Equilibrium states of the pressure function for products of matrices

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.Comment: 12 pages. To appear in DCD

Temporal and Spectral Correlations of Cyg X-1

Temporal and spectral properties of X-ray rapid variability of Cyg X-1 are studied by an approach of correlation analysis in the time domain on different time scales. The correlation coefficients between the total intensity in 2-60 keV and the hardness ratio of 13-60 keV to 2-6 keV band on the time scale of about 1 ms are always negative in all states. For soft states, the correlation coefficients are positive on all the time scales from about 0.01 s to 100 s, which is significantly different with that for transition and low states. Temporal structures in high energy band are narrower than that in low energy band in quite a few cases. The delay of high energy photons relative to low energy ones in the X-ray variations has also been revealed by the correlation analysis. The implication of observed temporal and spectral characteristics to the production region and mechanism of Cyg X-1 X-ray variations is discussed.Comment: 17 pages, 6 figures included, to appear in Ap

Phase field crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures

The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zeroth-mode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.Comment: 26 pages, 5 figures; submitted to Phys. Rev.

A note on the sign (unit root) ambiguities of Gauss sums in index 2 and 4 cases

Recently, the explicit evaluation of Gauss sums in the index 2 and 4 cases have been given in several papers (see [2,3,7,8]). In the course of evaluation, the sigh (or unit root) ambiguities are unavoidably occurred. This paper presents another method, different from [7] and [8], to determine the sigh (unit root) ambiguities of Gauss sums in the index 2 case, as well as the ones with odd order in the non-cyclic index 4 case. And we note that the method in this paper are more succinct and effective than [8] and [7]

SIMPle Dark Matter: Self-Interactions and keV Lines

We consider a simple supersymmetric hidden sector: pure SU(N) gauge theory. Dark matter is made up of hidden glueballinos with mass $m_X$ and hidden glueballs with mass near the confinement scale $\Lambda$. For $m_X \sim 1\,\text{TeV}$ and $\Lambda \sim 100\,\text{MeV}$, the glueballinos freeze out with the correct relic density and self-interact through glueball exchange to resolve small-scale structure puzzles. An immediate consequence is that the glueballino spectrum has a hyperfine splitting of order $\Lambda^2 / m_X \sim 10\,\text{keV}$. We show that the radiative decays of the excited state can explain the observed 3.5 keV X-ray line signal from clusters of galaxies, Andromeda, and the Milky Way.Comment: v1: 6 pages, 2 figures; v2: added references, published version; v3: note adde

Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) block code. This tradeoff is precisely known in the case of i.i.d. Rayleigh-fading, for T>= n_t+n_r-1 where T is the number of time slots over which coding takes place and n_t,n_r are the number of transmit and receive antennas respectively. For T < n_t+n_r-1, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA) based ST codes having the non-vanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n. For the rectangular, T > n_t case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all T>= n_t is the same as that previously known to hold for T >= n_t + n_r -1.Comment: Revised submission to IEEE Transactions on Information Theor

Emergent Calabi-Yau Geometry

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the partition function of the topological string theory by relating the Ronkin function of the characteristic polynomial of the crystal melting model to the holomorphic 3-form on the corresponding Calabi-Yau manifold.Comment: 4 pages; v2: revised discussion on wall crossing; v3: typos corrected, published versio