Stochastic Ornstein–Uhlenbeck Capacitors

We introduce and study a class of random capacitor systems which are both charged and discharged stochastically. A capacitor is fed by a random inflow with stationary and independent increments. Discharging occurs according to a Markovian rate which is linear in the capacitors level. The resulting capacitor dynamics are Markovian, stochastically cyclic, and regenerative. We coin these systems Lévy-charged Ornstein–Uhlenbeck capacitors. Various random quantities associated with these systems are analyzed, including: the time-to- discharge; the duration of the charging cycle; the trajectory and the peak height of the capacitor level during a charging cycle; and, the capacitors stationary equilibrium level. Furthermore, we show that there are sharp distinctions between these capacitor systems and corresponding standard Lévy-driven Ornstein–Uhlenbeck systems.     

The maximal process of nonlinear shot noise

In the nonlinear shot noise system-model shots’ statistics are governed by general Poisson processes, and shots’ decay-dynamics are governed by general nonlinear differential equations. In this research we consider a nonlinear shot noise system and explore the process tracking, along time, the system’s maximal shot magnitude. This ‘maximal process’ is a stationary Markov process following a decay-surge evolution; it is highly robust, and it is capable of displaying both a wide spectrum of statistical behaviors and a rich variety of random decay-surge sample-path trajectories. A comprehensive analysis of the maximal process is conducted, including its Markovian structure, its decay-surge structure, and its correlation structure. All results are obtained analytically and in closed-form.     

Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

Lorenzian analysis of infinite Poissonian populations and the phenomena of Paretian ubiquity

The Lorenz curve is a universally calibrated statistical tool measuring quantitatively the distribution of wealth within human populations. We consider infinite random populations modeled by inhomogeneous Poisson processes defined on the positive half-line—the randomly scattered process-points representing the wealth of the population-members (or any other positive-valued measure of interest such as size, mass, energy, etc.). For these populations the notion of “macroscopic Lorenz curve” is defined and analyzed, and the notion of “Lorenzian fractality” is defined and characterized. We show that the only non-degenerate macroscopically observable Lorenz curves are power-laws manifesting Paretian statistics—thus providing a universal “Lorenzian explanation” to the ubiquitous appearance of Paretian probability laws in nature.     

Ergodic behavior of diffusions with random jumps from the boundary

We consider a diffusion process on D⊂R^d$D\subset {\mathbb{R}}^d$, which upon hitting ∂D$\partial D$, is redistributed in D$D$ according to a probability measure depending continuously on its exit point. We prove that the distribution of the process converges exponentially fast to its unique invariant distribution and characterize the exponent as the spectral gap for a differential operator that serves as the generator of the process on a suitable function space.     

Polymyxin-coated Au and carbon nanotube electrodes for stable [NiFe]-hydrogenase film voltammetry

We report on the use of polymyxin (PM), a cyclic cationic lipodecapeptide, as an electrode modifier for studying protein film voltammetry (PFV) on Au and single-walled carbon nanotube (SWNT) electrodes. Pretreating the electrodes with PM allows for the subsequent immobilization of an active submonolayer of [NiFe]-hydrogenase from Allochromatium vinosum ( Av H2ase). Probed by cyclic voltammetry (CV), the adsorbed enzyme exhibits characteristic electrocatalytic behavior that is stable for several hours under continuous potential cycling. An unexpected feature of the immobilization procedure is that the presence of chloride ions is a prerequisite for obtaining electrocatalytic activity. Atomic force microscopy (AFM) relates the observed catalytic activity to enzymatic adsorption at the PM/Au(111) surface, and a combination of concentration-dependent CV and AFM is used to investigate the interaction between the enzyme and the PM layer.     

Gated Polling Systems with Lévy Inflow and Inter-Dependent Switchover Times: A Dynamical-Systems Approach

We study asymmetric polling systems where: (i) the incoming workflow processes follow general Lévy-subordinator statistics; and, (ii) the server attends the channels according to the gated service regime, and incurs random inter-dependentswitchover times when moving from one channel to the other. The analysis follows a dynamical-systems approach: a stochastic Poincaré map, governing the one-cycle dynamics of the polling system is introduced, and its statistical characteristics are studied. Explicit formulae regarding the evolution of the mean, covariance, and Laplace transform of the Poincaré map are derived. The forward orbit of the maps transform – a nonlinear deterministic dynamical system in Laplace space – fully characterizes the stochastic dynamics of the polling system. This enables us to explore the long-term behavior of the system: we prove convergence to a (unique) steady-state equilibrium, prove the equilibrium is stationary, and compute its statistical characteristics.     

Short propositional refutations for dense random 3CNF formulas

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n^1.5−ε)$\mathrm{\Omega }(n^{1.5-\epsilon })$ clauses (Chvátal and Szemerédi [14], Ben-Sasson and Wigderson [10]). On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with Ω(n²/log?n)$\mathrm{\Omega }(n^2/\log ?n)$ clauses, shown by Beame et al. [6]. In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC⁰${\mathit{T}\mathit{C}}^0$ formulas (i.e., TC⁰${\mathit{T}\mathit{C}}^0$-Frege proofs) for random 3CNF formulas with n   variables and Ω(n^1.4)$\mathrm{\Omega }(n^{1.4})$ clauses.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.