Fluctuations of an Evaporating Black Hole from Back Reaction of Its Hawking Radiation: Questioning a Premise in Earlier Work

This paper delineates the first steps in a systematic quantitative study of the spacetime fluctuations induced by quantum fields in an evaporating black hole. We explain how the stochastic gravity formalism can be a useful tool for that purpose within a low-energy effective field theory approach to quantum gravity. As an explicit example we apply it to the study of the spherically-symmetric sector of metric perturbations around an evaporating black hole background geometry. For macroscopic black holes we find that those fluctuations grow and eventually become important when considering sufficiently long periods of time (of the order of the evaporation time), but well before the Planckian regime is reached. In addition, the assumption of a simple correlation between the fluctuations of the energy flux crossing the horizon and far from it, which was made in earlier work on spherically-symmetric induced fluctuations, is carefully analyzed and found to be invalid. Our analysis suggests the existence of an infinite amplitude for the fluctuations of the horizon as a three-dimensional hypersurface. We emphasize the need for understanding and designing operational ways of probing quantum metric fluctuations near the horizon and extracting physically meaningful information. Dedicated to Rafael Sorkin on the occasion of his 60th birthday.     

Synthesis of 5-azaadenine derivatives from N-(1,2,4-triazol-5-yl)amidines

A method is proposed for the synthesis of N-substituted 5-azaadenines. The condensation of N-(1,2,4-triazol-5-yl)amidines with trichloroacetonitrile gives trichloromethyl derivatives of 1,2,4-triazolo[1,5-a]-1,3,5-triazines, which are converted by the action of primary or secondary amines into N-alkylamino- or N,N-dialkylamino-1,2,4-triazolo-[1,5-a]-1,3,5-triazines (5-azaadenines).Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 1, pp. 241–243, January, 1991.     

Asymptotics of Dirichlet Spectrum on Some Class of Noncompact Domains

We investigate the asymptotic behaviour of the counting function of Dirichlet eigenvalues on some class of noncompact manifolds. We prove that in cases when the volume or the perimeter (the volume of the boundary) of the manifold is infinite, some additional (nonclassical) terms appear in the precise asymptotics. The coefficients at the classical terms in those are regularized in some special way volume (resp. perimeter) of the manifold.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

Second order three boundary value problem in Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral

Existence theorems and some properties of solutions set of three boundary value second order differential equations and inclusions in Banach spaces are obtained under Henstock, respectively Henstock-Kurzweil-Pettis integrability assumptions. Our results extend those obtained by Azzam, Castaing and Thibault in the Bochner integrability setting and in the Pettis integrability one. The continuity of the (unique) solution with respect to a parameter in the single-valued case is also studied.     

On some periodic solutions of singularly perturbed parabolic equations

We present results from the theory of singular perturbations and, in particular, from a new branch of this theory (contrast alternating-type structures). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 359–369, March, 2007.     

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.     

Evaluation of the Lazarus-Leblond constants in the asymptotic model of the interfacial wavy crack

The paper addresses the problem of a semi-infinite plane crack along the interface between two isotropic half-spaces. Two methods of solution have been considered in the past: Lazarus and Leblond [1998a. Three-dimensional crack-face weight functions for the semi-infinite interface crack-I: variation of the stress intensity factors due to some small perturbation of the crack front. J. Mech. Phys. Solids 46, 489-511, 1998b. Three-dimensional crack-face weight functions for the semi-infinite interface crack-II: integrodifferential equations on the weight functions and resolution J. Mech. Phys. Solids 46, 513-536] applied the “special” method by Bueckner [1987. Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three space. Int. J. Solids Struct. 23, 57-93] and found the expression of the variation of the stress intensity factors for a wavy crack without solving the complete elasticity problem; their solution is expressed in terms of the physical variables, and it involves five constants whose analytical representation was unknown; on the other hand, the “general” solution to the problem has been recently addressed by Bercial-Velez et al. [2005. High-order asymptotics and perturbation problems for 3D interfacial cracks. J. Mech. Phys. Solids 53, 1128-1162], using a Wiener-Hopf analysis and singular asymptotics near the crack front.The main goal of the present paper is to complete the solution to the problem by providing the connection between the two methods. This is done by constructing an integral representation for Lazarus-Leblond's weight functions and by deriving the closed form representations of Lazarus-Leblond's constants.