Pub Date : 2022-10-13DOI: 10.4153/s0008439523000322
Samya Kumar Ray, S. Sarkar
In this article, we study the following question asked by Michael Hartz in a recent paper cite{Hartz}: textit{which operator spaces satisfy the column-row property?} We provide a complete classification of the column-row property for non-commutative $L_{p}$-spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the column-row property and discuss their existence and non-existence for various natural examples of operator spaces.
{"title":"A note on the column-row property","authors":"Samya Kumar Ray, S. Sarkar","doi":"10.4153/s0008439523000322","DOIUrl":"https://doi.org/10.4153/s0008439523000322","url":null,"abstract":"In this article, we study the following question asked by Michael Hartz in a recent paper cite{Hartz}: textit{which operator spaces satisfy the column-row property?} We provide a complete classification of the column-row property for non-commutative $L_{p}$-spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the column-row property and discuss their existence and non-existence for various natural examples of operator spaces.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48389717","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 : 2022-10-11DOI: 10.4153/s0008439523000206
Carlo Collari
A result of Corfield, Sati, and Schreiber asserts that $mathfrak{gl}_n$-weight systems associated to the defining representation are quantum states. In this short note we extend this result to all $mathfrak{gl}_n$-weight systems corresponding to labeling by symmetric and exterior powers of the defining representation.
{"title":"A note on weight systems which are quantum states","authors":"Carlo Collari","doi":"10.4153/s0008439523000206","DOIUrl":"https://doi.org/10.4153/s0008439523000206","url":null,"abstract":"A result of Corfield, Sati, and Schreiber asserts that $mathfrak{gl}_n$-weight systems associated to the defining representation are quantum states. In this short note we extend this result to all $mathfrak{gl}_n$-weight systems corresponding to labeling by symmetric and exterior powers of the defining representation.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48425175","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 : 2022-10-10DOI: 10.4153/S0008439522000601
P. Pollack, Akash Singha Roy
Abstract We investigate the leading digit distribution of the kth largest prime factor of n (for each fixed $k=1,2,3,dots $ ) as well as the sum of all prime factors of n. In each case, we find that the leading digits are distributed according to Benford’s law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.
{"title":"Benford behavior and distribution in residue classes of large prime factors","authors":"P. Pollack, Akash Singha Roy","doi":"10.4153/S0008439522000601","DOIUrl":"https://doi.org/10.4153/S0008439522000601","url":null,"abstract":"Abstract We investigate the leading digit distribution of the kth largest prime factor of n (for each fixed \u0000$k=1,2,3,dots $\u0000 ) as well as the sum of all prime factors of n. In each case, we find that the leading digits are distributed according to Benford’s law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"626 - 642"},"PeriodicalIF":0.6,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43181377","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 : 2022-09-22DOI: 10.4153/s0008439523000516
Zhe Li, Shanwen Wang
In this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra $mathbb {H}$ , we show that, for metapletic–orthogonal dual pair over $mathbb {R}$ and the symplectic–orthogonal dual pair over quaternion algebra $mathbb {H}$ , the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.
{"title":"Compatibility of theta lifts and tempered condition","authors":"Zhe Li, Shanwen Wang","doi":"10.4153/s0008439523000516","DOIUrl":"https://doi.org/10.4153/s0008439523000516","url":null,"abstract":"\u0000 In this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra \u0000 \u0000 \u0000 \u0000$mathbb {H}$\u0000\u0000 \u0000 , we show that, for metapletic–orthogonal dual pair over \u0000 \u0000 \u0000 \u0000$mathbb {R}$\u0000\u0000 \u0000 and the symplectic–orthogonal dual pair over quaternion algebra \u0000 \u0000 \u0000 \u0000$mathbb {H}$\u0000\u0000 \u0000 , the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45492197","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 : 2022-09-21DOI: 10.4153/S0008439523000073
D. McCoy
Abstract In this note, we exhibit concrete examples of characterizing slopes for the knot $12n242$ , also known as the $(-2,3,7)$ -pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the nonconstructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.
{"title":"Characterizing slopes for the \u0000$(-2,3,7)$\u0000 -pretzel knot","authors":"D. McCoy","doi":"10.4153/S0008439523000073","DOIUrl":"https://doi.org/10.4153/S0008439523000073","url":null,"abstract":"Abstract In this note, we exhibit concrete examples of characterizing slopes for the knot \u0000$12n242$\u0000 , also known as the \u0000$(-2,3,7)$\u0000 -pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the nonconstructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"937 - 950"},"PeriodicalIF":0.6,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46972539","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 : 2022-09-02DOI: 10.4153/s0008439523000449
S. Rodney, S. F. MacDonald
Given a $sigma$-finite measure space $(X,mu)$, a Young function $Phi$, and a one-parameter family of Young functions ${Psi_q}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $fin L^Phi(X,mu)$ to satisfy [ lim_{qrightarrow infty}|f|_{L^{Psi_q}(X,mu)}=C|f|_{L^infty(X,mu)}. ] The constant $C$ is independent of $f$ and depends only on the family ${Psi_q}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.
{"title":"Families of Young Functions and Limits of Orlicz Norms","authors":"S. Rodney, S. F. MacDonald","doi":"10.4153/s0008439523000449","DOIUrl":"https://doi.org/10.4153/s0008439523000449","url":null,"abstract":"Given a $sigma$-finite measure space $(X,mu)$, a Young function $Phi$, and a one-parameter family of Young functions ${Psi_q}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $fin L^Phi(X,mu)$ to satisfy [ lim_{qrightarrow infty}|f|_{L^{Psi_q}(X,mu)}=C|f|_{L^infty(X,mu)}. ] The constant $C$ is independent of $f$ and depends only on the family ${Psi_q}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42500604","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 : 2022-08-24DOI: 10.4153/s0008439523000309
N. Ilten, Oscar Lautsch
Abstract We show that a sufficiently general hypersurface of degree d in $mathbb {P}^n$ admits a toric Gröbner degeneration after linear change of coordinates if and only if $dleq 2n-1$ .
{"title":"TORIC DEGENERATIONS OF LOW DEGREE HYPERSURFACES","authors":"N. Ilten, Oscar Lautsch","doi":"10.4153/s0008439523000309","DOIUrl":"https://doi.org/10.4153/s0008439523000309","url":null,"abstract":"Abstract We show that a sufficiently general hypersurface of degree d in \u0000$mathbb {P}^n$\u0000 admits a toric Gröbner degeneration after linear change of coordinates if and only if \u0000$dleq 2n-1$\u0000 .","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46603486","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 : 2022-08-19DOI: 10.4153/S0008439522000510
Adam Brown-Sarre, Mumtaz Hussain
Abstract Let $r=[a_1(r), a_2(r),ldots ]$ be the continued fraction expansion of a real number $rin mathbb R$ . The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let $(t_1, ldots , t_m)in mathbb R_+^m$ , and let $Psi :mathbb {N}rightarrow (1,infty )$ be a function such that $Psi (n)to infty $ as $nto infty $ . We calculate the Hausdorff dimension of the set of all $ (x, y)in [0,1)^2$ such that $$ begin{align*} maxleft{prod_{i=1}^ma_{n+i}^{t_i}(x), prod_{i=1}^ma_{n+i}^{t_i}(y)right} geq Psi(n) end{align*} $$ is satisfied for all $ngeq 1$ .
{"title":"A note on the relative growth of products of multiple partial quotients in the plane","authors":"Adam Brown-Sarre, Mumtaz Hussain","doi":"10.4153/S0008439522000510","DOIUrl":"https://doi.org/10.4153/S0008439522000510","url":null,"abstract":"Abstract Let \u0000$r=[a_1(r), a_2(r),ldots ]$\u0000 be the continued fraction expansion of a real number \u0000$rin mathbb R$\u0000 . The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let \u0000$(t_1, ldots , t_m)in mathbb R_+^m$\u0000 , and let \u0000$Psi :mathbb {N}rightarrow (1,infty )$\u0000 be a function such that \u0000$Psi (n)to infty $\u0000 as \u0000$nto infty $\u0000 . We calculate the Hausdorff dimension of the set of all \u0000$ (x, y)in [0,1)^2$\u0000 such that \u0000$$ begin{align*} maxleft{prod_{i=1}^ma_{n+i}^{t_i}(x), prod_{i=1}^ma_{n+i}^{t_i}(y)right} geq Psi(n) end{align*} $$\u0000 is satisfied for all \u0000$ngeq 1$\u0000 .","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"544 - 552"},"PeriodicalIF":0.6,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47148957","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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