Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.Comment: 6 pages, 4 figures, revtex

Professor C. N. Yang and Statistical Mechanics

Professor Chen Ning Yang has made seminal and influential contributions in many different areas in theoretical physics. This talk focuses on his contributions in statistical mechanics, a field in which Professor Yang has held a continual interest for over sixty years. His Master's thesis was on a theory of binary alloys with multi-site interactions, some 30 years before others studied the problem. Likewise, his other works opened the door and led to subsequent developments in many areas of modern day statistical mechanics and mathematical physics. He made seminal contributions in a wide array of topics, ranging from the fundamental theory of phase transitions, the Ising model, Heisenberg spin chains, lattice models, and the Yang-Baxter equation, to the emergence of Yangian in quantum groups. These topics and their ramifications will be discussed in this talk.Comment: Talk given at Symposium in honor of Professor C. N. Yang's 85th birthday, Nanyang Technological University, Singapore, November 200

Random walk generated by random permutations of {1,2,3, ..., n+1}

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.Comment: text shortened, new results added, appearing in J. Phys.

Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,d\nu_\lambda) and L^2_h(C^4_+,d\tilde\nu_\lambda) of square integrable holomorphic functions with scale dimension \lambda and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a \lambda-extension of Schwinger's Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L^2_h(D_4,d\nu_\lambda). SMT is related to MacMahon's Master Theorem (MMT) and an extension of both in terms of Louck's SU(N) solid harmonics is also provided for completeness. Convergence conditions are also studied.Comment: LaTeX, 40 pages, three new Sections and six new references added. To appear in ACH

Exact expressions for correlations in the ground state of the dense O(1) loop model

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries ''open''. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O$(1)$ loop model and the XXZ spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA

Initial results from a realtime FRB search with the GBT

We present the data analysis pipeline, commissioning observations and initial results from the GREENBURST fast radio burst (FRB) detection system on the Robert C. Byrd Green Bank Telescope (GBT) previously described by Surnis et al. which uses the 21~cm receiver observing commensally with other projects. The pipeline makes use of a state-of-the-art deep learning classifier to winnow down the very large number of false positive single-pulse candidates that mostly result from radio frequency interference. In our observations totalling 156.5 days so far, we have detected individual pulses from 20 known radio pulsars which provide an excellent verification of the system performance. We also demonstrate, through blind injection analyses, that our pipeline is complete down to a signal-to-noise threshold of 12. Depending on the observing mode, this translates to peak flux sensitivities in the range 0.14--0.89~Jy. Although no FRBs have been detected to date, we have used our results to update the analysis of Lawrence et al. to constrain the FRB all-sky rate to be $1140^{+200}_{-180}$ per day above a peak flux density of 1~Jy. We also constrain the source count index $\alpha=0.83\pm0.06$ which indicates that the source count distribution is substantially flatter than expected from a Euclidean distribution of standard candles (where $\alpha=1.5$). We discuss this result in the context of the FRB redshift and luminosity distributions. Finally, we make predictions for detection rates with GREENBURST, as well as other ongoing and planned FRB experiments.Comment: 9 pages, 7 figures, submitted to MNRA

The breakthrough listen search for intelligent life: 1.1–1.9 GHz observations of 692 nearby stars

We report on a search for engineered signals from a sample of 692 nearby stars using the Robert C. Byrd Green Bank Telescope, undertaken as part of the Breakthrough Listen Initiative search for extraterrestrial intelligence. Observations were made over 1.1−1.9 GHz (L band), with three sets of five-minute observations of the 692 primary targets, interspersed with five-minute observations of secondary targets. By comparing the “ON” and “OFF” observations we are able to identify terrestrial interference and place limits on the presence of engineered signals from putative extraterrestrial civilizations inhabiting the environs of the target stars. During the analysis, eleven events passed our thresholding algorithm, but a detailed analysis of their properties indicates they are consistent with known examples of anthropogenic radio frequency interference. We conclude that, at the time of our observations, none of the observed systems host high-duty-cycle radio transmitters emitting between 1.1 and 1.9 GHz with an Equivalent Isotropic Radiated Power of ∼ 1013 W, which is readily achievable by our own civilization. Our results suggest that fewer than ∼ 0.1% of the stellar systems within 50 pc possess the type of transmitters searched in this survey