Bijections and symmetries for the factorizations of the long cycle

We study the factorizations of the permutation $(1,2,...,n)$ into $k$ factors of given cycle types. Using representation theory, Jackson obtained for each $k$ an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases $k=2,3$ Schaeffer and Vassilieva gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of $(1,2,...,n)$ into $k$ factors for all $k$. We thereby obtain refinements of Jackson's formulas which extend the cases $k=2,3$ treated by Morales and Vassilieva. Our bijections are described in terms of "constellations", which are graphs embedded in surfaces encoding the transitive factorizations of permutations

Separation probabilities for products of permutations

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of the random permutation of $\{1,2,...,n\}$ obtained as product of two uniformly random $n$-cycles

The Stellar Mass Fundamental Plane: The virial relation and a very thin plane for slow-rotators

Early-type galaxies -- slow and fast rotating ellipticals (E-SRs and E-FRs) and S0s/lenticulars -- define a Fundamental Plane (FP) in the space of half-light radius $R_e$, enclosed surface brightness $I_e$ and velocity dispersion $\sigma_e$. Since $I_e$ and $\sigma_e$ are distance-independent measurements, the thickness of the FP is often expressed in terms of the accuracy with which $I_e$ and $\sigma_e$ can be used to estimate sizes $R_e$. We show that: 1) The thickness of the FP depends strongly on morphology. If the sample only includes E-SRs, then the observed scatter in $R_e$ is $\sim 16\%$, of which only $\sim 9\%$ is intrinsic. Removing galaxies with $M_*<10^{11}M_\odot$ further reduces the observed scatter to $\sim 13\%$ ($\sim 4\%$ intrinsic). The observed scatter increases to the $\sim 25\%$ usually quoted in the literature if E-FRs and S0s are added. If the FP is defined using the eigenvectors of the covariance matrix of the observables, then the E-SRs again define an exceptionally thin FP, with intrinsic scatter of only $5\%$ orthogonal to the plane. 2) The structure within the FP is most easily understood as arising from the fact that $I_e$ and $\sigma_e$ are nearly independent, whereas the $R_e-I_e$ and $R_e-\sigma_e$ correlations are nearly equal and opposite. 3) If the coefficients of the FP differ from those associated with the virial theorem the plane is said to be `tilted'. If we multiply $I_e$ by the global stellar mass-to-light ratio $M_*/L$ and we account for non-homology across the population by using S\'ersic photometry, then the resulting stellar mass FP is less tilted. Accounting self-consistently for $M_*/L$ gradients will change the tilt. The tilt we currently see suggests that the efficiency of turning baryons into stars increases and/or the dark matter fraction decreases as stellar surface brightness increases.Comment: 13 pages, 9 figures, 3 tables, accepted for publication in MNRA

Fully Frustrated Ising System on a 3D Simple Cubic Lattice: Revisited

Using extensive Monte Carlo simulations, we clarify the critical behaviour of the 3 dimensional simple cubic Ising Fully Frustrated system. We find two transition temperatures and two long range ordered phases. Within the present numerical accuracy, the transition at higher temperature is found to be second order and we have extracted the standard critical exponent using finite size scaling method. On the other hand, the transition at lower temperature is found to be first order. It is argued that entropy plays a major role on determining the low temperature state.Comment: 14 pages 14 figures iop style include

A radio-polarisation and rotation measure study of the Gum Nebula and its environment

The Gum Nebula is 36 degree wide shell-like emission nebula at a distance of only 450 pc. It has been hypothesised to be an old supernova remnant, fossil HII region, wind-blown bubble, or combination of multiple objects. Here we investigate the magneto-ionic properties of the nebula using data from recent surveys: radio-continuum data from the NRAO VLA and S-band Parkes All Sky Surveys, and H-alpha data from the Southern H-Alpha Sky Survey Atlas. We model the upper part of the nebula as a spherical shell of ionised gas expanding into the ambient medium. We perform a maximum-likelihood Markov chain Monte-Carlo fit to the NVSS rotation measure data, using the H-halpha data to constrain average electron density in the shell $n_e$. Assuming a latitudinal background gradient in RM we find $n_e=1.3^{+0.4}_{-0.4} {\rm cm}^{-3}$, angular radius $\phi_{\rm outer}=22.7^{+0.1}_{-0.1} {\rm deg}$, shell thickness $dr=18.5^{+1.5}_{-1.4} {\rm pc}$, ambient magnetic field strength $B_0=3.9^{+4.9}_{-2.2} \mu{\rm G}$ and warm gas filling factor $f=0.3^{+0.3}_{-0.1}$. We constrain the local, small-scale (~260 pc) pitch-angle of the ordered Galactic magnetic field to $+7^{\circ}\lesssim\wp\lesssim+44^{\circ}$, which represents a significant deviation from the median field orientation on kiloparsec scales (~-7.2$^{\circ}$). The moderate compression factor X=6.0\,^{+5.1}_{-2.5} at the edge of the H-alpha shell implies that the 'old supernova remnant' origin is unlikely. Our results support a model of the nebula as a HII region around a wind-blown bubble. Analysis of depolarisation in 2.3 GHz S-PASS data is consistent with this hypothesis and our best-fitting values agree well with previous studies of interstellar bubbles.Comment: 33 pages, 16 figures. Accepted by The Astrophysical Journa

Decomposition of homogeneous polynomials with low rank

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset \mathbb{P}^{{m+d\choose d}-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1, ..., M_r$ is $F=M_1^d+... + M_r^d$ with $r>s$. We show that if $s+r\leq 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.Comment: final version. Math. Z. (to appear

Crustal Velocity Structure in Italy from Analysis of Regional Seismic Waveforms

In this paper, we use regional seismic waveforms recorded by the recently-installed Istituto Nazionale di Geofisica e Vulcanologia (INGV) national network and the Mediterranean Very Broadband Seismographic Network (MedNet) stations to develop one-dimensional (1-D) crustal velocity models for the Italian peninsula. About 55,000 P -wave and 35,000 S -wave arrival times from 4,727 events are used to derive average seismic parameters in the crust and uppermost mantle. We define four regions, according to geological constraints and recent travel-time tomography results. Based on the average seismic parameters, we combine broadband seismic waveforms and travel-times of regional phases to model crustal structures for the four regions by applying the genetic algorithm. Our results indicate smooth velocity gradients with depth beneath the Apennines, and a deep Moho beneath the central Alps. Green’s functions from the regionalized 1-D velocity models are used to determine source depths and focal mechanisms for 37 events with magnitude larger than 3.5 by a grid search technique. Our results show that normal and strike-slip faulting source mechanisms dominate the Apenninic belt and most thrust faulting events occur in the Adriatic sea and the outer margin of the northern Apennines