Pub Date : 2022-02-28DOI: 10.4153/s0008414x23000470
Franco Giovenzana, Luca Giovenzana, C. Onorati
We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface.
{"title":"On the period of Li, Pertusi and Zhao’s symplectic variety","authors":"Franco Giovenzana, Luca Giovenzana, C. Onorati","doi":"10.4153/s0008414x23000470","DOIUrl":"https://doi.org/10.4153/s0008414x23000470","url":null,"abstract":"We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"34 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88012401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-15DOI: 10.4153/S0008414X22000062
M. A. Hernández Cifre, Miriam Tárraga, J. Yepes Nicolás
Abstract In this paper, we consider the family of nth degree polynomials whose coefficients form a log-convex sequence (up to binomial weights), and investigate their roots. We study, among others, the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every $nin mathbb {N}$ , $ngeq 2$ . Dual Steiner polynomials of star bodies are a particular case of them, and so we derive, as a consequence, further properties for their roots.
{"title":"On the roots of polynomials with log-convex coefficients","authors":"M. A. Hernández Cifre, Miriam Tárraga, J. Yepes Nicolás","doi":"10.4153/S0008414X22000062","DOIUrl":"https://doi.org/10.4153/S0008414X22000062","url":null,"abstract":"Abstract In this paper, we consider the family of nth degree polynomials whose coefficients form a log-convex sequence (up to binomial weights), and investigate their roots. We study, among others, the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every \u0000$nin mathbb {N}$\u0000 , \u0000$ngeq 2$\u0000 . Dual Steiner polynomials of star bodies are a particular case of them, and so we derive, as a consequence, further properties for their roots.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"53 1","pages":"470 - 493"},"PeriodicalIF":0.7,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73332105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.4153/s0008414x22000049
{"title":"CJM volume 74 issue 1 Cover and Front matter","authors":"","doi":"10.4153/s0008414x22000049","DOIUrl":"https://doi.org/10.4153/s0008414x22000049","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"29 1","pages":"f1 - f3"},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83223078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.4153/s0008414x22000050
{"title":"CJM volume 74 issue 1 Cover and Back matter","authors":"","doi":"10.4153/s0008414x22000050","DOIUrl":"https://doi.org/10.4153/s0008414x22000050","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"11 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84407891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.4153/S0008414X22000517
J. Bell, Daniel Smertnig
Abstract A multivariate, formal power series over a field K is a Bézivin series if all of its coefficients can be expressed as a sum of at most r elements from a finitely generated subgroup $G le K^*$ ; it is a Pólya series if one can take $r=1$ . We give explicit structural descriptions of D-finite Bézivin series and D-finite Pólya series over fields of characteristic $0$ , thus extending classical results of Pólya and Bézivin to the multivariate setting.
{"title":"D-finite multivariate series with arithmetic restrictions on their coefficients","authors":"J. Bell, Daniel Smertnig","doi":"10.4153/S0008414X22000517","DOIUrl":"https://doi.org/10.4153/S0008414X22000517","url":null,"abstract":"Abstract A multivariate, formal power series over a field K is a Bézivin series if all of its coefficients can be expressed as a sum of at most r elements from a finitely generated subgroup \u0000$G le K^*$\u0000 ; it is a Pólya series if one can take \u0000$r=1$\u0000 . We give explicit structural descriptions of D-finite Bézivin series and D-finite Pólya series over fields of characteristic \u0000$0$\u0000 , thus extending classical results of Pólya and Bézivin to the multivariate setting.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"18 1","pages":"1745 - 1779"},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82399972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.4153/s0008414x23000044
Thomas Gotfredsen, Sophie Emma Zegers
. We investigate quantum lens spaces, C ( L 2 n +1 q ( r ; m )), introduced by Brzezi´nski-Szyma´nski as graph C ∗ -algebras. We give a new description of C ( L 2 n +1 q ( r ; m )) as graph C ∗ -algebras amending an error in the original paper by Brzezi´nski-Szyma´nski. Furthermore, for n ≤ 3, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r . This builds upon the work of Eilers, Restorff, Ruiz and Sørensen.
{"title":"On The Classification and Description of Quantum Lens Spaces as Graph algebras","authors":"Thomas Gotfredsen, Sophie Emma Zegers","doi":"10.4153/s0008414x23000044","DOIUrl":"https://doi.org/10.4153/s0008414x23000044","url":null,"abstract":". We investigate quantum lens spaces, C ( L 2 n +1 q ( r ; m )), introduced by Brzezi´nski-Szyma´nski as graph C ∗ -algebras. We give a new description of C ( L 2 n +1 q ( r ; m )) as graph C ∗ -algebras amending an error in the original paper by Brzezi´nski-Szyma´nski. Furthermore, for n ≤ 3, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r . This builds upon the work of Eilers, Restorff, Ruiz and Sørensen.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88128550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-28DOI: 10.4153/S0008414X22000554
Laurent M'enard
Abstract We derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $beta = 1/2$ . Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$ . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$ . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.
{"title":"Percolation probability and critical exponents for site percolation on the UIPT","authors":"Laurent M'enard","doi":"10.4153/S0008414X22000554","DOIUrl":"https://doi.org/10.4153/S0008414X22000554","url":null,"abstract":"Abstract We derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is \u0000$beta = 1/2$\u0000 . Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like \u0000$n^{-1/7}$\u0000 . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like \u0000$n^{-4/3}$\u0000 . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"138 1","pages":"1869 - 1903"},"PeriodicalIF":0.7,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74100157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-20DOI: 10.4153/S0008414X22000384
M. Jury, R. Martin, E. Shamovich
Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.
{"title":"Noncommutative rational Clark measures","authors":"M. Jury, R. Martin, E. Shamovich","doi":"10.4153/S0008414X22000384","DOIUrl":"https://doi.org/10.4153/S0008414X22000384","url":null,"abstract":"Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over \u0000$mathbb {C} ^d$\u0000 is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital \u0000$C^*$\u0000 -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"409 1","pages":"1393 - 1445"},"PeriodicalIF":0.7,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79833105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-31DOI: 10.4153/S0008414X22000645
Zhuofeng He, Sihan Wei
For every one-sided shift space $X$ over a finite alphabet, left special elements are those points in $X$ having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner $C^*$-algebra $mathcal{O}_X$ has nuclear dimension 1 when $X$ is minimal and the number of left special elements in $X$ is finite. This is done by describing thoroughly the cover of $X$ which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.
{"title":"A NOTE ON THE NUCLEAR DIMENSION OF CUNTZ-PIMSNER C*-ALGEBRAS ASSOCIATED WITH MINIMAL SHIFT SPACES","authors":"Zhuofeng He, Sihan Wei","doi":"10.4153/S0008414X22000645","DOIUrl":"https://doi.org/10.4153/S0008414X22000645","url":null,"abstract":"For every one-sided shift space $X$ over a finite alphabet, left special elements are those points in $X$ having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner $C^*$-algebra $mathcal{O}_X$ has nuclear dimension 1 when $X$ is minimal and the number of left special elements in $X$ is finite. This is done by describing thoroughly the cover of $X$ which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81741920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-10DOI: 10.4153/S0008414X22000487
M. Bousquet-M'elou
Abstract In the past $20$ years, the enumeration of plane lattice walks confined to a convex cone—normalized into the first quadrant—has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of these results deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps. More recently, similar questions have been raised for nonconvex cones, typically the three-quadrant cone $mathcal {C} = { (i,j) : i geq 0 text { or } j geq 0 }$ . They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in $mathcal {C}$ , which can be seen as the first level of difficulty beyond quadrant problems. This collection consists of diagonally symmetric models in ${-1, 0,1}^2setminus {(-1,1), (1,-1)}$ . Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte’s notion of invariants, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model ${ nearrow , nwarrow , swarrow , searrow }$ , which is D-finite. The three algebraic models are those of the Kreweras trilogy, $mathcal S={nearrow , leftarrow , downarrow }$ , $mathcal S^*={rightarrow , uparrow , swarrow }$ , and $mathcal Scup mathcal S^*$ . Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in $mathcal S$ is an explicit rational function in the quadrant generating function with steps in $mathscr S:= {(j-i,j): (i,j) in mathcal S}$ . We derive various exact or asymptotic corollaries, including an explicit algebraic description of a positive harmonic function in $mathcal C$ for the (reverses of the) five models that are at least D-finite.
{"title":"Enumeration of three-quadrant walks via invariants: some diagonally symmetric models","authors":"M. Bousquet-M'elou","doi":"10.4153/S0008414X22000487","DOIUrl":"https://doi.org/10.4153/S0008414X22000487","url":null,"abstract":"Abstract In the past \u0000$20$\u0000 years, the enumeration of plane lattice walks confined to a convex cone—normalized into the first quadrant—has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of these results deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps. More recently, similar questions have been raised for nonconvex cones, typically the three-quadrant cone \u0000$mathcal {C} = { (i,j) : i geq 0 text { or } j geq 0 }$\u0000 . They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in \u0000$mathcal {C}$\u0000 , which can be seen as the first level of difficulty beyond quadrant problems. This collection consists of diagonally symmetric models in \u0000${-1, 0,1}^2setminus {(-1,1), (1,-1)}$\u0000 . Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte’s notion of invariants, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model \u0000${ nearrow , nwarrow , swarrow , searrow }$\u0000 , which is D-finite. The three algebraic models are those of the Kreweras trilogy, \u0000$mathcal S={nearrow , leftarrow , downarrow }$\u0000 , \u0000$mathcal S^*={rightarrow , uparrow , swarrow }$\u0000 , and \u0000$mathcal Scup mathcal S^*$\u0000 . Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in \u0000$mathcal S$\u0000 is an explicit rational function in the quadrant generating function with steps in \u0000$mathscr S:= {(j-i,j): (i,j) in mathcal S}$\u0000 . We derive various exact or asymptotic corollaries, including an explicit algebraic description of a positive harmonic function in \u0000$mathcal C$\u0000 for the (reverses of the) five models that are at least D-finite.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"29 12 1","pages":"1566 - 1632"},"PeriodicalIF":0.7,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82730586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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