Universality of Long-Range Correlations in Expansion-Randomization Systems

We study the stochastic dynamics of sequences evolving by single site mutations, segmental duplications, deletions, and random insertions. These processes are relevant for the evolution of genomic DNA. They define a universality class of non-equilibrium 1D expansion-randomization systems with generic stationary long-range correlations in a regime of growing sequence length. We obtain explicitly the two-point correlation function of the sequence composition and the distribution function of the composition bias in sequences of finite length. The characteristic exponent $\chi$ of these quantities is determined by the ratio of two effective rates, which are explicitly calculated for several specific sequence evolution dynamics of the universality class. Depending on the value of $\chi$, we find two different scaling regimes, which are distinguished by the detectability of the initial composition bias. All analytic results are accurately verified by numerical simulations. We also discuss the non-stationary build-up and decay of correlations, as well as more complex evolutionary scenarios, where the rates of the processes vary in time. Our findings provide a possible example for the emergence of universality in molecular biology.Comment: 23 pages, 15 figure

SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $\ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semi-algebraic programs can be found by solving a single semi-definite programming problem (SDP). We achieve the results by using tools from semi-algebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we outline how the derived results can be applied to show that robust SOS-convex optimization problems under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers the open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting

Canonical form of master equations and characterization of non-Markovianity

Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.Comment: v2: Significant update, with many new results and one new author. 12 pages; v3: Minor clarifications, to appear in PRA; v4: matches published versio

Static and Dynamic Spectroscopy of (Al,Ga)As/GaAs Microdisk Lasers with Interface Fluctuation Quantum Dots

We have studied the steady state and dynamic optical properties of semiconductor microdisk lasers whose active region contains interface fluctuation quantum dots in GaAs/(Ga,Al)As quantum wells. Steady-state measurements of the stimulated emission via whispering gallery modes yield a quality factor $Q \sim 5600$ and a coupling constant $\beta \sim 0.09$. The broad gain spectrum produces mode hopping between spectrally adjacent whispering gallery modes as a function of temperature and excitation power. Time- and energy-resolved photoluminescence measurements show that the emission rise and decay rates increase significantly with excitation power. Marked differences are observed between the radiative decay rates in processed and unprocessed samples.Comment: To appear in Phys. Rev.

Disclination Asymmetry in Two-Dimensional Nematic Liquid Crystals with Unequal Frank Constants

The behavior of a thin film of nematic liquid crystal with unequal Frank constants is discussed. Distinct Frank constants are found to imply unequal core energies for $+1/2$ and $-1/2$ disclinations. Even so, a topological constraint is shown to ensure that the bulk densities of the two types of disclinations are the same. For a system with free boundary conditions, such as a liquid membrane, unequal core energies simply renormalize the Gaussian rigidity and line tension.Comment: RevTex forma