A Differentiation Theory for It\^o's Calculus

A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to It\^o's integral calculus? From It\^o's definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.Comment: 10 pages, 9/9 papers from my 2000-2006 collection. I proved these results and more earlier in 2004. I generalize this theory in upcoming articles. I also apply this theory in upcoming article

Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for H\"olderian almost complex structures

If $X$ is an almost complex manifold, with an almost complex structure $J$ of class \CC^\alpha, for some $\alpha >0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class \CC^{1,\alpha} in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. For complex manifolds it is the crucial point in Royden's proof of the equivalence of the two standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for structures that are smooth enough. In [I-R], it is shown that \CC^{1,\alpha} regularity of $J$ is enough. Here we show the following: Theorem. There exists an almost complex structure $J$ of class \CC^{1\over 2} on the unit bidisc \D^2\subset \C^2, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle.Comment: 5 page

Tungsten resonance integrals and Doppler coefficients First quarterly progress report, Jul. - Sep. 1965

Resonance integrals and Doppler coefficients of samples of natural tungsten, tungsten isotopes, and uranium oxide tungsten fue

Tungsten resonance integrals and Doppler coefficients Third quarterly report, Jan. - Mar. 1966

Reactivities, Doppler coefficients, and resonance integrals for tungsten isotope

Old, strong continental lithosphere with weak Archaean margin at - 1.8 Ga, Kaapvaal Craton, South Africa

Low elastic strength of ancient lithosphere based on flexural analyses has been interpreted to reflect elevated regional geothermal gradients in response to higher global heat production in the past. Here we present a flexural analysis of Archean/Palaeoproterozoic sediment cover along the western margin of the Archaean Kaapvaal craton based on seismic stratigraphy. Our results show that between ~1.93 and ~1.75 Ga, the Archaean margin of the craton had an effective elastic thickness of 7.5 to 10km compared to its present day value of 60 to 70km. Because the Kaapvaal craton had already stabilized by ~2.7 Ga and was underlain by 150 to 300km thick strong mantle lithosphere, it is unlikely that the relatively thin elastic thickness along this old margin reflects a change in secular cooling of the Earth. Instead, we interpret the low elastic strength to be a transient marginal tectonic effect similar to that recorded along modern continental margins

Quantum graphs as holonomic constraints

We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in R^2 with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex.Comment: 28 pages, 3 figure

Dynamic topography produced by lower crustal flow against rheological strength heterogeneities bordering the Tibetan Plateau

Dynamic stresses developed in the deep crust as a consequence of flow of weak lower crust may explain anomalously high topography and extensional structures localized along orogenic plateau margins. With lubrication equations commonly used to describe viscous flow in a thin-gap geometry, we model dynamic stresses associated with the obstruction of lower crustal channel flow due to rheological heterogeneity. Dynamic stresses depend on the mean velocity (Ū), viscosity (µ) and channel thickness (h), uniquely through the term µŪ/h^2. These stresses are then applied to the base of an elastic upper crust and the deflection of the elastic layer is computed to yield the predicted dynamic topography. We compare model calculations with observed topography of the eastern Tibetan Plateau margin where we interpret channel flow of the deep crust to be inhibited by the rigid Sichuan Basin. Model results suggest that as much 1500 m of dynamic topography across a region of several tens to a hundred kilometres wide may be produced for lower crustal material with a viscosity of 2 × 10^(18) Pa s flowing in a 15 km thick channel around a rigid cylindrical block at an average rate of 80 mm yr^(−1)

New Classes of Potentials for which the Radial Schrodinger Equation can be solved at Zero Energy

Given two spherically symmetric and short range potentials $V_0$ and V_1 for which the radial Schrodinger equation can be solved explicitely at zero energy, we show how to construct a new potential $V$ for which the radial equation can again be solved explicitely at zero energy. The new potential and its corresponding wave function are given explicitely in terms of V_0 and V_1, and their corresponding wave functions \phi_0 and \phi_1. V_0 must be such that it sustains no bound states (either repulsive, or attractive but weak). However, V_1 can sustain any (finite) number of bound states. The new potential V has the same number of bound states, by construction, but the corresponding (negative) energies are, of course, different. Once this is achieved, one can start then from V_0 and V, and construct a new potential \bar{V} for which the radial equation is again solvable explicitely. And the process can be repeated indefinitely. We exhibit first the construction, and the proof of its validity, for regular short range potentials, i.e. those for which rV_0(r) and rV_1(r) are L^1 at the origin. It is then seen that the construction extends automatically to potentials which are singular at r= 0. It can also be extended to V_0 long range (Coulomb, etc.). We give finally several explicit examples.Comment: 26 pages, 3 figure

Clean milk : essential requirements from production to consumption

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