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

The Penn State - Torun Centre for Astronomy Planet Search stars. II. Lithium abundance analysis of the Red Giant Clump sample

Using the sample of 348 stars from the PennState-Torun Centre for Astronomy Planet Search, for which uniformly determined atmospheric parameters are available, with chemical abundances and rotational velocities presented here, we investigate various channels of Li enrichment in giants. Our work is based on the HET/HRS spectra. The A(Li) was determined from the 670.8nm line, while we use a more extended set of lines for alpha-elements abundances. In a series of K-S tests, we compare Li-rich giants with other stars in the sample. We also use available IR photometric and kinematical data in search for evidence of mass-loss. We investigate properties of the most Li-abundant giants in more detail by using multi-epoch precise radial velocities. We present Li and alpha-elements abundances, as well as vsini for 348 stars. We detected Li in 92 stars, of which 82 are giants. 11 of them show significant Li abundance A(Li)>1.4 and 7 of them are Li-overabundant objects, according to criterion of A(Li)>1.5 and their location on HR diagram, including two giants with Li abundances close to meteoritic level. For another 271 stars, upper limits of A(Li) are presented. We show that Li-rich giants are among the most massive stars from our sample and show larger than average effective temperatures. They are indistinguishable from the complete sample in terms of their distribution of luminosity, [Fe/H], vsini, and alpha-elements abundances. Our results do not point out to one specific Li enrichment mechanism operating in our sample of giants. On the contrary, in some cases, we cannot identify fingerprints of any of known scenarios. We show, however, that the 4 most Li-rich giant in our sample either have low-mass companions or have RV variations at the level of ~100 m/s, which strongly suggests that the presence of companions is an important factor in the Li-enrichment processes in giants.Comment: Accepted for publication in A&A, 13 figures, 11 tables, 26 page

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

Coevolution of dynamical states and interactions in dynamic networks

We explore the coupled dynamics of the internal states of a set of interacting elements and the network of interactions among them. Interactions are modeled by a spatial game and the network of interaction links evolves adapting to the outcome of the game. As an example we consider a model of cooperation, where the adaptation is shown to facilitate the formation of a hierarchical interaction network that sustains a highly cooperative stationary state. The resulting network has the characteristics of a small world network when a mechanism of local neighbor selection is introduced in the adaptive network dynamics. The highly connected nodes in the hierarchical structure of the network play a leading role in the stability of the network. Perturbations acting on the state of these special nodes trigger global avalanches leading to complete network reorganization.Comment: 4 pages, 5 figures, for related material visit http:www.imedea.uib.es/physdept

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