In this article, we conjecture exact estimates for the Weyl-invariant Opdam-Cherednik hypergeometric functions. We prove the conjecture for the root system $A_n$ and for all rank 1 cases. We provide other evidence that the conjecture might be true in general.
{"title":"Sharp estimates for the hypergeometric functions related to root systems of type $A$ and of rank $1$","authors":"P. Graczyk, P. Sawyer","doi":"10.4064/cm8893-5-2023","DOIUrl":"https://doi.org/10.4064/cm8893-5-2023","url":null,"abstract":"In this article, we conjecture exact estimates for the Weyl-invariant Opdam-Cherednik hypergeometric functions. We prove the conjecture for the root system $A_n$ and for all rank 1 cases. We provide other evidence that the conjecture might be true in general.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42062519","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}
Let L be a finite Galois field extension of K with Galois group G. We decompose any idempotent 2-cocycle f using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra Af . We specialize the results in case f is the corresponding idempotent 2-cocycle fr for some semilinear map r ∶ G → Ω, where Ω is a multiplicative monoid with minimum element.
{"title":"Decomposition of idempotent 2-cocycles","authors":"C. Lamprakis, Th. Theohari-Apostolidi","doi":"10.4064/cm8720-3-2022","DOIUrl":"https://doi.org/10.4064/cm8720-3-2022","url":null,"abstract":"Let L be a finite Galois field extension of K with Galois group G. We decompose any idempotent 2-cocycle f using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra Af . We specialize the results in case f is the corresponding idempotent 2-cocycle fr for some semilinear map r ∶ G → Ω, where Ω is a multiplicative monoid with minimum element.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43974197","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}
. The relation between tempered distributions and measures is analysed and clar-ified. While this is straightforward for positive measures, it is surprisingly subtle for signed or complex measures.
{"title":"A note on tempered measures","authors":"M. Baake, Nicolae Strungaru","doi":"10.4064/cm8796-7-2022","DOIUrl":"https://doi.org/10.4064/cm8796-7-2022","url":null,"abstract":". The relation between tempered distributions and measures is analysed and clar-ified. While this is straightforward for positive measures, it is surprisingly subtle for signed or complex measures.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46846170","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}
We show that the spaces of transfinite words, namely ordinal-indexed words, over a Noetherian space, is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its dimension and on its stature.
{"title":"Infinitary Noetherian constructions II. Transfinite words and the regular subword topology","authors":"J. Goubault-Larrecq, Simon Halfon, Aliaume Lopez","doi":"10.4064/cm8793-3-2023","DOIUrl":"https://doi.org/10.4064/cm8793-3-2023","url":null,"abstract":"We show that the spaces of transfinite words, namely ordinal-indexed words, over a Noetherian space, is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its dimension and on its stature.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46364799","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}
Let Λ n be a radical square zero Nakayama algebra with n simple modules and Γ n the Auslander algebra of Λ n . We calculate the number j τ - tilt Γ n j of τ -tilting modules and the number j s τ - tilt Γ n j of support τ -tilting modules over Γ n . In particular, we prove that there are recurrence relations: from which the exact values of j τ - tilt Γ n j and j s τ - tilt Γ n j are derived. Mathematics
{"title":"On the number of $tau $-tilting modules over the Auslander algebras of radical square zero Nakayama algebras","authors":"Hanpeng Gao, Zongzhen Xie, Zhaoyong Huang","doi":"10.4064/cm8474-7-2021","DOIUrl":"https://doi.org/10.4064/cm8474-7-2021","url":null,"abstract":"Let Λ n be a radical square zero Nakayama algebra with n simple modules and Γ n the Auslander algebra of Λ n . We calculate the number j τ - tilt Γ n j of τ -tilting modules and the number j s τ - tilt Γ n j of support τ -tilting modules over Γ n . In particular, we prove that there are recurrence relations: from which the exact values of j τ - tilt Γ n j and j s τ - tilt Γ n j are derived. Mathematics","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70140884","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}
I. Assani, R. S. Hallyburton, S. McMahon, S. Schmidt, C. Schoone
{"title":"New estimates on the Brunel operator","authors":"I. Assani, R. S. Hallyburton, S. McMahon, S. Schmidt, C. Schoone","doi":"10.4064/cm8557-5-2021","DOIUrl":"https://doi.org/10.4064/cm8557-5-2021","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70141226","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}
{"title":"Structure Lie operator on real hypersurfaces of complex two-plane Grassmannians","authors":"Yaning Wang","doi":"10.4064/cm8558-1-2022","DOIUrl":"https://doi.org/10.4064/cm8558-1-2022","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70141287","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}
. Let X be a linear lattice, let x ∈ X + , and let τ be a locally solid topology on X . We present four conditions equivalent to the τ -compactness of the order interval [0 , x ] in X , including the following ones: (i) there is a set S and an affine homeomorphism of [0 , x ] onto the Tychonoff cube [0 , 1] S which preserves order; (ii) C x , the set of components of x , is τ -compact and [0 , x ] is order σ -complete. In the special case where X is a Banach lattice and τ is its norm topology, another equivalent condition is: (iii) C x is weakly compact and [0 , x ] is order σ -complete.
{"title":"Compactness of order intervals in a locally solid linear lattice","authors":"Z. Lipecki","doi":"10.4064/cm8624-11-2021","DOIUrl":"https://doi.org/10.4064/cm8624-11-2021","url":null,"abstract":". Let X be a linear lattice, let x ∈ X + , and let τ be a locally solid topology on X . We present four conditions equivalent to the τ -compactness of the order interval [0 , x ] in X , including the following ones: (i) there is a set S and an affine homeomorphism of [0 , x ] onto the Tychonoff cube [0 , 1] S which preserves order; (ii) C x , the set of components of x , is τ -compact and [0 , x ] is order σ -complete. In the special case where X is a Banach lattice and τ is its norm topology, another equivalent condition is: (iii) C x is weakly compact and [0 , x ] is order σ -complete.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70141698","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}
. Writing p 1 ( n ) < · · · < p r ( n ) for the distinct prime divisors of a given integer n ≥ 2 and letting, for a fixed λ ∈ (0 , 1] , U λ ( n ) := # { j ∈ { 1 , . . . , r − 1 } : log p j ( n ) / log p j +1 ( n ) < λ } , we recently proved that U λ ( n ) /r ∼ λ for almost all integers n ≥ 2 . Now, given λ ∈ (0 , 1) and p ∈ ℘ , the set of prime numbers, let B λ ( p ) := { q ∈ ℘ : λ < log q log p < 1 /λ } and consider the arithmetic function u λ ( n ) := # { p | n : ( n/p, B λ ( p )) = 1 } . Here, we prove that (cid:80) n ≤ x ( u λ ( n ) − λ 2 log log n ) 2 = ( C + o (1)) x log log x as x → ∞ , where C is a positive constant which depends only on λ , and thereafter we consider the case of shifted primes. Finally, we study a new function V ( n ) which counts the number of divisors of n with large neighbour spacings and establish the mean value of V ( n ) and of V 2 ( n ) .
． 写作p 1 (n ) < · · · < p r (n)为《a distinct prime divisors赐予整数n≥2和放,for a固定λ∈(0,1),Uλ(n ) := # { j∈{1,。。j p, r−1}日志:p (n) / log j + 1 (n)的<λ,我们最近proved that Uλ(n) - r∼λ为几乎所有integers n≥2。现在,赐予λ∈p(0, 1)和∈℘素数之设置,让Bλ(p): q ={∈℘:λ< q p < 1 /λ的日志和日志认为《arithmetic功能uλ(n ) := # { p | n: p (n -, B型λ(p) = 1}。这里,我们证明那cid: 80) n≤x (uλ(n)−λ2 log log n) = (C + o(1)美国对数log x x x→∞,哪里只有C是一个积极、康斯坦哪种depends onλ,我们和thereafter认为《shifted凯斯质数。最后，我们研究了一个新的功能V (n)，这包括了V (n)和V (n)值的数。
{"title":"Consecutive neighbour spacings\u0000between the prime divisors of an integer","authors":"J. De Koninck, I. Kátai","doi":"10.4064/cm8745-2-2022","DOIUrl":"https://doi.org/10.4064/cm8745-2-2022","url":null,"abstract":". Writing p 1 ( n ) < · · · < p r ( n ) for the distinct prime divisors of a given integer n ≥ 2 and letting, for a fixed λ ∈ (0 , 1] , U λ ( n ) := # { j ∈ { 1 , . . . , r − 1 } : log p j ( n ) / log p j +1 ( n ) < λ } , we recently proved that U λ ( n ) /r ∼ λ for almost all integers n ≥ 2 . Now, given λ ∈ (0 , 1) and p ∈ ℘ , the set of prime numbers, let B λ ( p ) := { q ∈ ℘ : λ < log q log p < 1 /λ } and consider the arithmetic function u λ ( n ) := # { p | n : ( n/p, B λ ( p )) = 1 } . Here, we prove that (cid:80) n ≤ x ( u λ ( n ) − λ 2 log log n ) 2 = ( C + o (1)) x log log x as x → ∞ , where C is a positive constant which depends only on λ , and thereafter we consider the case of shifted primes. Finally, we study a new function V ( n ) which counts the number of divisors of n with large neighbour spacings and establish the mean value of V ( n ) and of V 2 ( n ) .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70143108","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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