Sign variation, the Grassmannian, and total positivity

International audience The totally nonnegative Grassmannian is the set of $k$-dimensional subspaces $V$ of ℝ$n$ whose nonzero Plücker coordinates (i.e. $k × k$ minors of a $k × n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some generic perturbation of $V$ change sign fewer than $l − k + 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the positroid cell of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids. La grassmannienne totalement non négative est l’ensemble des sous-espaces $V$ de ℝ$n$ de dimension $k$ dont coordonnées plückeriennes non nulles (mineurs de l’ordre $k$ d’une matrice $k × n$ dont les lignes engendrent $V$) ont toutes le même signe. La positivité totale a beaucoup été étudiée durant les deux dernières décennies d’une perspective algébrique, combinatoire, et topologique, mais a pris naissance dans la théorie analytique des oscillations. C’est dans ce contexte que Gantmakher et Krein (1950) et Schoenberg et Whitney (1951) ont indépendamment démontré qu’un sous-espace $V$ est totalement non négatif ssi chaque vecteur dans $V$, lorsque considéré comme une séquence de $n$ nombres et dont on ignore les zéros, change de signe moins de $k$ fois. Nous généralisons ce résultat, démontrant que les vecteurs dans $V$ changent de signe moins de $l$ fois ssi certaines séquences des coordonnées plückeriennes d’une perturbation générique de $V$ changent de signe moins de $l − k + 1$ fois. Un algorithme construisant une telle perturbation générique est obtenu. De plus, nous déterminons la cellule positroïde de chaque $V$ totalement non négatif à partir des données de signe des vecteurs dans $V$. Ces résultats sont valides pour les matroïdes orientés.

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Triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ and Tropical Oriented Matroids

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2947 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Suho Oh ◽

Hwanchul Yoo

Keyword(s):

Oriented Matroids ◽

Oriented Matroid ◽

Nous Proposons ◽

Combinatorial Objects ◽

New Class ◽

Nous Montrons ◽

International Audience ◽

Tropical Polytopes

International audience Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Develin et Sturmfels ont montré que les triangulations de $\Delta_{n-1} \times \Delta_{d-1}$ peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de $\Delta_{n-1} \times \Delta_{d-1}$. Dans cet article, nous montrons que toute triangulation de $\Delta_{n-1} \times \Delta_{d-1}$ encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d'objets combinatoires qui peuvent décrire toutes les subdivisions d'une plus grande classe de polytopes.

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The Bibliometric Analysis of Czech Geography in the Scopus Database

Geografie ◽

10.37040/geografie2012117010052 ◽

2012 ◽

Vol 117 (1) ◽

pp. 52-71 ◽

Cited By ~ 2

Author(s):

Artur Bajerski ◽

Tadeusz Siwek

Keyword(s):

Bibliometric Analysis ◽

Restricted Range ◽

Geographical Research ◽

Geographical Society ◽

The Past ◽

Exchange Of Information ◽

International Audience ◽

Regional Basis ◽

Scopus Database

The analysis focuses on two journals: Geografie (published by the Czech Geographical Society in Czech and English) and Moravian Geographical Reports (only in English). The analysis demonstrates that the scope of both journals is only regional, due to their relatively restricted range of authors and citations: the first periodical functions mainly within Bohemia, with some overlap into other Czech regions, while the second is active primarily in Moravia, overlapping somewhat into Slovakia and Poland. Despite their status as premier Czech geographical periodicals, both journals serve mainly as conduits for the exchange of information among academics on a regional basis. Important papers presenting the results of Czech geographical research to a wide international audience are rarely featured in these journals; such research is usually published as monographs, as has been the case in the past. This paper lists the most frequently cited Czech geographers and interdisciplinary citations – especially in and from sociology and economics papers.

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Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3374 ◽

2005 ◽

Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽

Author(s):

Shuji Kijima ◽

Tomomi Matsui

Keyword(s):

Probability Function ◽

Discrete Distribution ◽

Integral Point ◽

Dimensional Simplex ◽

Perfect Sampling ◽

The Past ◽

International Audience ◽

Hit And Run ◽

Multivariate Discrete Distribution ◽

Dimensional Integral

International audience In this paper, we are concerned with random sampling of an n dimensional integral point on an $(n-1)$ dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two "hit-and-run'' chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of alternating inequalities and show that a logarithmic separable concave function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in $\textit{O}(n^2 \textit{ln}(Kɛ^{-1}))$, namely $(1/2)n(n-1) \textit{ln}(K ɛ^{-1})$, where $K$ is the side length of the simplex and $ɛ (0<ɛ<1)$ is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler is bounded by $\textit{O}(n^3 \textit{ln}(Kn))$.

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"Medusa's Smirk" (1975) by Mihaela Miroiu

Aspasia ◽

10.3167/asp.2019.130110 ◽

2019 ◽

Vol 13 (1) ◽

pp. 147-150

Author(s):

Maria Bucur

Keyword(s):

Sexual Violence ◽

Political Scientist ◽

Traumatic Memories ◽

The Past ◽

International Audience ◽

The One ◽

Past Half ◽

Half Decade

Over the past half decade, philosopher and political scientist Mihaela Miroiu published a series of short autobiographical stories that were eventually collected in a book, Cumintea mea de femeie [With my woman’s mind] (Bucharest: Cartea românească, 2017), which was reviewed in Aspasia (vol. 12) in 2018. While the whole volume deserves an international audience, I have selected the story “Medusa’s Smirk,” for translation because it sheds light on a topic little known, yet extremely important, in the lives of many women: sexual violence. Discussing sexual violence was a taboo topic under communism, and many women suppressed their traumatic memories of violence both seen and experienced. Yet accounts such as the one shared below have circulated orally and deserve further attention from scholars. For another relevant account, see http://www.publicseminar.org/2017/12/sex-in-the-time-of-communism/.

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Total positivity for cominuscule Grassmannians

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3633 ◽

2008 ◽

Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽

Author(s):

Thomas Lam ◽

Lauren Williams

Keyword(s):

Total Positivity ◽

Pattern Avoidance ◽

Young Diagrams ◽

Negative Part ◽

Schubert Cell ◽

Combinatorial Objects ◽

Preference Functions ◽

International Audience ◽

Orthogonal Grassmannians

International audience In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively. Dans cet article nous schtroumpfons la combinatoire de la partie non-négative $(G/P)_{\geq 0}$ d'une Grassmannienne cominuscule. Pour chaque Grassmannienne de ce type nous définissons les Le-diagrammes ― certains remplissages de diagrammes de Young généralisés en bijection avec les cellules de $(G/P)_{\geq 0}$. Dans les cas classiques, nous décrivons les Le-diagrammes explicitement en termes d'évitement de certains motifs. Aussi nous définissons un jeu sur les diagrammes, avec lequel on peut réduire un diagramme arbitraire à un Le-diagramme. Nous donnons les résultats énumératifs et nous relions nos Le-diagrammes à d'autres objets combinatoires. Étonnamment, les cellules non-négatives dans la cellule de Schubert ouverte des Grassmanniennes orthogonales impaires et paires sont essentiellement en bijection avec les fonctions de préférence et les fonctions de préférence atomiques.

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The structure of communication problems in cellular automata

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2977 ◽

2011 ◽

Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽

Author(s):

Raimundo Briceño ◽

Pierre-Etienne Meunier

Keyword(s):

Cellular Automata ◽

Communication Complexity ◽

Intrinsic Universality ◽

The Past ◽

Communication Problems ◽

International Audience ◽

Simulation Relations

International audience Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various "communication problems'', or "questions'' one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations.

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Counting quadrant walks via Tutte's invariant method (extended abstract)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6416 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Olivier Bernardi ◽

Mireille Bousquet-Mélou ◽

Kilian Raschel

Keyword(s):

Differential Equation ◽

Functional Equation ◽

Linear Differential Equation ◽

Generating Function ◽

Algebraic Approach ◽

Rational Functions ◽

The Past ◽

Lattice Walks ◽

International Audience ◽

Linear Differential

Extended abstract presented at the conference FPSAC 2016, Vancouver. International audience In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

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Total positivity for the Lagrangian Grassmannian

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6392 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Rachel Karpman

Keyword(s):

Symplectic Form ◽

Total Positivity ◽

Combinatorial Structure ◽

Schubert Varieties ◽

Flag Varieties ◽

Young Diagrams ◽

Lagrangian Grassmannian ◽

Planar Network ◽

International Audience ◽

Isotropic Subspaces

International audience The positroid decomposition of the Grassmannian refines the well-known Schubert decomposition, and has a rich combinatorial structure. There are a number of interesting combinatorial posets which index positroid varieties,just as Young diagrams index Schubert varieties. In addition, Postnikov’s boundary measurement map gives a family of parametrizations for each positroid variety. The domain of each parametrization is the space of edge weights of a weighted planar network. The positroid stratification of the Grassmannian provides an elementary example of Lusztig’s theory of total non negativity for partial flag varieties, and has remarkable applications to particle physics.We generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian, the moduli space of maximal isotropic subspaces with respect to a symplectic form

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Latitudinal Gradient in Urban Pressure and Socio-Environmental Quality: The “Peninsula Effect” in Italy

Land ◽

10.3390/land9040126 ◽

2020 ◽

Vol 9 (4) ◽

pp. 126 ◽

Cited By ~ 2

Author(s):

Bernardino Romano ◽

Lorena Fiorini ◽

Chiara Di Dato ◽

Vanessa Tomei

Keyword(s):

Urban Development ◽

Environmental Quality ◽

Continental Margin ◽

Latitudinal Gradient ◽

Southern Italy ◽

Final Part ◽

Behavioral Characteristics ◽

The Past ◽

International Audience

The purpose of this work is to synthesize, for an international audience, certain fundamental elements that characterize the Italian peninsular territory, through the use of a biogeographical model known as the “peninsula effect” (PE). Just as biodiversity in peninsulas tends to change, diverging from the continental margin, so do some socio-economic and behavioral characteristics, for which it is possible to detect a progressive and indisputable variation depending on the distance from the continental mass. Through the use of 14 indicators, a survey was conducted on the peninsular sensitivity (which in Italy is also latitudinal) of as many phenomena. It obtained confirmation results for some of them, well known as problematic for the country, but contradictory results for others, such as those related to urban development. In the final part, the work raises a series of questions, also showing how peninsular Italy, and in particular Central–Southern Italy, is not penalized so dramatically by its geography and morphology as many political and scientific opinions suggest. The result is a very ambiguous image of Italy, in which the country appears undoubtedly uniform in some aspects, while the PE is very evident in others; it is probably still necessary to investigate, without relying on simplistic and misleading equations, the profound reasons for some phenomena that could be at the basis of less ephemeral rebalancing policies than those practiced in the past.

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Understanding Basava: history, hagiography and a modern Kannada drama

Bulletin of the School of Oriental and African Studies ◽

10.1017/s0041977x00013793 ◽

1998 ◽

Vol 61 (2) ◽

pp. 228-261 ◽

Cited By ~ 8

Author(s):

Julia Leslie

Keyword(s):

Twentieth Century ◽

Religious Experience ◽

Twelfth Century ◽

Religious Studies ◽

Social Implications ◽

Religious Context ◽

The Past ◽

International Audience ◽

Primary Focus ◽

New Religion

This paper is a meditation from a religious studies perspective on the twelfth-century Vīraśaiva saint, Basava (c. 1105–68); that is, its primary focus is religious experience rather than literary evaluation or the historicity of the past. By exploring a variety of sources—ancient and modern, fact and fiction—and by making connections with urgent twentieth-century concerns, it seeks to bring into focus the religious aspirations and social implications of Basava's world. Part 1 is derived from history and hagiography. It provides an outline of Vīraśaiva belief and practice, and then proceeds to discuss the religious context of twelfth-century Karnataka, the debate regarding the origins of this ‘new’ religion, and a key inscription in the debate. It ends with a summary of the tradition's account of Basava's life. Part 2 focuses on a play written in Kannada (Taledaṇḍa, ‘Death by beheading’, 1990) and then rewritten in English for a pan-Indian and international audience (Talé-Daṇḍa: a play, 1993), in both cases by Girish Karnad. Karnad is not the first playwright to focus on Basava, and he will not be the last. In the preface to the Kannada version, he explains that ‘it becomes inevitable for every Kannadiga to return, like a tongue that returns again and again to a painful tooth, to the victories and agonies of that period.’ Karnad's dramatization of Basava's catastrophic final year is discussed in the context of the historical and hagiographical material considered in Part 1.

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