336 research outputs found
Stanley's work on unimodality
No full textThis paper surveys Stanley's work on unimodality, and its impact. It also poses some open problems that arise naturally from his work in this area
Some open problems on Coxeter groups and unimodality
No full textIn this paper I present some open problems on Coxeter groups
and unimodality, together with the main partial results, and computational
evidence, that are known about them
Comparing products of Schur functions and quasisymmetric functions
No full textThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (p. 71-74).In this thesis a conjecture of Okounkov, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity of certain differences of products of Schur functions are proved. In the first part of the work a combinatorial method is developed that allows to prove weaker versions of those conjectures. In the second part a recent result of Rhoades and Skandera is used to provide a proof of actual Schur positivity results. Several further generalizations are stated and proved. In particular, an intriguing log-concavity property of Schur functions is observed. In addition, a stronger conjecture is stated in language of alcoved polytops. A weaker version of this conjecture is proved using a characterization of Klyachko cone and the theory of Temperley-Lieb immanants.by Pavlo Pylyavskyy.Ph.D
Electrical Networks and Electrical Lie Theory of Classical Types.
No full textIn this thesis we investigate the structure of electrical Lie algebras of finite Dynkin type. These Lie algebras were introduced by Lam-Pylyavskyy in the study of circular planar electrical networks. Among these electrical Lie algebras, the Lie group corresponding to type A electrical Lie algebra acts on such networks via some combinatorial operations studied by Curtis-Ingerman-Morrow and Colin de Verdi`ere-Gitler-Vertigan. Lam-Pylyavskyy studied the type A electrical Lie algebra of even rank in detail, and gave a conjecture for the dimension of electrical Lie algebras of finite Dynkin types. We prove this conjecture for all classical Dynkin types, that is, A, B, C, and D. Furthermore, we are able to explicitly describe the structure of some electrical Lie algebras of classical types as the semisimple product of the symplectic Lie algebra with its finite dimensional irreducible representations.
We then introduce mirror symmetric circular planar electrical networks as the mirror symmetric subset of circular planar electrical networks studied by Curtis-Ingerman-Morrow [CIM] and Colin de Verdi`ere-Gitler-Vertigan [dVGV]. These mirror symmetric networks can be viewed as the type B generalization of circular planar electrical networks. We show that most of the properties of circular planar electrical
viii
networks are well inherited by these mirror symmetric electrical networks. In particular, the type B electrical Lie algebra has an infinitesimal action on such networks. Inspired by Lam [Lam], the space of mirror symmetric circular planar electrical net- works can be compactified using mirror symmetric cactus networks, which admit a stratification indexed by mirror symmetric matchings on [4n]. The partial order on the mirror symmetric matchings emerging from mirror symmetric electrical networks is dual to a subposet of affine Bruhat order of type C. We conjecture that this partial order is the closure partial order of the stratification of mirror symmetric cactus networks.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113393/1/yisu_1.pd
Het'man Pavlo Skoropads'kyi and Foundation of Ukrainian Academy of Sciences
No full textАвтор намагається з’ясувати, яку роль відіграв гетьман Павло Скоропадський у справі заснування Української академії наук та причини, що призвели до затьмарення ролі глави Української Держави в «академічному процесі». Аналізуються самооцінка Павла Скоропадського, інвективи Миколи Василенка та Володимира Вернадського щодо його ролі в заснуванні УАН, а також офіційна документація та публічні акти 1918 р. Автор статті доходить висновку, що П.Скоропадський має бути залучений до кола «батьків-засновників» УАН.The author is trying to find out which role hetman Pavlo Skoropads’kyi performed in the case of foundation of Ukrainian Academy of Sciences and the causes that led to diminishing of the part of Ukrainian state’s head in «academic process». The article analyzes Pavlo Skoropads’kyi’s self-assessment, Mykola Vasylenko’s and Volodymyr Vernads’kyi’s invectives as to his role in foundation of UAS, and official documents and public acts of 1918. The author of the article comes to the conclusion that P.Skoropads’kyi has to be enlisted to the circle of founders of UAS
Quivers with subadditive labelings: classification and integrability
No full textAbstract
Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg’s definition for undirected graphs. In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity. In this paper, we classify all quivers with subadditive labelings. We conjecture them to exhibit a certain form of integrability, namely, as the T-system dynamics proceeds, the values at each vertex satisfy a linear recurrence. Conversely, we show that every quiver integrable in this sense is necessarily one of the 19 items in our classification. For the quivers of type
A
^
⊗
A
we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called Goncharov–Kenyon Hamiltonians. We also consider tropical T-systems of type
A
^
⊗
A
and explain how affine slices exhibit solitonic behavior, i.e. soliton resolution and speed conservation. Throughout, we conjecture how the results in the paper are expected to generalize from
A
^
⊗
A
to all other quivers in our classification
Analytical Review of the Biobibliography of Pavlo Zahrebelnyi
No full textThe purpose of the article is to conduct a critical study of the bibliography of Pavlo Zahrebelnyi in the indexes contained in the collection of V. I. Vernadsky National Library of Ukraine (herein after referred to as VNLU) and available in the Internet environment, to analyze the selected bibliographic indexesusing modern methodological developments; to evaluate and provide evidence of their scientific, informational, and cultural and historical value. Methodology. The study used survey-bibliographic, analytical-critical, historical, and comparative scientific methods. Scientific novelty. For the first time, a study of the bibliography, divided into many editions in the form of indexes, was carried out; the artistic and literary orientation of Pavlo Zahrebelnyi’s works was taken into account using new theoretical and methodological principles. The article meets the current tasks of modern bibliographic studies. Conclusions. The works by Pavlo Zahrebelnyi are known outside Ukraine, have been translated into many languages of the world, and have been repeatedly filmed. Not all of them have become outstanding artistic achievements of Ukrainian prose, but all of them have been actively perceived by readers, discussed by critics, and in one way or another influenced the development of the modern literary process. The critical analysis of the indexes logically led the author of the article to the opinion that the compilation of acomplete consolidated bibliographic index is still waiting for its author. The publication of the writer’s private diaries is still ahead, and there is information on the Internet about holding scientific conferences dedicated to the work of Pavlo Zahrebelnyi and, as a result, the publication of articles in periodicals and collections
A2-web immanants
No full textAbstractWe describe the rank 3 Temperley–Lieb–Martin algebras in terms of Kuperberg’s A2-webs. We define consistent labelings of webs and use them to describe the coefficients of decompositions into reduced webs. We introduce web immanants, inspired by Temperley–Lieb immanants of Rhoades and Skandera. We show that web immanants are positive when evaluated on totally positive matrices and describe some further properties
Pavlo Galagan (1853-1869) in the sources of personal origin
No full textУ статті поглиблено уявлення про родинно-генеалогічну історію сімейства Ґалаґанів у другій половині XIX ст. Автор створює «біографічний портрет» Павла Ґалаґана, останнього прямого нащадка роду, який набув високого соціального статусу на початку XVIII ст. У статті
Павло Ґалаґан охарактеризований у контексті життєдіяльності свого соціального середовища на основі інтерпретації змісту джерел особового походження (мемуари, щоденникові записи, листи тощо), створених ним самим та близькими до нього людьми.В статье расширено представление о семейно-генеалогической истории рода Галаганов, представителей украинской элиты казацко-старшинского происхождения, во второй половине XIX в. Автор создает «биографический портрет» Павла Галагана, последнего прямого потомка рода,
который приобрел высокий социальный статус в начале XVIII в. В статье Павел Галаган охарактеризован в контексте жизнедеятельности своей социальной среды на основе интерпретации содержания источников личного происхождения (мемуары, дневниковые записи, письма и т. п.), созданных им самим и близкими к нему людьми.The article explains deep understanding of the family and genealogical history of some families of Ukrainian
elite of Cossack origin in the second half of the 19th century.
Pavlo Galagan, as a personality, who is the last direct descendant of the Cossack Hetman origin who acquired
high social status at the beginning of the 18th century, remains only as a “memorial name”. Grygoriy Galagan
and Catherine Galagan (Kochubey), his parents, to commemorate his son after his premature death, often
mention him in the context of the university foundation. The author creates a “biographical portrait”
of Pavlo Galagan and interprets information about the socio-cultural life of the Galagan family, describes
the attitude of the representatives of Ukrainian elite of the 19th century to the problems of education, religion,
Ukrainian identity etc.
The article describes Pavlo Galagan’s social life in the context of the environment based on the interpretation
of the content sources of personal origin (memoirs, diaries, letters, etc.) created by him and people close to
him. The main focus is devoted to memories of Pavlo Galagan’s father and mother created for the College
“Yearbook” of P. Galagan, published by A. Stepovych in 1896, correspondence, in particular a pile of letters from
father to his son dated the fi rst half of 1860s, travel notes and P. Galagan’s diary, created upon the impressions
from the fi rst trip abroad in 1866.
The author emphasizes that Pavlo Galagan origin and ideological priorities are formed, on the one hand, upon
well-established traditions of training of young elite, on the other hand, according to the liberal-democratic
beliefs and “Ukrainophile” preferences of his parents
Linear Recurrences for Cylindrical Networks
No full textAbstract
We prove a general theorem that gives a linear recurrence for tuples of paths in every cylindrical network. This can be seen as a cylindrical analog of the Lindström–Gessel–Viennot theorem. We illustrate the result by applying it to Schur functions, plane partitions, and domino tilings.</jats:p
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