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Journal articles on the topic 'Manifolds (Mathematics) Nonlinear theories'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

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 (December 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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2

Avramidi, Ivan G., and Giampiero Esposito. "Gauge Theories on Manifolds with Boundary." Communications in Mathematical Physics 200, no. 3 (February 1, 1999): 495–543. http://dx.doi.org/10.1007/s002200050539.

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3

Dimofte, Tudor, Davide Gaiotto, and Sergei Gukov. "Gauge Theories Labelled by Three-Manifolds." Communications in Mathematical Physics 325, no. 2 (December 15, 2013): 367–419. http://dx.doi.org/10.1007/s00220-013-1863-2.

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4

Wei, Shihsuh Walter. "The balance between existence and nonexistence theorems in differential geometry." Tamkang Journal of Mathematics 32, no. 1 (March 31, 2001): 61–88. http://dx.doi.org/10.5556/j.tkjm.32.2001.370.

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We discuss the delicate balance between existence and nonexistence theorems in differential geometry. Studying their interplay yields some information about $ p $-harmonic maps, $ p $-SSU manifolds, geometric $ k_p $-connected manifolds, minimal hypersurfaces and Gauss maps, and manifolds admitting essential positive supersolutions of certain nonlinear PDE. As an application of the theory developed, we obtain a topological theorem for minimal submanifolds in complete manifolds with nonpositive sectional curvature.

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5

Monnier, Samuel. "Topological field theories on manifolds with Wu structures." Reviews in Mathematical Physics 29, no. 05 (April 12, 2017): 1750015. http://dx.doi.org/10.1142/s0129055x17500155.

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We construct invertible field theories generalizing abelian prequantum spin Chern–Simons theory to manifolds of dimension [Formula: see text] endowed with a Wu structure of degree [Formula: see text]. After analyzing the anomalies of a certain discrete symmetry, we gauge it, producing topological field theories whose path integral reduces to a finite sum, akin to Dijkgraaf–Witten theories. We take a general point of view where the Chern–Simons gauge group and its couplings are encoded in a local system of integral lattices. The Lagrangian of these theories has to be interpreted as a class in a generalized cohomology theory in order to obtain a gauge invariant action. We develop a computationally friendly cochain model for this generalized cohomology and use it in a detailed study of the properties of the Wu Chern–Simons action. In the 3-dimensional spin case, the latter provides a definition of the “fermionic correction” introduced recently in the literature on fermionic symmetry protected topological phases. In order to construct the state space of the gauged theories, we develop an analogue of geometric quantization for finite abelian groups endowed with a skew-symmetric pairing. The physical motivation for this work comes from the fact that in the [Formula: see text] case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with [Formula: see text] supersymmetry, as will be discussed elsewhere.

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6

Park, Jae-Suk. "Semi-Classical Quantum Fields Theories and Frobenius Manifolds." Letters in Mathematical Physics 81, no. 1 (June 21, 2007): 41–59. http://dx.doi.org/10.1007/s11005-007-0165-z.

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7

Cattaneo, Alberto S., Pavel Mnev, and Nicolai Reshetikhin. "Perturbative Quantum Gauge Theories on Manifolds with Boundary." Communications in Mathematical Physics 357, no. 2 (December 5, 2017): 631–730. http://dx.doi.org/10.1007/s00220-017-3031-6.

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8

SONG, YI, and STEPHEN P. BANKS. "DYNAMICAL SYSTEMS ON THREE MANIFOLDS PART II: THREE-MANIFOLDS, HEEGAARD SPLITTINGS AND THREE-DIMENSIONAL SYSTEMS." International Journal of Bifurcation and Chaos 17, no. 06 (June 2007): 2085–95. http://dx.doi.org/10.1142/s0218127407018233.

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The global behavior of nonlinear systems is extremely important in control and systems theory since the usual local theories will only provide information about a system in some neighborhood of an operating point. Away from that point, the system may have totally different behavior and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on two- and three-manifolds and show that it is possible to obtain systems with "arbitrarily strange" behavior, i.e. arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.

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9

Wu, Siye. "Topological quantum field theories on manifolds with a boundary." Communications in Mathematical Physics 136, no. 1 (February 1991): 157–68. http://dx.doi.org/10.1007/bf02096795.

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10

Dung, Nguyen Thac, Pham Duc Thoan, and Nguyen Dang Tuyen. "Liouville theorems for nonlinear elliptic equations on Riemannian manifolds." Journal of Mathematical Analysis and Applications 496, no. 1 (April 2021): 124803. http://dx.doi.org/10.1016/j.jmaa.2020.124803.

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11

Kelnhofer, Gerald. "Abelian gauge theories on compact manifolds and the Gribov ambiguity." Journal of Mathematical Physics 49, no. 5 (May 2008): 052302. http://dx.doi.org/10.1063/1.2909197.

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12

BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (August 2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.

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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explanation of the physical world, essentially depends on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, it still plays a fundamental role in non-Abelian gauge theories and in superstring theory, thanks to which a great variety of new mathematical structures has emerged. The scope of this presentation is to highlight the importance of these mathematical structures for theoretical physics. A very interesting hypothesis is that the global topological properties of the manifold's model of space–time play a major role in quantum field theory (QFT) and that, consequently, several physical quantum effects arise from the nonlocal changing metrical and topological structure of these manifold. Thus the unification of general relativity and quantum theory require some fundamental breakthrough in our understanding of the relationship between space–time and quantum process. In particular the superstring theories lead to the guess that the usual structure of space–time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections, and curvature in explaining the essential laws of nature. Kaluza–Klein theories and more remarkably superstring theory showed that space–time symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. This essentially means, in our view, that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain class of (presumably) nonsmooth manifolds. In the second part of this paper, we address the subject of topological quantum field theories (TQFTs), which constitute a remarkably important meeting ground for physicists and mathematicians. TQFTs can be used as a powerful tool to probe geometry and topology in low dimensions. Chern–Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of QFTs which can be exactly (nonperturbatively) and explicitly solved. Abelian Chern–Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link (i.e. the union of a finite number of disjoint knots). In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the well-known Jones polynomial. Powerful methods for complete analytical and nonperturbative computation of these knot and link invariants have been developed. From these invariants for unoriented and framed links in S3, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish–Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern–Simons field theory. Even perturbative analysis of Chern–Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. In Donaldson–Witten theory perturbative methods have proved their relations to Donaldson invariants. Nonperturbative methods have been applied after the work by Seiberg and Witten on N = 2 supersymmetric Yang–Mills theory. The outcome of this application is a totally unexpected relation between Donaldson invariants and a new set of topological invariants called Seiberg–Witten invariants. Not only in mathematics, Chern–Simons theories find important applications in three- and four-dimensional quantum gravity also. Work on TQFT suggests that a quantum gravity theory can be formulated in three-dimensional space–time. Attempts have been made in the last years to formulate a theory of quantum gravity in four-dimensional space–time using "spin networks" and "spin foams". More generally, the developments of TQFTs represent a sort of renaissance in the relation between geometry and physics. The most important (new) feature of present developments is that links are being established between quantum physics and topology. Maybe this link essentially rests on the fact that both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. One very interesting example is the super-symmetric quantum mechanics theory, which has a deep geometric meaning. In the Witten super-symmetric quantum mechanics theory, where the Hamiltonian is just the Hodge–Laplacian (whereas the quantum Hamiltonian corresponding to a classical particle moving on a Riemannian manifold is just the Laplace–Beltrami differential operator), differential forms are bosons or fermions depending on the parity of their degrees. Witten went to introduce a modified Hodge–Laplacian, depending on a real-valued function f. He was then able to derive the Morse theory (relating critical points of f to the Betti numbers of the manifold) by using the standard limiting procedures relating the quantum and classical theories. Super-symmetric QFTs essentially should be viewed as the differential geometry of certain infinite-dimensional manifolds, including the associated analysis (e.g. Hodge theory) and topology (e.g. Betti numbers). A further comment is that the QFTs of interest are inherently nonlinear, but the nonlinearities have a natural origin, e.g. coming from non-Abelian Lie groups. Moreover there is usually some scaling or coupling parameter in the theory which in the limit relates to the classical theory. Fundamental topological aspects of such a quantum theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit. This means that such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory. In the last decade much effort has been done to use these QFTs as a conceptual tool to suggest new mathematical results. In particular, they have led to spectacular progress in our understanding of geometry in low dimensions. It is most likely no accident that the usual QFTs can only be renormalized in (space–time) dimensions ≤4, and this is precisely the range in which difficult phenomena arise leading to deep and beautiful theories (e.g. the work of Thurston in three dimensions and Donaldson in four dimensions). It now seems clear that the way to investigate the subtleties of low-dimensional manifolds is to associate to them suitable infinite-dimensional manifolds (e.g. spaces of connections) and to study these by standard linear methods (homology, etc.). In other words we use QFT as a refined tool to study low-dimensional manifolds.

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13

DE WIT, B., and A. VAN PROEYEN. "HIDDEN SYMMETRIES, SPECIAL GEOMETRY AND QUATERNIONIC MANIFOLDS." International Journal of Modern Physics D 03, no. 01 (March 1994): 31–47. http://dx.doi.org/10.1142/s0218271894000058.

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The moduli space of the Calabi-Yau three-folds, which play a role as superstring ground states, exhibits the same special geometry that is known from nonlinear sigma models in N=2 supergravity theories. We discuss the symmetry structure of special real, complex and quaternionic spaces. Maps between these spaces are implemented via dimensional reduction. We analyze the emergence of extra and hidden symmetries. This analysis is then applied to homogeneous special spaces and the implications for the classification of homogeneous quaternionic spaces are discussed.

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14

Lukas, Andre, and Challenger Mishra. "Discrete Symmetries of Complete Intersection Calabi–Yau Manifolds." Communications in Mathematical Physics 379, no. 3 (September 24, 2020): 847–65. http://dx.doi.org/10.1007/s00220-020-03838-6.

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Abstract In this paper, we classify non-freely acting discrete symmetries of complete intersection Calabi–Yau manifolds and their quotients by freely-acting symmetries. These non-freely acting symmetries can appear as symmetries of low-energy theories resulting from string compactifications on these Calabi–Yau manifolds, particularly in the context of the heterotic string. Hence, our results are relevant for four-dimensional model building with discrete symmetries and they give an indication which symmetries of this kind can be expected from string theory. For the 1695 known quotients of complete intersection manifolds by freely-acting discrete symmetries, non-freely-acting, generic symmetries arise in 381 cases and are, therefore, a relatively common feature of these manifolds. We find that 9 different discrete groups appear, ranging in group order from 2 to 18, and that both regular symmetries and R-symmetries are possible.

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15

Eguchi, Tohru, Yuji Sugawara, and Satoshi Yamaguchi. "Supercoset CFT’s for String Theories on Non-compact Special Holonomy Manifolds." Annales Henri Poincaré 4, S1 (December 2003): 93–95. http://dx.doi.org/10.1007/s00023-003-0908-z.

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16

Aref'eva, I. Ya, and I. V. Volovich. "Manifolds of constant negative curvature as vacuum solutions in Kaluza-Klein and superstring theories." Theoretical and Mathematical Physics 64, no. 2 (August 1985): 866–71. http://dx.doi.org/10.1007/bf01017969.

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17

Mokhov, O. I. "Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds." Theoretical and Mathematical Physics 152, no. 2 (August 2007): 1183–90. http://dx.doi.org/10.1007/s11232-007-0101-5.

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18

FORGER, MICHAEL, CORNELIUS PAUFLER, and HARTMANN RÖMER. "THE POISSON BRACKET FOR POISSON FORMS IN MULTISYMPLECTIC FIELD THEORY." Reviews in Mathematical Physics 15, no. 07 (September 2003): 705–43. http://dx.doi.org/10.1142/s0129055x03001734.

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We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

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19

Bonacina, Giuseppe, Maurizio Martellini, and Jeanette Nelson. "Generalized link-invariants on 3-manifolds ?h � [0, 1] from Chern-Simons gauge and gravity theories." Letters in Mathematical Physics 23, no. 4 (December 1991): 279–86. http://dx.doi.org/10.1007/bf00398825.

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20

Gliklikh, Yuri E., and Andrei V. Obukhovskii. "On a two-point boundary value problem for second-order differential inclusions on Riemannian manifolds." Abstract and Applied Analysis 2003, no. 10 (2003): 591–600. http://dx.doi.org/10.1155/s1085337503209027.

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We consider second-order differential inclusions on a Riemannian manifold with lower semicontinuous right-hand sides. Several existence theorems for solutions of two-point boundary value problem are proved to be interpreted as controllability of special mechanical systems with control on nonlinear configuration spaces. As an application, a statement of controllability under extreme values of controlling force is obtained.

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21

RICHARD, S., and R. TIEDRA DE ALDECOA. "SPECTRAL ANALYSIS AND TIME-DEPENDENT SCATTERING THEORY ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS." Reviews in Mathematical Physics 25, no. 02 (March 2013): 1350003. http://dx.doi.org/10.1142/s0129055x13500037.

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We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with the Eisenbud–Wigner time delay is finally presented. Our analysis mainly differs from the existing literature on the choice of a simpler comparison dynamics as well as on the complementary use of time-dependent and stationary scattering theories.

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22

Zeng, Fanqi. "Hamilton type gradient estimates for a general type of nonlinear parabolic equations on Riemannian manifolds." AIMS Mathematics 6, no. 10 (2021): 10506–22. http://dx.doi.org/10.3934/math.2021610.

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<abstract><p>In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $\end{document} </tex-math></disp-formula></p> <p>on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.</p></abstract>

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23

ZHU, Xiaobao. "Gradient estimates and liouville theorems for linear and nonlinear parabolic equations on riemannian manifolds." Acta Mathematica Scientia 36, no. 2 (March 2016): 514–26. http://dx.doi.org/10.1016/s0252-9602(16)30017-0.

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24

Wang, Wen. "Complement of gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds." Mathematical Methods in the Applied Sciences 40, no. 6 (August 4, 2016): 2078–83. http://dx.doi.org/10.1002/mma.4121.

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25

Jost, Jürgen, and Shing-Tung Yau. "A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry." Acta Mathematica 170, no. 2 (1993): 221–54. http://dx.doi.org/10.1007/bf02392786.

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26

Klyachin, Aleksey, and Vladimir Klyachin. "Research in the Field of Geometric Analysis at Volgograd State University." Mathematical Physics and Computer Simulation, no. 2 (August 2020): 5–21. http://dx.doi.org/10.15688/mpcm.jvolsu.2020.2.1.

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This article discusses the main directions of research in geometric analysis, which were conducted and are being carried out by the scientific mathematical school of Volgograd State University. The results of the founder of the scientific school, Doctor of Physics and Mathematics, Professor Vladimir Mikhailovich Miklyukov and his students are summarized. These results concern the solution of a number of problems in the field of quasiconformal flat mappings and mappings with bounded distortion of surfaces and Riemannian manifolds, the theory of minimal surfaces and surfaces of prescribed mean curvature, surfaces of zero mean curvature in Lorentz spaces, as well as problems associated with the study of the stability of such surfaces. In addition, the results of the study of various classes of triangulations — an object that appears at the junction of research in the field of geometric analysis and computational mathematics — are noted. Besides, this review discusses papers that use the Fourier decomposition method for solutions of the Laplace — Beltrami equations and the stationary Schr¨odinger equation with respect to the eigenfunctions of the corresponding boundary value problems. In particular, the authors give the results on finding capacitive characteristics that allowed for the first time to formulate and prove the criteria for the fulfillment of various theorems of Liouville type and the solvability of boundary value problems on model and quasimodel Riemannian manifolds. The role of the equivalent function method is also indicated in the study of such problems on manifolds of a fairly general form. In addition to this, the article gives an overview of the results concerning estimates of calculating error integral functionals and convergence of piecewise polynomial solutions of nonlinear variational type equations: minimal surface equations, equilibrium equations capillary surface and equations of biharmonic functions.

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Saberi, Ingmar, and Brian R. Williams. "Twisted characters and holomorphic symmetries." Letters in Mathematical Physics 110, no. 10 (August 3, 2020): 2779–853. http://dx.doi.org/10.1007/s11005-020-01319-4.

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Abstract We consider holomorphic twists of arbitrary supersymmetric theories in four dimensions. Working in the BV formalism, we rederive classical results characterizing the holomorphic twist of chiral and vector supermultiplets, computing the twist explicitly as a family over the space of nilpotent supercharges in minimal supersymmetry. The BV formalism allows one to work with or without auxiliary fields, according to preference; for chiral superfields, we show that the result of the twist is an identical BV theory, the holomorphic $$\beta \gamma $$ β γ system with superpotential, independent of whether or not auxiliary fields are included. We compute the character of local operators in this holomorphic theory, demonstrating agreement of the free local operators with the usual index of free fields. The local operators with superpotential are computed via a spectral sequence and are shown to agree with functions on a formal mapping space into the derived critical locus of the superpotential. We consider the holomorphic theory on various geometries, including Hopf manifolds and products of arbitrary pairs of Riemann surfaces, and offer some general remarks on dimensional reductions of holomorphic theories along the $$(n-1)$$ ( n - 1 ) -sphere to topological quantum mechanics. We also study an infinite-dimensional enhancement of the flavor symmetry in this example, to a recently studied central extension of the derived holomorphic functions with values in the original Lie algebra, that generalizes the familiar Kac–Moody enhancement in two-dimensional chiral theories.

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Rejzner, Kasia, and Michele Schiavina. "Asymptotic Symmetries in the BV-BFV Formalism." Communications in Mathematical Physics 385, no. 2 (April 5, 2021): 1083–132. http://dx.doi.org/10.1007/s00220-021-04061-7.

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AbstractWe show how to derive asymptotic charges for field theories on manifolds with “asymptotic” boundary, using the BV-BFV formalism. We also prove that the conservation of said charges follows naturally from the vanishing of the BFV boundary action, and show how this construction generalises Noether’s procedure. Using the BV-BFV viewpoint, we resolve the controversy present in the literature, regarding the status of large gauge transformation as symmetries of the asymptotic structure. We show that even though the symplectic structure at the asymptotic boundary is not preserved under these transformations, the failure is governed by the corner data, in agreement with the BV-BFV philosophy. We analyse in detail the case of electrodynamics and the interacting scalar field, for which we present a new type of duality to a sourced two-form model.

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Jost, Jürgen, and Shing-Tung Yau. "Erratum to: A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry." Acta Mathematica 173, no. 2 (1994): 307. http://dx.doi.org/10.1007/bf02398438.

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Huang, Wentao, Chengcheng Cao, and Dongping He. "Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow." International Journal of Bifurcation and Chaos 31, no. 02 (February 2021): 2150018. http://dx.doi.org/10.1142/s0218127421500188.

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In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

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Feehan, Paul M. N., and Manousos Maridakis. "Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 765 (August 1, 2020): 35–67. http://dx.doi.org/10.1515/crelle-2019-0029.

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AbstractWe prove several abstract versions of the Łojasiewicz–Simon gradient inequality for an analytic function on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, that of the well-known infinite-dimensional version of the gradient inequality due to Łojasiewicz [S. Łojasiewicz, Ensembles semi-analytiques, (1965), Publ. Inst. Hautes Etudes Sci., Bures-sur-Yvette. LaTeX version by M. Coste, August 29, 2006 based on mimeographed course notes by S. Łojasiewicz, https://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf] and proved by Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525–571]. We prove that the optimal exponent of the Łojasiewicz–Simon gradient inequality is obtained when the function is Morse–Bott, improving on similar results due to Chill [R. Chill, On the Łojasiewicz–Simon gradient inequality, J. Funct. Anal. 201 2003, 2, 572–601], [R. Chill, The Łojasiewicz–Simon gradient inequality in Hilbert spaces, Proceedings of the 5th European-Maghrebian workshop on semigroup theory, evolution equations, and applications 2006, 25–36], Haraux and Jendoubi [A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 2007, 3, 449–470], and Simon [L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lect. Math. ETH Zürich, Birkhäuser, Basel 1996]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for harmonic maps, preprint 2019, https://arxiv.org/abs/1903.01953], we apply our abstract gradient inequalities to prove Łojasiewicz–Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities generalize those of Kwon [H. Kwon, Asymptotic convergence of harmonic map heat flow, ProQuest LLC, Ann Arbor 2002; Ph.D. thesis, Stanford University, 2002], Liu and Yang [Q. Liu and Y. Yang, Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds, Ark. Mat. 48 2010, 1, 121–130], Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525–571], [L. Simon, Isolated singularities of extrema of geometric variational problems, Harmonic mappings and minimal immersions (Montecatini 1984), Lecture Notes in Math. 1161, Springer, Berlin 1985, 206–277], and Topping [P. M. Topping, Rigidity in the harmonic map heat flow, J. Differential Geom. 45 1997, 3, 593–610]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions, preprint 2019, https://arxiv.org/abs/1510.03815v6; to appear in Mem. Amer. Math. Soc.], we prove Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang–Mills energy function due to the first author [P. M. N. Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang–Mills gradient flow, preprint 2016, https://arxiv.org/abs/1409.1525v4] for base manifolds of arbitrary dimension and due to Råde [J. Råde, On the Yang–Mills heat equation in two and three dimensions, J. reine angew. Math. 431 1992, 123–163] for dimensions two and three.

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32

Lazaroiu, C. I., and C. S. Shahbazi. "Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds." Reviews in Mathematical Physics 30, no. 05 (May 31, 2018): 1850012. http://dx.doi.org/10.1142/s0129055x18500125.

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We give the global mathematical formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields on a Lorentzian four-manifold, for the generalized situation when the duality structure of the Abelian gauge theory is described by a flat symplectic vector bundle [Formula: see text] defined over the scalar manifold [Formula: see text]. The construction uses a taming of [Formula: see text], which we find to be the correct mathematical object globally encoding the inverse gauge couplings and theta angles of the “twisted” Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of [Formula: see text] to the bundle [Formula: see text] and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over [Formula: see text] and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows, in particular, that a generalization of part of the mathematical structure familiar from [Formula: see text] supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry.

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Branding, Volker. "Nonlinear Dirac Equations, Monotonicity Formulas and Liouville Theorems." Communications in Mathematical Physics 372, no. 3 (November 13, 2019): 733–67. http://dx.doi.org/10.1007/s00220-019-03608-z.

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Abstract We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

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34

Caccese, E. "On some involution theorems on twofold Poisson manifolds." Letters in Mathematical Physics 15, no. 3 (April 1988): 193–200. http://dx.doi.org/10.1007/bf00398587.

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Ding, Lu. "Positive mass theorems for higher dimensional Lorentzian manifolds." Journal of Mathematical Physics 49, no. 2 (February 2008): 022504. http://dx.doi.org/10.1063/1.2830803.

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36

Luskin, Mitchell, and George R. Sell. "Approximation theories for inertial manifolds." ESAIM: Mathematical Modelling and Numerical Analysis 23, no. 3 (1989): 445–61. http://dx.doi.org/10.1051/m2an/1989230304451.

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Guddat, J., H. Th Jongen, and J. Rueckmann. "On stability and stationary points in nonlinear optimization." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 28, no. 1 (July 1986): 36–56. http://dx.doi.org/10.1017/s033427000000518x.

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This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).

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38

Richard, Edouard, and Jean C. Vivalda. "Mathematical Analysis of Stability and Drift Behavior of Hydraulic Cylinders Driven by a Servovalve." Journal of Dynamic Systems, Measurement, and Control 124, no. 1 (May 18, 2001): 206–13. http://dx.doi.org/10.1115/1.1433482.

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Hydraulic cylinders are commonly used and many works deal with modeling and control of such devices. This article deals with the stability properties of hydraulic cylinders drift in various situations. The study is based on a classic nonlinear model of these physical systems. The cases of system models without leakages and models with cylinder leakages or servovalve leakages are distinguished and lead to distinct behaviors. The stability properties are proven by various mathematical arguments such as first integrals, Lyapunov theorems, LASALLE invariance principle, BARBALAT’s lemma, and the center manifold theory.

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Weinberger, Shmuel. "Fixed-point theories on noncompact manifolds." Journal of Fixed Point Theory and Applications 6, no. 1 (September 7, 2009): 15–25. http://dx.doi.org/10.1007/s11784-009-0112-y.

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Cao, Yalong, and Naichung Conan Leung. "Orientability for gauge theories on Calabi–Yau manifolds." Advances in Mathematics 314 (July 2017): 48–70. http://dx.doi.org/10.1016/j.aim.2017.04.030.

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Ohta, Shin-ichi. "Nonlinear geometric analysis on Finsler manifolds." European Journal of Mathematics 3, no. 4 (May 1, 2017): 916–52. http://dx.doi.org/10.1007/s40879-017-0143-7.

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Yamada, Shinichi, and Yoshiharu Kato. "Reflection principles for synthetic theories of smooth manifolds." Nonlinear Analysis: Theory, Methods & Applications 30, no. 8 (December 1997): 5135–46. http://dx.doi.org/10.1016/s0362-546x(96)00153-8.

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NG, S. K., and P. E. CAINES. "Nonlinear Filtering in Rieznnnnifln Manifolds." IMA Journal of Mathematical Control and Information 2, no. 1 (1985): 25–36. http://dx.doi.org/10.1093/imamci/2.1.25.

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Iwaniec, T., C. Scott, and B. Stroffolini. "Nonlinear Hodge theory on manifolds with boundary." Annali di Matematica Pura ed Applicata 177, no. 1 (December 1999): 37–115. http://dx.doi.org/10.1007/bf02505905.

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Bugajska, Krystyna. "Gauge theories on open spin space-time manifolds." International Journal of Theoretical Physics 26, no. 7 (July 1987): 637–47. http://dx.doi.org/10.1007/bf00670574.

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Byeon, Jaeyoung, and Junsang Park. "Singularly perturbed nonlinear elliptic problems on manifolds." Calculus of Variations and Partial Differential Equations 24, no. 4 (October 18, 2005): 459–77. http://dx.doi.org/10.1007/s00526-005-0339-4.

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FORGER, MICHAEL, and LEANDRO G. GOMES. "MULTISYMPLECTIC AND POLYSYMPLECTIC STRUCTURES ON FIBER BUNDLES." Reviews in Mathematical Physics 25, no. 09 (October 2013): 1350018. http://dx.doi.org/10.1142/s0129055x13500189.

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We propose new definitions of the concepts of a multisymplectic structure and of a polysymplectic structure which extend previous ones so as to cover the cases that are of interest in mathematical physics: they are tailored to apply to fiber bundles, rather than just manifolds, and at the same time they are sufficiently specific to allow us to prove Darboux theorems for the existence of canonical local coordinates. A key role is played by the notion of "symbol" of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, thus clarifying the relation between these two closely related but not identical concepts.

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48

Fan, Jinyan. "Duality theories in nonlinear semidefinite programming." Applied Mathematics Letters 18, no. 9 (September 2005): 1068–73. http://dx.doi.org/10.1016/j.aml.2004.09.017.

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Fomenko, A. T., and N. V. Krylov. "Nonlinear analysis on Manifolds: Monge-Amp�re equations." Acta Applicandae Mathematicae 8, no. 2 (February 1987): 206–10. http://dx.doi.org/10.1007/bf00046714.

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FUJII, KAZUYUKI, HIROSHI OIKE, and TATSUO SUZUKI. "UNIVERSAL YANG–MILLS ACTION ON FOUR-DIMENSIONAL MANIFOLDS." International Journal of Geometric Methods in Modern Physics 03, no. 07 (November 2006): 1331–40. http://dx.doi.org/10.1142/s0219887806001740.

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The usual action of the Yang–Mills theory is given by the quadratic form of curvatures of a principal G bundle defined on four-dimensional manifolds. The nonlinear generalization which is known as the Born–Infeld action has been given. In this paper we give another nonlinear generalization on four-dimensional manifolds and call it a universal Yang–Mills action. The advantage of our model is that the action splits automatically into two parts consisting of self-dual and anti-self-dual directions, that is, we have automatically the self-dual and anti-self-dual equations without solving the equations of motion as in usual case. Our method may be applicable to recent non-commutative Yang–Mills theories studied widely.

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