Convex Hull

Cooperative control of multirobot systems (MRSs) has earned significant research interest over the past two decades due to its potential applications in multidisciplinary engineering problems. In contrast to a single specialized robot, the MRS can be designed to offer flexibility, reconfigurability, robustness to faults, and cost-effectiveness in solving complex and challenging tasks. In this article, we aim to develop a unified cluster formation containment coordination framework for networked robots that can be decomposed into two layers containing the leaders and the followers. According to the proposed methodology, the leader robots are first distributed into a set of distinct and nonoverlapping clusters depending on the positions and priorities of the targets exploiting a game-theoretic rule. Then, they are steered to attain the desired formations around the corresponding targets. Subsequently, the follower robots are made to converge into the convex hull spanned by the leaders of the individual clusters. A prototype search and rescue operation is considered to highlight the usefulness of the proposed coordination framework. Furthermore, real-time hardware experiments were conducted on miniature mobile robots to validate the feasibility of the theoretical results.

This paper addresses the distributed formation-containment (DFC) problem for multiple Euler-Lagrange systems with model uncertainties via output feedback in both constant and time-varying formation cases. First, a novel definition of the DFC problem is proposed using a two-layer framework. Since only parts of the followers can acquire the states of the dynamic leader, we design a distributed finite-time sliding-mode estimator to obtain accurate estimations of the desired position and velocity for each agent. Next, to deal with the absence of velocity sensors, we propose two DFC control laws combined with the high-gain observer for the leaders and the followers, respectively, while the time-varying formation in the first layer and the leader-based containment in the second layer can be achieved. Further, the adaptive neural networks are applied to deal with the model uncertainties due to their superior approximation capability. The uniform ultimate boundedness of all the state errors can be guaranteed by Lyapunov stability theory. In addition, a unified framework is given which can be transformed to four other basic distributed problems. Finally, simulation examples are presented to illustrate the feasibility of the theoretical results.

Cooperative control of multirobot systems (MRSs) has earned significant research interest over the past two decades due to its potential applications in multidisciplinary engineering problems. In contrast to a single specialized robot, the MRS can be designed to offer flexibility, reconfigurability, robustness to faults, and cost-effectiveness in solving complex and challenging tasks. In this article, we aim to develop a unified cluster formation containment coordination framework for networked robots that can be decomposed into two layers containing the leaders and the followers. According to the proposed methodology, the leader robots are first distributed into a set of distinct and nonoverlapping clusters depending on the positions and priorities of the targets exploiting a game-theoretic rule. Then, they are steered to attain the desired formations around the corresponding targets. Subsequently, the follower robots are made to converge into the convex hull spanned by the leaders of the individual clusters. A prototype search and rescue operation is considered to highlight the usefulness of the proposed coordination framework. Furthermore, real-time hardware experiments were conducted on miniature mobile robots to validate the feasibility of the theoretical results.

The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the illumination numbers of 2-illuminable spiky balls as well as centrally symmetric cap bodies. In particular, we prove the Illumination Conjecture for centrally symmetric cap bodies in sufficiently large dimensions by showing that any d-dimensional centrally symmetric cap body can be illuminated by < 2 d directions in Euclidean d-space for all d ≥ 20. Furthermore, we strengthen the latter result for 1-unconditionally symmetric cap bodies.

This note presents some new information on how the minimum distance of the generalized toric code corresponding to a fixed set of integer lattice points S ⊂ R 2 varies with the base field. The main results show that in some cases, over sufficiently large fields, the minimum distance of the code corresponding to a set S will be the same as that of the code corresponding to the convex hull conv(S). In an example, we will also discuss a [49, 28] generalized toric code over F 8 , better than any previously known code according to M. Grassl's online tables, as of September 2011.

A walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W , and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W . In the resulting interval convexity, a set S ⊂ V (G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.

The inverse problem under consideration is to reconstruct the characteristic of scatterer from the scattering E field. Dynamic differential evolution (DDE) and selfadaptive dynamic differential evolution (SADDE) are stochastic-type optimization approach that aims to minimize a cost function between measurements and computer-simulated data. These algorithms are capable of retrieving the location, shape, and permittivity of the dielectric cylinder in a slab medium made of lossless materials. The finite-difference timedomain (FDTD) is employed for the analysis of the forward scattering. The comparison is carried out under the same conditions of initial population of candidate solutions and number of iterations. Numerical results indicate that SADDE outperforms the DDE a little in terms of reconstruction accuracy.

We develop efficient algorithms for problems in computational geometry-convex hull, smallest enclosing box, ECDF, two-set dominance, maximal points, all-nearest neighbor, and closest-pair-on the OTIS-Mesh optoelectronic computer. We also demonstrate the algorithms for computing convex hull and prefix sum with condition on a multi-dimensional mesh, which are used to compute convex hull and ECDF respectively. We show that all these problems can be solved in ¡ £¢ ¥¤ ¦ ¨ § time even with ¦ ¨© inputs.

We consider the P3-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v ∈ H(S \ {v}) for all v in S. The rank rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.

Special issue PRIMA 2013 We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic con...

The generalization of classical results about convex sets in R n to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P 3 -convexity on graphs. in R. The P 3 -convex hull of a set of vertices is the smallest P 3 -convex set containing it. The P 3 -Radon number r(G) of a graph G is the smallest integer r such that every set R of r vertices of G has a partition R = R 1 ∪R 2 such that the P 3 -convex hulls of R 1 and R 2 intersect. We prove that r(G) ≤ 2 3 (n(G) + 1) + 1 for every connected graph G and characterize all extremal graphs.

The evaluation of geometric defects is necessary in order to maintain the integrity of structures over time. These assessments are designed to detect damages of structures and ideally help inspectors to estimate the remaining life of structures. Current methodologies for monitoring structural systems, while providing useful information about the current state of a structure, are limited in the monitoring of defects over time and in linking them to predictive simulation. This paper presents a new approach to the predictive modeling of geometric defects. A combination of segmented from point clouds are parametrized using the convex hull algorithm to extract features from detected defects, and a stochastic dynamic model is then adapted to these features to model the evolution of the hull over time. Describing a defect in terms of its parameterized hull enables consistent temporal tracking for predictive purposes, while implicitly reducing data dimensionality and complexity as well. In ...

Structured optimization problems are ubiquitous in fields like data science and engineering. The goal in structured optimization is using a prescribed set of points, called atoms, to build up a solution that minimizes or maximizes a given function. In the present paper, we want to minimize a black-box function over the convex hull of a given set of atoms, a problem that can be used to model a number of real-world applications. We focus on problems whose solutions are sparse, i.e., solutions that can be obtained as a proper convex combination of just a few atoms in the set, and propose a suitable derivative-free inner approximation approach that nicely exploits the structure of the given problem. This enables us to properly handle the dimensionality issues usually connected with derivative-free algorithms, thus getting a method that scales well in terms of both the dimension of the problem and the number of atoms. We analyze global convergence to stationary points. Moreover, we show that, under suitable assumptions, the proposed algorithm identifies a specific subset of atoms with zero weight in the final solution after finitely many iterations. Finally, we report numerical results showing the effectiveness of the proposed method.

In this paper, we address the problem of covering points with orthogonally convex polygons. In particular, given a point set of size n on the plane, we aim at finding if there exists an orthogonally convex polygon such that each edge of the polygon covers exactly one point and each point is covered by exactly one edge. We show that if such a polygon exists, it may not be unique. We propose an O (n log n) algorithm to construct such a polygon if it exists, or else report the non-existence in the same time bound. We also extend our algorithm to count all such polygons without hindering the overall time complexity. Finally, we show how to construct all k such polygons in O (n log n + kn) time. All the proposed algorithms are fast and practical.

WC present lower bounds on the number of rounds required to solve a decision problem in the parallel algebraic decision tree model. More specifically, we show that any parallel algorithm in the fixed degree algebraic decision tree model that answers membership queries in M' CR" using p processors, requires s2(log 1 WI/H log( p;'n)) rounds where 1 W 1 is the number of connected components of W. This implies non-trivia1 lower bounds for parallel algorithms that use a superlinear number of processors, namely, that the speed-up obtainable in such cases is not proportional to the number of processors. We further prove a similar result for the average case complexity. We give applications of this result to various fundamental problems in computational geometry like convex-hull construction and trapezoidal decomposition and also present algorithms with matching upper bounds. The algorithms extend Reif and Sen's work in parallel computational geometry to the sublogarithmic time range based on recent progress in padded-sorting. A corollary of our result strengthens the known lower-bound of parallel sorting from the parallel comparison tree model to the more powerful bounded-degree decision tree.

In this paper we focus on the problem of designing very fast parallel algorithms for the convex hull and the vector maxima problems in three dimensions that are output-size sensitive. Our algorithms achieve Oðlog log 2 n log hÞ parallel time and optimal Oðn log hÞ work with high probability in the CRCW PRAM where n and h are the input and output size, respectively. These bounds are independent of the input distribution and are faster than the previously known algorithms. We also present an optimal speed-up (with respect to the input size only) sublogarithmic time algorithm that uses superlinear number of processors for vector maxima in three dimensions.

In this paper we present a truly practical and provably optimal O(n logh) time outputsensitive algorithm for the planar convex hull problem. The basic algorithm is similar to the algorithm presented in Chan, Snoeyink and Yap 2] where the median-nding step is replaced by an approximate median. We analyze two such schemes and show that for both methods, the algorithm runs in expected O(n log h) time. The expected number of comparisons can be made smaller than 5n logh for the upper-hull. We further show that the probability of deviation from expected running time approaches 0 rapidly with increasing values of n and h for any input. Our experiments suggest that this algorithm is a practical alternative to the worstcase O(n log n) algorithms like Graham's and especially faster for small output-sizes. Our approach bears some resemblance to a recent algorithm of Wenger 13] but our analysis is substantially di erent. The planar convex hull problem is perhaps the most studied problem in computational geometry and a large body of literature deals with computing convex hulls. Graham 5] was the rst to present an O(n log n) worst-case time algorithm. This algorithm is optimal as Yao 14] showed that (n log n) is the lower bound of the convex hull problem for the worst-case input. Some simple algorithms have O(n) expected time for known distributions of points such as uniform in a box, normal, etc. The rst output-sensitive algorithm was proposed by Chand and Kapur 3]. The two-dimensional version of their algorithm is known as the rope fence method and was independently reported by Jarvis 6]. The rope fence method takes O(nh) time to compute h extreme edges of the convex hull. Kirkpatrick and Seidel 8] proved an (n logh) lower bound when both input and output sizes are considered, so Yao's lower-bound is a special case when log h 2 (log n). They also proposed an O(n log h) optimal algorithm based on the prune-and-search technique developed by Dyer 4] and Megiddo 9]. However, it has high constants and is considered prohibitively complicated for implementation. Very recently, in 1], two O(n log h) algorithms have been proposed. One uses the linear-time median nding algorithm and the other uses a clever grouping technique. Although the latter algorithm does not have any expensive median-nding step it relies on a sophisticated logarithmic time tangent-nding routine.

In this paper we focus on the problem of designing very fast parallel algorithms for the planar convex hull problem that achieve the optimal O(n log H) work-bound for input size n and output size H. Our algorithms are designed for the arbitrary CRCW PRAM model. We first describe a very simple ©(log n log H) time optimal deterministic algorithm for the planar hulls which is an improvement over the previously known ~(log 2 n) time algorithm for small outputs. For larger values of H, we can achieve a running time of ©(log n log log n) steps with optimal work. We also present a fast randomized algorithm that runs in expected time ©(log H-log log n) and does optimal O(nlogH) work. For logH = ~2(log log n), we can achieve the optimal running time of O(log H) while simultaneously keeping the work optimal. When log H is o(log n), our results improve upon the previously best known ©(log n) expected time randomized algorithm of Ghouse and Goodrich. The randomized algorithms do not assume any input distribution and the running times hold with high probability.

In order to express a polyhedron as the (Minkowski) sum of a polytope and a polyhedral cone, Motzkin (1936) made a transition from the polyhedron to a polyhedral cone. Based on his excellent idea, we represent a set by a characteristic cone. By using this representation, we then reach four main results: (i) expressing a closed convex set containing no line as the direct sum of the convex hull of its extreme points and conical hull of its extreme directions, (ii) establishing a convex programming (CP) based framework for determining a maximal element-an element with the maximum number of positive components-of a convex set, (iii) developing a linear programming problem for finding a relative interior point of a polyhedron, and (iv) proposing two procedures for the identification of a strictly complementary solution in linear programming.

In order to express a polyhedron as the (Minkowski) sum of a polytope and a polyhedral cone, Motzkin (1936) made a transition from the polyhedron to a polyhedral cone. Based on his excellent idea, we represent a set by a characteristic cone. By using this representation, we then reach four main results: (i) expressing a closed convex set containing no line as the direct sum of the convex hull of its extreme points and conical hull of its extreme directions, (ii) establishing a convex programming (CP) based framework for determining a maximal element-an element with the maximum number of positive components-of a convex set, (iii) developing a linear programming problem for finding a relative interior point of a polyhedron, and (iv) proposing two procedures for the identification of a strictly complementary solution in linear programming.

Let S be a set of n &gt; 2 points in the plane whose convex hull has perimeter t. Given a number P ≥ t, we study the following problem: Of all curves of perimeter P that enclose S, which is the curve that encloses the maximum area? In this paper, we give a complete characterization of this curve. We show that there are cases where this curve cannot be computed exactly and provide an O(n logn)-time algorithm to obtain an approximation of this curve, with arbitrary precision, having the same combinatorial structure.

Given a planar straight-line graph G = (V, E) in R 2 , a circumscribing polygon of G is a simple polygon P whose vertex set is V , and every edge in E is either an edge or an internal diagonal of P . A circumscribing polygon is a polygonization for G if every edge in E is an edge of P . We prove that every arrangement of n disjoint line segments in the plane has a subset of size Ω( √ n) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R 3 . We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.

This paper generalizes the linear sector in the classical absolute stability theory to a sector bounded by concave/convex functions. This generalization allows more flexible or more specific description of the nonlinearity and will thus reduce the conservatism in the estimation of the domain of attraction. We introduce the notions of generalized sector and absolute contractive invariance for estimating the domain of attraction of the origin. Necessary and sufficient conditions are identified under which an ellipsoid is absolutely contractively invariant. In the case that the sector is bounded by piecewise linear concave/convex functions, these conditions can be exactly stated as linear matrix inequalities. Moreover, if we have a set of absolutely contractively invariant (ACI) ellipsoids, then their convex hull is also ACI and inside the domain of attraction. We also present optimization technique to maximize the absolutely contractively invariant ellipsoids for the purpose of estimating the domain of attraction. The effectiveness of the proposed method is illustrated with examples.

In Indonesia, plastic garbage bottles are the most common sort of waste. Given that waste is expected to grow annually, managing plastic waste is a major challenge. The results of the study were achieved by comparing the reference, which was a collection of manually created contour images, with 50 sets of vortex images with different forms and vortex areas as experimental objects. The results indicate that the suggested approach reports a mean error of 2.84%, a correlation coefficient of 0.9965, and a root mean square error of 0.2903 when compared to the manual extraction method. These findings imply that the extract area determined by the procedure outlined in this research is more accurate and nearer to the actual values. The proposed method can therefore be used in place of the traditional process for investigating cooling parameters through manual testing. With measurement values mean absolute percentage error (MAPE)=121,842, mean absolute deviation (MAD)=20,140, and mean squared deviation (MSD)=776,712, the trend analysis of plastic bottles for autoregressive integrated moving average (ARIMA) modeling leads to the conclusion that the waste from plastic bottles will continue to rise annually and that efforts must be made to address this trend with knowledge and waste recycling technology. Plastic that is advantageous to industry and society.

The World Health Organization (WHO) has identified coronavirus disease (COVID-19), as a global pandemic due to its quick global spread to more than 183 countries. Many countries have used movement control orders (MCO) and high alert levels to halt the spread. The primary goal of this research is to provide a geofencing architecture that is specially tailored to the MCO's standard requirement for monitoring an individual's whereabouts during a lockdown. Whenever an individual tests Corona positive, Geofencing uses technology to notify an anticipated network of people who may be affected and to enable traceability for potential patients. Computational techniques such as Delaunay triangulation (inpolygon) and triangle weight characterization (inside polygon) are applied to analyze the geographical boundary in which the patient is isolated. Convex hull, on the other hand, is a better technique than computational algorithms. It is considered the best mathematical technique because it takes the least amount of time (0.014985 sec) to detect the patient within the geofence layer and has the lowest standard deviation when compared with the other computational techniques.

Benchmarking plays a relevant role in performance analysis, and statistical methods can be fruitfully exploited for its aims. While clustering, regression, and frontier analysis may serve some benchmarking purposes, we propose to consider archetypal analysis as a suitable technique. Archetypes are extreme points that synthesize data and that, in our opinion, can be profitably used as benchmarks. That is, they may be viewed as key reference performers in the comparison process. We suggest a three‐step data driven benchmarking procedure, which enables users: (i) to identify some reference performers, (ii) to analyze their features, (iii) to compare observed performers with them. An exploratory point of view is preferred, and graphical devices are adopted throughout the procedure. Finally, our approach is presented by means of an illustrative example based on The Times league table of the world top 200 universities. Copyright © 2008 John Wiley & Sons, Ltd.

The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

We consider planar geometric models given by an explicit boundary of 0 en) algebraic curve segments of maximum degree d. We present an 0 en"dO(l) time algorithm to compute its convex hull and an 0 (nlog logn + K) . dO(l) time algorithms to compute various decompositions of an object, where K is the characteristic number of this object. Both operations, besides being solutions to interesting computational geometry problems, prove useful in motion planning with planar geometric models. ...

A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is a set of non-crossing straight line segments with endpoints in S. Given a set of red points and a set of blue points in the plane, the red/blue spanning tree problem is to find a geometric spanning tree for red points and a geometric spanning tree for blue points such that the number of crossing points of the two trees is a minimum. If no three points are collinear, we show that the minimum number of crossing points is completely determined by the number of maximal red (or blue) chains on the convex hull of all red points and blue points. We design an optimal algorithm for constructing a geometric spanning tree of all the red points and a geometric spanning tree of all the blue points with the minimum number of crossing points. If collinear points are allowed, we prove that the problem of deciding whether there exists a geometric spanning path of all the red points and a geometric spanning path of all the blue points without crossing is NP-complete.

This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point simply by "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure to take expected time close to O(nl/(d+l)) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.

Ten points in the plane numbered {1 ′ , 2 ′ , … , 10 ′ } are arbitrarily combined in pairs denoted {1,. . ,5}. Pairs of the points denoted {1,. . ,5} serve as the vertices of the graph. The vertices are "friends", when all of the points belonging to the both of the vertices are located on the same straight line. In this case, they are connected with brown link. The vertices are "strangers", when the points related to these vertices do not belong to the same straight line (at least one of the points is out of the straight line). In this case, they are connected with the violet link. The emerging bi-colored, complete, semitransitive graph will contain at least one monochromatic triangle; whatever, is the location of the points on the plane and whatever is their partition into pairs. 3D generalization of the suggested approach is discussed.

Main results of this paper were obtained together with Alexander Nagaev, with whom the first author had collaborated for more than 35 years, until Alexander's tragic death in 2005. Since then, we have gathered strength and finalised this paper, strongly feeling Alexander's absence -our memories of him will stay with us forever.

It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d , its convex hull V (t) = conv{ B(s), s ≤ t} with probability 1 for each t > 0 contains 0 as an interior point (see ). We also know that the winding number of a typical path of a 2-dimensional BM is equal to +∞. The aim of this article is to show that these properties aren't specifically "Brownian", but hold for a much larger class of d-dimensional self similar processes. This class contains in particular d-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Levy processes.

made the conjecture that, for n ≥ 3, any set of 2 n-2 + 1 points in the plane, in general position, contains n points in convex position. A computer-based proof of this conjecture for n = 6 appeared in of Szekeres and Peters. The aim of this paper is to give a proof of the conjecture for n = 6, without the use of computers, under the restriction that the convex hull of the point set is a pentagon.

Cahier n°2003-002 Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet de G est muni d'un poids. Le problème du sous-graphe 2-arêtes connexe de poids minimum dans G (2ECSP), est de trouver un sous-graphe 2-arêtes connexe de G tel que la somme des poids sur ses sommets et ses arête soit minimum. Le 2ECSP généralise le problème, bien connu, du sousgraphe Steiner 2-arêtes connexe. Dans cet article, l'enveloppe convexe des vecteurs d'incidences des solutions de 2ECSP est étudiée. Une formulation naturelle du problème par un programme linéaire en nombres entiers est premièrement établie. Il est aussi montré que la relaxation ne suffit pas pour décrire l'enveloppe convexe associée au 2ECSP même dans une classe restreinte comme celle des graphes série-parallèles. Une nouvelle classe d'inégalités valides pour le 2ECSP est introduite. Il est monté qu'une sousclasse de ces inégalités peut être séparée en temps polynomial. Comme conséquence, ces nouvelles inégalités peuvent être séparées en temps polynomial dans la classe des graphes série-parallèles.

We present a general method for constructing effective field theories for non-relativistic superfluids, generalizing the previous approaches of Greiter, Witten, and Wilczek, and Son and Wingate to the case of several superfluids in solution. We investigate transport in mixtures with broken parity and find a parity odd "Hall drag" in the presence of independent motion as well as a pinning of mass, charge, and energy to sites of nonzero relative velocity. Both effects have a simple geometric interpretation in terms of the signed volumes and directed areas of various sub-complexes of a "velocity polyhedron": the convex hull formed by the endpoints of the velocity vectors of a superfluid mixture. We also provide a simple quasi-one-dimensional model that exhibits non-zero Hall drag.

We present a general method for constructing effective field theories for non-relativistic superfluids, generalizing the previous approaches of Greiter, Witten, and Wilczek, and Son and Wingate to the case of several superfluids in solution. We investigate transport in mixtures with broken parity and find a parity odd "Hall drag" in the presence of independent motion as well as a pinning of mass, charge, and energy to sites of nonzero relative velocity. Both effects have a simple geometric interpretation in terms of the signed volumes and directed areas of various sub-complexes of a "velocity polyhedron": the convex hull formed by the endpoints of the velocity vectors of a superfluid mixture. We also provide a simple quasi-one-dimensional model that exhibits non-zero Hall drag.

A polytope plays a central role in different areas of mathematics. It is used quite heavily in applied fields of mathematics, such as medical imaging and robotics, geometric modeling. A polytope has many users in modern science such as computer graphics, optimization, and search engine. It is intended for a broad audience of mathematically inclined. Therefore in this paper, we shall take a polytope with one kind of application, which is known as a Stasheff polytope. it has been applied in: moduli spaces, Erhart polynomial, physics-chemistry, and Hopf algebras. In this paper, a finite graph G with tubing is taken, the nodes of the graph are reduce to points in and the convex hull for them are simplex, permutahedron and associahedron (Stasheff polytope ) are studied. Also, a fan graph to the painted tree is also taken and reduces its nodes to points in . The converse of this result is also given with different examples to consult our results.

We consider regret minimization in repeated games with non-convex loss functions. Minimizing the standard notion of regret is computationally intractable. Thus, we define a natural notion of regret which permits efficient optimization and generalizes offline guarantees for convergence to an approximate local optimum. We give gradient-based methods that achieve optimal regret, which in turn guarantee convergence to equilibrium in this framework.