In this work, we use Lorentz invariance violation (LIV) introduced as a generic modification to particle dispersion relations to study some consequences of single photon emission, known as vacuum Cherenkov radiation, and photon decay processes in cosmic and gamma rays. These processes are forbidden in a Lorentz invariant theory but allowed under the hypothesis of LIV. We show that the emission ...

Let D be the sheaf of analytic differential operators of n variables x1, . . . , xn. We consider a left ideal I of D generated by {l1, . . . , lm} which are in the Weyl algebra D = C〈x1, . . . , xn, ∂1, . . . , ∂n〉. If no confusion arises, we also denote by I the left ideal D ·{l1, . . . , ln} in D. Assume that D/I is holonomic. It was proved by M.Kashiwara that the germs of the k-th extension ...

where m is the normalized Lebesgue measure on T. By the Fatou Theorem, Kf can also be defined on T by its non-tangential boundary values. After such an extension one can view the Cauchy integral as a “transform”, i.e., an operator in L(m) which sends f into the boundary values of Kf . It is well known, and not difficult to see, that the Cauchy transform is unbounded for p = 1 or p = ∞. However,...

This paper proposes a stable combination test, which is natural extension of Cauchy tests by Liu and Xie (2020). Similarly to the our test simple compute, enjoys good sizes, has asymptotically optimal powers even when individual are not independent. finding supported both in theory finite samples.

In this paper, the Schwarz boundary value problem (BVP) for the inhomogeneous Cauchy-Riemann equation in a triangle is investigated explicitly. Firstly, by the technique of parquetingreflection and the Cauchy-Pompeiu representation formula a modified Cauchy-Schwarz representation formula is obtained. Then, the solution of the Schwarz BVP is explicitly solved. In particular, the boundary behavio...

(for example, the wave operator acting in the exterior of a smooth obstacle). After extending the coefficients of P smoothly across <9M, we can view P as an operator on some open extension M of M. The problem is easily solved in the two cases where no null bicharacteristics tangent to dT*M are present. When dM is everywhere elliptic with respect to P , classical theory implies that the desired ...

We propose a new family of symmetric unimodal distributions on the circle which contains the uniform, von Mises, cardioid and wrapped Cauchy distributions, amongst others, as special cases. The basic form of the densities of this family is very simple, although its normalisation constant involves an associated Legendre function. The family of distributions can also be derived by conditioning an...

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study PlemeljCalderón-Seeley-Bojarski...

In this paper we consider the Cauchy problem for gravity water-wave equations, in a domain with flat bottom and arbitrary space dimension. We prove that if data are of size ? analytic functions which have holomorphic extension strip ?, then solution exists up to time C/? having at t ??C??t.