1,343 research outputs found
Airy, Beltrami, Maxwell, Morera, Einstein and Lanczos potentials revisited
The main purpose of this paper is to revisit the well known potentials,
called stress functions, needed in order to study the parametrizations of the
stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional
elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) and G. Morera
(1892) for 3-dimensional elasticity, finally by A. Einstein (1915) for
4-dimensional elasticity, both with a variational procedure introduced by C.
Lanczos (1949,1962) in order to relate potentials to Lagrange multipliers.
Using the methods of Algebraic Analysis, namely mixing differential geometry
with homological algebra and combining the double duality test involved with
the Spencer cohomology, we shall be able to extend these results to an
arbitrary situation with an arbitrary dimension n. We shall also explain why
double duality is perfectly adapted to variational calculus with differential
constraints as a way to eliminate the corresponding Lagrange multipliers. For
example, the canonical parametrization of the stress equations is just
described by the formal adjoint of the n2(n2 -- 1)/12 components of the
linearized Riemann tensor considered as a linear second order differential
operator but the minimum number of potentials needed in elasticity theory is
equal to n(n -- 1)/2 for any minimal parametrization. Meanwhile, we can provide
all the above results without even using indices for writing down explicit
formulas in the way it is done in any textbook today. The example of
relativistic continuum mechanics with n = 4 is provided in order to prove that
it could be strictly impossible to obtain such results without using the above
methods. We also revisit the possibility (Maxwell equations of electromag-
netism) or the impossibility (Einstein equations of gravitation) to obtain
canonical or minimal parametrizations for various other equations of physics.
It is nevertheless important to notice that, when n and the algorithms
presented are known, most of the calculations can be achieved by using
computers for the corresponding symbolic computations. Finally, though the
paper is mathematically oriented as it aims providing new insights towards the
mathematical foundations of elasticity theory and mathematical physics, it is
written in a rather self-contained way
Spacetime deployments parametrized by gravitational and electromagnetic fields
On the basis of a "Punctual" Equivalence Principle of the general relativity
context, we consider spacetimes with measurements of conformally invariant
physical properties. Then, applying the Pfaff theory for PDE to a particular
conformally equivariant system of differential equations, we make explicit the
dependence of any kind of function describing a "spacetime deployment", on
n(n+1) parametrizing functions, denoting by n the spacetime dimension. These
functions, appearing in a linear differential Spencer sequence and determining
gauge fields of spacetime deformations relatively to a "substrat spacetime",
can be consistently ascribed to unified electromagnetic and gravitational
fields, at any spacetime dimensions n greater or equal to 4.Comment: 26 pages, LaTeX2e, file macro "suppl.sty", correction in the
definition of germs and local ring
A hierarchy of parametrizing varieties for representations
The primary purpose is to introduce and explore projective varieties,
, parametrizing the full collection of those
modules over a finite dimensional algebra which have dimension vector
. These varieties extend the smaller varieties previously studied by the
author; namely, the projective varieties encoding those modules with dimension
vector which, in addition, have a preassigned top or radical layering.
Each of the is again partitioned by the action
of a linear algebraic group, and covered by certain
representation-theoretically defined affine subvarieties which are stable under
the unipotent radical of the acting group. A special case of the pertinent
theorem served as a cornerstone in the work on generic representations by
Babson, Thomas, and the author. Moreover, applications are given to the study
of degenerations
"Not not bad" is not "bad": A distributional account of negation
With the increasing empirical success of distributional models of
compositional semantics, it is timely to consider the types of textual logic
that such models are capable of capturing. In this paper, we address
shortcomings in the ability of current models to capture logical operations
such as negation. As a solution we propose a tripartite formulation for a
continuous vector space representation of semantics and subsequently use this
representation to develop a formal compositional notion of negation within such
models.Comment: 9 pages, to appear in Proceedings of the 2013 Workshop on Continuous
Vector Space Models and their Compositionalit
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