Proceedings of the International Geometry Center最新文献

{"title":"On the Koebe Quarter Theorem for Polynomials","authors":"J. Dillies, D. Dmitrishin, A. Smorodin, A. Stokolos","doi":"10.15673/tmgc.v14i3.2057","DOIUrl":"https://doi.org/10.15673/tmgc.v14i3.2057","url":null,"abstract":"The Koebe One Quarter Theorem states that the range of any Schlicht function contains the centered disc of radius 1/4 which is sharp due to the value of the Koebe function at −1. A natural question is finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials. In particular, it was asked in [7] whether Suffridge polynomials [15] are optimal. For polynomials of degree 1 and 2 that is obviously true. It was demonstrated in [10] that Suffridge polynomials of degree 3 are not optimal and a promising alternative family of polynomials was introduced. These very polynomials were actually discovered earlier independently by M. Brandt [3] and D. Dimitrov [9]. In the current article we reintroduce these polynomials in a natural way and make a far-reaching conjecture that we verify for polynomials up to degree 6 and with computer aided proof up to degree 52. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82471213","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":"Nonpositive curvature foliations on 3-manifolds with bounded total absolute curvature of leaves","authors":"D. Bolotov","doi":"10.15673/TMGC.V11I4.1307","DOIUrl":"https://doi.org/10.15673/TMGC.V11I4.1307","url":null,"abstract":"In this paper we introduce a new class of foliations on Rie-mannian 3-manifolds, called B-foliations, generalizing the class of foliations of non-negative curvature. The leaves of B-foliations have bounded total absolute curvature in the induced Riemannian metric. We describe several topological and geometric properties of B-foliations and the structure of closed oriented 3-dimensional manifolds admitting B-foliations with non-positive curvature of leaves.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83632859","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":"Специальные классы псевдоримановых пространств с f-структурой, допускающих 2F-планарные отображения","authors":"Надежда Григорьевна Коновенко, Ирина Николаевна Курбатова","doi":"10.15673/TMGC.V11I4.1304","DOIUrl":"https://doi.org/10.15673/TMGC.V11I4.1304","url":null,"abstract":"В статье изучаются 2F-планарные отображения псевдоримановых пространств, снабженных аффинорной структурой определенного типа. Понятие 2F-планарного отображения аффинносвязных и римановых пространств было введено в рассмотрение Р.Дж. Кадемом. В его работах исследовались общие вопросы теории 2F-планарных отображений аффинносвязных и римановых пространств, снабженных аффинорной структурой. В частности, он доказал, что такое отображение по необходимости сохраняет аффинорную структуру. Мы рассматриваем 2F-планарное отображение псевдоримановых пространств с абсолютно параллельной  f-структурой. Ранее мы доказали, что  псевдориманово пространство с абсолютно параллельной f-структурой представляет собой произведение двух псевдоримановых пространств, одно из которых - келерово; класс псевдоримановых пространств с абсолютно параллельной  f-структурой замкнут относительно рассматриваемых отображений; при условии ковариантного постоянства аффинора f-структуры в отображаемых пространствах  нетривиальные 2F-планарные отображения могут быть трех типов: полные и канонические I,II типа; в зависимости от типа 2F-планарное отображение индуцирует на компонентах произведения отображаемых пространств геодезическое, голоморфно-проективное или аффинное отображение. \u0000В настоящей статье продолжается исследование 2F-планарного отображения псевдоримановых пространств с абсолютно параллельной f-структурой. Для всех типов этого отображения (основного и канонических I  и II ) строятся геометрические объекты, инвариантные относительно рассматриваемых отображений: неоднородный объект ( типа параметров Томаса в теории геодезических отображений римановых пространств)  и тензорный  (типа тензора голоморфно-проективной кривизны в теории аналитически-планарных отображений келеровых многообразий).  Выделены классы пространств (2F-плоские, 2F(I)- и 2F(II)-плоские), допускающих 2F-планарное отображение.  Для них выявлена структура тензора Римана и доказаны аналоги теоремы Бельтрами из теории геодезических отображений. Найдены метрики  2F-, 2F(I)-  и 2F(II)-плоских пространств   в специальной системе координат.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87847220","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":"Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere","authors":"A. Kravchenko, S. Maksymenko","doi":"10.15673/tmgc.v11i4.1306","DOIUrl":"https://doi.org/10.15673/tmgc.v11i4.1306","url":null,"abstract":"Let $M$ be a compact two-dimensional manifold and, $f in C^{infty}(M, R)$ be a Morse function, and $Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{infty}$, and by $S(f)={hin D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $hin S(f)$ induces an automorphism of the graph $Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. \u0000In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere.  \u0000The present paper is devoted to the case $M = S^2$. In this situation $Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 to R$ whose fixed subtree $Fix(G)$ consists of more than one point.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88277892","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":"Deformations of smooth functions on 2-torus","authors":"Bohdan Feshchenko","doi":"10.15673/tmgc.v12i3.1528","DOIUrl":"https://doi.org/10.15673/tmgc.v12i3.1528","url":null,"abstract":"Let $f$ be a Morse function on a smooth compact surface $M$ and $mathcal{S}'(f)$ be the group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the Kronrod-Reeb graph of $f$ induced by elements from $mathcal{S}'(f)$, and $Delta'$ be the subgroup of $mathcal{S}'(f)$ consisting of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group $pi_0mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. The groups $pi_0Delta'(f)$ and $G(f)$ encode ``combinatorially trivial'' and ``combinatorially nontrivial'' counterparts of $pi_0mathcal{S}'(f)$ respectively. In the paper we compute groups $pi_0mathcal{S}'(f)$, $G(f)$, and $pi_0Delta'(f)$ for Morse functions on $2$-torus $T^2$.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89346702","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":"Three spectra problem for Stieltjes string equation and Neumann conditions","authors":"A. Dudko, V. Pivovarchik","doi":"10.15673/TMGC.V12I1.1367","DOIUrl":"https://doi.org/10.15673/TMGC.V12I1.1367","url":null,"abstract":"Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74121544","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":"Про структуру матриць над областю головних ідеалів відносно перетворення подібності","authors":"Володимир Прокіп","doi":"10.15673/TMGC.V12I1.1368","DOIUrl":"https://doi.org/10.15673/TMGC.V12I1.1368","url":null,"abstract":"В статті дослiджується структура матриць над областю головних iдеалiв вiдносно перетворення подiбностi. В другому розділі наведено допоміжні результати. В цьому розділі вказано трикутну формуматрицi відносно перетворення подібності, мінімальний многочлен якої розкладається в добуток різних лінійних множників. В розділі 3 доведено, що форма Хессенберга матриці A з незвідним мінімальним квадратичним многочленом m(λ) є блочно-трикутна матриця з блоками вимірності 2х2 на головній діагоналі та з характеристичними многочленами m(λ). У четвертому розділі доведено, що матриця A із мінімальним многочленом m (λ) = (λ-α) (λ-β), α ≠ β подібна нижній блочно-трикутній матриці, діагональними блоками якої є діагональні матриці з елементами α i β на головних діагоналях відповідно. Як наслідок вказано канонічну форму інволютивної матриці над кільцем цілих чисел відносно перетворень подібності.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"115 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82483018","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 the integrability problem for systems of partial differential equations in one unknown function, I","authors":"A. Kumpera","doi":"10.15673/TMGC.V11I4.1305","DOIUrl":"https://doi.org/10.15673/TMGC.V11I4.1305","url":null,"abstract":"We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"326 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76638085","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 the integrability problem for systems of partial differential equations in one unknown function, II","authors":"A. Kumpera","doi":"10.15673/tmgc.v12i1.1366","DOIUrl":"https://doi.org/10.15673/tmgc.v12i1.1366","url":null,"abstract":"We continue here our discussion of Part I, [18], by examining the local equivalence problem for partial differential equations and illustrating it with some examples, since almost any integration process or method is actually a local equivalence problem involving a suitable model. We terminate the discussion by inquiring on non-integrable Pfaffian systems and on their integral manifolds of maximal dimension.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74403020","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":"A metrizable Lawson semitopological semilattice with non-closed partial order","authors":"T. Banakh, S. Bardyla, A. Ravsky","doi":"10.15673/tmgc.v13i3.1756","DOIUrl":"https://doi.org/10.15673/tmgc.v13i3.1756","url":null,"abstract":"We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $le_X,={(x,y)in Xtimes X:xy=x}$ is not closed in $Xtimes X$. This resolves a problem posed earlier by the authors.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82160972","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}