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The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870), the Maxwell/Weyl equations (1873,1918) are among the most famous partial differential equations that can be found today in any textbook dealing {\it separately and/or successively} with elasticity theory, continuum mechanics, thermodynamics, electromagnetism and electrodynamics. Over a manifold of dimension $n$, their respective numbers are $n, n(n-1)/2, 1, n$ with a total of $(n+1)(n+2)/2$, that is $15$ when $n= 4$ for space-time. As a matter of fact, this is just the number of parameters of the Lie group of conformal transformations with $n$ translations, $n(n-1)/2$ rotations, $1$ dilatation and $n$ highly non-linear elations introduced by E. Cartan in $1922$. The purpose of this short but difficult paper is to prove that the form of these equations only depends on the structure of the conformal group for $n\geq 1$ arbitrary because they are described {\it as a whole} by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained for the first time by totally new differential geometric methods. Meanwhile, these equations can be all parametrized by the adjoint of the second Spencer operator through $ n(n^2 - 1)(n+2)/4$ potentials.This result brings the need to revisit the mathematical foundations of Electromagnetism and Gauge Theory according to a clever but rarely quoted paper of H. Poincar\'{e} (1901)., Comment: In classical Gauge Theory, the group U(1) is not acting on space-time when describing Maxwell equations. On the contrary, in this new approach, a Lie group of transformations is considered as a Lie pseudogroup of transformations used in order to construct differential sequences and their adjoint sequences. arXiv admin note: text overlap with arXiv:2007.01710, arXiv:1504.04118

According to a quite clever but never acknowledged work of E. Vessiot that won the prize of the Acad\'{e}mie des Sciences in 1904, " Differential Galois Theory " (DGT) has mainly to do with the study of " Principal Homogeneous Spaces " (PHS) for finite groups ( classical Galois theory), algebraic groups (Picard-Vessiot theory) and algebraic pseudogroups (Drach-Vessiot theory). The corresponding automorphic differential extension are such that $ dim_K(L)< \infty $, transcendence degree $ trd(L/K)< \infty $ and $ trd(L/K)=\infty $ with $ diff trd(L/K)< \infty $ respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry in order to revisit DGT, pointing out the deep confusion between {\it prime differential ideals} ( Defined by J.-F. Ritt in 1930) and {\it maximal ideals} thad has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we use Hopf algebras in order to study the structure of the algebraic Lie pseudogroups involved, namely Lie pseudogroups defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time in order to illustrate these results. This paper is also paying a tribute to Prof. A. Bialynicki-Birula on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group $G$ acting on itself is the simplest example of a PHS. If $G$ is defined over a field $K$ and we introduce the algebraic extension $L=K(G)$, then there is a Galois correspondence between the intermediate fields $K \subset K' \subset L$ and the subgroups $e \subset G' \subset G $ provided that $K'$ is stable under a Lie algebra $\Delta $ of invariant derivations of $L/K$. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without {\it any} way to use parameters., Comment: The comparison of the paper written by J. J. Kovacic in 2005 with our second Gordon and Breach book (1983) needs no comment on the anteriority of using Hopf algebras in DGT.(See arXiv: 1710.08122now published in the book "New Mathematical Methods for Physics" (NOVA, New York, 2018)

The first finite length differential sequence, now called {\it Janet sequence}, has been introduced by Janet in 1920. Thanks to the first book of Pommaret in 1978, this algorithmic approach has been extended by Gerdt, Blinkov, Zharkov, Seiler and others who introduced Janet and Pommaret bases in computer algebra. After 1990, new intrinsic tools have been developed in homological algebra with the definition of {\it extension differential modules} through the systematic use of {\it double differential duality} (Zbl 1079.93001). If an operator ${\cal{D}}_1$ generates the compatibility conditions (CC) of an operator ${\cal{D}}$, then the {\it adjoint operator} $ad( {\cal{D}})$ may not generate the CC of $ad({\cal{D}}_1)$. Equivalently, an operator ${\cal{D}}$ with coefficients in a differential field $K$ can be parametrized by an operator ${\cal{D}}_{-1}$ iff the differential module $M$ defined by ${\cal{D}}$ is torsion-free, that is $t(M)={ext}^1_D(N,D) = 0$ when $N$ is the differential module defined by $ad( {\cal{D}})$ and $D$ is the ring of differential operators with coefficients in $K$. Also $R = hom_K(M,K)$ is a differential module for the Spencer operator $d:R \rightarrow T^* \otimes R$, first introduced by Macaulay in 1916 with {\it inverse systems}. When ${\cal{D}}$ is the self-adjoint Einstein operator, it is not evident that $t(M)\neq 0$ is generated by the Weyl tensor having only to do with the group of conformal transformations. Gravitational waves are not coherent with these results because the stress-functions parametrizing the Cauchy = ad (Killing) operator have nothing to do with the metric, {\it exactly like the Airy or Maxwell functions in elasticity}. Similarly, the Cauchy operator has nothing to do with any contraction of the Bianchi operator., Comment: This paper also provides a new domain of application for computer algebra. arXiv admin note: text overlap with arXiv:2101.03959

Lie pseudogroups are groups of transformations solutions of systems of ordinary (OD) or partial differential (PD) equations. The purpose of this paper is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of OD or PD equations and Lie pseudogroups to engineering (elasticity, electromagnetism) or mathematical physics (general relativity, gauge theory) and their couplings (piezoelectricity, photoelasticity). The work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is still not known today. Using differential duality in the linear framework, the adjoint of the Spencer operator for the group of conformal transformations provides the Cosserat equations, the Maxwell equations and the Weyl equations on equal footing. Such a result allows to unify the finite elements of engineering sciences but also leads to deep contradictions in the case of gravitational waves. Indeed, the Beltrami operator (1892) which is parametrizing the Cauchy operator of elasticity by means of 6 stress functions is nothing else than the self-adjoint Einstein operator (1915) in dimension 3 for the deformation of the metric which is parametrizing the div operator induced from the Bianchi identities. The same confusion between the Cauchy and div operators is existing on space-time as the Cauchy operator can be parametrized by the adjoint of the Ricci operator. Accordingly, the foundations of engineering and mathematical physics must be revisited within this new framework, though striking it may sometimes look like., Comment: This paper is an elementary summary of a more technical paper just appeared as a chapter of a book on gravitational waves: https://www.intechopen.com/online-first/1119249 (https://doi.org/10.5992/intechopen.1000851). arXiv admin note: substantial text overlap with arXiv:1707.09763, arXiv:1802.02430, arXiv:1706.04105

Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian geometry. We also discovered in the meantime that, in our first book of 1978, we had already used a new way for studying the various compatibility conditions (CC) of an operator that may not be necessarily formally integrable (FI) in order to construct canonical formally exact differential sequences on the jet level. The purpose of this paper is to prove that the combination of these two facts clearly shows the specific importance of the Spencer operator and the Spencer $\delta$-cohomology, totally absent from mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide for the first time the purely intrinsic interpretation of the respective numbers of successive first, second, third and higher order generating CC. However, if they of course agree with the linearized Killing operator over the Minkowski metric, they largely disagree with recent publications on the respective numbers of generating CC for the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular the only symbol existing in the literature which is 2-ayclic witout being of finite type, contrary to the conformal situation., Comment: This paper is achieving the work we started in arXiv:1805.11958 and arXiv:2010.07001 (now published) on the search of the generating compatibility conditions of the Killing operator for the Schwarzschild and Kerr metrics by means of new intrinsic homological methods based on a systematic use of the Spencer operator

When a differential field $K$ having $n$ commuting derivations is given together with two finitely generated differential extensions $L$ and $M$ of $K$, an important problem in differential algebra is to exhibit a common differential extension $N$ in order to define the new differential extensions $L\cap M$ and the smallest differential field $(L,M)\subset N$ containing both $L$ and $M$. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules $L$ and $M$ over the non-commutative ring ring $D=K[d_1,... ,d_n]=K[d]$ of differential operators with coefficients in $K$, we may similarly look for a differential module $N$ containing both $L$ and $M$ in order to define $L\cap M$ and $L+M$. This is {\it exactly} the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a {\it built-in} property of a control system, not depending on the choice of inputs and outputs. The main purpose of this paper is to revisit control theory by showing the specific importance of the two previous problems and the part plaid by $N$ in both cases for the parametrization of the control system. The essential tool will be the study of {\it differential correspondences}, a modern name for what was called {\it B\"{a}cklund problem} during the last century, namely the study of elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The difficulty is to revisit {\it differential homological algebra} by using non-commutative localization. Finally, when $M$ is a $D$-module, this paper is using for the first time the fact that the system $R=hom_K(M,K)$ is a $D$-module for the Spencer operator acting on sections., Comment: Up to the knowledge of the author, this paper is using for the first time the Spencer operator in order to avoid behaviors, trajectories and signal spaces in the study of linear OD or PD control systems with variable coefficients

After reading such a question, any mathematician will say that, according to a well known result of L.P. Eisenhart found in 1926, the answer is " One " of course, namely the only constant allowing to describe the so-called " {\it constant riemannian curvature} " condition. The purpose of this paper is to prove the contrary by studying the case of two dimensional riemannian geometry in the light of an old work of E. Vessiot published in 1903 but {\it still totally unknown today} after more than a century. In fact, we shall compute locally the {\it Vessiot structure equations} and prove that there are indeed " Two " {\it Vessiot structure constants} satisfying a single {\it linear Jacobi condition} showing that one of them must vanish while the other one must be equal to the known one. This result depends on deep mathematical reasons in the formal theory of Lie pseudogroups, which are involving both the Spencer $\delta$-cohomology and diagram chasing in homological algebra. Another similar example will illustrate and justify this comment out of the classical tensorial framework of the famous " {\it equivalence problem} ". The case of contact transformations will also be studied. Though it is quite unexpected, we shall reach the conclusion that the mathematical foundations of both classical and conformal riemannian geometry must be revisited. We have treated the case of conformal geometry in a recent arXiv preprint., Comment: In memory of the Russian mathematician V.P. GERDT who died from COVID in January 2021. He compared Grobner, Janet and Pommaret bases in computer algebra and supervised in Aachen the first PhD thesis existing on the algorithmic aspect of the Vessiot structure equations

When ${\cal{D}}:\xi \rightarrow \eta$ is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that ${\cal{D}}\xi=\eta$ implies ${\cal{D}}_1\eta=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ when the ground manifold has dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a more difficult " inverse problem " is to look for an operator ${\cal{D}}: \xi \rightarrow \eta$ having the generating CC ${\cal{D}}_1\eta=0$. If this is possible, that is when the differential module defined by ${\cal{D}}_1$ is torsion-free, one shall say that the operator ${\cal{D}}_1$ is parametrized by ${\cal{D}}$ and there is no relation in general between ${\cal{D}}$ and ${\cal{D}}_2$. The parametrization is said to be " minimum " if the differential module defined by ${\cal{D}}$ has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchy stress operator in arbitrary dimension $n$ has attracted many famous scientists (G.B. Airy in 1863 for $n=2$, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for $n=3$, A. Einstein in 1915 for $n=4$) . This paper proves that all these works are using the Einstein operator and not the Ricci operator. As a byproduct, they are all based on a confusion between the so-called $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator which is the formal adjoint of the Killing operator ${\cal{D}}$ parametrizing the Riemann operator ${\cal{D}}_1$ for an arbitrary $n$. Like the Michelson and Morley experiment, it is an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment., Comment: The purpose of this paper is to prove that Einstein equations had already been exhibited by E. Beltrami 20 years before A. Einstein and to solve the minimum parametrization problem in arbitrary dimension

A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M) , Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper, we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S($m$) and K($m,a$) are depending on constant parameters in such a way that S $\rightarrow $ M when $m \rightarrow 0$ and K $\rightarrow$ S when $ a \rightarrow 0$, the CC of S do not provide the CC of M when $m \rightarrow 0$ while the CC of K do not provide the CC of S when $a\rightarrow 0$. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the {\it prolongation/projection} (PP) procedure, we provide for the first time all the CC for K in an intrinsic way, showing that they only depend on the underlying Killing algebras and that the role played by the Spencer operator is crucial. We get K$<$S$<$M with $2 < 4 < 10$ for the Killing algebras and explain why the formal search of the CC for M, S or K are strikingly different, even though each Spencer sequence is isomorphic to the tensor product of the Poincar\'{e} sequence for the exterior derivative by the corresponding Lie algebra., Comment: This paper achieves the formal study of the Kerr metric that we had only sketched in J. of Modern Physics, 9 (2018) 1970-2007 (arXiv:1805.11958v2)

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the physical foundations of both electromagnetism (EM) and gravitation, with the only experimental need to measure the EM constant in vacuum and the gravitational constant. With a manifold of dimension $n$, the difficulty is to deal with the $n$ nonlinear transformations that have been called "elations" by E. Cartan in 1922. Using the fact that dimension $n=4$ has very specific properties for the computation of the Spencer cohomology, we prove that there is no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, ...) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. In the sudy of gravitation, the dimension $n=4$ also allows to have a conformal factor defined everywhere but at the central attractive mass and the inversion law of the subgroupoid made by strict second order jets transforms attraction into repulsion., Comment: These new nonlinear results question the mathematical foundations of both elasticity theory, gauge theory and general relativity

Since 100 years or so, it has been usually accepted that the " conformal group " could be defined in an arbitrary dimension n as the group of transformations preserving a non degenerate flat metric up to a nonzero invertible point depending factor called " conformal factor ". However, when n > 2, it is a finite dimensional Lie group of transformations with n translations, n(n-1)/2 rotations, 1 dilatation and n nonlinear transformations called " elations " , that is a total of (n+1)(n+2)/2 transformations. Because of the Michelson-Morley experiment, the conformal group of space-time with 15 parameters is well known as the biggest group of invariance of the constitutive law of electromagnetism (EM) in vacuum, even though the two sets of field and induction Maxwell equations are respectively invariant by any local invertible transformation. As this last generic number is also well defined and becomes equal to 3 for n=1 or 6 for n=2, the purpose of this paper is to use modern mathematical tools such as the Spencer operator on systems of OD or PD equations, both with its restriction to their symbols leading to the Spencer -cohomology, in order to provide a unique striking definition that could be valid for any n. The concept of a " finite type " system is crucial for such a new definition.

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:1212.4590

The last invited lecture published in $1962$ by Lanczos on his potential theory is never quoted because it is in french. Comparing it with a commutative diagram in a recently published paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the formal theory of Lie pseudogroups through differential double duality and the construction of finite length differential sequences for Lie operators. We use the fact that a differential module $M$ defined by an operator ${\cal{D}}$ with coefficients in a differential field $K$ has vanishing first and second differential extension modules if and only if its adjoint differential module $N=ad(M)$ defined by the adjoint operator $ad({\cal{D}})$ is reflexive, that is $ad({\cal{D}})$ can be parametrized by the operator $ad({\cal{D}}_1)$ when ${\cal{D}}_1$ generates the compatibilty conditions (CC) of ${\cal{D}}$ while $ad({\cal{D}}_1)$ can be parametrized by $ad({\cal{D}}_2)$ when ${\cal{D}}_2$ generates the CC of ${\cal{D}}_1$. We provide an explicit description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their respective adjoint operators., Comment: This paper provides the explicit computations of all the parametrizing operators known to exist according to arXiv:1803.09610

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)

In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the {\it Vessiot structure equations}. In all these cases, they discovered that the {\it compatibility conditions} (CC) for the corresponding Killing operator were involving {\it a mixture of both second order and third order CC} and their idea has been to exhibit only a {\it minimal number of generating ones}. However, even if they exhibited a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. Using the formal theory of systems of partial differential equations and Lie pseudogroups, the purpose of this difficult computational paper is to provide new intrinsic differential and homological methods involving the Spencer operator in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools are now available as computer algebra packages., Comment: In this v2 we improve the search of generating compatibility conditions for the Killing operators used in general relativity and the corresponding differential sequences, by means of new differential homological methods. The existence of a partition 10=4+4+2 of the Ricci tensor questions the mathematical coherence of Einstein equations with group theory and formal integrability

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) 699-728

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 figure

During the last months or so we had the opportunity to read two papers trying to relate the study of Macaulay (1916) inverse systems with the so-called Riquier (1910)-Janet (1920) initial conditions for the integration of linear analytic systems of partial differential equations. One paper has been written by F. Piras (1998) and the other by U. Oberst (2013), both papers being written in a rather algebraic style though using quite different techniques. It is however evident that the respective authors, though knowing the computational works of C. done during the first half of the last century in a way not intrinsic at all, are not familiar with the formal theory of systems of ordinary or partial differential equations developped by D.C. Spencer (1912-2001) and coworkers around 1965 in an intrinsic way, in particular with its application to the study of differential modules in the framework of algebraic analysis. As a byproduct, the first purpose of this paper is to establish a close link between the work done by F. S. Macaulay (1862-1937) on inverse systems in 1916 and the well-known Cartan-K{\"a}hler theorem (1934). The second purpose is also to extend the work of Macaulay to the study of arbitrary linear systems with variable coefficients. The reader will notice how powerful and elegant is the use of the Spencer operator acting on sections in this general framework. However, we point out the fact that the literature on differential modules mostly only refers to a complex analytic structure on manifolds while the Spencer sequences have been created in order to study any kind of structure on manifolds defined by a Lie pseudogroup of transformations, not just only complex analytic ones. Many tricky explicit examples illustrate the paper, including the ones provided by the two authors quoted but in a quite different framework.