Combinatorial Alexander Duality -- a Short and Elementary Proof

Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof.Comment: 7 pages, 2 figure; v3: the sign function was simplifie

Skateboard Injuries in a Campus Community

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67018/2/10.1177_000992286700600421.pd

DEVELOPMENT AND PRODUCTION OF URANIUM 10 wt.% MOLYBDENUM AS-CAST SLUGS FOR HNPF PHYSICS EXPERIMENT

A multiple casting process was developed for the production of 0.590 plus or minus 0.005-in. diam. by 12-in. long solid right cylinders. The process involves vacuum induction melting in MgZrO/sub 3/ coated graphite crucibles and casting into coated graphite molds. Thirty-eight hundred fifty 90 wt.% U of 1.5 wt.% enrichment-10 wt.% Mo as-cast pins were produced for HNPF Exponential Experiments. (auth

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

Statistical mechanics in the context of special relativity

In the present effort we show that $S_{\kappa}=-k_B \int d^3p (n^{1+\kappa}-n^{1-\kappa})/(2\kappa)$ is the unique existing entropy obtained by a continuous deformation of the Shannon-Boltzmann entropy $S_0=-k_B \int d^3p n \ln n$ and preserving unaltered its fundamental properties of concavity, additivity and extensivity. Subsequently, we explain the origin of the deformation mechanism introduced by $\kappa$ and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter $\kappa$ which results to depend on the light speed $c$ and reduces to zero as $c \to \infty$ recovering in this way the ordinary statistical mechanics and thermodynamics. The novel statistical mechanics constructed starting from the above entropy, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data. PACS number(s): 05.20.-y, 51.10.+y, 03.30.+p, 02.20.-aComment: 17 pages (two columns), 5 figures, RevTeX4, minor typing correction

Probability Distribution Function of the Order Parameter: Mixing Fields and Universality

We briefly review the use of the order parameter probability distribution function as a useful tool to obtain the critical properties of statistical mechanical models using computer Monte Carlo simulations. Some simple discrete spin magnetic systems on a lattice, such as Ising, general spin-$S$ Blume-Capel and Baxter-Wu, $Q$-state Potts, among other models, will be considered as examples. The importance and the necessity of the role of mixing fields in asymmetric magnetic models will be discussed in more detail, as well as the corresponding distributions of the extensive conjugate variables.Comment: 14 pages, 13 figures, accepted for publication (Computer Physics Communications

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

Combinatorial Markov chains on linear extensions

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in terms of discrete time Markov chain

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