arXiv subjects

Jiří Benedikt

Publications and source records attributed to Jiří Benedikt.

4 recordsLinked to original sources

arXiv2025

Revisiting Functional Derivatives in Multi-object Tracking

Probability generating functionals (PGFLs) are efficient and powerful tools for tracking independent objects in clutter. It was shown that PGFLs could be used for the elegant derivation of practical multi-object tracking algorithms, e.g., the probability hypothesis density (PHD) filter. However, derivations using PGFLs use the so-called functional derivatives whose definitions usually appear too complicated or heuristic, involving Dirac delta ``functions''. This paper begins by comparing different definitions of functional derivatives and exploring their relationships and implications for practical applications. It then proposes a rigorous definition of the functional derivative, utilizing straightforward yet precise mathematics for clarity. Key properties of the functional derivative are revealed and discussed.

eess.SY↗

arXiv2023

Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

We show that the parabolic equation $u_t + (-Δ)^s u = q(x) |u|^{α-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times (\mathbb{R}^N \setminus Ω)$, where $Ω$ is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided $α\in (0,1)$ and the nonnegative bounded weight function $q$ is separated from zero on an open subset of $Ω$. This fact contrasts with the (super)linear case $α\geq 1$ in which the only bounded finite energy solution is identically zero.

math.AP↗

arXiv2022

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Using symmetrization techniques, we show that, for every $N \geq 2$, any second eigenfunction of the fractional Laplacian in the $N$-dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.

math.AP↗

arXiv2022

On the Strong Comparison Principle for Degenerate Elliptic Problems with Convection

The weak and strong comparison principles, respectively, are investigated for quasi-linear elliptic boundary value problems with the $p$-Laplacian in one space dimension. We treat the degenerate case of $2 < p < \infty$ and allow also for the nontrivial convection velocity $b\colon [-1,1]\to \mathbb{R}$ in the underlying domain $(-1,1)$. We establish the weak comparison principle under a rather general, natural sufficient condition on the convection velocity, $b(x)$, and the reaction function, $φ(x,u)$. Furthermore, we establish also the strong comparison principle under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the strong comparison principle and discuss their applications to an interesting classical problem of fluid flow in porous medium, seepage flow of fluids in inclined bed. Our methods are based on a mixture of classical and new techniques.

math.AP↗