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.

Discussion on the Homology Theory of Lie Algebras

Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).

Killing Operator for the Kerr Metric

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: Jq(E) →F=F0 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 D1: F0 →F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) 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 Rq 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.

Nonlinear Conformal Gravitation

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.

Differential temporal expression of matrix metalloproteinases following sciatic nerve crush

We previously performed transcriptome sequencing and found that genes for matrix metalloproteinases（MMPs）,such as MMP7 and 12,seem to be highly upregulated following peripheral nerve injury,and may be involved in nerve repair.In the present study,we systematically determined the expression levels of MMPs and their regulators at 1,4,7 and 14 days after sciatic nerve crush injury.The number of differentially expressed genes was elevated at 4 and 7 days after injury,but decreased at 14 days after injury.Among the differentially expressed genes,those most up-regulated showed fold changes of more than 214,while those most down-regulated exhibited fold changes of more than 2-10.Gene sequencing showed that,at all time points after injury,a variety of MMP genes in the “Inhibition of MMPs” pathway were up-regulated,and their inhibitor genes were down-regulated.Expression of key up-and down-regulated genes was verified by quantitative real-time polymerase chain reaction analysis and found to be consistent with transcriptome sequencing.These results suggest that MMP-related genes are strongly involved in the process of peripheral nerve regeneration.

Homological Solution of the Lanczos Problems in Arbitrary Dimension

When D is a linear partial differential operator of any order, a direct problem is to look for an operator D1 generating the compatibility conditions (CC) D1η = 0 of Dξ = η. Conversely, when D1 is given, an inverse problem is to look for an operator D such that its CC are generated by D1 and we shall say that D1 is parametrized by D = D0. We may thus construct a differential sequence with successive operators D, D1, D2, ..., each operator parametrizing the next one. Introducing the formal adjoint ad() of an operator, we have but ad (Di-1) may not generate all the CC of ad (Di). When D = K [d1, ..., dn] = K [d] is the (non-commutative) ring of differential operators with coefficients in a differential field K, then D gives rise by residue to a differential module M over D while ad (D) gives rise to a differential module N =ad (M) over D. The differential extension modules with ext0(M) = homD (M, D) only depend on M and are measuring the above gaps, independently of the previous differential sequence, in such a way that ext1 (N) = t (M) is the torsion submodule of M. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension n and to prove for the first time that the extension modules highly depend on the Vessiot structure constants c. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint Beltrami = ad (Riemann) which can indeed be parametrized by the operator Lanczos = ad (Bianchi) in arbitrary dimension, “one step further on to the right” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when D is a Lie operator of finite type, we shall prove that . It will follow that the Riemann-Lanczos and Weyl-Lanczos problems just amount to prove such a result for i = 1,2 and arbitrary n when D is the classical or conformal Killing operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.

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.

ISOLATION OF HUMAN TRANSCRIPTS EXPRESSED IN 16HBE CELLS RELATED TO CHLOROPHYLLIN ANTITRANSFORMING ACTIVITY AGAINST ANTI-BPDE BY cDNA REPRESENTATIONAL DIFFERENCE ANALYSIS

Objective: To analyze the differentially expressed cDNA sequences related to chlorophyllin (CHL) mediated inhibition of malignant transformation of human bronchia1 epithelial cell line (16HBE). Methods: 16HBE cells treated with chlorophyllin and anti-BPDE were conducted as tester, 16HBE cells treated only with anti-BPDE were conducted as driver, and cDNA representational difference analysis (cDNA RDA) was used to compare the differential gene expression between the two kinds of cells. The cDNA fragments were ligated to pGEM-T vector and transformed into JM109 bacteria. The plasmid DNA was sequenced and compared with database in GenBank by BLASTN. Results: Among the 5 cloned cDNA sequences, three were novel and were registered in dbEST database, two showed sequence homology to alpha-enolase and a newly found gene ribosomal protein S18/S6-like. Conclusion: These 5 cDNA sequences might play important roles in antitransforming effect of chlorophyllin.

The Genetic Structure and Diversity of Repomucenus curvicornis Inhabiting Liaoning Coast Based on Mitochondrial COⅠ Gene and Control Region

[Object] This study was conducted to explore the genetic diversity and structure of the wild Repomucenus curvicornis inhabiting Liaoning Coast, China. [Method] The mitochondrial COⅠ gene and control region（CR） were PCR amplified from the wild R. curvicornis populations from the Liaodong Bay（n=22） and the northern Yellow Sea（n=18）, sequenced and analyzed for genetic diversity. [Result] The contents of A, T, C and G of 624 bp COⅠ gene were 24.09%, 31.04%, 25.28%, and 19.59%, and those of 460 bp CR fragment were 32.96%, 32.80%, 14.86% and 19.38%, respectively. The total number of variable sites, average number of nucleotide differences（ k）, haplotype diversity（H） and nucleotide diversity（π） based on COⅠ gene were 38, 4.67,（0.96±0.02） and（0.007 5±0.004 2）, and those based on CR fragment were 26, 3.35,（0.97 ±0.02） and（0.007 3±0.004 3）, respectively. Based on mitochondrial COⅠ gene and CR, the genetic diversity of Liaodong Bay population was lower than that of the northern Yellow Sea population. The AMOVA analysis based on CR fragments revealed almost significant genetic divergence between the Liaodong Bay and the northern Yellow Sea populations, while there was no significant genetic divergence based on COⅠ gene. The results showed that CR and COⅠ gene are effective molecular markers for detecting the genetic diversity of R. curvicornis population, while CR is more reliable than COⅠ gene in detecting the genetic structure. [Conclusion] CR is an appropriate marker for genetic analysis of marine fish population.

Nonlinear Conformal Electromagnetism

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.