The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.

Geometry and dynamics link form, function, and evolution of finch beaks

Darwin’s finches are a classic example of adaptive radiation, exemplified by their adaptive and functional beak morphologies. To quantify their form, we carry out a morphometric analysis of the three-dimensional beak shapes of all of Darwin’s finches and find that they can be fit by a transverse parabolic shape with a curvature that increases linearly from the base toward the tip of the beak. The morphological variation of beak orientation, aspect ratios, and curvatures allows us to quantify beak function in terms of the elementary theory of machines, consistent with the dietary variations across finches. Finally, to explain the origin of the evolutionary morphometry and the developmental morphogenesis of the finch beak, we propose an experimentally motivated growth law at the cellular level that simplifies to a variant of curvature-driven flow at the tissue level and captures the range of observed beak shapes in terms of a simple morphospace. Altogether, our study illuminates how a minimal combination of geometry and dynamics allows for functional form to develop and evolve.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$ , generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$ . In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$ . There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$ , \[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \] where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$ . Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$ . To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$ -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Sustainable coastal tourism-an overview

An elementary theory of sustainable coastal tourism mainly depends on the crystal clear water, healthy ecosystem, and well preserved coastal environment. Coastal tourism has drawn worldwide attention and has become extremely competitive as everybody tries to increase their profit in terms of beach visitors, both domestic and international [Joseph, and Pakkeerappa,2015]. Usually, any coastal area growth is always looked at from a business perspective, while the environmental aspect is mostly neglected. This strategy leads towards impractical beachfront development, which has different negative ramifications, on the climate and society. It is uneconomical as it destroys the actual ecological structures, such as the beach scenery, the biodiversity, and the biological system in the ocean and on the land. Also the travel industry exercise at the seaside territory destroys the beach. This study summarizes environmental impacts of coastal Tourism and the sustainable options to make it eco-friendly. The sustainable solutions were suggested for better coastal conservation. Also the study points out the future crisis pertaining to the latest Environmental Impact Assessment (EIA) 2020 by the government of India. Keywords—Coastal, Sustainable, tour, biodiversity, CRZ, EIA

Effects of Variable Punch Speed and Blank Holder Force in Warm Superplastic Deep Drawing Process

A cylindrical deep drawing test was conducted for the purpose of improving the drawability, product accuracy, and quality in warm deep drawing using a superplastic material with large strain rate dependence. Then, the effects of blank holding force (BHF) and punch speed (SPD) on the flange wrinkle behavior and wall thickness distribution were investigated by experiments and theoretical analysis. A Zn-22Al-0.5Cu-0.01Mg alloy superplastic material SPZ2 with a sheet thickness of 1 mm was employed as the experimental material, and a cylindrical deep drawing experiment with the drawing ratio (DR) of 3.1 and 5 was performed at 250 °C. A good agreement was qualitatively obtained between the elementary theory on the flange wrinkle limit, the fracture limit, and the experimental results. In addition, the authors examined each variable BHF and SPD method obtained from the theory and experimentally demonstrated that the variable BHF method has a great effect on uniform wall thickness distribution and that variable SPD has a great effect on shortening the processing time for superplastic materials. Furthermore, the authors demonstrated the effectiveness of the variable BHF/SPD deep drawing method that varies both BHF and SPD simultaneously.

Overview

This chapter presents the basic designs and working principles of STM and AFM, as well as an elementary theory of tunneling and the imaging mechanism of atomic resolution. Three elementary theories of tunneling are presented: the one-dimensional Schrödinger’s equation in vacuum, the semi-classical approximation, and the Landauer formalism. The relation between the decay constant and the work function, and a general expression of tunneling conductance versus tip-sample distance are derived. A brief summary of experimental facts on the mechanism of atomic resolution STM and AFM is presented, which leads to a picture of interplay between the atomic states of the tip and the sample, as well as the role of partial covalent bonds formed between those electronic states. Four illustrative applications are presented, including imaging self-assembed molecules on solid-liquid interfaces, electrochemical STM, catalysis research, and atom manipulation.

Ehrenfeucht-Fraïssé games without identity

This note defines Ehrenfeucht-Fraïssé games where identity is not present in the basic language. The formulation is applied to show that there is no elementary theory in the language of one binary relation that exactly characterizes models in which the relation is the identity relation.