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Sobhy Elsayed Ibrahim - One of the best experts on this subject based on the ideXlab platform.
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The Spectra of General Differential Operators in the Direct Sum Spaces
Czechoslovak Mathematical Journal, 2004Co-Authors: Sobhy Elsayed IbrahimAbstract:In this paper, the general ordinary quasi-differential expression M _pof n -th order with complex coefficients and its Formal Adjoint M _p ^+ on any finite number of intervals I _ p =( a _ p , b _ p ), p = 1,..., N , are considered in the setting of the direct sums of L _ wp ^2 ( a _ p , b _ p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.
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On the spectra of non-selfAdjoint differential operators and their Adjoints in direct sum spaces
International Journal of Mathematics and Mathematical Sciences, 2003Co-Authors: Sobhy Elsayed IbrahimAbstract:The general ordinary quasidifferential expression Mp of nth order, with complex coefficients and its Formal Adjoint M + p on any finite number of intervals Ip = (ap ,b p), p = 1 ,...,N , are considered in the setting of the direct sums of L 2p (ap ,b p)-spaces of functions defined on each of the separate intervals. And a number of results concerning the location of the point spectra and regularity fields of general differential operators generated by such expressions are obtained.
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on the boundary conditions characterizing a general differential operators with a countable number of singular points
Italian journal of pure and applied mathematics, 2001Co-Authors: Sobhy Elsayed IbrahimAbstract:The general ordinary quasi-differential expression M of nth order with complex coefficients and its Formal Adjoint M + are considered over a region (a, b) on the real line R, -∞ < a < b < oo, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur ony at the ends, restrictions of the maximal operator generated by M in L 2 w (a, b) which are regularly solvable with respect to the minimal operators T 0 (M) and T 0 (M + ). In addition to direct sums of regularly solvable operators defined on the separate subintervals, there are other regularly solvable restrictions of the maximal operator which involve linking the various intervals together in interface like style.
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The spectra of well-posed operators
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 1995Co-Authors: Sobhy Elsayed IbrahimAbstract:In this paper, the general ordinary quasidifferential expression M of n th order, with complex coefficients, and its Formal Adjoint M − are considered. It is shown in the case of two singular endpomts and when all solutions of the equation and the Adjoint equation are in (the limit-circle case) that all well-posed extensions of the minimal operator T 0 ( M ) have resolvents which are Hilbert Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all of the standard essential spectra to be empty. These results extend those for the Formally symmetric expression M studied in [ 1 ] and [ 14 ], and also extend those proved in [ 8 ] for one singular endpoint.
Jean-françois Pommaret - One of the best experts on this subject based on the ideXlab platform.
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A mathematical comment on gravitational waves
arXiv: Mathematical Physics, 2017Co-Authors: Jean-françois PommaretAbstract:In classical General Relativity, the way to exhibit the equations for the gravitational waves is based on two "tricks" allowing to transform the Einstein equations after linearizing them over the Minkowski metric. With specific notations used in the study of {\it Lie pseudogroups} of transformations of an $n$-dimensional manifold, let $\Omega=({\Omega}\_{ij}={\Omega}\_{ji})$ be a perturbation of the non-degenerate metric $\omega=({\omega}\_{ij}={\omega}\_{ji})$ with $det(\omega)\neq 0$ and call ${\omega}^{-1}=({\omega}^{ij}={\omega}^{ji})$ the inverse matrix appearing in the Dalembertian operator $\Box = {\omega}^{ij}d\_{ij}$. The first idea is to introduce the linear transformation ${\bar{\Omega}}\_{ij}={\Omega}\_{ij}-\frac{1}{2}{\omega}\_{ij}tr(\Omega)$ where $tr(\Omega)={\omega}^{ij}{\Omega}\_{ij}$ is the {\it trace} of $\Omega$, which is invertible when $n\geq 3$. The second important idea is to notice that the composite second order linearized Einstein operator $\bar{\Omega} \rightarrow \Omega \rightarrow E=(E\_{ij}=R\_{ij} - \frac{1}{2}{\omega}\_{ij}tr(R))$ where $\Omega \rightarrow R=(R\_{ij}=R\_{ji})$ is the linearized Ricci operator with trace $tr(R)={\omega}^{ij}R\_{ij}$ is reduced to $\Box {\bar{\Omega}}\_{ij}$ when ${\omega}^{rs}d\_{ri}{\bar{\Omega}}\_{sj}=0$. The purpose of this short but striking paper is to revisit these two results in the light of the {\it differential duality} existing in Algebraic Analysis, namely a mixture of differential geometry and homological agebra, providing therefore a totally different interpretation. In particular, we prove that the above operator $\bar{\Omega} \rightarrow E$ is nothing else than the Formal Adjoint of the Ricci operator $\Omega \rightarrow R$ and that the map $\Omega \rightarrow \bar{\Omega}$ is just the Formal Adjoint (transposed) of the defining tensor map $R \rightarrow E$. Accordingly, the Cauchy operator (stress equations) can be directly parametrized by the Formal Adjoint of the Ricci operator and the Einstein operator is no longer needed.
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AIRY, BELTRAMI, MAXWELL, MORERA, EINSTEIN AND LANCZOS POTENTIALS REVISITED
2015Co-Authors: Jean-françois PommaretAbstract: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.
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CLAUSIUS/COSSERAT/MAXWELL/WEYL EQUATIONS: THE VIRIAL THEOREM REVISITED
2015Co-Authors: Jean-françois PommaretAbstract:In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincaré introduced a duality principle in analytical mechanics in order to study lagrangians invariant under the action of a Lie group of transformations. In 1909, the brothers E. and F. Cosserat discovered another approach for studying the same problem though using quite different equations. In 1916, H. Weyl considered again the same problem for the conFormal group of transformations, obtaining at the same time the Maxwell equations and an additional specific equation also involving the trace of the impulsion-energy tensor. Finally, having in mind the space-time formulation of electromagnetism and the Maurer-Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills in 1954 as a way to introduce in physics the differential geometric methods available at that time, independently of any group action, contrary to all the previous approaches. The main purpose of this paper is to revisit the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the Formal theory of systems of partial differential equations and Lie pseudogroups, mostly developped by D.C Spencer and coworkers around 1970. In particular, we justify and extend the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are nothing else but the Formal Adjoint of the Spencer operator appearing in the canonical Spencer sequence for the conFormal group of space-time and are thus totally dependent on the group action. The duality principle also appeals to the Formal Adjoint of a linear differential operator used in differential geometry and to the extension modules used in homological algebra.
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CLAUSIUS/COSSERAT/MAXWELL/WEYL EQUATIONS: THE VIRIAL THEOREM REVISITED
arXiv: Differential Geometry, 2015Co-Authors: Jean-françois PommaretAbstract:In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincare introduced a duality principle in analytical mechanics in order to study lagrangians invariant under the action of a Lie group of transformations. In 1909, the brothers E. and F. Cosserat discovered another approach for studying the same problem though using quite different equations. In 1916, H. Weyl considered again the same problem for the conFormal group of transformations, obtaining at the same time the Maxwell equations and an additional specific equation also involving the trace of the impulsion-energy tensor. Finally, having in mind the space-time formulation of electromagnetism and the Maurer-Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills in 1954 as a way to introduce in physics the differential geometric methods available at that time, independently of any group action, contrary to all the previous approaches. The main purpose of this paper is to revisit the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the Formal theory of systems of partial differential equations and Lie pseudogroups, mostly developped by D.C Spencer and coworkers around 1970. In particular, we justify and extend the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are nothing else but the Formal Adjoint of the Spencer operator appearing in the canonical Spencer sequence for the conFormal group of space-time and are thus totally dependent on the group action. The duality principle also appeals to the Formal Adjoint of a linear differential operator used in differential geometry and to the extension modules used in homological algebra.
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THE MATHEMATICAL FOUNDATIONS OF GENERAL RELATIVITY REVISITED
2013Co-Authors: Jean-françois PommaretAbstract:The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the Formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide. The paper is therefore divided into three parts corresponding to the different Formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms appearing in the " Riemann tensor " according to the " Vessiot structure equations " must not be identified with the quadratic terms appearing in the well known " Cartan structure equations " for Lie groups and a similar comment can be done for the " Weyl tensor ". In particular, " curvature+torsion" (Cartan) must not be considered as a generalization of "curvature alone" (Vessiot). Roughly, Cartan and followers have not been able to " quotient down to the base manifold ", a result only obtained by Spencer in 1970 through the "nonlinear Spencer sequence" but in a way quite different from the one followed by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS SPENCER: The " Ricci tensor " only depends on the nonlinear transformations (called " elations " by Cartan in 1922) that describe the "difference " existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conFormal group of space-time (15 parameters). It can be defined by a canonical splitting, that is to say without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly, the Spencer sequence for the conFormal Killing system and its Formal Adjoint fully describe the Cosserat/Maxwell/Weyl theory but General Relativity is not coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be " parametrized ", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of mathematical physics must be revisited within this Formal framework, though striking it may look like for certain apparently well established theories such as electromagnetism and general relativity. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.
Endre Süli - One of the best experts on this subject based on the ideXlab platform.
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A posteriori error analysis for stabilised finite element approximations of transport problems
Computer Methods in Applied Mechanics and Engineering, 2000Co-Authors: Paul Houston, Rolf Rannacher, Endre SüliAbstract:We develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments. Two alternative dual problems are considered: one is based on the Formal Adjoint of the hyperbolic differential operator, the other on the transposition of the bilinear form for the stabilised finite element method. We show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes.
Pommaret J. -f. - One of the best experts on this subject based on the ideXlab platform.
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Differential Homological Algebra and General Relativity
'Scientific Research Publishing Inc.', 2019Co-Authors: Pommaret J. -f.Abstract:In 1916, F.S. Macaulay developed specific localization techniques for dealing with "unmixed polynomial ideals" in commutative algebra, transforming them into what he called "inverse systems" of partial differential equations. In 1970, D.C. Spencer and coworkers studied the Formal theory of such systems, using methods of homological algebra that were giving rise to "differential homological algebra", replacing unmixed polynomial ideals by "pure differential modules". The use of "extension modules" and "differential double duality" is essential for such a purpose. In particular, 0-pure differential modules are torsion-free and admit an "absolute parametrization" by means of arbitrary potential like functions. In 2012, we have been able to extend this result to arbitrary pure modules, introducing a "relative parametrization" where the potentials should satisfy compatible "differential constraints". We recently discovered that General Relativity is just a way to parametrize the Cauchy stress equations by means of the Formal Adjoint of the Ricci operator in order to obtain a "minimum parametrization" by adding sufficiently many compatible differential constraints, exactly like the Lorenz condition in electromagnetism. These unusual purely mathematical results are illustrated by many explicit examples and even strengthen the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory.Comment: This paper strengthens the doubts we had recently about the existence and origin of gravitational waves. arXiv admin note: substantial text overlap with arXiv:1212.459
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From Elasticity to Electromagnetism: Beyond the Mirror
2018Co-Authors: Pommaret J. -f.Abstract:The first purpose of this short but striking paper is to revisit Elasticity (EL) and Electromagnetism (EM) by comparing the structure of these two theories and examining with details their well known couplings, in particular piezoelectricity and photoelasticity. Despite the strange Helmholtz and Mach-Lippmann analogies existing between them, no classical technique may provide a common setting. However, unexpected arguments discovered independently by the brothers E. and F. Cosserat in 1909 for EL and by H. Weyl in 1918 for EM are leading to construct a new differential sequence called Spencer sequence in the framework of the Formal theory of Lie pseudogroups and to introduce it for the conFormal group of space-time with 15 parameters. Then, all the previous explicit couplings can be deduced abstractly and one must just go to a laboratory in order to know about the coupling constants on which they are depending, like in the Hooke or Minkowski constitutive relations existing respectively in EL or EM separately. We finally provide a new combined experimental and theoretical proof of the fact that any 1-form with value in the second order jets (elations) of the conFormal group of space-time can be uniquely decomposed into the direct sum of the Ricci tensor R and the electromagnetic field F. This result questions the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT). In particular, the Einstein operator (6 terms) must be thus replaced by the Formal Adjoint of the Ricci operator (4 terms only) in the study of gravitational waves.Comment: Summary Note of lectures given at the Albert Einstein Institute (AEI Berlin/Potsdam, october 23-27, 2017), 21 pages including 1 figur
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Computer Algebra and Lanczos Potential
2018Co-Authors: Pommaret J. -f.Abstract:We found in 2016 a few results on the mathematical structure of the conFormal Killing differential sequence in arbitrary dimension $n$, in particular the rank and order changes of the successive differential operators for $n=3,n=4$ or $n\geq 5$. They were so striking that we did not dare to publish them before our former PhD student A. Quadrat (INRIA) could confirm them while using new computer algebra packages that he developped for studying extension modules in differential homological algebra. In the meantime, as a complementary result, we found in 2017 the "missing link" justifying the doubts we had since a long time on the origin and existence of Gravitational Waves in General Relativity. In both cases, the main tool is the explicit computation of certain extension modules for the classical or conFormal Killing differential sequences. These results therefore lead to revisit the work of C. Lanczos and successors on the existence of a parametrization of the Riemann or Weyl operators and their respective Formal Adjoint operators. We also provide an example showing how these extension modules are depending on the structure constants appearing in the Vessiot structure equations (1903), still not acknowledged after one century even though they generalize the constant Riemannian curvature integrability condition of L.P. Eisenhart (1926) for the Killing equations. The present paper is written from a lecture gven at the recent 24 th conference on Applications of Computer Algebra (ACA 2018) held in Santiago de Compostela, Spain, june 18-22, 2018.Comment: The paper is written from a lecture given at the recent 24th conference on Applications of Computer Algebra (ACA 2018) held in Santiago de Compostela, Spain, june 18-22, 2018 (See also ACA 2009 for other applications
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Homological Solution of the Riemann-Lanczos and Weyl-Lanczos Problems in Arbitrary Dimension
'Scientific Research Publishing Inc.', 2018Co-Authors: Pommaret J. -f.Abstract:When ${\cal{D}}$ is a linear partial differential operator of any order, a direct problem is to look for an operator ${\cal{D}}_1$ generating the compatibility conditions (CC) ${\cal{D}}_1\eta=0$ of ${\cal{D}}\xi=\eta$. We may thus construct a differential sequence with successive operators ${\cal{D}},{\cal{D}}_1,{\cal{D}}_2, ...$, where each operator is generating the CC of the previous one. Introducing the Formal Adjoint $ad( )$, we have ${\cal{D}}_i\circ {\cal{D}}_{i-1}=0 \Rightarrow ad({\cal{D}}_{i-1}) \circ ad({\cal{D}}_i)=0$ but $ad({\cal{D}}_{i-1})$ may not generate all the CC of $ad({\cal{D}}_i)$. When $D=K[d_1,...,d_n]=K[d]$ is the (non-commutative) ring of differential operators with coefficients in a differential field $K$, it gives rise by residue to a differential module $M$ over $D$. The homological extension modules $ext^i(M)=ext^i_D(M,D)$ with $ext^0(M)=hom_D(M,D)$ only depend on $M$ and are measuring the above gaps, independently of the previous differential sequence.The purpose of this rather technical paper is to compute them for certain Lie operators involved in the Formal theory of Lie pseudogroups in arbitrary dimension $n$. In particular, we prove that the extension modules highly depend on the Vessiot structure constants $c$. When one is dealing with a Lie group of transformations or, equivalently, when ${\cal{D}}$ is a Lie operator of finite type, then we shall prove that $ext^i(M)=0, \forall 0\leq i \leq n-1$. It will follow that the Riemann-Lanczos and Weyl-Lanczos problems just amount to prove such a result for $i=2$ and arbitrary $n$ when ${\cal{D}}$ is the Killing or conFormal Killing operator. We finally prove that ${ext}^i(M)=0, \forall i\geq 1$ for the Lie operator of infinitesimal contact transformations with arbitrary $n=2p+1$. Most of these new results have been checked by means of computer algebra.Comment: This paper is largely improving the former arXiv:1512.05982 now published in Journal of Modern Physics, 7 (2016) 699-72
Paul Houston - One of the best experts on this subject based on the ideXlab platform.
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A posteriori error analysis for stabilised finite element approximations of transport problems
Computer Methods in Applied Mechanics and Engineering, 2000Co-Authors: Paul Houston, Rolf Rannacher, Endre SüliAbstract:We develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments. Two alternative dual problems are considered: one is based on the Formal Adjoint of the hyperbolic differential operator, the other on the transposition of the bilinear form for the stabilised finite element method. We show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes.