Area versus Length Distribution for Closed Random Walks

Using a connection between the $q$-oscillator algebra and the coefficients of the high temperature expansion of the frustrated Gaussian spin model, we derive an exact formula for the number of closed random walks of given length and area, on a hypercubic lattice, in the limit of infinite number of dimensions. The formula is investigated in detail, and asymptotic behaviours are evaluated. The area distribution in the limit of long loops is computed. As a byproduct, we obtain also an infinite set of new, nontrivial identities.Comment: 17 page

X-ray Isophote Shapes and the Mass of NGC 3923

We present analysis of the shape and radial mass distribution of the E4 galaxy NGC 3923 using archival X-ray data from the ROSAT PSPC and HRI. The X-ray isophotes are significantly elongated with ellipticity e_x=0.15 (0.09-0.21) (90% confidence) for semi-major axis a\sim 10h^{-1}_70 kpc and have position angles aligned with the optical isophotes within the estimated uncertainties. Applying the Geometric Test for dark matter, which is independent of the gas temperature profile, we find that the ellipticities of the PSPC isophotes exceed those predicted if M propto L at a marginal significance level of 85% (80%) for oblate (prolate) symmetry. Detailed hydrostatic models of an isothermal gas yield ellipticities for the gravitating matter, e_mass=0.35-0.66 (90% confidence), which exceed the intensity weighted ellipticity of the R-band optical light, = 0.30 (e_R^max=0.39). We conclude that mass density profiles with rho\sim r^{-2} are favored over steeper profiles if the gas is essentially isothermal (which is suggested by the PSPC spectrum) and the surface brightness in the central regions (r<~15") is not modified substantially by a multi-phase cooling flow, magnetic fields, or discrete sources. We argue that these effects are unlikely to be important for NGC 3923. (The derived e_{mass} range is very insensitive to these issues.) Our spatial analysis also indicates that the allowed contribution to the ROSAT emission from a population of discrete sources with Sigma_x propto Sigma_R is significantly less than that indicated by the hard spectral component measured by ASCA.Comment: 14 pages (6 figures), To Appear in MNRA

Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation

Principal Component Analysis of the Time- and Position-Dependent Point Spread Function of the Advanced Camera for Surveys

We describe the time- and position-dependent point spread function (PSF) variation of the Wide Field Channel (WFC) of the Advanced Camera for Surveys (ACS) with the principal component analysis (PCA) technique. The time-dependent change is caused by the temporal variation of the $HST$ focus whereas the position-dependent PSF variation in ACS/WFC at a given focus is mainly the result of changes in aberrations and charge diffusion across the detector, which appear as position-dependent changes in elongation of the astigmatic core and blurring of the PSF, respectively. Using >400 archival images of star cluster fields, we construct a ACS PSF library covering diverse environments of the $HST$ observations (e.g., focus values). We find that interpolation of a small number ($\sim20$) of principal components or ``eigen-PSFs'' per exposure can robustly reproduce the observed variation of the ellipticity and size of the PSF. Our primary interest in this investigation is the application of this PSF library to precision weak-lensing analyses, where accurate knowledge of the instrument's PSF is crucial. However, the high-fidelity of the model judged from the nice agreement with observed PSFs suggests that the model is potentially also useful in other applications such as crowded field stellar photometry, galaxy profile fitting, AGN studies, etc., which similarly demand a fair knowledge of the PSFs at objects' locations. Our PSF models, applicable to any WFC image rectified with the Lanczos3 kernel, are publicly available.Comment: Accepted to PASP. To appear in December issue. Figures are degraded to meet the size limit. High-resolution version can be downloaded at http://acs.pha.jhu.edu/~mkjee/acs_psf/acspsf.pd

Asymptotics of characters of symmetric groups, genus expansion and free probability

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups S_q for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q^{-1/2} converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to two-dimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.Comment: version 2: change of title; the section on Gaussian fluctuations was moved to a subsequent paper [Piotr Sniady: "Gaussian fluctuations of characters of symmetric groups and of Young diagrams" math.CO/0501112

Efficient Local Comparison Of Images Using Krawtchouk Descriptors

It is known that image comparison can prove cumbersome in both computational complexity and runtime, due to factors such as the rotation, scaling, and translation of the object in question. Due to the locality of Krawtchouk polynomials, relatively few descriptors are necessary to describe a given image, and this can be achieved with minimal memory usage. Using this method, not only can images be described efficiently as a whole, but specific regions of images can be described as well without cropping. Due to this property, queries can be found within a single large image, or collection of large images, which serve as a database for search. Krawtchouk descriptors can also describe collections of patches of 3D objects, which is explored in this paper, as well as a theoretical methodology of describing nD hyperobjects. Test results for an implementation of 3D Krawtchouk descriptors in GNU Octave, as well as statistics regarding effectiveness and runtime, are included, and the code used for testing will be published open source in the near future