Novel FEM Approach for the Analysis of Cylindrically Symmetric Photonic Devices

A novel scheme based on a 2-D finite element method (2-D-FEM) for the frequency domain, in cylindrical coordinates in conjunction with the perfectly matched layers (PML), is proposed and validated here. This scheme permits the analysis and simulation of photonic devices, including discontinuities along the propagation direction. Also, the present approach takes into account the dispersive nature of metals at optical wavelengths.     

Drug Design Inspired by Nature: Crystallographic Detection of an Auto‐Tailored Protease Inhibitor Template

De novo drug discovery is still a challenge in the search for potent and selective modulators of therapeutically relevant target proteins. Here, we disclose the unexpected discovery of a peptidic ligand 1 by X‐ray crystallography, which was auto‐tailored by the therapeutic target MMP‐13 through partial self‐degradation and subsequent structure‐based optimization to a highly potent and selective β‐sheet peptidomimetic inhibitor derived from the endogenous tissue inhibitors of metalloproteinases (TIMPs). The incorporation of non‐proteinogenic amino acids in combination with a cyclization strategy proved to be key for the de novo design of TIMP peptidomimetics. The optimized cyclic peptide 4 (ZHAWOC7726) is membrane permeable with an IC₅₀ of 21 nm for MMP‐13 and an attractive selectivity profile with respect to a polypharmacology approach including the anticancer targets MMP‐2 (IC₅₀: 170 nm ) and MMP‐9 (IC₅₀: 140 nm ).     

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

We present a variation-of-constants formula for functional differential equations of the form

$$\dot y = \mathcal{L}\left( t \right)y_t + f\left( {y_t,t} \right),\;y_{t_0}= \varphi $$

, where \(\mathcal{L}\) is a bounded linear operator and φ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t \(t \mapsto f\left( {y_t,t} \right)\) is Kurzweil integrable with t in an interval of ?, for each regulated function y. This means that t \(t \mapsto f\left( {y_t,t} \right)\) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J.Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type

$$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x} \right],x\left( {{t_0}} \right) = \tilde x$$

and the solutions of the perturbed Cauchy problem

$$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x + F\left( {x,t} \right)} \right],x\left( {{t_0}} \right) = \tilde x$$

Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form

$$\dot y = \mathcal{L}\left( t \right)y_t,\;y_{t_0} = \varphi$$

, where \(\mathcal{L}\) is a bounded linear operator and φ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.

     

On the dominance of roots of characteristic equations for neutral functional differential equations

We present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type, to be simple and dominant. This knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations. A simple characteristic equation, arisen from the study of delay equations with small delay, is analyzed in greater detail.     

Large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order

The spectral theory for linear autonomous neutral functional differential equations (FDE) yields explicit formulas for the large time behaviour of solutions. Our results are based on resolvent computations and Dunford calculus, applied to establish explicit formulas for the large time behaviour of solutions of FDE. We investigate in detail a class of two-dimensional systems of FDE.