Purification of hyaluronidase as an anticancer agent inhibiting CD44

Hyaluronidase (Hyal) can be employed to accomplish a diversity of complications related to hyaluronic acid (HA). Hyal contains some classes of catalysts that cleave HA. This enzyme is detected in several human tissues as well as in animal venoms, pathogenic organisms and cancers. Destructive cancer cells regularly increase the CD44 receptor existing in a cell membrane. This receptor acts as an exact receptor for HA, and HA is recognized to motivate the migration, spread, attack and metastasis of cancer cells. Nearly all of the methods used to purify Hyal are highly costly and not proper for industrial applications. This survey aims to review different methods of Hyal purification, which acts as an anticancer agent by degrading HA in tissues and thus inhibiting the CD44–HA interaction. Hyal can be successfully employed in the management of cancer, which is associated with HA–CD44. This review has described different methods for Hyal purification to prepare an origin to develop a novel purification technique for this highly appreciated protein. Using multiple columns is not applicable for the purification of Hyal and thus cannot be used at the industrial level. It is better to use affinity chromatography of anti-Hyal for Hyal with one-step purification.     

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.     

Properties of G-martingales with finite variation and the application to G-Sobolev spaces

As is known, if $B={\left(B_t\right)}_{t\in [0,T]}$ is a $G$-Brownian motion, a process of form ${\int }_0^t{\eta }_{s}d{〈B〉}_s?{\int }_0^{t}2G\left({\eta }_s\right)ds$, $\eta \in M_G^1(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale cannot be form of ${\int }_0^t{\eta }_{s}ds$ or ${\int }_0^t{\gamma }_{s}d{〈B〉}_s$, $\eta ,\gamma \in M_G^1(0,T)$, which implies that the decomposition for generalized $G$-It44; processes is unique: For arbitrary $\zeta \in H_G^1(0,T)$, $\eta \in M_G^1(0,T)$ and non-increasing $G$-martingales $K,L$, if ${\int }_0^t{\zeta }_{s}d{B}_s+{\int }_0^t{\eta }_{s}ds+K_t=L_t,t\in [0,T],$then we have $\eta \equiv 0$, $\zeta \equiv 0$ and$K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).     

Non-existence of extremals for the Adimurthi–Druet inequality

The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a $L^2$-type perturbation, quantified by $\alpha \in [0,{\lambda }_1)$, where ${\lambda }_1$ is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3], [10], [14], [19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches ${\lambda }_1$. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as $\alpha \to {\lambda }_1$.     

Mean curvature flow with free boundary – Type 2 singularities

In this paper we give sufficient conditions that guarantee the mean curvature flow with free boundary on a pinched cylinder develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.     

A fresh approach to the Paley–Wiener theorem for Mellin transforms and the Mellin–Hardy spaces

Here we give a new approach to the Paley–Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of polar‐analytic function in the Mellin setting and Mellin–Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.     

Diffusions of multiplicative cascades

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. For Gaussian independent increment processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Hölder exponent in the Wasserstein distance on measures. We also discuss applications of this process to the model of tree polymers.     

Nonexistence of global solutions to new ordinary differential inequality and applications to nonlineardispersive equations

A new ordinary differential inequality without global solutions is proposed. Comparison with similar differential inequalities in the well‐known concavity method is performed. As an application, finite time blow up of the solutions to nonlinear Klein–Gordon equation and generalized Boussinesq equation is proven. The initial energy is arbitrary high positive. The structural conditions on the initial data generalize the assumptions used in the literature for the time being. Copyright © 2015 John Wiley & Sons, Ltd.     

Central limit theorems for realized volatility under hitting times of an irregular grid

We consider a continuous semi-martingale sampled at hitting times of an irregular grid. The goal of this work is to analyze the asymptotic behavior of the realized volatility under this rather natural observation scheme. This framework strongly differs from the well understood situations when the sampling times are deterministic or when the grid is regular. Indeed, neither Gaussian approximations nor symmetry properties can be used. In this setting, as the distance between two consecutive barriers tends to zero, we establish central limit theorems for the normalized error of the realized volatility. In particular, we show that there is no bias in the limiting process.