When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1:</su...When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1:</sub>η→ξ such that Dξ = η implies D<sub>1</sub>η = 0. When D is involutive, the procedure provides successive first-order involutive operators D<sub>1</sub>,...,D<sub>n </sub>when the ground manifold has dimension n. Conversely, when D<sub>1</sub> is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D<sub>1</sub>η = 0. If this is possible, that is when the differential module defined by D<sub>1</sub> is “torsion-free”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D<sub>1</sub> is parametrized by D. The parametrization is said to be “minimum” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “built in” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D<sub>1</sub> is formally surjective, controllability just amounts to the formal injectivity of ad(D<sub>1</sub>), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, 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 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4). We 展开更多
We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusio...We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms： a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfiirth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.展开更多
When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) &ra...When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) →F=F<sub>0</sub> that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: F<sub>0</sub> →F<sub>1</sub>. When D is involutive, that is when the corresponding system R<sub>q</sub> = ker (Φ) is involutive, this procedure provides successive first order involutive operators D<sub>1</sub>, ..., D<sub>n</sub>. Though D<sub>1</sub> οD = 0 implies ad (D) οad(D<sub>1</sub>) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D<sub>1</sub>) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R<sub>q</sub> is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.展开更多
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 framework, they used the nonli...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 framework, they used 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 linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, 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 also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.展开更多
The first purpose of this striking but difficult paper is to revisit the mathematical foundations of Elasticity (EL) and Electromagnetism (EM) by comparing the structure of these two theories and examining with detail...The first purpose of this striking but difficult paper is to revisit the mathematical foundations of Elasticity (EL) and Electromagnetism (EM) by comparing the structure of these two theories and examining with details their 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 pseudo groups 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 and separately in EL or EM. 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 and the electromagnetic field. 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 adjoint of the Ricci operator (4 terms only) in the study of gravitational waves.展开更多
In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed that aim at coupling an atomistic model （discrete mechan...In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed that aim at coupling an atomistic model （discrete mechanics） with a macroscopic model （continuum mechanics）. We provide here a theoretical basis for such a coupling in a one-dimensional setting, in the case of convex energy.展开更多
When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em&...When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> 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 <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-展开更多
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 ...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 successively. 1) VESSIOT VERSUS CARTAN: 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. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 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 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 and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the g展开更多
This paper is an introduction to the modelling of viscoelastic fluids, with an emphasis on micromacro （or multiscale） models. Some elements of mathematical and numerical analysis are provided. These notes closely fo...This paper is an introduction to the modelling of viscoelastic fluids, with an emphasis on micromacro （or multiscale） models. Some elements of mathematical and numerical analysis are provided. These notes closely follow the lectures delivered by the second author at the Chinese Academy of Science during the Workshop ＂Stress Tensor Effects on Fluid Mechanics＂ in January 2010.展开更多
When D: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">...When D: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ</span></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em><em><span style="white-space:nowrap;"></span></em><em></em></span> </span>is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ</span> </span></em></span></span>such that <span style="white-space:nowrap;">D<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ</span></span></em></span>=<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em></span></span> implies <span style="white-space:nowrap;">D<sub>1</sub><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em></span>=0</span>. When D is involutive, the procedure provides successive first order involutive operators D1, ..., D<sub>n</sub>, when the ground manifold has dimension <em>n</em>, a result first found by M. Janet as early as in 1920, in a footnote. However, the link between this “Janet sequence” and the “Spencer sequence” first found by the author of this paper in 1978 is still not acknowledged. Conversely, when D<sub>1</sub> is given, a more difficult “inverse problem” is to look for an operator D: <span style="white-space:nowrap;"><span style="white-space:nowrap;">展开更多
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 know...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 (DOI:10.4236/jmp.2018.910125) 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(<em>m</em>) and K(<em>m</em>, <em>a</em>) are depending on constant parameters in such a way that S <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0 and K<span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><span style="white-space:nowrap;"><span style="white-space:nowrap;"></span></span> S when <em>a</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0, the CC of S do not provide the CC of M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 0 while the CC of K do not provide the CC of S when a <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 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 <em>prolongation/projection</em> (PP) procedure,展开更多
When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em...When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em>compatibility conditions </em>(CC) <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub><em>1</em></sub><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span> =</span><sub></sub> 0 of <em>D</em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">ξ </span></em></span></span>= <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span>. Conversely, when <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is given, an <em>inverse problem</em> is to look for an operator <span style="white-space:normal;"><em>D</em></span> such that its CC are generated by <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> and we shall say that <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is <em>parametrized</em> by <em>D</em> = <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>0</sub></span>. We may thus construct a differential sequence with successive operators <em>D</em>, <em>D</em><sub>1</sub>, <em>D</em><sub>2</sub>, ..., each operator parametrizing the next one. Introducing the<em> formal adjoint ad</em>() of an operator, we have <img src="Edit_ecbb631c-2896-4dad-8234-cacd5504f138.png" alt="" />but <span style="white-space:nowrap;"><em>ad</em> (<em>D</em><sub><em>i</em>-1</sub>)</span> may not generate <em>all</em> the CC of <em>ad </em>(<em>D</em><sub>i</sub>). When <em>D </em>= <em>K</em> [d<sub>1</sub>, ..., d<sub>n</sub>] = <em>K </em>[<em>d</em>] is the (non-commutative) ring of differential operators wit展开更多
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 differentia...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 “differential 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 differential modules, introducing a “relative parametrization” where the potentials should satisfy compatible “differential constraints”. We recently noticed 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. In order to make this difficult paper rather self-contained, these unusual purely mathematical results are illustrated by many explicit examples, two of them dealing with contact transformations, and even strengthening the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory. They also bring additional doubts on the origin and existence of gravitational waves.展开更多
We devise hybrid high-order(HHO)methods for the acoustic wave equation in the time domain.We frst consider the second-order formulation in time.Using the Newmark scheme for the temporal discretization,we show that the...We devise hybrid high-order(HHO)methods for the acoustic wave equation in the time domain.We frst consider the second-order formulation in time.Using the Newmark scheme for the temporal discretization,we show that the resulting HHO-Newmark scheme is energy-conservative,and this scheme is also amenable to static condensation at each time step.We then consider the formulation of the acoustic wave equation as a frst-order system together with singly-diagonally implicit and explicit Runge-Kutta(SDIRK and ERK)schemes.HHO-SDIRK schemes are amenable to static condensation at each time step.For HHO-ERK schemes,the use of the mixed-order formulation,where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns,is key to beneft from the explicit structure of the scheme.Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order O(hk+1)in the H1-norm and of order O(h^(k+2))in the L^(2)-norm.Moreover,test cases on wave propagation in heterogeneous media indicate the benefts of using high-order methods.展开更多
The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingl...The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.展开更多
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 identi...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 GB book of 1978, we had already used a new way for studying the 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 δ-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 a few resolutions in which the orders of the successive operators may go “up and down” surprisingly, like in the conformal situation for various dimensions.展开更多
In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar...In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) 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 Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered 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. The purpose of this difficult computational paper is to provide differential and homological methods 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, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.展开更多
The purpose of this paper is to revisit the well known potentials, also called stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-d...The purpose of this paper is to revisit the well known potentials, also 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) 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 components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed is equal to for any minimal parametrization, the Einstein parametrization being “in between” with potentials. We provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today, but it could be strictly impossible to obtain them without using the above methods. We also revisit the possibility (Maxwell equations of electromagnetism) or the impossibility (Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various 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 ins展开更多
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 ...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 mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. 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 thus 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, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.展开更多
In this paper,a multiscale problem arising in material science is considered.The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient,in a sense made precisely in ...In this paper,a multiscale problem arising in material science is considered.The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient,in a sense made precisely in the body of the text.The homogenized limit is then computed by using a perturbation approach.This computation requires repeatedly solving a corrector-like equation for various configurations of the material.For this purpose,the reduced basis approach is employed and adapted to the specific context.The authors perform numerical tests that demonstrate the efficiency of the approach.展开更多
文摘When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1:</sub>η→ξ such that Dξ = η implies D<sub>1</sub>η = 0. When D is involutive, the procedure provides successive first-order involutive operators D<sub>1</sub>,...,D<sub>n </sub>when the ground manifold has dimension n. Conversely, when D<sub>1</sub> is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D<sub>1</sub>η = 0. If this is possible, that is when the differential module defined by D<sub>1</sub> is “torsion-free”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D<sub>1</sub> is parametrized by D. The parametrization is said to be “minimum” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “built in” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D<sub>1</sub> is formally surjective, controllability just amounts to the formal injectivity of ad(D<sub>1</sub>), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, 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 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4). We
文摘We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms： a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfiirth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.
文摘When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) →F=F<sub>0</sub> that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: F<sub>0</sub> →F<sub>1</sub>. When D is involutive, that is when the corresponding system R<sub>q</sub> = ker (Φ) is involutive, this procedure provides successive first order involutive operators D<sub>1</sub>, ..., D<sub>n</sub>. Though D<sub>1</sub> οD = 0 implies ad (D) οad(D<sub>1</sub>) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D<sub>1</sub>) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R<sub>q</sub> is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.
文摘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 framework, they used 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 linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, 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 also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.
文摘The first purpose of this striking but difficult paper is to revisit the mathematical foundations of Elasticity (EL) and Electromagnetism (EM) by comparing the structure of these two theories and examining with details their 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 pseudo groups 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 and separately in EL or EM. 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 and the electromagnetic field. 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 adjoint of the Ricci operator (4 terms only) in the study of gravitational waves.
文摘In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed that aim at coupling an atomistic model （discrete mechanics） with a macroscopic model （continuum mechanics）. We provide here a theoretical basis for such a coupling in a one-dimensional setting, in the case of convex energy.
文摘When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> 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 <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-
文摘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 successively. 1) VESSIOT VERSUS CARTAN: 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. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 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 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 and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the g
文摘This paper is an introduction to the modelling of viscoelastic fluids, with an emphasis on micromacro （or multiscale） models. Some elements of mathematical and numerical analysis are provided. These notes closely follow the lectures delivered by the second author at the Chinese Academy of Science during the Workshop ＂Stress Tensor Effects on Fluid Mechanics＂ in January 2010.
文摘When D: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ</span></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em><em><span style="white-space:nowrap;"></span></em><em></em></span> </span>is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ</span> </span></em></span></span>such that <span style="white-space:nowrap;">D<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ</span></span></em></span>=<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em></span></span> implies <span style="white-space:nowrap;">D<sub>1</sub><span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;">η</span></span></em></span>=0</span>. When D is involutive, the procedure provides successive first order involutive operators D1, ..., D<sub>n</sub>, when the ground manifold has dimension <em>n</em>, a result first found by M. Janet as early as in 1920, in a footnote. However, the link between this “Janet sequence” and the “Spencer sequence” first found by the author of this paper in 1978 is still not acknowledged. Conversely, when D<sub>1</sub> is given, a more difficult “inverse problem” is to look for an operator D: <span style="white-space:nowrap;"><span style="white-space:nowrap;">
文摘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 (DOI:10.4236/jmp.2018.910125) 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(<em>m</em>) and K(<em>m</em>, <em>a</em>) are depending on constant parameters in such a way that S <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0 and K<span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><span style="white-space:nowrap;"><span style="white-space:nowrap;"></span></span> S when <em>a</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0, the CC of S do not provide the CC of M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 0 while the CC of K do not provide the CC of S when a <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 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 <em>prolongation/projection</em> (PP) procedure,
文摘When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em>compatibility conditions </em>(CC) <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub><em>1</em></sub><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span> =</span><sub></sub> 0 of <em>D</em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">ξ </span></em></span></span>= <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span>. Conversely, when <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is given, an <em>inverse problem</em> is to look for an operator <span style="white-space:normal;"><em>D</em></span> such that its CC are generated by <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> and we shall say that <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is <em>parametrized</em> by <em>D</em> = <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>0</sub></span>. We may thus construct a differential sequence with successive operators <em>D</em>, <em>D</em><sub>1</sub>, <em>D</em><sub>2</sub>, ..., each operator parametrizing the next one. Introducing the<em> formal adjoint ad</em>() of an operator, we have <img src="Edit_ecbb631c-2896-4dad-8234-cacd5504f138.png" alt="" />but <span style="white-space:nowrap;"><em>ad</em> (<em>D</em><sub><em>i</em>-1</sub>)</span> may not generate <em>all</em> the CC of <em>ad </em>(<em>D</em><sub>i</sub>). When <em>D </em>= <em>K</em> [d<sub>1</sub>, ..., d<sub>n</sub>] = <em>K </em>[<em>d</em>] is the (non-commutative) ring of differential operators wit
文摘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 “differential 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 differential modules, introducing a “relative parametrization” where the potentials should satisfy compatible “differential constraints”. We recently noticed 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. In order to make this difficult paper rather self-contained, these unusual purely mathematical results are illustrated by many explicit examples, two of them dealing with contact transformations, and even strengthening the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory. They also bring additional doubts on the origin and existence of gravitational waves.
基金The authors would like to thank L.Guillot(CEA/DAM)for insightful discussions and CEA/DAM for partial fnancial support.EB was partially supported by the EPSRC grants EP/P01576X/1 and EP/P012434/1.
文摘We devise hybrid high-order(HHO)methods for the acoustic wave equation in the time domain.We frst consider the second-order formulation in time.Using the Newmark scheme for the temporal discretization,we show that the resulting HHO-Newmark scheme is energy-conservative,and this scheme is also amenable to static condensation at each time step.We then consider the formulation of the acoustic wave equation as a frst-order system together with singly-diagonally implicit and explicit Runge-Kutta(SDIRK and ERK)schemes.HHO-SDIRK schemes are amenable to static condensation at each time step.For HHO-ERK schemes,the use of the mixed-order formulation,where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns,is key to beneft from the explicit structure of the scheme.Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order O(hk+1)in the H1-norm and of order O(h^(k+2))in the L^(2)-norm.Moreover,test cases on wave propagation in heterogeneous media indicate the benefts of using high-order methods.
文摘The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.
文摘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 GB book of 1978, we had already used a new way for studying the 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 δ-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 a few resolutions in which the orders of the successive operators may go “up and down” surprisingly, like in the conformal situation for various dimensions.
文摘In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) 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 Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered 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. The purpose of this difficult computational paper is to provide differential and homological methods 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, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.
文摘The purpose of this paper is to revisit the well known potentials, also 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) 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 components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed is equal to for any minimal parametrization, the Einstein parametrization being “in between” with potentials. We provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today, but it could be strictly impossible to obtain them without using the above methods. We also revisit the possibility (Maxwell equations of electromagnetism) or the impossibility (Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various 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 ins
文摘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 mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. 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 thus 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, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.
基金Project supported by EOARD(European Office of Aerospace Research and Development) (No.FA865510-C-4002)
文摘In this paper,a multiscale problem arising in material science is considered.The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient,in a sense made precisely in the body of the text.The homogenized limit is then computed by using a perturbation approach.This computation requires repeatedly solving a corrector-like equation for various configurations of the material.For this purpose,the reduced basis approach is employed and adapted to the specific context.The authors perform numerical tests that demonstrate the efficiency of the approach.
