Transactions of A Razmadze Mathematical Institute最新文献

{"title":"Boundary value problems of statics of thermoelasticity of bodies with microstructure and microtemperatures","authors":"L. Giorgashvili,&nbsp;S. Zazashvili","doi":"10.1016/j.trmi.2017.09.002","DOIUrl":"10.1016/j.trmi.2017.09.002","url":null,"abstract":"<div><p>The paper deals with boundary value problems of statics of the thermoelasticity theory of isotropic microstretch materials with microtemperatures and microdilatations. For the system of differential equations of equilibrium the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 30-57"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.09.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45811543","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":"The stability of orthotropic shells of revolution, close to cylindrical ones, with an elastic filler, under the action of torsion, normal pressure and temperature","authors":"Sergo Kukudzhanov","doi":"10.1016/j.trmi.2017.10.005","DOIUrl":"10.1016/j.trmi.2017.10.005","url":null,"abstract":"<div><p>The paper investigates the stability of orthotropic shells of revolution which are by their form close to cylindrical ones, with an elastic filler, under the action of torques, external pressure and temperature. The shell is assumed to be thin and elastic. Temperature is uniformly distributed in the shell body. The filler is simulated by an elastic base. The shells of positive and negative Gaussian curvature are considered. Formulas for finding critical loadings and corresponding forms of stability loss are derived.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 64-72"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.10.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47568972","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":"On the internal tensor structures of the fibration T(Lm(Vn))","authors":"Gocha Todua","doi":"10.1016/j.trmi.2017.07.003","DOIUrl":"10.1016/j.trmi.2017.07.003","url":null,"abstract":"<div><p>In the present paper we consider some internal tensor structures. It is proved that if on the space <span><math><mo>Lm</mo><mrow><mo>(</mo><mi>V</mi><mi>n</mi><mo>)</mo></mrow></math></span> there are tensor fields <span><math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>β</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span> defining almost a dual structure, then there exist such lifts of these tensor fields which represent either almost a binary structure, or almost a complex structure on <span><math><mi>T</mi><mrow><mo>(</mo><mo>Lm</mo><mrow><mo>(</mo><mi>V</mi><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 126-132"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.07.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48242431","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":"Generalized semi-open and pre-semiopen sets via ideals","authors":"Bishwambhar Roy ,&nbsp;Ritu Sen","doi":"10.1016/j.trmi.2017.08.003","DOIUrl":"10.1016/j.trmi.2017.08.003","url":null,"abstract":"<div><p>In this paper we have introduced a new type of sets termed as <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-open sets which unifies semiopen sets, <span><math><mi>β</mi></math></span>-open sets and discussed some of its properties. We have also introduced another type of weak open sets termed as <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>μ</mi></mrow></msub></mrow></msub></math></span>-open sets depending on a GT as well as an ideal on a topological space. Finally the concept of weakly <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>μ</mi></mrow></msub></mrow></msub></math></span>-open sets are investigated.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 95-100"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.08.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45211659","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":"Pointwise multipliers for inhomogeneous multi-parameter Besov and Triebel–Lizorkin spaces associated with mixed homogeneities","authors":"Jian Tan","doi":"10.1016/j.trmi.2017.11.002","DOIUrl":"10.1016/j.trmi.2017.11.002","url":null,"abstract":"<div><p>In this article, we obtain pointwise multipliers on inhomogeneous multi-parameter Besov and Triebel–Lizorkin spaces associated with mixed homogeneities.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 101-114"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.11.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46574431","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":"Jagged non-zero submatrix data structure","authors":"Giga Chalauri ,&nbsp;Vakhtang Laluashvili ,&nbsp;Koba Gelashvili","doi":"10.1016/j.trmi.2017.10.002","DOIUrl":"10.1016/j.trmi.2017.10.002","url":null,"abstract":"<div><p>On the basis of C language matrix having rows of different length, we havedeveloped a new storage format for rectangular matrix. It stores non-zero entries, their column indices and is called jagged non-zero sub-matrix data structure or simply <em>jnz-format</em>.</p><p>In case of simple applications, when the only requirement from the format is to ensure the serial algorithm of multiplying matrix by vector (e.g. conjugate gradient (CG) method), two following issues are experimentally studied: </p><ul><li><span>•</span><span><p>For what amount of zero-entries do we accept the rectangular matrix as sparse, with respect to used memory and speed;</p></span></li><li><span>•</span><span><p>What should the <em>jnz-format</em>’s interface look like.</p></span></li></ul>Determining the interface is comparatively laborious; <em>jnz-format</em> is compared to two approved formats—CRS and <em>Mapped Matrix</em>. In comparisons, CRS format is considered by using two different implementations, whilst <em>jnz</em> and <em>Mapped Matrix</em> —by using one. In comparisons, we use <em>jnz</em> and CRS formats with our own simple interface implementations and CRS and <em>Mapped Matrix</em> with <em>boost</em>’s library interfaces and implementations. Experiments’ results show <em>jnz</em> format’s prospect and visible advantage of the relatively easy interface.<p>All the material regarding experiments can be seen at <span>https://github.com/vakho10/Sparse-Storage-Formats</span><svg><path></path></svg>.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 7-14"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.10.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42646774","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":"On the structure of constituents of finite independent families of convex bodies in R2 and R3 spaces","authors":"Tengiz Tetunashvili","doi":"10.1016/j.trmi.2017.10.004","DOIUrl":"10.1016/j.trmi.2017.10.004","url":null,"abstract":"<div><p>Properties of certain families of subsets of Euclidean spaces are established. Using the established properties theorems concerning the structure of constituents of finite independent families of convex bodies in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> spaces are proved.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 115-125"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.10.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47607352","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":"Potential operators in modified Morrey spaces defined on Carleson curves","authors":"I.B. Dadashova ,&nbsp;C. Aykol ,&nbsp;Z. Cakir ,&nbsp;A. Serbetci","doi":"10.1016/j.trmi.2017.09.004","DOIUrl":"10.1016/j.trmi.2017.09.004","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In this paper we study the potential operator &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; in the modified Morrey space &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and the spaces &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; defined on Carleson curves &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We prove that for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; the potential operator &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is bounded from the modified Morrey space &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; if and in the case of infinite curve only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and from the spaces &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; if and in the case of infinite curve only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;. Furthermore, for the limiting case &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; we show that if &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is an infinite Carleson curve, then the modified potential operator &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is bounded from &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 15-29"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.09.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42524048","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":"On some methods of extending invariant and quasi-invariant measures","authors":"A. Kirtadze ,&nbsp;N. Rusiashvili","doi":"10.1016/j.trmi.2017.08.002","DOIUrl":"10.1016/j.trmi.2017.08.002","url":null,"abstract":"<div><p>In the present paper an approach to some questions in the theory of invariant (quasi-invariant) measures is discussed. It is useful in certain situations, where given topological groups or topological vector spaces are equipped with various nonzero <span><math><mi>σ</mi></math></span>-finite left invariant (left quasi-invariant) measures.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 58-63"},"PeriodicalIF":0.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.08.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44678714","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":"The loop cohomology of a space with the polynomial cohomology algebra","authors":"Samson Saneblidze","doi":"10.1016/j.trmi.2017.07.002","DOIUrl":"10.1016/j.trmi.2017.07.002","url":null,"abstract":"<div><p>Given a simply connected space <span><math><mi>X</mi></math></span> with polynomial cohomology <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∗</mo></mrow></msup><mspace></mspace><mrow><mo>(</mo><mi>X</mi><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>,</mo></math></span> we calculate the loop cohomology algebra <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mi>X</mi><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span> by means of the action of the Steenrod cohomology operation <span><math><mi>S</mi><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>.</mo></math></span> This calculation uses an explicit construction of the minimal Hirsch filtered model of the cochain algebra <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>.</mo></math></span> As a consequence we obtain that <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mi>X</mi><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span> is the exterior algebra if and only if <span><math><mi>S</mi><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is multiplicatively decomposable on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>;</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>.</mo></math></span> The last statement in fact contains a converse of a theorem of A. Borel (Switzer, 1975, Theorem 15.60).</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"171 3","pages":"Pages 389-395"},"PeriodicalIF":0.2,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.07.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80886817","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}