Pub Date : 2021-09-28DOI: 10.17323/1609-4514-2023-23-2-133-167
B. Bychkov, V. Gorbounov, A. Kazakov, D. Talalaev
We refine the result of T. Lam cite{L} on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $mathrm{Gr}_{geq 0}(n-1,2n)$ by proving first that the image lands in $mathrm{Gr}(n-1,V)subset mathrm{Gr}(n-1,2n)$ where $Vsubset mathbb{R}^{2n}$ is a certain subspace of dimension $2n-2$. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian $mathrm{LG}(n-1,V)subset mathrm{Gr}(n-1,V)$. As it is well known $mathrm{LG}(n-1)$ can be identified with $mathrm{Gr}(n-1,2n-2)cap mathbb{P} L$ where $Lsubset bigwedge^{n-1}mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, moreover it is the space of the fundamental representation of the symplectic group $Sp(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of $E_n$ out of $mathrm{Gr}(n-1,2n)$ found in cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in cite{L} and representation theory of the symplectic group.
{"title":"Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups","authors":"B. Bychkov, V. Gorbounov, A. Kazakov, D. Talalaev","doi":"10.17323/1609-4514-2023-23-2-133-167","DOIUrl":"https://doi.org/10.17323/1609-4514-2023-23-2-133-167","url":null,"abstract":"We refine the result of T. Lam cite{L} on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $mathrm{Gr}_{geq 0}(n-1,2n)$ by proving first that the image lands in $mathrm{Gr}(n-1,V)subset mathrm{Gr}(n-1,2n)$ where $Vsubset mathbb{R}^{2n}$ is a certain subspace of dimension $2n-2$. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian $mathrm{LG}(n-1,V)subset mathrm{Gr}(n-1,V)$. As it is well known $mathrm{LG}(n-1)$ can be identified with $mathrm{Gr}(n-1,2n-2)cap mathbb{P} L$ where $Lsubset bigwedge^{n-1}mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, moreover it is the space of the fundamental representation of the symplectic group $Sp(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of $E_n$ out of $mathrm{Gr}(n-1,2n)$ found in cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in cite{L} and representation theory of the symplectic group.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48299106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-18DOI: 10.17323/1609-4514-2021-21-3-493-506
M. Corrêa, Fernando C. Lourenço, D. Machado, Antonio M. Ferreira
{"title":"On Gauss–Bonnet and Poincaré–Hopf Type Theorems for Complex ∂ -Manifolds","authors":"M. Corrêa, Fernando C. Lourenço, D. Machado, Antonio M. Ferreira","doi":"10.17323/1609-4514-2021-21-3-493-506","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-3-493-506","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43162502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-21DOI: 10.17323/1609-4514-2023-23-2-169-203
Jonathan Godin, C. Rousseau
We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension 1 (i.e. a double fixed point) under conjugacy. These generic unfolding depend on one real parameter. The classification is done by assigning to each such germ a weak and a strong modulus, which are unfoldings of the modulus assigned to the antiholomorphic parabolic point. The weak and the strong moduli are unfoldings of the 'Ecalle-Voronin modulus of the second iterate of the germ which is a real unfolding of a holomorphic parabolic point. A preparation of the unfolding allows to identify one real analytic canonical parameter and any conjugacy between two prepared generic unfoldings preserves the canonical parameter. We also solve the realisation problem by giving necessary and sufficient conditions for a strong modulus to be realized. This is done simultaneously with solving the probem of the existence of an antiholomorphic square root to a germ of generic analytic unfolding of a holomorphic parabolic germ. As a second application we establish the condition for the existence of a real analytic invariant curve.
{"title":"Analytic Classification of Generic Unfoldings of Antiholomorphic Parabolic Fixed Points of Codimension 1","authors":"Jonathan Godin, C. Rousseau","doi":"10.17323/1609-4514-2023-23-2-169-203","DOIUrl":"https://doi.org/10.17323/1609-4514-2023-23-2-169-203","url":null,"abstract":"We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension 1 (i.e. a double fixed point) under conjugacy. These generic unfolding depend on one real parameter. The classification is done by assigning to each such germ a weak and a strong modulus, which are unfoldings of the modulus assigned to the antiholomorphic parabolic point. The weak and the strong moduli are unfoldings of the 'Ecalle-Voronin modulus of the second iterate of the germ which is a real unfolding of a holomorphic parabolic point. A preparation of the unfolding allows to identify one real analytic canonical parameter and any conjugacy between two prepared generic unfoldings preserves the canonical parameter. We also solve the realisation problem by giving necessary and sufficient conditions for a strong modulus to be realized. This is done simultaneously with solving the probem of the existence of an antiholomorphic square root to a germ of generic analytic unfolding of a holomorphic parabolic germ. As a second application we establish the condition for the existence of a real analytic invariant curve.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45521221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.17323/1609-4514-2021-21-1-175-190
T. Szarek, A. Zdunik
{"title":"The Central Limit Theorem for Iterated Function Systems on the Circle","authors":"T. Szarek, A. Zdunik","doi":"10.17323/1609-4514-2021-21-1-175-190","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-1-175-190","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42051548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-15DOI: 10.17323/1609-4514-2022-22-2-265-294
A. Blokh, L. Oversteegen, Anastasia Shepelevtseva, V. Timorin
. The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices . We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and agrees with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei.
{"title":"Modeling Core Parts of Zakeri Slices I","authors":"A. Blokh, L. Oversteegen, Anastasia Shepelevtseva, V. Timorin","doi":"10.17323/1609-4514-2022-22-2-265-294","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-2-265-294","url":null,"abstract":". The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices . We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and agrees with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42341577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-18DOI: 10.17323/1609-4514-2022-22-3-427-450
A. Dimca, Gabriel Sticlaru
We show that two questions raised by Brian Harbourne related to the bounded negativity conjecture and singular plane curves have a negative answer in some cases. For rational curves having only ordinary singularities, this question is shown to be related to strong new bounds on the number of singularities of multiplicity greater or equal to 3 such a curve may have. This fact suggests a conjecture on the non-existence of rational curves of degree d > 8 having only ordinary triple points as singularities. We also give lower bounds for theH-constant H(C) in terms of the maximal multiplicity of the singularities of C, or when C has only singularities of type As with 1 ≤ s ≤ 5 and D4.
{"title":"On the Bounded Negativity Conjecture and Singular Plane Curves","authors":"A. Dimca, Gabriel Sticlaru","doi":"10.17323/1609-4514-2022-22-3-427-450","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-3-427-450","url":null,"abstract":"We show that two questions raised by Brian Harbourne related to the bounded negativity conjecture and singular plane curves have a negative answer in some cases. For rational curves having only ordinary singularities, this question is shown to be related to strong new bounds on the number of singularities of multiplicity greater or equal to 3 such a curve may have. This fact suggests a conjecture on the non-existence of rational curves of degree d > 8 having only ordinary triple points as singularities. We also give lower bounds for theH-constant H(C) in terms of the maximal multiplicity of the singularities of C, or when C has only singularities of type As with 1 ≤ s ≤ 5 and D4.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44739856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-4-789-805
S. Louboutin
{"title":"On Ennola's Conjecture on Non-Galois Cubic Number Fields with Exceptional Units","authors":"S. Louboutin","doi":"10.17323/1609-4514-2021-21-4-789-805","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-4-789-805","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67825838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-4-767-788
Davi Lima, C. Matheus, C. Moreira, Sandoel Vieira
{"title":"M ∖ L Near 3","authors":"Davi Lima, C. Matheus, C. Moreira, Sandoel Vieira","doi":"10.17323/1609-4514-2021-21-4-767-788","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-4-767-788","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67825957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-2-413-426
N. G. Pavlova, A. O. Remizov
{"title":"Smooth Local Normal Forms of Hyperbolic Roussarie Vector Fields","authors":"N. G. Pavlova, A. O. Remizov","doi":"10.17323/1609-4514-2021-21-2-413-426","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-413-426","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67826036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-2-287-323
A. Dosi
{"title":"The Spectrum of a Module along Scheme Morphism and Multi-Operator Functional Calculus","authors":"A. Dosi","doi":"10.17323/1609-4514-2021-21-2-287-323","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-287-323","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67825132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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