Journal articles: 'Differential geometry – Global differential geometry – Global differential geometry'

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Banchoff, Thomas F., and S. S. Chern. "Global Differential Geometry." American Mathematical Monthly 98, no. 7 (1991): 669. http://dx.doi.org/10.2307/2324949.

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Banchoff, Thomas F. "Global Differential Geometry. Editor, S. S. Chern." American Mathematical Monthly 98, no. 7 (1991): 669–71. http://dx.doi.org/10.1080/00029890.1991.11995775.

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Huhtanen, Marko. "Differential geometry of matrix inversion." MATHEMATICA SCANDINAVICA 107, no. 2 (2010): 267. http://dx.doi.org/10.7146/math.scand.a-15155.

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Essentially, there exists just the dimension segregating (square) matrix subspaces. In view of algebraic operations, this quantity is not particularly descriptive. For differential geometric information on matrix inversion, the second fundamental form is found for the set of inverses of the invertible elements of a matrix subspace. Several conditions for this form to vanish are given, such as being equivalent to a Jordan subalgebra. Global measures of curvature are introduced in terms of an analogy of the Nash fiber.

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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Mechanical Design 117, B (1995): 87–92. http://dx.doi.org/10.1115/1.2836475.

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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.

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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Vibration and Acoustics 117, B (1995): 87–92. http://dx.doi.org/10.1115/1.2838681.

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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.

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Simon, Udo. "Global uniqueness for ovaloids in Euclidean and affine differential geometry." Tohoku Mathematical Journal 44, no. 3 (1992): 327–34. http://dx.doi.org/10.2748/tmj/1178227299.

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Hale, J. K., and W. Z. Huang. "Global Geometry of the Stable Regions for Two Delay Differential Equations." Journal of Mathematical Analysis and Applications 178, no. 2 (1993): 344–62. http://dx.doi.org/10.1006/jmaa.1993.1312.

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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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Malta, Iaci, Nicolau C. Saldanha, and Carlos Tomei. "Morin singularities and global geometry in a class of ordinary differential operators." Topological Methods in Nonlinear Analysis 10, no. 1 (1997): 137. http://dx.doi.org/10.12775/tmna.1997.026.

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SARDANASHVILY, G. "GEOMETRY OF CLASSICAL HIGGS FIELDS." International Journal of Geometric Methods in Modern Physics 03, no. 01 (2006): 139–48. http://dx.doi.org/10.1142/s0219887806001065.

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In gauge theory, Higgs fields are responsible for spontaneous symmetry breaking. In classical gauge theory on a principal bundle P, a symmetry breaking is defined as the reduction of a structure group of this principal bundle to a subgroup H of exact symmetries. This reduction takes place if and only if there exists a global section of the quotient bundle P/H. It is a classical Higgs field. A metric gravitational field exemplifies such a Higgs field. We summarize the basic facts on the reduction in principal bundles and geometry of Higgs fields. Our goal is the particular covariant differential in the presence of a Higgs field.

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Bibikov, Pavel, and Alexander Malakhov. "On classification problems in the theory of differential equations: Algebra + geometry." Publications de l'Institut Math?matique (Belgrade) 103, no. 117 (2018): 33–52. http://dx.doi.org/10.2298/pim1817033b.

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We study geometric and algebraic approaches to classification problems of differential equations. We consider the so-called Lie problem: provide the point classification of ODEs y?? = F(x, y). In the first part of the paper we consider the case of smooth right-hand side F. The symmetry group for such equations has infinite dimension, so classical constructions from the theory of differential invariants do not work. Nevertheless, we compute the algebra of differential invariants and obtain a criterion for the local equivalence of two ODEs y?? = F(x, y). In the second part of the paper we develop a new approach to the study of subgroups in the Cremona group. Namely, we consider class of differential equations y?? = F(x, y) with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of C2, which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others.

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Chiu, Hung-Lin, and Pak Tung Ho. "Global Differential Geometry of Curves in Three-Dimensional Heisenberg Group and CR Sphere." Journal of Geometric Analysis 29, no. 4 (2018): 3438–69. http://dx.doi.org/10.1007/s12220-018-00122-x.

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Guo, Zhaolu, Gang Liu, Dahai Li, and Shenwen Wang. "Self-adaptive differential evolution with global neighborhood search." Soft Computing 21, no. 13 (2016): 3759–68. http://dx.doi.org/10.1007/s00500-016-2029-x.

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Llibre, Jaume, and Dana Schlomiuk. "The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order." Canadian Journal of Mathematics 56, no. 2 (2004): 310–43. http://dx.doi.org/10.4153/cjm-2004-015-2.

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AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

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Catino, Giovanni, Lorenzo Mazzieri, and Samuele Mongodi. "Rigidity of gradient Einstein shrinkers." Communications in Contemporary Mathematics 17, no. 06 (2015): 1550046. http://dx.doi.org/10.1142/s0219199715500467.

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In this paper, we consider a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63] and studied in [H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B27(2) (2006) 121–142] and we classify noncompact gradient shrinkers with bounded non-negative sectional curvature.

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Urban, Zbyněk, and Jana Volná. "Exactness of Lepage 2-forms and globally variational differential equations." International Journal of Geometric Methods in Modern Physics 16, supp02 (2019): 1950106. http://dx.doi.org/10.1142/s0219887819501068.

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The exactness equation for Lepage [Formula: see text]-forms, associated with variational systems of ordinary differential equations on smooth manifolds, is analyzed with the aim to construct a concrete global variational principle. It is shown that locally variational systems defined by homogeneous functions of degree [Formula: see text] are automatically globally variational. A new constructive method of finding a global Lagrangian is described for these systems, which include for instance the geodesic equations in Riemann and Finsler geometry.

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Dirkes, Rebecca K., Nathan C. Winn, Thomas J. Jurrissen та ін. "Global estrogen receptor-α knockout has differential effects on cortical and cancellous bone in aged male mice". FACETS 5, № 1 (2020): 328–48. http://dx.doi.org/10.1139/facets-2019-0043.

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Estrogen receptor-α knockout (ERKO) in female rodents results in bone loss associated with increased osteocyte sclerostin expression; whether this also occurs in males is unknown. Here, we examined the effects of ERKO on femoral cortical geometry, trabecular microarchitecture, and osteocyte sclerostin expression of the femur and lumbar vertebrae. At 14 months of age, male ERKO and wild-type (WT) littermates ( n = 6 per group) were sacrificed, and femora and vertebra were collected. Cortical geometry and trabecular microarchitecture were assessed via micro-computed tomography; osteocyte sclerostin expression was assessed via immunohistochemistry. ANCOVA with body weight was used to compare ERKO and WT for cortical geometry; t-tests were used for all other outcomes. Regardless of skeletal site, ERKO mice had greater trabecular bone volume and trabecular number and decreased trabecular separation compared with WT. In the femoral diaphysis, ERKO had lower total area, cortical area, and cortical thickness compared with WT. The percentage of sclerostin+ osteocytes was increased in ERKO animals in cortical bone but not in cancellous bone of the femur or the lumbar vertebrae. In conclusion, ERKO improved trabecular microarchitecture in aged male mice, but negatively altered femoral cortical geometry associated with a trend towards increased cortical sclerostin expression.

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ARTÉS, JOAN C., JAUME LLIBRE, and DANA SCHLOMIUK. "THE GEOMETRY OF QUADRATIC DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS OF SECOND ORDER." International Journal of Bifurcation and Chaos 16, no. 11 (2006): 3127–94. http://dx.doi.org/10.1142/s0218127406016720.

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Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass [Formula: see text] which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class [Formula: see text] also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for [Formula: see text], 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class [Formula: see text] reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

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He, Yang Jun, and Gui Jun Zhang. "Global Optimization of Tersoff Clusters Using Differential Evolution with Inexact Line Search." Applied Mechanics and Materials 48-49 (February 2011): 565–68. http://dx.doi.org/10.4028/www.scientific.net/amm.48-49.565.

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Differential Evolution with Inexact Line Search (DEILS) is proposed to determination of the ground-state geometry of atom clusters. DEILS algorithm adopts probabilistic inexact line search method in acceptance rule of differential evolution to accelerate the convergence as the region of global minimum is approached. More realistic many-body potential energy functions, namely the Tersoff and Tersoff-like semi-empirical potentials for silicon, are considered. Numerical studies indicate that the new algorithm is considerably faster and more reliable than original differential evolution algorithm, especially for large-scale global optimization problem of MBP6/Si(C). Moreover, some ground-state solutions, which are superior to the known best solution given in literature, are reported.

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Zhu, Jinghao, Shangrui Zhao, and Guohua Liu. "Solution to Singular Optimal Control by Backward Differential Flow." Journal of Control Science and Engineering 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/678679.

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This paper presents a backward differential flow for solving singular optimal control problems. By using Krotov equivalent transformation, the cost functional is converted to a class of global optimization problems. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. The proposed method is also used in solving a class of variational problems. Some examples are illustrated.

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Hitchin, Nigel J. "Shiing-Shen Chern. 26 October 1911 — 3 December 2004." Biographical Memoirs of Fellows of the Royal Society 60 (January 2014): 75–85. http://dx.doi.org/10.1098/rsbm.2014.0018.

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Shiing-Shen Chern was a towering figure in mathematics, both for his contributions to differential geometry and as a source of inspiration and encouragement for all mathematicians, and particularly those in China. Born in the final year of the Qing dynasty, and educated at a time when China was only beginning to set up Western-style universities, he lived to preside over the 2002 International Congress of Mathematicians in Beijing. He was a co-founder of the Mathematical Sciences Research Institute in Berkeley and its first Director in 1981; he also set up the Nankai Institute for Mathematics in 1985. His contributions to differential geometry were of foundational importance for the global viewpoint that developed in the postwar years, and the mathematical tools he introduced are now the common currency in geometry, topology and even aspects of theoretical physics.

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Aripov, R. G., and D. Khadzhiev. "The complete system of global differential and integral invariants of a curve in Euclidean geometry." Russian Mathematics 51, no. 7 (2007): 1–14. http://dx.doi.org/10.3103/s1066369x07070018.

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Zapata-Carratalá, Carlos. "Jacobi geometry and Hamiltonian mechanics: The unit-free approach." International Journal of Geometric Methods in Modern Physics 17, no. 12 (2020): 2030005. http://dx.doi.org/10.1142/s0219887820300056.

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We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function [Formula: see text]. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

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Kusano, Takaŝi, and William F. Trench. "Global existence of solutions of mixed sublinear-superlinear differential equations." Hiroshima Mathematical Journal 16, no. 3 (1986): 597–606. http://dx.doi.org/10.32917/hmj/1206130310.

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Khalil, Salma A., Mohammed A. Basheer, and Tarig A. Abdelhaleem. "Types of Derivatives: Concepts and Applications (II)." Journal of Mathematics Research 9, no. 1 (2017): 50. http://dx.doi.org/10.5539/jmr.v9n1p50.

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The notion of differential geometry is known to have played a fundamental role in unifying aspects of the physics of particles and fields, and have completely transformed the study of classical mechanics.In this paper we applied the definitions and concepts which we defined and derived in part (I) of our paper: Types of Derivatives: Concepts and Applications to problems arising in Geometry and Fluid Mechanics using exterior calculus. We analyzed this problem, using the geometrical formulation which is global and free of coordinates.

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Sun, Gaoji, Bai Yang, Zuqiao Yang, and Geni Xu. "An adaptive differential evolution with combined strategy for global numerical optimization." Soft Computing 24, no. 9 (2019): 6277–96. http://dx.doi.org/10.1007/s00500-019-03934-3.

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Ghosh, Arka, Swagatam Das, and Asit Kr Das. "A simple two-phase differential evolution for improved global numerical optimization." Soft Computing 24, no. 8 (2020): 6151–67. http://dx.doi.org/10.1007/s00500-020-04750-w.

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Akter, Sayeda Irin, Md Shahriar Mahmud, Md Kamrujjaman, and Hazrat Ali. "Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations." GANIT: Journal of Bangladesh Mathematical Society 40, no. 1 (2020): 28–42. http://dx.doi.org/10.3329/ganit.v40i1.48193.

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Numerical analysis is the area of mathematics that creates, analyzes, and implements algorithms for solving numerically the problems from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. Till now, numerous numerical methods have been introduced. Spectral method is one of those techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform (FFT). This study presents some of the fundamental ideas of spectral method. Orthogonal basis are used to establish computational algorithm. The accuracy and efficiency of proposed model are discussed. This manuscript estimates for the error between the exact and approximated discrete solutions. This paper shows that, grid points for polynomial spectral methods should lie approximately in a minimal energy configuration associated with inverse linear repulsion between points. The wave equation, linear and non-linear boundary value problems are solved using spectral method on the platform of MATLAB language. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 28-42

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Torbati, Mehdi, Tanmay P. Lele, and Ashutosh Agrawal. "Ultradonut topology of the nuclear envelope." Proceedings of the National Academy of Sciences 113, no. 40 (2016): 11094–99. http://dx.doi.org/10.1073/pnas.1604777113.

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The nuclear envelope is a unique topological structure formed by lipid membranes in eukaryotic cells. Unlike other membrane structures, the nuclear envelope comprises two concentric membrane shells fused at numerous sites with toroid-shaped pores that impart a “geometric” genus on the order of thousands. Despite the intriguing architecture and vital biological functions of the nuclear membranes, how they achieve and maintain such a unique arrangement remains unknown. Here, we used the theory of elasticity and differential geometry to analyze the equilibrium shape and stability of this structure. Our results show that modest in- and out-of-plane stresses present in the membranes not only can define the pore geometry, but also provide a mechanism for destabilizing membranes beyond a critical size and set the stage for the formation of new pores. Our results suggest a mechanism wherein nanoscale buckling instabilities can define the global topology of a nuclear envelope-like structure.

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Mustafa, Octavian G., and Yuri V. Rogovchenko. "Global Existence and Asymptotic Behavior of Solutions of Nonlinear Differential Equations." Funkcialaj Ekvacioj 47, no. 2 (2004): 167–86. http://dx.doi.org/10.1619/fesi.47.167.

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Bouzahir, Hassane, Honglian You, and Rong Yuan. "Global Attractor for Some Partial Functional Differential Equations with Infinite Delay." Funkcialaj Ekvacioj 54, no. 1 (2011): 139–56. http://dx.doi.org/10.1619/fesi.54.139.

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Bertolin, Cristiana. "G-fonctions et cohomologie des hypersurfaces singulières." Bulletin of the Australian Mathematical Society 55, no. 3 (1997): 353–83. http://dx.doi.org/10.1017/s0004972700034043.

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Our object of study is the arithmetic of the differential modules (l) (l ∈ ℕ – {0}), associated by Dwork's theory to a homogeneous polynomial f (λ,X) with coefficients in a number field. Our main result is that (1) is a differential module of type G, c'est-à-dire, a module those solutions are G-functions. For the proof we distinguish two cases: the regular one and the non regular one.Our method gives us an effective upper bound for the global radius of (l), which doesn't depend on “l” but only on the polynomial f (λ,X). This upper bound is interesting because it gives an explicit estimate for the coefficients of the solutions of (l).In the regular case we know there is an isomorphism of differential modules between (1) and a certain De Rham cohomology group, endowed with the Gauss-Manin connection, c'est-à-dire, our module “comes from geometry”. Therefore our main result is a particular case of André's theorem which assert that at least in the regular case, all modules coming from geometry are of type G.

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ESPOSITO, GIAMPIERO, DIEGO N. PELLICCIA, and FRANCESCO ZACCARIA. "GRIBOV PROBLEM FOR GAUGE THEORIES: A PEDAGOGICAL INTRODUCTION." International Journal of Geometric Methods in Modern Physics 01, no. 04 (2004): 423–41. http://dx.doi.org/10.1142/s0219887804000216.

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The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a nonlinear differential equation, and the various solutions of such a nonlinear equation represent different gauge configurations known as Gribov copies. Their occurrence (lack of global cross-sections from the point of view of differential geometry) is called Gribov ambiguity, and is here presented within the framework of a global approach to quantum field theory. We first give a simple (standard) example for the SU(2) group and spherically symmetric potentials, then we discuss this phenomenon in general relativity, and recent developments, including lattice calculations.

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Iwamiya, Toshiyuki. "Global existence of mild solutions to semilinear differential equations in Banach spaces." Hiroshima Mathematical Journal 16, no. 3 (1986): 499–530. http://dx.doi.org/10.32917/hmj/1206130305.

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Takei, Yoshitsugu. "The geometry of bicharacteristics and the global existence of holomorphic solutions of systems of linear differential equations." Journal of Mathematics of Kyoto University 31, no. 3 (1991): 845–73. http://dx.doi.org/10.1215/kjm/1250519734.

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Katsikadelis, John T. "A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions." Engineering Analysis with Boundary Elements 32, no. 5 (2008): 353–67. http://dx.doi.org/10.1016/j.enganabound.2007.09.001.

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Jin, Tian, Heliang Yuan, Keck-Voon Ling, Honglei Qin, and Jianrong Kang. "Differential Kalman Filter Design for GNSS Open Loop Tracking." Remote Sensing 12, no. 5 (2020): 812. http://dx.doi.org/10.3390/rs12050812.

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Global navigation satellite system (GNSS) positioning in an urban environment is in need for accurate, reliable and robust positioning. Unfortunately, conventional closed-loop tracking fails to meet the demand. The open loop tracking shows improved robustness, however, the precision is unsatisfactory. We propose a differential Kalman filter for open loop, of which the measurement vector contains the differential values of open loop navigation results between adjacent epochs. The differential Kalman filter makes use of the satellite geometry (i.e., spatial domain) and motion relationship (i.e., temporal domain) to filter frequency and code phase estimations of conventional open loop tracking. The improved performances of this architecture have been analyzed theoretically and demonstrated by road tests in an urban environment. The proposed architecture shows more than 50% accuracy improvement than the conventional open-loop tracking architecture.

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Resca, Lorenzo G., and Nicholas A. Mecholsky. "Geometry and Geodesy on the Primary Visual Cortex as a Surface of Revolution." Mathematical and Computational Applications 25, no. 4 (2020): 64. http://dx.doi.org/10.3390/mca25040064.

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Biological mapping of the visual field from the eye retina to the primary visual cortex, also known as occipital area V1, is central to vision and eye movement phenomena and research. That mapping is critically dependent on the existence of cortical magnification factors. Once unfolded, V1 has a convex three-dimensional shape, which can be mathematically modeled as a surface of revolution embedded in three-dimensional Euclidean space. Thus, we solve the problem of differential geometry and geodesy for the mapping of the visual field to V1, involving both isotropic and non-isotropic cortical magnification factors of a most general form. We provide illustrations of our technique and results that apply to V1 surfaces with curve profiles relevant to vision research in general and to visual phenomena such as ‘crowding’ effects and eye movement guidance in particular. From a mathematical perspective, we also find intriguing and unexpected differential geometry properties of V1 surfaces, discovering that geodesic orbits have alternative prograde and retrograde characteristics, depending on the interplay between local curvature and global topology.

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Causa, Flavia, and Giancarmine Fasano. "Improved In-Flight Estimation of Inertial Biases through CDGNSS/Vision Based Cooperative Navigation." Sensors 21, no. 10 (2021): 3438. http://dx.doi.org/10.3390/s21103438.

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This paper discusses the exploitation of a cooperative navigation strategy for improved in-flight estimation of inertial sensors biases on board unmanned aerial vehicles. The proposed multi-vehicle technique is conceived for a “chief” Unmanned Aerial Vehicle (UAV) and relies on one or more deputy aircrafts equipped with Global Navigation Satellite System (GNSS) antennas for differential positioning which also act as features for visual tracking. Combining carrier-phase differential GNSS and visual estimates, it is possible to retrieve accurate inertial-independent attitude information, thus potentially enabling improved bias estimation. Camera and carrier-phase differential GNSS measurements are integrated within a 15 states extended Kalman filter. Exploiting an ad hoc developed numerical environment, the paper analyzes the performance of the cooperative approach for inertial biases estimation as a function of number of deputies, formation geometry and distances, and absolute and relative dynamics. It is shown that exploiting two deputies it is possible to improve biases estimation, while a single deputy can be effective if changes of relative geometry and dynamics are also considered. Experimental proofs of concept based on two multi-rotors flying in formation are presented and discussed. The proposed framework is applicable beyond the domain of small UAVs.

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Liess, Otto. "Microlocal regularity for global solutions of partial differential equations with polynomial coefficients." Annals of Global Analysis and Geometry 5, no. 1 (1987): 23–51. http://dx.doi.org/10.1007/bf00140753.

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Philos, Ch G., I. K. Purnaras, and P. Ch Tsamatos. "Global Solutions Approaching Lines at Infinity to Second Order Nonlinear Delay Differential Equations." Funkcialaj Ekvacioj 50, no. 2 (2007): 213–59. http://dx.doi.org/10.1619/fesi.50.213.

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Khadjiev, Djavvat, and Ömer Pekşen. "The complete system of global integral and differential invariants for equi-affine curves." Differential Geometry and its Applications 20, no. 2 (2004): 167–75. http://dx.doi.org/10.1016/j.difgeo.2003.10.005.

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LØVGREN, ALF EMIL, YVON MADAY, and EINAR M. RØNQUIST. "GLOBAL C1 MAPS ON GENERAL DOMAINS." Mathematical Models and Methods in Applied Sciences 19, no. 05 (2009): 803–32. http://dx.doi.org/10.1142/s0218202509003632.

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In many contexts, there is a need to construct C1 maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C1. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are discussed and compared: the transfinite interpolation proposed by Gordon and Hall,12 the pseudo-harmonic extension proposed by Gordon and Wixom,13 a new generalization of the Gordon–Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean-valued extension proposed by Floater.8

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Poulter, Steven L., Yutaka Kosaki, David J. Sanderson, and Anthony McGregor. "Spontaneous object-location memory based on environmental geometry is impaired by both hippocampal and dorsolateral striatal lesions." Brain and Neuroscience Advances 4 (January 2020): 239821282097259. http://dx.doi.org/10.1177/2398212820972599.

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We examined the role of the hippocampus and the dorsolateral striatum in the representation of environmental geometry using a spontaneous object recognition procedure. Rats were placed in a kite-shaped arena and allowed to explore two distinctive objects in each of the right-angled corners. In a different room, rats were then placed into a rectangular arena with two identical copies of one of the two objects from the exploration phase, one in each of the two adjacent right-angled corners that were separated by a long wall. Time spent exploring these two objects was recorded as a measure of recognition memory. Since both objects were in different locations with respect to the room (different between exploration and test phases) and the global geometry (also different between exploration and test phases), differential exploration of the objects must be a result of initial habituation to the object relative to its local geometric context. The results indicated an impairment in processing the local geometric features of the environment for both hippocampus and dorsolateral striatum lesioned rats compared with sham-operated controls, though a control experiment showed these rats were unimpaired in a standard object recognition task. The dorsolateral striatum has previously been implicated in egocentric route-learning, but the results indicate an unexpected role for the dorsolateral striatum in processing the spatial layout of the environment. The results provide the first evidence that lesions to the hippocampus and dorsolateral striatum impair spontaneous encoding of local environmental geometric features.

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Gerbet, Daniel, and Klaus Röbenack. "An Algebraic Approach to Identifiability." Algorithms 14, no. 9 (2021): 255. http://dx.doi.org/10.3390/a14090255.

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This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and observability. Employing techniques from algebraic geometry such as polynomial ideals and Gröbner bases, local as well as global results are derived. The methods are illustrated on some example systems.

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Araújo, Gabriel. "Global regularity and solvability of left-invariant differential systems on compact Lie groups." Annals of Global Analysis and Geometry 56, no. 4 (2019): 631–65. http://dx.doi.org/10.1007/s10455-019-09682-9.

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Nassar, H., A. Lebée, and L. Monasse. "Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2197 (2017): 20160705. http://dx.doi.org/10.1098/rspa.2016.0705.

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Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

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Zou, Feng, Debao Chen, Renquan Lu, Suwen Li, and Lehui Wu. "Teaching–learning-based optimization with differential and repulsion learning for global optimization and nonlinear modeling." Soft Computing 22, no. 21 (2017): 7177–205. http://dx.doi.org/10.1007/s00500-017-2722-4.

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Semmelmann, Uwe, and Gregor Weingart. "The Weitzenböck machine." Compositio Mathematica 146, no. 2 (2010): 507–40. http://dx.doi.org/10.1112/s0010437x09004333.

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AbstractWeitzenböck formulas are an important tool in relating local differential geometry to global topological properties by means of the so-called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non-symmetric holonomy groups. We explicitly construct a basis of the space of Weitzenböck formulas. This classification allows us to find customized Weitzenböck formulas for applications such as eigenvalue estimates or Betti number estimates.

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Nigsch, E. A., and J. A. Vickers. "Nonlinear generalized functions on manifolds." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2244 (2020): 20200640. http://dx.doi.org/10.1098/rspa.2020.0640.

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In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.

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Journal articles on the topic 'Differential geometry – Global differential geometry – Global differential geometry'

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