1,720,995 research outputs found
Une relation entre nombre de points entiers, volumes des faces et degré du discriminant des polytopes entiers non singuliers
Get PDFWe present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than [dim P / 2].Fil: Dickenstein, Alicia Marcela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Nill, Benjamin. Case Western Reserve University; Estados UnidosFil: Vergne, Michèle. Institut de mathématiques de Jussieu; Franci
Flag Matroids: Algebra and Geometry
No full textMatroids are ubiquitous in modern combinatorics. As discovered by Gel’fand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: representable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer, general matroids correspond to classes in the K-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. In this review we describe all these constructions in detail, and moreover we generalise some of them to polymatroids. More precisely, we study the class of flag matroids and their relations to flag varieties. In this way, we obtain an analogue of the Tutte polynomial for flag matroids
A SIMPLE COMBINATORIAL CRITERION FOR PROJECTIVE TORIC MANIFOLDS WITH DUAL DEFECT
Get PDFWe show that any smooth lattice polytope P with codegree greater or equal than (dim(P) + 3)/2 (or equivalently, with degree smaller than dim(P)/2), defines a dual defective projective toric manifold. This implies that P is Q-normal (in the terminology of [11]) and answers partially an adjunction-theoretic conjecture by BeltramettiSommese (see [5],[4],[11]). Also, it follows from [24] that smooth lattice polytopes with this property are precisely strict Cayley polytopes, which completes the answer in [11] of a question in [1] for smooth polytopes.Fil: Dickenstein, Alicia Marcela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; ArgentinaFil: Nill, Benjamin. University of Georgia; Estados Unido
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Enumeration and sparsity in algebraic geometry
Get PDFDie vorliegende Dissertation behandelt Fragen in der enumerativen algebraischen
Geometrie, der kommutativen Algebra, der algebraischen Statistik und der Theorie
der Gitterpolytope. Allen gemeinsam sind Verbindungen zur Kombinatorik.
In Kapitel 1 geht es um das Problem der charakteristischen Zahlen für kubische Hy-
perflächen im projektiven Raum. Wir berechnen unter anderem die Anzahl kubischer
Flächen tangential zu 19 Geraden im P3 und die Anzahl kubischer Dreifaltigkeiten
tangential zu 34 Geraden im P4. Unsere Resultate ermöglichen es prinzipiell, die
analogen Fragen in beliebiger Dimension zu beantworten. Dies ist in Teilen gemein-
same Arbeit mit Mara Belotti, Alessandro Danelon und Claudia Fevola.
In Kapitel 2 bestimmen wir die minimale freie Auflösung des Ideals aller (n − 1)-
Minoren einer spärlich besetzten generischen symmetrischen n × n Matrix. Als
Anwendung berechnen wir die erste nicht-triviale charakteristische Zahl für alle
Familien spärlich besetzter Quadriken ohne diagonale Nullen. Dies ist gemeinsame
Arbeit mit Jiahe Deng.
In Kapitel 3 studieren wir eine gemeinsame Verallgemeinerung von ungerichteten
Gaußschen graphischen Modellen und Kovarianzmodellen, bei denen wir Nullen
in sowohl der Kovarianzmatrix als auch der Konzentrationsmatrix erlauben. Wir
beweisen Strukturresultate für diese Modelle, z.B. Kriterien für Glattheit, Schranken
an die Dimension, implizierte Nullen und Blockstrukturen. Dies ist gemeinsame
Arbeit mit Tobias Boege, Thomas Kahle und Frank Röttger.
Kapitel 4 handelt von symmetrischen Idealen; dies sind Ideale in einem Polynom-
ring, die invariant unter allen Permutationen der Variablen sind. Wir beweisen,
dass Ideale, die von dem Orbit eines generischen homogenen Polynoms erzeugt
werden, in einem präzisen Sinn das gr¨oßtm¨ogliche Radikal und die kleinstmögliche
Verschwindungsmenge besitzen.
Kapitel 5 ist ein Beitrag zur lokalen Ehrhart-Theorie. Wir studieren dünne Polytope,
also Gitterpolytope, deren lokales h∗-Polynom verschwindet. In Dimension 3 klassi-
fizieren wir dünne Gitterpolytope vollständig und in beliebiger Dimension liefern wir
eine Charakterisierung dünner Gorensteinpolytope. Dies ist gemeinsame Arbeit mit
Christopher Borger und Benjamin Nill.The present thesis deals with questions in enumerative algebraic geometry, commu-
tative algebra, algebraic statistics and the theory of lattice polytopes. All of them
share a combinatorial flavor.
Chapter 1 is about characteristic numbers for cubic hypersurfaces in projective space.
For instance, we explicitly compute the number of cubic surfaces tangent to 19 lines
in P3 and the number of cubic threefolds tangent to 34 lines in P4. In principle, our
results allow us to answer the analogous question in arbitrary dimensions. This is
partly joint work with Mara Belotti, Alessandro Danelon and Claudia Fevola.
Chapter 2 provides the minimal free resolution of the ideal of (n − 1)-minors of a
sparse generic symmetric n × n matrix. As an application, we compute the first
non-trivial characteristic number for all families of sparse quadrics without diagonal
zeros. This is joint work with Jiahe Deng.
In Chapter 3 we study a common generalization of undirected Gaussian graphical
models and covariance models, allowing for zeros in the covariance and the concen-
tration matrix simultaneously. We prove structural results like smoothness criteria,
dimension bounds, implied zeros and block structures. This is joint work with Tobias
Boege, Thomas Kahle and Frank R¨ottger.
Chapter 4 is about symmetric ideals, i.e., ideals in a polynomial ring invariant under
all permutations of the variables. We prove that ideals generated by the orbit of a
general homogeneous polynomial have, in a precise sense, the largest possible radical
and the smallest possible vanishing set.
Chapter 5 is a contribution to local Ehrhart theory. We study thin polytopes,
i.e., lattice polytopes whose local h∗-polynomial vanishes. We provide a complete
classification in dimension 3 and a characterization of thin Gorenstein polytopes in
arbitrary dimension. This is joint work with Christopher Borger and Benjamin Nill
Mixed lattice polytope theory with a view towards sparse polynomial systems
Get PDFDiese Arbeit befasst sich mit verschiedenen gemischten Fragestellungen aus der
Gitterpolytoptheorie. Damit sind Fragestellungen gemeint, die sich auf Tupel von
Gitterpolytopen beziehen und für die somit sowohl die Struktur der einzelnen Polytope
als auch deren Lage zueinander eine Rolle spielen. Eine zentrale Motivation hierfür
ist der berühmte Satz von Bernstein-Khovanskii-Kushnirenko, der die Anzahl an
Lösungen eines polynomiellen Gleichungssystems durch das gemischte Volumen des
Tupels der Newton-Polytope der Polynome beschränkt. Ziel dieser Arbeit ist es,
verschiedene Probleme an der Grenze zwischen algebraischer und diskreter Geometrie
aus dem Blickwinkel einer gemischten Gitterpolytoptheorie zu behandeln und damit
sowohl die Relevanz dieses Gebietes zu illustrieren als auch die Entwicklung der
Grundlagen auf diesem Feld voranzutreiben.
Im ersten Teil der Arbeit führen wir grundlegende Begriffe und Notationen ein.
Im zweiten Teil präsentieren wir Resultate über die Cayley-Summe eines Tupels
von Gitterpolytopen, welche wir an verschiedenen anderen Stellen in dieser Arbeit
benötigen.
Im dritten Teil widmen wir uns der gemischten Diskriminanten eines Tupels von
ganzzahligen Punktkonfigurationen. Diese ist ein Polynom, das enkodiert unter
welchen Bedingungen ein polynomielles Gleichungssystem bestimmte mehrfache
Nullstellen hat. Wir geben eine hinreichende kombinatorische Bedingung für die
Existenz dieser gemischten Diskrimanten und beweisen damit eine Vermutung von
Cattani et. al.
Der vierte Teil ist der Entwicklung eines Algorithmus für die Klassifikation von
Tripeln von Gitterpolytopen im R3 mit gegebenem gemischten Volumen gewidmet.
Nach dem Satz von BKK ist dies äquivalent zu der Klassifikation von generischen
Systemen trivariater Polynome mit gegebener Anzahl von Lösungen. Anhand einer
Implementierung dieses Algorithmus erhalten wir eine vollständige Klassifikation
dieser Tripel mit gemischtem Volumen höchstens vier.
Im fünften Teil dieser Arbeit untersuchen wir Tupel von Gitterpolytopen, deren
gemischter Grad höchstens eins ist. Wir zeigen, dass es in jeder Dimension abgesehen
von einer gut verstandenen Familie von Tupeln nur endlich viele exzeptionelle Tupel
mit gemischtem Grad eins gibt. Des Weiteren klassifizieren wir solche Tupel in
Dimension drei vollständig.
Im sechsten und letzten Teil dieser Arbeit zeigen wir eine obere Schranke an das
Volumen der Minkowski-Summe eines Tupels konvexer Körper, dessen gemischtes
Volumen gegeben ist. Unsere Schranke ist asymptotisch scharf und in den Spezialfällen
von Dimension zwei und drei finden wir darüber hinaus eine scharfe exakte Schranke.This work treats several mixed questions in the theory of lattice polytopes. By that
we mean questions that are in terms of tuples of lattice polytopes and for which one
has to consider not only the structure of the single lattice polytopes in the tuple,
but also their alignment with respect to each other. Central motivation for treating
such questions comes from the famous Bernstein-Khovanskii-Kushnirenko theorem.
This result bounds the number of solutions of a polynomial system by the mixed
volume of the tuple of Newton polytopes of the polynomials. The scope of this work
is to treat different problems at the intersection of algebraic and discrete geometry
from the point of view of a mixed lattice polytope theory. In the course of this we
illustrate the relevance of this field of research and make progress in the development
of its foundations.
The first chapter is dedicated to the introduction of basic concepts and notation.
In the second chapter we present results about the Cayley sum of a tuple of lattice
polytopes that we make use of in several parts of this work.
The third chapter deals with the mixed discriminant of a tuple of point configurations,
which is a polynomial that encodes the conditions for a system of polynomial
equations to have a multiple root. We prove a sufficient combinatorial condition for
the existence of the mixed discriminant and employ this to solve a conjecture by
Cattani et. al.
In the fourth chapter we present an algorithm for the classification of triples of
lattice polytopes in R3 with a given mixed volume. By the BKK-theorem, this is
equivalent to the classification of generic systems of trivariate polynomials with a
given number of solutions. Via this algorithm, we obtain a complete classification of
triples of lattice polytopes with mixed volume at most four.
The fifth chapter treats tuples of lattice polytopes whose mixed degree is at
most one. We show that, in dimension at least four, there exist only finitely many
exceptional tuples of mixed degree one that are not part of a well-understood family.
We furthermore present a complete classification of such tuples in dimension three.
Finally, in chapter six we prove an upper bound on the volume of the Minkowksi
sum of a tuple of convex bodies in terms of its mixed volume. Our bound is
asymptotically sharp. In dimensions two and three we furthermore prove an exact
sharp bound
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
- …
