Analyticity of the Planar Limit of a Matrix Model

Using Chebyshev polynomials combined with some mild combinatorics, we provide an alternative approach to the analytical and formal planar limits of a random matrix model with a 1-cut potential V. For potentials ${V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n}$ , as a power series in all a _n , the formal Taylor expansion of the analytic planar limit is exactly the formal planar limit. In the case V is analytic in infinitely many variables {a _n } _n ≥ 1 (on the appropriate spaces), the planar limit is also an analytic function in infinitely many variables and we give quantitative versions of where this is defined. Particularly useful in enumerative combinatorics are the gradings of ${V,V_{t}(x)=x^{2}/2-\sum_{n\ge1}a_{n}t^{n/2}x^{n}/n}$ and ${V_{t}(x)=x^{2}/2-\sum_{n\ge3}a_{n}t^{n/2 -1}x^{n}/n}$ . The associated planar limits F(t) as functions of t count planar diagram sorted by the number of edges respectively faces. We point out a method of computing the asymptotic of the coefficients of F(t) using the combination of the wzb method and the resolution of singularities. This is illustrated in several computations revolving around the important extreme potential ${V_{t}(x)=x^{2}/2+\log(1-\sqrt{t}x)}$ and its variants. This particular example gives a quantitative and sharp answer to a conjecture of ’t Hooft’s, which states that if the potential is analytic, the planar limit is also analytic.     

On the Existence of Trace for Elements of \mathbb{C}_p

Let T be a transcendental element of and the orbit of T. On we have a Haar measure . The goal of this paper is to characterize all the elements of for which the integral , called the trace of T, is well defined.Presented by A. Verschoren     

A new entropy upper bound

Entropy, conditional entropy and mutual information for discrete-valued random variables play important roles in information theory (see Ash (1965) [8] and Cover and Thomas (2006) [9]). Our purpose within this work is to present a strong upper bound for the classical Shannon entropy, refining recent results from the literature. For this purpose we have considered the work of Simic (2009) [4], where new entropy bounds based on a new refinement of Jensen’s inequality are presented. Our work improves the basic result of Simic through a stronger refinement of Jensen’s inequality which is then applied to information theory.     

View factors in a finite length axysymmetric cylindrical annulus enclosure

View factor – defined as fraction of total outgoing radiation from surface 1 intercepted by surface 2 – is a key concept in solving the radiative heat transfer in systems of surfaces. Analytical forms of the view factor have been developed by means of integration for many conventional systems of 2D and 3D surfaces in various spatial positions. However, little investigation has been carried out for situations in which there is no complete visibility between the surfaces involved in radiative exchange due to configuration of 3D surfaces or due to presence of interposing object that obstruct partially the mutual visibility.Radiative heat exchange was modeled in a finite length cylindrical annulus consisting of non-isothermal diffusely-emitting surfaces. The canonical definition of the view factor was extended by means of an analytical approach to both 3D and 2D surfaces with shading effects (incomplete mutual visibility) and numerical algorithms were developed in order to calculate view factors values and test view factor algebra rules. The system considered in this paper was an axysymmetric cylindrical annulus – frequently used in many heat transfer applications. View factor values for various surface elements were calculated by means of numerical integration. View factor algebra rules – summation and reciprocity – were verified.     

Diffraction phase microscopy with white light

We present white light diffraction phase microscopy (wDPM) as a quantitative phase imaging method that combines the single shot measurement benefit associated with off-axis methods, high temporal phase stability associated with common path geometries, and high spatial phase sensitivity due to the white light illumination. We propose a spatiotemporal filtering method that pushes the limit of the pathlength sensitivity to the subangstrom level at practical spatial and temporal bandwidths. We illustrate the utility of wDPM with measurements on red blood cell morphology and HeLa cell growth over 18 hours.     

Extinction of polarized light in ferrofluids with different magnetic particle concentrations

The magnetic field intensity and nanoparticle concentration dependence of the polarized light extinction in a ferrofluid made of magnetite particles stabilized with technical grade oleic acid dispersed in transformer oil was experimentally investigated. The magnetically induced optical anisotropy, i.e. the dichroism divided by concentration, was found to decrease with increasing sample concentration from 2% to 8%. The magnetically induced change in the optical extinction of light polarized at 54.74° with respect to the magnetic field direction was found to be positive for the less concentrated sample (2%) and negative for the samples with 4% and 8% magnetic nanoparticle concentrations, the more negative the higher the concentration and field intensity. Based on the theoretically proven fact [11] that the particle orientation mechanism has no effect on the extinction of light polarized at 54.74° with respect to the field direction, we analyzed the experimental findings in the frames of the agglomeration and long-range pair correlations theories for the magnetically induced optical anisotropy in ferrofluids. We developed a theoretical model in the approximation of single scattering for the optical extinction coefficient of a ferrofluid with magnetically induced particle agglomeration. The model predicts the existence of a polarization independent component of the optical extinction coefficient that is experimentally measurable at 54.74° polarization angle. The change in the optical extinction of light polarized at 54.74° is positive if only the formation of straight n-particle chains is considered and may become negative in the hypothesis that the longer chains degenerate to more isotropic structures (polymer-like coils, globules or bundles of chains). The model for the influence on the light absorption of the long-range pair correlations, published elsewhere, predicts that the change in the optical extinction of light polarized at 54.74° is always negative, the more negative the higher the magnetic field intensity and particle concentration.     

Hydroxy Functional Acrylates: Enzymatic Synthesis and Free Radical Polymerization

Hydroxy functional acrylates were prepared via transacylation of methyl acrylate with 1,2,6-hexanetriol catalyzed by Candida antarctica lipase B. Under the applied reaction conditions, 70% of the substrate was converted into hydroxy functional acrylates. In a subsequent step, after filtration of the enzyme, the monomer mixture containing methyl acrylate and the new monomers was polymerized via free radical polymerization resulting in a functional copolymer with M_n = 14500 g/mol.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.