where U, V are scalar functions defined on R. The model was termed the Cauchy matrix model because of the shape of the coupling term. Similarly to the case of the Hermitean one-matrix models for which the spectral statistics is expressible in terms of appropriate orthogonal polynomials [3], this two-matrix model is solvable with the help of a new family of biorthogonal polynomials named the Cau...

The refractive indices of E7 liquid-crystal mixture were measured at six visible and two infrared sl=1.55 and 10.6 mmd wavelengths at different temperatures, using Abbe and wedged cell refractometer methods, respectively. The experimental data of the visible wavelengths fit the extended Cauchy equations well. Using the extended Cauchy equations, we can extrapolate the refractive indices of E7 t...

We study the interior electrovacuum region of axisymmetric and stationary black holes with surrounding matter and find that there exists always a regular inner Cauchy horizon inside the black hole, provided the angular momentum J and charge Q of the black hole do not vanish simultaneously. In particular, we derive an explicit relation for the metric on the Cauchy horizon in terms of that on the...

We study the interior electrovacuum region of axisymmetric and stationary black holes with surrounding matter and find that there exists always a regular inner Cauchy horizon inside the black hole, provided the angular momentum J and charge Q of the black hole do not vanish simultaneously. In particular, we derive an explicit relation for the metric on the Cauchy horizon in terms of that on the...

We prove that the space-time developments of generic solutions of the vacuum constraint Einstein equations do not possess any global or local Killing vectors, when Cauchy data are prescribed on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that non-exi...

Absfrset. The notion of the Cauchy indexof a real rational scalar function is generalized todefinc the Cauchy index of a real rational symmetric matrix in terms of the behavior of the matrix at real singularities of its elements. Alternative characterizations are obtained which flow from iepresenrations of the real rational matrix using a Laurent series, a matrix fraction description. and a sta...

Abstract. It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent joint work [PM16], the author proved a surprising multivariate generalization of the above facts. Fix m > 1 and let Σ be a m × m positive semi-definite...

Hyperbolic equations and inequalities in octants with the lateral Cauchy data at coordinate planes are consdered. Lipschitz stability estimate is established in the case when both the inhomogeneous right hand side and (unknown) initial conditions at {t = 0} have a finite support. This is the first stability estimate for such a Cauchy problem in an infinite domain. Refocusing of time reversal fi...

in this article, the effect of cells' radius on the behavior of wavelength switching optical filter andthe effect of the radius of the optical filters' key characteristics such as wavelength resonance onan optical filter based on photonic crystals, have been investigated. currently, the most commonapplied mechanism for designing optical filter based on photonic crystals is using twome...

We consider the Cauchy problem for Laplacian. Using the single layer representation, we obtain an equivalent system of boundary integral equations. We show the singular values of the ill-posed Cauchy operator decay exponentially, which means that a small error is exponentially amplified in the solution of the Cauchy problem. We show the decaying rate is dependent on the geometry of the domain, ...