Definability of derivations in the reducts of differentially closed fields

Let $\mathcal{F}=(F;+,\cdot,0,1,D)$ be a differentially closed field. We consider the question of definability of the derivation $D$ in reducts of $\mathcal{F}$ of the form $\mathcal{F}_{R}=(F;+,\cdot,0,1,P)_{P \in R}$ where $R$ is a collection of definable sets in $\mathcal{F}$. We give examples and non-examples and establish some criteria for definability of $D$. Finally, using the tools developed in the paper we prove that under the assumption of inductiveness of $Th(\mathcal{F}_{R})$ model completeness is a necessary condition for definability of $D$. This can be seen as part of a broader project where one is interested in finding Ax-Schanuel type inequalities (or predimension inequalities) for differential equations.Comment: 34 pages. New results and clarifications in some proofs adde

Ax-Schanuel for Linear Differential Equations

We generalise the exponential Ax-Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by J. Kirby and C. Crampin we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax-Schanuel inequalities are adequate for them.Comment: 24 pages. Minor changes in Sections 1-5. Section 6 has been significantly modifie

Cosmo++: An Object-Oriented C++ Library for Cosmology

This paper introduces a new publicly available numerical library for cosmology, Cosmo++. The library has been designed using object-oriented programming techniques, and fully implemented in C++. Cosmo++ introduces a unified interface for using most of the frequently used numerical methods in cosmology. Most of the features are implemented in Cosmo++ itself, while a part of the functionality is implemented by linking to other publicly available libraries. The most important features of the library are Cosmic Microwave Background anisotropies power spectrum and transfer function calculations, likelihood calculations, parameter space sampling tools, sky map simulations, and mask apodization. Cosmo++ also includes a few mathematical tools that are frequently used in numerical research in cosmology and beyond. A few simple examples are included in Cosmo++ to help the user understand the key features. The library has been fully tested, and we describe some of the important tests in this paper. Cosmo++ is publicly available at http://cosmo.grigoraslanyan.comComment: Paper updated to reflect the updates in version 2.0 of Cosmo+

Greedy Set Cover Estimations

More precise estimation of the greedy algorithm complexity for a special case of the set cover problem is given in this paper

Sorting Algorithms with Restrictions

Sorting is one of the most used and well investigated algorithmic problem [1]. Traditional postulation supposes the sorting data archived, and the elementary operation as comparisons of two numbers. In a view of appearance of new processors and applied problems with data streams, sorting changed its face. This changes and generalizations are the subject of investigation in the research below

Adequate Predimension Inequalities in Differential Fields

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation and its analogue for the differential equation of the $j$-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the $j$-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In a joint work with Sebastian Eterovi\'c and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the $j$-function, that is, the Ax-Schanuel inequality for the $j$-function is adequate.Comment: 50 pages. Some revisions, a new section added for $j$ and its derivative

Exotic strange multibaryon states searches with Λ\Lambda- hyperon and Ks0K^0_s- meson systems in p+A collisions at momentum 10 GeV/c

Review for exotic strange multibaryon states were obseved in the effective mass spectra of: 1)$\Lambda \pi^+$, $\Lambda \pi^-$, $\Lambda p$, $\Lambda p p$, and $\Lambda K^0_S$, $K^0_S\pi^{\pm}$ and $K^0_Sp$ subsystems.The invariant mass of $\Lambda \pi^+$ and $K^0_S\pi^{\pm}$ spectra has observed well known $\Sigma^{*+}$(1385) and $K^{*\pm}$(892) resonances. The width of $\Sigma^{*-}(1385)$ for p+A reaction is two time larger than that presented in PDG. The cross section of $\Xi^-\to \Lambda \pi^-$ is 7-8 times larger than expected geometrical cross section in p+propane interaction. A few events detected on the photographs of the propane bubble chamber, were interpreted as S=-2 light and heavy $H^{0,+}$ dibaryons.Comment: 10 pages, 16 figures, Proc. Int. Conference Hadron Structure,2-7 September, Bratislava, 200

The preliminary result from spectra of Ks0π−K^0_s \pi^- in reaction p+propane at 10 GeV/c

The experimental data from 2m propane bubble chamber have been analyzed to search for scalar meson $\kappa(800)$ in a $K^0_s\pi$ decay mode for the reaction p+$C_3H_8$ at 10 GeV/c. The $K^0_s\pi^-$ invariant mass spectrum has shown resonant structures with $M_{K^0_s\pi^-}$=730, 900 and $\Gamma$=143, 48 MeV/$c^2$, respectively. The statistical significance are estimated to be of 14.2$\sigma$ and 4.2$\sigma$, respectively. The peak in M(900) is identified as reflection from the well known resonance with mass of 892 MeV/c$^2$.Comment: 8 pages, 4 figures, LVE and LPP seminar(JINR

On the dielectric susceptibility calculation in the incommensurate phase of K_2SeO_4

It is shown that the thermodynamic potential of the domain-like incommensurate (IC) phase of the K_2SeO_4 crystal should be supplemented with a term, taking into account the local, Lorentz electric field. The latter qualitatively changes the result of calculation of the dielectric susceptibility for this IC structure by Nattermann and Trimper, J. Phys. C: Solid State Phys. 14, 1603, (1981), and gives phase transition to the ferroelectric IC phase obtained by Aslanyan, Phys. Rev. B 70, 024102, (2004).Comment: 5 pages, 1 figur

Inclusive production of Λ\Lambda, Ks0K^0_s and exotic narrow resonances for systems Ks0pK_s^0 p, Ks0ΛK_s^0 \Lambda, Λp\Lambda p from p+propane interactions at 10 GeV/c

Experimental data from the 2m propane bubble chamber for production of $\Lambda$, $K^0_s$ have been used to search of exotic baryon states, in the $K_s^0 p$, $K_s^0 \Lambda$ and $\Lambda p$ decay mode for the reaction p+propane at 10 GeV/c. The estimation of experimental inclusive cross sections for $\Lambda$ and $K^0_s$ production in the p$^{12}C$ collision is equal to $\sigma_{\Lambda}$= 13.3$\pm$1.7 mb and $\sigma_{K^0_s}$= 3.8$\pm$0.6 mb, respectively. The measured $\Lambda /\pi^+$ ratio from pC reaction is equal to (5.3$\pm0.8)*10^{-2}$. The experimental $\Lambda /\pi^+$ ratio from the pC reaction is approximately two times larger than the $\Lambda /\pi^+$ ratio simulated by FRITIOF model from the pC reaction. The invariant mass spectrum $\Lambda K^0_s$ registered narrow peaks in regions of 1750 and 1795 MeV/$c^2$. The statistical significance of these peaks has been estimated as 5.6 and 3.3 S.D., respectively. These would be candidates for the $N^0$ or the $\Xi^0$ pentaquark states. The $pK^0_s$ invariant mass spectrum shows resonant structures with $M_{K_s^0 p}$=1540, 1613, 1821 MeV/$c^2$. The statistical significance of these peaks have been estimated as 5.5,4.8 and 5.0 s.d., respectively. The invariant mass spectrum S=-1 $\Lambda p$ observed a narrow peaks at 2100, 2175,2285 and 2353 MeV/$c^2$. Their excess above background by the second method is 6.9, 4.9, 3.8 and 2.9 S.D., respectively.Comment: 6 pages, 3 figures, Proc. workshop on "Quark-Gluon-Plasma Thermalization",Vienna, Austria, August 10-12, 200