From Control Theory to Gravitational Waves

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 D1:η→ξ such that Dξ = η implies D1η = 0. When D is involutive, the procedure provides successive first-order involutive operators D1,...,Dn when the ground manifold has dimension n. Conversely, when D1 is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D1η = 0. If this is possible, that is when the differential module defined by D1 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 D1 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 D1 is formally surjective, controllability just amounts to the formal injectivity of ad(D1), 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 prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized and the comparison needs no comment. As a byproduct, they are all based on a confusion between the so-called div operator D2 induced from the Bianchi operator and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D1 for an arbitrary n. This purely mathematical result deeply questions the origin and existence of gravitational waves, both with the mathematical foundations of general relativity. As a matter of fact, this new framework provides a totally open domain of applications for computer algebra as the quoted test can be studied by means of Pommaret bases and related recent packages.

增湿情况重塑黄土非饱和渗透系数的测定方法研究

对浸润锋前进法、Boltzmann变换法和瞬态剖面法3种直接方法在确定增湿情况下重塑黄土非饱和渗透系数的适宜性进行了研究,并对其进行了改进。首先应用重塑黄土进行了一维水平入渗试验,对前2种方法进行了改进,检验了改进后的效果,比较了这2种改进方法、瞬态剖面法和间接法所得非饱和渗透系数的差异,然后应用Hydrus软件对一维入渗试验进行了数值模拟,获得了采用不同方法所得非饱和渗透参数时预测浸润锋迁移距离和时间关系的准确性,分析了前2种方法所采用假定的合理性并对瞬态剖面法测点布置进行了改进和效果检验。室内试验结果表明浸润锋前进法采用改进的水力梯度计算公式后,不同吸力下渗透系数波动的现象减少;Boltzmann变换法采用新的变量λ*后,各测点λ*-θ关系的统一性增加;由于测点间距较大,瞬态剖面法偏离其他方法所得渗透系数很大。数值模拟结果表明采用2种改进方法所得的非饱和渗透参数均能较好地预测浸润锋迁移距离和时间的关系;浸润锋前进法采用的水分剖面平行移动假定以及Boltzmann变换方法采用的Boltzmann变量与体积含水率关系唯一假定,均需要在入渗一定距离后才能得到基本满足;采用改进后的测点布置位置和间距后,瞬态剖面法所得不同吸力下的非饱和渗透系数与前2种改进方法所得的差距减少,但仍不如前2种改进方法更接近于输入值。

基于等含水率试验结果的非饱和原状黄土本构模型

基于国内通常进行的非饱和原状黄土等含水率试验规律,在Barcelona模型基础上,建立了考虑非饱和原状黄土结构性和水敏性两个主要特性的本构模型,并检验了等含水率等向压缩试验和三轴压缩试验中该模型的力学响应。结果表明该模型能反映等向压缩试验中不同含水率试样在结构屈服应力之前压缩线趋向重合,屈服应力之后压缩线相互趋近的特性;中等含水率试样三轴压缩试验中初始弹性模量随着围压先减少后增大的特性,e-lgp空间路径在低围压情况呈现明显转折的现象。模型共11个参数,均能从国内常用的等含水率试验获取。

Effect of temperature on soil-water characteristics and hysteresis of compacted Gaomiaozi bentonite

Laboratory tests under different constraint conditions were carried out to obtain the soil-water retention curves(SWRCs) of highly-compacted confined/unconfined Gaomiaozi(GMZ) bentonite at 20,40 and 80 ℃,respectively. The effect of temperature on the soil-water characteristics of the highly-compacted GMZ bentonite was analyzed. The results show that the water retention capacity of the highly-compacted GMZ bentonite decreases as the temperature increases under unconfined and confined conditions. At a certain temperature,the constraint conditions have little influence on the water retention capacity of the compacted bentonite at high suction,but the water retention capacity of the confined specimen is lower than that of the unconfined specimen at low suction. Under unconfined conditions,the hysteretic behaviour of the compacted bentonite decreases with increasing temperature. At high suction(>4 MPa) ,the hysteretic behaviour of the unconfined bentonite tends to increase with the decrease of the suction. In summary,the hysteretic behaviour of the compacted bentonite is not significant.

The Mathematical Foundations of Elasticity and Electromagnetism Revisited

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.

Generating Compatibility Conditions and General Relativity

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.

Minimum Parametrization of the Cauchy Stress Operator

When D: ξ→η is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: η→ξ such that Dξ=η implies D1η=0. When D is involutive, the procedure provides successive first order involutive operators D1, ..., Dn, when the ground manifold has dimension n, 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 D1 is given, a more difficult “inverse problem” is to look for an operator D: ξ→η having the generating CC D1η=0. If this is possible, that is when the differential module defined by D1 is torsion-free, one shall say that the operator D1 is parametrized by D and there is no relation in general between D and D2. The parametrization is said to be “minimum” if the differential module defined by D has a vanishing differential rank and is thus a torsion module. The solution of this problem, first found by the author of this paper in 1995, is still not acknowledged. As for the applications of the “differential double duality” theory to standard equations of physics (Cauchy and Maxwell equations can be parametrized while Einstein equations cannot), we do not know other references. When erator in arbitrary dimension=1 as in control theory, the fact that controllability is a “built in” property of a control system, amounting to the existence of a parametrization and thus not depending on the choice of inputs and outputs, even with variable coefficients, is still not acknowledged by engineers. The parametrization of the Cauchy stress operator in arbitrary dimension n has nevertheless attracted, “separately” and without any general “guiding line”, 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 ). The aim of this paper is to solve the minimum parametrization problem in arbitrary dimension and to apply it through effective methods that could even be achieved by using computer algebra. Meanwhile, we prove that all these works are using the Einstein operator which is self-adjoint and not the Ricci operator, a fact showing that the Einstein operator, which cannot be parametrized, has already been exhibited by Beltrami more than 20 years before Einstein. As a byproduct, they are all based on the same confusion between the so-called div operator induced from the Bianchi operator D2 and the Cauchy operator which is the formal adjoint of the Killing operator D parametrizing the Riemann operator D1 for an arbitrary n. We prove that this purely mathematical result deeply questions the origin and existence of gravitational waves. We also present the similar motivating situation met in the study of contact structures when n = 3. Like the Michelson and Morley experiment, it is thus an open historical problem to know whether Einstein was aware of these previous works or not, but the comparison needs no comment.

Why Gravitational Waves Cannot Exist

The purpose of this short but difficult paper is to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT) in the light of a modern approach to nonlinear systems of ordinary or partial differential equations, using new methods from Differential Geometry (D.C. Spencer, 1970), Differential Algebra (J.F. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). The main idea is to identify the differential indeterminates of Ritt and Kolchin with the jet coordinates of Spencer, in order to study Differential Duality by using only linear differential operators with coefficients in a differential field K. In particular, the linearized second order Einstein operator and the formal adjoint of the Ricci operator are both parametrizing the 4 first order Cauchy stress equations but cannot themselves be parametrized. In the framework of Homological Algebra, this result is not coherent with the vanishing of a certain second extension module and leads to question the proper origin and existence of gravitational waves. As a byproduct, we also prove that gravitation and electromagnetism only depend on the second order jets (called elations by E. Cartan in 1922) of the system of conformal Killing equations because any 1-form with value in the bundle of elations can be decomposed uniquely into the direct sum (R, F) where R is a section of the Ricci bundle of symmetric covariant 2-tensors and the EM field F is a section of the vector bundle of skew-symmetric 2-tensors. No one of these purely mathematical results could have been obtained by any classical approach. Up to the knowledge of the author, it is also the first time that differential algebra in a modern setting is applied to study the specific algebraic feature of most equations to be found in mathematical physics, particularly in GR.

Differential Homological Algebra and General Relativity

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.

Organic aerosol molecular composition and gas–particle partitioning coefficients at a Mediterranean site(Corsica)

Molecular speciation of atmospheric organic matter was investigated during a short summer field campaign performed in a citrus fruit field in northern Corsica（June 2011）. Aimedat assessing the performance on the field of newly developed analytical protocols, this work focuses on the molecular composition of both gas and particulate phases and provides an insight into partitioning behavior of the semi-volatile oxygenated fraction. Limonene ozonolysis tracers were specifically searched for, according to gas chromatography–mass spectrometry（GC–MS） data previously recorded for smog chamber experiments. A screening of other oxygenated species present in the field atmosphere was also performed. About sixty polar molecules were positively or tentatively identified in gas and/or particle phases. These molecules comprise a wide range of branched and linear, mono and di-carbonyls（C_3–C7）,mono and di-carboxylic acids（C_3–C_18）, and compounds bearing up to three functionalities.Among these compounds, some can be specifically attributed to limonene oxidation and others can be related to α- or β-pinene oxidation. This provides an original snapshot of the organic matter composition at a Mediterranean site in summer. Furthermore, for compounds identified and quantified in both gaseous and particulate phases, an experimental gas/particle partitioning coefficient was determined. Several volatile products, which are not expected in the particulate phase assuming thermodynamic equilibrium, were nonetheless present in significant concentrations. Hypotheses are proposed to explain these observations, such as the possible aerosol viscosity that could hinder the theoretical equilibrium to be rapidly reached.