arXiv subjects

Marijan Marković

Publications and source records attributed to Marijan Marković.

2 recordsLinked to original sources

arXiv2021

On synchronization in Kuramoto models on spheres

We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto models always satisfies the same ODE, regardless of the dimension. The derivation of equations for real order parameters in Kuramoto models on spheres is based on recently unveiled connections of these models with geometries of unit balls. Simulations of the system with several hundreds of oscillators yield perfect fits with the theoretical predictions, that are obtained by solving equations for the order parameter.

nlin.AO↗

arXiv2020

Best constants in inequalities involving analytic and co-analytic projections and Riesz theorem for various function spaces

\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$. Interestingly, for this range of $s$ there holds the appropriate vector-valued inequality with the same constant. Also, we obtain the right asymptotic of the constants for large $s$. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. As a consequence of inequalities we have proved in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. In particular, slightly general version of Stout's theorem for Lumer Hardy spaces is obtained by a new approach. \end{abstract}

math.FA↗