scholarly journals The freeness of Ish arrangements

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Daisuke Suyama ◽  
Shuhei Tsujie
Keyword(s):  
Catalan Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Free Arrangement ◽  
International Audience ◽  
New Interpretation ◽  
Necessary And Sufficient ◽  
Shi Arrangement

International audience The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q; t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free L’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres $q; t$-Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre.

scholarly journals The competition number of a generalized line graph is at most two

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Boram Park ◽  
Yoshio Sano
Keyword(s):  
Graph Theory ◽  
Necessary Conditions ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Competition Number

Graph Theory International audience In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

scholarly journals Defining amplituhedra and Grassmann polytopes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Steven N. Karp
Keyword(s):  
Dimensional Subspace ◽  
Equivalent Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positivity Condition ◽  
Yang Mills ◽  
Cyclic Polytopes ◽  
Plücker Coordinates ◽  
International Audience ◽  
Necessary And Sufficient

International audience The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).

scholarly journals Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Shonda Gosselin ◽  
Andrzej Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Uniform Hypergraph ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
International Audience ◽  
Complete Multipartite ◽  
Necessary And Sufficient

Combinatorics International audience A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.

Local distinguishability of quantum states and the distillation of entanglement

2003 ◽  
Vol 3 (3) ◽  
pp. 203-210
Author(s):  
P.-X. Chen ◽  
C.-Z. Li
Keyword(s):  
Information Entropy ◽  
Quantum States ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximal Yield ◽  
Pure States ◽  
New Interpretation ◽  
Necessary And Sufficient ◽  
Special Protocol

This paper tries to probe the relation between the local distinguishability of orthogonal quantum states and the distillation of entanglement. An new interpretation for the distillation of entanglement and the distinguishability of orthogonal quantum states in terms of information is given, respectively. By constraining our discussion on a special protocol we give a necessary and sufficient condition for the local distinguishability of the orthogonal pure states, and gain the maximal yield of the distillable entanglement. It is shown that the information entropy, the locally distinguishability of quantum states and the distillation of entanglement are closely related.

scholarly journals Balanced labellings of affine permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Hwanchul Yoo ◽  
Taedong Yun
Keyword(s):  
Symmetric Functions ◽  
Simple Algorithm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reduced Decompositions ◽  
Finite Case ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Affine Permutations ◽  
Necessary And Sufficient

International audience We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.

scholarly journals Nearly recombining processes and the calculation of expectations

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Imme Van den Berg ◽  
Elsa Amaro
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Nonstandard Analysis ◽  
Conditional Variance ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Original Process ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Ito's Lemma

International audience In the context of Nonstandard Analysis, we study stochastic difference equations with infinitesimal time-steps. In particular we give a necessary and sufficient condition for a solution to be nearly-equivalent to a recombining stochastic process. The characterization is based upon a partial differential equation involving the trend and the conditional variance of the original process. An analogy with Ito’s Lemma is pointed out. As an application we obtain a method for approximation of expectations, in terms of two ordinary differential equations, also involving the trend and the conditional variance of the original process, and of Gaussian integrals. Dans le contexte de l’Analyse Nonstandard, nous étudions des équations différentielles stochastiques avec des pas infiniment petits. En particulier, nous formulons une condition nécessaire et suffisante pourqu’une solution soit presque-équivalente à un processus stochastique recombinant. La caractérisation est donnée par une équation aux dérivées partielles de la tendance et de la variance conditionnelle du processus de départ. Nous indiquons une analogie avec le Lemme d’Ito. Nous appliquons cette caractérisation au problème de la détermination d’espérances pour le processus de départ. En fait, on obtient une approximation infinitésimale en resolvant deux équations différentielles ordinaires, également de la tendance et de la variance conditionnelle de ce processus, et en calculant une intégrale de Gauss.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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