Felix Alexandrovich Berezin and his work

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page

On a formula of Gammelgaard for Berezin-Toeplitz quantization

We give a proof of a slightly refined version of Gammelgaard's graph theoretic formula for Berezin-Toeplitz quantization on (pseudo-)Kaehler manifolds. Our proof has the merit of giving an alternative approach to Karabegov-Schlichenmaier's identification theorem. We also identify the dual Karabegov-Bordemann-Waldmann star product.Comment: 18 page

Weyl invariant polynomial and deformation quantization on Kahler manifolds

Given a polynomial P of partial derivatives of the Kahler metric, expressed as a linear combination of directed multigraphs, we prove a simple criterion in terms of the coefficients for $P$ to be an invariant polynomial, i.e. invariant under the transformation of coordinates. As applications, we prove an explicit composition formula for covariant differential operators under a canonical basis, also known as invariant differential operators in the case of bounded symmetric domains. We also prove a general explicit formula of star products on Kahler manifolds.Comment: 17 page

Toeplitz operators on symplectic manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.Comment: 40 page

An explicit formula for the Berezin star product

We prove an explicit formula of the Berezin star product on Kaehler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Englis' work on the asymptotic expansion of the Laplace integral.Comment: 19 pages, to appear in Lett. Math. Phy

Two-photon phosphorescence lifetime imaging of cells and tissues using a long-lived cyclometallated (NpyridylCphenylNpyridyl)-C-boolean AND-N-boolean AND Pt(II) complex

Using a combination of multiphoton excitation, confocal scanning, TCSPC and beam blanking in conjunction with a cyclometallated Npyridyl^Cphenyl^Npyridyl Pt(II) complex (1) with a long luminescence lifetime, we demonstrate lifetime mapping of living cells and histological tissue sections over a time-frame of 50 microseconds, using a laser on/off “beam blanking” approach. This method of performing phosphorescence lifetime imaging microscopy (PLIM) represents an order of magnitude enhancement of the two-photon time-resolved emission imaging microscopy (TP-TREM) method, where in order to achieve a longer imaging window, the excitation laser repetition rate was reduced by cavity dumping [Chem. Sci., 2014, 5, 879]. The method complements and expands other existing imaging methodologies by enabling simultaneous PLIM and FLIM (fluorescence lifetime imaging microscopy – recorded between beam blanking), whilst maintaining essential sub-micron spatial resolution. We demonstrate how the Pt(II) complex can be used to distinguish between cell nuclei and matrix proteins on the basis of emission lifetime, in both structured and homogeneous tissue sections; whilst also revealing how the Pt(II) emission lifetime varies with tissue matrix composition. The proposed imaging approach can be used in conjunction with any biocompatible emissive probe with a long emission lifetime – exemplified here by (1) – and for an array of fluorescent/phosphorescent labels, where discrimination is lifetime-based

Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization

For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given

Direct electrochemistry of Heme Proteins on Electrodes Modified with Didodecyldimethyl Ammonium Bromide and Carbon Black

A novel matrix based on commercially available carbon black (CB) N220 and didodecyldimethyl ammonium bromide (DDAB) was shown to be a reliable support for direct electron transfer reactions between screen printed electrode (SPE) and Fe(III)-heme proteins. Cytochrome c(cytc), myoglobin (Mb), horseradish peroxidase (HRP) and cytochromes P450 (CYP 51A1, CYP 3A4, CYP 2B4) generated well-shaped cyclic voltammograms on SPE/CB/ DDAB electrodes (both in solution and in immobilized state). The attractive performance characteristics of CB modified electrodes are advantageous over single-walled carbon nanotubes (SW CNT) based ones. The achieved direct electrochemistry of heme proteins on CB/DDAB-modified electrodes provided successful elaboration of the immunosensor for cardiac Mb. The immunosensor showed applicability for diagnostics of myocardial infarction displaying significant difference in cardiac Mb content of human blood plasma samples taken from the corresponding patients

Families of exact solutions to Vasiliev's 4D equations with spherical, cylindrical and biaxial symmetry

We provide Vasiliev's four-dimensional bosonic higher-spin gravities with six families of exact solutions admitting two commuting Killing vectors. Each family contains a subset of generalized Petrov Type-D solutions in which one of the two so(2) symmetries enhances to either so(3) or so(2,1). In particular, the spherically symmetric solutions are static and we expect one of them to be gauge-equivalent to the extremal Didenko-Vasiliev solution given in arXiv:0906.3898. The solutions activate all spins and can be characterized either via generalized electric and magnetic charges defined asymptotically in weak-field regions or via the values of fully higher-spin gauge-invariant observables given by on-shell closed zero-forms. The solutions are obtained by combining the gauge-function method with separation of variables in twistor space via expansion of the Weyl zero-form in Di-Rac supersingleton projectors times deformation parameters in a fashion that is suggestive of a generalized electromagnetic duality.Comment: v1: 77 pages; 3 tables; 23 pages of appendices. v2: 5 additional pages of clarifications, typos correcte