Nanotechnology for Electroanalytical Biosensors of Reactive Oxygen and Nitrogen Species

Over the past several decades, nanotechnology has contributed to the progress of biomedicine, biomarker discovery, and the development of highly sensitive electroanalytical / electrochemical biosensors for in vitro and in vivo monitoring, and quantification of oxidative and nitrosative stress markers like reactive oxygen species (ROS) and reactive nitrogen species (RNS). A major source of ROS and RNS is oxidative stress in cells, which can cause many human diseases, including cancer. Therefore, the detection of local concentrations of ROS (e. g. superoxide anion radical; O₂^•−) and RNS (e. g. nitric oxide radical; NO^• and its metabolites) released from biological systems is increasingly important and needs a sophisticated detection strategy to monitor ROS and RNS in vitro and in vivo. In this review, we discuss the nanomaterials‐based ROS and RNS biosensors utilizing electrochemical techniques with emphasis on their biomedical applications.     

Positional number systems with digits forming an arithmetic progression

A novel digit system that arises in a natural way in a graph-theoretical problem is studied. It is defined by a set of positive digits forming an arithmetic progression and, necessarily, a complete residue system modulo the base b. Since this is not enough to guarantee existence of a digital representation, the most significant digit is allowed to come from an extended set. We provide explicit formulæ for the j th digit in such a representation as well as for the length. Furthermore, we study digit frequencies and average lengths, thus generalising classical results for the base-b representation. For this purpose, an appropriately adapted form of the Mellin-Perron approach is employed.     

Essential countability of treeable equivalence relations

We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E   is essentially countable if and only if there is no continuous embedding of _E1${\mathbb{E}}_1$ into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of X(X-Y)(X-(T+x)Y)+Y³=1+xT(1-T)X(X-Y)(X-(T+\xi)Y)+Y^3=1+\xi T(1-T) over \Bbb C[T]{\Bbb C}[T] for all x ? \Bbb C\xi\in{\Bbb C} .     

Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error ||u^*-u_h^*||_L²(G)\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)} decreases like N ⁻¹, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N ^−3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results.     

Flat fronts and stability for the porous medium equation

This work is concerned with the equation ?_tρ = Δ_xρ^m, m > 1, known as the porous medium equation. It shows stability of the pressure of solutions close to flat travelling wave fronts in the homogeneous Lipschitz sense that is in a way optimal for the treatment of the equation. This is the first result of this type and implies global regularity estimates for any number of derivatives of the pressure. Consequences include smoothness, analyticity in temporal and tangential directions, and analyticity of the interface between empty and occupied regions.     

Bayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous within-class variances

A Bayesian inference for a linear Gaussian random coefficient regression model with inhomogeneous within-class variances is presented. The model is motivated by an application in metrology, but it may well find interest in other fields. We consider the selection of a noninformative prior for the Bayesian inference to address applications where the available prior knowledge is either vague or shall be ignored. The noninformative prior is derived by applying the Berger and Bernardo reference prior principle with the means of the random coefficients forming the parameters of interest. We show that the resulting posterior is proper and specify conditions for the existence of first and second moments of the marginal posterior. Simulation results are presented which suggest good frequentist properties of the proposed inference. The calibration of sonic nozzle data is considered as an application from metrology. The proposed inference is applied to these data and the results are compared to those obtained by alternative approaches.     

Product formulas and convolution structure for Fourier-Bessel series

One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter ,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases and . As a consequence, a positive convolution structure is established for . The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder.