Integral Representations of the Macdonald Symmetric Functions

Multiple-integral representations of the (skew-)Macdonald symmetric functions are obtained. Some bosonization schemes for the integral representations are also constructed.Comment: LaTex 21page

Long-lived metal complexes open up microsecond lifetime imaging microscopy under multiphoton excitation: from FLIM to PLIM and beyond

Lifetime imaging microscopy with sub-micron resolution provides essential understanding of living systems by allowing both the visualisation of their structure, and the sensing of bio-relevant analytes in vivo using external probes. Chemistry is pivotal for the development of the next generation of bio-tools, where contrast, sensitivity, and molecular specificity facilitate observation of processes fundamental to life. A fundamental limitation at present is the nanosecond lifetime of conventional fluorescent probes which typically confines the sensitivity to sub-nanosecond changes, whilst nanosecond background autofluorescence compromises the contrast. High-resolution visualization with complete background rejection and simultaneous mapping of bio-relevant analytes including oxygen – with sensitivity orders of magnitude higher than that currently attainable – can be achieved using time-resolved emission imaging microscopy (TREM) in conjunction with probes with microsecond (or longer) lifetimes. Yet the microsecond timescale has so far been incompatible with available multiphoton excitation/detection technologies. Here we realize for the first time microsecond-imaging with multiphoton excitation whilst maintaining the essential sub-micron spatial resolution. The new method is background-free and expands available imaging and sensing timescales 1000-fold. Exploiting the first engineered water-soluble member of a family of remarkably emissive platinum-based, microsecond-lived probes amongst others, we demonstrate (i) the first instance of background-free multiphoton-excited microsecond depth imaging of live cells and histological tissues, (ii) over an order-of-magnitude variation in the probe lifetime in vivo in response to the local microenvironment. The concept of two-photon TREM can be seen as “FLIM + PLIM” as it can be used on any timescale, from ultrafast fluorescence of organic molecules to slower emission of transition metal complexes or lanthanides/actinides, and combinations thereof. It brings together transition metal complexes as versatile emissive probes with the new multiphoton-excitation/microsecond-detection approach to create a transformative framework for multiphoton imaging and sensing across biological, medicinal and material sciences

The generalised scaling function: a systematic study

We describe a procedure for determining the generalised scaling functions $f_n(g)$ at all the values of the coupling constant. These functions describe the high spin contribution to the anomalous dimension of large twist operators (in the $sl(2)$ sector) of ${\cal N}=4$ SYM. At fixed $n$, $f_n(g)$ can be obtained by solving a linear integral equation (or, equivalently, a linear system with an infinite number of equations), whose inhomogeneous term only depends on the solutions at smaller $n$. In other words, the solution can be written in a recursive form and then explicitly worked out in the strong coupling regime. In this regime, we also emphasise the peculiar convergence of different quantities ('masses', related to the $f_n(g)$) to the unique mass gap of the $O(6)$ nonlinear sigma model and analyse the first next-to-leading order corrections.Comment: Latex version, journal version (with explanatory appendices and more references

A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric

Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law on the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers and answers a question by Beck and Zaslavsky.Regal Entertainment Group (Competitive Earmarked Research Grants 600703)Regal Entertainment Group (Competitive Earmarked Research Grants 600506)Regal Entertainment Group (Competitive Earmarked Research Grants 600608

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.