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Journal articles on the topic 'Local and global topology of singularity'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Meeks III, William H., Joaquín Pérez, and Antonio Ros. "Structure theorems for singular minimal laminations." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 763 (2020): 271–312. http://dx.doi.org/10.1515/crelle-2018-0036.

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AbstractWe apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of {\mathbb{R}^{3}}. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in {\mathbb{R}^{3}}, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.

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CAZALS, FRÉDÉRIC, and MARC POUGET. "DIFFERENTIAL TOPOLOGY AND GEOMETRY OF SMOOTH EMBEDDED SURFACES: SELECTED TOPICS." International Journal of Computational Geometry & Applications 15, no. 05 (2005): 511–36. http://dx.doi.org/10.1142/s0218195905001816.

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The understanding of surfaces embedded in E3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of core concepts from differential topology applied to smooth embedded surfaces. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of points induced by the contact of the surface with a sphere. This classification is further used to define ridges and their properties, and to recall the stratification properties of the medial axis. Finally, properties of the medial axis are used to present sufficient conditions ensuring that two embedded surfaces are ambient isotopic. From a theoretical perspective, we expect this survey to ease the access to intricate notions scattered over several sources. From a practical standpoint, we hope it will be useful for those interested in certified approximations of smooth surfaces.

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3

Liu, Jikai, Jingjing Yan, and Huangchao Yu. "Stress-constrained topology optimization for material extrusion polymer additive manufacturing." Journal of Computational Design and Engineering 8, no. 3 (2021): 979–93. http://dx.doi.org/10.1093/jcde/qwab028.

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Abstract This paper presents a comprehensive numerical and experimental study on stress-constrained topology optimization for Fused Deposition Modeling (FDM) additive manufacturing. The qp method is employed to avoid the singularity issue of stress-constrained problems. The P-norm function with stability transformation is adopted to build the global stress constraint with iterative corrections to eliminate the gap between the maximum local stress and the P-norm stress. The Heaviside projection is employed to generate clear-cut 0–1 designs. Two benchmark examples have been studied with the numerical algorithm. Experiments are performed on the topologically optimized MBB beam to investigate the impact of the FDM process parameters, including deposition path direction, building direction, and slicing layer height, on the resulted structural strength. The stress-constrained designs without and with Heaviside projection are comparatively tested with experiments. The stress-minimization designs subject to different P-norm parameters are compared both numerically and experimentally. Experiments show that the deposition path direction and the building direction evidently affect the derived structural strength. Moreover, overthin structural members may severely degrade the structural strength due to manufacturing and loading uncertainties.

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4

Kotlyar, Victor V., Sergey S. Stafeev, and Anton G. Nalimov. "Sharp Focusing of a Hybrid Vector Beam with a Polarization Singularity." Photonics 8, no. 6 (2021): 227. http://dx.doi.org/10.3390/photonics8060227.

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The key result of this work is the use of the global characteristics of the polarization singularities of the entire beam as a whole, rather than the analysis of local polarization, Stokes and Poincare–Hopf indices. We extend Berry’s concept of the topological charge of scalar beams to hybrid vector beams. We discuss tightly focusing a new type of nth-order hybrid vector light field comprising n C-lines (circular polarization lines). Using a complex Stokes field, it is shown that the field polarization singularity index equals n/2 and does not preserve in the focal plane. The intensity and Stokes vector components in the focal plane are expressed analytically. It is theoretically and numerically demonstrated that at an even n, the intensity pattern at the focus is symmetrical, and instead of C-lines, there occur C-points around which axes of polarization ellipses are rotated. At n = 4, C-points characterized by singularity indices 1/2 and ‘lemon’-type topology are found at the focus. For an odd source field order n, the intensity pattern at the focus has no symmetry, and the field becomes purely vectorial (with no elliptical polarization) and has n V-points, around which linear polarization vectors are rotating.

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5

Li, Piyu, Lei Mou, and Yanqin Xu. "Paratopological groups: Local versus global." Topology and its Applications 286 (December 2020): 107424. http://dx.doi.org/10.1016/j.topol.2020.107424.

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6

Anandan, J. "Global Topology and Local Violation of Discrete Symmetries." Physical Review Letters 81, no. 7 (1998): 1363–66. http://dx.doi.org/10.1103/physrevlett.81.1363.

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7

Vathy-Fogarassy, Agnes, and Janos Abonyi. "Local and global mappings of topology representing networks." Information Sciences 179, no. 21 (2009): 3791–803. http://dx.doi.org/10.1016/j.ins.2009.07.001.

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8

Beheshti, Zahra, Siti Mariyam Shamsuddin, and Sarina Sulaiman. "Fusion Global-Local-Topology Particle Swarm Optimization for Global Optimization Problems." Mathematical Problems in Engineering 2014 (2014): 1–19. http://dx.doi.org/10.1155/2014/907386.

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In recent years, particle swarm optimization (PSO) has been extensively applied in various optimization problems because of its structural and implementation simplicity. However, the PSO can sometimes find local optima or exhibit slow convergence speed when solving complex multimodal problems. To address these issues, an improved PSO scheme called fusion global-local-topology particle swarm optimization (FGLT-PSO) is proposed in this study. The algorithm employs both global and local topologies in PSO to jump out of the local optima. FGLT-PSO is evaluated using twenty (20) unimodal and multimodal nonlinear benchmark functions and its performance is compared with several well-known PSO algorithms. The experimental results showed that the proposed method improves the performance of PSO algorithm in terms of solution accuracy and convergence speed.

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9

Ge, Song, and Guoliang Fan. "Topology-aware non-rigid point set registration via global–local topology preservation." Machine Vision and Applications 30, no. 4 (2019): 717–35. http://dx.doi.org/10.1007/s00138-019-01024-w.

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10

Clapsadle, R. A., and R. H. Schelp. "Local edge colorings that are global." Journal of Graph Theory 18, no. 4 (1994): 389–99. http://dx.doi.org/10.1002/jgt.3190180409.

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11

Donelan, P. S. "Singularity-theoretic methods in robot kinematics." Robotica 25, no. 6 (2007): 641–59. http://dx.doi.org/10.1017/s0263574707003748.

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SUMMARYThe significance of singularities in the design and control of robot manipulators is well known, and there is an extensive literature on the determination and analysis of singularities for a wide variety of serial and parallel manipulators—indeed such an analysis is an essential part of manipulator design. Singularity theory provides methodologies for a deeper analysis with the aim of classifying singularities, providing local models and local and global invariants. This paper surveys applications of singularity-theoretic methods in robot kinematics and presents some new results.

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12

Khamitova, Daniya R., Vladislav A. Blatov, Lucia Carlucci, Gianfranco Ciani, and Davide M. Proserpio. "Local and global topology of two-dimensional structural groups." Acta Crystallographica Section A Foundations of Crystallography 65, a1 (2009): s306. http://dx.doi.org/10.1107/s0108767309093490.

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13

HERNÁNDEZ, DANIEL J., LUIS NÚÑEZ-BETANCOURT та EMILY E. WITT. "Local 𝔪-adic constancy of F-pure thresholds and test ideals". Mathematical Proceedings of the Cambridge Philosophical Society 164, № 2 (2017): 285–95. http://dx.doi.org/10.1017/s0305004117000196.

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AbstractIn this paper, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the 𝔪-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.

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14

ASOREY, M., A. IBORT, and G. MARMO. "GLOBAL THEORY OF QUANTUM BOUNDARY CONDITIONS AND TOPOLOGY CHANGE." International Journal of Modern Physics A 20, no. 05 (2005): 1001–25. http://dx.doi.org/10.1142/s0217751x05019798.

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We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold M with regular boundary Γ=∂M. The space ℳ of self-adjoint extensions of the covariant Laplacian on M is shown to have interesting geometrical and topological properties which are related to the different topological closures of M. In this sense, the change of topology of M is connected with the nontrivial structure of ℳ. The space ℳ itself can be identified with the unitary group [Formula: see text] of the Hilbert space of boundary data [Formula: see text]. This description, is shown to be equivalent to the classical von Neumann's description in terms of deficiency index subspaces, but it is more efficient and explicit because it is given only in terms of the boundary data, which are the natural external inputs of the system. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, [Formula: see text] (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary conditions reaches the Cayley submanifold 𝒞_. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space ℳ is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self-adjoint boundary conditions, the space 𝒞_ can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold 𝒞_ is dual of the Maslov class of ℳ. The phenomena are illustrated with some simple low dimensional examples.

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15

Crespi, Vincent H. "Relations between global and local topology in multiple nanotube junctions." Physical Review B 58, no. 19 (1998): 12671. http://dx.doi.org/10.1103/physrevb.58.12671.

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16

Afuni, Ahmad. "Local Regularity for the Harmonic Map and Yang–Mills Heat Flows." Journal of Geometric Analysis 31, no. 10 (2021): 9677–707. http://dx.doi.org/10.1007/s12220-021-00624-1.

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AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

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17

Mond, David. "Looking at bent wires – -codimension and the vanishing topology of parametrized curve singularities." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 2 (1995): 213–22. http://dx.doi.org/10.1017/s0305004100073060.

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Projecting a knot onto a plane – or, equivalently, looking at it through one eye – one sees a more or less complicated plane curve with a number of crossings (‘nodes’); viewing it from certain positions, some other more complicated singularities appear. If one spends a little time experimenting, looking at the knot from different points of view, then provided the knot is generic, one can convince oneself that there is only a rather short list of essentially distinct local pictures (singularities) – see Fig. 3 below. All singularities other than nodes are unstable: by moving one's eye slightly, one can make them break up into nodes. For each type X the following two numbers can easily be determined experimentally:1. the codimension in ℝ3 of the set View(X) of centres of projection (viewpoints) for which a singularity of type X appears, and2. the maximum number of nodes n into which the singularity X splits when the centre of projection is moved.

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18

Saji, Kentaro, and Masatomo Takahashi. "Singularities of smooth mappings with patterns." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 137, no. 2 (2007): 415–30. http://dx.doi.org/10.1017/s030821050500123x.

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We study smooth mappings with patterns which given by certain divergence diagrams of smooth mappings. The divergent diagrams of smooth mappings can be regard as smooth mappings from manifolds with singular foliations. Our concerns are generic differential topology and generic smooth mappings with patterns. We give a generic semi-local classification of surfaces with singularities and patterns as an application of singularity theory.

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19

STRUZIK, ZBIGNIEW R. "DETERMINING LOCAL SINGULARITY STRENGTHS AND THEIR SPECTRA WITH THE WAVELET TRANSFORM." Fractals 08, no. 02 (2000): 163–79. http://dx.doi.org/10.1142/s0218348x00000184.

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We present a robust method of estimating the effective strength of singularities (the effective Hölder exponent) locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima (WTMM) tree. In addition, we illustrate the direct estimation of the scaling spectrum of the effective singularity strength, and we link it to the established partition function-based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer-generated and real-life time series.

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20

Bertram, E., P. Erds, P. Horák, J. Širáň, and Z. S. Tuza. "Local and global average degree in graphs and multigraphs." Journal of Graph Theory 18, no. 7 (1994): 647–61. http://dx.doi.org/10.1002/jgt.3190180702.

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21

Frank, J., P. Cheeseman, and J. Stutz. "When Gravity Fails: Local Search Topology." Journal of Artificial Intelligence Research 7 (December 1, 1997): 249–81. http://dx.doi.org/10.1613/jair.445.

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Local search algorithms for combinatorial search problems frequently encounter a sequence of states in which it is impossible to improve the value of the objective function; moves through these regions, called plateau moves, dominate the time spent in local search. We analyze and characterize plateaus for three different classes of randomly generated Boolean Satisfiability problems. We identify several interesting features of plateaus that impact the performance of local search algorithms. We show that local minima tend to be small but occasionally may be very large. We also show that local minima can be escaped without unsatisfying a large number of clauses, but that systematically searching for an escape route may be computationally expensive if the local minimum is large. We show that plateaus with exits, called benches, tend to be much larger than minima, and that some benches have very few exit states which local search can use to escape. We show that the solutions (i.e., global minima) of randomly generated problem instances form clusters, which behave similarly to local minima. We revisit several enhancements of local search algorithms and explain their performance in light of our results. Finally we discuss strategies for creating the next generation of local search algorithms.

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22

Bonet-Revés, Carles, and Tere M-Seara. "Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems." Discrete and Continuous Dynamical Systems 36, no. 7 (2016): 3545–601. http://dx.doi.org/10.3934/dcds.2016.36.3545.

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23

Radzikowski, Marek J., and Rainer Verch. "A local-to-global singularity theorem for quantum field theory on curved space-time." Communications in Mathematical Physics 180, no. 1 (1996): 1–22. http://dx.doi.org/10.1007/bf02101180.

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24

Cavoretto, Roberto, Alessandra De Rossi, and Hanli Qiao. "Topology analysis of global and local RBF transformations for image registration." Mathematics and Computers in Simulation 147 (May 2018): 52–72. http://dx.doi.org/10.1016/j.matcom.2017.10.010.

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25

Saha, Punam K., and Azriel Rosenfeld. "Local and global topology preservation in locally finite sets of tiles." Information Sciences 137, no. 1-4 (2001): 303–11. http://dx.doi.org/10.1016/s0020-0255(01)00107-4.

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26

París, J., F. Navarrina, I. Colominas, and M. Casteleiro. "Topology optimization of continuum structures with local and global stress constraints." Structural and Multidisciplinary Optimization 39, no. 4 (2008): 419–37. http://dx.doi.org/10.1007/s00158-008-0336-2.

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27

Du, Quan, Alexei Livshits, Agnieszka Kwiatek, Makkuni Jayaram, and Alexander Vologodskii. "Protein-induced Local DNA Bends Regulate Global Topology of Recombination Products." Journal of Molecular Biology 368, no. 1 (2007): 170–82. http://dx.doi.org/10.1016/j.jmb.2007.02.010.

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28

LEE, H. K., K. I. GOH, B. KAHNG, and D. KIM. "INTERNET DATA PACKET TRANSPORT: FROM GLOBAL TOPOLOGY TO LOCAL QUEUEING DYNAMICS." International Journal of Bifurcation and Chaos 17, no. 07 (2007): 2485–90. http://dx.doi.org/10.1142/s0218127407018555.

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We study structural feature and evolution of the Internet at the autonomous systems level. Extracting relevant parameters for the growth dynamics of the Internet topology, we construct a toy model for the Internet evolution, which includes the ingredients of multiplicative stochastic evolution of nodes and edges and adaptive rewiring of edges. The model reproduces successfully structural features of the Internet at a fundamental level. We also introduce a quantity called the load as the capacity of node needed for handling the communication traffic and study its time-dependent behavior at the hubs across years. The load at hub increases with network size N as ~ N1.8. Finally, we study data packet traffic in the microscopic scale. The average delay time of data packets in a queueing system is calculated, in particular, when the number of arrival channels is scale-free. We show that when the number of arriving data packets follows a power law distribution, ~ n-λ, the queue length distribution decays as n1-λ and the average delay time at the hub diverges as ~ N(3-λ)/(γ-1) in the N → ∞ limit when 2 < λ < 3γ being the network degree exponent.

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29

Machon, Thomas, and Gareth P. Alexander. "Global defect topology in nematic liquid crystals." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2191 (2016): 20160265. http://dx.doi.org/10.1098/rspa.2016.0265.

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We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in R 3 , showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.

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30

Zhong, Penghong. "Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on S2." Advances in Mathematical Physics 2018 (August 1, 2018): 1–15. http://dx.doi.org/10.1155/2018/5370349.

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The singularity solution for the inhomogeneous Landau-Lifshitz (ILL) equation without damping term in n-dimensional space was investigated. The implicit singularity solution was obtained for the case where the target space is on S2. This solution can be classified into four types that cover the global and local solutions. An estimation of the energy density of one of these types indicates its exact decay rate, which allows a global solution with finite initial energy under n>3. Analysis of the four aperiodic solutions indicates that energy gaps that are first contributions to the literature of ILL will occur for particular coefficient settings, and these are shown graphically.

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31

HOBBS, C. A., and H. M. OSINGA. "BIFURCATIONS OF THE GLOBAL STABLE SET OF A PLANAR ENDOMORPHISM NEAR A CUSP SINGULARITY." International Journal of Bifurcation and Chaos 18, no. 08 (2008): 2207–22. http://dx.doi.org/10.1142/s021812740802166x.

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The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J0 where the Jacobian of the map is singular. The critical locus, denoted J1, is the image of J0. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set Ws of a fixed point or periodic orbit interacts with J1 near such a cusp point C1. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve Ws is tangent to J1 exactly at C1. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

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32

Duston, Christopher Levi. "Using cosmic strings to relate local geometry to spatial topology." International Journal of Modern Physics D 26, no. 04 (2017): 1750033. http://dx.doi.org/10.1142/s021827181750033x.

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In this paper, we will discuss how cosmic strings can be used to bridge the gap between the local geometry of our spacetime model and the global topology. The primary tool is the theory of foliations and surfaces, and together with observational constraints, we can isolate several possibilities for the topology of the spatial section of the observable universe. This implies that the discovery of cosmic strings would not just be significant for an understanding of structure formation in the early universe, but also for the global properties of the spacetime model.

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33

Liu, Dedao, Louis N. S. Chiu, Chris Davies, and Wenyi Yan. "A post-processing method to remove stress singularity and minimize local stress concentration for topology optimized designs." Advances in Engineering Software 145 (July 2020): 102815. http://dx.doi.org/10.1016/j.advengsoft.2020.102815.

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34

Wang, Jiqiang. "Vibration distribution in complex dynamical systems: Local topology design for global performance?" Mechanical Systems and Signal Processing 151 (April 2021): 107394. http://dx.doi.org/10.1016/j.ymssp.2020.107394.

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35

Alexandrin, Yurii Semenovich, Sergei Aleksandrovich Tuktarov, and Vasilii Vasil'evich Chedrik. "STRUCTURAL DESIGN OF HELICOPTER WING BASED ON TOPOLOGY AND GLOBAL-LOCAL OPTIMIZATION." TsAGI Science Journal 48, no. 1 (2017): 81–96. http://dx.doi.org/10.1615/tsagiscij.2017020620.

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36

Karuza, Elisabeth A., Sharon L. Thompson-Schill, and Danielle S. Bassett. "Local Patterns to Global Architectures: Influences of Network Topology on Human Learning." Trends in Cognitive Sciences 20, no. 8 (2016): 629–40. http://dx.doi.org/10.1016/j.tics.2016.06.003.

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37

Prince, Russell. "Local or global policy? Thinking about policy mobility with assemblage and topology." Area 49, no. 3 (2016): 335–41. http://dx.doi.org/10.1111/area.12319.

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38

Fu, Yu-Hsiang, Chung-Yuan Huang, and Chuen-Tsai Sun. "Using global diversity and local topology features to identify influential network spreaders." Physica A: Statistical Mechanics and its Applications 433 (September 2015): 344–55. http://dx.doi.org/10.1016/j.physa.2015.03.042.

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39

Timsit, Youri, and Péter Várnai. "Helical Chirality: a Link between Local Interactions and Global Topology in DNA." PLoS ONE 5, no. 2 (2010): e9326. http://dx.doi.org/10.1371/journal.pone.0009326.

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40

Benedini Riul, P., and R. Oset Sinha. "A relation between the curvature ellipse and the curvature parabola." Advances in Geometry 19, no. 3 (2019): 389–99. http://dx.doi.org/10.1515/advgeom-2019-0002.

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Abstract At each point in an immersed surface in ℝ4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. Recently, at the singular point of a corank 1 singular surface in ℝ3, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in ℝ4 to ℝ3 in a tangent direction, corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in ℝ4 at a certain point to the geometry of the projection of the surface to ℝ3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.

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41

Windels, Sam F. L., Noël Malod-Dognin, and Nataša Pržulj. "Graphlet Laplacians for topology-function and topology-disease relationships." Bioinformatics 35, no. 24 (2019): 5226–34. http://dx.doi.org/10.1093/bioinformatics/btz455.

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Abstract Motivation Laplacian matrices capture the global structure of networks and are widely used to study biological networks. However, the local structure of the network around a node can also capture biological information. Local wiring patterns are typically quantified by counting how often a node touches different graphlets (small, connected, induced sub-graphs). Currently available graphlet-based methods do not consider whether nodes are in the same network neighbourhood. To combine graphlet-based topological information and membership of nodes to the same network neighbourhood, we generalize the Laplacian to the Graphlet Laplacian, by considering a pair of nodes to be ‘adjacent’ if they simultaneously touch a given graphlet. Results We utilize Graphlet Laplacians to generalize spectral embedding, spectral clustering and network diffusion. Applying Graphlet Laplacian-based spectral embedding, we visually demonstrate that Graphlet Laplacians capture biological functions. This result is quantified by applying Graphlet Laplacian-based spectral clustering, which uncovers clusters enriched in biological functions dependent on the underlying graphlet. We explain the complementarity of biological functions captured by different Graphlet Laplacians by showing that they capture different local topologies. Finally, diffusing pan-cancer gene mutation scores based on different Graphlet Laplacians, we find complementary sets of cancer-related genes. Hence, we demonstrate that Graphlet Laplacians capture topology-function and topology-disease relationships in biological networks. Availability and implementation http://www0.cs.ucl.ac.uk/staff/natasa/graphlet-laplacian/index.html Supplementary information Supplementary data are available at Bioinformatics online.

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LI, XUELONG, ZHONGHUI LI, ENYUAN WANG, et al. "PATTERN RECOGNITION OF MINE MICROSEISMIC AND BLASTING EVENTS BASED ON WAVE FRACTAL FEATURES." Fractals 26, no. 03 (2018): 1850029. http://dx.doi.org/10.1142/s0218348x18500299.

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A microseismic (MS) monitoring system in a mine can monitor the MS signals generated by coal rock rupture and blasting waves and can distinguish the two types of waves more clearly to monitor and analyze the rupture and evolution process of coal rock. According to the nonlinearity characteristics of the waveform, the fractal characteristics of a mine’s MS and blasting waves are analyzed by simple fractal and multifractal theory, and the simple fractal dimension [Formula: see text] and multifractal parameters are obtained, respectively. Results show that the simple fractal dimension [Formula: see text] reflects the complexity and frequency structure of the wave. The simple fractal dimension [Formula: see text] of a blasting wave is larger than that of a mine MS wave, which indicates that the blasting wave is relatively complex with higher frequency, while the mine MS wave is relatively simple with lower frequency. However, the simple fractal dimension [Formula: see text] can only describe the wave integrity features, not the local features. The multifractal parameters can describe the local characteristics of the wave more finely, and the multifractal spectrum describes the probability information of the singularity exponent [Formula: see text]. The singularity exponential range and multifractal spectral width [Formula: see text] of the blasting wave are smaller than those of the mine MS wave, which indicates that the probability measure of distribution unevenness and the degree of partial parameter fluctuation of the blasting wave are more severe than those of the mine MS wave. Wave signal analysis based on simple fractal and multifractal methods can not only obtain the characteristics of the wave strength and spectral structure but also other important information, such as local singularity. Therefore, it is possible to more clearly and conspicuously identify mine MS and blasting waves, so that coal rock rupture can be monitored more accurately.

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Henry, Teague R., Kelly A. Duffy, Marc D. Rudolph, Mary Beth Nebel, Stewart H. Mostofsky, and Jessica R. Cohen. "Bridging global and local topology in whole-brain networks using the network statistic jackknife." Network Neuroscience 4, no. 1 (2020): 70–88. http://dx.doi.org/10.1162/netn_a_00109.

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Whole-brain network analysis is commonly used to investigate the topology of the brain using a variety of neuroimaging modalities. This approach is notable for its applicability to a large number of domains, such as understanding how brain network organization relates to cognition and behavior and examining disrupted brain network organization in disease. A benefit to this approach is the ability to summarize overall brain network organization with a single metric (e.g., global efficiency). However, important local differences in network structure might exist without any corresponding observable differences in global topology, making a whole-brain analysis strategy unlikely to detect relevant local findings. Conversely, using local network metrics can identify local differences, but are not directly informative of differences in global topology. Here, we propose the network statistic (NS) jackknife framework, a simulated lesioning method that combines the utility of global network analysis strategies with the ability to detect relevant local differences in network structure. We evaluate the NS jackknife framework with a simulation study and an empirical example comparing global efficiency in children with attention-deficit/hyperactivity disorder (ADHD) and typically developing (TD) children. The NS jackknife framework has been implemented in a public, open-source R package, netjack, available at https://cran.r-project.org/package=netjack .

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Xue, Jian, Shengwang Hao, Rong Yang, Ping Wang, and Yilong Bai. "Localization of deformation and its effects on power-law singularity preceding catastrophic rupture in rocks." International Journal of Damage Mechanics 29, no. 1 (2019): 86–102. http://dx.doi.org/10.1177/1056789519857121.

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Three distinct length scales are involved in the deformation evolution and catastrophic rupture of heterogeneous rocks in general: two essential ones are the specimen size macroscopically and the grain size at micro-scale respectively, the other is the emerging localized band of deformation and damage. The band initiates almost nearby the peak load, and the rupture eventually occurs afterwards within the localized band. In this paper, we report that with the evolution of concentrated high strain and damage in the localized band, a power-law singularity emerges within the localized band preceding the eventual rupture. The localization of deformation imposes a spatial non-uniqueness on the power-law singularity, and then leads to a trans-scale characteristic of the power-law singularity. Based on this characteristic, it is demonstrated that the singularity presented by the global response of a whole specimen comes from the singularity of local response in the localized band. The localization and the power-law singularity are associated precursory events, spatially and temporally, respectively, before macroscopic rupture. In particular, based on the power-law singularity exhibited in the zonal areas near or across the rupture surface, a prediction of the occurrence time of catastrophic rupture can be made accordingly. This provides a practically helpful approach to the prediction of rupture, merely by means of monitoring the zonal areas adjacent to the localized band.

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Wang, Jin-Liang, Zhi-Chun Yang, Tingwen Huang, and Mingqing Xiao. "Local and global exponential synchronization of complex delayed dynamical networks with general topology." Discrete & Continuous Dynamical Systems - B 16, no. 1 (2011): 393–408. http://dx.doi.org/10.3934/dcdsb.2011.16.393.

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Lin, Chia-Ying, Chun-Ching Hsiao, Po-Quan Chen, and Scott J. Hollister. "Interbody Fusion Cage Design Using Integrated Global Layout and Local Microstructure Topology Optimization." Spine 29, no. 16 (2004): 1747–54. http://dx.doi.org/10.1097/01.brs.0000134573.14150.1a.

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Furter, J. E., and J. C. Eilbeck. "Analysis of bifurcations in reaction–diffusion systems with no-flux boundary conditions: the Sel'kov model." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 2 (1995): 413–38. http://dx.doi.org/10.1017/s0308210500028109.

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A plot of the bifurcation diagram for a two-component reaction-diffusion equation with no-flux boundary conditions reveals an intricate web of competing stable and unstable states. By studying the one-dimensional Sel'kov model, we show how a mixture of local, global and numerical analysis can make sense of several aspects of this complex picture. The local bifurcation analysis, via the power of singularity theory, gives us a framework to work in. We can then fill in the details with numerical calculations, with the global analytic results fixing the outline of the solution set. Throughout, we discuss to what extent our results can be applied to other models.

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Külske, Christof, and Daniel Meißner. "Stable and Metastable Phases for the Curie–Weiss–Potts Model in Vector-Valued Fields via Singularity Theory." Journal of Statistical Physics 181, no. 3 (2020): 968–89. http://dx.doi.org/10.1007/s10955-020-02615-y.

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Abstract We study the metastable minima of the Curie–Weiss Potts model with three states, as a function of the inverse temperature, and for arbitrary vector-valued external fields. Extending the classic work of Ellis and Wang (Stoch Process Appl 35(1):59–79, 1990) and Wang (Stoch Process Appl 50(2):245–252, 1994) we use singularity theory to provide the global structure of metastable (or local) minima. In particular, we show that the free energy has up to four local minimizers (some of which may at the same time be global) and describe the bifurcation geometry of their transitions under variation of the parameters.

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Chojecki, Przemyslaw. "Weak local-global compatibility and ordinary representations." Rendiconti del Seminario Matematico della Università di Padova 139 (June 6, 2018): 159–83. http://dx.doi.org/10.4171/rsmup/139-5.

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Wolf, Joseph A. "Local and global homogeneity for three obstinate spheres." Differential Geometry and its Applications 76 (June 2021): 101757. http://dx.doi.org/10.1016/j.difgeo.2021.101757.

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