The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/∞ system revisited: finiteness,summability, long range dependence,and reverse engineering

We explore M/G/∞ systems ‘fed’ by Poissonian inflows with infinite arrival rates. Three processes – corresponding to the system's state, workload, and queue-size – are studied and analyzed. Closed form formulae characterizing the system's stationary structure and correlation structure are derived. And, the issues of queue finiteness, workload summability, and Long Range Dependence are investigated. We then turn to devise a ‘reverse engineering’ scheme for the design of the system's correlation structure. Namely: how to construct an M/G/∞ system with a pre-desired ‘target’ workload/queue auto-covariance function. The ‘reverse engineering’ scheme is applied to various examples, including ones with infinite queues and non-summable workloads. AMS Subject Classifications Primary: 60K25; Secondary: 60G55, 60G10     

Stereoselective syntheses of 1,4-dideoxy-1,4-imino-octitols and novel tetrahydroxyindolizidines

A new route for the preparation of four new indolizidines, (1R,2S,6S,7S,8aS)- and (1R,2S,6R,7R,8aS)-1,2,6,7-tetrahydroxyindolizidine (30 and 32) and (1S,2R,7S,8S,8aR)- and (1S,2R,7R,8R,8aR)-1,2,7,8-tetrahydroxyindolizidine (44 and 46), is reported. The synthesis is based on Knoevenagel homologation of the readily available enantiomerically pure pyrrolidin-carbaldehydes 13 and 37followed by asymmetric dihydroxylation of the subsequent alkenyl pyrrolidines and cyclization of the corresponding imino-octitols. The new indolizidines and their precursors (imino-octitols 20, 25, 26) and indolizidinones 28a and 28b have been tested for inhibitory activities toward 26 glycosidases. The enzymatic inhibition of trans-7-hydroxy-d-(-)-swainsonine (44) toward alpha-mannosidases is similar to that described for trans-7-hydroxy-l-(+)-swainsonine (11b) toward naringinase (alpha-l-rhamnosidase from Penicillium decumbens).     

DFT analysis: Fe4 cluster and Fe(110) surface interaction studies with pyrrole,furan, thiophene,and selenophene molecules

DFT studies of both the Fe₄ cluster and the Fe(110) surface interaction with pyrrole, furan, thiophene, and selenophene showed that selenophene forms a stabler adsorbate iron complex than the other heterocyclic molecules; this is consistent with the binding energy data that were calculated between the Fe cluster and the Fe(110) surface with the heterocycles. Furthermore, when the adsorption of the compounds with the iron cluster was analyzed by molecular orbital studies, the orbitals of selenophene overlapped more strongly with the Fe atom than that of the other molecules. In TD-DFT, the π → π* peak observed for the molecules disappeared when they formed complexes, and there appeared a charge transfer band (ligand to metal), thus confirming the coordination of these molecules with the cluster. The data suggest that the chemisorption is an exothermic process.     

Hydrophobic Self‐Assembly Affords Robust Noncovalent Polymer Isomers

In covalent polymerization, a single monomer can result in different polymer structures due to positional, geometric, or stereoisomerism. We demonstrate that strong hydrophobic interactions result in stable noncovalent polymer isomers that are based on the same covalent unit (amphiphilic perylene diimide). These isomers have different structures and electronic/photonic properties, and are stable in water, even upon prolonged heating at 100 °C. Such combination of covalent‐like stability together with structural/functional variation is unique for noncovalent polymers, substantially advancing their potential as functional materials.     

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.     

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 Pietra term structures of financial assets

This paper explores an elemental connection between call options-the most commonly tradable financial derivatives, implied volatility term structures-critical “market information” emanating from call-option prices, and the Pietra index-a quantitative economic measure of societal egalitarianism. Our study: (i) unveils an intrinsic “Pietra structure” of call-option prices; (ii) introduces the notion of the “Pietra term structures” of financial assets; (iii) describes the probabilistic meaning of the Pietra term structures; (iv) establishes an explicit nonlinear one-to-one mapping between the Pietra term structures and the implied volatility term structures of financial assets. The results presented in this paper provide a deep insight into the econophysics of call options and implied volatility term structures.     

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.     

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.