Jordan curves and funnel sections

We study the case when solution of an ODE at a given initial condition fail to be unique and investigate what are the possible time-1 sections of the `solution funnel'. Along the way we give construction of a natural complete metric on the space of Jordan curves and prove that the generic Jordan curve is nowhere pierceable by arcs of finite length.Comment: 22 pages, 16 figure

On a model of forced axisymmetric flows

In this work, we consider a model of forced axisymmetric flows which is derived from the inviscid Boussinesq equations. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. We show that these flows give rise to an unusual Monge-Ampere equations for which we prove the existence and the uniqueness of a variational solution. We take advantage of these Monge-Ampere equations and construct a solution to the model

On periodic solutions in the Whitney's inverted pendulum problem

In the book `What is Mathematics?' Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. The first formal proof based on the idea of Courant and Robbins was published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the movement of the carriage on the line is periodic. In the present paper we slightly improve the Polekhin's theorem by lowering the regularity class of the motion and we prove a theorem on the existence of a periodic solution if the carriage moves periodically on the plane

Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity

Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences diffusion at a rate proportional to 1/ν, which is much faster than what would be produced by the static fluid if its viscosity is 0<ν≪1. This phenomenon is analyzed rigorously using the linear PDE governing the evolution of the solute. It is shown that the solution can be split into two pieces, an approximate solution and a remainder term. The approximate solution is governed by an infinite-dimensional system of ODEs that possesses a finite-dimensional center manifold, on which the dynamics correspond to diffusion at a rate proportional to 1/ν. The remainder term is shown to decay at a rate that is much faster than the leading order behavior of the approximate solution. This is proven using a spectral decomposition in Fourier space and a hypocoercive estimate to control the intermediate Fourier modes.https://arxiv.org/abs/1804.06916First author draf

Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity

Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences diffusion at a rate proportional to $1/\nu$, which is much faster than what would be produced by the static fluid if its viscosity is $0 < \nu \ll 1$. This phenomenon is analyzed rigorously using the linear PDE governing the evolution of the solute. It is shown that the solution can be split into two pieces, an approximate solution and a remainder term. The approximate solution is governed by an infinite-dimensional system of ODEs that possesses a finite-dimensional center manifold, on which the dynamics correspond to diffusion at a rate proportional to $1/\nu$. The remainder term is shown to decay at a rate that is much faster than the leading order behavior of the approximate solution. This is proven using a spectral decomposition in Fourier space and a hypocoercive estimate to control the intermediate Fourier modes.Comment: 37 pages, 0 figure

The place of Zeno’s paradox

This paper begins by examining the recent history of interpretations of one of Zeno’s paradoxes of motion, the paradox of dichotomy. It then returns to the record of antiquity to ask how Aristotle ‘solved’ the paradox and what decisions about place and motion were assumed in that solution. After appealing to Heidegger’s readings of the Aristotelian text, the paper then proceeds to offer an entirely original interpretation of Zeno’s paradox of dichotomy, which has important implications for a contemporary understanding of motion and place (rather than space). Instead, the paradox is read as a provocation to ‘see’ something which Zeno, it would appear, believed was ‘missing’, or had been forgotten and had disappeared, and to review all over again what Parmenides might have meant in his claim that being is one, singular, and indivisible

A comparative analysis of the value of information in a continuous time market model with partial information: the cases of log-utility and CRRA

We study the question what value an agent in a generalized Black-Scholes model with partial information attributes to the complementary information. To do this, we study the utility maximization problems from terminal wealth for the two cases partial information and full information. We assume that the drift term of the risky asset is a dynamic process of general linear type and that the two levels of observation correspond to whether this drift term is observable or not. Applying methods from stochastic filtering theory we derive an analytical tractable formula for the value of information in the case of logarithmic utility. For the case of constant relative risk aversion (CRRA) we derive a semianalytical formula, which uses as an input the numerical solution of a system of ODEs. For both cases we present a comparative analysis