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.

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.

Modified Ureterosigmoidostomy

Objective To introduce an operation procedure and evaluate the continence diversion results of the modified ureterosigmoidostomy after radical cystectomy. Methods Fourteen cases of bladder cancer or prostate carcinoma were operated on with modified Sigma pouch from Feb, 1998 to Dec, 1999. A longitudinal incision about 25 cm on the sigmoid wall was done to form a low pressure pouch. The vertex of the new pouch was fixed to sacrum. Both ends of ureters were anastomosed side to side and to form a big nipple and inserted into the top of pouch for 2 to 3 centimeters.Results It took about sixty five minutes to create a new low pressure pouch after radical cystectomy. Early complication of was found in two cases postoperatively, and cured with temporary colonostomy. Hydronephrosis and hypokalemia in one patient were cured by percutaneous anterograde ureter dilatation with balloon and oral replacement of potassium salt. All patients displayed urinary continence. No symptomatic renal infection or hypercholoraemic acidosis occurred. Conclusion Modified ureterosigmoidostomy is a safe procedure of urinary diversion and provides a big volume, low intravesical pressure pouch. The patients are free from the troublesome urine bag, intermittert catheterization, and upper urinary tracts are protected effectively. The quality of life is satisfied.