The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

Straightforward Synthesis of Mn3O4/ZnO/Eu2O3-Based Ternary Heterostructure Nano-Photocatalyst and Its Application for the Photodegradation of Methyl Orange and Methylene Blue Dyes

Zinc oxide-ternary heterostructure Mn₃O₄/ZnO/Eu₂O₃ nanocomposites were successfully prepared via waste curd as fuel by a facile one-pot combustion procedure. The fabricated heterostructures were characterized utilizing XRD, UV–Visible, FT-IR, FE-SEM, HRTEM and EDX analysis. The photocatalytic degradation efficacy of the synthesized ternary nanocomposite was evaluated utilizing model organic pollutants of methylene blue (MB) and methyl orange (MO) in water as examples of cationic dyes and anionic dyes, respectively, under natural solar irradiation. The effect of various experimental factors, viz. the effect of a light source, catalyst dosage, irradiation time, pH of dye solution and dye concentration on the photodegradation activity, was systematically studied. The ternary Mn₃O₄/ZnO/Eu₂O₃ photocatalyst exhibited excellent MB and MO degradation activity of 98% and 96%, respectively, at 150 min under natural sunlight irradiation. Experiments further conclude that the fabricated nanocomposite exhibits pH-dependent photocatalytic efficacy, and for best results, concentrations of dye and catalysts have to be maintained in a specific range. The prepared photocatalysts are exemplary and could be employed for wastewater handling and several ecological applications.     

Blocks of monodromy groups in complex dynamics

Motivated by a problem in complex dynamics, we examine the block structure of the natural action of iterated monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.     

Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds

We study the renormalized area functional ${\mathcal{A}}We study the renormalized area functional A{\mathcal{A}} in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3-manifolds (and somewhat more broadly, Poincaré-Einstein spaces). Our main results include an explicit formula for the renormalized area of such a minimal surface Y as an integral of local geometric quantities, as well as formul? for the first and second variations of A{\mathcal{A}} which are given by integrals of global quantities over the asymptotic boundary loop γ of Y. All of these formul? are also obtained for a broader class of nonminimal surfaces. The proper setting for the study of this functional (when the ambient space is hyperbolic) requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary. We show that this moduli space is a smooth Banach manifold and develop a \mathbbZ{\mathbb{Z}} -valued degree theory for the natural map taking a minimal surface to its boundary curve. We characterize the nondegenerate critical points of A{\mathcal{A}} for minimal surfaces in \mathbbH³{\mathbb{H}^3} , and finally, discuss the relationship of A{\mathcal{A}} to the Willmore functional.     

Synthesis,Spectra Characterization,Electrochemical Behavior of Silyl Derivatives of Ferrocene

Ferrocenyl‐substituted silanes via the reaction between mono and dilithio‐ferrocene with trimethyl, trietheyl, vinyldimethyl or t‐butyldimethyl chlorosilanes is prepared. The electrochemical behaviors of these compounds were investigated by cyclic voltammetry. This work describes the electrochemical and electrocatalytic properties of carbon ceramic electrode (CCE) modified with ferrocene composites as a new electrocatalyst material.     

Short-time existence for the network flow

This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard-Reitich, Mantegazza-Novaga-Tortorelli and eventually, in the most general case of irregular networks by Ilmanen-Neves-Schulze. Although the present paper proves a result similar to the one in Ilmanen et al., the method here provides substantially more detailed information about how an irregular network “resolves” into a regular one. Either approach relies on the existence of self-similar expanding solutions found in Mazzeo and Saez. As a precursor to the main theorem, we also prove an unexpected regularity result for the mixed Cauchy-Dirichlet boundary problem for the linear heat equation on a manifold with boundary.     

Elliptic theory of differential edge operators I

Examples of edge operators include Laplacians on asymptotically flat and asymptotically hyperbolic manifolds. Edge operators also arise in boundary problems around higher condimension boundaries. This paper is concerned with the analysis of general elliptic edge operators with constant indicide roots. We determine when such an operator has a distributional asymptotic expansion. Conditions are given to guarantee that the coefficients of this expansion are smooth. In Part I of this paper we only study the case when the operator is semi-Fredholm. Part II will examine edge operators with infinite dimensional kernel and cokernel, as well as develop the theory of Poisson edge operators.     

BF3-OEt2 Catalyzed C3-Alkylation of Indole: Synthesis of Indolylsuccinimidesand Their Cytotoxicity Studies

A simple and efficient BF₃-OEt₂ promoted C3-alkylation of indole has been developed to obtain3-indolylsuccinimidesfrom commercially available indoles and maleimides, with excellent yields under mild reaction conditions. Furthermore, anti-proliferative activity of these conjugates was evaluated against HT-29 (Colorectal), Hepg2 (Liver) and A549 (Lung) human cancer cell lines. One of the compounds, 3w, having N,N-Dimethylatedindolylsuccinimide is a potent congener amongst the series with IC₅₀ value 0.02 µM and 0.8 µM against HT-29 and Hepg2 cell lines, respectively, and compound 3i was most active amongst the series with IC₅₀ value 1.5 µM against A549 cells. Molecular docking study and mechanism of reaction have briefly beendiscussed. This method is better than previous reports in view of yield and substrate scope including electron deficient indoles.