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In the present paper, we consider the following fractional double phase problem with nonlocal reaction:
\begin{align*}\qquad \quad \left\{\begin{array}{lll}(- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N,\end{array}\right\}\end{align*}
where ɛ is apositive parameter, 
$0 \lt s \lt 1$, 
$2 \leqslant p \lt q \lt \min\{2p, N / s\}$, 
$0 \lt \mu \lt sp$, 
$(- \Delta)_t^s$ 
$(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term 
$f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential 
$V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.
We design a scheme for laser-inertial odometry and mapping with bundle adjustment (BA-LIOM), which can greatly mitigate the problem of undesired ground warping due to sparsity of laser scans and significantly reduce odometry drift. Specifically, an Inertial measurement unit (IMU)-assisted adaptive voxel map initialization algorithm is proposed and elaborately integrated with the existing framework LIO-SAM, allowing for accurate registration in the beginning of the localization and mapping process. In addition, to accommodate to fast-moving and structure-less scenarios, we design a tightly coupled odometry, which jointly optimizes both the IMU preintegration constraints and scan matching with adaptive voxel maps. The voxels (edge and plane, respectively) are updated with BA optimization. And then the accurate mapping result is obtained by performing local BA. The proposed BA-LIOM is thoroughly assessed using datasets collected from multiple platforms over a variety of environments. Experimental results show the superiority of BA-LIOM over the state-of-the-art methods in robustness and precision, especially for large-scale scenarios. BA-LIOM improves the accuracy of localization by 
$61\%$ and 
$73\%$ on the buildings and lawn datasets, respectively, and has a 
$29\%$ accuracy improvement over LIO-SAM on the KITTI datasets. A supplementary video can be accessed at https://youtu.be/5l4ZFhTc2sw.
An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with
• 
$\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal 
$\kappa \geq \aleph _2$,
• 
$\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $.
Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that
• 
$\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with 
$\neg \mathsf {CH}$.
To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.
