Relevant bibliographies by topics / Strictly hyperbolic operator

Academic literature on the topic 'Strictly hyperbolic operator'

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Published: 14 March 2023

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Journal articles on the topic "Strictly hyperbolic operator"

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Esposito, Giampiero. "A parametrix for quantum gravity?" International Journal of Geometric Methods in Modern Physics 13, no. 05 (April 21, 2016): 1650060. http://dx.doi.org/10.1142/s0219887816500602.

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In the 60s, DeWitt discovered that the advanced and retarded Green functions of the wave operator on metric perturbations in the de Donder gauge make it possible to define classical Poisson brackets on the space of functionals that are invariant under the action of the full diffeomorphism group of spacetime. He therefore tried to exploit this property to define invariant commutators for the quantized gravitational field, but the operator counterpart of such classical Poisson brackets turned out to be a hard task. On the other hand, in the mathematical literature, it is by now clear that, rather than inverting exactly an hyperbolic (or elliptic) operator, it is more convenient to build a quasi-inverse, i.e. an inverse operator up to an operator of lower order which plays the role of regularizing operator. This approximate inverse, the parametrix, which is, strictly, a distribution, makes it possible to solve inhomogeneous hyperbolic (or elliptic) equations. We here suggest that such a construction might be exploited in canonical quantum gravity provided one understands what is the counterpart of classical smoothing operators in the quantization procedure. We begin with the simplest case, i.e. fundamental solution and parametrix for the linear, scalar wave operator; the next step are tensor wave equations, again for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear Einstein equations are studied, relying upon the well-established Choquet-Bruhat construction, according to which the fifth derivatives of solutions of a nonlinear hyperbolic system solve a linear hyperbolic system. The latter is solved by means of Kirchhoff-type formulas, while the former fifth-order equations can be solved by means of well-established parametrix techniques for elliptic operators. But then the metric components that solve the vacuum Einstein equations can be obtained by convolution of such a parametrix with Kirchhoff-type formulas. Some basic functional equations for the parametrix are also obtained, that help in studying classical and quantum version of the Jacobi identity.
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BYTSENKO, ANDREI A., and SERGIO ZERBINI. "SEMICLASSICAL APPROXIMATION FOR A CLASS OF QUANTUM p-BRANE MODELS." Modern Physics Letters A 08, no. 17 (June 7, 1993): 1573–84. http://dx.doi.org/10.1142/s0217732393001318.

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A class of quantum p-brane models is considered. An action that is linear in the determinant of the world-metric and equivalent to Nambu-Goto-Dirac p-brane action is proposed. A semiclassical approximation to path integral quantization is presented for the bosonic sector when the closed p-brane sweeps out an n=(p+1) dimensional compact hyperbolic manifold Hn/Γ, Γ being a strictly hyperbolic subgroup of isometries of the Lobachevsky space Hn. The computation of the related Laplace operator determinant is presented.
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Allilueva, A. I., S. Yu Dobrokhotov, S. A. Sergeev, and A. I. Shafarevich. "New representations of the Maslov canonical operator and localized asymptotic solutions for strictly hyperbolic systems." Doklady Mathematics 92, no. 2 (September 2015): 548–53. http://dx.doi.org/10.1134/s1064562415050129.

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Korzyuk, V. I., and Nguyen Van Vinh. "A MIXED PROBLEM FOR THE FOUR-ORDER ONE-DIMENSIONAL HYPERBOLIC EQUATION WITH PERIODIC CONDITIONS." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 54, no. 2 (July 1, 2018): 135–48. http://dx.doi.org/10.29235/1561-2430-2018-54-2-135-148.

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This article considers a classical solution of the boundary problem for the four-order strictly hyperbolic equation with four different characteristics. Note that the well-posed statement of mixed problems for hyperbolic equations not only depends on the number of characteristics, but also on their location. The operator appearing in the equation involves a composition of first-order differential operators. The equation is defined in the half-strip of two independent variables. There are Cauchy’s conditions at the domain bottom and periodic conditions at other boundaries. Using the method of characteristics, the analytic solution of the considered problem is obtained. The uniqueness of the solution is proved. We have also noted that the solution in the whole given domain is a composition of the solutions obtained in some subdomains. Thus, for the obtained classical solution to possess required smoothness, the values of these piecewise solutions, as well as their derivatives up to the fourth order must coincide at the boundary of these subdomains. A classical solution is understood as a function that is defined everywhere at all closure points of a given domain and has all classical derivatives entering the equation and the conditions of the problem.
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Nielsen, Frank. "The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain." Entropy 22, no. 9 (September 12, 2020): 1019. http://dx.doi.org/10.3390/e22091019.

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We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
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Cicognani, Massimo. "The Cauchy Problem for Strictly Hyperbolic Operators with Non-Absolutely Continuous Coefficients." Tsukuba Journal of Mathematics 27, no. 1 (June 2003): 1–12. http://dx.doi.org/10.21099/tkbjm/1496164556.

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Vasil'ev, V. A. "SHARPNESS AND THE LOCAL PETROVSKIĬ CONDITION FOR STRICTLY HYPERBOLIC OPERATORS WITH CONSTANT COEFFICIENTS." Mathematics of the USSR-Izvestiya 28, no. 2 (April 30, 1987): 233–73. http://dx.doi.org/10.1070/im1987v028n02abeh000880.

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Sogge, Christopher D. "On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators." Transactions of the American Mathematical Society 304, no. 2 (February 1, 1987): 733. http://dx.doi.org/10.1090/s0002-9947-1987-0911093-5.

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Gramchev, Todor V., and Peter R. Popivanov. "Singularities of the Solutions of Non-Correct Mixed Problems for Second Order Strictly Hyperbolic Operators." Mathematische Nachrichten 121, no. 1 (1985): 53–60. http://dx.doi.org/10.1002/mana.19851210107.

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Reiche, Sönke, and Benjamin Berkels. "Automated stacking of seismic reflection data based on nonrigid image matching." GEOPHYSICS 83, no. 3 (May 1, 2018): V171—V183. http://dx.doi.org/10.1190/geo2017-0189.1.

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Stacking of multichannel seismic reflection data is a crucial step in seismic data processing, usually leading to the first interpretable seismic image. Stacking is preceded by traveltime correction, in which all events contained in a common-midpoint (CMP) gather are corrected for their offset-dependent traveltime increase. Such corrections are often based on the assumption of hyperbolic traveltime curves, and a best fit hyperbola is usually sought for each reflection by careful determination of stacking velocities. However, assuming hyperbolic traveltime curves is not accurate in many situations, e.g., in the case of strongly curved reflectors, large offset-to-target-ratios, or strong anisotropy. Here, we found that an underlying model parameterizing the shape of the traveltime curve is not a strict necessity for producing high-quality stacks. Based on nonrigid image-matching techniques, we developed an alternative way of stacking, both independent of a reference velocity model and any prior assumptions regarding the shape of the traveltime curve. Mathematically, our stacking operator is based on a variational approach that transforms a series of seismic traces contained within a CMP gather into a common reference frame. Based on the normalized crosscorrelation and regularized by penalizing irregular displacements, time shifts are sought for each sample to minimize the discrepancy between a zero-offset trace and traces with larger offsets. Time shifts are subsequently exported as a data attribute and can easily be converted to stacking velocities. To demonstrate the feasibility of this approach, we apply it to simple and complex synthetic data and finally to a real seismic line. We find that our new method produces stacks of equal quality and velocity models of slightly better quality compared with an automated, hyperbolic traveltime correction and stacking approach for complex synthetic and real data cases.
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Book chapters on the topic "Strictly hyperbolic operator"

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Hörmander, Lars. "The Strictly Hyperbolic Cauchy Problem." In The Analysis of Linear Partial Differential Operators III, 385–415. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-49938-1_8.

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Colombini, Ferruccio, and Daniele Del Santo. "Strictly Hyperbolic Operators and Approximate Energies." In Analysis and Applications — ISAAC 2001, 253–77. Boston, MA: Springer US, 2003. http://dx.doi.org/10.1007/978-1-4757-3741-7_17.

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"The strictly hyperbolic Cauchy problem — construction of a parametrix." In Microlocal Analysis for Differential Operators, 67–76. Cambridge University Press, 1994. http://dx.doi.org/10.1017/cbo9780511721441.007.

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"Particle flow and invariant algebra of a semi-strictly hyperbolic system; coordinate invariance of Opψxm." In The Technique of Pseudodifferential Operators, 282–309. Cambridge University Press, 1995. http://dx.doi.org/10.1017/cbo9780511569425.011.

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