1,154 research outputs found
The Symmetries of Nature
The study of the symmetries of nature has fascinated scientists for eons. The application of the formal mathematical description of
symmetries during the last century has produced many breakthroughs in
our understanding of the substructure of matter. In this talk, a number
of these advances are discussed, and the important role that George
Sudarshan played in their development is emphasize
Detectability of Cosmic Topology in Flat Universes
Recent observations seem to indicate that we live in a universe whose spatial
sections are nearly or exactly flat. Motivated by this we study the problem of
observational detection of the topology of universes with flat spatial
sections. We first give a complete description of the diffeomorphic
classification of compact flat 3-manifolds, and derive the expressions for the
injectivity radii, and for the volume of each class of Euclidean 3-manifolds.
There emerges from our calculations the undetectability conditions for each
(topological) class of flat universes. To illustrate the detectability of flat
topologies we construct toy models by using an assumption by Bernshtein and
Shvartsman which permits to establish a relation between topological typical
lengths to the dynamics of flat models.Comment: 17 pages, 1 figure, latex2e. New references added. Inserted
clarifying points. To appear in Phys. Lett. A (2003) in the present for
Using geometric algebra to create differentiable models for optimizing camera-based optical metrology systems
In the design process of camera-based optical metrology systems numerous intricate and seemingly distinct optimization tasks emerge.A frequently occurring but crucial task in design or calibration is to optimize the spatial degrees of freedom of system components. Of course, modelling the poses of rigid bodies is long solved using rotation matrices and translation vectors, but when it comes to optimizing, this choice of model gets quite tedious to handle. Useful concepts such as homogeneous coordinates or (dual) quaternions have been introduced to overcome this, which however – lacking a unified framework – can quickly become difficult to maintain. As an alternative, in this contribution it is shown how the unifying methods of geometric algebra can be used as an advantage for gradient-based optimization of camera-based optical metrology and imaging systems – and how this can be done in a generalized way for seemingly different objectives with respect to system design and calibration
Riemann-Cartan Geometry of nonlinear dislocation mechanics
We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold – where the body is stress free – is a Weitzenbock manifold, i.e. a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance
Spencer Operator and Applications: From Continuum Mechanics to Mathematical physics
The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely
used in mathematics today and, up to our knowledge, has never been used in
engineering applications or mathematical physics. The main purpose of this
paper, an extended version of a lecture at the second workshop on Differential
Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is
to prove that the use of the Spencer operator constitutes the common secret of
the three following famous books published about at the same time in the
beginning of the last century, though they do not seem to have anything in
common at first sight as they are successively dealing with elasticity theory,
commutative algebra, electromagnetism and general relativity: (C) E. and F.
COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S.
MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University
Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922,
1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for
studying control identifiability and of (C)+(W) for the group theoretical
unification of finite elements in engineering sciences, recovering in a purely
mathematical way well known field-matter coupling phenomena (piezzoelectricity,
photoelasticity, streaming birefringence, viscosity, ...). As a byproduct and
though disturbing it could be, we shall prove that these unavoidable new
diferential and homological methods contradict the mathematical foundations of
both engineering (continuum mechanics,electromagnetism) and mathematical (gauge
theory, general relativity) physics.Comment: Though a few of the results presented are proved in the recent
references provided, the way they are combined with others and patched
together around the three books quoted is new. In view of the importance of
the full paper, the present version is only a summary of the definitive
version to appear later on. Finally, the reader must not forget that "each
formula" appearing in this new general framework has been used explicitly or
implicitly in (C), (M) and (W) for a mechanical, mathematical or physical
purpos
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