This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the tent map, are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.

https://www.jstor.org/stable/10.14321/realanalexch.37.1.0001

The Takagi Function: a Survey

The Takagi function (x) is a continuous non-dierentiable function introduced by Teiji Takagi in 1903. It has appeared in a surprising number of dierent mathematical contexts, including mathematical analysis, probability theory and number theory. This paper surveys properties of this function.

The Takagi function and its properties

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Grun for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

In this paper, the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered. The surface heat flux history of a heated conducting body will be identified. The approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data. Some numerical examples will also be presented.

An inverse heat conduction problem with a nonlinear source term

In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points. A.~M.~Amp\`ere even tried to give a theoretical justificati ...

/pdf/continuous-nowhere-differentiable-functions-2kanl3dr25.pdf

Continuous Nowhere Differentiable Functions

An inverse problem for a nonlinear diffusion equation

It is well known that the number v is transcendental; the proof is complicated. However, there are several elementary proofs that X and v 2 are irrational; see for example [1]. Using a simple trick, we will easily show that T4 is irrational, which of course implies that X and 'T2 are also irrational. THEOREM. 'T4 is irrational. Proof. Suppose v 4 is rational. Then there are positive integers r and s such that 'i4 = 4r/s. Let a = v (1 + i)/2, let p denote an integer of the form 1 + 4jrs(j = 1, 2,. .. ), and let 1p3 f (x4 -1P (1) IP = a4p-3sPj ( _ 1)! (ea(x-1) e-a(x-l) + ieia(x-l) ie-ia(x-l)) dx.

http://www.jstor.org/stable/pdfplus/2323598.pdf

On the Continuity of van der Waerden's Function in the Hölder Sense

The determination of an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured at the boundary will be considered. We also design a new method for identifying mesh points and seek a member of admissible coefficients which minimize a given error functional.

Abdullah Shidfar

Papers

An inverse problem for a nonlinear diffusion equation

On the Continuity of van der Waerden's Function in the Hölder Sense

An inverse problem for a nonlinear diffusion equation

Nonlinear parabolic problems

An inverse problem for a linear diffusion equation with non-linear boundary condition