Interacting tachyon Fermi gas

We consider a system of many fermionic tachyons coupled to a scalar, pseudoscalar, vector and pseudovector fields. The scalar and pseudoscalar fields are responsible for the effective mass, while the pseudovector field is similar to ordinary electromagnetic field. The action of vector field $\omega_\mu$ results in tachyonic dispersion relation $\varepsilon_p=\sqrt{p^2+g^2\omega_0^2-hpg\omega_0-g\vec \sigma \cdot \nabla \omega_0-m^2} -g\vec \sigma \cdot \vec \omega$ that depends on helicity $h$ and spin $\vec \sigma$. We apply the mean field approximation and find that there appears a vector condensate with finite average $$ depending on the tachyon density. The pressure and energy density of a many-tachyon system include the mean-field energy $=\sqrt{p^2+hpng^2/M^2+n^2g^4/M^4-m^2}$ which is real when the particle number density exceeds definite threshold which is $n>mM^2/g^2$ for right-handed and $n>\frac 2{\sqrt{3}}mM^2/g^2$ for left-handed tachyons, while all tachyons are subluminal at high density. There is visible difference in the properties of right-handed and left-handed tachyons. Interaction via the vector field $\omega_0$ may lead to stabilization of tachyon matter if its density is large enough.Comment: 13 pages, 2 figure

Long time behaviour of random walks on the integer lattice

We consider an irreducible finite range random walk on the $d$-dimensional integer lattice and study asymptotic behaviour of its transition function $p(n; x)$. In particular, for simple random walk our asymptotic formula is valid as long as $n (n - |x|_1)^{-2}$ tends to zero

Superluminal neutrino energy spectrum of OPERA and MINOS

We analyze the velocity dependence on energy of superluminal neutrino recorded by the OPERA and MINOS collaborations and manage to approximate the energy spectrum by a power law E=p+Cp^a where parameters must be taken in the range a=0.40--1.18 and C=1.5x10^{-5}--4.15x10^{-4} (momentum and energy are expressed in GeV). This rough estimation is constrained by the errors of measurements, and new experimental data are requested.Comment: 6 pages, no figures [v2 - misprints corrected

Specific heat and entropy of tachyon Fermi gas

We consider an ideal Fermi gas of tachyons and derive a low temperature expansion of its thermodynamical functions. The tachyonic specific heat is linear dependent on temperature $C_V=\epsilon_Fk_FT$ and formally coincides with the specific heat of electron gas if the tachyon Fermi energy is defined as $\epsilon_F=\sqrt{k_F-m^2}$.Comment: 10 page

Pointwise ergodic theorems for some thin subsets of primes

We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ \varphi_1(n)\} < \psi(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where $p > 1$ and $r > 2$

Analytic theory of discontinuities in current-carrying cosmic strings

We formulate an analytic method to study the discontinuities in superconducting cosmic strings. Equations of discontinuities and conditions of their existence are derived from the intrinsic and extrinsic equations of motion. It is the fundamental for research of particular solutions, associated with kinks, cusps and shocks.Comment: 12 pages, no figure

Heat kernel and Green function estimates on affine buildings

We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function

Dirac tachyons and antitachyons in many-particle system

A consistent description of charged many-tachyon Fermi system is developed. Tachyons and antitachyons have the same chemical potential \mu+=\mu- because the axial coupling constant g+=g- is invariant under the charge conjugation, in contrast to reversion of the electric charge e+=-e-. The axial density n5= is incorporated in the thermodynamical functions instead of which is not associated with any conserved quantity. The number of tachyons and antitachyons are undefined but it is possible to estimate their difference and establish a link between the total electric charge density $en$ and $n5$.Comment: 16 pages, 1 figur

Stability of hot tachyon gas

We consider a tachyon gas that obeys Maxwell-Boltzmann statistics. The sound speed is always subluminal and it tends to the limiting minimum value $c_s=1/\sqrt{2}$ in non-relativistic gas (at low temperature), decreasing monotonously with the growth of temperature and attaining ultra-relativistic limit $c_s=1/$ at high temperature. The hot tachyon gas always satisfies the causality.Comment: 11 pages, 3 figure

Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors

We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition.Comment: 34 page