Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

@article{Pommaret2012SpencerOA,
  title={Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"},
  author={J. F. Pommaret},
  journal={arXiv: Analysis of PDEs},
  year={2012}
}
  • J. Pommaret
  • Published 24 February 2011
  • Mathematics
  • arXiv: Analysis of PDEs
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… 
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  • J. Pommaret
  • Mathematics
    Multidimens. Syst. Signal Process.
  • 2015
TLDR
The purpose of this paper is to replace unmixed polynomial ideals by pure differential modules defined by linear multidimensional systems through the use of “involution” and a ‘relative localization” leading to a “relative parametrization”.
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References

SHOWING 1-10 OF 73 REFERENCES
Macaulay inverse systems revisited
Parametrization of Cosserat equations
The solution space of many systems of ordinary differential (OD) or partial differential (PD) equations in engineering or mathematical physics “can/cannot” be parameterized by a certain number of
"Sur une forme nouvelle des ´ equations de la M´ ecanique"
We present in modern language the contents of the famous note published by Henri Poincare in 1901 "Sur une forme nouvelle des ´ equations de la Mecanique", in which he proves that, when a Lie algebra
Théorie des Corps déformables
THE authors, who are well known by their writings on general elastic theory, here reprint in separate form an appendix contributed by them to M. Chwolson's “Traité de Physique.” The kinematical and
The Theory of Relativity
THE earlier portion of Dr. Carmichael's book is a reprint of the first edition, which received notice in NATURE for March 12, 1914. The later pages, which are grouped together under one large chapter
An Introduction to Homological Algebra
An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext
MAGNETIC MONOPOLES, FIBER BUNDLES, AND GAUGE FIELDS
The reports in this monograph have shown great enthusiasm and exuberance for the unification of various interactions through the concept of gauge fields. I would like to emphasize a point that has
La théorie de la relativité restreinte
(2) kinematics and optics of Special Relativity ; (3) relativistic electromagnetic theory, including a discussion of the invariance and conservation of electric charge ; (4) a long chapter, covering
Group interpretation of coupling phenomena
SummaryIt is well known that group theory interferes with constitutive (Hooke, Fourier, Minkowski) laws and coupling phenomena (piezoelectricity, thermoelasticity, photoelasticity, thermoelectricity)
SOME RESEARCHES ON GAUGE THEORIES OF GRAVITATION
It is suggested that the geometrical properties of space-time with some gauge symmetries can be determined by the physical properties of matter with the same gauge symmetries. With this principle
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