Can points of bounded orbits surround points of unbounded orbits ?

We show a somewhat surprising result: if $E$ is a disk in the plane $\mathbb R^2$, then there is a homeomorphism $h:\mathbb R^2\rightarrow\mathbb R^2$ such that, for every $x\in\partial E$, the orbit $O(x, h)$ is bounded, but for every $y\in {\rm Int}(E)$, the orbit $O(y, h)$ is doubly divergent. To prove this, we define a class of homeomorphisms on the square $[-1, 1]^2$, called normally rising homeomorphisms, and show that a normally rising homeomorphism can have very complex $\omega$-limit sets and $\alpha$-limt sets, though the homeomorphism itself looks very simple.Comment: 18 pages. Comments are welcom

Some extensions of the Brouwer fixed point theorem

We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in $\mathbb R^n$ to $\mathbb R^n$ with $n\geq 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset\partial X$, $f(Y)\subset X$ and $\partial X-Y$ is an $(f, X)$-blockading set. For $n\geq 2$, let $D$ be an $n$-ball in $X$ and $Y$ be an $(n-1)$-ball in $\partial X$. Relying on the result just mentioned, we show the existence of a fixed point of $f$, if $D$ and $Y$ are well placed and behave well under $f$, and ${\rm deg}(f_D)=-{\rm deg}(f_{\partial Y})$, where $f_D=f|D: D \rightarrow \mathbb{R}^n$ and $f_{\partial Y}=f|\partial Y: \partial Y \rightarrow \partial Y$. The degree ${\rm deg}(f_D)$ of $f_D$ is explicitly defined and some elementary properties of which are investigated. These results extend the Brouwer fixed point theorem

A new construction of counterexamples to the bounded orbit conjecture

The bounded orbit conjecture says that every homeomorphism on the plane with each of its orbits being bounded must have a fixed point. Brouwer's translation theorem asserts that the conjecture is true for orientation preserving homeomorphisms, but Boyles' counterexample shows that it is false for the orientation reversing case. In this paper, we give a more comprehensible construction of counterexamples to the conjecture. Roughly speaking, we construct an orientation reversing homeomorphisms $f$ on the square $J^2=[-1, 1]^2$ with $\omega(x, f)=\{(-1. 1), (1, 1)\}$ and $\alpha(x, f)=\{(-1. -1), (1, -1)\}$ for each $x\in (-1, 1)^2$. Then by a semi-conjugacy defined by pushing an appropriate part of $\partial J^2$ into $(-1, 1)^2$, $f$ induces a homeomorphism on the plane, which is a counterexample

Structures of R(f)−P(f)‾R(f)-\overline{P(f)} for graph maps ff

Let $G$ be a graph and $f: G\rightarrow G$ be a continuous map. We establish a structure theorem which describes the structures of the set $R(f)-\overline{P(f)}$, where $R(f)$ and $P(f)$ are the recurrent point set and the periodic point set of $f$ respectively. Roughly speaking, the set $R(f)-\overline{P(f)}$ is covered by finitely many pairwise disjoint $f$-invariant open sets $U_{1\,},\,\cdots,\,U_{n\,}$; each $U_i$ contains a unique minimal set $W_i$ which absorbs each point of $U_i$; each $W_i$ lies in finitely many pairwise disjoint circles each of which is contained in a connected closed set; all of these connected closed sets are contained in $U_i$ and permutated cyclically by $f$. As applications of the structure theorem, several known results are improved or reproved

PV-SSD: A Projection and Voxel-based Double Branch Single-Stage 3D Object Detector

LIDAR-based 3D object detection and classification is crucial for autonomous driving. However, inference in real-time from extremely sparse 3D data poses a formidable challenge. To address this issue, a common approach is to project point clouds onto a bird's-eye or perspective view, effectively converting them into an image-like data format. However, this excessive compression of point cloud data often leads to the loss of information. This paper proposes a 3D object detector based on voxel and projection double branch feature extraction (PV-SSD) to address the problem of information loss. We add voxel features input containing rich local semantic information, which is fully fused with the projected features in the feature extraction stage to reduce the local information loss caused by projection. A good performance is achieved compared to the previous work. In addition, this paper makes the following contributions: 1) a voxel feature extraction method with variable receptive fields is proposed; 2) a feature point sampling method by weight sampling is used to filter out the feature points that are more conducive to the detection task; 3) the MSSFA module is proposed based on the SSFA module. To verify the effectiveness of our method, we designed comparison experiments

Improvement of earthquake and tsunami early warning service capabilities in Japan

In recent years, in order to improve the intelligent level of earthquake prevention and disaster reduction, the Japan Meteorological Agency has continuously improved the functions of the earthquake phenomena observation system (EPOS) and the intelligent prediction level of tsunami, and has made efforts to improve the earthquake and tsunami early warning service capabilities. In December 2022, it began to release follow-up earthquake early warning information in Hokkaido and Sanriku regions, and in February 2023, it began to release earthquake early warning information of long-term ground motion levels. This article aims to provide reference for the research and construction of earthquake early warning systems in China

The coastal ocean response to the global warming acceleration and hiatus

本研究揭示了全球沿岸对目前全球气候变暖减缓背景下的不同的区域响应，有助于理解全球增暖减缓现象的潜在机理，并为近期发生在各个海岸地区的极端热和极端冷的事件提供了更好的理解，有助于灾害防治和生态保护的政策制定。【Astract】Coastlines are fundamental to humans for habitation, commerce, and natural resources. Many coastal ecosystem disasters, caused by extreme sea surface temperature (SST), were reported when the global climate shifted from global warming to global surface warming hiatus after 1998. The task of understanding the coastal SST variations within the global context is an urgent matter. Our study on the global coastal SST from 1982 to 2013 revealed a significant cooling trend in the low and mid latitudes (31.4% of the global coastlines) after 1998, while 17.9% of the global coastlines changed from a cooling trend to a warming trend concurrently. The trend reversals in the Northern Pacific and Atlantic coincided with the phase shift of Pacific Decadal Oscillation and North Atlantic Oscillation, respectively. These coastal SST changes are larger than the changes of the global mean and open ocean, resulting in a fast increase of extremely hot/cold days, and thus extremely hot/cold events. Meanwhile, a continuous increase of SST was detected for a considerable portion of coastlines (46.7%) with a strengthened warming along the coastlines in the high northern latitudes. This suggests the warming still continued and strengthened in some regions after 1998, but with a weaker pattern in the low and mid latitudes.该研究工作获得了中国国家自然基金委、福建省和美国NOAA-Sea Grant 的资助

Copper Active Sites in Metal–Organic Frameworks Advance CO2 Adsorption and Photocatalytic Conversion

The photocatalytic reduction of CO2 into high-value chemicals utilizing solar energy represents a sustainable approach to mitigating greenhouse gas emissions and advancing renewable chemical production. Recently, copper-based metal–organic frameworks (Cu-MOFs) have been extensively researched for their potential in photocatalytic CO2 reduction, due to their high affinity for capturing CO2, the presence of unsaturated Cu sites, and their advantageous photochemical properties. In this review, we first provide an overview of Cu active sites in the secondary building units (SBUs) of MOFs, focusing on their selective adsorption of CO2 gas and analyzing the mechanisms of the multi-electron transfer processes involved in Cu-based photocatalytic reduction of CO2. Ultimately, this article outlines the existing obstacles and suggests potential avenues for future research