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Автор: Grafiati
Опубліковано: 28 липня 2022
Оновлено: 8 серпня 2022

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1

Lau, Chun Pong, Chun Pang Yung, and Lok Ming Lui. "Image Retargeting via Beltrami Representation." IEEE Transactions on Image Processing 27, no. 12 (December 2018): 5787–801. http://dx.doi.org/10.1109/tip.2018.2858146.

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2

Belkin, Mikhail, and Partha Niyogi. "Laplacian Eigenmaps for Dimensionality Reduction and Data Representation." Neural Computation 15, no. 6 (June 1, 2003): 1373–96. http://dx.doi.org/10.1162/089976603321780317.

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One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.
3

Sato, N., and M. Yamada. "Local representation and construction of Beltrami fields." Physica D: Nonlinear Phenomena 391 (April 2019): 8–16. http://dx.doi.org/10.1016/j.physd.2019.02.003.

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4

Lam, Ka Chun, Tsz Ching Ng, and Lok Ming Lui. "Multiscale Representation of Deformation via Beltrami Coefficients." Multiscale Modeling & Simulation 15, no. 2 (January 2017): 864–91. http://dx.doi.org/10.1137/16m1056614.

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5

Alonso-Orán, Diego, Antonio Córdoba, and Ángel D. Martínez. "Integral representation for fractional Laplace–Beltrami operators." Advances in Mathematics 328 (April 2018): 436–45. http://dx.doi.org/10.1016/j.aim.2018.01.014.

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6

Lui, Lok Ming, Ka Chun Lam, Tsz Wai Wong, and Xianfeng Gu. "Texture Map and Video Compression Using Beltrami Representation." SIAM Journal on Imaging Sciences 6, no. 4 (January 2013): 1880–902. http://dx.doi.org/10.1137/120866129.

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7

OLIVIER, D., and G. VALENT. "MULTIPLICATIVE RENORMALIZABILITY AND THE LAPLACE-BELTRAMI OPERATOR." International Journal of Modern Physics A 06, no. 06 (March 10, 1991): 955–76. http://dx.doi.org/10.1142/s0217751x91000526.

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For some rank 1 non-linear σ models we prove that a necessary and sufficient condition of multiplicative renormalizability for composite fields is that they should be eigenfunctions of the coset Laplace-Beltrami operator. These eigenfunctions span the irreducible representation space of the isometry group and may be finite- or infinite-dimensional. The zero mode of the Laplace-Beltrami operator plays a particular role since it is not renormalized at all.
8

Sato, N., and M. Yamada. "Local representation and construction of Beltrami fields II.solenoidal Beltrami fields and ideal MHD equilibria." Physica D: Nonlinear Phenomena 400 (December 2019): 132142. http://dx.doi.org/10.1016/j.physd.2019.06.008.

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9

Chan, Hei-Long, Shi Yan, Lok-Ming Lui, and Xue-Cheng Tai. "Topology-Preserving Image Segmentation by Beltrami Representation of Shapes." Journal of Mathematical Imaging and Vision 60, no. 3 (October 12, 2017): 401–21. http://dx.doi.org/10.1007/s10851-017-0767-8.

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10

Barletta, Elisabetta, Sorin Dragomir, and Francesco Esposito. "Beltrami Equations on Rossi Spheres." Mathematics 10, no. 3 (January 25, 2022): 371. http://dx.doi.org/10.3390/math10030371.

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Beltrami equations L¯t(g)=μ(·,t)Lt(g) on S3 (where Lt, |t|<1, are the Rossi operators i.e., Lt spans the globally nonembeddable CR structure H(t) on S3 discovered by H. Rossi) are derived such that to describe quasiconformal mappings f:S3→N⊂C2 from the Rossi sphere S3,H(t). Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions gt such that gt−v∈WF1,2S3,θ with v∈CR∞S3,H(0).
11

Yamaleev, Robert M. "Coulomb Potential as Metrics of Hyperbolic Space." International Journal of Emerging Technology and Advanced Engineering 10, no. 10 (October 25, 2020): 6–8. http://dx.doi.org/10.46338/ijetae1020_02.

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Representation of the Coulomb field as a metrics of the hyperbolic space within the Beltrami- Poincare model of hyperbolic geometry is done. In order to connect the potential of the electric field with the length of a geodesic line a new characteristics of the Coulomb field, the entropy, is introduced.
12

Barabanov, Nikita E., and Abraham A. Ungar. "Differential Geometry and Binary Operations." Symmetry 12, no. 9 (September 16, 2020): 1525. http://dx.doi.org/10.3390/sym12091525.

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We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.
13

Mackay, Tom G., and Akhlesh Lakhtakia. "On electromagnetics of an isotropic chiral medium moving at constant velocity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2078 (October 3, 2006): 397–418. http://dx.doi.org/10.1098/rspa.2006.1770.

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A medium which is an isotropic chiral medium from the perspective of a co-moving observer is a Faraday chiral medium (FCM) from the perspective of a non-co-moving observer. The Tellegen constitutive relations for this FCM are established. By an extension of the Beltrami field concept, these constitutive relations are exploited to show that plane wave propagation is characterized by four generally independent wavenumbers. This FCM can support negative phase velocity at certain translational velocities and with certain wavevectors, even though the corresponding isotropic chiral medium does not. The constitutive relations and Beltrami-like fields are also used to develop a convenient spectral representation of the dyadic Green functions for the FCM.
14

Balmaseda, Aitor, Fabio Di Cosmo, and Juan Manuel Pérez-Pardo. "On Z -Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits." Symmetry 11, no. 8 (August 14, 2019): 1047. http://dx.doi.org/10.3390/sym11081047.

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An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the Laplace–Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace–Beltrami operator on an infinite set of intervals, Ω , constituting a quantum circuit, which are invariant under a given action of the group Z . A study of the different unitary representations of the group Z on the space of square integrable functions on Ω is performed and the corresponding Z -invariant self-adjoint extensions of the Laplace–Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples.
15

Bojarski, B., V. Gutlyanskii, and V. Ryazanov. "On existence and representation of solutions for general degenerate Beltrami equations." Complex Variables and Elliptic Equations 59, no. 1 (July 10, 2013): 67–75. http://dx.doi.org/10.1080/17476933.2013.795955.

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16

Radnell, David, Eric Schippers, and Wolfgang Staubach. "Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces." Communications in Contemporary Mathematics 19, no. 01 (November 24, 2016): 1650025. http://dx.doi.org/10.1142/s0219199716500255.

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Consider a Riemann surface of genus [Formula: see text] bordered by [Formula: see text] curves homeomorphic to the unit circle, and assume that [Formula: see text]. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in [Formula: see text] and [Formula: see text], and furthermore that the space of [Formula: see text] differentials in [Formula: see text] decomposes as a direct sum of infinitesimally trivial differentials and [Formula: see text] harmonic [Formula: see text] differentials. Thus the tangent space of this Teichmüller space is given by [Formula: see text] harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of [Formula: see text] harmonic Beltrami differentials.
17

Katz, D. B., and B. A. Kats. "Integral Representations for Solutions of Some Types of the Beltrami Equations." Russian Mathematics 62, no. 3 (March 2018): 18–22. http://dx.doi.org/10.3103/s1066369x18030039.

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18

Alsnayyan, A. M. A., and B. Shanker. "Laplace-Beltrami based multi-resolution shape reconstruction on subdivision surfaces." Journal of the Acoustical Society of America 151, no. 3 (March 2022): 2207–22. http://dx.doi.org/10.1121/10.0009851.

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The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics (MHs) provide the means to smoothly interpolate data on a manifold and are highly effective, specifically as it relates to geometry representation and editing; MHs form a natural basis for multi-resolution representation (and editing) of complex surfaces and functions defined therein. In this paper, we seek to develop the framework to exploit the benefits of MHs for shape reconstruction. To this end, a highly compressible, multi-resolution shape reconstruction scheme using MHs is developed. The method relies on subdivision basis sets to construct boundary element isogeometric methods for analysis and surface finite elements to construct MHs. This technique is paired with the volumetric source reconstruction method to determine an initial starting point. The examples presented highlight efficacy of the approach in the presence of noisy data, including a significant reduction in the number of degrees of freedom for complex objects, accuracy of reconstruction, and multi-resolution capabilities.
19

Green, Christopher C., and Jonathan S. Marshall. "Green's function for the Laplace–Beltrami operator on a toroidal surface." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2149 (January 8, 2013): 20120479. http://dx.doi.org/10.1098/rspa.2012.0479.

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Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.
20

Barabanov, Nikita E. "Isomorphism of Binary Operations in Differential Geometry." Symmetry 12, no. 10 (October 3, 2020): 1634. http://dx.doi.org/10.3390/sym12101634.

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We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.
21

Şeremet, Victor. "New Thermoelastic Green’s Functions by Using a New Integral Representation of Beltrami–Michel Equations." Journal of Engineering Mechanics 141, no. 11 (November 2015): 04015044. http://dx.doi.org/10.1061/(asce)em.1943-7889.0000940.

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22

Admal, Nikhil Chandra, and E. B. Tadmor. "The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation." Journal of the Mechanics and Physics of Solids 93 (August 2016): 72–92. http://dx.doi.org/10.1016/j.jmps.2016.03.016.

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23

Klimentov, S. B. "Representations of the “second kind” for the hardy classes of solutions to the Beltrami equation." Siberian Mathematical Journal 55, no. 2 (March 2014): 262–75. http://dx.doi.org/10.1134/s0037446614020098.

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24

Majdalani, Joseph. "On the generalized Beltramian motion of the bidirectional vortex in a conical cyclone." Physics of Fluids 34, no. 3 (March 2022): 036604. http://dx.doi.org/10.1063/5.0083740.

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This work presents an exact solution of Euler's incompressible equations in the context of a bidirectional vortex evolving inside a conically shaped cyclonic chamber. The corresponding helical flowfield is modeled under inviscid conditions assuming constant angular momentum. By leveraging the axisymmetric nature of the problem, a steady-state solution of the generalized Beltramian type is obtained directly from first principles, namely, from the Bragg–Hawthorne equation in spherical coordinates. The resulting stream function representation enables us to fully describe the ensuing swirl-dominated motion including its fundamental flow characteristics. After identifying an isolated singularity that appears at a cone divergence half-angle of 63.43°, two piecewise formulations are provided that correspond to either fluid injection or extraction at the top section of the conical cyclone. In this process, analytical expressions are readily retrieved for the three velocity components, vorticity, and pressure. Other essential flow indicators, such as the theoretically preferred mantle orientation, the empirically favored locus of zero vertical velocity, the maximum polar and axial velocities, the crossflow velocity, and other such terms, are systematically deduced. Results are validated using limiting process verifications and comparisons to both numerical and experimental measurements. The subtle differences between the present model and a strictly Beltramian flowfield are also highlighted and discussed. The conically cyclonic configuration considered here is relevant to propulsive devices, such as vortex-fired liquid rocket engines with tapered walls; meteorological phenomena, such as tornadoes, dust devils, and fire whirls; and industrial contraptions, such as cyclonic flow separators, collectors, centrifuges, boilers, vacuum cleaners, cement grinders, and so on.
25

Friedman, Michael. "Paper, Plaster, Strings: Exploratory Material Mathematical Models between the 1860s and 1930s." Perspectives on Science 29, no. 4 (July 2021): 436–67. http://dx.doi.org/10.1162/posc_a_00378.

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Abstract Does the materiality of a three-dimensional model have an effect on how this model operates in an exploratory way, how it prompts discovery of new mathematical results? Material mathematical models were produced and used during the second half of the nineteenth century, visualizing mathematical objects, such as curves and surfaces—and these were produced from a variety of materials: paper, cardboard, plaster, strings, wood. However, the question, whether their materiality influenced the status of these models—considered as exploratory, technical, or representational—was hardly touched upon. This article aims to approach this question by investigating two case studies: Beltrami’s paper models vs. Dyck’s plaster ones of the hyperbolic plane; and Chisini’s string models of braids vs. Artin’s and Moishezon’s algebraization of these braids. These two case studies indicate that materiality might have a decisive role in how the model was taken into account mathematically: either as an exploratory or rather as a technical or pedagogical object.
26

Akhmed-Zaki, D. Zh, O. N. Turar, and D. V. Lebedev. "FRAGMENTED ALGORITHM FOR ADAPTED GRID CONSTRUCTION." BULLETIN Series of Physics & Mathematical Sciences 71, no. 3 (September 30, 2020): 37–44. http://dx.doi.org/10.51889/2020-3.1728-7901.05.

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The paper describes the use of a computational fragments management automation system called LuNA (Language for Numerical Algorithms) for a fragmented solution for the problem of constructing an adaptive structured grid. The main idea behind LuNA is to use a high-level representation of an application's algorithm to enable it to automatically execute on hybrid node multicomputers without low-level programming by the researcher. The constructed computational grid structure is adapted to the values of the given function by solving the system of differential equations by the finite difference method. The inverse Beltrami equation is used as a differential meshing method. Several tests were carried out on a supercomputer using the described fragmented algorithm, including on a large number of threads for a mesh of 500 million nodes.
27

Kartashov, E. M. "Model representations of heat shock in terms of dynamic thermal elasticity." Russian Technological Journal 8, no. 2 (April 14, 2020): 85–108. http://dx.doi.org/10.32362/2500-316x-2020-8-2-85-108.

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This article is devoted to mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to the specific conditions of intensive heating and cooling of solids. A scheme is proposed for deriving the compatibility equation in voltages for dynamic problems, which generalizes the well-known Beltrami-Mitchell relation for quasistatic cases. The proposed relation can be used to consider numerous special cases in the theory of thermal shock in Cartesian coordinates for both bounded canonical bodies and partially bounded ones. As a detailed study, the latter case was considered under conditions of abrupt temperature heating and cooling, thermal heating and cooling, and medium heating and cooling. Numerical experiments were carried out, and the wave nature of the propagation of thermoelastic waves was described. The effect of relaxation of the solid boundary on sudden heating and sudden cooling, which has been little studied in thermomechanics, is described. It is established that this effect influences maximum of internal temperature stresses, which depend on the parameters characterizing the elastic and thermal properties of materials, as well as the heating time and cooling time. A “compatibility equation” in displacements was proposed to study the problem of thermal shock in cylindrical and spherical coordinate systems in bodies with a radial heat flow and central symmetry. The formulation of a generalized problem in the theory of thermal shock is formulated, which is of practical and theoretical interests for many areas of science and technology.
28

Barber, Timothy A., and Joseph Majdalani. "On the Beltramian motion of the bidirectional vortex in a conical cyclone." Journal of Fluid Mechanics 828 (September 12, 2017): 708–32. http://dx.doi.org/10.1017/jfm.2017.494.

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In this work, an exact Eulerian model is used to describe the steady-state motion of a bidirectional vortex in a conical chamber. This particular model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The corresponding bulk motion is assumed to be non-reactive, rotational, inviscid and incompressible. Then, following Bloor & Ingham (J. Fluid Mech., vol. 178, 1987, pp. 507–519), the spherical Bragg–Hawthorne equation is used to construct a mathematical model that connects the solution to the swirl number and the cone divergence angle. Consequently, a self-similar formulation is obtained independently of the cone’s finite body length. This enables us to characterize the problem using closed-form approximations of the principal flow variables. Among the cyclonic parameters of interest, the mantle divergence angle and the maximum cross-flow velocity are obtained explicitly. The mantle consists of a spinning cone that separates the circumferential inflow region from the central outflow. This interfacial layer bisects the fluid domain at approximately 60 per cent of the cone’s divergence half-angle. Its accurate determination is proven asymptotically using two different criteria, one being preferred by experimentalists. Finally, recognizing that the flow in question is of the Beltramian type, results are systematically described over a range of cone angles and spatial locations in both spherical and cylindrical coordinates; they are also compared to available experimental and numerical data.
29

Fisanov, V. V. "Representations of the Beltrami Fields in an Isotropic Chiral Medium with the Drude–Born–Fedorov Constitutive Relations." Russian Physics Journal 55, no. 9 (February 2013): 1022–27. http://dx.doi.org/10.1007/s11182-013-9916-8.

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30

Tomek, Lukáš, and Karol Mikula. "Discrete duality finite volume method with tangential redistribution of points for surfaces evolving by mean curvature." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 6 (October 18, 2019): 1797–840. http://dx.doi.org/10.1051/m2an/2019040.

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We propose a new discrete duality finite volume method for solving mean curvature flow of surfaces in ℝ3. In the cotangent scheme, which is widely used discretization of Laplace–Beltrami operator, a two-dimensional surface is usually approximated by a triangular mesh. In the cotangent scheme the unknowns are the vertices of the triangulation. A finite volume around each vertex is constructed as a surface patch bounded by a piecewise linear curve with nodes in the midpoints of the neighbouring edges and a representative point of each adjacent triangle. The basic idea of our new approach is to include the representative points into the numerical scheme as supplementary unknowns and generalize discrete duality finite volume method from ℝ2 to 2D surfaces embedded in ℝ3. To improve the quality of the mesh we use an area-oriented tangential redistribution of the grid points. We derive the numerical scheme for both closed surfaces and surfaces with boundary, and present numerical experiments. Surface evolution models are applied to construction of minimal surfaces with given set of boundary curves.
31

Zhang, Dan, and Kang Wang. "A Synthetic Feature Skull Descriptor for 3D Skull Similarity Measurement." Mathematical Problems in Engineering 2019 (November 30, 2019): 1–12. http://dx.doi.org/10.1155/2019/8083504.

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3D skull similarity measurement is a challenging and meaningful task in the fields of archaeology, forensic science, and anthropology. However, it is difficult to correctly and directly measure the similarity between 3D skulls which are geometric models with multiple border holes and complex topologies. In this paper, based on the synthetic feature method, we propose a novel 3D skull descriptor, synthetic wave kernel distance distribution (SWKDD) constructed by the laplace–beltrami operator. By defining SWKDD, we obtain a concise global skull representation method and transform the complex 3D skull similarity measurement into a simple 1D vector similarity measurement. First, we give the definition and calculation of SWKDD and analyse its properties. Second, we represent a framework for 3D skull similarity measurement using the SWKDD of 3D skulls and details of the calculation steps involved. Finally, we validate the effectiveness of our proposed method by calculating the similarity measurement of 3D skulls based on the real craniofacial database.
32

Bashiri, Fereshteh S., Reihaneh Rostami, Peggy Peissig, Roshan M. D’Souza, and Zeyun Yu. "An Application of Manifold Learning in Global Shape Descriptors." Algorithms 12, no. 8 (August 16, 2019): 171. http://dx.doi.org/10.3390/a12080171.

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With the rapid expansion of applied 3D computational vision, shape descriptors have become increasingly important for a wide variety of applications and objects from molecules to planets. Appropriate shape descriptors are critical for accurate (and efficient) shape retrieval and 3D model classification. Several spectral-based shape descriptors have been introduced by solving various physical equations over a 3D surface model. In this paper, for the first time, we incorporate a specific manifold learning technique, introduced in statistics and machine learning, to develop a global, spectral-based shape descriptor in the computer graphics domain. The proposed descriptor utilizes the Laplacian Eigenmap technique in which the Laplacian eigenvalue problem is discretized using an exponential weighting scheme. As a result, our descriptor eliminates the limitations tied to the existing spectral descriptors, namely dependency on triangular mesh representation and high intra-class quality of 3D models. We also present a straightforward normalization method to obtain a scale-invariant and noise-resistant descriptor. The extensive experiments performed in this study using two standard 3D shape benchmarks—high-resolution TOSCA and McGill datasets—demonstrate that the present contribution provides a highly discriminative and robust shape descriptor under the presence of a high level of noise, random scale variations, and low sampling rate, in addition to the known isometric-invariance property of the Laplace–Beltrami operator. The proposed method significantly outperforms state-of-the-art spectral descriptors in shape retrieval and classification. The proposed descriptor is limited to closed manifolds due to its inherited inability to accurately handle manifolds with boundaries.
33

Fist, Andrew, and Joseph Majdalani. "On the swirling Trkalian mean flow field in solid rocket motors." Journal of Fluid Mechanics 824 (July 5, 2017): 265–85. http://dx.doi.org/10.1017/jfm.2017.342.

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In this work, an exact Euler solution is derived under the fundamental contingencies of axisymmetric, steady, rotational, incompressible, single-phase, non-reactive and inviscid fluid, which also stand behind the ubiquitously used mean flow profile named ‘Taylor–Culick.’ In comparison with the latter, which proves to be complex lamellar, the present model is derived in the context of a Trkalian flow field, and hence is capable of generating a non-zero swirl component that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate for flow configurations where the bulk gaseous motion is driven to swirl. From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg–Hawthorne equation, which has been repeatedly shown to possess sufficient latitude to reproduce several existing profiles such as Taylor–Culick’s as special cases. Throughout this study, the fundamental properties of the present model are considered and discussed in the light of existing flow approximations. Consistent with the original Taylor–Culick mean flow motion, the Trkalian velocity is seen to exhibit both axial and tangential components that increase linearly with the distance from the headwall, and a radial component that remains axially invariant. Furthermore, the Trkalian model is shown to form a subset of the Beltramian class of solutions for which the velocity and vorticity vectors are not only parallel but also directly proportional. This characteristic feature is interesting, as it stands in sharp contrast to the complex-lamellar nature of the Taylor–Culick motion, where the velocity and vorticity vectors remain orthogonal. By way of verification, a numerical simulation is carried out using a finite-volume solver, thus leading to a favourable agreement between theoretical and numerical predictions.
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Reznikov, Andre. "A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces." Forum Mathematicum 27, no. 3 (January 1, 2015). http://dx.doi.org/10.1515/forum-2012-0185.

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AbstractWe consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace–Beltrami operator on a compact hyperbolic Riemann surface. We obtain uniform bounds for such periods as the corresponding eigenvalue tends to infinity. We use methods from the theory of automorphic functions and, in particular, the uniqueness of the corresponding invariant functionals on irreducible unitary representations of PGL
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Basteiro, Pablo, Janine Elfert, Johanna Erdmenger, and Haye Hinrichsen. "Fractional Klein-Gordon equation on AdS2+1." Journal of Physics A: Mathematical and Theoretical, July 20, 2022. http://dx.doi.org/10.1088/1751-8121/ac82d5.

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Abstract We propose a covariant definition of the fractional Klein-Gordon equation with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional Klein-Gordon equation on AdS2+1, computing the explicit kernel representation of the fractional Laplace-Beltrami operator as well as the two-point propagator of the fractional Klein-Gordon equation. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information.
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Kirchbach, M., T. Popov, and J. A. Vallejo. "Color confinement at the boundary of the conformally compactified AdS5." Journal of High Energy Physics 2021, no. 9 (September 2021). http://dx.doi.org/10.1007/jhep09(2021)171.

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Abstract The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of AdS5 spacetime, topologically equivalent to the closed manifold S1× S3, by considering the coupling of two massless opposite charges on it. Taking the interaction potential as the analog of Coulomb interaction (derived from a fundamental solution of the S3 Laplace-Beltrami operator), a conformal S1× S3 metric deformation is proposed, such that free motion on the deformed metric is equivalent to motion on the round metric in the presence of the interaction potential. We give explicit expressions for the generators of the conformal algebra in the representation induced by the metric deformation.By identifying the charge as the color degree of freedom in QCD, and the two charges system as a quark-anti-quark system, we argue that the associated conformal wave operator equation could provide a realistic quantum mechanical description of the simplest QCD system, the mesons.Finally, we discuss the possibility of employing the compactification radius, R, as an- other scale along ΛQCD, by means of which, upon reparametrizing Q2c2 as (Q2c2+ħ2c2/R2), a perturbative treatment of processes in the infrared could be approached.

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