Four Dimensional Orbit Spaces of Compact Coregular Linear Groups

All four dimensional orbit spaces of compact coregular linear groups have been determined. The results are obtained through the integration of a universal differential equation, that only requires as input the number of elements of an integrity basis of the ideal of polynomial invariants of the linear group. Our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking at the classical level.Comment: 23 pages, plain LaTeX, a review that has not distributed regularl

Flat Bases of Invariant Polynomials and P-matrices of E7 and E8

Let $G$ be a compact group of linear transformations of an Euclidean space $V$. The $G$-invariant $C^\infty$ functions can be expressed as $C^\infty$ functions of a finite basic set of $G$-invariant homogeneous polynomials, called an integrity basis. The mathematical description of the orbit space $V/G$ depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semi-definiteness conditions of the $P$-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of $G$-invariant homogeneous polynomials forming an integrity basis is not unique, so it is not unique the mathematical description of the orbit space too. If $G$ is an irreducible finite reflection group, Saito et al. in 1980 characterized some special basic sets of $G$-invariant homogeneous polynomials that they called {\em flat}. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups $E_7$ and $E_8$. In this paper the flat basic sets of invariant homogeneous polynomials of $E_7$ and $E_8$ and the corresponding $P$-matrices are determined explicitly. Using the results here reported one is able to determine easily the $P$-matrices corresponding to any other integrity basis of $E_7$ or $E_8$. From the $P$-matrices one may then write down the equations and inequalities defining the orbit spaces of $E_7$ and $E_8$ relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups $E_7$ and $E_8$ or one of the Lie groups $E_7$ and $E_8$ in their adjoint representations.Comment: 14 page

Progressive damage and rupture in polymers

Progressive damage, which eventually leads to failure, is ubiquitous in biological and synthetic polymers. The simplest case to consider is that of elastomeric materials, which can undergo large reversible deformations with negligible rate dependence. In this paper, we develop a theory for modeling progressive damage and rupture of such materials. We extend the phase-field method, which is widely used to describe the damage and fracture of brittle materials, to elastomeric materials undergoing large deformations. A central feature of our theory is the recognition that the free energy of elastomers is not entirely entropic in nature---there is also an energetic contribution from the deformation of the bonds in the chains. It is the energetic part in the free energy which is the driving force for progressive damage and fracture

The macro-environment for liquid biofuels in the US mass media, science and government

The purpose of this study is to investigate under which dimensions the macro-environment for liquid biofuels has been structured during time, respectively by science, mass media, and government in Germany, and how these three social expressions related to each other. Research was carried out on German official government documents, mass media news, and scientific papers on the topic ‘liquid biofuels’. Text Mining was used to extract knowledge from their content. The results indicate that in configurating the macro-environment for liquid biofuels there is some degree of proximity between media and government, less between media and science, and the least between government and scienc

A SMARC Effect for Loudness

Various reports suggest that the pitch height of musical tones may be represented along a mental space, with lower pitch heights represented on the left or lower sectors and higher pitch heights represented on the right or upper sectors of the mental space. Given that in Western languages the loudness of tones is often addressed spatially, with loud sounds referred to as \u201chigh\u201d and quiet sounds referred to as \u201clow,\u201d here we investigated whether loudness might also have a spatial representation. Participants judged whether a tone was louder or quieter than a reference tone, by pressing two keys: one at the top and the other at the bottom of a response box. Participants were faster in a situation where they pressed the key at the top to report louder sounds, and the key at the bottom to report quieter sounds, than vice versa. This result supports the view that loudness, like other types of magnitudes, might be represented spatially

The macro-environment for liquid biofuels in the German science, mass, media and government

The purpose of this study is to investigate under which dimensions the macro-environment for liquid biofuels has been structured during time, respectively by science, mass media, and government in Germany, and how these three social expressions related to each other. Research was carried out on German official government documents, mass media news, and scientific papers on the topic 'liquid biofuels'. Text Mining was used to extract knowledge from their content. The results indicate that in configurating the macro-environment for liquid biofuels there is some degree of proximity between media and government, less between media and science, and the least between government and science