Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke

Structural and functional characterization of proteins adsorbed on hydrophilized polylactide-co-glycolide microfibers

Rajesh Vasita, Dhirendra S KattiDepartment of Biological Sciences and Bioengineering, Indian Institute of Technology, Kanpur, Uttar Pradesh, IndiaBackground: Hydrophobic biopolymers such as polylactide-co-glycolide (PLGA, 85:15) have been extensively explored as scaffolding materials for tissue engineering applications. More recently, electrospun microfiber-based and nanofiber-based scaffolds of PLGA have received increased attention because they act as physical mimics of the fibrillar extracellular matrix. However, the hydrophobicity of the PLGA microfiber surface can limit its use in biomedical applications. Therefore, in a previous study, we fabricated Pluronic® F-108 (PF-108)-blended PLGA microfibrous scaffolds that alleviated the hydrophobicity associated with PLGA by enriching the surface of microfibers with the ethylene oxide units present in PF-108.Methods: In this study, we report the influence of the extent of surface enrichment of PLGA microfibers on their interaction with two model proteins, ie, bovine serum albumin (BSA) and lysozyme. BSA and lysozyme were adsorbed onto PLGA microfiber meshes (unmodified and modified) and studied for the amount, secondary structure conformation, and bioactivity of released protein.Results: Irrespective of the type of protein, PF-108-blended PLGA microfibers showed significantly greater protein adsorption and release than the unblended PLGA samples. However, in comparison with BSA, lysozyme showed a 7–9-fold increase in release. The Fourier transform infrared spectroscopy studies for secondary structure determination demonstrated that irrespective of type of microfiber surface (unblended or blended), adsorbed BSA and lysozyme did not show any significant change in secondary structure (α-helical content) as compared with BSA and/or lysozyme in the free powder state. Further, the bioactivity assay of lysozyme released from blended PLGA microfiber meshes demonstrated 80%–85% bioactivity, indicating that the process of adsorption did not significantly affect biological activity. Therefore, this study demonstrated that the decreased hydrophobicity of blended PLGA microfibrous meshes not only improved the amount of protein adsorbed (lysozyme and BSA) but also maintained the secondary structure and bioactivity of the adsorbed proteins.Conclusion: Modulating the hydrophobicity of PLGA via blending with PF-108 could be a viable strategy to improve its interaction with proteins and subsequent cell interaction in tissue engineering applications.Keywords: microfiber, protein adsorption, electrospinnin

The universe in a soap film

The value of the cosmological constant is one of the major puzzles of modern cosmology: it is tiny but nonzero. Sorkin predicted, from the Causet approach to quantum gravity, that the cosmological constant has quantum fluctuations. The predicted order of magnitude of the fluctuations agrees with the subsequently observed value of the cosmological constant. We had earlier developed an analogy between the cosmological constant of the universe and the surface tension of fluid membranes. Here we demonstrate by computer simulations that the surface tension of a fluid membrane has statistical fluctuations stemming from its discrete molecular structure. Our analogy enables us to view these numerical experiments as probing a small and fluctuating cosmological constant. Deriving insights from our analogy, we show that a fluctuating cosmological constant is a generic feature of quantum gravity models and is far more general than the specific context in which it was originally proposed. We pursue and refine the idea of a fluctuating cosmological constant and work towards making further testable predictions. © 2009 IOP Publishing Ltd