diff --git "a/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt" "b/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/19E0T4oBgHgl3EQfugE5/content/tmp_files/load_file.txt" @@ -0,0 +1,1199 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf,len=1198 +page_content='Generic transversality of radially symmetric stationary solutions stable at infinity for parabolic gradient systems Emmanuel Risler January 9, 2023 This paper is devoted to the generic transversality of radially symmetric stationary solutions of nonlinear parabolic systems of the form ∂tw(x, t) = −∇V �w((x, t)) � + ∆xw(x, t) , where the space variable x is multidimensional and unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It is proved that, generically with respect to the potential V , radially symmetric stationary solutions that are stable at infinity (in other words, that approach a minimum point of V at infinity in space) are transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' as a consequence, the set of such solutions is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This result can be viewed as the extension to higher space dimensions of the generic elementarity of symmetric standing pulses, proved in a companion paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It justifies the generic character of the discreteness hypothesis concerning this set of stationary solutions, made in another companion paper devoted to the global behaviour of (time dependent) radially symmetric solutions stable at infinity for such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 35K57, 37C20, 37C29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Key words and phrases: parabolic gradient systems, radially symmetric stationary solutions, generic transversality, Morse–Smale theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02605v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='AP] 6 Jan 2023 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 An insight into the main result .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Radially symmetric stationary solutions stable at infinity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Differential systems governing radially symmetric stationary solutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Transversality of radially symmetric stationary solutions stable at infinity 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 The space of potentials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 Main result .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 Key differences with the generic transversality of standing pulses in space dimension one .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 9 2 Preliminary properties 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Transversality of homogeneous radially symmetric stationary solutions stable at infinity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Additional properties close to the origin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Additional properties close to infinity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 14 3 Tools for genericity 15 4 Generic transversality among potentials that are quadratic past a given radius 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Notation and statement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Reduction to a local statement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Proof of the local statement (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Setting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Equivalent characterizations of transversality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 23 5 Proof of the main results 24 2 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 An insight into the main result The purpose of this paper is to prove the generic transversality of radially symmetric stationary solutions stable at infinity for gradient systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) ∂tw(x, t) = −∇V �w((x, t)) � + ∆xw(x, t) , where time variable t is real, space variable x lies in the spatial domain Rdsp with dsp an integer not smaller than 2, the state function (x, t) �→ w(x, t) takes its values in Rdst with dst a positive integer, and the nonlinearity is the gradient of a scalar potential function V : Rdst → R, which is assumed to be regular (of class at least C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' An insight into the main result of this paper (Theorem 1 on page 8) is provided by the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For a generic potential V , the following conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' every radially symmetric stationary solution stable at infinity of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) is robust with respect to small perturbations of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the set of all such solutions is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The discreteness stated in conclusion 2 of this corollary is a required assumption for the main result of [4], which describes the global behaviour of radially symmetric (time dependent) solutions stable at infinity for the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 provides a rigorous proof that this assumption holds generically with respect to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This paper can be viewed as a supplement of the article [1], which is devoted to the generic transversality of bistable travelling fronts and standing pulses stable at infinity for parabolic systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) in (unbounded) space dimension one, and which provides a rigorous proof of the genericity of similar assumptions made in [2, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The ideas, the nature of the results, and the scheme of the proof are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Radially symmetric stationary solutions stable at infinity A function u : [0, +∞) → Rdst, r �→ u(r) defines a radially symmetric stationary solution of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) if and only if it satisfies, on (0, +∞), the (non-autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) ¨u(r) = −dsp − 1 r ˙u(r) + ∇V �u(r) � , where ˙u and ¨u stand for the first and second derivatives of r �→ u(r), together with the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) ˙u(r) → 0 as r → 0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Observe that, in this case, u(·) is actually the restriction to [0, +∞) of an even function in C3(R, Rd st) which is a solution (on R) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) (the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) ensures 3 that equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) still makes sense and holds at r equals 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In other words, provided that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) holds, it is equivalent to assume that system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) holds on (0, +∞) or on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By abuse of language, the terminology radially symmetric stationary solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) will refer, all along the paper, to functions u : [0, +∞) → Rdst satisfying these conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) (even if, formally, it is rather the function Rd sp → Rd st, x �→ u �|x| � that fits with this terminology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us denote by Σmin(V ) the set of nondegenerate (local or global) minimum points of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' with symbols, Σmin(V ) = �u ∈ Rdst : ∇V (u) = 0 and D2V (u) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Throughout all the paper, the words minimum point will be used to denote a local or global minimum point of a (potential) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A (global) solution (0, +∞) → Rdst, r �→ u(r), of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) (in particular a radially symmetric stationary solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1)) is said to be stable at infinity if u(r) approaches a point of Σmin(V ) as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If this point of Σmin(V ) is denoted by u∞, then the solution is said to be stable close to u∞ at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u∞ in Σmin(V ), let SV, u∞ denote the set of the radially symmetric stationary solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) that are stable close to u∞ at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With symbols, SV, u∞ = �u : [0, +∞) → Rdst : u satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) and u(r) −−−−→ r→+∞ u∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let S0 V, u∞ = � u(0) : u ∈ SV, u∞ � , and let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) SV = � u∞∈Σmin(V ) SV, u∞ and S0 V = � u∞∈Σmin(V ) S0 V, u∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following statement is an equivalent (simpler) formulation of conclusion 2 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For a generic potential V , the subset S0 V of Rdst is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Differential systems governing radially symmetric stationary solutions The second-order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) is equivalent to the (non-autonomous) 2dst- dimensional first order differential differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) � � � � � ˙u = v ˙v = −dsp − 1 r v + ∇V (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Introducing the auxiliary variables τ and c defined as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) τ = log(r) and c = 1 r , 4 the previous 2dst-dimensional differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) is equivalent to each of the following two 2dst + 1-dimensional autonomous differential systems: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) � � � � � � � uτ = rv vτ = −(dsp − 1)v + r∇V (u) rτ = r , and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) � � � � � � � ur = v vr = −(dsp − 1)cv + ∇V (u) cr = −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Integrating the third equations of systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) yields r = r0eτ−τ0 and 1 c − 1 c0 = r − r0 , and the parameters τ0 and c0 (which determine in each case the origin of “time”) do not matter in principle, since those systems are autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' However, if the “initial conditions” r0 and c0 are positive (which is true for the solutions that describe radially symmetric stationary solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1)), it is natural to choose, in each case, the origins of time according to equalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6), that is : τ0 = ln(r0) and c0 = 1 r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Properties close to origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) is relevant to provide an insight into the limit system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) as r goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The subspace R2dst × {0} (r equal to 0) is invariant by the flow of this system, and the system reduces on this invariant subspace to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) � uτ = 0 vτ = −(dsp − 1)v , see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u0 in Rdst, the point (u0, 0Rdst, 0) is an equilibrium of sys- tem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' let us denote by W u, 0 V (u0) the (one-dimensional) unstable manifold of this equilibrium, for this system, let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) W u, 0, + V (u0) = W u, 0 V (u0) ∩ �R2dst × (0, +∞) � , and let W u, 0, + V = � u0∈Rdst W u, 0, + V (u0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The subspace (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) Ssym = Rdst × {0Rdst} × {0} of R2dst+1 can be seen as the higher space dimension analogue of the symmetry (reversibil- ity) subspace Rdst × {0Rdst} of R2dst (which is relevant for symmetric standing pulses in space dimension 1, see [1] and subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the set W u, 0, + V can be seen as the unstable manifold of this subspace Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1: Dynamics of the (equivalent) differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) (for r nonnegative finite) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) (for c = 1/r nonnegative finite) in Rdst × Rdst × [0, +∞] (this domain is three-dimensional if dst is equal to 1, as on the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For the limit differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) in the subspace r = 0 (in green), the trajectories are vertical and the solutions converge towards the horizontal u-axis, defined as Ssym in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11), and which is the higher space dimensional analogue of the symmetry subspace for symmetric standing pulses in space dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The point u∞ is a local minimum point of V , so that the point (u∞, 0Rdst) is a hyperbolic equilibrium for the limit differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12) in the subspace c = 0 ⇐⇒ r = +∞ (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) are autonomous, but the quantity r (the quantity c) goes monotonously from 0 to +∞ (from +∞ to 0) for all the solutions in the subspace r > 0 ⇐⇒ c > 0, so that those solutions can be parametrized with r (with c) as time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The unstable manifold W u, 0, + V (u0) is one-dimensional and is a transverse intersection between the unstable set W u, 0, + V of the subspace {r = 0, v = 0Rdst} and the centre stable manifold W cs, ∞, + V (u∞) of the equilibrium (u∞, 0Rdst, c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To prove the generic transversality of this intersection is the main goal of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The dotted red curve is the projection onto the (u, r)-subspace of this intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The part of W cs, ∞, + V (u∞) which is displayed on the figure can also be seen as the local centre stable manifold W cs, ∞, + loc, V, ε1, c1(u∞) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) (with u∞ equal to the point u∞,1 introduced there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 6 wo u sym om L 0 (8m) loc, V, E1, C1 C1 L0= u 8 E1Properties close to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) is relevant to provide an insight into the limit system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The subspace R2dst × {0} of R2dst+1 (c equal to 0, or in other words r equal to +∞) is invariant by the flow of this system, and the system reduces on this invariant subspace to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12) � ur = v vr = ∇V (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u∞ in Σmin(V ), the point (u∞, 0Rdst, 0) is an equilibrium of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' let us consider its global centre-stable manifold in R2dst × (0, +∞), defined as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13) W cs, ∞, + V (u∞) = � (u0, v0, c0) ∈ R2dst × (0, +∞) : the solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) with initial condition (u0, v0, c0) at “time” r0 = 1/c0 is defined up to +∞ and goes to (u∞, 0, 0) as r goes to +∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set W cs, ∞, + V (u∞) is a dst + 1-dimensional submanifold of R2dst × (0, +∞) (see subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Radially symmetric stationary solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the involution ι : R2dst × (0, +∞) → R2dst × (0, +∞) , (u, v, r) �→ (u, v, 1/r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following lemma, proved in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1, formalizes the correspondence between the radially symmetric stationary solutions stable at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) and the manifolds defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let u∞ be a point of Σmin(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A (global) solution [0, +∞) → Rdst, r �→ u(r) of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) belongs to SV, u∞ if and only if its trajectory (in R2dst × (0, +∞)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='14) ��u(r), ˙u(r), r � : r ∈ (0, +∞) � belongs to the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) W u, 0, + V ∩ ι−1�W cs, ∞, + V (u∞) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Transversality of radially symmetric stationary solutions stable at infinity Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let u∞ be a point of Σmin(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A radially symmetric stationary solution stable close to u∞ at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) (in other words, a function u of SV, u∞) is said to be transverse if the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) is transverse, in R2dst × (0, +∞), along the trajectory (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The natural analogue of radially symmetric stationary solutions stable at infinity when space dimension dsp is equal to 1 are symmetric standing pulses stable at infinity (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 of [1]), and the natural analogue for such pulses of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 above 7 is their elementarity, not their transversality (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' However, the transversality of a symmetric standing pulse (when the space dimension dsp equals 1) makes little sense in higher space dimension, because of the singularity at r equals 0 for the differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5), or because of the related fact that the subspace {r = 0} is invariant for the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For that reason, the adjective transverse (not elementary) is chosen to qualify the property considered in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 The space of potentials For the remaining of the paper, let us take and fix an integer k not smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the space Ck+1 b (Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' R) of functions Rd → R of class Ck+1 which are bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' as well as their derivatives of order not larger than k + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' equipped with the norm ∥W∥Ck+1 b = max α multi-index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' |α|≤k+1 ∥∂|α| uαW∥L∞(Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='R) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and let us embed the larger space Ck+1(Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' R) with the following topology: for V in this space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' a basis of neighbourhoods of V is given by the sets V + O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' where O is an open subset of Ck+1 b (Rdst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' R) embedded with the topology defined by ∥·∥Ck+1 b (which can be viewed as an extended metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For comments concerning the choice of this topology, see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 Main result The following generic transversality statement is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Theorem 1 (generic transversality of radially symmetric stationary solutions stable at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists a generic subset G of � Ck+1(Rdst, R), ∥·∥Ck+1 b � such that, for every potential function V in G, every radially symmetric stationary solution stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Theorem 1 can be viewed as the extension to higher space dimensions (for radially symmetric solutions) of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 of [1] (which is concerned with elementary standing pulses stable at infinity in space dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' A short comparison between these two results and their proofs is provided in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For more comments and a short historical review on transversality results in similar contexts, see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 of the same reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The core of the paper (section 4) is devoted to the proof of the conclusions of Theorem 1 among potentials which are quadratic past a certain radius (defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2)), as stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The extension to general potentials of Ck+1 b (Rdst, R) is carried out in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As in [1] (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8 of that reference), the same arguments could be called upon to prove that the following additional conclusions hold, generically with respect to the potential V : 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' for every minimum point of V , the smallest eigenvalue of D2V at this minimum point is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' every radially symmetric stationary solution stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) approaches its limit at infinity tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 Key differences with the generic transversality of standing pulses in space dimension one Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 lists the key differences between the proof of the generic elementarity of symmetric standing pulses carried out in [1], and the proof of the generic transversality of radially symmetric stationary solutions carried out in the present paper (implicitly, the other steps/features of the proofs are similar or identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The state dimension, which is simply denoted by d in [1], is here denoted by dst in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Some of the notation/rigour is lightened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Symmetric standing pulse Radially symmetric stationary solution Critical point at infinity critical point e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' E = (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0Rdst ) minimum point u∞ Symmetry subspace Ssym {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' v) ∈ R2dst : v = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' r) ∈ R2dst+1 : (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' r) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst Differential system governing the profiles autonomous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' conservative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' regular at Ssym non-autonomous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dissipative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' singular at reversibility subspace Direction of the flow E → Ssym Ssym → u∞ Invariant manifold at infinity W u(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst − m(e) W cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' +(u∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst + 1 Invariant manifold at symmetry subspace none W u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' dimension dst + 1 Transversality W u(E) ⋔ Ssym W cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' +(u∞) ⋔ W u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' + Transversality of spatially homogeneous solutions irrelevant Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Interval Ionce (values reached only once) “anywhere” close to Ssym M (departure set of Φ) parametrization of ∂W u loc, V (E) and time, dimension dst − m(e) Ssym and W cs, ∞, + loc (u∞) at r = N, dimension 2dst N (arrival set of Φ) R2dst R2dst × R2dst W (target manifold) Ssym diagonal of N dim(M) − codim(W) −m(e) 0 Condition to be fulfilled by perturbation W � DΦ(W) �� (0, ψ)� ̸= 0 � DΦu(W) �� (φ, ψ)� ̸= 0 Perturbation W, case 3 precluded W(u0) ̸= 0 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1: Formal comparison between the generic elementarity of symmetric standing pulses (space dimension 1) proved in [1], and the generic transversality of radially symmetric stationary solutions (higher space dimension dsp) proved in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Here are a few additional comments about these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 9 Concerning the critical point at infinity, u∞ is assumed (here) to be a minimum point, whereas (in [1]) the Morse index of e is any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Indeed, if the Morse index m(u∞) of u∞ was positive, then the dimension of the centre-stable manifold W cs, ∞, + V (u∞) would be equal to dst + m(u∞) + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' as a consequence, proving the transversality of the intersection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) in that case would require more stringent regularity assumptions on V (see hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1]) while nothing particularly useful could be derived from this transversality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' On the other hand, assuming that u∞ is a minimum point allows to view its local centre-stable manifold as a graph (u, c) �→ v (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4), which is slightly simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Concerning the interval Ionce providing values u reached “only once” by the profile (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3), the proof of the present paper takes advantage of the dissipation to find a convenient interval close to the “departure point” u0, as was done in [1] for travelling fronts (whereas, for standing pulse, the interval is to be found “anywhere”, thanks to the conservative nature of the differential system governing the profiles, see conclusion 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Concerning the function Φ to which Sard–Smale theorem is applied in the present paper, both manifolds W u, 0, + and W cs, ∞, +(u∞) depend on the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' However, the transversality of an intersection between these two manifolds can be seen as the transversality of the image of Φ with the (fixed) diagonal of R2dst × R2dst, for a function Φ combining the parametrization of these two manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This trick, which is the same as in [1] for travelling fronts, allows to apply Sard–Smale theorem to a function Φ with a fixed arrival space N containing a fixed target manifold W (in this case the diagonal of N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By contrast, for symmetric standing pulses in [1], since the subspace Ssym involved in the transverse intersection is fixed, the previous trick is unnecessary and the setting is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Finally, a technical difference occurs in “case 3” of the proof that the degrees of freedom provided by perturbing the potential allow to reach enough directions in the arrival state of Φ (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6, which is the core of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In [1], case 3 is shown to lead to a contradiction, not only for symmetric standing pulses, but also for asymmetric ones and for travelling fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Here, such a contradiction does not seem to occur (or at least is more difficult to prove), but this has no harmful consequence: a suitable perturbation of the potential can still be found in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2 Preliminary properties 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Let V denote a potential function in Ck+1(Rdst, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (0, +∞) → Rdst, r �→ u(r) denote a (global) solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2), assumed to be stable close to some point u∞ of Σmin(V ) at infinity (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 follows from the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The derivative ˙u(r) goes to 0 as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the Hamiltonian function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) HV : R2dst → R , (u, v) �→ v2 2 − V (u) , and, for every r in (0, +∞), let h(r) = HV �u(r), ˙u(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) that, for every r in (0, +∞), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) ˙h(r) = −dsp − 1 r ˙u(r)2 , thus the function h(·) decreases, and it follows from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) of the Hamiltonian that this function converges, as r goes to +∞, towards a finite limit h∞ which is not smaller than −V (u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us proceed by contradiction and assume that h∞ is larger than −V (u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then, it follows again from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) of the Hamiltonian that the quantity ˙u(r)2 con- verges towards the positive quantity 2 �h∞ + V (u∞) � as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As a consequence, it follows from equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) that h(r) goes to −∞ as r goes to +∞, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Transversality of homogeneous radially symmetric stationary solutions stable at infinity Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every potential function V in Ck+1(Rdst, R) and for every nonde- generate minimum point u∞ of V , the constant function [0, +∞) → Rdst , r �→ u∞ , which defines an (homogeneous) radially symmetric stationary solution stable at infinity for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) , is transverse (in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let V denote a function in Ck+1(Rdst, R) and u∞ denote a nondegenerate minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The function [0, +∞) → Rdst, r �→ u∞ is a (constant) solution of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5), and the linearization of this differential system around this solution reads (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) ¨u = −dsp − 1 r ˙u + D2V (u∞) · u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (0, +∞) → Rdst, r �→ u(r) denote a nonzero solution of this differential system, and, for every r in (0, +∞), let v(r) = ˙u(r) and U(r) = �u(r), v(r) � and q(r) = u(r)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 11 Then (omitting the dependency on r), ˙q = u · ˙u and ¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1 r ˙q + D2V (u∞) · (u, u) , so that d dr �rdsp−1 ˙q(r) � = rdsp−1 � ¨q + dsp − 1 r ˙q � = rdsp−1� ˙u2 + D2V (u∞) · (u, u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since r �→ u(r) was assumed to be nonzero, it follows that the quantity rdsp−1 ˙q(r) is strictly increasing on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To prove the intended conclusion, let us proceed by contradiction and assume that, for every r in (0, +∞), �u(r), v(r), r � belongs: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' to the tangent space T(u∞,0Rdst ,r)W u, 0, + V (u∞), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and to the tangent space T(u∞,0Rdst ,r) � ι−1�W cs, ∞, + V (u∞) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6), let us introduce the auxiliary variables τ (equal to log(r)) and c (equal to 1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With this notation, system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) � � � � � � � uτ = rv vτ = −(dsp − 1)v + rD2V (u∞) · u rτ = r , and to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) � � � � � � � ur = v vr = −(dsp − 1)cv + D2V (u∞) · u cr = −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Assumptions 1 and 2 above yield the following conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In view of the limit of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) as r goes to 0+, it follows from assumption 1 that there exists δu0 in Rdst such that �u(r), v(r) � goes to (δu0, 0Rdst) as r goes to 0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and in view of the limit of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) as c goes to 0+, it follows from assumption 2 that �u(r), v(r) � goes to (0Rdst, 0Rdst), at an exponential rate, as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from these two conclusions that the quantity rdsp−1 ˙q(r) goes to 0 as r goes to 0+ and as r goes to +∞, a contradiction with the fact (observed above) that this quantity is strictly increasing with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Additional properties close to the origin Let V denote a potential function in Ck+1(Rdst, R) and let u0 be a point in Rdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us recall (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) that the unstable manifold W u, 0 V (u0) of the equilibrium (u0, 0Rdst, 0) for the autonomous differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7)) is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As a consequence there exists a unique solution r �→ u(r) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) such that the image of the map r �→ �u(r), ˙u(r), r) lies in the intersection W u, 0, + V (u0) of this unstable manifold with the half-space where r is positive (this intersection was defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' or, in other words, such that �u(r), ˙u(r) � goes to (u0, 0) as r goes to 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This solution is defined on some (maximal) interval (0, rmax), where rmax is either a finite quantity or +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following lemma provides properties of this solution that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To ease its statement, let us assume that rmax is equal to +∞ (only this case will turn out to be relevant), and let us consider the continuous extension of u(·) to the interval [0, +∞) (and let us still denote by u(·) this continuous extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If u(·) is not identically equal to u0 (in other words, if u0 is not a critical point of V ), then there exists a positive quantity ronce such that, denoting by Ionce the interval [0, ronce), the following conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the function ˙u(·) does not vanish on Ionce, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' and, for every r∗ in Ionce and r in [0, +∞), u(r) = u(r∗) =⇒ r = r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The linearized system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) at the equilibrium (u0, 0Rdst, 0) reads: d dτ � � � δu δv δr � � � = � � � 0 0 0 0 −(dsp − 1) ∇V (u0) 0 0 1 � � � � � � δu δv δr � � � , thus the tangent space at (u0, 0Rdst, 0) to W u, 0 V (u0) (the unstable eigenspace of the matrix of this system) is spanned by the vector �0, ∇V (u0)/dsp, 1 �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' it follows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) ˙u(r) = r dsp ∇V (u0) �1 + or→0+(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus, if r0 is a sufficiently small positive quantity, then ˙u(·) does not vanish on (0, r0] (so that conclusion 1 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 holds provided that ronce is not larger than r0), and the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) [0, r0] → Rdst , r �→ u(r) is a C1-diffeomorphism onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For r in [0, +∞), let us denote �u(r), ˙u(r) � by U(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the decrease (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) of the Hamiltonian, there exists a quantity ronce in (0, r0) such that, for every r∗ in [0, ronce), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) HV �U(r0) � < −V �u(r∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 13 Take r∗ in [0, ronce] and r in [0, +∞), and let us assume that u(r) equals u(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If r was larger than r0 then it would follow from the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) of the Hamiltonian, its decrease (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2), and inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) that −V �u(r) � ≤ HV �U(r) � ≤ HV �U(r0) � < −V �u(r∗) � , a contradiction with the equality of u(r) and u(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus r is not larger than r0, and it follows from the one-to-one property of the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) that r must be equal to r∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' conclusion 2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 thus holds, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Additional properties close to infinity Let V1 denote a potential function in Ck+1(Rdst, R) and u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the implicit function theorem, there exists a (small) neighbourhood νrobust(V1, u1,∞) of Vquad-R and a Ck-function V �→ u∞(V ) defined on νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following proposi- tion is nothing but the local centre-stable manifold theorem applied to the equilibrium �u∞(V ), 0Rdst, 0 � of the (autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8), for V close to V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Addi- tional comments and references concerning local stable/centre/unstable manifolds are provided in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 (local centre-stable manifold at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exist a neighbourhood ν of V1 in Ck+1(Rdst, R), included in νrobust(V1, u1,∞), such that, if ε1 and c1 denote sufficiently small positive quantities, then, for every V in ν, there exists a Ck-map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) wcs, ∞ loc, V : BRdst(u1,∞, ε1) × [0, c1] → Rdst , (u, c) �→ wcs, ∞ loc, V (u, c) , such that, for every (u0, v0, c0) in BRdst(u1,∞, ε1) × Rdst × [0, c1], the following two statements are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' v = wcs, ∞ loc, V (u, c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' the solution r �→ �u(r), v(r), c(r) � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) with initial condi- tion (u0, v0, c0) at time r0 = 1/c0 is defined up to +∞, remains in BRdst(u1,∞, ε1)× Rdst × [0, c1] of all r larger than r0, and goes to �u∞(V ), 0Rdst, 0 � as r goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In particular, wcs, ∞ loc, V �u∞(V ), 0 � is equal to 0Rdst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In addition, the map BRdst(u1,∞, ε1) × [0, c1] × ν → Rdst , (u, c, V ) �→ wcs, ∞ loc, V (u, c) is of class Ck (with respect to u and c and V ), and, for every V in ν, the graph of the differential at �u∞(V ), 0) of the map (u, c) �→ wcs, ∞ loc, V (u, c) is equal to the centre-stable subspace of the linearization at �u∞(V ), 0Rdst, 0 � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 14 Let us denote by W cs, ∞, + loc, V, ε1, c1 �u∞(V ) � the graph of the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) (restricted to positive values of c), see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' with symbols, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) W cs, ∞, + loc, V, ε1, c1 �u∞(V ) � = ��u, wcs, ∞ loc, V (u, c), c � : (u, c) ∈ BRdst(u1,∞, ε1) × (0, c1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set defines a local centre-manifold (restricted to positive values of c) for the equilib- rium �u∞(V ), 0Rdst, 0 � of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Its uniqueness (for positive values of c) is ensured by the dynamics of the centre component c, which, according to the third equation of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8), decreases to 0 (see figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The global centre-stable manifold W cs, ∞, + V �u∞(V ) � already defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13) can be redefined as the points of R2dst ×(0, +∞) that eventually reach the local centre manifold W cs, ∞, + loc, V, ε1, c1 �u∞(V ) � when they are transported by the flow of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If the state dimension dst is equal to 1, then a calculation shows that wcs, ∞ loc, V (u, c) = − �u − u∞(V ) � �� V ′′�u∞(V ) � + dsp − 1 2 c + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' � , where “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' ” stands for higher order terms in u − u∞(V ) and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In particular the quantity ∂c∂uwcs, ∞ loc, V �u∞(V ), 0 � is equal to the (negative) quantity −(dsp − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The display of the local centre-stable manifold at infinity on figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 fits with the sign of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3 Tools for genericity Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) Vfull = Ck+1(Rdst, R) , and, for a positive quantity R, let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) Vquad-R = � V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = u2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us recall the notation SV introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every positive quantity R and for every potential V in Vquad-R, the following conclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The flow defined by the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) (governing radially symmetric stationary solutions of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1)) is global (that is, every solution is defined on (0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every u in SV , the following bound holds: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) sup r∈(0,+∞) |u(r)| < R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let V be in Vquad-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) of Vquad-R, there exists a positive quantity K such that, for every u in Rdst, |∇V (u)| ≤ K + |u| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As a consequence, the following inequalities hold for the right-hand side of the first order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5): ���� � v, −dsp − 1 r v + ∇V (u) ����� ≤ |v| + dsp − 1 r |v| + K + |u| ≤ K + � 2 + dsp − 1 r � |(u, v)| , and this bound prevents the solution from blowing up in finite time, which proves conclusion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Now, take a function u in SV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us still denote by u(·) the continuous extension of this solution to [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in [0, +∞), let q(r) = u(r)2 2 and Q(r) = rdsp−1 ˙q(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then (omitting the dependency on r), ˙q = u · ˙u and ¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1 r ˙q + u · ∇V (u) , so that ˙Q = rdsp−1 � ¨q + dsp − 1 r ˙q � = rdsp−1� ˙u2 + u · ∇V (u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) of Vquad-R, there exists a positive quantity δ (sufficiently small) so that, for every w in Rdst, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) |w| ≥ R − δ =⇒ w · ∇V (w) ≥ w2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us proceed by contradiction and assume that supr∈(0,+∞) |u(r)| is not smaller than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since u(·) is stable at infinity and since the critical points of V belong to the open ball BRdst(0, R − δ), it follows that the set �r ∈ [0, +∞) : |u(r)| ≥ R � is nonempty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' let rout denote the minimum of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For the same reason, the set �r ∈ (rout, +∞) : |u(r)| < R − δ � is also nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let rback denote the infimum of this last set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from these definitions that rback is larger than rout and that, for every r in (rout, rback), according to inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) ˙Q(r) ≥ rdsp−1 � ˙u2(r) + u2(r) 2 � > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 16 If on the one hand rout equals 0 then |u(0)| is not smaller than R and, since Q(0) equals 0, it follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) that Q(·) is positive on (0, rback), so that the same is true for ˙q(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus q(·) is strictly increasing on [0, rback] and |u(rback)| must be larger than |u(rout)|, a contradiction with the definition of rback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If on the other hand rout is positive, then |u(rout)| is equal to R and ˙q(rout) is nonnegative so that the same is true for Q(rout), and it again follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) that Q(·) is positive on (0, rback), yielding the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Conclusion 2 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every positive quantity R and every potential V in Vquad-R, let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) SV : (0, +∞)2 × R2dst → R2dst , �(rinit, r), (uinit, vinit) � �→ SV �(rinit, r), (uinit, vinit) � denote the (globally defined) flow of the (non-autonomous) differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) for this potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In other words, for every rinit in (0, +∞) and (uinit, vinit) in R2dst, the function (0, +∞) → R2dst , r �→ SV �(rinit, r1), (uinit, vinit) � is the solution of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) for the initial condition (uinit, vinit) at r equals rinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3, the flow SV may be extended to the larger set (0, +∞)2 × R2dst ∪ [0, +∞)2 × Rdst × {0Rdst} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' according to this extension, for every u0 in Rdst, the solution taking its values in the (one-dimensional) unstable manifold W u, 0, + V (u0) reads: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) [0, +∞) → Rdst , r �→ SV �(0, r), (u0, 0Rdst) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4 Generic transversality among potentials that are quadratic past a given radius 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Notation and statement Let us recall the notation SV and SV, u∞ introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists a generic subset of Vquad-R such that, for every potential V in this subset, every radially symmetric stationary solution stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) (in other words, every u in SV ) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Reduction to a local statement Let V1 denote a potential function in Vquad-R and u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the implicit function theorem, there exists a (small) neighbour- hood νrobust(V1, u1,∞) of Vquad-R and a Ck-function u∞(·) defined on νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following local generic transversality statement yields Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 (as shown below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 17 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists a neighbourhood νV1, u1,∞ of V1 in νrobust(V1, u1,∞) and a generic subset νV1, u1,∞, gen of νV1, u1,∞ such that, for every V in νV1, u1,∞, gen, every radially symmetric stationary solution stable close to u∞(V ) at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) (in other words, every u in SV, u∞(V )) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 yields Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us denote by Vquad-R-Morse the dense open subset of Vquad-R defined by the Morse property: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1) Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are nondegenerate} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let V1 denote a potential function in Vquad-R-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to the Morse property its minimum points are isolated and since V1 is in Vquad-R they belong to the open ball BRd(0, R), so that those minimum points are in finite number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Assume that Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With the notation of this proposition, let us consider the following two intersections, at each time over all minimum points u1,∞ of V1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) νV1 = � νV1, u1,∞ and νV1, gen = � νV1, u1,∞, gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since those are finite intersections, νV1 is still a neighbourhood of V1 in Vquad-R and the set νV1, gen is still a generic subset of νV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This shows that the set {V ∈ Vquad-R-Morse : every u in SV, u∞(V ) is transverse} is locally generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 of [1] as in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of this reference shows that this local genericity implies the global genericity stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1, which is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Proof of the local statement (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 Setting For the remaining part of this section, let us fix a potential function V1 in Vquad-R and a nondegenerate minimum point u1,∞ of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let ν be a neighbourhood of V1 in Vquad-R, included in νrobust(V1, u1,∞), and let ε1 and c1 be positive quantities, with ν and ε1 and c1 small enough so that the conclusions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let r1 = 1/c1 and M = Rdst × BRdst(u1,∞, ε1) and Λ = ν , and N = (R2dst)2 and W = {(A, B) ∈ N : A = B} , thus W is the diagonal of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let N denote an integer not smaller than r1, and let us consider the functions Φu : Rdst × Λ → R2dst , (u0, V ) �→ SV �(0, N), (u0, 0Rdst) � , and Φcs : BRdst(u1,∞, ε1) × Λ → R2dst , (uN, V ) �→ �uN, wcs, ∞ loc, V (uN, 1/N) � , and the function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3) Φ : M × Λ → N , (m, V ) = (u0, uN, V ) �→ �Φu(u0, V ), Φcs(uN, V ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 Equivalent characterizations of transversality Let us consider the set SΛ,u1,∞,N = �(V, u) : V ∈ Λ and u ∈ SV, u∞(V ) and u(N) ∈ BRdst(u1,∞, ε1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) Φ−1(W) → SΛ,u1,∞,N , (u0, u, V ) �→ � V, r �→ SV �(0, r), (u0, 0Rdst �� is well defined and one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The image by Φ of a point (u0, uN, V ) of M × Λ belongs to the diagonal W of N if and only if Φu(u0, V ) equals Φcs(uN, V ), and in this case the function u : r �→ SV �(0, r), (u0, 0Rdst � belongs to SV, u∞(V ) and u(N) (which is equal to uN) belongs to BRdst(u1,∞, ε1), so that (V, u) belongs to SΛ,u1,∞,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4) above is thus well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Now, for every (V, u) in SΛ,u1,∞,N, if we denote by u0 the limit limr→0+ u(r) and by uN the vector u(N), then (u0, uN, V ) is the only possible antecedent of (V, u) by the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In addition, SV �(0, N), (u0, 0Rdst) � = �uN, ˙u(N) � , and since u(r) goes to u∞(V ) as r goes to +∞, the vector �u(N), ˙u(N), 1/N � must belong to the centre-stable manifold W cs, ∞, + V �u∞(V ) � of u∞(V ), so that, according to the definition of wcs, ∞ loc, V , ˙u(N) = wcs, ∞ loc, V �u(N), 1/N � , and this yields the equality between Φu(u0, V ) and Φcs(uN, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus Φ(V, u) belongs to W and (u0, uN, V ) belongs to Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every potential function V in Λ, the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The image of the function M → N, m �→ Φ(m, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2, for every V in Λ, the constant function r �→ u∞(V ), which belongs to SV , is already (a priori) known to be transverse, therefore only nonconstant solutions matter in statement 2 of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider (m2, V2) in M × Λ such that Φ(m2, V2) is in W, let (u2,0, u2,N) denote the components of m2, and let r �→ u2(r) and r �→ U2(r) denote the functions satisfying, for all r in [0, +∞), U2(r) = �u2(r), ˙u2(r) � = SV �(0, r), (u2,0, 0Rdst � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the map ∆Φ : M → R2dst , (u0, uN) �→ Φu(u0, V2) − Φcs(uN, V2) , 19 and let us write, only for this proof, DΦ and DΦu and DΦcs and D(∆Φ) for the differentials of Φ and Φu and Φcs and ∆Φ at (m2, V2) and with respect to all variables in M (but not with respect to V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5, the transversality of u2 is defined as the transversality of the intersection W u, 0, + V2 ∩ ι−1� W cs, ∞, + V2 �u∞(V2) �� along the trajectory of U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This transversality can be considered at a single point, no matter which, of the trajectory U2 �(0, +∞) �, in particular at the point Φu(u2,0, V2) which is equal to Φcs�u2(N), V 2 �, and is equivalent to the transversality of the dst-dimensional manifolds W u, 0, + V2 ∩ �R2dst × {N} � and ι−1� W cs, ∞, + V2 �u∞(V2) �� ∩ �R2dst × {N} � in R2dst ×{N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It is therefore equivalent to the surjectivity of the map D(∆Φ) (statement (B) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' On the other hand, the image of the function M → N, m �→ Φ(m, V2) is transverse at Φ(m, V2) to the diagonal W of N if and only if the image of DΦ contains a complementary space of this diagonal (statement (A) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 below)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 is a consequence of the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' (A) The image of DΦ contains a complementary subspace of the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' (B) The map D(∆Φ) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If statement (A) holds, then, for every (α, β) in N, there exist γ in R2dst and δm in Tm2M such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) (γ, γ) + DΦ · δm = (α, β) , so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) D(∆Φ) · δm = α − β , and statement (B) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Conversely, if statement (B) holds, then, for every (α, β) in N, there exists δm in Tm2M such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) holds, and as a consequence, if (δu0, δuN) denote the components of δm, then α − DΦu(δu0) is equal to β − DΦcs(δuN), and if this vector is denoted by γ, then equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5) holds, and this shows that statement (A) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As explained above, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5, and is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] The function Φ is as regular as the flow SV , thus of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from the definitions of M and N and W that dim(M) − codim(W) = (dst + dst) − 2dst = 0 , so that hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] For every V in Vquad-R, let us recall the notation SV introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) for the flow of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Take (m2, V2) in the set Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (u2,0, u2,N) denote the components of m2, and, for every r in (0, +∞), let us write U2(r) = �u2(r), v2(r) � = SV2 �(0, r), (u2,0, 0Rdst) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us write DΦ and DΦu and DΦcs for the full differentials (with respect to arguments m in M and V in Λ) of the three functions Φ and Φu and Φcs respectively at the points �u2,0, u2,N, V2 �, �u2,0, V2 � and �u2,N, V2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] amounts to prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) im(DΦ) + W = N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If u2(·) is constant (that is, identically equal to u∞(V2)), then equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus, let us assume that u2(·) is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case, equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every nonzero vector (φ2, ψ2) in R2dst, there exists a function W in Ck+1 b (Rdst, R) such that supp(W) ⊂ BRd(0, R) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) and �DΦu · (0, 0, W) �� (φ2, ψ2) � ̸= 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) and DΦcs · (0, 0, W) = 0R2dst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) Proof that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) shows that the orthogonal complement, in R2dst, of the directions that can be reached by DΦu·(0, 0, W) for potentials W satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) is reduced to 0R2dst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' in other words, all directions of R2dst can be reached by that means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This shows that im(DΦ) ⊃ R2dst × {0R2dst} , and since the subspace at the right-hand side of this inclusion is transverse to W in R4dst, this proves equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7) (and shows that hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] is fulfilled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let (φ2, ψ2) denote a nonzero vector in R2dst, let W be a function in Ck+1 b (Rdst, R) satisfying the inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) supp(W) ⊂ BRd(0, R) \\ BRdst(u1,∞, ε1) , and observe that inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='8) and equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='10) follow from this inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the linearization of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2), for the potential V2, around the solution r �→ U2(r): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12) d dr � δu(r) δv(r) � = � 0 id D2V2 �u2(r) � −dsp−1 r � � δu(r) δv(r) � , 21 and let T(r, r′) denote the family of evolution operators obtained by integrating this linearized di���erential system between r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from the variation of constants formula that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13) DΦu · (0, 0, W) = � N −∞ T(r, N) � 0, ∇W �u2(r) �� dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in (0, +∞), let T ∗(r, N) denote the adjoint operator of T(r, N), and let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='14) �φ(r), ψ(r) � = T ∗(r, N) · (φ2, ψ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='13), inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) reads � N −∞ �� 0, ∇W �u2(r) �� ��� T ∗(r, N) · (φ2, ψ2) � dr ̸= 0 , or equivalently (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) � N −∞ ∇W �u2(r) � · ψ(r) dr ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Due to the expression of the linearized differential system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='12), (φ, ψ) is a solution of the adjoint linearized system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) � ˙φ(r) ˙ψ(r) � = − � 0 D2V2 �u2(r) � id −dsp−1 r � � φ(r) ψ(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 (and since u2(·) was assumed to be nonconstant), there exists positive quantity ronce such that, if we denote by Ionce the interval (0, ronce], then ˙u2(·) does not vanish on Ionce, and, for all r∗ in Ionce and r in R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='17) u2(r) = u2(r∗) =⇒ r = r∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In addition, up to replacing ronce by a smaller positive quantity, it may be assumed that the following conclusions hold: u2(Ionce) ∩ BRdst(u1,∞, ε1) = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' To complete the proof three cases have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' There exists r∗ in Ionce such that ψ(r∗) is not collinear to ˙u2(r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case, the construction of a potential function W satisfying inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) (and thus the conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) is the same as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If case 1 does not occur, then ψ(r) is collinear to ˙u2(r), and since ˙u2(·) does not vanish on Ionce, there exists a C1-function α : Ionce → R such that, for every r in Ionce, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='18) ψ(r) = α(r) ˙u2(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The next cases 2 and 3 differ according to whether the function α(·) is constant or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 22 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in Ionce, equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='18) holds for some nonconstant function α(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case there exists r∗ in Ionce such that ˙α(r∗) is nonzero, and again the construction of a potential function W satisfying inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11) and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='9) (and thus the conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6) is the same as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every r in Ionce, ψ(r) = α ˙u2(r) for some real (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this case the quantity α cannot be 0 or else, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='18), both φ(·) and ψ(·) would identically vanish on Ionce and thus on (0, +∞), a contradiction with the assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus, without loss of generality, we may assume that α is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' If supp(W) is included in a sufficiently small neighbourhood of u2,0, then W(·) vanishes on u2 �[ronce, N] � and the integral on the left-hand side of inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) reads � ronce 0 ∇W �u2(r) � · ˙u2(r) dr = W �u2(ronce) � − W(u2,0) = −W(u2,0) , so that inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='15) holds as soon as W(u2,0) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By contrast with the proof of the generic elementarity of standing pulses in [1], case 3 above cannot be easily precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Indeed, let us assume that, for every r in Ionce, ψ(r) is equal to α ˙u2(r) for some nonzero (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Without loss of generality, we may assume that α is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then, it follows from the second equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) that, still for every r in Ionce (omitting the dependency on r), φ = dsp − 1 r ψ − ˙ψ = dsp − 1 r ˙u2 − ¨u2 = 2(dsp − 1) r ˙u2 − ∇V2(u2) , and it follows from the first equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='16) that −D2V2(u2) ˙u2 = ˙φ = −2(dsp − 1) r2 ˙u2 + 2(dsp − 1) r ¨u2 − D2V2(u2) ˙u2 , and thus, after simplification, ¨u2 = 1 r ˙u2 , or equivalently ˙u2 = r dsp ∇V (u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As illustrated by equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6), this last equality indeed holds if ∇V2 is constant on the set u2(Ionce).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Case 3 can therefore not be a priori precluded, and if it may be argued that this case is “unlikely” (non generic), the direct argument provided above in this case is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By contrast, in [1] for standing pulses in space dimension one (dsp equal to 1), this case could not occur because ψ was assumed to be nonzero on the symmetry subspace, defined here as {(v, r) = (0Rdst, 0)}, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5 Conclusion As seen in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3, hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1] is fulfilled for the function Φ defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3), and since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='6 yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='7), hypothesis 2 of this 23 theorem is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The conclusion of this theorem ensures that there exists a generic subset Λgen, N of Λ such that, for every V in Λgen, N, the image of the function M → N, m �→ Φ(m, V ) is transverse to the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' According to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4, it follows that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' The set Λgen = � N∈N, N≥r0 Λgen, N is still a generic subset of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every V in Λgen and every u in SV, u∞(V ), since u(r) goes to u∞(V ) as r goes to +∞, there exists N such that u(N) is in BRdst(u1,∞, ε1), and according to the previous statements u is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In other words, the conclusions of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 hold with: νV1, u1,∞ = ν = Λ and νV1, u1,∞, gen = Λgen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 5 Proof of the main results Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 shows the genericity of the property considered in Theorem 1, but only inside the space Vquad-R of the potentials that are quadratic past some radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In this section, the arguments will be adapted to obtain the genericity of the same property in the space Vfull (that is Ck+1(Rdst, R)) of all potentials, endowed with the extended topology (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' They are identical to those of section 9 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us recall the notation SV introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4), and, for every positive quantity R, let us consider the set SV,R = � u ∈ SV : sup r∈[0,+∞) |u(r)| ≤ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Exactly as shown in subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 of [1], Theorem 1 follows from the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' For every positive quantity R, there exists a generic subset Vfull-⋔-S-R of Vfull such that, for every potential V in this subset, every radially symmetric stationary solution stable at infinity in SV,R is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let R denote a positive quantity, let V1 denote a potential function in Vquad-(R+1), and let u1,∞ denote a nondegenerate minimum point of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the neigh- bourhood νV1, u1,∞ of V1 in Vquad-(R+1) provided by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 for these objects, together with the quantities ε1, c1, and r1 introduced in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Up to replacing νV1, u1,∞ by its interior, we may assume that it is open in Vquad-(R+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As in sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1, let us consider an integer N not smaller than r1, and the same function Φ : M × Λ → N as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Here is the sole difference with the setting of sub-subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1: by contrast with the non-compact set M defining the departure set of Φ, let us consider the compact subset MN defined as: MN = BRdst(0Rdst, N) × BRdst(u1,∞, ε1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Thus the integer N now serves two purposes: the “time” (radius) at which the intersection between unstable and centre-stable manifolds is considered, and the radius of the ball 24 containing the departure points of the unstable manifolds that are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' These purposes are independent (two different integers instead of the single integer N may as well be introduced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the set: OV1,u1,∞,N = � V ∈ νV1, u1,∞ : Φ(MN, V ) is transverse to W in N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' As shown in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4, this set OV1,u1,∞,N is made of the potential functions V in νV1, u1,∞ such that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) and u(0) is in BRdst(0Rdst, N), is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set contains the generic subset νV1, u1,∞, gen = Λgen of νV1, u1,∞ and is therefore generic (thus, in particular, dense) in νV1, u1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' By comparison with νV1, u1,∞, gen, the additional feature of this set OV1,u1,∞,N is that it is open: exactly as in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='2 of [1], this openness follows from the intrinsic openness of a transversality property and the compactness of MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us make the additional assumption that the potential V1 is a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Then, the set of minimum points of V1 is finite and depends smoothly on V in a neighbourhood νrobust(V1) of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Intersecting the sets νV1, u1,∞ and OV1,u1,∞,N above over all the minimum points u1,∞ of V1 provides an open neighbourhood νV1 of V1 and an open dense subset OV1,N of νV1 such that, for all V in νV1, every radially symmetric stationary solution stable close to a minimum point of V at infinity, and equal at origin to some point of BRdst(0Rdst, N), is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Denoting by int(A) the interior of a set A and using the notation of subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1], let us introduce the sets ˜νV1 = res−1 R,∞ ◦ resR,(R+1)(νV1) , and ˜OV1,N = res−1 R,∞ ◦ resR,(R+1)(OV1,N) , and ˜Oext V1,N = ˜OV1,N ⊔ int �Vfull \\ ˜νV1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' It follows from these definitions that ˜Oext V1,N is a dense open subset of Vfull (for more details, see Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Since Vquad-(R+1) is a separable space, it is second-countable, and can be covered by a countable number of sets of the form νV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' With symbols, there exists a countable family (V1,i)i∈N of potentials of Vquad-(R+1)-Morse so that Vquad-(R+1)-Morse = � i∈N νV1,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Let us consider the set Vfull-⋔-S-R = Vfull-Morse ∩ � � � (i,N)∈N2 ˜Oext V1,i,N � � , where Vfull-Morse is the set of potentials in Vfull which are Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' This set is a countable intersection of dense open subsets of Vfull, and is therefore a generic subset of Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' And, for every potential V in this set Vfull-⋔-S-R, every radially symmetric stationary solution stable at infinity in SV,R is transverse (for more details, see Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 25 As already mentioned at the beginning of this section, Theorem 1 follows from Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Finally, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='1 follows from Theorem 1 (for more details, see subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='4 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Acknowledgements This paper owes a lot to numerous fruitful discussions with Romain Joly, about both its content and the content of the companion paper [1] written in collaboration with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Joly and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2023), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02095 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3, 5, 7–10, 14, 18, 20–26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global behaviour of bistable solutions for gradient systems in one unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02002 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='01221 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global behaviour of radially symmetric solutions stable at infinity for gradient systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='02134 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Risler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' “Global relaxation of bistable solutions for gradient systems in one unbounded spatial dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' In: arXiv (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 1–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' arXiv: 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='00804 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' Emmanuel Risler Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69621 Villeurbanne, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content=' emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='risler@insa-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'} +page_content='fr 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E0T4oBgHgl3EQfugE5/content/2301.02605v1.pdf'}