Emergence of order in selection-mutation dynamics

We characterize the time evolution of a d-dimensional probability distribution by the value of its final entropy. If it is near the maximally-possible value we call the evolution mixing, if it is near zero we say it is purifying. The evolution is determined by the simplest non-linear equation and contains a d times d matrix as input. Since we are not interested in a particular evolution but in the general features of evolutions of this type, we take the matrix elements as uniformly-distributed random numbers between zero and some specified upper bound. Computer simulations show how the final entropies are distributed over this field of random numbers. The result is that the distribution crowds at the maximum entropy, if the upper bound is unity. If we restrict the dynamical matrices to certain regions in matrix space, for instance to diagonal or triangular matrices, then the entropy distribution is maximal near zero, and the dynamics typically becomes purifying.Comment: 8 pages, 8 figure

New spectral relations between products and powers of isotropic random matrices

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed non-hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide an evidence that the result holds also for isotropic orthogonal ensembles (IOE).Comment: 8 pages, 3 figures (in version 2 we added a figure and discussion on finite size effects for isotropic orthogonal ensemble

Heavy Hadrons and QCD Instantons

Heavy hadrons are analyzed in a random and dilute gas of instantons. We derive the instanton-induced interactions between heavy and light quarks at next to leading order in the heavy quark mass and in the planar approximation, and discuss their effects on the hadronic spectrum. The role of these interactions in the formation of exotic hadrons is also discussed.Comment: 26 pages, REVTeX, 2 tables, 5 figures, uses FEYNMAN.st

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical Physics and Information Theory," September 26-30, 2010, Krakow, Polan

A Proof of Tarski’s Fixed Point Theorem by Application of Galois Connections

Two examples of Galois connections and their dual forms are considered. One of them is applied to formulate a criterion when a given subset of a complete lattice forms a complete lattice. The second, closely related to the first, is used to prove in a short way the Knaster-Tarski’s fixed point theore

On a Possibility to Determine the Sign of the Polarized Gluon Distribution

We investigate the possibility to draw conclusions on the sign of the spin-dependent gluon distribution, $\Delta G(x, Q^2)$, from existing polarized DIS data. The spin-dependent parton distributions $\Delta u_v, \Delta d_v, \Delta {\bar u}, \Delta {\bar d}, \Delta {s}$, and $\Delta G$ are constructed in the framework of a phenomenological procedure taking into account some assumptions on signs of valence and sea parton distributions motivated by 't Hooft's mechanism of quark-quark interaction induced by instantons. The axial gluon anomaly and data on integral quark contributions to the proton spin, $\Delta \tilde u, \Delta \tilde d$, and $\Delta \tilde s$, are also taken into account. Predictions for the $x$- and $Q^2$-dependencies of the polarized proton and neutron structure functions, $g_1^p$ and $g_1^n$, are compared to experimental data. It is shown that the neutron structure function, $g_1^n$, is especially sensitive to the sign of $\Delta G(x, Q^2)$. The results of our analysis supports the conclusion that this sign should be positive.Comment: 14 pages, latex, 12 figure

A simple toy model for effective restoration of chiral symmetry in excited hadrons

A simple solvable toy model exhibiting effective restoration of chiral symmetry in excited hadrons is constructed. A salient feature is that while physics of the low-lying states is crucially determined by the spontaneous breaking of chiral symmetry, in the high-lying states the effects of chiral symmetry breaking represent only a small correction. Asymptotically the states approach the regime where their properties are determined by the underlying unbroken chiral symmetry.Comment: This is the published version of this paper. Note that the title has changed from earlier versions as has the abstract. The emphasis is slightly different from previous versions but the essential physical content is the sam

The Penn State - Toru\'n Centre for Astronomy Planet Search stars IV. Dwarfs and the complete sample

Our knowledge of the intrinsic parameters of exoplanets is as precise as our determinations of their stellar hosts parameters. In the case of radial velocity searches for planets, stellar masses appear to be crucial. But before estimating stellar masses properly, detailed spectroscopic analysis is essential. With this paper we conclude a general spectroscopic description of the Pennsylvania-Torun Planet Search (PTPS) sample of stars. We aim at a detailed description of basic parameters of stars representing the complete PTPS sample. We present atmospheric and physical parameters for dwarf stars observed within the PTPS along with updated physical parameters for the remaining stars from this sample after the first Gaia data release. We used high resolution (R=60 000) and high signal-to-noise-ratio (S/N=150-250) spectra from the Hobby-Eberly Telescope and its High Resolution Spectrograph. Stellar atmospheric parameters were determined through a strictly spectroscopic local thermodynamic equilibrium analysis (LTE) of the equivalent widths of FeI and FeII lines. Stellar masses, ages, and luminosities were estimated through a Bayesian analysis of theoretical isochrones. We present $T_{eff}$, log$g$ , [Fe/H], micrturbulence velocities, absolute radial velocities, and rotational velocities for 156 stars from the dwarf sample of PTPS. For most of these stars these are the first determinations. We refine the definition of PTPS subsamples of stars (giants, subgiants, and dwarfs) and update the luminosity classes for all PTPS stars. Using available Gaia and Hipparcos parallaxes, we redetermine the stellar parameters (masses, radii, luminosities, and ages) for 451 PTPS stars. The complete PTPS sample of 885 stars is composed of 132 dwarfs, 238 subgiants, and 515 giants, of which the vast majority are of roughly solar mass.Comment: 11 pages, 10 figures, Accepted for publication in Astronomy & Astrophysic