These are detailed notes for a lecture on "Sharp restriction theory" which I presented as part of my "Agregação em Matemática" in Instituto Superior Técnico, Lisboa, Portugal (9-10 February, 2023).
{"title":"Sharp Restriction Theory","authors":"Diogo Oliveira E Silva","doi":"10.46298/cm.12415","DOIUrl":"https://doi.org/10.46298/cm.12415","url":null,"abstract":"These are detailed notes for a lecture on \"Sharp restriction theory\" which I presented as part of my \"Agregação em Matemática\" in Instituto Superior Técnico, Lisboa, Portugal (9-10 February, 2023).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"32 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141350079","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}
Artem Lopatin, Carlos Arturo Rodriguez Palma, Liming Tang
In this paper we investigate weak polynomial identities for the Weyl algebra $mathsf{A}_1$ over an infinite field of arbitrary characteristic. Namely, we describe weak polynomial identities of the minimal degree, which is three, and of degrees 4 and 5. We also describe weak polynomial identities is two variables.
{"title":"Weak polynomial identities of small degree for the Weyl algebra","authors":"Artem Lopatin, Carlos Arturo Rodriguez Palma, Liming Tang","doi":"10.46298/cm.13107","DOIUrl":"https://doi.org/10.46298/cm.13107","url":null,"abstract":"In this paper we investigate weak polynomial identities for the Weyl algebra\u0000$mathsf{A}_1$ over an infinite field of arbitrary characteristic. Namely, we\u0000describe weak polynomial identities of the minimal degree, which is three, and\u0000of degrees 4 and 5. We also describe weak polynomial identities is two\u0000variables.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"49 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140440144","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}
We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.
{"title":"A complete invariant for doodles on a 2-sphere","authors":"Jacob Mostovoy","doi":"10.46298/cm.12893","DOIUrl":"https://doi.org/10.46298/cm.12893","url":null,"abstract":"We define a complete invariant for doodles on a 2-sphere which takes values\u0000in series of chord diagrams of certain type. The coefficients at the diagrams\u0000with $n$ chords are finite type invariants of doodles of order at most $2n$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"71 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140505258","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}
Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``$preceq$-morphisms'' preserving a surpassing relation $preceq$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.
{"title":"Lie pairs","authors":"Letterio Gatto, Louis Rowen","doi":"10.46298/cm.12413","DOIUrl":"https://doi.org/10.46298/cm.12413","url":null,"abstract":"Extending the theory of systems, we introduce a theory of Lie semialgebra\u0000``pairs'' which parallels the classical theory of Lie algebras, but with a\u0000``null set'' replacing $0$. A selection of examples is given. These Lie pairs\u0000comprise two categories in addition to the universal algebraic definition, one\u0000with ``weak Lie morphisms'' preserving null sums, and the other with\u0000``$preceq$-morphisms'' preserving a surpassing relation $preceq$ that\u0000replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt)\u0000Theorem in these three categories.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"61 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139451033","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}
These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agregac{c}~ao em Matem'atica e Applicac{c}~oes (University of Beira Interior, Covilh~a, Portugal, 13-14/03/2023).
这些是我在Agregac{c}~ao em Matem'atica e Applicac{c}~oes (University of Beira Interior, Covilh~a, Portugal, 13-14/03/2023)中关于“非结合代数结构:分类和结构”讲座的详细笔记。
{"title":"Non-associative algebraic structures: classification and structure","authors":"Ivan Kaygorodov","doi":"10.46298/cm.11419","DOIUrl":"https://doi.org/10.46298/cm.11419","url":null,"abstract":"These are detailed notes for a lecture on \"Non-associative Algebraic Structures: Classification and Structure\" which I presented as a part of my Agregac{c}~ao em Matem'atica e Applicac{c}~oes (University of Beira Interior, Covilh~a, Portugal, 13-14/03/2023).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"18 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135821112","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}
We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.
{"title":"Alternating Roots of Polynomials over Cayley-Dickson Algebras","authors":"Adam Chapman, Ilan Levin","doi":"10.46298/cm.11514","DOIUrl":"https://doi.org/10.46298/cm.11514","url":null,"abstract":"We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136376603","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}
In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.
{"title":"Well-Rounded ideal lattices of cyclic cubic and quartic fields","authors":"Dat T. Tran, Nam H. Le, Ha T. N. Tran","doi":"10.46298/cm.11138","DOIUrl":"https://doi.org/10.46298/cm.11138","url":null,"abstract":"In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135824339","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}
In this editorial survey we introduce the special issue of the journal Communications in Mathematics on the topic in the title of the article. Our main goal is to briefly outline some of the main aspects of this important area at the intersection of theory and applications, providing the context for the articles showcased in this special issue.
{"title":"Euclidean lattices: theory and applications","authors":"Lenny Fukshansky, Camilla Hollanti","doi":"10.46298/cm.11596","DOIUrl":"https://doi.org/10.46298/cm.11596","url":null,"abstract":"In this editorial survey we introduce the special issue of the journal Communications in Mathematics on the topic in the title of the article. Our main goal is to briefly outline some of the main aspects of this important area at the intersection of theory and applications, providing the context for the articles showcased in this special issue.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135824342","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}
It is shown that the variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. The Gr"obner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4. Furthermore, we demonstrate that every transposed Poisson algebra is F-manifold. We verify that the special identities of GD-algebras hold in transposed Poisson algebras. Finally, we propose a conjecture stating that every transposed Poisson algebra is special, i.e., can be embedded into a differential Poisson algebra.
{"title":"Some generalizations of the variety of transposed Poisson algebras","authors":"B. K. Sartayev","doi":"10.46298/cm.11346","DOIUrl":"https://doi.org/10.46298/cm.11346","url":null,"abstract":"It is shown that the variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. The Gr\"obner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4. Furthermore, we demonstrate that every transposed Poisson algebra is F-manifold. We verify that the special identities of GD-algebras hold in transposed Poisson algebras. Finally, we propose a conjecture stating that every transposed Poisson algebra is special, i.e., can be embedded into a differential Poisson algebra.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293860","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}
A relaxation in the tropical sandpile model is a process of deforming a tropical hypersurface towards a finite collection of points. We show that, in the one-dimensional case, a relaxation terminates after a finite number of steps. We present experimental evidence suggesting that the number of such steps obeys a power law.
{"title":"Relaxation in one-dimensional tropical sandpile","authors":"Mikhail Shkolnikov","doi":"10.46298/cm.10483","DOIUrl":"https://doi.org/10.46298/cm.10483","url":null,"abstract":"A relaxation in the tropical sandpile model is a process of deforming a tropical hypersurface towards a finite collection of points. We show that, in the one-dimensional case, a relaxation terminates after a finite number of steps. We present experimental evidence suggesting that the number of such steps obeys a power law.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957987","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：