arXiv subjects
Publications and source records attributed to Omid Amini.
We introduce higher rank inner products on real and complex vector spaces and study their corresponding Voronoi tilings. We use the framework to describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings of discrete sets.
Soccer, also known as football in some parts of the world, involves two teams of eleven players whose objective is to score more goals than the opposing team. To simulate this game and attract scientists from all over the world to conduct research and participate in an annual computer-based soccer world cup, Soccer Simulation 2D (SS2D) was one of the leagues initiated in the RoboCup competition. In every SS2D game, two teams of 11 players and one coach connect to the RoboCup Soccer Simulation Server and compete against each other. Over the past few years, several C++ base codes have been employed to control agents' behavior and their communication with the server. Although C++ base codes have laid the foundation for the SS2D, developing them requires an advanced level of C++ programming. C++ language complexity is a limiting disadvantage of C++ base codes for all users, especially for beginners. To conquer the challenges of C++ base codes and provide a powerful baseline for developing machine learning concepts, we introduce Pyrus, the first Python base code for SS2D. Pyrus is developed to encourage researchers to efficiently develop their ideas and integrate machine learning algorithms into their teams. Pyrus base is open-source code, and it is publicly available under MIT License on GitHub
The RoboCup competitions hold various leagues, and the Soccer Simulation 2D League is a major one among them. Soccer Simulation 2D (SS2D) match involves two teams, including 11 players and a coach, competing against each other. The players can only communicate with the Soccer Simulation Server during the game. This paper presents the latest research of the CYRUS soccer simulation 2D team, the champion of RoboCup 2021. We will explain our denoising idea powered by long short-term memory networks (LSTM) and deep neural networks (DNN). The CYRUS team uses the CYRUS2D base code that was developed based on the Helios and Gliders bases.
Soccer Simulation 2D League is one of the major leagues of RoboCup competitions. In a Soccer Simulation 2D (SS2D) game, two teams of 11 players and one coach compete against each other. Several base codes have been released for the RoboCup soccer simulation 2D (RCSS2D) community that have promoted the application of multi-agent and AI algorithms in this field. In this paper, we introduce "Cyrus2D Base", which is derived from the base code of the RCSS2D 2021 champion. We merged Gliders2D base V2.6 with the newest version of the Helios base. We applied several features of Cyrus2021 to improve the performance and capabilities of this base alongside a Data Extractor to facilitate the implementation of machine learning in the field. We have tested this base code in different teams and scenarios, and the obtained results demonstrate significant improvements in the defensive and offensive strategy of the team.
The aim of this paper is to introduce a certain number of tools and results suitable for the study of valuations of higher rank on function fields of algebraic varieties. This will be based on a study of higher rank quasi-monomial valuations taking values in the lexicographically ordered group R^k. We prove a duality theorem that gives a geometric realization of higher rank quasi-monomial valuations as tangent cones of dual cone complexes. Using this duality, we provide an analytic description of quasi-monomial valuations as multi-directional derivative operators on tropical functions. We consider moreover a refined notion of tropicalization in which we remember the initial terms of power series on each cone of a dual complex, and prove a tropical analogue of the weak approximation theorem in number theory by showing that any compatible collection of initial terms on cones of a dual cone complex is the refined tropicalization of a rational function in the function field of the variety. Endowing the value group R^k with its Euclidean topology, we study then a natural topology on spaces of higher rank valuations that we call the tropical topology. By using the approximation theorem we provide an explicit description of the tropical topology on tangent cones of dual cone complexes. Finally, we show that tangent cones of dual complexes provide a notion of skeleton in higher rank non-archimedean geometry. That is, generalizing the picture in rank one to higher rank, we construct retraction maps to tangent cones of dual cone complexes, and use them to obtain limit formulae in which we reconstruct higher rank non-archimedian spaces with their tropical topology as the projective limit of their higher rank skeleta. We conjecture that these higher rank skeleta provide appropriate bases for the study of variations of Newton-Okounkov bodies.
In this report, we briefly present the technical procedure and simulation steps for the 2D soccer simulation of team Cyrus. We emphasize on this document on how the prediction of teammates' behavior is performed. In our proposed method, the agent receives the noisy inputs from the server, and predicts the ball holder full state behavior. Taking advantage of this approach for choosing the optimal view angle shows 11.30% improvement on the expected win rate.
Soccer Simulation 2D League is one of the major leagues of RoboCup competitions. In a Soccer Simulation 2D (SS2D) game, two teams of 11 players and one coach compete against each other. The players are only allowed to communicate with the server that is called Soccer Simulation Server. This paper introduces the previous and current research of the CYRUS soccer simulation team, the champion of RoboCup 2021. We will present our idea about improving Unmarking Decisioning and Positioning by using Pass Prediction Deep Neural Network. Based on our experimental results, this idea proven to be effective on increasing the winning rate of Cyrus against opponents.
The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions. As a byproduct of our approach, we obtain other results of independent interest. In particular, we introduce higher rank canonical compactifications of fans and polyhedral spaces and use them to define the moduli space of higher rank tropical curves. Moreover, we develop the first steps of a function theory in higher rank non-Archimedean, hybrid, and tame analysis. Furthermore, we establish the convergence of the Laplace operator on metric graphs toward the tropical Laplace operator on limit tropical curves in the corresponding moduli spaces, leading to new perspectives in operator theory on metric graphs. Our result on the Arakelov Green function is inspired by the works of several authors, in particular those of Faltings, de Jong, Wentworth and Wolpert, and solves a long-standing open problem arising from the Arakelov geometry of Riemann surfaces. The hybrid layered behavior close to the boundary of moduli spaces is expected to be a broad phenomenon and will be explored in our forthcoming work.
The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of smooth tropical varieties. Among the resulting applications, we prove the Hodge isomorphism theorem which asserts that the Chow rings of smooth unimodular tropical fans are isomorphic to the tropical cohomology rings of their corresponding canonical compactifications, and prove a slightly weaker statement for any unimodular fan. We furthermore introduce a notion of shellability for tropical fans and show that shellable tropical fans are smooth and thus enjoy all the nice homological properties of smooth tropical fans. Several other interesting properties for tropical fans are shown to be shellable. Finally, we obtain a generalization, both in the tropical and in the classical setting, of the pioneering work of Feichtner-Yuzvinsky and De Concini-Procesi on the cohomology ring of wonderful compactifications of complements of hyperplane arrangements. The results in this paper form the basis for our subsequent works on Hodge theory for tropical and non-Archimedean varieties.
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and the first two that preceded it are the first in a series aimed to explore this new approach. In Part I, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to its subgraphs. In Part II, we described the arrangements of toric varieties associated to the tilings of Part I in several ways: using normal fans, as unions of orbits, by equations and as degenerations of tori. In the present Part III, we show how these combinatorial and toric frameworks allow us to describe all stable limits of a family of line bundles along a degenerating family of curves. Our main result asserts that the collection of all these limits is parametrized by a connected 0-dimensional closed substack of the Artin stack of all torsion-free rank-one sheaves on the limit curve. Moreover, we thoroughly describe this closed substack and all the closed substacks that arise in this way as certain torus quotients of the arrangements of toric varieties of Part II determined by the Voronoi tilings of Euclidean spaces studied in Part I.
We show that the Voronoi cells of the lattice of integer flows of a finite connected graph $G$ in the quadratic vector space of real valued flows have the following very precise combinatorics: the face poset of a Voronoi cell is isomorphic to the poset of strongly connected orientations of subgraphs of $G$. This confirms a conjecture of Caporaso and Viviani {Torelli Theorem For Graphs and Tropical Curves, Duke Math. J. 153(1) (2010), 129-171}.
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This paper and its second and third companion parts are the first in a series aimed to explore this new approach. In the present article, we set up the combinatorial framework and show how graphs with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to certain of its subgraphs. We further provide a description of the combinatorial structure of these polytopes and the way they are glued together in the tiling. In the second part of the series, we describe the arrangements of toric varieties associated to these tilings. These results will be of use in the third part to achieve our goal of describing all stable limits of a family of line bundles along a degenerating family of curves.
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and its first and third part companion parts are the first in a series aimed to explore this new approach. In the first part, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by polytopes associated to the graph itself and to its subgraphs. In this part, we describe the arrangements of toric varieties associated to these tilings. Roughly speaking, the normal fan to each polytope in the tiling corresponds to a toric variety, and these toric varieties are glued together in an arrangement according to how the polytopes meet. We provide a thorough description of these toric arrangements from different perspectives: by using normal fans, as unions of torus orbits, by describing the (infinitely many) polynomial equations defining them in products of doubly infinite chains of projective lines, and as degenerations of algebraic tori. These results will be of use in the third part to achieve our goal of describing all stable limits of a family of line bundles along a degenerating family of curves.
This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
In this paper we prove that the cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. On the way to establish these results, we introduce and prove other results of independent interest. This includes a generalization of the results of Adiprasito-Huh-Katz, Hodge theory for combinatorial geometries, to any unimodular quasi-projective fan having the same support as the Bergman fan of a matroid, a tropical analog for Bergman fans of the pioneering work of Feichtner-Yuzvinsky on cohomology of wonderful compactifications (treated in a separate paper, recalled and used here), a combinatorial study of the tropical version of the Steenbrink spectral sequence, a treatment of Kahler forms in tropical geometry and their associated Hodge-Lefschetz structures, a tropical version of the projective bundle formula, and a result in polyhedral geometry on the existence of quasi-projective unimodular triangulations of polyhedral spaces.
To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargava's factorials to a natural combinatorial setting suitable for studying questions around generalized factorials. We discuss several basic aspects of the framework in this paper. In particular, we relate the growth of the sequence of logarithmic factorials associated to a tree to the transience of the random walk and the existence of a harmonic measure on the tree, obtain an equidistribution theorem for factorial-determining-sequences of subsets of local fields, and provide a factorial-based characterization of the branching number of infinite trees. Our treatment is based on a local weighting process in the tree which gives an effective way of constructing the factorial sequence.
We prove results describing the structure of a Chow ring associated to a product of graphs, which arises from the Gross-Schoen desingularization of a product of regular proper semi-stable curves over discrete valuation rings. By the works of Johannes Kolb and Shou-Wu Zhang, this ring controls the behavior of the non-Archimedean height pairing on products of smooth proper curves over non-Archimedean fields. We provide a complete description of the degree map, and prove vanishing results affirming a conjecture of Kolb, which, combined with his work, leads to an analytic formula for the arithmetic intersection number of adelic metrized line bundles on products of curves over complete discretely valued fields.
Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fresán, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory. Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a description of the connected components of this graph, and use this to prove our result on boundedness of the variations.