Integral pentagon relations for 3d superconformal indices

The superconformal index of a three-dimensional supersymmetric field theory can be expressed in terms of basic hypergeometric integrals. By comparing the indices of dual theories, one can find new integral identities for basic hypergeometric integrals. Some of these integral identities have the form of the pentagon identity which can be interpreted as the 2-3 Pachner move for triangulated 3-manifolds.Comment: 9 pages. Based on arXiv:1309.2195 with new results and comments. Presented at String-Math conference, Edmonton, Canada, June 9-13, 2014; v2: minor corrections and comments adde

Basic hypergeometry of supersymmetric dualities

We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic proofs and physical interpretations of the presented identities.Comment: 25 pages, v2: minor corrections and comment

Integrable lattice spin models from supersymmetric dualities

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories.Comment: 35 pages, preliminary versio

A new pentagon identity for the tetrahedron index

Recently Kashaev, Luo and Vartanov, using the reduction from a four-dimensional superconformal index to a three-dimensional partition function, found a pentagon identity for a special combination of hyperbolic Gamma functions. Following their idea we have obtained a new pentagon identity for a certain combination of so-called tetrahedron indices arising from the equality of superconformal indices of dual three-dimensional N=2 supersymmetric theories and give a mathematical proof of it.Comment: 13 pages, v2: we added a new section with the proof of the identity, misprints correcte

Comments on the multi-spin solution to the Yang-Baxter equation and basic hypergeometric sum/integral identity

We present a multi-spin solution to the Yang-Baxter equation. The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parameterized in terms of the basic hypergeometric functions. We obtain this solution from a non-trivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional N=2 Seiberg dual theories.Comment: 8 pp, based on a talk given at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, Czech Republic, 06-10 June, 2017; v2: minor change

Neuroloogia ja neurofüsioloogia areng põhjalas ühe elukäigu taustal

Eesti Arst 2012; 91(7):380–38

Keelekasutuse aluspõhimõtted Eesti Vabariigi põhiseaduses ja nende rakendamine

Artiklis vaadeldakse keelekasutust sätestavaid norme Eesti Vabariigi põhiseaduses, et välja selgitada, mil määral tagavad need eesti keele kasutamise kõigis olulisemates eluvaldkondades; kas need on piisavad eesti keele arendamiseks ja kaitsmiseks ning isikute keeleõiguste tagamiseks.Põhiseaduse keelesätete analüüs näitab, et keelekasutuse põhimõtted on põhiseaduses kirjas piisava selguse ning üksikasjalikkusega, võimaldades täita põhiseaduse preambulis seatud eesmärki tagada eesti rahvuse, keele ja kultuuri säilimine läbi aegade. Eesti keele kasutamine on tagatud põhiseaduslike institutsioonide tegevuses ning enamikus eluvaldkondades. Mõnes valdkonnas (kohalike omavalitsuste asjaajamine, haridus, meditsiin) jätab aga põhiseadusega antud eesti keele kasutamise võimaluste rakendamine soovida.Neid puudujääke on üritatud korvata riikliku keelejärelevalvega, kuid keelepoliitika heitlikkust, õigusaktides esinevaid puudujääke ja keeleoskuse probleemidest möödavaatamist ei ole võimalik järelevalvemeetmetega parandada. Põhiseadust aluseks võttes tuleks parandada seadusi ja rakendusakte.Abstract. Ilmar Tomusk: The constitution of the Republic of Estonia on the basic principles of language use, and their application. The article examines the rules of language use provided by the Constitution of the Republic of Estonia. The aim of the article is to find out to what extent the rules of language use ensure the use of the Estonian language in all key areas of life, whether they are sufficient for the development and protection of the Estonian language, and whether they guarantee the linguistic rights of individuals.The analysis of the language provisions of the Constitution shows that the key principles of language use are worded with sufficient clarity that they meet the objective set in the preamble of the Constitution to guarantee the preservation of the Estonian people, the Estonian language and the Estonian culture through the ages. The use of the Estonian language is guaranteed in constitutional institutions and also in most areas of life.In some areas (e.g. the administration of local governments, education and health care), however, the implementation of the main constitutional principles of language use on the legislative level is insufficient.Deficiencies in legislation have been compensated for by state language supervision. But the volatility of language policy, deficiencies in language policy planning and the ignoring of language proficiency problems cannot be compensated for by state language supervision. Language legislation should be aligned with the main constitutional principles of language use.Keywords: Constitution; official language; language of public administration; local government; education; health care; state language supervisio