scholarly journals Identifying the Maximum Concentration of Results in Bilateral Sports Competitions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1293
Author(s):  
Antonio Avila-Cano ◽  
Amparo Ruiz-Sepulveda ◽  
Francisco Triguero-Ruiz
Keyword(s):  
Stability Condition ◽  
Scoring System ◽  
Maximum Concentration ◽  
Concentration Level ◽  
Maximum Level ◽  
Competitive Balance ◽  
Sports Economics ◽  
Relevant Question ◽  
European Football ◽  
The Stability

There are situations in which a monopoly solution cannot be reached. In these cases, which situation represents the maximum concentration (minimum competitive balance)? It is a relevant question, given that in sports economics, measuring the competitive balance of a league is done through normalized indices. These indices require that the maximum level of concentration be known. Until now, the distribution of results that generates the maximum level of concentration has been identified in the literature as Complete cascade distribution. However, if the scoring system used does not fulfil the stability condition, which implies that the total number of points of a championship is constant, it can be demonstrated that the Complete cascade distribution does not generate the maximum level of concentration. This is the case, for example, with major European football leagues. In this article, we constructed a perfectly unbalanced distribution, which we called a Truncated cascade distribution. This distribution generates the maximum concentration level. Therefore, if we do not use Truncated cascade distribution, there is an overestimation of the concentration measured with normalized indices. Then, the calculated competitive balance will be wrong, that is, underestimated. We provided a spreadsheet for identifying this distribution.

scholarly journals Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1447
Author(s):  
Jose P. Suárez ◽  
Agustín Trujillo ◽  
Tania Moreno
Keyword(s):  
Data Structure ◽  
Stability Condition ◽  
Open Problem ◽  
Integer Arithmetic ◽  
Exact Number ◽  
Similarity Class ◽  
The Stability ◽  
High Level ◽  
The Cost

Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

Density evolution, thresholds and the stability condition for non-binary LDPC codes

2005 ◽  
Vol 152 (6) ◽  
pp. 1069 ◽  
Author(s):  
V. Rathi ◽  
R. Urbanke
Keyword(s):  
Stability Condition ◽  
Ldpc Codes ◽  
Density Evolution ◽  
The Stability

Computation of the Stability Condition

1976 ◽  
pp. 104-130
Author(s):  
J. E. Marsden ◽  
M. McCracken
Keyword(s):  
Stability Condition ◽  
The Stability

On multitype branching processes in a random environment

1981 ◽  
Vol 13 (3) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

Characterization of the dissipative structure for the symmetric hyperbolic system with non-symmetric relaxation

2021 ◽  
Vol 18 (01) ◽  
pp. 195-219
Author(s):  
Yoshihiro Ueda
Keyword(s):  
Hyperbolic System ◽  
Stability Condition ◽  
Dissipative Structure ◽  
General System ◽  
Partial Result ◽  
Symmetric Hyperbolic System ◽  
Standard Type ◽  
Classical Stability ◽  
The Stability

This paper is concerned with the dissipative structure for the linear symmetric hyperbolic system with non-symmetric relaxation. If the relaxation matrix of the system has symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called Classical Stability Condition in this paper, and Umeda, Kawashima and Shizuta in 1984 analyzed the dissipative structure of the standard type. On the other hand, Ueda, Duan and Kawashima in 2012 and 2018 focused on the system with non-symmetric relaxation, and got the partial result which is the extension of known results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, our purpose of this paper is to extend the stability theory introduced by Shizuta and Kawashima in 1985 and Umeda, Kawashima and Shizuta in 1984 for our general system.

Kartellrechtliche Zulässigkeit von Superligen im Fußball

2021 ◽  
Author(s):  
Dennis Cukurov
Keyword(s):  
Conflict Of Interest ◽  
Competition Law ◽  
Competitive Balance ◽  
Prevention Measures ◽  
Before And After ◽  
European Football ◽  
The Creation ◽  
Super League

The creation of a European football super league is becoming more and more likely. Some top clubs want to introduce such a league without involving the UEFA. The UEFA wants to prevent this in order to keep its tournaments free of competition. This conflict of interest is an example of the more general tension between competition law and sport. The author examines not only UEFA’s prevention measures, but also possible cooperation between the clubs. Among other things, he addresses two topics that have been insufficiently discussed so far, the concept of legitimate objective within the meaning of the so-called Meca-Medina test and the competitive balance before and after the creation of a super league, and argues for the implementation of a “more Europe” approach in European football.

The driving forces of competitive balance in European football

2018 ◽  
pp. 101-132
Author(s):  
N. David Pifer
Keyword(s):  
Driving Forces ◽  
Competitive Balance ◽  
European Football
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