On the density of the odd values of the partition function, II: An infinite conjectural framework

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that $p(n)$ is odd exactly $50\%$ of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality. A striking consequence is that, under suitable existence conditions, if any $t$-multipartition function is odd with positive density and $t\not \equiv 0$ (mod 3), then $p(n)$ is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today. Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.Comment: 14 pages. To appear in the J. of Number Theor

Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions

We present an analytic study of the Potts model partition function on two different types of self-similar lattices of triangular shape with non integer Hausdorff dimension. Both types of lattices analyzed here are interesting examples of non-trivial thermodynamics in less than two dimensions. First, the Sierpinski gasket is considered. It is shown that, by introducing suitable geometric coefficients, it is possible to reduce the computation of the partition function to a dynamical system, whose variables are directly connected to (the arising of) frustration on macroscopic scales, and to determine the possible phases of the system. The same method is then used to analyse the Hanoi graph. Again, dynamical system theory provides a very elegant way to determine the phase diagram of the system. Then, exploiting the analysis of the basins of attractions of the corresponding dynamical systems, we construct various examples of self-similar lattices with more than one critical temperature. These multiple critical temperatures correspond to crossing phases with different degrees of frustration.Comment: 16 pages, 12 figures, 1 table; title changed, references and discussion on multiple transitions adde

On the density of the odd values of the partition function

The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo 2. Our main result will relate the densities, say $\delta_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $\delta_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $\delta_1>0$; that is, $p(n)$ itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that $p(n)$ is odd for $\sqrt{x}$ values of $n\le x$. In general, we conjecture that $\delta_t=1/2$ for all $t$ odd, i.e., that similarly to the case of $p(n)$, all multipartition functions are in fact equidistributed modulo 2. Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu.Comment: Several changes with respect to the 2015 version. 18 pages. To appear in the Annals of Combinatoric

A Schwarz Method for the Magnetotelluric Approximation of Maxwell's equations

The magnetotelluric approximation of the Maxwell's equations is used to model the propagation of low frequency electro-magnetic waves in the Earth's subsurface, with the purpose of reconstructing the presence of mineral or oil deposits. We propose a classical Schwarz method for solving this magnetotelluric approximation of the Maxwell equations, and prove its convergence using maximum principle techniques. This is not trivial, since solutions are complex valued, and we need a new result that the magnetotelluric approximations satisfy a maximum modulus principle for our proof. We illustrate our analysis with numerical experiments.Comment: 9 pages, 3 figure

Possible phases of two coupled n-component fermionic chains

A two-leg ladder with $n$-component fermionic fields in the chains has been considered using an analytic renormalization group method. The fixed points and possible phases have been determined for generic filling as well as for a half-filled system and for the case when one of the subbands is half filled. A weak-coupling Luttinger-liquid phase and several strong-coupling gapped phases have been found. In the Luttinger liquid phase, for the most general spin dependence of the couplings, all $2n$ modes have different velocities if the interband scattering processes are scaled out, while $n$ doubly degenerate modes appear if the interband scattering processes remain finite. The role of backward-scattering, charge-transfer and umklapp processes has been analysed using their bosonic form and the possible phases are characterized by the number of gapless modes. As a special case the SU($n$) symmetric Hubbard ladder has been investigated numerically. It was found that this model does not scale to the Luttinger liquid fixed point. Even for generic filling gaps open up in the spectrum of the spin or charge modes, and the system is always insulator in the presence of umklapp processes

A measure of tripartite entanglement in bosonic and fermionic systems

We describe an efficient theoretical criterion suitable for the evaluation of the tripartite entanglement of any mixed three-boson or -fermion state, based on the notion of the entanglement of particles for bipartite systems of identical particles. Our approach allows one to quantify the accessible amount of quantum correlations in the systems without any violation of the local particle number superselection rule. A generalization of the tripartite negativity is here applied to some correlated systems including the continuous-time quantum walks of identical particles (both for bosons and fermions) and compared with other criteria recently proposed in the literature. Our results show the dependence of the entanglement dynamics upon the quantum statistics: the bosonic bunching results into a low amount of quantum correlations while Fermi-Dirac statistics allows for higher values of the entanglement.Comment: 19 pages, 3 figure

Learned Belief-Propagation Decoding with Simple Scaling and SNR Adaptation

We consider the weighted belief-propagation (WBP) decoder recently proposed by Nachmani et al. where different weights are introduced for each Tanner graph edge and optimized using machine learning techniques. Our focus is on simple-scaling models that use the same weights across certain edges to reduce the storage and computational burden. The main contribution is to show that simple scaling with few parameters often achieves the same gain as the full parameterization. Moreover, several training improvements for WBP are proposed. For example, it is shown that minimizing average binary cross-entropy is suboptimal in general in terms of bit error rate (BER) and a new "soft-BER" loss is proposed which can lead to better performance. We also investigate parameter adapter networks (PANs) that learn the relation between the signal-to-noise ratio and the WBP parameters. As an example, for the (32,16) Reed-Muller code with a highly redundant parity-check matrix, training a PAN with soft-BER loss gives near-maximum-likelihood performance assuming simple scaling with only three parameters.Comment: 5 pages, 5 figures, submitted to ISIT 201

Quieting the Sharholders\u27 Voice: Empirical Evidence of Pervasive Bundling in Proxy Solicitations

The integrity of shareholder voting is critical to the legitimacy of corporate law. One threat to this process is proxy “bundling,” or the joinder of more than one separate item into a single proxy proposal. Bundling deprives shareholders of the right to convey their views on each separate matter being put to a vote and forces them to either reject the entire proposal or approve items they might not otherwise want implemented. In this Paper, we provide the first comprehensive evaluation of the anti-bundling rules adopted by the Securities and Exchange Commission (“SEC”) in 1992. While we find that the courts have carefully developed a framework for the proper scope and application of the rules, the SEC and proxy advisory firms have been less vigilant in defending this instrumental shareholder right. In particular, we note that the most recent SEC interpretive guidance has undercut the effectiveness of the existing rules, and that, surprisingly, proxy advisory firms do not have well-defined heuristics to discourage bundling. Building on the theoretical framework, this Article provides the first large-scale empirical study of bundling of management proposals. We develop four possible definitions of impermissible bundling and, utilizing a data set of over 1,300 management proposals, show that the frequency of bundling in our sample ranges from 6.2 percent to 28.8 percent (depending on which of the four bundling definitions is used). It is apparent that bundling occurs far more frequently than indicated by prior studies. We further examine our data to report the items that are most frequently bundled and to analyze the proxy advisors’ recommendations and the voting patterns associated with bundled proposals. This Article concludes with important implications for the SEC, proxy advisors, and institutional investors as to how each party can more effectively deter impermissible bundling and thus better protect the shareholder franchise

Spine-sheath layer radiative interplay in subparsec-scale jets and the TeV emission from M87

Simple one-zone homogeneous synchrotron self-Compton models have severe difficulties in explaining the TeV emission observed in the radiogalaxy M87. Also the site of the TeV emission region is uncertain: it could be the unresolved jet close to the nucleus, analogously to what proposed for blazars, or an active knot, called HST-1, tens of parsec away. We explore the possibility that the TeV emission of M87 is produced in the misaligned subpc scale jet. We base our modelling on a structured jet, with a fast spine surrounded by a slower layer. In this context the main site responsible for the emission of the TeV radiation is the layer, while the (debeamed) spine accounts for the emission from the radio to the GeV band: therefore we expect a more complex correlation with the TeV component than that expected in one-zone scenarios, in which both components are produced by the same region. Observed from small angles, the spine would dominate the emission, with an overall Spectral Energy Distribution close to those of BL Lac objects with a synchrotron peak located at low energy (LBLs).Comment: 5 pages, 2 figures. Accepted for publication in MNRAS Letter

Homogeneous and inhomogeneous contributions to the luminescence linewidth of point defects in amorphous solids: Quantitative assessment based on time-resolved emission spectroscopy

The article describes an experimental method that allows to estimate the inhomogeneous and homogeneous linewidths of the photoluminescence band of a point defect in an amorphous solid. We performed low temperature time-resolved luminescence measurements on two defects chosen as model systems for our analysis: extrinsic Oxygen Deficient Centers (ODC(II)) in amorphous silica and F+ 3 centers in crystalline Lithium Fluoride. Measurements evidence that only defects embedded in the amorphous matrix feature a dependence of the radiative decay lifetime on the emission energy and a time dependence of the first moment of the emission band. A theoretical model is developed to link these properties to the structural disorder typical of amorphous solids. Specifically, the observations on ODC(II) are interpreted by introducing a gaussian statistical distribution of the zero phonon line energy position. Comparison with the results obtained on F+ 3 crystalline defects strongly confirms the validity of the model. By analyzing experimental data within this frame, we obtain separate estimations of the homogenous and inhomogeneous contributions to the measured total linewidth of ODC(II), which results to be mostly inhomogeneous.Comment: 8 pages, 4 figure