and is ended up to yN. This chain has some conceptual interpretations of the elasticity of the global left ventricular motion/function and the regional LV fiber arrangement/movement. Response: Yes, it is correct. Question: 3—Embed observed data to a high dimensional space Response: Yes, it is correct. Question: 4—Image processing on yi′ s and geodesic distance between yi′ s Observed data y1, y2,…,yN are embed to a meshed surface sized number of pixels of an image. Translation, rotation and pure deformation of y1, y2,…,yN are occurred/and studied at this surface utilizing the mathematical elasticity theory. A distance metric is defined that computes distances between observed points. Medical interpretations of observed points over the time should be checked/and stated clearly at your manuscripts. Images are not usual images but also are images from the left ventricle. Motion (displacement and velocity), deformation (strain and strain rate) and torsion of each myocardial sample have to be extracted during a cardiac cycle in the mentioned surface. These are used on the creation of graph G referred to their papers. Response: In this stage, we have just computed the distance between the two images in a cardiac cycle using non-rigid image registration regardless of relationship between images over the time. Therefore by this method, the motion and longitudinal deformation (short-axis views are needed to get for radial and circumferential deformations) of LV myocardium between all of two images in a cardiac cycle are extracted (for example, between end systole image and end diastole image, etc.). Then, in other stage of our method (calculation of the matrix of pairwise geodesic distance and apply multidimensional scaling on the resulting geodesic distance matrix to construct the low dimensional data points xi), motion and longitudinal deformation of LV myocardium during a cardiac cycle are extracted. Question: 5—An Isomap f The main tool at this method is the function “f” which is an isometric map [2, 3]. I will be interested to know the structure/formula or relationship of this isometric function which has most probably a lot of practical information of the left ventricular function and structure. I mean, how we can get a good representation of such a this function “f” for heart science study? Is it “f” known as an algebraic function? What information has been coded to fibers (f−1) of this isometric function “f”? Response: Isomap algorithm is one of the most popular sets of nonlinear dimensionality reduction (NLDR) algorithms. This algorithm attempts to extract low-dimensional data points from input points in a high-dimensional space in such a way that pairwise geodesic distances (the distance between two points measured over the manifold) are preserved. So that nearby and far points in high-dimensional space map to nearby and far points in low-dimensional space. This method is frequently used for visualization of medial image set. Detail of this method is described in Tenenbaum et al. and Borg et al. [3, 4]. It should be noted that we do not change the structure or formula of isomap algorithm. We only define a new image distance function based on image registration in calculation of geodesic distance between yi′ s in isomap algorithm. With this modification, Isomap algorithm is specified for assessment of the left ventricular function and structure. However, precise mathematical descriptions of f and (f−1) applied to echocardiography images and using the other NLDR algorithms need further researches in the following of this article. Question: 6—A reconstructed curve on a 2D manifold space & hidden data xi′ s; f(yi) = xi Isometric function results in a reconstructive curve crossing from hidden data (xi′ s) in a surface. It’s natural that some information (maybe clinical information) would betransferred to these hidden points like the curve of LV volume changes and so on and so on. A main question is: what are these new points xi′ s exactly (medically point of view)? What the other data has been come out from these hidden points? I think all of these questions back to gain a good understanding of isometric function “f”. Response: These points xi′ s will probably enable physicians to diagnose and follow up many cardiac structures and functions that also open doors to much more research at imaging cardiology in the future. References 1. AlizadehSani Z, Shalbaf A, Behnam H, Shalbaf R: Automatic computation of left ventricularvolume changes over a cardiac cycle from echocardiography images by nonlinear dimensionality reduction. J Digit Imaging : July, 2014 2. Tenenbaum JB, de Silva V, Langford JC: global geometric framework for nonlinear dimensionalityreduction. Science 290:2319–2323, 2000. Reprint available online: http://web.mit.edu/cocosci/Papers/sci_reprint.pdf 3. Ledesma-Carbayo MJ, Kybic J, Desco M, et al.: Spatio-temporal nonrigid registration for ultrasoundcardiac motion estimation. IEEE Trans Med Imaging 24:1113–1126, 2005.
- Published
- 2015
10. Contribution to the Discussion of the Paper 'Geodesic Monte Carlo on Embedded Manifolds'
- Author
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Daniel Simpson
- Subjects
Statistics and Probability ,Geodesic ,Monte Carlo method ,Mathematical analysis ,Calculus ,Statistics, Probability and Uncertainty ,Mathematics - Published
- 2014
11. Volumes of hyperbolic 3-manifolds. Notes on a paper of Gabai, Meyerhoff, and Milley
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T. H. Marshall and Gaven Martin
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Pure mathematics ,Geodesic ,Hyperbolic group ,Hyperbolic 3-manifold ,Mathematical analysis ,Hyperbolic manifold ,Geometry and Topology ,Weeks manifold ,Mathematics ,Closed geodesic - Abstract
We present a new approach and improvements to the recent results of Gabai, Meyerhoff and Milley concerning tubes and short geodesics in hyperbolic 3 3 -manifolds. We establish the following two facts: if a hyperbolic 3 3 -manifold admits an embedded tubular neighbourhood of radius r 0 > 1.32 r_0>1.32 about any closed geodesic, then its volume exceeds that of the Weeks manifold. If the shortest geodesic of M M has length less than ℓ 0 > 0.1 \ell _0>0.1 , then its volume also exceeds that of the Weeks manifold.
- Published
- 2003
12. Rhino horns and paper cups: Deceptive similarities between natural and human designs
- Author
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Steven Vogel
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Technology ,Geodesic ,Computer science ,General Medicine ,Biological evolution ,Conical surface ,Mechanics ,Biological Evolution ,General Biochemistry, Genetics and Molecular Biology ,Biomechanical Phenomena ,Classical mechanics ,Equipment and Supplies ,Animals ,Humans ,Natural (music) ,Point (geometry) ,General Agricultural and Biological Sciences ,Mechanical devices - Abstract
One cannot assume that resemblances between the mechanical devices of human technology and those produced by the evolutionary process reflect either specific copying of nature by people or some particular point of functional superiority. A third alternative is that the two mechanical contexts derive quite different advantages from a given arrangement. While this latter might appear unlikely, one can argue that it underlies such things as the use of conical shapes, helical tensile structures, spheres and cylinders, beams and columns of relatively low torsional stiffness, and geodesic shells.
- Published
- 2000
13. Contribution to the Discussion of the Paper ‘Geodesic Monte Carlo on Embedded Manifolds’
- Author
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Babak Shahbaba, Shiwei Lan, and Jeffrey Streets
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Statistics and Probability ,Simplex ,Geodesic ,Monte Carlo method ,Riemannian geometry ,Riemannian manifold ,Statistics::Computation ,Hybrid Monte Carlo ,symbols.namesake ,symbols ,Calculus ,Probability distribution ,Statistics, Probability and Uncertainty ,Gibbs sampling ,Mathematics - Abstract
We would like to start by congratulating Byrne and Girolami for writing such a thoughtful and extremely interesting paper. This is in fact a worthy addition to other high impact papers recently published by Professor Girolami’s lab in this field. The common theme of these papers is to use geometrically motivated methods to improve efficiency of sampling algorithms. In their seminal paper, Girolami & Calderhead (2011) proposed a novel Hamiltonian Monte Carlo (HMC) method, called Riemannian manifold HMC, that exploits the Riemannian geometry of the target distribution to improve standard HMC’s efficiency by automatically adapting to the local structure. Although this is a natural and beautiful idea, there are significant computational difficulties, which arise in effectively implementing this algorithm. In contrast, in this current contribution, Byrne and Girolami focused on special probability distributions, which give rise to particularly nice Riemannian geometries. In particular, the examples under consideration described in Section 4 allow for closed-form solutions to the geodesic equation, which can be used to reduce computational cost of geometrically motivated Monte Carlo methods. Although the proposed splitting algorithm is quiet interesting, we initially doubted its impact because Riemannian metrics with closed-form geodesics are extremely rare. However, we are now convinced that this approach will likely see application beyond what is outlined herein. For example, we believe that this approach can be used to improve computational efficiency of sampling algorithms when the parameter space is constrained. The standard HMC algorithm needs to evaluate each proposal to ensure that it is within the boundaries imposed by the constraints. Alternatively, as discussed by Neal (2011), one could modify standard HMC so the sampler bounces back after hitting the boundaries. In Appendix A, Byrne and Girolami discussed this approach for geodesic updates on the simplex. In many cases, a constrained parameter space can be bijectively mapped to a unit ball
- Published
- 2014
14. Curved paper rectification for digital camera document images by shape from parallel geodesics using continuous dynamic programming
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Satoshi Naoi, Misako Suwa, Takebe Hiroaki, Jun Sun, and K. Fujimoto
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Surface (mathematics) ,business.product_category ,Ruled surface ,Geodesic ,business.industry ,Iterative method ,Computer science ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Parallel ,Rectification ,Distortion ,Curve fitting ,Computer vision ,Artificial intelligence ,business ,Digital camera - Abstract
Methods to rectify distortion of digital camera document images of curved papers have become important for camera-based image recognition. In this paper we propose a novel distortion rectification method based on "shape from parallel geodesies." This method considers the following features: parallel lines corresponding to character strings or ruled lines of tables on extended surface become parallel geodesies on a curved paper surface and a smoothly curved paper can be modeled by a ruled surface, that is, a sweep surface of rulings. The projected geodesies and the projected rulings exist in the input image derived from perspective transformation. The presented method extracts the projected geodesies, estimates the projected rulings in the input image, estimates the ruled surface that models the curved paper, and generates the corrected image, in this order. It can estimate the ruled surface model directly by numerical operations of differentiation, integration and matrix inversion without any iterative calculation. We also report on experiments that show the effectiveness of the proposed method.
- Published
- 2007
15. Contribution to the Discussion of the Paper Geodesic Monte Carlo on Embedded Manifolds
- Author
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John T. Kent
- Subjects
Statistics and Probability ,Geodesic ,Monte Carlo method ,Mathematical analysis ,Calculus ,Statistics, Probability and Uncertainty ,Mathematics - Published
- 2014
16. Erratum to the Paper 'A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold'
- Author
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Wei Zhao
- Subjects
Pure mathematics ,Geodesic ,General Mathematics ,Geometry ,Finsler manifold ,Upper and lower bounds ,Mathematics - Abstract
We correct two clerical errors made in the paper ”A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold”.
- Published
- 2014
17. Special Invited Paper: Geodesics And Spanning Tees For Euclidean First-Passage Percolaton
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Charles M. Newman and C. Douglas Howard
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Statistics and Probability ,Spanning tree ,Geodesic ,82D30 ,random surface ,First passage percolation ,Poisson process ,Infimum and supremum ,Euclidean distance ,Combinatorics ,random metric ,firstpassage percolation ,60K35 ,Lattice (order) ,minimal spanning tree ,shape theorem ,Almost surely ,combinatorial optimization ,60G55 ,Statistics, Probability and Uncertainty ,First-hitting-time model ,geodesic ,Mathematics ,60F10 - Abstract
The metric $D_{\alpha}(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $\mathbb{R}^d$ , defined as the infimum of $(\sum_i |q_i - q_{i+1}|^{\alpha})^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $|·|$ denotes Euclidean distance) has nontrivial geodesics when $\alpha>1$. The cases $1< \alpha
- Published
- 2001
18. Bounded-Diameter Tree-Decompositions
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Berger, Eli and Seymour, Paul
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- 2024
- Full Text
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19. Vector-Algebra Algorithms to Draw the Curve of Alignment, the Great Ellipse, the Normal Section, and the Loxodrome.
- Author
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Meyer, Thomas H.
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ALGORITHMS ,ELLIPSOIDS ,GEODESICS ,GEODESY - Abstract
This paper recasts four geodetic curves—the great ellipse, the normal section, the loxodrome, and the curve of alignment—into a parametric form of vector-algebra formula. These formulas allow these curves to be drawn using simple, efficient, and robust algorithms. The curve of alignment, which seems to be quite obscure, ought not to be. Like the great ellipse and the loxodrome, and unlike the normal section, the curve of alignment from point A to point B (both on the same ellipsoid) is the same as the curve of alignment from point B to point A. The algorithm used to draw the curve of alignment is much simpler than any of the others and its shape is quite similar to that of the geodesic, which suggests it would be a practical surrogate when drawing these curves. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. CORRIGENDUM TO: "SHARP Lp HARDY TYPE AND UNCERTAINTY PRINCIPLE INEQUALITIES ON THE SPHERE".
- Author
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ABOLARINWA, ABIMBOLA, RAUF, KAMILU, and SONGTING YIN
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GEODESIC equation ,TECHNOLOGY ,EQUILIBRIUM ,VARIATIONAL inequalities (Mathematics) ,ITERATIVE methods (Mathematics) - Abstract
The aim of this note is to put straight the inaccuracy in Theorem 1 of the paper mentioned in the title (J. Math. Ineq. 13 (4), 1011-1022). The inaccuracy was due to an omission of a factor in the proof of the theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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21. Further Comments on Some Comments on a Paper by Bradley Efron
- Author
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A. P. Dawid
- Subjects
Statistics and Probability ,Discrete mathematics ,Hellinger distance ,62B10 ,Fisher information ,Geodesic ,Mathematical analysis ,53C40 ,symbols.namesake ,Bhattacharya distance ,symbols ,Bhattacharyya distance ,Geometry of distribution space ,Statistics, Probability and Uncertainty ,geodesic ,Mathematics - Abstract
The intrinsic geometry of a space of distributions is represented by means of an embedding in Hilbert space, which has been previously studied.
- Published
- 1977
22. ON STRESS OF A VERTEX IN A GRAPH.
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Bhargava, K., Dattatreya, N. N., and Rajendra, R.
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REGULAR graphs ,GEODESICS - Abstract
The stress of a vertex in a graph is the number of geodesics passing through it (A. Shimbel, 1953). A graph is k-stress regular if stress of each of its vertices is k. In this paper, we investigate some results and compute stress of vertices in some standard graphs and give a characterization of graphs with all vertices of zero stress except for one. Also we compute stress of vertices in graphs of diameter 2 and in the corona product Km 0G. Further we prove that any strongly regular graph is stress regular and characterize k-stress regular graphs for k = 0; 1; 2. [ABSTRACT FROM AUTHOR]
- Published
- 2023
23. Automatic 3D neuron tracing using all-path pruning
- Author
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Hanchuan Peng, Fuhui Long, and Gene Myers
- Subjects
Statistics and Probability ,Theoretical computer science ,Computational complexity theory ,Neurite ,Geodesic ,Computer science ,Image processing ,Tracing ,computer.software_genre ,Biochemistry ,Imaging, Three-Dimensional ,Voxel ,Image Processing, Computer-Assisted ,Animals ,Ismb/Eccb 2011 Proceedings Papers Committee July 17 to July 19, 2011, Vienna, Austria ,Molecular Biology ,Neurons ,Brain ,Original Papers ,Bioimaging ,Computer Science Applications ,Computational Mathematics ,Computational Theory and Mathematics ,Shortest path problem ,Path (graph theory) ,computer ,Algorithm ,Algorithms ,Software ,Pruning (morphology) - Abstract
Motivation: Digital reconstruction, or tracing, of 3D neuron structures is critical toward reverse engineering the wiring and functions of a brain. However, despite a number of existing studies, this task is still challenging, especially when a 3D microscopic image has low signal-to-noise ratio (SNR) and fragmented neuron segments. Published work can handle these hard situations only by introducing global prior information, such as where a neurite segment starts and terminates. However, manual incorporation of such global information can be very time consuming. Thus, a completely automatic approach for these hard situations is highly desirable. Results: We have developed an automatic graph algorithm, called the all-path pruning (APP), to trace the 3D structure of a neuron. To avoid potential mis-tracing of some parts of a neuron, an APP first produces an initial over-reconstruction, by tracing the optimal geodesic shortest path from the seed location to every possible destination voxel/pixel location in the image. Since the initial reconstruction contains all the possible paths and thus could contain redundant structural components (SC), we simplify the entire reconstruction without compromising its connectedness by pruning the redundant structural elements, using a new maximal-covering minimal-redundant (MCMR) subgraph algorithm. We show that MCMR has a linear computational complexity and will converge. We examined the performance of our method using challenging 3D neuronal image datasets of model organisms (e.g. fruit fly). Availability: The software is available upon request. We plan to eventually release the software as a plugin of the V3D-Neuron package at http://penglab.janelia.org/proj/v3d. Contact: pengh@janelia.hhmi.org
- Published
- 2011
24. Quantitative phenotyping of leaf margins in three dimensions, demonstrated on KNOTTED and TCP trangenics in Arabidopsis
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Shahaf Armon, Eran Sharon, Naomi Ori, and Osnat Yanai
- Subjects
Current (mathematics) ,Geodesic ,Physiology ,growth ,Arabidopsis ,Plant Science ,Biology ,Curvature ,leaf shape ,Zea mays ,Measure (mathematics) ,Dexamethasone ,lobes ,Botany ,differential geometry ,Plant Proteins ,Homeodomain Proteins ,Models, Genetic ,Waviness ,Function (mathematics) ,Plants, Genetically Modified ,waviness ,Plant Leaves ,Phenotype ,Differential geometry ,curvature ,Biological system ,Research Paper ,Transcription Factors ,Geodesic curvature - Abstract
Summary Three-dimensional geometry of leaf margins is an important shape characteristic to distinguish different leaf phenotypes. Novel geometrical methods were defined, measured, and used to quantify waviness and lobiness of leaves., The geometry of leaf margins is an important shape characteristic that distinguishes among different leaf phenotypes. Current definitions of leaf shape are qualitative and do not allow quantification of differences in shape between phenotypes. This is especially true for leaves with some non-trivial three-dimensional (3D) configurations. Here we present a novel geometrical method novel geometrical methods to define, measure, and quantify waviness and lobiness of leaves. The method is based on obtaining the curve of the leaf rim from a 3D surface measurement and decomposing its local curvature vector into the normal and geodesic components. We suggest that leaf waviness is associated with oscillating normal curvature along the margins, while lobiness is associated with oscillating geodesic curvature. We provide a way to integrate these local measures into global waviness and lobiness quantities. Using these novel definitions, we analysed the changes in leaf shape of two Arabidopsis genotypes, either as a function of gene mis-expression induction level or as a function of time. These definitions and experimental methods open the way for a more quantitative study of the shape of leaves and other growing slender organs.
- Published
- 2014
25. On the existence of geodesics connecting Lagrangian graphs in Euclidean space
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Xu, Yiyan
- Published
- 2024
- Full Text
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26. Topological properties of convex order in Wasserstein metric spaces
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Ju, Hongbing, Wang, Feng, and Wu, Hongguang
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- 2024
- Full Text
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27. Paths of means for positive operators with strongly unitarily equivalent supports
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Fujii, Jun Ichi
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- 2024
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28. Angular Location of the nth Einstein Ring at Large n
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Minwalla, Spandan
- Published
- 2024
- Full Text
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29. Self-organizing mappings on the flag manifold with applications to hyper-spectral image data analysis.
- Author
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Ma, Xiaofeng, Kirby, Michael, and Peterson, Chris
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SELF-organizing maps ,IMAGE analysis ,DATA analysis ,VECTOR spaces ,SEQUENCE spaces ,GEODESICS - Abstract
A flag is a nested sequence of vector spaces. The type of the flag encodes the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a fixed type in a fixed vector space. This paper provides the mathematical framework necessary for implementing self-organizing mappings on flag manifolds. Flags arise implicitly in many data analysis contexts including wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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30. Stress and Tension of Generalized Complements of Graphs.
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V., Madhumitha K., Nayak, Swati, and D'Souza, Sabitha
- Subjects
- *
GEODESICS - Abstract
Centrality measures are scalar values given to each node in the graph to quantify its importance based on an assumption. Stress and tension are the centrality index based on shortest path. In this paper, stress and tension of generalized complements of some standard graphs are calculated by counting the number of geodesics of different length. [ABSTRACT FROM AUTHOR]
- Published
- 2024
31. On Translation Curves and Geodesics in Sol 1 4.
- Author
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Erjavec, Zlatko and Maretić, Marcel
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GEODESICS ,HOMOGENEOUS spaces ,COLLOIDS ,LIE groups ,SOLVABLE groups - Abstract
A translation curve in a homogeneous space is a curve such that for a given unit vector at the origin, translation of this vector is tangent to the curve in its every point. Translation curves coincide with geodesics in most Thurston spaces, but not in twisted product Thurston spaces. Moreover, translation curves often seem more intuitive and simpler than geodesics. In this paper, we determine translation curves in Sol 1 4 space. Their curvature properties are discussed and translation spheres are presented. Finally, characterization of geodesics in Sol 1 4 space is given. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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32. New Generalization of Geodesic Convex Function.
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Almutairi, Ohud Bulayhan and Saleh, Wedad
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CONVEX functions ,GENERALIZATION ,GEODESICS ,RIEMANNIAN manifolds - Abstract
As a generalization of a geodesic function, this paper introduces the notion of the geodesic φ E -convex function. Some properties of the φ E -convex function and geodesic φ E -convex function are established. The concepts of a geodesic φ E -convex set and φ E -epigraph are also given. The characterization of geodesic φ E -convex functions in terms of their φ E -epigraphs, are also obtained. [ABSTRACT FROM AUTHOR]
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- 2023
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33. Parallel Translations for a Left Invariant Spray.
- Author
-
Xu, Ming
- Abstract
In this paper, we study the left invariant spray geometry on a connected Lie group. Using the technique of invariant frames, we find the ordinary differential equations on the Lie algebra describing for a left invariant spray structure the linearly parallel translations along a geodesic and the nonlinearly parallel translations along a smooth curve. In these equations, the connection operator plays an important role. Using parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. In particular, the Riemannian curvture appears in both a double Lie derivative along the spray vector and brackets between smooth vector fields induced by the connection operator. We propose two questions in left invariant spray geometry. One question generalizes Landsberg Problem in Finsler geometry, and the other concerns the restricted holonomy group. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. Geodesics and Translation Curves in Sol 0 4.
- Author
-
Erjavec, Zlatko
- Subjects
GEODESICS ,COLLOIDS ,LIE groups ,SOLVABLE groups - Abstract
A translation curve in a Thurston space is a curve such that for given unit vector at the origin, the translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces, translation curves coincide with geodesic lines. However, this does not hold for Thurston spaces equipped with twisted product. In these spaces, translation curves seem more intuitive and simpler than geodesics. In this paper, geodesics and translation curves in Sol 0 4 space are classified and the curvature properties of translation curves are investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. Geodesics and natural complex magnetic trajectories on tangent bundles.
- Author
-
ABBASSI, Mohamed Tahar Kadaoui and AMRI, Noura
- Subjects
TANGENT bundles ,GEODESICS ,MAGNETIC fields ,RIEMANNIAN manifolds - Abstract
In this paper, we investigate geodesics of the tangent bundle TM of a Riemannian manifold (M, g) endowed with an arbitrary pseudo-Riemannian g -natural metric of Kaluza-Klein type. Then considering a class of naturally defined almost complex structures on TM, constructed by V. Oproiu, we construct a class of magnetic fields and we characterize the corresponding magnetic curves on TM, when (M, g) is a space form. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Umbilical Foliations on a Riemannian Manifold
- Author
-
André O. Gomes
- Subjects
Geodesic ,Quantitative Biology::Tissues and Organs ,Applied Mathematics ,Mathematical analysis ,Short paper ,Field (mathematics) ,Riemannian manifold ,Combinatorics ,Mathematics (miscellaneous) ,Differential geometry ,Unit vector ,GEOMETRIA DIFERENCIAL ,Foliation (geology) ,Mathematics::Differential Geometry ,Sectional curvature ,Mathematics - Abstract
In this short paper, we will prove the following Theorem: Let \(\mathcal {F}\) be a codimension-one foliation of a complete and connected Riemannian manifold \(M^{n+1}\) with constant sectional curvature \(c {\leq} 0\). Suppose that \(\mathcal {F} \) is transversely orientable, i.e., there exists an unit vector field \(N \epsilon {\mathfrak{X}}(M)\) such that N is normal to the leaves of \(\mathcal {F} \). Suppose that N is geodesic. Then, if the foliation \(\mathcal {F} \) has an umbilical leaf, every leaf of foliation \(\mathcal {F} \) must be umbilical.
- Published
- 2008
37. Automatic reconstruction of 3D neuron structures using a graph-augmented deformable model
- Author
-
Deniz Atasoy, Scott M. Sternson, Hanchuan Peng, and Zongcai Ruan
- Subjects
Statistics and Probability ,Geodesic ,Computer science ,Image processing ,Biochemistry ,Fuzzy logic ,Mice ,Imaging, Three-Dimensional ,Image Processing, Computer-Assisted ,Animals ,Computer vision ,Caenorhabditis elegans ,Molecular Biology ,Neurons ,business.industry ,Ismb 2010 Conference Proceedings July 11 to July 13, 2010, Boston, Ma, Usa ,Original Papers ,Bioimaging ,Graph ,Computer Science Applications ,Euclidean distance ,Computational Mathematics ,Computational Theory and Mathematics ,Shortest path problem ,Graph (abstract data type) ,Drosophila ,Artificial intelligence ,business ,Algorithm - Abstract
Motivation: Digital reconstruction of 3D neuron structures is an important step toward reverse engineering the wiring and functions of a brain. However, despite a number of existing studies, this task is still challenging, especially when a 3D microscopic image has low single-to-noise ratio and discontinued segments of neurite patterns. Results: We developed a graph-augmented deformable model (GD) to reconstruct (trace) the 3D structure of a neuron when it has a broken structure and/or fuzzy boundary. We formulated a variational problem using the geodesic shortest path, which is defined as a combination of Euclidean distance, exponent of inverse intensity of pixels along the path and closeness to local centers of image intensity distribution. We solved it in two steps. We first used a shortest path graph algorithm to guarantee that we find the global optimal solution of this step. Then we optimized a discrete deformable curve model to achieve visually more satisfactory reconstructions. Within our framework, it is also easy to define an optional prior curve that reflects the domain knowledge of a user. We investigated the performance of our method using a number of challenging 3D neuronal image datasets of different model organisms including fruit fly, Caenorhabditis elegans, and mouse. In our experiments, the GD method outperformed several comparison methods in reconstruction accuracy, consistency, robustness and speed. We further used GD in two real applications, namely cataloging neurite morphology of fruit fly to build a 3D ‘standard’ digital neurite atlas, and estimating the synaptic bouton density along the axons for a mouse brain. Availability: The software is provided as part of the V3D-Neuron 1.0 package freely available at http://penglab.janelia.org/proj/v3d Contact: pengh@janelia.hhmi.org
- Published
- 2010
38. Geodesies and isocline distributions in tangent bundles of nonflat Lorentzian-Heisenberg spaces.
- Author
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ALTUNBAŞ, Murat
- Subjects
TANGENT bundles ,GEODESY - Abstract
Let (H
3 ,g1 ) and (H3 ,g2 ) be the Lorentzian-Heisenberg spaces with nonflat metrics g1 and g2 , and (TH3 ,g1 s ), (TH3 ,g2 s ) be their tangent bundles with the Sasaki metric, respectively. In the present paper, we find nontotally geodesic distributions in tangent bundles by using lifts of contact forms from the base manifold H3 . We give examples for totally geodesic but not isocline distributions. We study the geodesics of tangent bundles by considering horizontal and natural lifts of geodesics of the base manifold H3 . We also investigate more general classes of geodesics which are not obtained from horizontal and natural lifts of geodesics. [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
39. Geometric properties of normal submanifolds
- Author
-
Meléndez, Josué and Hernández, Mario
- Published
- 2020
- Full Text
- View/download PDF
40. Asymptotic relative equilibrium in the n-body problem: Relativistic application in the Poincaré upper half-plane.
- Author
-
Ortiz, Ruben Dario Ortiz, Caro, Mario Enrique Almanza, Perez, Alejandra Patricia Guzman, and Ramirez, Ana Magnolia Marin
- Subjects
- *
MANY-body problem , *CENTER of mass , *EQUILIBRIUM , *GEODESICS , *GAUSSIAN curvature , *SPHERICAL projection - Abstract
In this paper, we study the n -body problem in the Poincaré upper half-plane ℍ R 2 , where the radius R of the Poincaré disk is fixed. We introduce a new potential to derive the condition for hyperbolic relative equilibria on ℍ R 2 . We analyze the relative equilibrium of positive masses moving along geodesics under the SL (2 , ℝ) group. This result is utilized to establish the existence of relative equilibria for the n -body problem on ℍ R 2 for n = 2 and n = 3. We revisit previously known results and uncover new qualitative findings on relative equilibria that are not evident in an extrinsic context. Additionally, we provide a simple expression for the center of mass of a system of point particles on a two-dimensional surface with negative constant Gaussian curvature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Semi-Continuity of Skeletons in Two-Manifold and Discrete Voronoi Approximation.
- Author
-
Liu, Yong-Jin
- Subjects
VORONOI polygons ,COMPUTER vision ,IMAGE processing ,ARTIFICIAL intelligence ,PATTERN matching - Abstract
The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a two-manifold \mathcal M
, based on a geodesic metric. We present a formal definition of the skeleton $S(\Omega )$ in S(\Omega )$ distinct from its Euclidean counterpart in . We further prove that for a shape sequence \lbrace \Omega _i\rbrace$ in \mathcal M gives a good approximation to the skeleton $S(\Omega )$ . Examples of skeleton computation in topography and brain morphometry are illustrated. [ABSTRACT FROM AUTHOR]- Published
- 2015
- Full Text
- View/download PDF
42. New Hadamard-type inequality for new class of geodesic convex functions.
- Author
-
Saleh, Wedad
- Subjects
CONVEX functions ,GEODESICS ,LEBESGUE integral ,INTEGRAL inequalities ,SPECIAL functions ,RIEMANNIAN manifolds - Abstract
This paper aims to introduce the concept of (E, μ, κ)-convex function by using special inequality. Hadamard integral inequality for this new class of geodesic convex function in the case of Lebesgue and Sugeno integrals is given. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
43. On conformal complex Finsler metrics.
- Author
-
Li, Hongjun, Qiu, Chunhui, Xia, Hongchuan, and Zhong, Guozhu
- Abstract
In this paper, we study conformal transformations in complex Finsler geometry. We first prove that two weakly Kähler Finsler metrics cannot be conformal. Moreover, we give a necessary and sufficient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal weakly Kähler Finsler. Finally, we discuss conformal transformations of a strongly pseudoconvex complex Finsler metric, which preserve the geodesics, holomorphic S curvatures and mean Landsberg tensors. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
44. Hermite-Hadamard type integral inequalities for uniformly convex functions by using Riemann-Liouville fractional integral transformations with respect to geodesic in Hadamard space.
- Author
-
Barsam, Hasan, Sayyari, Yamin, and Ramezani, Sayyed Mehrab
- Subjects
INTEGRAL inequalities ,HERMITE polynomials ,CONVEX functions ,INTEGRAL transforms ,GEODESICS - Abstract
In this paper, we investigate the Hermite-Hadamard inequality for the Riemann-Liouville integral transformations for uniformly convex functions from a geodesic perspective in the Hadamard spaces. This inequality is widely used in some fractional integral approximations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
45. Quantitative Behavior Of Non-Integrable Systems. IV.
- Author
-
Beck, J., Chen, W. W. L., and Yang, Y.
- Subjects
ORBITS (Astronomy) ,DYNAMICAL systems - Abstract
In this paper, there are two sections. In Section 7, we simplifythe eigenvalue-based surplus shortline method for arbitrary finite polysquaretranslation surfaces. This makes it substantially simpler to determine the irregularityexponents of some infinite orbits, and quicker to find the escape rate toinfinity of some orbits in some infinite models. In Section 8, our primary goalis to extend the surplus shortline method, both this eigenvalue-based version aswell as the eigenvalue-free version, for application to a large class of 2-dimensionalflat dynamical systems beyond polysquares, including all Veech surfaces, and establishtime-quantitative equidistribution and time-quantitative superdensity ofsome infinite orbits in these new systems. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
46. Surface pencil with a common adjoint curve.
- Author
-
BAYRAM, Ergin
- Subjects
CURVES ,PENCILS ,CURVATURE - Abstract
In the present paper, we construct surfaces possessing an adjoint curve of a given space curve as an asymptotic curve, geodesic or line of curvature. We obtain conditions for ruled surfaces and developable ones. Finally, we present illustrative examples to show the validity of the present method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
47. 1-Dimensional intrinsic persistence of geodesic spaces.
- Author
-
Virk, Žiga
- Subjects
GEODESIC spaces ,COMPACT spaces (Topology) ,FILTERS & filtration ,POINT set theory - Abstract
Given a compact geodesic space X , we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of X to obtain what we call a persistence object. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and closed, Rips and Čech induced persistences. Amongst other results, we prove that a Rips critical point c corresponds to an isometrically embedded circle of length 3 c , that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor 3 / 4. The theory describes geometric properties of the underlying space encoded and extractable from persistence. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
48. Differential Geometry of Magnetic and Killing Magnetic Trajectories in de Sitter 3-Space
- Author
-
Iqbal, Zafar
- Published
- 2022
- Full Text
- View/download PDF
49. Particle dynamics and shadow of a regular non-minimal magnetic black hole.
- Author
-
Al-Badawi, Ahmad and Owaidat, M. Q.
- Subjects
- *
BLACK holes , *PARTICLE dynamics , *GAUGE field theory , *PARTICLE motion , *LYAPUNOV exponents , *ORBITS (Astronomy) - Abstract
In this paper, we study the dynamics of a test particle around a regular black hole (BH) in a non-minimal Einstein-Yang-Mills (EYM) theory and examine the BH shadow. The EYM theory is a non-minimally coupled theory in which curvature couples to non-Abelian gauge fields. We investigate particle motion with parameters in EYM BH for massless and massive particles. This work provides the horizon structure, photon radius and inner stable circular orbit (ISCO) of a mass particle with EYM BH parameters. An analysis is provided of the effective potential as well as the possible orbits for test particles under various EYM BH parameters values. In timelike radial geodesics, we find that for smaller values of magnetic charge, particles following a timelike radial geodesic are more hasty in EYM BH, and hence arrive at the center faster than those traveling a Schwarzschild BH geodesic. However, at larger values of the magnetic charge, the inverse effect is observed. The effect of model parameters is investigated in order to study the ISCO, photon radius, orbit stability (Lyapunov exponent), and effective force on particles for the BH in the EYM theory. Finally, we investigate the BH shadow. We find that higher magnetic charge values and non-minimal coupling parameters result in smaller shadow radius values. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. Affine differential geometry analysis of human arm movements
- Author
-
Amir Handzel and Tamar Flash
- Subjects
Original Paper ,Time Factors ,General Computer Science ,Geodesic ,Movement ,Affine differential geometry ,Motion Perception ,Motion (geometry) ,Curvature ,Proprioception ,Models, Biological ,Biomechanical Phenomena ,Differential geometry ,Moving frame ,Control theory ,Euclidean geometry ,Arm ,Humans ,Point (geometry) ,Algorithm ,Mathematics ,Psychomotor Performance ,Computer Science(all) ,Biotechnology - Abstract
Humans interact with their environment through sensory information and motor actions. These interactions may be understood via the underlying geometry of both perception and action. While the motor space is typically considered by default to be Euclidean, persistent behavioral observations point to a different underlying geometric structure. These observed regularities include the “two-thirds power law”, which connects path curvature with velocity, and “local isochrony”, which prescribes the relation between movement time and its extent. Starting with these empirical observations, we have developed a mathematical framework based on differential geometry, Lie group theory and Cartan’s moving frame method for the analysis of human hand trajectories. We also use this method to identify possible motion primitives, i.e., elementary building blocks from which more complicated movements are constructed. We show that a natural geometric description of continuous repetitive hand trajectories is not Euclidean but equi-affine. Specifically, equi-affine velocity is piecewise constant along movement segments, and movement execution time for a given segment is proportional to its equi-affine arc-length. Using this mathematical framework, we then analyze experimentally recorded drawing movements. To examine movement segmentation and classification, the two fundamental equi-affine differential invariants—equi-affine arc-length and curvature are calculated for the recorded movements. We also discuss the possible role of conic sections, i.e., curves with constant equi-affine curvature, as motor primitives and focus in more detail on parabolas, the equi-affine geodesics. Finally, we explore possible schemes for the internal neural coding of motor commands by showing that the equi-affine framework is compatible with the common model of population coding of the hand velocity vector when combined with a simple assumption on its dynamics. We then discuss several alternative explanations for the role that the equi-affine metric may play in internal representations of motion perception and production.
- Published
- 2006
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