Anomalous is ubiquitous

Brownian motion is widely considered the quintessential model of diffusion processes—the most elemental random transport processes in Science and Engineering. Yet so, examples of diffusion processes displaying highly non-Brownian statistics–commonly termed “Anomalous Diffusion” processes–are omnipresent both in the natural sciences and in engineered systems. The scientific interest in Anomalous Diffusion and its applications is growing exponentially in the recent years. In this Paper we review the key statistics of Anomalous Diffusion processes: sub-diffusion and super-diffusion, long-range dependence and the Joseph effect, Lévy statistics and the Noah effect, and 1/f noise. We further present a theoretical model–generalizing the Einstein–Smoluchowski diffusion model–which provides a unified explanation for the prevalence of Anomalous Diffusion statistics. Our model shows that what is commonly perceived as “anomalous” is in effect ubiquitous.     

The first synthesis of substituted azepanes mimicking monosaccharides: a new class of potent glycosidase inhibitors

The synthesis of the first examples of seven-membered ring iminoalditols, molecules displaying an extra hydroxymethyl substituent on their seven-membered ring compared to the previously reported polyhydroxylated azepanes, has been achieved from d-arabinose in 10 steps using RCM of a protected N-allyl-aminohexenitol as a key step. While the (2R,3R,4R)-2-hydroxymethyl-3,4-dihydroxy-azepane 10, a seven-membered ring analogue of fagomine, is a weak inhibitor of glycosidases, the (2R,3R,4R,5S,6S)-2-hydroxymethyl-3,4,5,6-tetrahydroxy-azepane 9 selectively inhibits green coffee bean alpha-galactosidase in the low micromolar range (Ki = 2.2 muM) despite a D-gluco relative configuration.     

Benford’s law: A Poisson perspective

Benford’s law is a counterintuitive statistical law asserting that the distribution of leading digits, taken from a large ensemble of positive numerical values that range over many orders of scale, is logarithmic rather than uniform (as intuition suggests). In this paper we explore Benford’s law from a Poisson perspective, considering ensembles of positive numerical values governed by Poisson-process statistics. We show that this Poisson setting naturally accommodates Benford’s law and: (i) establish a Poisson characterization and a Poisson multidigit-extension of Benford’s law; (ii) study a system-invariant leading-digit distribution which generalizes Benford’s law, and establish a Poisson characterization and a Poisson multidigit-extension of this distribution; (iii) explore the universal emergence of the system-invariant leading-digit distribution, couple this universal emergence to the universal emergence of the Weibull and Fréchet extreme-value distributions, and distinguish the special role of Benford’s law in this universal emergence; (iv) study the continued-fractions counterpart of the system-invariant leading-digit distribution, and establish a Poisson characterization of this distribution; and (v) unveil the elemental connection between the system-invariant leading-digit distribution and its continued-fractions counterpart. This paper presents a panoramic Poisson approach to Benford’s law, to its system-invariant generalization, and to its continued-fractions counterpart.     

From micro-correlations to macro-correlations

Random vectors with a symmetric correlation structure share a common value of pair-wise correlation between their different components. The symmetric correlation structure appears in a multitude of settings, e.g. mixture models. In a mixture model the components of the random vector are drawn independently from a general probability distribution that is determined by an underlying parameter, and the parameter itself is randomized. In this paper we study the overall correlation of high-dimensional random vectors with a symmetric correlation structure. Considering such a random vector, and terming its pair-wise correlation “micro-correlation”, we use an asymptotic analysis to derive the random vector’s “macro-correlation” : a score that takes values in the unit interval, and that quantifies the random vector’s overall correlation. The method of obtaining macro-correlations from micro-correlations is then applied to a diverse collection of frameworks that demonstrate the method’s wide applicability.     

How random is a random vector?

Over 80 years ago Samuel Wilks proposed that the “generalized variance” of a random vector is the determinant of its covariance matrix. To date, the notion and use of the generalized variance is confined only to very specific niches in statistics. In this paper we establish that the “Wilks standard deviation” –the square root of the generalized variance–is indeed the standard deviation of a random vector. We further establish that the “uncorrelation index” –a derivative of the Wilks standard deviation–is a measure of the overall correlation between the components of a random vector. Both the Wilks standard deviation and the uncorrelation index are, respectively, special cases of two general notions that we introduce: “randomness measures” and “independence indices” of random vectors. In turn, these general notions give rise to “randomness diagrams”—tangible planar visualizations that answer the question: How random is a random vector? The notion of “independence indices” yields a novel measure of correlation for Lévy laws. In general, the concepts and results presented in this paper are applicable to any field of science and engineering with random-vectors empirical data.     

Temporal generation of power-law distributions: A universal ‘oligarchy mechanism’

We present a universal mechanism for the temporal generation of power-law distributions with arbitrary integer-valued exponents.     

Synthesis and α‐Mannosidase Inhibitory Evaluation of (2R,3R,4S)‐ and (2S,3R,4S)‐2‐(Aminomethyl)pyrrolidine‐3,4‐diol Derivatives

The synthesis of 46 derivatives of (2R,3R,4S)‐2‐(aminomethyl)pyrrolidine‐3,4‐diol is reported (Scheme 1 and Fig. 3), and their inhibitory activities toward α‐mannosidases from jack bean (B) and almonds (A) are evaluated (Table). The most‐potent inhibitors are (2R,3R,4S)‐2‐{[([1,1′‐biphenyl]‐4‐ylmethyl)amino]methyl}pyrrolidine‐3,4‐diol ( 3fs ; IC₅₀(B)=5 μM , K_i=2.5 μM ) and (2R,3R,4S)‐2‐{[(1R)‐2,3‐dihydro‐1H‐inden‐1‐ylamino]methyl}pyrrolidine‐3,4‐diol ( 3fu ; IC₅₀(B)=17 μM , K_i=2.3 μM ). (2S,3R,4S)‐2‐(Aminomethyl)pyrrolidine‐3,4‐diol ( 6 , R?H) and the three 2‐(N‐alkylamino)methyl derivatives 6fh, 6fs , and 6f are prepared (Scheme 2) and found to inhibit also α‐mannosidases from jack bean and almonds (Table). The best inhibitor of these series is (2S,3R,4S)‐2‐{[(2‐thienylmethyl)amino]methyl}pyrrolidine‐3,4‐diol ( 6o ; IC₅₀(B)=105 μM , K_i=40 μM ). As expected (see Fig. 4), diamines 3 with the configuration of α‐D ‐mannosides are better inhibitors of α‐mannosidases than their stereoisomers 6 with the configuration of β‐D ‐mannosides. The results show that an aromatic ring (benzyl, [1,1′‐biphenyl]‐4‐yl, 2‐thienyl) is essential for good inhibitory activity. If the C‐chain that separates the aromatic system from the 2‐(aminomethyl) substituent is longer than a methano group, the inhibitory activity decreases significantly (see Fig. 7). This study shows also that α‐mannosidases from jack bean and from almonds do not recognize substrate mimics that are bulky around the O‐glycosidic bond of the corresponding α‐D ‐mannopyranosides. These observations should be very useful in the design of better α‐mannosidase inhibitors.     

From Polling to Snowplowing

We study the limiting behavior of gated polling systems, as their dimension (the number of queues) tends to infinity, while the system's total incoming workflow and total switchover time (per cycle) remain unchanged. The polling systems are assumed asymmetric, with incoming workflow obeying general Lévy statistics, and with general inter-dependent switchover times. We prove convergence, in law, to a limiting polling system on the circle. The derivation is based on an asymptotic analysis of the stochastic Poincaré maps of the polling systems. The obtained polling limit is identified as a snowplowing system on the circle—whose evolution, steady-state equilibrium, and statistics have been recently investigated and are known.     

Paretian Poisson Processes

Many random populations can be modeled as a countable set of points scattered randomly on the positive half-line. The points may represent magnitudes of earthquakes and tornados, masses of stars, market values of public companies, etc. In this article we explore a specific class of random such populations we coin ‘Paretian Poisson processes’. This class is elemental in statistical physics—connecting together, in a deep and fundamental way, diverse issues including: the Poisson distribution of the Law of Small Numbers; Paretian tail statistics; the Fréchet distribution of Extreme Value Theory; the one-sided Lévy distribution of the Central Limit Theorem; scale-invariance, renormalization and fractality; resilience to random perturbations.