Low-energy expansion formula for one-dimensional Fokker-Planck and Schr\"odinger equations with asymptotically periodic potentials

We consider one-dimensional Fokker-Planck and Schr\"odinger equations with a potential which approaches a periodic function at spatial infinity. We extend the low-energy expansion method, which was introduced in previous papers, to be applicable to such asymptotically periodic cases. Using this method, we study the low-energy behavior of the Green function.Comment: author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretica

A multi-dimensional SRBM: Geometric views of its product form stationary distribution

We present a geometric interpretation of a product form stationary distribution for a $d$-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The $d$-dimensional SRBM data can be equivalently specified by $d+1$ geometric objects: an ellipse and $d$ rays. Using these geometric objects, we establish necessary and sufficient conditions for characterizing product form stationary distribution. The key idea in the characterization is that we decompose the $d$-dimensional problem to $\frac{1}{2}d(d-1)$ two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of [14]. A $d$-station tandem queue example is presented to illustrate how the product form can be obtained using our characterization. Drawing the two-dimensional results in [1,7], we discuss potential optimal paths for a variational problem associated with the three-station tandem queue. Except Appendix D, the rest of this paper is almost identical to the QUESTA paper with the same title

The Emergence of Scaling in Sequence-based Physical Models of Protein Evolution

It has recently been discovered that many biological systems, when represented as graphs, exhibit a scale-free topology. One such system is the set of structural relationships among protein domains. The scale-free nature of this and other systems has previously been explained using network growth models that, while motivated by biological processes, do not explicitly consider the underlying physics or biology. In the present work we explore a sequence-based model for the evolution protein structures and demonstrate that this model is able to recapitulate the scale-free nature observed in graphs of real protein structures. We find that this model also reproduces other statistical feature of the protein domain graph. This represents, to our knowledge, the first such microscopic, physics-based evolutionary model for a scale-free network of biological importance and as such has strong implications for our understanding of the evolution of protein structures and of other biological networks.Comment: 20 pages (including figures), 4 figures, to be submitted to PNA

The BAR approach for multiclass queueing networks with SBP service policies

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the "interchange of limits" approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being "tight." These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem