scholarly journals Intersections of Amoebas

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Martina Juhnke-Kubitzke ◽  
Timo De Wolff
Keyword(s):  
Upper Bound ◽  
Connected Components ◽  
Algebraic Varieties ◽  
Connected Component ◽  
Absolute Value ◽  
Algebraic Torus ◽  
International Audience ◽  
Natural Way ◽  
Generalized Order

International audience Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been inten- sively studied during the last years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of n hypersurfaces in (C∗)n, which are genuine supersets of amoebas given by non-hypersurface vari- eties. Our main results are amoeba analogs of Bernstein's Theorem and Be ́zout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the order map for hypersur- face amoebas can be generalized in a natural way to intersections of amoebas. We show that, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still 1-to-1.

scholarly journals A Symmetric Banzhaf Cooperation Value for Games with a Proximity Relation among the Agents

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1196
Author(s):  
Inés Gallego ◽  
Julio R. Fernández ◽  
Andrés Jiménez-Losada ◽  
Manuel Ordóñez
Keyword(s):  
Common Good ◽  
A Priori ◽  
Group Symmetry ◽  
Connected Components ◽  
Banzhaf Value ◽  
Connected Component ◽  
Communication Graph ◽  
Political Situation ◽  
Proximity Relation ◽  
Natural Way

A cooperative game represents a situation in which a set of agents form coalitions in order to achieve a common good. To allocate the benefits of the result of this cooperation there exist several values such as the Shapley value or the Banzhaf value. Sometimes it is considered that not all communications between players are feasible and a graph is introduced to represent them. Myerson (1977) introduced a Shapley-type value for these situations. Another model for cooperative games is the Owen model, Owen (1977), in which players that have similar interests form a priori unions that bargain as a block in order to get a fair payoff. The model of cooperation introduced in this paper combines these two models following Casajus (2007). The situation consists of a communication graph where a two-step value is defined. In the first step a negotiation among the connected components is made and in the second one players inside each connected component bargain. This model can be extended to fuzzy contexts such as proximity relations that consider leveled closeness between agents as we proposed in 2016. There are two extensions of the Banzhaf value to the Owen model, because the natural way loses the group symmetry property. In this paper we construct an appropriate value to extend the symmetric option for situations with a proximity relation and provide it with an axiomatization. Then we apply this value to a political situation.

scholarly journals Mean-square upper bound of Hecke $L$-functions on the critical line.

2003 ◽  
Vol Volume 26 ◽  
Author(s):  
A Sankaranarayanan
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Critical Line ◽  
Congruence Subgroup ◽  
Mean Square ◽  
Absolute Value ◽  
International Audience ◽  
The Mean ◽  
The Absolute

International audience We prove the upper bound for the mean-square of the absolute value of the Hecke $L$-functions (attached to a holomorphic cusp form) defined for the congruence subgroup $\Gamma_0 (N)$ on the critical line uniformly with respect to its conductor $N$.

scholarly journals Representations on Hessenberg Varieties and Young's Rule

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Nicholas Teff
Keyword(s):  
Symmetric Group ◽  
Bruhat Order ◽  
Irreducible Representations ◽  
Connected Components ◽  
Algebraic Varieties ◽  
Hessenberg Varieties ◽  
Young's Rule ◽  
Kostka Numbers ◽  
International Audience ◽  
Open Question

International audience We combinatorially construct the complex cohomology (equivariant and ordinary) of a family of algebraic varieties called regular semisimple Hessenberg varieties. This construction is purely in terms of the Bruhat order on the symmetric group. From this a representation of the symmetric group on the cohomology is defined. This representation generalizes work of Procesi, Stembridge and Tymoczko. Here a partial answer to an open question of Tymoczko is provided in our two main result. The first states, when the variety has multiple connected components, this representation is made up by inducing through a parabolic subgroup of the symmetric group. Using this, our second result obtains, for a special family of varieties, an explicit formula for this representation via Young's rule, giving the multiplicity of the irreducible representations in terms of the classical Kostka numbers. Nous construisons la cohomologie complexe (équivariante et ordinaire) d'une famille de variétés algébriques appelées variétés régulières semisimples de Hessenberg. Cette construction utilise exclusivement l'ordre de Bruhat sur le groupe symétrique, et on en déduit une représentation du groupe symétrique sur la cohomologie. Cette représentation généralise des résultats de Procesi, Stembridge et Tymoczko. Nous offrons ici une réponse partielle à une question de Tymoczko grâce à nos deux résultats principaux. Le premier déclare que lorsque la variété a plusieurs composantes connexes, cette représentation s'obtient par induction à travers un sous-groupe parabolique du groupe symétrique. Nous en déduisons notre deuxième résultat qui fournit, pour une famille spéciale de variétés, une formule explicite pour cette représentation par la règle de Young, et donne ainsi la multiplicité des représentations irréductibles en termes des nombres classiques de Kostka.

scholarly journals On an exponential sum involving the Möbius function

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
H. Maier ◽  
A Sankaranarayanan
Keyword(s):  
Upper Bound ◽  
Exponential Sum ◽  
Möbius Function ◽  
Absolute Value ◽  
Mobius Function ◽  
International Audience ◽  
The Absolute

International audience In this paper we study the upper bound for the absolute value of the exponential sum related to the Möbius function unconditionally and present some interesting applications also.

The size of the connected components of excursion sets of χ2, t and F fields

1999 ◽  
Vol 31 (03) ◽  
pp. 579-595 ◽  
Author(s):  
J. Cao
Keyword(s):  
Positron Emission Tomography ◽  
Two Dimensions ◽  
Emission Tomography ◽  
Connected Components ◽  
Gaussian Fields ◽  
Connected Component ◽  
Test Statistic ◽  
Excursion Sets ◽  
Positron Emission ◽  
Excursion Set

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

scholarly journals Partitions of an Integer into Powers

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Tree Structure ◽  
Dynamical Model ◽  
International Audience ◽  
Partitions Of Integers ◽  
Natural Way

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Dental Images’ Segmentation Using Threshold Connected Component Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Vincent Majanga ◽  
Serestina Viriri
Keyword(s):  
Deep Learning ◽  
Region Of Interest ◽  
Component Analysis ◽  
Connected Components ◽  
Connected Component ◽  
Low Contrast ◽  
Connected Component Analysis ◽  
Feature Representations ◽  
Learning Techniques ◽  
Performance Results

Recent advances in medical imaging analysis, especially the use of deep learning, are helping to identify, detect, classify, and quantify patterns in radiographs. At the center of these advances is the ability to explore hierarchical feature representations learned from data. Deep learning is invaluably becoming the most sought out technique, leading to enhanced performance in analysis of medical applications and systems. Deep learning techniques have achieved great performance results in dental image segmentation. Segmentation of dental radiographs is a crucial step that helps the dentist to diagnose dental caries. The performance of these deep networks is however restrained by various challenging features of dental carious lesions. Segmentation of dental images becomes difficult due to a vast variety in topologies, intricacies of medical structures, and poor image qualities caused by conditions such as low contrast, noise, irregular, and fuzzy edges borders, which result in unsuccessful segmentation. The dental segmentation method used is based on thresholding and connected component analysis. Images are preprocessed using the Gaussian blur filter to remove noise and corrupted pixels. Images are then enhanced using erosion and dilation morphology operations. Finally, segmentation is done through thresholding, and connected components are identified to extract the Region of Interest (ROI) of the teeth. The method was evaluated on an augmented dataset of 11,114 dental images. It was trained with 10 090 training set images and tested on 1024 testing set images. The proposed method gave results of 93 % for both precision and recall values, respectively.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

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