We enrich the class of power-constructible functions, introduced in [CCRS23], to a class $\mathcal {C}^{\mathcal {M,F}}$ of algebras of functions which contains all complex powers of subanalytic functions and their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form, we deduce information on the asymptotics and on the loci of integrability of the functions of $\mathcal {C}^{\mathcal {M,F}}$. We furthermore identify a subclass $\mathcal {C}^{\mathbb {C},\mathcal {F}}$ of $\mathcal {C}^{\mathcal {M,F}}$, which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This class is also stable under parametric integration, under taking pointwise and $\text {L}^p$-limits and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in $\mathcal {C}^{\mathbb {C},\mathcal {F}}$.

We study a nonlinear Beltrami equation $f_\theta =\sigma \,|f_r|^m f_r$ in polar coordinates $(r,\theta ),$ which becomes the classical Cauchy–Riemann system under $m=0$ and $\sigma =ir.$ Using the isoperimetric technique, various lower estimates for $|f(z)|/|z|, f(0)=0,$ as $z\to 0,$ are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.

In analogy to classical spherical t-design points, we introduce the concept of t-design curves on the sphere. This means that the line integral along a t-design curve integrates polynomials of degree t exactly. For low degrees, we construct explicit examples. We also derive lower asymptotic bounds on the lengths of t-design curves. Our main results prove the existence of asymptotically optimal t-design curves in the Euclidean $2$-sphere and the existence of t-design curves in the d-sphere.

The aim of this paper is to analyze a class of random processes which models the motion of a particle on the real line with random velocity and subject to the action of friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and moment generating functions of the position reached by the particle at time t > 0. We are able to derive the explicit probability distributions in a few cases. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we consider randomly varying time. Essentially, we consider two different types of random time, namely Bessel and gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, exponential). For the random processes built by means of these compositions, we derive the probability distributions for a fixed number of Poisson events. Some remarks on possible extensions to random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.

The paper is dealing with determination of the integral ${{\int }_{\gamma }}\,f$ along the fractal arc $\gamma $ on the complex plane by terms of polynomial approximations of the function $f$. We obtain inequalities for polynomials and conditions of integrability for functions from the Hölder, Besov and Slobodetskii spaces.