Transactions of A Razmadze Mathematical Institute最新文献

{"title":"On ∗ and Ψ operators in topological spaces with ideals","authors":"Shyamapada Modak,&nbsp;Md. Monirul Islam","doi":"10.1016/j.trmi.2018.07.002","DOIUrl":"10.1016/j.trmi.2018.07.002","url":null,"abstract":"<div><p>The paper concerns operators in ideal topological spaces. Some characterizations of Hayashi–Samuel spaces, <span><math><mi>Ψ</mi></math></span>-<span><math><mi>C</mi></math></span> sets and semi-open sets of <span><math><mo>∗</mo></math></span>-topology are investigated. Continuity and decomposition are also part of this paper.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 3","pages":"Pages 491-497"},"PeriodicalIF":0.2,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2018.07.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48299323","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":"Generated sets of the complete semigroup binary relations defined by semilattices of the finite chains","authors":"Omari Givradze,&nbsp;Yasha Diasamidze,&nbsp;Nino Tsinaridze","doi":"10.1016/j.trmi.2018.08.004","DOIUrl":"10.1016/j.trmi.2018.08.004","url":null,"abstract":"<div><p>In this article, we study generated sets of the complete semigroups of binary relations defined by <span><math><mi>X</mi></math></span>-semilattices unions of the finite chains. We found uniquely irreducible generating set for the given semigroups.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 3","pages":"Pages 378-387"},"PeriodicalIF":0.2,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2018.08.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41820524","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":"Besov continuity for global operators on compact Lie groups: The critical case p=q=∞.","authors":"Duván Cardona","doi":"10.1016/j.trmi.2018.08.001","DOIUrl":"10.1016/j.trmi.2018.08.001","url":null,"abstract":"<div><p>In this note, we study the mapping properties of global pseudo-differential operators with symbols in Ruzhansky–Turunen classes on Besov spaces <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></math></span> The considered classes satisfy Fefferman type conditions of limited regularity.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 3","pages":"Pages 354-360"},"PeriodicalIF":0.2,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2018.08.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55644079","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":"Connections between a system of forward–backward SDEs and backward stochastic PDEs related to the utility maximization problem","authors":"Michael Mania ,&nbsp;Revaz Tevzadze","doi":"10.1016/j.trmi.2018.08.003","DOIUrl":"https://doi.org/10.1016/j.trmi.2018.08.003","url":null,"abstract":"<div><p>Connections between a system of Forward–Backward SDEs derived in Horst et al., (2014) and Backward Stochastic PDEs (Mania and Tevzadze, 2010) related to the utility maximization problem are established. Besides, we derive another version of Forward–Backward SDE of the same problem and prove the existence of solution.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 3","pages":"Pages 429-439"},"PeriodicalIF":0.2,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2018.08.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91770822","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":"Study residuated lattice via some elements","authors":"R. Tayebi Khorami ,&nbsp;A. Borumand Saeid","doi":"10.1016/j.trmi.2018.02.003","DOIUrl":"10.1016/j.trmi.2018.02.003","url":null,"abstract":"<div><p>In this paper, the notions of distributive, standard and neutral elements in residuated lattices were introduced and relationships between them were investigated. Also we study the sets of distributive, standard and neutral elements in residuated lattices. Then we show that under some conditions, the sets of distributive, standard and neutral elements in residuated lattices become a <span><math><mi>M</mi><mi>T</mi><mi>L</mi></math></span>-algebra. Finally, special elements of type 2 in residuated lattices were introduced.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 2","pages":"Pages 238-250"},"PeriodicalIF":0.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2018.02.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41717249","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":"Classes of pseudo BL-algebras with right Boolean lifting property","authors":"B. Barani nia ,&nbsp;A. Borumand Saeid","doi":"10.1016/j.trmi.2017.09.003","DOIUrl":"10.1016/j.trmi.2017.09.003","url":null,"abstract":"<div><p>In this paper, we define the right Boolean lifting property (left Boolean lifting property) RBLP (LBLP) for pseudo BL-algebra to be the property that all Boolean elements can be lifted modulo every right filter (left filter) and next we study the behavior of RBLP (LBLP) with respect to direct products of pseudo BL-algebra. We introduce some conditions, which turn out to be a strengthening and a weakling of RBLP (LBLP) respectively and which open new ways of approaching the study of the RBLP (LBLP) in pseudo BL-algebras.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 2","pages":"Pages 146-163"},"PeriodicalIF":0.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.09.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55644356","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":"Stationary distribution and extinction of a three-species food chain stochastic model","authors":"Yonggang Ma ,&nbsp;Qimin Zhang","doi":"10.1016/j.trmi.2017.12.002","DOIUrl":"10.1016/j.trmi.2017.12.002","url":null,"abstract":"<div><p>In this paper, we investigate long-time behaviour of a stochastic three-species food chain model. By Markov semigroups theory, we prove that the densities of this model can converge to an invariant density or can converge weakly to a singular measure in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> under appropriate conditions. Further, several sufficient conditions for the extinction of the three species were obtained. Finally, numerical simulations are carried out to illustrate our theoretical results.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 2","pages":"Pages 251-264"},"PeriodicalIF":0.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.12.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48141373","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 uniqueness theorem for cohomologies on the category of polyhedral pairs","authors":"L. Mdzinarishvili","doi":"10.1016/j.trmi.2018.03.002","DOIUrl":"10.1016/j.trmi.2018.03.002","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a topological space and &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be a direct system of all compact subsets &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, directed by inclusions. For any homology theory &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; the groups &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∣&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; constitute a direct system, and the maps &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; define a homomorphism &lt;span&gt;&lt;math&gt;&lt;msub&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;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⟶&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/p&gt;&lt;p&gt;As is known (Theorem 4.4.6, Spanier, 1966), for the singular homology, the homomorphism &lt;span&gt;&lt;math&gt;&lt;msub&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;/msub&gt;&lt;/math&gt;&lt;/span&gt; is an isomorphism &lt;span&gt;&lt;span&gt;&lt;span&gt;(1)&lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&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;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⟶&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mo&gt;⟶&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;Using the isomorphism &lt;span&gt;(1)&lt;/span&gt;, it is proved that for the homologies having compact support &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; there is the uniqueness theorem on the category of polyhedral pairs (Theorem 4.8.14, Spanier, 1966).&lt;/p&gt;&lt;p&gt;Since the singular homology theory is a homology theory with compact supports, the uniqueness theorem connects all homology theories having compact supports with the singular homology theory.&lt;/p&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; be a cohomology theory. The groups &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;ms","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 2","pages":"Pages 265-275"},"PeriodicalIF":0.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2018.03.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47324196","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":"Weighted Hardy-type inequalities involving convex function for fractional calculus operators","authors":"Sajid Iqbal ,&nbsp;Josip Pečarić ,&nbsp;Lars-Erik Persson ,&nbsp;Zivorad Tomovski","doi":"10.1016/j.trmi.2017.12.001","DOIUrl":"10.1016/j.trmi.2017.12.001","url":null,"abstract":"<div><p>The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer fractional derivative and fractional integral operator with generalized Mittag-Leffler function in its kernel. We also discuss one dimensional cases of our related results. As a special case of our general results we obtain the results of Iqbal et al. (2017). Moreover, the refinement of Hardy-type inequalities for Hilfer fractional derivative is also included.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 2","pages":"Pages 205-222"},"PeriodicalIF":0.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.12.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44194067","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":"Some aspects of picture fuzzy set","authors":"Palash Dutta,&nbsp;Silpashree Ganju","doi":"10.1016/j.trmi.2017.10.006","DOIUrl":"10.1016/j.trmi.2017.10.006","url":null,"abstract":"<div><p>Picture fuzzy set (PFS) is a recently developed tool to deal with uncertainty which is a direct extension of intuitionistic fuzzy set (IFS) that can model uncertainty in such situations involving more answers of these types: yes, abstain, no. In this paper, (<span><math><mi>α</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>β</mi></math></span>)-cut and strong (<span><math><mi>α</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>β</mi></math></span>)-cut of PFS have been defined and decomposition theorems of PFS are proved. Later on extension principle for PFS has been defined and studied some of its properties. Finally, picture fuzzy arithmetic based on extension principle has been performed with examples.</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 2","pages":"Pages 164-175"},"PeriodicalIF":0.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.10.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47581340","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}