Improved Polynomial Remainder Sequences for Ore Polynomials

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients

Automated Generation of Non-Linear Loop Invariants Utilizing Hypergeometric Sequences

Analyzing and reasoning about safety properties of software systems becomes an especially challenging task for programs with complex flow and, in particular, with loops or recursion. For such programs one needs additional information, for example in the form of loop invariants, expressing properties to hold at intermediate program points. In this paper we study program loops with non-trivial arithmetic, implementing addition and multiplication among numeric program variables. We present a new approach for automatically generating all polynomial invariants of a class of such programs. Our approach turns programs into linear ordinary recurrence equations and computes closed form solutions of these equations. These closed forms express the most precise inductive property, and hence invariant. We apply Gr\"obner basis computation to obtain a basis of the polynomial invariant ideal, yielding thus a finite representation of all polynomial invariants. Our work significantly extends the class of so-called P-solvable loops by handling multiplication with the loop counter variable. We implemented our method in the Mathematica package Aligator and showcase the practical use of our approach.Comment: A revised version of this paper is published in the proceedings of ISSAC 201

Invariant Generation for Multi-Path Loops with Polynomial Assignments

Program analysis requires the generation of program properties expressing conditions to hold at intermediate program locations. When it comes to programs with loops, these properties are typically expressed as loop invariants. In this paper we study a class of multi-path program loops with numeric variables, in particular nested loops with conditionals, where assignments to program variables are polynomial expressions over program variables. We call this class of loops extended P-solvable and introduce an algorithm for generating all polynomial invariants of such loops. By an iterative procedure employing Gr\"obner basis computation, our approach computes the polynomial ideal of the polynomial invariants of each program path and combines these ideals sequentially until a fixed point is reached. This fixed point represents the polynomial ideal of all polynomial invariants of the given extended P-solvable loop. We prove termination of our method and show that the maximal number of iterations for reaching the fixed point depends linearly on the number of program variables and the number of inner loops. In particular, for a loop with m program variables and r conditional branches we prove an upper bound of m*r iterations. We implemented our approach in the Aligator software package. Furthermore, we evaluated it on 18 programs with polynomial arithmetic and compared it to existing methods in invariant generation. The results show the efficiency of our approach

Critical Kinetic Plasma Processes In Relativistic Astrophysics

Plasma astrophysics deals with collective plasma processes in astrophysical scenarios. As observational astronomy pushes towards unprecedented resolutions in space and time, the focus of theoretical research necessarily ventures towards a description of the plasma microphysics. On microphysical scales the plasma is pervasively collisionless and the magnetohydrodynamic approximation breaks down. Consequently theoretical concepts rely on a kinetic plasma description as the most sophisticated plasma model. The present work discusses some fundamental kinetic plasma processes in relativistic astrophysics: Fast Magnetic Reconnection (FMR) associated with discontinuities in the magnetic field topology, and the Coupled Two-Stream-Weibel instability (CTW) in the wake of collisionless shocks. Both processes are ubiquitous in astrophysical sites, prevail over competing plasma modes because of dominant growth rates, experience significant relativistic modifications, and develop essential features solely in the highly non-linear regime. The computational representation invokes the entire 6D phase space. These characteristics distinguish FMR and the CTW as distinctively critical processes. FMR and the CTW are studied here in the framework of self-consistent, relativistic and fully electromagnetic Particle-In-Cell (PIC) simulations. Typical scenarios comprise ensembles of 10^9 particles and endure for several 10^4 time steps. The computational task is challenging and completely in the realm of the massively parallelized architectures of state-of-the-art supercomputers. We present the first self-consistent 3D simulations of FMR in relativistic pair plasma. Focusing on the mechanism of particle acceleration we show that the highly dynamic evolution of the current sheet in the non-linear regime is the essential stage. Therein non-stationary acceleration zones arise in the superposition of the relativistic tearing and the relativistic drift kink mode as competing current sheet instabilities. Though the topology of electromagnetic fields is highly turbulent, the FMR process shows the remarkable quality to generate smooth and stable power-laws in the particle distribution function (PDF) out of an initial Maxwellian. The upper PDF cut-off in relativistic energy is determined by the ratio of light to Alfven velocity c/v_A. The power-law index assumes s~-1 within the reconnection X-zone irrespective of parameter variations. Intriguingly the power-law index appears as the universal characteristic of the source process. The associated synchrotron spectra provide a valid description of the extremely hard spectra and rapid variabilities of `Flat Spectrum Radio Quasars'. Conceptual Gamma-Ray Burst (GRB) synchrotron emission models depend on a plasma process which ensures efficient magnetic field generation. The CTW converts bulk-kinetic energy of counter-streaming plasma shells into Weibel magnetic fields. Pivoted by the linear analysis of the CTW, the PIC simulations confirm the correspondence between saturation magnetic fields and bulk-kinetic energy. Plasma shell collisions in GRBs are either associated with internal or external shocks. As direct consequence of the energy dependence the CTW evolves from a complex 3D topology in internal collisions towards quasi-2D, Weibel-dominated conformalizations at the higher external shock energies. The PIC results prove that the Weibel fields are sufficiently strong to sustain synchrotron emission scenarios, particularly in external shocks. By determining the first lifetime limits we show that Weibel fields are also sufficiently long-lived with respect to typical synchrotron cooling times. We further identify the stability-limiting diffusion process as of `Bohm'-type, i.e. the diffusion coefficient exhibits the T/B-dependence and herewith represents a conservative stability criterion. The CTW generates stable power-law spectra in the magnetic fields implying power-law shaped PDFs as self-similar solutions for diffusive particle scattering. This suggests a universal power-law index as the characteristic of the CTW process. Imposing a magnetic guide field of well-defined strength suppresses the Weibel contributions of the CTW in favour of the electrostatic Two-Stream instability (TSI). The pulsar magnetosphere is the paradigmatic scenario in which we discuss the mechanism of Coherent Collisionless Bremsstrahlung (CCB) triggered by the TSI. The PIC simulations show that the CCB mechanism provides a valid description of the phenomenon of `Giant Radio Pulses' as recently observed from the Crab pulsar

On deformation of electron holes in phase space

This Letter shows that for particularly shaped background particle distributions momentum exchange between phase space holes and the distribution causes acceleration of the holes along the magnetic field. In the particular case of a non-symmetric ring distribution (ring with loss cone) this acceleration is nonuniform in phase space being weaker at larger perpendicular velocities thus causing deformation of the hole in phase space.Comment: Original MS in EPL style, 1 Figur

High-energy emission from the pulsar striped wind: a synchrotron model for gamma-ray pulsars

(abridged) Gamma-ray pulsars constitute a class of high and very high-energy emitters for which the known population is steadily increasing thanks to the Fermi/Large Area Telescope. In this paper, their gamma-ray luminosity and spectral features are explained in the framework of synchrotron radiation from particles located in the stripe of the pulsar wind. Apart from radiative losses, particles are also subject to a constant re-acceleration and reheating for instance by a magnetic reconnection induced electric field. The high-energy luminosity scales as $L_\gamma \approx 2\times10^{26} \textrm{W} \, (L_{\rm sd}/10^{28} \textrm{W})^{1/2} \, (P/1 \textrm{s})^{-1/2}$ where $L_{\rm sd}$ is the pulsar spindown luminosity and $P$ its period. From this relation, we derive important parameters of pulsar magnetosphere and wind theories. Indeed, we find bulk Lorentz factor of the wind scaling as $\Gamma_{\rm v} \approx 10 \, \tau_{\rm rec}^{1/5} \, (L_{\rm sd}/10^{28} \textrm{W})^{1/2}$, pair multiplicity $\kappa$ related to the magnetization parameter $\sigma$ by $\kappa\,\sigma \, \tau_{\rm rec}^{1/5} \approx 10^8$, and efficiency $\eta$ of spin-down luminosity conversion into particle kinetic energy according to the relation $\eta\,\sigma\approx1$. A good guess for the associated reconnection rate is then $\tau_{\rm rec} \approx 0.5 \, (L_{\rm sd}/10^{28} \textrm{W})^{-5/12}$. Finally, pulses in gamma-rays are visible only if $L_{\rm sd}/P\gtrsim 10^{27} \textrm{W/s}$. This model differs from other high-energy emission mechanisms because it makes allowance not only for rotational kinetic energy release but also for an additional reservoir of energy anchored to the magnetic field of the stripe and released for instance by some magnetic reconnection processes.Comment: 5 pages; 2 figures; accepted by MNRA