Lattice structure on bounded homomorphisms between topological lattice rings

Suppose $X$ is a locally solid lattice ring. It is known that there are three classes of bounded group homomorphisms on $X$ whose topological structures make them again topological rings. In this note, we consider lattice structure on them; more precisely, we show that, under some mild assumptions, they are locally solid lattice rings.Comment: 8 pages. To appear in Vladikavkaz mathematical journal. arXiv admin note: text overlap with arXiv:1811.0429

On the Reality of the Wavefunction

Within the Ontological Models Framework (OMF), Pusey, Barrett, and Rudolph (PBR) have given an argument by which they claimed that the epistemic view on the wavefunction should be ruled out. This study highlights an incorrect conclusion in PBR's arguments, which was made due to inadequacies in the definitions of OMF. To be precise, OMF models the ontology of the preparation procedure, but it does not model the ontology of the measurement device. Such an asymmetric treatment becomes problematic, in scenarios in which measurement devices have a quantum nature. Modifying the OMF's definition such that the ontology of the measurement device becomes included, we will see how PBR's result disappears.Comment: 20 pages, 6 figure

Shear Waves, Sound Waves On A Shimmering Horizon

In the context of the so called ``membrane paradigm'' of black holes/branes, it has been known for sometime that the dynamics of small fluctuations on the stretched horizon can be viewed as corresponding to diffusion of a conserved charge in simple fluids. To study shear waves in this context properly, one must define a conserved stress tensor living on the stretched horizon. Then one is required to show that such a stress tensor satisfies the corresponding constitutive relations. These steps are missing in a previous treatment of the shear perturbations by Kovtun, Starinets and Son. In this note, we fill the gap by prescribing the stress tensor on the stretched horizon to be the Brown and York (or Balasubramanian-Kraus (BK) in the AdS/CFT context) holographic stress tensor. We are then able to show that such a conserved stress tensor satisfies the required constitutive relation on the stretched horizon using Einstein equations. We read off the shear viscosity from the constitutive relations in two different channels, shear and sound. We find an expression for the shear viscosity in both channels which are equal, as expected. Our expression is in agreement with a previous membrane paradigm formula reported by Kovtun, Starinets and Son.Comment: McGill Universit

Lattice of Integer Flows and Poset of Strongly Connected Orientations

We show that the Voronoi cells of the lattice of integer flows of a finite connected graph $G$ in the quadratic vector space of real valued flows have the following very precise combinatorics: the face poset of a Voronoi cell is isomorphic to the poset of strongly connected orientations of subgraphs of $G$. This confirms a recent conjecture of Caporaso and Viviani {Torelli Theorem For Graphs and Tropical Curves, Duke Math. J. 153(1) (2010), 129-171}. We also prove an analogue theorem for the lattice of integer cuts.Comment: 16 page

Disorder in Gauge/Gravity Duality, Pole Spectrum Statistics and Random Matrix Theory

In condensed-matter, level statistics has long been used to characterize the phases of a disordered system. We provide evidence within the context of a simple model that in a disordered large-N gauge theory with a gravity dual, there exist phases where the nearest neighbor spacing distribution of the unfolded pole spectra of generic two-point correlators is Poisson. This closely resembles the localized phase of the Anderson Hamiltonian. We perform two tests on our statistical hypothesis. One is based on a statistic defined in the context of Random Matrix Theory, the so-called $\bar{\Delta_3}$, or spectral rigidity, proposed by Dyson and Mehta. The second is a $\chi$-squared test. In our model, the results of both tests are consistent with the hypothesis that the pole spectra of two-point functions can be at least in two distinct phases; first a regular sequence and second a completely uncorrelated sequence with a Poisson nearest neighbor spacing distribution.Comment: 5 page

Chern-Simons terms in the 3D Weyl semi-metals

Based on some theoretical arguments, it has been suggested that electromagnetic response of 3D Weyl semi-metals with non-zero chiral- chemical potential may have a Chern-Simons term, $\frac{1}{2}k_\mu \epsilon^{\mu\nu\rho\sigma}F_{\nu\rho}A_\sigma$, in their effective action for the gauge field. An independent numerical study has shown that such a term is absent in a similar system. In this paper, we investigate the non-equilibrium and equilibrium response of 3D Weyl semi-metals. We argue that the controversy in literature stems from the difference in response of these two distinct states. We then develop a method to deal with well-known ambiguities in quantum electrodynamics in 3D (QED$_{3+1}$) with non-zero chiral-chemical potential and calculate the Chern-Simons term unambiguously. We find that time-like Chern-Simons term can exist in non-equilibrium conditions. We observe that there does not exist any chiral-magnetic effect in equilibrium and anomalous Hall effect replaces it.Comment: six pages, two figure

A few remarks on bounded homomorphisms acting on topological lattice groups and topological rings

Suppose $G$ is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on $G$ which have topological structures. In this paper, our attempt is to assign lattice structures on them. More precisely, we use of a version of the remarkable Riesz-Kantorovich formulae and Fatou property for bounded order bounded homomorphisms to allocate the desired structures. Moreover, we show that unbounded convergence on a locally solid lattice group is topological and we investigate some applications of it. Also, some necessary and sufficient conditions for completeness of different types of bounded group homomorphisms between topological rings have been obtained, as well.Comment: 9 pages. Some results have been added. The title has changed to be more effective. Submitte

Bessel Type Orthogonality For Hermite Polynomials

It is shown that Hermite polynomials satisfy a Bessel type orthogonality relation, based on the zeros of a single index Hermite polynomial and with a finite integration interval. Because of the role of non-symmetric zeros in the final relation, its applicability covers Hermite polynomials $P_n(x)$ with $n\ge 3$.Comment: Five pages including title pag

Reduced Divisors and Embeddings of Tropical Curves

Given a divisor $D$ on a tropical curve $\Gamma$, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system $|D|$. This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the linear system, and in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points on tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve $C$ and a divisor $D$ of non-negative rank on $C$, reduced divisors equivalent to $D$ define a morphism from $C$ to the complete linear system $|D|$, which is described in terms of Wronskians.Comment: Final version (to appear in Trans. Amer. Math. Soc.), 29 pages, 2 figure

Monte Carlo Simulation of The Adjoint Coulomb Gas

Monte Carlo simulation results for unitary matrix quantum mechanics, describing two-dimensional Yang-Mills theory coupled to a finite density of non-dynamical quarks (adjoint Coulomb gas), are presented. We characterize the deconfining transition in this model, by measuring the Polyakov Loop Susceptibility and employing finite-size scaling analysis. We provide evidence that the phase transition is first-order. Our results are consistent with the outcome of earlier large-$N$ studies of the model.Comment: 17 pages, 9 figures, 3 references adde