scholarly journals Links in the complex of weakly separated collections

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Suho Oh ◽  
David Speyer
Keyword(s):  
Short Note ◽  
Cluster Algebras ◽  
Combinatorial Objects ◽  
The Face ◽  
International Audience ◽  
Plabic Graphs ◽  
Positive Integers ◽  
Do So

International audience Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.

scholarly journals Partition-theoretic formulas for arithmetic densities, II

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Ken Ono ◽  
Robert Schneider ◽  
Ian Wagner
Keyword(s):  
Dirichlet Series ◽  
First Principles ◽  
Prime Numbers ◽  
Integer Partitions ◽  
Root Of Unity ◽  
International Audience ◽  
Positive Integers ◽  
Limiting Values ◽  
Original Theorem ◽  
Do So

International audience In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

scholarly journals Newton Polytopes of Cluster Variables of Type $A_n$

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Adam Kalman
Keyword(s):  
Type A ◽  
Laurent Expansion ◽  
Cluster Algebras ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Affine Hull ◽  
Cluster Variable ◽  
Nous Montrons ◽  
The Face ◽  
International Audience

International audience We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph. Nous étudions polytopes de Newton des variables amassées dans les algèbres amassées de type A, dont les variables sont indexés par les diagonales et les côtés d’un polygone. Nos principaux résultats comprennent une description explicite de l’enveloppe affine et facettes du polytope de Newton du développement de Laurent de toutes variables amassées. En particulier, nous montrons que tout monôme Laurent dans un développement de Laurent de variable amassée de type A correspond à un sommet du polytope de Newton. Nous décrivons aussi le treillis des facesde chaque polytope de Newton via un isomorphisme avec le treillis des sous-graphes élémentaires du “snake graph” qui est associé.

scholarly journals Positivity in Coefficient-Free Rank Two Cluster Algebras

10.37236/187 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
G. Dupont
Keyword(s):  
Short Note ◽  
Rational Functions ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Free Rank ◽  
Positive Integers

Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over ${\Bbb Z}$ and $x_m, m \in {\Bbb Z}$ be rational functions defined by $x_{m-1}x_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}x_{m+1}=x_m^c+1$ if $m$ is even. In this short note, we prove that for any $m,k \in {\Bbb Z}$, $x_k$ can be expressed as a substraction-free Laurent polynomial in ${\Bbb Z}[x_m^{\pm 1},x_{m+1}^{\pm 1}]$. This proves Fomin-Zelevinsky's positivity conjecture for coefficient-free rank two cluster algebras.

Challenging Gender Hierarchies in Narratives of the Nation: Representations of Women in Zintgraff and the Battle of Mankon and Hard Choice

Imbizo ◽  
2020 ◽  
Vol 11 (2) ◽  
Author(s):  
Naomi Epongse Nkealah ◽  
Olutoba Gboyega Oluwasuji
Keyword(s):  
The Other ◽  
Literary Texts ◽  
Gender Differentiation ◽  
Hard Choice ◽  
Other Hand ◽  
Gender Hierarchies ◽  
The Face ◽  
The One ◽  
Representations Of Women ◽  
Do So

Ideas of nationalisms as masculine projects dominate literary texts by African male writers. The texts mirror the ways in which gender differentiation sanctions nationalist discourses and in turn how nationalist discourses reinforce gender hierarchies. This article draws on theoretical insights from the work of Anne McClintock and Elleke Boehmer to analyse two plays: Zintgraff and the Battle of Mankon by Bole Butake and Gilbert Doho and Hard Choice by Sunnie Ododo. The article argues that women are represented in these two plays as having an ambiguous relationship to nationalism. On the one hand, women are seen actively changing the face of politics in their societies, but on the other hand, the means by which they do so reduces them to stereotypes of their gender.

Empire and the Johannine Epistles

2017 ◽  
Vol 114 (4) ◽  
pp. 542-557
Author(s):  
Kenneth L. Waters
Keyword(s):  
Jesus Christ ◽  
The Face ◽  
Mission Field ◽  
The Church ◽  
Do So

In what ways are the Johannine Epistles a response to empire ideology and propaganda? These Epistles proclaim a more complete and correct cosmology, a greater Savior and soteriology, a better pedagogy, a truer doctrine, a sounder koinōnia, and a more nurturing paterfamilias; moreover, they do so while indicting schismatics, who, in the view of the elder, represent the face of the empire. Although the resurrection and ascension of Jesus Christ drive the elder’s witness and ministry, he must still shape his message to counter the encroachment of empire in the church and on the mission field.

scholarly journals Leviathan, or the Rudder of Public Health

2017 ◽  
Vol 2 (Suppl. 1) ◽  
pp. 1-8
Author(s):  
Denis Horgan ◽  
Walter Ricciardi
Keyword(s):  
Public Health ◽  
Big Data ◽  
New Technologies ◽  
Personalised Medicine ◽  
Member States ◽  
Reform Agenda ◽  
The World ◽  
The Face ◽  
The Eu ◽  
Do So

In the world of modern health, despite the fact that we've been blessed with amazing advances of late - the advent of personalised medicine is just one example - “change” for most citizens seems slow. There are clear discrepancies in availability of the best care for all, the divisions in access from country to country, wealthy to poor, are large. There are even discrepancies between regions of the larger countries, where access often varies alarmingly. Too many Member States (with their competence for healthcare) appear to be clinging stubbornly to the concept of “one-size-fits-all” in healthcare and often stifle advances possible through personalised medicine. Meanwhile, the legislative arena encompassing health has grown big and unwieldy in many respects. And bigger is not always better. The health advances spoken of above, an increased knowledge on the part of patients, the emergence of Big Data and more, are quickly changing the face of healthcare in Europe. But healthcare thinking across the EU isn't changing fast enough. The new technologies will certainly speak for themselves, but only if allowed to do so. Acknowledging that, this article highlights a positive reform agenda, while explaining that new avenues need to be explored.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

scholarly journals Savaris et al (2021) erroneously interpreted their regressions

2021 ◽  
Author(s):  
Carlos Góes
Keyword(s):  
Time Series ◽  
Short Note ◽  
Mortality Rates ◽  
Specific Time ◽  
Weighted Averages ◽  
Staying At Home ◽  
Do So ◽  
At Home

Savaris et al. (2021) aim at "verifying if staying at home had an impact on mortality rates." This short note shows that the methodology they have applied in their paper does not allow them to do so. An estimated coefficient β≈0 does not imply that there is no association between the variables in either country. Rather, their pairwise difference regressions are computing coefficients that are weighted-averages of region-specific time series regressions, such that it is possible that the association is significant in both regions but their weighted-averages is close to zero. Therefore, the results do not back up the conclusions of the paper.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

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