Finiteness of simple holomorphic eta quotients of a given weight

We provide a simplified proof of Zagier's conjecture / Mersmann's theorem which states that of any particular weight, there are only finitely many holomorphic eta quotients, none of which is an integral rescaling of another eta quotient or a product of two holomorphic eta quotients other than 1 and itself.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1602.02835, arXiv:1602.03087, arXiv:1602.0281

Factorization of holomorphic eta quotients

We conjecture the occurrence of a certain type of factor of a holomorphic eta quotient whenever it is reducible and we prove this conjecture for all prime power levels. In particular, this also implies that rescaling and Atkin-Lehner involutions of irreducible holomorphic eta quotients of prime power levels are irreducible. We also show that there are finitely many simple holomorphic eta quotients of a given level and provide a bound on the weights of such eta quotients of a given level. Finally, we construct an infinite family of irreducible holomorphic eta quotients of prime power levels

Gravitational wave memory for a class of static and spherically symmetric spacetimes

This article aims at comparing gravitational wave memory effect in a Schwarzschild spacetime with that of other compact objects with static and spherically symmetric spacetime, with the purpose of proposing a procedure for differentiating between various compact object geometries. We do this by considering the relative evolution of two nearby test geodesics with in different backgrounds in the presence and absence of a gravitational wave pulse and comparing them. Memory effect due to a gravitational wave would ensure that there is a permanent effect on each spacetime and the corresponding geodesic evolution, being metric dependent, would display distinct results in each case. For a complete picture, we have considered both displacement and velocity memory effect in each geometry.Comment: 21 pages, 14 figure

Cosmic Acceleration and the notion of Alternative Vacuum in Nash Theory

We argue that Nash theory, a quadratic theory of Gravity, can describe a late-time cosmic acceleration without any exotic matter or cosmological constant. The observational viability of an exact cosmological solution of Nash theory is adjudged using a Markov chain Monte Carlo simulation and JLA+OHD+BAO data sets. Departures from standard {\Lambda}CDM cosmology are noted and analyzed. We prove that the Nash vacuum dynamics is equivalent to the dynamics of an Einstein vacuum plus a self-interacting Higgs scalar field, only if a mild evolution of the Higgs Vacuum Expectation Value is allowed. This leads to variations in the mass scales of fundamental particles and the fine structure constant. The variations are found to fit nicely with the analysis of molecular absorption spectra from a series of Quasars.Comment: 10 pages, 3 tables, 8 figures, comments and suggestions are welcom

A new formulation of Galilean electrodynamics

In this paper, we discuss Galilean electrodynamics in detail. We construct the two limits of Galilean electrodynamics from Maxwell's theory in the potential formalism for both contravariant and covariant vectors which are now distinct entities. Field equations are derived and their internal consistency is shown. The entire analysis is then performed in terms of electric and magnetic fields for both covariant and contravariant components. Duality transformations and their connection with boost symmetry are discussed which reveal a rich structure. Next we consider the gauge symmetry, construct the Noether currents and show their on-shell conservation. We also discuss the shift symmetry under which the Lagrangian is invariant, where the corresponding currents are off-shell conserved. At the end we analyse the Gailean electrodynamics by including sources for both contravariant and covariant sectors.Comment: 30 pages, 7 table

New structural insights of alkanethiol self-assembled monolayers on the Au(111): a molecular dynamics and density functional theory study

Self-assembled monolayers (SAMs) of alkanethiol molecules have been widely studied over the last three decades because of their diverse applications in the biomedical, nanotechnology, surface science, and electronics. It is also regarded as the model system to study the binding of organic molecules on the metal surfaces via thiol functional group. The robustness of the SAM structure combined with the ease of preparation makes it an ideal candidate for both fundamental and applied research. The structure of the alkanethiol SAMs on the Au(111) substrate is determined by the interplay between the alkyl chain packing and the interaction at the Au-S interface. Although our understanding of the SAM structure has significantly advanced over the last 35 years, an unambiguous atomic description of the Au-S interface and its influence on the chain packing remains elusive. In order to have better control of the SAM structure for different applications, we require a better understanding of the alkanethiol monolayer. In this work, we use a reductionist approach to determine the preferred head group positions driven by the chain packing, and by the interaction at the Au-S interface, separately. We use molecular dynamics (MD) to study the chain packing of the dense phase saturation coverage decanethiol SAM, and density functional theory (DFT) to study the interaction at the Au-S interface using an isolated methanethiol adsorbate. Alkane chains prefer a close-packed structure for the efficient interlocking of the methylene groups that minimizes the energy of the system. The molecular plane adjusts its orientation (molecular twist) depending on the spacing and the symmetry of the head groups to achieve a close packing of the chains. We first constrain the head groups at the high symmetry (√3 × √3)R30° sites to study the preferred combination of the molecular twists. We use this result as our baseline to study the effect of chain packing on the head group offset from the (√3 × √3)R30° sites. The position of the head groups also depends on the interaction at the Au-S interface. The preferred sites are determined by the tetrahedral coordination and the sp3 hybridization of the sulfur head groups. Relaxation and reconstruction (involve adatoms and/or surface vacancies) of the substrate also has significant influence on the preferred adsorption sites. We begin by determining the preferred positions of the head group on the unrelaxed substrate driven by the interaction at the Au-S interface alone. We then use the unrelaxed substrate as our reference to study the effect of substrate relaxation on the head group positions. To simulate a realistic model of the technologically interesting long-chain dense-phase alkanethiol SAM, we need to combine the effect of the chain packing and the interaction at the Au-S interface. Currently, we do not have a site-dependent force field for the Au-S interface to simulate the SAM structure using MD. On the other hand, the size of the problem is computationally too large for the DFT method. We demonstrated an approach to bridge the computational gap by using the atomic structure at the Au-S interface predicted by the DFT to study the effect on the chain packing. Using our DFT results, we predicted the symmetry of the adsorption site dependent dihedral force fields that can be used in MD to improve the prediction of the SAM structure