A high-order nonconservative approach for hyperbolic equations in fluid dynamics

It is well known, thanks to Lax-Wendroff theorem, that the local conservation of a numerical scheme for a conservative hyperbolic system is a simple and systematic way to guarantee that, if stable, a scheme will provide a sequence of solutions that will converge to a weak solution of the continuous problem. In [1], it is shown that a nonconservative scheme will not provide a good solution. The question of using, nevertheless, a nonconservative formulation of the system and getting the correct solution has been a long-standing debate. In this paper, we show how get a relevant weak solution from a pressure-based formulation of the Euler equations of fluid mechanics. This is useful when dealing with nonlinear equations of state because it is easier to compute the internal energy from the pressure than the opposite. This makes it possible to get oscillation free solutions, contrarily to classical conservative methods. An extension to multiphase flows is also discussed, as well as a multidimensional extension

Unitarity of Minkowski non-local theories made explicit

In this work we explicitly show that the perturbative unitarity of analytic infinite derivative (AID) scalar field theories can be achieved using a modified prescription for computing scattering amplitudes. The crux of the new prescription is the analytic continuation of a result obtained in the Euclidean signature to the Minkowski external momenta. We explicitly elaborate an example of a non-local $\phi^4$ model for various infinite derivative operators. General UV properties of amplitudes in non-local theories are discussed.Comment: 16 pages, 7 figure

Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history

UV graviton scattering and positivity bounds from IR dispersion relations

Scattering amplitudes mediated by graviton exchange display IR singularities in the forward limit. This obstructs standard application of positivity bounds based on twice subtracted dispersion relations. Such divergences can be cancelled only if the UV limit of the scattering amplitude behaves in a specific way, which implies a very non-trivial connection between the UV and IR behaviors of the amplitude. We show that this relation can be expressed in terms of an integral transform, obtaining analytic results when $t \log{s}\rightarrow 0$. Carefully applying this limit to dispersion relations, we find that infinite arc integrals, which are usually taken to vanish, can give a non-trivial contribution in the presence of gravity, unlike in the case of finite negative $t$. This implies that gravitational positivity bounds cannot be trusted unless the size of this contribution is estimated in some way, which implies assumptions on the UV completion of gravitational interactions. We discuss the relevance of these findings in the particular case of QED coupled to gravity.Comment: 20 pages, 2 figure

Post-inflationary GW production in generic higher (infinite) derivative gravity

Gravity can be embedded into a renormalizable theory by means of adding quadratic in curvature terms. However, this at first leads to the presence of the Weyl ghost. It is possible to get rid of this ghost if the locality assumption is weakened and the propagator of the graviton is represented by an entire function of the d'Alembertian operator without new poles and zeros. Models of this type admit a cosmological solution describing the $R^2$, or Starobinsky, inflation. We study graviton production after inflation in this model and show that it is negligible despite the presence of the higher derivative operators which could potentially cause instabilities.Comment: We dedicate this paper to the memory of Valery Rubako

Positronium oscillations to Mirror World revisited

We present a calculation of the branching ratio of orthopositronium decay into an invisible mode, which is done in the context of Mirror World models, where ordinary positronium can disappear from our world due to oscillation into its mirror twin. In this revision we clarify some formulas and approximations used previously, correct them at some places, add new effects relevant for a feasible experiment and finally perform a combined analysis. We include into consideration various effects due to external magnetic and electric fields, collisions with cavity walls and scattering off gas atoms in the cavity. Oscillations of the Rydberg positroniums are also considered. To perform a numerical estimates in a realistic case we wrote computer code, which can be adopted in any experimental setup. Its work is illustrated with an example of a planned positronium experiment within the AEgIS project.Comment: 23 pages, 4 figures, typos corrected, references added, published versio