...für Differentialgleichungen von normalem hyperbolischem...

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ArticleEine neue Lösungsmethode für Differentialgleichungen von normalem hyperbolischem TypusDecember 1933Mathematische Annalen 107(1):400-419DOI: 10.1007/BF01448901Authors:

Myron Mathisson

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Request full-text PDFCitations (12)References (1)Hilberts Arbeiten über Integralgleichungen und unendliche GleichungssystemeChapterErnst HellingerHilberts Untersuchungen über Integralgleichungen sind aus dem Bestreben entstanden, mit einem einheitlichen theoretischen Ansatz einen möglichst großen Bereich der linearen Probleme der Analysis, darunter speziell die linearen Randwertaufgaben der gewöhnlichen und partiellen linearen Differentialgleichungen und die aus ihnen entspringenden Reihenentwicklungen, zu umfassen. Er verfolgt hier das gleiche Ziel auf einem weiter ausholenden und weiter führenden Wege, das er unmittelbar zuvor mit Methoden der Variationsrechnung (Dirichletsches Prinzip, s. Abh. 3 und 4) in Angriff genommen hatte. Charakteristisch für diesen neuen Ansatz und entscheidend für die Art seiner Durchführung ist das Bestreben, einheitliche Tatsachenkomplexe der Analysis als naturgemäße Ausdehnung algebraischer Tatsachen zu erkennen und ihre Herleitung in möglichst weitgehender methodischer Analogie zur Algebra zu vollziehen2. Solche Beziehungen waren bei einzelnen Problemen schon vielfach mehr oder weniger ausdrücklich zum mindesten als heuristisches Hilfsmittel zur Geltung gekommen; sie sollen in dem folgenden kurzen Überblick über das, was Hilbert an Gedanken und Resultaten als Material für seine Theorie vorfand, besonders hervorgehoben werden3.ViewShow abstractSingularities, Supports and LacunasArticleJan 1986G. F. D. DuffThese notes are a survey of three aspects of the modern theory of linear partial differential equations, and its generalization to the microlocal analysis of pseudodifferential operators. The first chapter is a study of the propagation of singularities of partial and pseudo differential equations, beginning with a sketch of the extensive background of pseudo differential and Fourier integral operators and wave front sets — the machinery of microlocal analysis in phase space. Selected results on equations with multiple characteristics are then discussed, in the involutive and non-involutive cases. The second chapter is a description of the work of C. Fefferman and others on the approximate simultaneous diagonalization of differential operators with variable coefficients, regarded as algebraic operators in phase space. The uncertainty principle, a title borrowed from Heisenberg’s quantum mechanics, limits the precision of this process, since a function and its Fourier transform cannot both have small supports. This area of investigation, which draws upon the full resources of microlocal analysis, appears to have interesting future prospects.ViewShow abstractReview: T. C. Hu, Integer programming and network flows, and Harold Greenberg, Integer programming, and R. S. Garfinkel and G. L. Nemhauser, Integer programming, and T. C. Hu and Stephen M. Robinson, Mathematical programming, and Harvey M. Salkin, Integer programming, Hamdy A. Taha, Integer programming: Theory, applications and computationsArticleJan 1978B AM MATH SOC

Ellis L. JohnsonViewReview: F. G. Friedlander, The wave equation on a curved space-timeArticleJan 1978B AM MATH SOCAvron DouglisViewAllowed gravitational Lagrangians imposing δ( t-r)/r as weak-field limit Green functionArticleFull-text availableOct 1995Il Nuovo CimentoJ. J. Giambiagi

Juan antonio Nieto

Osiris ObregonSummary A discussion is given on the restrictions imposed on generalized gravitational Lagrangians by requiring that the Green function,for their corresponding weak-field limit, be of the form δ(t-r)/r, which implies the very physical fact that waves propagate with only one velocity. It is shown that in four dimensions onlya Lagrangian proportional toR is allowed and for six dimensions it should be proportional toRΜΝ. No powers of combinations of the curvature tensors give a satisfactory theory for dimensionsn 6.ViewShow abstractCalculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light BeamArticleFeb 1972PHYS REV DChristian ImbertWiegrefe, Fedorov, Costa de Beauregard, and Schilling have discussed the transverse energy flux existing in total reflection of an elliptically polarized light beam, the latter two proposing formulas for the transverse shift of the reflected beam. We have calculated the transverse shift by an energy-flux-conservation argument similar to Kristoffel s and to Renard s in their deduction of the longitudinal Goos-Hänchen shift, thus obtaining a formula different from those of the previous authors. We have also tested experimentally the existence of the transverse shift, in the optimal case of circular polarization and quasilimit total reflection, by using two slightly different multiplying procedures. Our measurements definitely vindicate our own formula for the transverse shift against both Costa de Beauregard s and Schilling s. The relevance of our results in connection with noncollinearity of velocity and momentum of the spinning photon inside the evanescent wave is very briefly discussed.ViewShow abstractAn Explicit Determination of the Empty Space–Times on Which the Wave Equation Satisfies Huygens PrincipleArticleJan 1969

Raymond G. MclenaghanThe validity of Huygens principle in the sense of Hadamard s ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard s necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens principle.ViewShow abstractL effet inertial de spin du photon Th orie et preuve exp rimentaleArticleDec 2002C ImbertUne onde électromagnétique, dont les champs sont indépendants d une des coordonnées, peut avoir une composante non nulle du vecteur de Poynting suivant cette coordonnée. Les ondes planes ayant cette propriété sont d une part celles en propagation dans un milieu absorbant et d autre part les ondes évanescentes. Lorsqu un faisceau se réfléchit totalement sur un dioptre plan, il existe une onde de surface dans le milieu le moins réfringent. Si le faisceau incident est polarisé circulairement, le vecteur de Poynting de cette onde évanescente est en dehors du plan d incidence; ceci permet de prévoir un décalage, du faisceau réfléchi, dans la direction perpendiculaire à ce plan d incidence, d un côté ou de l autre suivant le sens de la polarisation circulaire. Cette translation, bien que petite (de l ordre de la demi-longueur d onde) a été mesurée et les résultats expérimentaux sont en accord avec les prévisions théoriques.ViewShow abstractThe problem of cauchy for linear, hyperbolic equations of second orderArticleMay 1954Comm Pure Appl MathAvron DouglisThe definition of hyperbolic equation by a prescribed vector field is introduced for linear differential equation of the second order. The Cauchy problem with prescribed boundary conditions is considered for such equations. The theorems of existence and uniqueness of a strong solution to the given problem are proved by the method of energy inequalities and mollifiers with variable step. Key words: hyperbolic equation, Cauchy problem, strong solution, energy inequality, mollifiersViewShow abstractSome new results on the validity of Huygens principle for the scalar wave equation on a curved space-timeChapter

John Carminati

Raymond G. MclenaghanIt is shown that every Petrov type N space-time on which the conformally invariant scalar wave equation satisfies Huygens principle is conformally related to a special complex recurrent space-time.ViewShow abstractEnergy-momentum quanta in Fresnel s evanescent waveArticleJan 1971INT J THEOR PHYSO. Costa de BeauregardC. ImbertJ. RicardFresnel s theory of the evanescent wave in total reflection entails that the propagation vectork and the momentum quantak have an imaginary component and, thus, a projection on the reflecting plane that is larger (in units such thatc=1) than the angular frequency and the energy quanta. We discuss the tachyon properties of these energy-momentum quanta and propose an experimental test using absorption or stimulated emission by an atomic or ionic beam. We then show that the Maxwell-Minkowski tensor (although certainly appropriate to discuss the macroscopic energy-momentum exchange between wave and diopter) does not describe adequately the energy-momentum density of the quanta in the evanescent wave, this stemming from its too remote connection with the generator i of space-time displacements. On the other hand de Broglie s energy-momentum tensorAk[i]Bis the density canonically associated with the generator of space-time displacements; we show that it describes quite satisfactorily both the energy fluxes (as measured through the longitudinal Goos-Hnchen and our new transverse shifts of the reflected beam in total reflection) and the momentum densities of the quanta inside the evanescent wave. Finally, we show that it is the gauge which is transverse in the diopter s rest frame that directly yields the physically measured energy fluxes. We take this fact as a new argument, strongly supported by experimental evidence, in favour of the physical reality of electromagnetic potentials.ViewShow abstractLe problème de M. Hadamard relatif à la diffusion des ondesArticleDec 1939ACTA MATH-DJURSHOLMMyron MathissonViewWir k6nnen voraussetzen, da~ das Element zwei zugeordnete Punkte A, A verbindet, denn die L~nge irgendeines Elements AC zwischen den NnJllnlen ist, bis auf hShere .GrSl~en, der L~nge von AA greichM MathissbnM. Mathissbn. Wir k6nnen voraussetzen, da~ das Element zwei zugeordnete Punkte A, A verbindet, denn die L~nge irgendeines Elements AC zwischen den NnJllnlen ist, bis auf hShere.GrSl~en, der L~nge von AA greich, wie man aus dem Dreieck A C-.~errechnet (A-C ist Nullinienelemeut mad zu A-4 orthogonal).Recommended publicationsDiscover moreArticleEine neue Integrationsmethode f�r Differentialgleichungen von normalem hyperbolischem TypusJanuary 1933[...]Myron MathissonRead moreArticleEine neue Lösungsmethode für die algebraische Gleichung vierten GradesJanuary 1960[...]Raimo LehtiRead moreArticleRandwertaufgaben bei partiellen Differentialgleichungen von hyperbolischem TypusJanuary 1909[...]A. MyllerRead moreArticleUntersuchung des Cauchy-Problems für Evolutionsgleichungen von hyperbolischem TypusJanuary 1964[...]S.Ya. YakubovRead moreDiscover the world s researchJoin ResearchGate to find the people and research you need to help your work.Join for free

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