{"title":"On Convex Bodies that are Characterizable by Volume Function","authors":"Ákos G. Horváth","doi":"10.1007/s40598-020-00132-0","DOIUrl":"10.1007/s40598-020-00132-0","url":null,"abstract":"<div><p>The “old-new” concept of a convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems connected with these functions whose serious investigation was closed before the “age of computers”. In this survey, we concentrate only on the three-dimensional case; we will mention the most important concepts, statements, and problems.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"6 1","pages":"1 - 20"},"PeriodicalIF":0.0,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00132-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41629324","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":"Invariant Spanning Trees for Quadratic Rational Maps","authors":"Anastasia Shepelevtseva, Vladlen Timorin","doi":"10.1007/s40598-019-00123-w","DOIUrl":"10.1007/s40598-019-00123-w","url":null,"abstract":"<div><p>We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes—the ivy graph.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 4","pages":"435 - 481"},"PeriodicalIF":0.0,"publicationDate":"2019-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00123-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44783173","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":"Faithful tropicalizations of elliptic curves using minimal models and inflection points","authors":"Paul Alexander Helminck","doi":"10.1007/s40598-019-00121-y","DOIUrl":"10.1007/s40598-019-00121-y","url":null,"abstract":"<div><p>We give an elementary proof of the fact that any elliptic curve <i>E</i> over an algebraically closed non-archimedean field <i>K</i> with residue characteristic <span>(ne {2,3})</span> and with <span>(v(j(E))<0)</span> admits a tropicalization that contains a cycle of length <span>(-v(j(E)))</span>. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves (<i>E</i>, <i>P</i>), where <i>E</i> has multiplicative reduction and <i>P</i> is an inflection point that reduces to the singular point on the reduction of <i>E</i>. We then follow the strategy as in Baker et al. (Algebraic Geom 3(1):63–105, 2016) and construct an embedding such that its tropicalization contains a cycle of length <span>(-v(j(E)))</span>. We call this a numerically faithful tropicalization. A key difference between this approach and the approach in Baker et al. (2016) is that we do not require any of the analytic theory on Berkovich spaces such as the <i>Poincaré–Lelong formula</i> or (Baker et al. 2016) to establish the numerical faithfulness of this tropicalization.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 4","pages":"401 - 434"},"PeriodicalIF":0.0,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00121-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45806812","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":"Solutions with Compact Time Spectrum to Nonlinear Klein–Gordon and Schrödinger Equations and the Titchmarsh Theorem for Partial Convolution","authors":"Andrew Comech","doi":"10.1007/s40598-019-00122-x","DOIUrl":"10.1007/s40598-019-00122-x","url":null,"abstract":"<div><p>We prove that finite energy solutions to the nonlinear Schrödinger equation and nonlinear Klein–Gordon equation which have the compact time spectrum have to be one-frequency solitary waves. The argument is based on the generalization of the Titchmarsh convolution theorem to partial convolutions.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 2-3","pages":"315 - 338"},"PeriodicalIF":0.0,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00122-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43898931","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":"On Osculating Framing of Real Algebraic Links","authors":"Grigory Mikhalkin, Stepan Orevkov","doi":"10.1007/s40598-019-00120-z","DOIUrl":"10.1007/s40598-019-00120-z","url":null,"abstract":"<div><p>For a real algebraic link in <span>({{mathbb {RP}}}^3)</span>, we prove that its encomplexed writhe (an invariant introduced by Viro) is maximal for a given degree and genus if and only if its self-linking number with respect to the framing by the osculating planes is maximal for a given degree.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 4","pages":"393 - 399"},"PeriodicalIF":0.0,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00120-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45026129","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":"Shifted Quantum Affine Algebras: Integral Forms in Type A","authors":"Michael Finkelberg, Alexander Tsymbaliuk","doi":"10.1007/s40598-019-00118-7","DOIUrl":"10.1007/s40598-019-00118-7","url":null,"abstract":"<div><p>We define an integral form of shifted quantum affine algebras of type <i>A</i> and construct Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized <i>K</i>-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type <i>A</i>. Finally, we obtain the rational (homological) analogues of the above results [proved earlier in Kamnitzer et al. (Proc Am Math Soc <b>146</b>(2):861–874, 2018a; On category <span>(mathcal {O})</span> for affine Grassmannian slices and categorified tensor products. arXiv:1806.07519, 2018b) via different techniques].</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 2-3","pages":"197 - 283"},"PeriodicalIF":0.0,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00118-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45543013","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":"Open WDVV Equations and Virasoro Constraints","authors":"Alexey Basalaev, Alexandr Buryak","doi":"10.1007/s40598-019-00115-w","DOIUrl":"10.1007/s40598-019-00115-w","url":null,"abstract":"<div><p>In their fundamental work, Dubrovin and Zhang, generalizing the Virasoro equations for the genus 0 Gromov–Witten invariants, proved the Virasoro equations for a descendent potential in genus 0 of an arbitrary conformal Frobenius manifold. More recently, a remarkable system of partial differential equations, called the open WDVV equations, appeared in the work of Horev and Solomon. This system controls the genus 0 open Gromov–Witten invariants. In our paper, for an arbitrary solution of the open WDVV equations, satisfying a certain homogeneity condition, we construct a descendent potential in genus 0 and prove an open analog of the Virasoro equations. We also present conjectural open Virasoro equations in all genera and discuss some examples.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 2-3","pages":"145 - 186"},"PeriodicalIF":0.0,"publicationDate":"2019-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00115-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48439898","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":"Triangulated Endofunctors of the Derived Category of Coherent Sheaves Which Do Not Admit DG Liftings","authors":"Vadim Vologodsky","doi":"10.1007/s40598-019-00114-x","DOIUrl":"10.1007/s40598-019-00114-x","url":null,"abstract":"<div><p>In, Rizzardo and Van den Bergh (An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves. arXiv:1410.4039, 2014) constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field <i>k</i> of characteristic 0 which is not of the Fourier–Mukai type. The purpose of this note is to show that if <span>({{,mathrm{{char}},}}k =p)</span> then there are very simple examples of such functors. Namely, for a smooth projective <i>Y</i> over <span>({{mathbb {Z}}}_p)</span> with the special fiber <span>(i: Xhookrightarrow Y)</span>, we consider the functor <span>(L i^* circ i_*: D^b(X) rightarrow D^b(X))</span> from the derived categories of coherent sheaves on <i>X</i> to itself. We show that if <i>Y</i> is a flag variety which is not isomorphic to <span>({{mathbb {P}}}^1)</span> then <span>(L i^* circ i_*)</span> is not of the Fourier–Mukai type. Note that by a theorem of Toen (Invent Math 167:615–667, 2007, Theorem 8.15) the latter assertion is equivalent to saying that <span>(L i^* circ i_*)</span> does not admit a lifting to a <span>({{mathbb {F}}}_p)</span>-linear DG quasi-functor <span>(D^b_{dg}(X) rightarrow D^b_{dg}(X))</span>, where <span>(D^b_{dg}(X))</span> is a (unique) DG enhancement of <span>(D^b(X))</span>. However, essentially by definition, <span>(L i^* circ i_*)</span> lifts to a <span>({{mathbb {Z}}}_p)</span>-linear DG quasi-functor.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 1","pages":"139 - 143"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00114-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47815315","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":"Laughlin states and gauge theory","authors":"Nikita Nekrasov","doi":"10.1007/s40598-019-00113-y","DOIUrl":"10.1007/s40598-019-00113-y","url":null,"abstract":"<div><p>Genus one Laughlin wavefunctions, describing the gas of interacting electrons on a two dimensional torus in the presence of a strong magnetic field, analytically continued in the filling fraction, are related to the partition functions of half-BPS surface defects in four dimensional <span>({{mathcal {N}}}=2)</span> supersymmetric gauge theory.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 1","pages":"123 - 138"},"PeriodicalIF":0.0,"publicationDate":"2019-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00113-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50455926","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":"Solutions of Polynomial Equations in Subgroups of (mathbb {F}_p^*)","authors":"Sergei Makarychev, Ilya Vyugin","doi":"10.1007/s40598-019-00112-z","DOIUrl":"10.1007/s40598-019-00112-z","url":null,"abstract":"<div><p>We present an upper bound on the number of solutions of an algebraic equation <span>(P(x,y)=0)</span> where <i>x</i> and <i>y</i> belong to the union of cosets of some subgroup of the multiplicative group <span>(kappa ^*)</span> of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927–1942, 2013) to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"5 1","pages":"105 - 121"},"PeriodicalIF":0.0,"publicationDate":"2019-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-019-00112-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50453462","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}