Combinatorial Identities Related to Eigen-Function Decompositions of Hill operators: Open Questions

We formulate three open questions related to enumerative combinatorics, which arise in the spectral analysis of Hill operators with trigonometric polynomial potentials.     

Phototransistor noise model based on noise measurements on PNP PIN phototransistors

A noise model for phototransistors in open base configuration is presented. The model was developed from the noise measurements on four different phototransistor designs. The extracted current gains from the noise model were verified by measured current gains from Gummel measurements. Furthermore the current noise spectral density was modeled and compared with the noise measurements. A difference of less than 12 % in the current gain was achieved. In addition to the extracted current gains also the most dominant shot noise terms including their values of each phototransistor are extracted.     

Kinetic Isotope Effect as a Tool To Investigate the Oxygen Reduction Reaction on Pt-based Electrocatalysts – Part II: Effect of Platinum Dispersion

We investigated the oxygen reduction reaction (ORR) mechanism on Pt nanoparticles (NPs) dispersed on several carbon blacks with various physicochemical properties (i. e. specific surface ranging from 80 to 900 m² g⁻¹, different graphitization degree, etc.). Using the kinetic isotope effect (KIE) along with various electrochemical characterizations, we determined that the rate determining step (RDS) of the ORR is a proton-independent step when the density of Pt NPs on the surface of the carbon support is high. Upon decrease of the density of Pt NPs on the surface, the RDS of the ORR starts involving a proton, as denoted by an increase of the KIE >1. This underlined the critical role played by the carbon support in the oxygen reduction reaction electrocatalysis by Pt supported on high surface area carbon.     

Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

Observations on toxicologically relevant arsenic in urine in adult offspring of families with Balkan Endemic Nephropathy and controls by batch hydride generation atomic absorption spectrometry

The aim of this study was to develop a method for the characterization of internal exposure to arsenic, which is thought to play a role in the development of a kidney disease, known as Balkan Endemic Nephropathy, typical for a district in Bulgaria, and to investigate whether the As body burden differs in the offspring versus control individuals. For this case study, an analytical procedure for the determination of toxicologically relevant arsenic (the sum of arsenite, arsenate, monomethylarsonate, and dimethylarsinate) in urine by batch-type hydride generation atomic absorption spectrometry was developed. Optimization experiments for levelling off the sensitivity of inorganic arsenic and its mono- and dimethylated species in dilute HCl–L-cysteine medium were performed. The limit of detection for hydride forming arsenic fraction was 0.5?ng As, i.e. 0.25?µg?L^?1 in 10?mL of 1?+?4 v/v diluted urine. The relative standard deviation was typically 1.5–1.8% for aqueous solution and 2–6% for urine samples at 1.0?µg?L^?1 As. The sample throughput rate was 15?h^?1. No statistical correlation and cross-correlation between individuals case-control and sex at 95% confidence were found: controls (n?=?99), mean 3.5?±?2.1 (SD), range 0.9–10.4, median 3.0?µg?L^?1 As and cases (n?=?102), mean 3.6?±?2.2 (SD), range 0.5–11.0, median 3.2?µg?L^?1 As. On the basis of this study, arsenic can be excluded as a factor involved in BEN development.     

On a Two-Variable Class of Bernstein–Szegő Measures

The one-variable Bernstein–Szegő theory for orthogonal polynomials on the real line is extended to a class of two-variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.      

q-Blossoming for analytic functions

Numerical Algorithms - We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic...     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Implicit standard Jacobi gives high relative accuracy

We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX ^T of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(ek(X)){O(\epsilon \kappa (X))} , where e{\epsilon} is the machine precision and k(X) o ||X||₂·||X^-1||₂{\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2} is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense. We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite) matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.