Related papers: Tube Volumes via Functional Equations
The notion of sprays introduced by Lapidus and his co-workers has proved useful in the context of fractal tube formulas. In the present note, we propose a more general concept of sprays, where we allow several generators to make them more…
In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong…
Tube formulas (by which we mean an explicit formula for the volume of an $\epsilon$-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of…
In this work, we provide a treatment of scaling functional equations in a general setting involving fractals arising from sufficiently nice self-similar systems in order to analyze the tube functions, tube zeta functions, and complex…
In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a…
Numerical tests of volume formulae are presented to compute efficiently the volume enclosed between flux surfaces for integrable 3D vector fields with various degrees of symmetry. In the process, a new case is proposed and tested.
Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of…
We show how to compute vacuum expectation values from derivative expansions of the vacuum wave functional. Such expansions appear to be valid only for slowly varying fields, but by exploiting analyticity in a complex scale parameter we can…
We consider a porous mediaum flow in which the gas is initially distributed in the exterior of an empty region (a hole) and study the final stage of the hole-filling process. Hole-filling is asymptotically described by a self-similar…
We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.
We use the self-similar tilings constructed by the second author in "Canonical self-affine tilings by iterated function systems" to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta…
We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced…
We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The…
Tube formulas refer to the study of volumes of $r$ neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a…
In a companion study \cite{patterson2020computing2D}, we present a numerical method for simulating 2D viscous flow through an open compliant closed channel, drive by pressure gradient. We consider the highly viscous regime, where fluid…
Analytic relations are derived for finite volume integrals over the radial distribution function of a fluid, so-called Kirkwood-Buff integrals. Closed form expressions are obtained for cubes and cuboids, the system shapes commonly employed…
The volume-of-tube formula was first introduced by Hotelling (1939), to solve significance of terms in nonlinear regression models. Since this pioneering paper, there has been significant work on extending the tube formula to more general…
We report on the numerical implementation of thin film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus…
A simple application of the semipositivity.
We present an efficient numerical scheme based on Monte Carlo integration to approximate statistical solutions of the incompressible Euler equations. The scheme is based on finite volume methods, which provide a more flexible framework than…