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

We start
recalling with critical eyes the mathematical methods used in gauge theory and
prove that they are not coherent with continuum mechanics, in particular the
analytical mechanics of rigid bodies (despite using the same group theoretical
methods) and the well known couplings existing between elasticity and
electromagnetism (piezzo electricity, photo elasticity, streaming
birefringence). The purpose of this paper is to avoid such contradictions by
using new mathematical methods coming from the formal theory of systems of
partial differential equations and Lie pseudo groups. These results finally
allow unifying the previous independent tentatives done by the brothers E. and
F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by
using respectively the group of rigid motions of space or the conformal group
of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra
and algebraic analysis. We insist on the fact that these results could not have
been obtained before 1975 as the corresponding tools were not known before.

Abstract:
In this article, we start by a review of the circle group？ [1] and its topology induced [1] by the quotient metric, which we later use to define a topological structure on the unit circle . Using points on？ under the complex exponential map, we can construct orthogonal projection operators. We will show that under this construction, we arrive at a topological group, denoted？ of projection matrices. Together with the induced topology, it will be demonstrated that？ is Hausdorff and Second Countable forming a topological manifold. Moreover, I will use an example of a group action on？ to generate subgroups of？.

Abstract:
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ？by using the rotation group [3] [4]. It will be proved that the group acts on elements of ？in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ？in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.

Abstract:
A randomized, parallel groups, controlled trial.Participants: Eight hundred low-income smokers in Switzerland (the less affluent third of the population, based on fiscal taxation).Intervention: A smoking cessation program including: (a) financial incentives given during 6？months; and (b) Internet-based counseling. Financial rewards will be offered for biochemically verified smoking abstinence after 1, 2, and 3？weeks and 1, 3, and 6？months, for a maximum of 1,500 CHF (1,250 EUR, 1,500 USD) for those abstinent at all time-points. All participants, including controls, will receive Internet-based, individually-tailored, smoking cessation counseling and self-help booklets, but there will be no in-person or telephone counseling, and participants will not receive medications. The control group will not receive financial incentives.Objective: To increase smoking cessation rates.Outcome: Smoking abstinence after 6 and 18？months, not contradicted by biochemical tests. We will assess relapse after the end of the intervention, to test whether 6-month effects translate into sustained abstinence 12？months after the incentives are withdrawn.Randomization: Will be done using sealed envelopes drawn by participants.Blinding: Is not possible in this context.Smoking prevention policies and interventions have been least effective in the least educated, low-income groups. Combining financial incentives and Internet-based counseling is an innovative approach that, if proven acceptable and effective, could be later implemented on a large scale at a reasonable cost, decrease health disparities, and save many lives.Current Controlled Trials ISRCTN04019434.

Abstract:
Agarose gel electrophoresis is a well established technique routinely used in clinical laboratories for screening protein abnormalities in various biological fluids (serum, urine, CSF). It is based on the principles of zone electrophoresis. Electrophoretograms are evaluated visually for the presence of quantitatively or qualitatively abnormal protein bands. The technique is used for electrophoresis of serum, urine, CSF proteins, enzymes (ALP, LDH and CK), lipoproteins and hemoglobin. Serum protein electrophoresis (SPE) is a very commonly used analytical method in clinical chemistry. Changes in the relative concentration of fractions allow easy recognition of pathological disorders associated with nephrotic syndrome, inflammatory reaction and hepatic diseases. SPE is a screening test for detecting the M component (MC). Immunofixation (IFE) with use of specific antisera allows detection of the type of MC. SPE is also a method for the quantification of MC and monitoring of disease that is essential for clinical evaluation and follow-up of patients with plasma cell disorders.

Abstract:
Leukocytes are believed to access sites of inflammation via cell adhesion molecule (CAM) mediated adhesion, activation and transmigration through the vascular endothelium. However, little is known about the mechanisms by which immune effector cells gain access to transplanted organ. The objective of this work was to evaluate a simple experimental model to characterize the migration of T lymphocytes to vascular tissue following transplantation. Brown Norway aortic allografts were transplanted orthotopically into Lewis recipients (n = 10). T cells were isolated from na ve or allo-exposed (activated) Lewis spleens using nylon wool columns. Cr51 labeled na ve T cells and In111 labeled activated T cells (15x106 of each) were injected into animals, 9 days after transplantation with or without anti- 4/ 2_monoclonal antibody (n = 5/group). All animals were then sacrificed on day 10 and evaluated by histology and counts. All allografts had significant infiltration of the adventitia by mononuclear cells at day 10 as compared to control aorta. There was an average of 220+33 Cr51 cpm and 224 + 30. In111 cpm as compared to control aorta (p<0.0001). Anti 4/ 2 mAb treatment resulted in significant reduction in labeled T cell infiltration with a 89% reduction in Cr51 cpm and 73% reduction in In111 cpm both approaching control levels. We have demonstrated that T lymphocytes infiltrate vascular tissue early after transplantation and appears to be mediated via 4/ 2 integrin mechanism. This represents direct evidence of the role of CAM in trafficking of T cells in transplanted allografts and the characterization of a model to evaluate this trafficking.

Abstract:
This article shows that the business cycle magnitude and the output gap strongly varied in the Lebanon between 1970 and 2008, the most important variations taking place from 1975 to 1990 corresponding to the period of the civil war. Nevertheless, a decrease in the discrepancy between these two indicators around the long-run trend can be observed since the beginning of 1990s. These reductions of the cyclic amplitude and output gap do not result from the end of the war in 1990 but rather from the general government final consumption expenditure and economic dependence rate. These two variables explain a mismatch between the observed GDP and the potential GP and consequently the occurrence of economic stability.

Abstract:
It has recently been shown that specific non-perturbative effects may lead to an explosive decay of flat direction condensates in supersymmetric theories. We confirm explicitly the efficiency of this process with lattice simulations: after few rotations of the condensates in their complex plane, most of their energy is quickly converted into inhomogeneous fluctuations. We then point out that this generates a gravitational wave background which depends on the inflaton sector and falls in the Hz-kHz frequency range today. We compute the resulting spectrum and study how it depends on the parameters. We show that these gravity waves can be observable by upcoming experiments like Advanced LIGO and depend crucially on (i) the initial VEV of flat directions when they start to oscillate, (ii) their soft SUSY-breaking mass and (iii) the reheat temperature of the universe. This signal could open a new observational window on inflation and low-energy supersymmetry.