Abstract:
The theory of valued difference fields $(K, \sigma, v)$ depends on how the valuation $v$ interacts with the automorphism $\sigma$. Two special cases have already been worked out - the isometric case, where $v(\sigma(x)) = v(x)$ for all $x\in K$, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where $v(\sigma(x)) > n\cdot v(x)$ for all $n\in\mathbb{N}$ and $x\in K^\times$ with $v(x) > 0$, has been worked out by Salih Azgin. In this paper we deal with a more general version, called the multiplicative case, where $v(\sigma(x)) = \rho\cdot v(x)$, where $\rho (> 0)$ is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for such a theory.

Abstract:
The current paper presents a new digital watermarking method through bit replacement technology, which stores mul-tiple copies of the same data that is to be hidden in a scrambled form in the cover image. In this paper an indigenous approach is described for recovering the data from the damaged copies of the data under attack by applying a majority algorithm to find the closest twin of the embedded information. A new type of non-oblivious detection method is also proposed. The improvement in performance is supported through experimental results which show much enhancement in the visual and statistical invisibility of hidden data.

Abstract:
In this paper we consider partial metric spaces in the sense of O'Neill. We introduce the notions of strong partial metric spaces and Cauchy functions. We prove a fixed point theorem for such spaces and functions that improves Matthews' contraction mapping theorem in two ways. First, the existence of fixed points now holds for a wider class of functions and spaces. Second, our theorem also allows for fixed points with nonzero self-distances. We also prove fixed point theorems for orbitally $r$-contractive and orbitally $\phi_r$-contractive maps. We then apply our results to give alternative proofs of some of the other known fixed point theorems in the context of partial metric spaces.

Abstract:
In this paper we combine the notions of partial metric spaces with negative distances, $G_p$-metric spaces and n-metric spaces together into one structure called the partial n-metric spaces. These are generalizations of all the said structures, and also generalize the notions of $G$-metric and $G_p$-metric spaces to arbitrary finite dimension. We prove Cauchy mapping theorems and other fixed point theorems for such spaces.

Abstract:
Topological insulators are non-trivial quantum states of matter which exhibit a gap in the electronic structure of their bulk form, but a gapless metallic electronic spectrum at the surface. Here, we predict a uniaxial strain induced electronic topological transition (ETT) from a band to topological insulating state in the rhombohedral phase (space group: R$\bar{3}$m) of As$_2$Te$_3$ ($\beta$-As$_2$Te$_3$) through \textit{first-principles} calculations including spin-orbit coupling within density functional theory. The ETT in $\beta$-As$_2$Te$_3$ is shown to occur at the uniaxial strain $\epsilon_{zz}$ = -0.05 ($\sigma_{zz}$=1.77 GPa), passing through a Weyl metallic state with a single Dirac cone in its electronic structure at the $\Gamma$ point. We demonstrate the ETT through band inversion and reversal of parity of the top of the valence and bottom of the conduction bands leading to change in the $\mathbb{Z}_2$ topological invariant $\nu_0$ from 0 to 1 across the transition. Based on its electronic structure and phonon dispersion, we propose ultra-thin films of As$_2$Te$_3$ to be promising for use in ultra-thin stress sensors, charge pumps and thermoelectrics.

Abstract:
Kikyo and Shelah showed that if $T$ is a theory with the Strict Order Property in some first-order language $\mathcal{L}$, then in the expanded language $\mathcal{L}_\sigma := \mathcal{L}\cup\{\sigma\}$ with a new unary function symbol $\sigma$, the bigger theory $T_\sigma := T\cup\{``\sigma \mbox{is an} \mathcal{L}\mbox{-automorphism''}\}$ does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory $T_\sigma$ as the base theory $T$ together with a ``restricted'' class of automorphisms, then $T_\sigma$ can have a model companion in $\mathcal{L}_\sigma$. We show this in the context of linear orders and ordered abelian groups.

Abstract:
In this paper, we give appropriate languages in which the theory of tame fields (of any characteristic) admits (relative) quantifier elimination.

Abstract:
A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.

Abstract:
The nontrivial electronic topology of a topological insulator is thus far known to display signatures in a robust metallic state at the surface. Here, we establish vibrational anomalies in Raman spectra of the bulk that signify changes in electronic topology: an E2 g phonon softens unusually and its linewidth exhibits an asymmetric peak at the pressure induced electronic topological transition (ETT) in Sb2Se3 crystal. Our first-principles calculations confirm the electronic transition from band to topological insulating state with reversal of parity of electronic bands passing through a metallic state at the ETT, but do not capture the phonon anomalies which involve breakdown of adiabatic approximation due to strongly coupled dynamics of phonons and electrons. Treating this within a four-band model of topological insulators, we elucidate how nonadiabatic renormalization of phonons constitutes readily measurable bulk signatures of an ETT, which will facilitate efforts to develop topological insulators by modifying a band insulator.