Tootfinder

Sources: Palmer Luckey is preparing to launch Erebor, a US bank backed by Joe Lonsdale and other tech investors to serve startups, including crypto businesses (Tabby Kinder/Financial Times)
https://www.ft.com/content/8c903f2e-42a6-496b-b098-ca733f340ffc

Peter Thiel joins tech billionaires backing new lender Erebor to rival Silicon Valley Bank
Start-up named after the dragon’s mountain in ‘Lord of the Rings’

Noch in Bayern. Muss ich ausnutzen.
#gmiatlichRadln #rausgeschafft

This also points at the likely answer to the headscratcher of where some ancient people of the region found the water they needed. https://wandering.shop/@adapalmer/114773151081694915

Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains
Lu Chen, Rou Jiang, Guozhen Lu, Maochun Zhu
https://arxiv.org/abs/2506.23076

Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains
In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:\[ S_Ω(λ,p)=\sup_{u\in H_{0}^{1}(Ω),\Vert\nabla u\Vert _{L^{2}\left( Ω\right) }\leq 1}\int_Ω\left( e^{4πu^{2}}-λ|u|^{p}\right) dx, \] where $1\leq p<\infty$ and $Ω$ is a bounded domain in $\mathbb{R}^2$. Our results demonstrate that there exists a threshold $λ^{\ast}(p)>0$ such that $S_Ω(λ,p)$ is attainable if $λ<λ^{\ast}(p)$, but unattainable if $λ>λ^{\ast}(p)$ when $p\in[1,2]$. For $p>…

Approximate Light Spanners in Planar Graphs
Hung Le, Shay Solomon, Cuong Than, Csaba D. T\'oth, Tianyi Zhang
https://arxiv.org/abs/2505.24825 https://

Approximate Light Spanners in Planar Graphs
In their seminal paper, Althöfer et al. (DCG 1993) introduced the {\em greedy spanner} and showed that, for any weighted planar graph $G$, the weight of the greedy $(1+ε)$-spanner is at most $(1+\frac{2}ε) \cdot w(MST(G))$, where $w(MST(G))$ is the weight of a minimum spanning tree $MST(G)$ of $G$. This bound is optimal in an {\em existential sense}: there exist planar graphs $G$ for which any $(1+ε)$-spanner has a weight of at least $(1+\frac{2}ε) \cdot w(MST(G))$. However, as an {\em a…

This https://arxiv.org/abs/2505.12311 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

Scene-Adaptive Motion Planning with Explicit Mixture of Experts and Interaction-Oriented Optimization
Despite over a decade of development, autonomous driving trajectory planning in complex urban environments continues to encounter significant challenges. These challenges include the difficulty in accommodating the multi-modal nature of trajectories, the limitations of single expert in managing diverse scenarios, and insufficient consideration of environmental interactions. To address these issues, this paper introduces the EMoE-Planner, which incorporates three innovative approaches. Firstly, …

On the maximum number of edges of outer k-planar graphs
Maximilian Pfister
https://arxiv.org/abs/2505.24490 https://arxiv.org/pdf/250…

On the maximum number of edges of outer k-planar graphs
We study the maximum number of straight-line segments connecting $n$ points in convex position in the plane, so that each segment intersects at most $k$ others. This question can also be framed as the maximum number of edges of an outer $k$-planar graph on $n$ vertices. We outline several approaches to tackle the problem with the best approach yielding an upper bound of $(\sqrt{2}+\varepsilon)\sqrt{k}n$ edges (with $\varepsilon \rightarrow 0$ for sufficiently large $k$). We further investigate …

"During his time at IBM, he wrote some of the first artificial intelligence software—his "geometry theorem machine" was the first advanced AI program, and the third AI program ever. It is a logical AI system that can prove theorems in planar geometry about parallel lines, congruence, and equality and inequality of segments and angles. Like Logic Theorist, it uses heuristics."

Planar Multiway Cut with Terminals on Few Faces
Sukanya Pandey, Erik Jan van Leeuwen
https://arxiv.org/abs/2506.23399 https://arxiv.o…

Planar Multiway Cut with Terminals on Few Faces
We consider the \textsc{Edge Multiway Cut} problem on planar graphs. It is known that this can be solved in $n^{O(\sqrt{t})}$ time [Klein, Marx, ICALP 2012] and not in $n^{o(\sqrt{t})}$ time under the Exponential Time Hypothesis [Marx, ICALP 2012], where $t$ is the number of terminals. A stronger parameter is the number $k$ of faces of the planar graph that jointly cover all terminals. For the related {\sc Steiner Tree} problem, an $n^{O(\sqrt{k})}$ time algorithm was recently shown [Kisfaludi-…

RealDrive: Retrieval-Augmented Driving with Diffusion Models
Wenhao Ding, Sushant Veer, Yuxiao Chen, Yulong Cao, Chaowei Xiao, Marco Pavone
https://arxiv.org/abs/2505.24808

RealDrive: Retrieval-Augmented Driving with Diffusion Models
Learning-based planners generate natural human-like driving behaviors by learning to reason about nuanced interactions from data, overcoming the rigid behaviors that arise from rule-based planners. Nonetheless, data-driven approaches often struggle with rare, safety-critical scenarios and offer limited controllability over the generated trajectories. To address these challenges, we propose RealDrive, a Retrieval-Augmented Generation (RAG) framework that initializes a diffusion-based planning po…