Luka Korkut

Generalized Fresnel integrals and fractal properties of related spirals

Abstract We obtain a new asymptotic expansion of generalized Fresnel integrals x ( t ) = ∫ 0 t cos q ( s ) d s for large t, where q ( s ) ∼ s p when s → ∞ , and p > 1 . The terms of the expansion are defined via a simple iterative algorithm. Using this we show that the box dimension of the related q-clothoid, also called the generalized Euler or Cornu spiral, is equal to d = 2 p / ( 2 p - 1 ) . Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. We also find oscillatory dimension of Fresnel integrals by studying the corresponding chirps.

BOX DIMENSION AND MINKOWSKI CONTENT OF THE CLOTHOID

We prove that the box dimension of the standard clothoid is equal to d = 4/3. Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. Oscillatory dimensions of component functions of the clothoid are also equal to 4/3.

Fractal properties of Bessel functions

A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory

Control of essential infimum and supremum of solutions of quasilinear elliptic equations

(x,\dot{x})

Some Qualitative Properties of Solutions of Quasilinear Elliptic Equations and Applications

in

On a class of nonlinear variational inequalities: High concentration of the graph of weak solution via its fractional dimension and Minkowski content

\mathbb{R}^2

Wavy spirals and their fractal connection with chirps

of a solution

Geometrical and fractal properties of a class of systems with spiral trajectories in R^3

x=x(t)

Geometrical properties of a class of systems with spiral trajectories in R^3

, assuming that

Box dimension and Minkowski content of generalized Euler spirals

(x,\dot{x})