Tootfinder

Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on $\mathbb{R}^N$: I. Global Solvability and Boundedness
Zulaihat Hassan, Wenxian Shen, Yuming Paul Zhang
https://arxiv.org/abs/2509.05494

Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on $\mathbb{R}^N$: I. Global Solvability and Boundedness
This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption: \begin{equation*} \begin{cases} u_t = m\nabla\cdot \left((\eps+u)^{m-1}\nabla u\right) - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{ in } (0,\infty)\times\mathbb{R}^N, \\ v_t = Δv - uv, & \text{ in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where $…

Arcs with increasing chords in $\mathbf{R}^d$
Adrian Dumitrescu, Zsolt L\'angi
https://arxiv.org/abs/2509.01580 https://arxiv.org/pdf/2509.01580…

Arcs with increasing chords in $\mathbf{R}^d$
A curve $γ$ that connects $s$ and $t$ has the increasing chord property if $|bc| \leq |ad|$ whenever $a,b,c,d$ lie in that order on $γ$. For planar curves, the length of such a curve is known to be at most $2π/3 \cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following: (I) The length of any $s-t$ curve with increasing chords in $\mathbf{R}^d$ is at most $2 \cdot \left( e/2 \cdot (d+4) \right)^{d-…