The object of this paper is the uniqueness for a d-dimensional Fokker-Planck type equation with inhomogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so-called Barenblatt’s solution of the fast diffusion equation which is the partial differential equation ∂tu = ∂ 2 xxu m with m ∈]0, 1[. Together...

We consider the cosmological models for the higher dimensional spacetime which includes the curvatures of our space as well as the curvatures of the internal space. We find that the condition for the integrability of the cosmological equations is that the total space-time dimensions are D = 10 or D = 11 which is exactly the conditions for superstrings or M-theory. We obtain analytic solutions w...

We relate a deterministic Kalman filter on semi-infinite interval to linear-quadratic tracking control model with unfixed initial condition.

For the sequence defined by a(n) = a(n−1)+gcd(n, a(n−1)) with a(1) = 7 we prove that a(n) − a(n − 1) takes on only 1s and primes, making this recurrence a rare “naturally occurring” generator of primes. Toward a generalization of this result to an arbitrary initial condition, we also study the limiting behavior of a(n)/n and a transience property of the evolution.

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.

Despite its central role in modern cosmology, doubts are often expressed as to whether cos-mological innation is really a falsiiable theory. We distinguish two facets of innation, one as a theory of initial conditions for the hot big bang and the other as a model for the origin of structure in the Universe. We argue that the latter can readily be excluded by observations, and that there are als...

We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearisation. The main difference with respect to the previous studies is that the linearisable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics...

Here we prove the necessary analytic results to construct a Morse theory for the YangMills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions (A, φ) converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the HarderNarasimhan-Seshadri...

This paper proposes a new technique to solve n-th order linear uncertain but bounded (interval) differential equations with interval initial conditions using the interval midpoint. First, the interval differential equation is solved in terms of the interval midpoint. This solution is then used to find the solution of the original interval differential equation. The method is discussed by consid...

Despite its central role in modern cosmology, doubts are often expressed as to whether cosmological inflation is really a falsifiable theory. We distinguish two facets of inflation, one as a theory of initial conditions for the hot big bang and the other as a model for the origin of structure in the Universe. We argue that the latter can readily be excluded by observations, and that there are a...