{"title":"Maximum Likelihood Degree of Surjective Rational Maps","authors":"Ilya Karzhemanov","doi":"10.1007/s40598-022-00207-0","DOIUrl":"10.1007/s40598-022-00207-0","url":null,"abstract":"<div><p>With any <i>surjective rational map</i> <span>(f: mathbb {P}^n dashrightarrow mathbb {P}^n)</span> of the projective space, we associate a numerical invariant (<i>ML degree</i>) and compute it in terms of a naturally defined vector bundle <span>(E_f longrightarrow mathbb {P}^n)</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 3-4","pages":"513 - 516"},"PeriodicalIF":0.0,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45581553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Classification of Generic Spherical Quadrilaterals","authors":"Andrei Gabrielov","doi":"10.1007/s40598-022-00204-3","DOIUrl":"10.1007/s40598-022-00204-3","url":null,"abstract":"<div><p>Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"9 2","pages":"151 - 203"},"PeriodicalIF":0.0,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00204-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42828660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cohomology Rings and Algebraic Torus Actions on Hypersurfaces in the Product of Projective Spaces and Bounded Flag Varieties","authors":"Grigory Solomadin","doi":"10.1007/s40598-022-00203-4","DOIUrl":"10.1007/s40598-022-00203-4","url":null,"abstract":"<div><p>In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"9 1","pages":"105 - 150"},"PeriodicalIF":0.0,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46354160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantitative Uncertainty Principles Related to Lions Transform","authors":"A. Achak, A. Abouelaz, R. Daher, N. Safouane","doi":"10.1007/s40598-022-00202-5","DOIUrl":"10.1007/s40598-022-00202-5","url":null,"abstract":"<div><p>We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 3-4","pages":"481 - 496"},"PeriodicalIF":0.0,"publicationDate":"2022-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43801122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-Sided Fundamental Theorem of Affine Geometry","authors":"Alexey Gorinov","doi":"10.1007/s40598-022-00201-6","DOIUrl":"10.1007/s40598-022-00201-6","url":null,"abstract":"<div><p>The fundamental theorem of affine geometry says that if a self-bijection <i>f</i> of an affine space of dimenion <i>n</i> over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then <i>f</i> of the expected type, namely <i>f</i> is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 3-4","pages":"469 - 480"},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41287513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Renormalization of Bicritical Circle Maps","authors":"Gabriela Estevez, Pablo Guarino","doi":"10.1007/s40598-022-00199-x","DOIUrl":"10.1007/s40598-022-00199-x","url":null,"abstract":"<div><p>A general <i>ansatz</i> in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of <i>bicritical circle maps</i>, which are <span>(C^3)</span> circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies <span>(C^{1+alpha })</span> rigidity for real-analytic bicritical circle maps with rotation number of <i>bounded type</i> (Corollary 1.1).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"9 1","pages":"69 - 104"},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49563940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Talia Blum, John R. Doyle, Trevor Hyde, Colby Kelln, Henry Talbott, Max Weinreich
{"title":"Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations","authors":"Talia Blum, John R. Doyle, Trevor Hyde, Colby Kelln, Henry Talbott, Max Weinreich","doi":"10.1007/s40598-022-00197-z","DOIUrl":"10.1007/s40598-022-00197-z","url":null,"abstract":"<div><p>A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 2","pages":"285 - 317"},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Catastrophe in Elastic Tensegrity Frameworks","authors":"Alexander Heaton, Sascha Timme","doi":"10.1007/s40598-021-00193-9","DOIUrl":"10.1007/s40598-021-00193-9","url":null,"abstract":"<div><p>We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called <i>catastrophe</i>, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the <i>catastrophe set</i>, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 3-4","pages":"423 - 443"},"PeriodicalIF":0.0,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49583252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The ({{mathbb {F}}}_p)-Selberg Integral","authors":"Richárd Rimányi, Alexander Varchenko","doi":"10.1007/s40598-021-00191-x","DOIUrl":"10.1007/s40598-021-00191-x","url":null,"abstract":"<div><p>We prove an <span>({{mathbb {F}}}_p)</span>-Selberg integral formula, in which the <span>({{mathbb {F}}}_p)</span>-Selberg integral is an element of the finite field <span>({{mathbb {F}}}_p)</span> with odd prime number <i>p</i> of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo <i>p</i>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 1","pages":"39 - 60"},"PeriodicalIF":0.0,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50510227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Equivalence Problem in Analytic Dynamics for 1-Resonance","authors":"Christiane Rousseau","doi":"10.1007/s40598-021-00196-6","DOIUrl":"10.1007/s40598-021-00196-6","url":null,"abstract":"<div><p>When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in a neighborhood of a singular point? The present paper, of a survey nature, presents a research program around this question. A way to answer is to use normal forms. However, there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper, we discuss the case of singularities for which the normalizing transformation is <i>k</i>-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic <i>k</i>-parameter families unfolding such singularities.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"9 1","pages":"1 - 39"},"PeriodicalIF":0.0,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-021-00196-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45454521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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