{"product_id":"9781470436834","title":"Quadratic Vector Equations on Complex Upper Half-Plane","description":"The authors consider the nonlinear equation $-\\frac 1m=z+Sm$ with a parameter $z$ in the complex upper half plane $\\mathbb H $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \\mathbb H$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\\mathbb R$. In a previous paper the authors qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur.In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any $z\\in \\mathbb H$, including the vicinity of the singularities.","brand":"Oskari Ajanki","offers":[{"title":"Default Title","offer_id":46776569364798,"sku":"9781470436834","price":0.0,"currency_code":"ZAR","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4357\/7662\/products\/9781470436834_2140b09c-f901-4d4a-9564-247054c2fa7e.jpg?v=1707476989","url":"https:\/\/exclusivebooks.co.za\/products\/9781470436834","provider":"Exclusive Books Online","version":"1.0","type":"link"}