On Invariance (2010-)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory", 2011)

"One of the most important skills you will need to acquire in order to use manifold theory effectively is an ability to switch back and forth easily between invariant descriptions and their coordinate counterparts." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The fact that manifolds do not come with any predetermined choice of coordinates is both a blessing and a curse. The flexibility to choose coordinates more or less arbitrarily can be a big advantage in approaching problems in manifold theory, because the coordinates can often be chosen to simplify some aspect of the problem at hand. But we pay for this flexibility by being obliged to ensure that any objects we wish to define globally on a manifold are not dependent on a particular choice of coordinates. There are generally two ways of doing this: either by writing down a coordinate-dependent definition and then proving that the definition gives the same results in any coordinate chart, or by writing down a definition that is manifestly coordinate-independent (often called an invariant definition)." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The invariance of physical laws with respect to position or orientation (i.e., the symmetry of space) gives rise to conservation laws for linear and angular momentum. Sometimes the implications of symmetry invariance are far more complicated or sophisticated than might at first be supposed; the invariance of the forces predicted by electromagnetic theory when measurements are made in observation frames moving uniformly at different speeds (inertial frames) was an important clue leading Einstein to the discovery of special relativity. With the advent of quantum mechanics, considerations of angular momentum and spin introduced new symmetry concepts into physics. These ideas have since catalyzed the modern development of particle theory." (George B Arfken et al, "Mathematical Methods for Physicists: A comprehensive guide", 2013)

"In Euclidean vector spaces, there is the concept of orthonormal bases. But the non-orthonormal bases are as legitimate as those which are. On the other hand, in the geometry of Euclidean spaces, orthonormal bases have a legitimacy that arbitrary bases do not. Why? Because in geometry the role of groups becomes paramount. Euclidean geometry is the study of figures and properties that are invariant under the Euclidean group, the basic figures being the orthonormal bases at all points. Affine geometry is the study of figures and properties invariant under the affine group, the basic figures being all vector bases at all points." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"One sometimes finds in the literature the statement that a tensor transforms in such and such way under a coordinate transformation. Tensors do not transform. They are invariants. Their components do, under changes of section of the frame bundle." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"We say that a set of quantities transforms vectorially under a group of transformations of the bases of a vector space if, under an element of the group, they transform like the components of vectors, whether contravariant or covariant. We say that some quantity is a scalar if it is an invariant under the transformations of a group. A quantity (respectively, a set of quantities) may be scalar (respectively vectorial) under the transformation of a group, while not being so under the transformations of another group." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)

"Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. 'Structure' has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Symmetries directly point to invariants, which pinpoint intrinsic properties of the data and of the background empirical domain of interest. As our data models change, so too do our perspectives on analysing data." (Fionn Murtagh, "Data Science Foundations: Geometry and Topology of Complex Hierarchic Systems and Big Data Analytics", 2018)