Characteristic Hypersurfaces Research Articles

Extension of Solutions to Holomorphic Partial Differential Equations

We derive sufficient conditions on a strictly pseudoconvex domain Ω and a linear, holomorphic partial differential operator P for the existence of solutions to Pu = 0 which are holomorphic in Ω near a point p in the boundary of Ω, but cannot be prolonged holomorphically across p. We first show how, given the existence of a supporting everywhere characteristic analytic hypersurface, a theorem of Tsuno can be used to construct the desired solutions. We then derive necessary conditions for the existence of such a hypersurface under the assumption that Ω is strictly pseudoconvex and simply characteristic at p with respect to P.

Local Propagation of Impulsive GravitationalWaves

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data. © 2015 Wiley Periodicals, Inc.

A Novel “Maximizing Kappa” Approach for Assessing the Ability of a Diagnostic Marker and Its Optimal Cutoff Value

Threshold-dependent accuracy measures such as true classification rates in ordered multiple-class (k > 3) receiver operating characteristic (ROC) hyper-surfaces have recently been used to assist with medical decision making. However, based on low power performance in some circumstances, we construct a new method that relies on the kappa coefficient to solve such diagnostic problems. Under the approach proposed in the present article, the statistics depend strongly on the cutoff threshold, which can be chosen to maximize the kappa statistics of true disease status and of the new biomarker. The Monte Carlo simulation results confirm the effectiveness of the proposed method in terms of its predictive power. The proposed design is then compared with the volume under the ROC hyper-surface by applying it to intracerebral hemorrhagic patients classified into five stroke classes using the National Institutes of Health Stroke Scale.

Propagation of singularities along characteristics of Maxwell's equations

We give a new proof that electromagnetic waves predict geometry, based on studying the propagation of singularities in first-order derivatives of generalized solutions to Maxwell's equations. As a byproduct, the growth of the intensity of the jumps in across a characteristic hypersurface is shown to be homogeneous of degree . We determine generalized solutions (whose first-order derivatives have jumps across a fixed characteristic line) to the initial value problem for Maxwell's equations in one space variable.

Acausality and nonunique evolution in generalized teleparallel gravity

We show the existence of physical superluminal modes and acausality in the Brans-Dicke type of extension of teleparallel gravity that includes F(T) gravity and teleparallel dark energy as special cases. We derive the characteristic hypersurface for the extra degrees of freedom in the theory, thereby showing that the time evolution is not unique and closed causal curves can appear. Furthermore, we present a concrete disastrous solution in Bianchi type I spacetime, in which the anisotropy in expansion can be any function of time, and thus anisotropy can emerge suddenly, a simple demonstration that the theory is physically problematic.

Solutions ramifiées à croissance lente de certaines équations de Fuchs quasi-linéaires

We consider a class of quasilinear fuchsian operators QQ of order m≥1m\textbackslashgeq 1, holomorphic in a neighborhood of the origin in Ct×Cnx\textbackslashmathbf\C\_\t\ \textbackslashtimes \textbackslashmathbf\C\_\x\^\n\, and having a simple characteristic hypersurface transverse to SS: t=0t=0. Under an assumption on the linear part of QQ, we construct solutions of the problem Qu=vQu=v in spaces of ramified functions of slow growth. The result is an extension of [15] to the quasilinear case.

An analysis of characteristics in nonlinear massive gravity

We study the Cauchy problem in a special case of nonlinear massive gravity. Despite being ghost free, it has recently been argued that the theory is inherently problematic due to the existence of superluminal shock waves. Furthermore, it is claimed that an acausal characteristic can arise for any choice of background. In order to further understand the causal structure of the theory, we carefully perform a detailed analysis of the characteristic equations and show that the theory does admit a well-posed Cauchy problem, i.e., there exists hypersurfaces that are not a characteristic hypersurface. Puzzles remain regarding the existence of a superluminal propagating mode in both the minimal ghost-free theory that we analyzed, as well as in the full nonlinear massive gravity. That is, our result should not be taken as any indication of the healthiness of the theory. We also give a detailed review of Cauchy–Kovalevskaya theorem and its application in the appendix, which should be useful for investigating causal structures of other theories of gravity.

General relativistic null-cone evolutions with a high-order scheme

We present a high-order scheme for solving the full non-linear Einstein equations on characteristic null hypersurfaces using the framework established by Bondi and Sachs. This formalism allows asymptotically flat spaces to be represented on a finite, compactified grid, and is thus ideal for far-field studies of gravitational radiation. We have designed an algorithm based on 4th-order radial integration and finite differencing, and a spectral representation of angular components. The scheme can offer significantly more accuracy with relatively low computational cost compared to previous methods as a result of the higher-order discretization. Based on a newly implemented code, we show that the new numerical scheme remains stable and is convergent at the expected order of accuracy.

On Two Aspects of the Painlevé Analysis

We use the Calogero equation to illustrate the following two aspects of the Painlevé analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painlevé test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Bäcklund transformation related to the Lax pair.

Long time existence of a class of contact discontinuities for second

The study of long time existence of classical smooth solutions for second order quasilinear wave equations has received much attention (see e.g. [1] and the references given there). Results were also obtained for continuous semilinear waves with gradient jumps on a characteristic hypersurface in [2] and for C1 quasilinear waves with second order derivatives jumps on a characteristic hypersurface in [3], [4]. In this paper we show how the methods and results of [2] can be extended to a class of continuous weak solutions, with gradient jumps on a characteristic hypersurface, for some second order quasilinear balance laws.

Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple

The aim of this paper is to construct a holomorphic solution, ramified around a simple characteristic hypersurface, for some linear Fuchsian equation of order $m\geq 1$. We consider an operator $L$, holomorphic in a neighborhood of the origin in ${\mathbb {C}}_t\times {\mathbb {C}}_x^n$, of the form $L=tA+B$ where $A$ and $B$ are linear partial differential operators of order $m$ and $m-1$, and where $A$ has a simple characteristic hypersurface transverse to $S:t=0$. Under an assumption linking the principal symbols of $A$ and $B$, the question is reduced to the study of an integro-differential Fuchsian equation with an additional variable $z$ that describes the universal covering of a pointed disk. It is an equation where terms like $t^lD_t^hD_x^\alpha (tD_t+1)^{-1}D_z^{-q}, l,h,q\in \mathbb {N}, \alpha \in \mathbb {N}^n$ with $l\leq 1$ and $h+|\alpha |\leq l+q$ appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.

Local Existence of a Solution of a Semilinear Wave Equation with Gradient in a Neighborhood of Initial Characteristic Hypersurfaces of a Lorentzian Manifold

We analyze an initial value problem for nonlinear wave equations with gradient in the second member, with data given on two transversely intersecting null hypersurfaces of a Lorentzian manifold. Existence and uniqueness of a solution is obtained in a (one-sided future) neighborhood of the initial data null hypersurfaces.

Uniqueness Results for Ill-Posed Characteristic Problems in Curved Space-Times

We prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski space-times, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern ill-posed Cauchy problems on bifurcate, characteristic hypersurfaces. In the case of the Kerr space-time, the hypersurface is precisely the event horizon of the black hole. The uniqueness theorem in this case, based on two Carleman estimates, is intimately connected to our strategy to prove uniqueness of the Kerr black holes among smooth, stationary solutions of the Einstein-vacuum equations, as formulated in [14].

Optimization of Restricted ROC Surfaces in Three-Class Classification Tasks

We have shown previously that an N-class ideal observer achieves the optimal receiver operating characteristic (ROC) hypersurface in a Neyman-Pearson sense. Due to the inherent complexity of evaluating observer performance even in a three-class classification task, some researchers have suggested a generally incomplete but more tractable evaluation in terms of a surface, plotting only the three "sensitivities." More generally, one can evaluate observer performance with a single sensitivity or misclassification probability as a function of two linear combinations of sensitivities or misclassification probabilities. We analyzed four such formulations including the "sensitivity" surface. In each case, we applied the Neyman-Pearson criterion to find the observer which achieves optimal performance with respect to each given set of "performance description variables" under consideration. In the unrestricted case, optimization with respect to the Neyman-Pearson criterion yields the ideal observer, as does maximization of the observer's expected utility. Moreover, during our consideration of the restricted cases, we found that the two optimization methods do not merely yield the same observer, but are in fact completely equivalent in a mathematical sense. Thus, for a wide variety of observers which maximize performance with respect to a restricted ROC surface in the Neyman-Pearson sense, that ROC surface can also be shown to provide a complete description of the observer's performance in an expected utility sense.

On Lars Hörmander's remark on the characteristic Cauchy problem

In 1990, L. Hörmander solved the Cauchy problem for the wave equation on a smooth spatially compact space–time, for data fixed on a Lipschitz and weakly spacelike hypersurface. He concluded his paper by a remark to the effect that his theorems should be valid for a Lipschitz metric. We extend his results to a Lipschitz metric for a spacelike hypersurface and to a metric whose regularity is intermediate between Lipschitz and C 1 for a totally characteristic hypersurface (the Goursat problem). To cite this article: J.-P. Nicolas, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x 0 + x n = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space \({{\mathcal S}'_{\partial_-}({\mathbb{R}}^n)}\) which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of \({{\mathcal S}'_{\partial_-}({\mathbb{R}}^n)}\) appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.

Restrictions on the three-class ideal observer's decision boundary lines

We are attempting to develop expressions for the coordinates of points on the three-class ideal observer's receiver operating characteristic (ROC) hypersurface as functions of the set of decision criteria used by the ideal observer. This is considerably more difficult than in the two-class classification task, because the conditional probabilities in question are not simply related to the cumulative distribution functions of the decision variables, and because the slopes and intercepts of the decision boundary lines are not independent; given the locations of two of the lines, the location of the third will be constrained depending on the other two. In this paper, we attempt to characterize those constraining relationships among the three-class ideal observer's decision boundary lines. As a result, we show that the relationship between the decision criteria and the misclassification probabilities is not one-to-one, as it is for the two-class ideal observer.

Problèmes de Goursat pour des systèmes semi-linéaires hyperboliques

For some semilinear hyperbolic systems of the second order with initial data on two intersecting characteristic hypersurfaces, and under a general hypothesis on the structure of the nonlinear terms, we prove that the solution of the so-called semilinear Goursat problem is defined, not only on a small neighbourhood of the intersection of these hypersurfaces, but also in a neighbourhood of the entire union of the hypersurfaces. To cite this article: D.E. Houpa, M. Dossa, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Removable singularities of holomorphic solutions of linear partial differential equations

In a complex domain V⊂Cn, let P be a linear holomorphic partial differential operator and K be its characteristic hypersurface. When the localization of P at K is a Fuchsian operator having a non-negative integral characteristic index, it is proved, under some conditions, that every holomorphic solution to Pu=0 in V∖K has a holomorphic extension in V. Besides, it is applied to the propagation of singularities for equations with non-involutive double characteristics.

Local existence of a solution of a semi-linear wave equation in a neighborhood of initial characteristic hypersurfaces

Dans cet article, nous nous placons dans l'espace de Minkowski R n+1 et nous nous interessons a une equation d'onde semi-lineaire avec donnees initiales sur des hypersurfaces caracteristiques. Nous prouvons l'existence et l'unicite d'une solution dans un voisinage dirige vers le futur d'un cote de ces hypersurfaces.