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Random Two-sided Matching
A random two-sided matching is a process by which members of two groups are matched to each other in a random way. It is often used in sports in order to match teams in knock-out tournaments. In this context, it is often called a draw, as it is implemented by drawing balls at random from a bowl, each ball representing the name of a team. Examples The UEFA Champions League, UEFA Europa League, and UEFA Conference League draw A random two-sided matching occurs in the UEFA Champions League Round of 16 and UEFA Europa League Round of 32. After some games are done within 8 groups, the group winner and the group runner-up proceed to the champions league. The UEFA rules say that each winner should be paired with a runner-up. Without further constraints, this problem could easily be solved by finding a random permutation of the winners. But UEFA rules impose two additional constraints: two teams from the same group cannot be paired, and two teams from the same association cannot be pai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knock-out Tournament
A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses: # One or more competitions held at a single venue and concentrated into a relatively short time interval. # A competition involving a number of matches, each involving a subset of the competitors, with the overall tournament winner determined based on the combined results of these individual matches. These are common in those sports and games where each match must involve a small number of competitors: often precisely two, as in most team sports, racket sports and combat sports, many card games and board games, and many forms of competitive debating. Such tournaments allow large numbers to compete against each other in spite of the restriction on numbers in a single match. These two senses are distinct. All golf tournaments meet the first definition, but while match play tournaments meet the second, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UEFA Champions League
The UEFA Champions League (UCL) is an annual club association football competition organised by the UEFA, Union of European Football Associations (UEFA) that is contested by List of top-division football clubs in UEFA countries, top-division European clubs. The competition begins with a Round-robin tournament, round robin league phase to qualify for the double-legged knockout rounds, and a single-leg final. It is the most-watched club competition in the world and the third most-watched football competition overall, behind only the FIFA World Cup and the UEFA European Championship. It is one of the most prestigious football tournaments in the world and the most prestigious club competition in European football, played by the national league champions (and, for some nations, one or more runners-up) of their national associations. Introduced 1955–56 European Cup, in 1955 as the European Champion Clubs' Cup (), and commonly known as the European Cup, it was initially a straigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UEFA Europa League
The UEFA Europa League (UEL), usually known simply as the Europa League, is an annual association football, football club competition organised since 1971 by the UEFA, Union of European Football Associations (UEFA) for eligible European football clubs. It is the second-tier competition of UEFA competitions, European club football, ranking below the UEFA Champions League and above the UEFA Conference League. Introduced in 1971 as the UEFA Cup, it replaced the Inter-Cities Fairs Cup. From the 2004–05 UEFA Cup, 2004–05 season a group stage was added before the knockout phase. The competition took on its current name in 2009–10 UEFA Europa League, 2009, following a change in format. The 2009 re-branding included a merge with the UEFA Intertoto Cup, producing an enlarged competition format, with an expanded group stage and a change in qualifying criteria. In the 2024–25 UEFA Europa League, 2024–25 season, the group stage was replaced with an expanded league phase of 36 te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Permutation
A random permutation is a sequence where any order of its items is equally likely at random, that is, it is a permutation-valued random variable of a set of objects. The use of random permutations is common in games of chance and in randomized algorithms in coding theory, cryptography, and simulation. A good example of a random permutation is the fair shuffling of a standard deck of cards: this is ideally a random permutation of the 52 cards. Computation of random permutations Entry-by-entry methods One algorithm for generating a random permutation of a set of size ''n'' uniformly at random, i.e., such that each of the ''n''! permutations is equally likely to appear, is to generate a sequence by uniformly randomly selecting an integer between 1 and ''n'' (inclusive), sequentially and without replacement ''n'' times, and then to interpret this sequence (''x''1, ..., ''x''''n'') as the permutation : \begin 1 & 2 & 3 & \cdots & n \\ x_1 & x_2 & x_3 & \cdots & x_n \\ \end, shown h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Football Association
A football association, also known as a football federation, soccer federation, or soccer association, is a governing body for association football. Many of them are members of the sport's regional bodies such as UEFA and CONMEBOL and the world governing body, FIFA. A small number have not yet applied for or been granted entry to these higher bodies. Below is a list of football associations for which there are articles. Asia Asian Football Confederation affiliated * Afghanistan Football Federation * Football Australia * Bahrain Football Association * Bangladesh Football Federation * Bhutan Football Federation * Football Association of Brunei Darussalam * Myanmar Football Federation * Football Federation of Cambodia * Chinese Football Association * Chinese Taipei Football Association * East Timor Football Federation * Guam Football Association * Hong Kong Football Association * All India Football Federation * Football Association of Indonesia * Football Federation Islami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conspiracy Theories
A conspiracy theory is an explanation for an event or situation that asserts the existence of a conspiracy (generally by powerful sinister groups, often political in motivation), when other explanations are more probable.Additional sources: * * * * The term generally has a negative connotation, implying that the appeal of a conspiracy theory is based in prejudice, emotional conviction, or insufficient evidence. A conspiracy theory is distinct from a conspiracy; it refers to a hypothesized conspiracy with specific characteristics, including but not limited to opposition to the mainstream consensus among those who are qualified to evaluate its accuracy, such as scientists or historians. Conspiracy theories tend to be internally consistent and correlate with each other; they are generally designed to resist Falsifiability, falsification either by evidence against them or a lack of evidence for them. They are reinforced by circular reasoning: both evidence against the conspiracy '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FIFA World Cup
The FIFA World Cup, often called the World Cup, is an international association football competition among the senior List of men's national association football teams, men's national teams of the members of the FIFA, Fédération Internationale de Football Association (FIFA), the sport's global governing body. The tournament has been held every four years since the 1930 FIFA World Cup, inaugural tournament in 1930, with the exception of 1942 and 1946 due to the Second World War. The reigning champions are Argentina national football team, Argentina, who won their third title at the 2022 FIFA World Cup, 2022 World Cup by defeating France national football team, France. The contest starts with the FIFA World Cup qualification, qualification phase, which takes place over the preceding three years to determine which teams qualify for the tournament phase. In the tournament phase, 32 teams compete for the title at venues within the host nation(s) over the course of about a month. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Random Assignment
Fair random assignment (also called probabilistic one-sided matching) is a kind of a fair division problem. In an ''assignment problem'' (also called '' house-allocation problem'' or '' one-sided matching''), there are ''m'' objects and they have to be allocated among ''n'' agents, such that each agent receives at most one object. Examples include the assignment of jobs to workers, rooms to housemates, dormitories to students, time-slots to users of a common machine, and so on. In general, a fair assignment may be impossible to attain. For example, if Alice and Batya both prefer the eastern room to the western room, only one of them will get it and the other will be envious. In the random assignment setting, fairness is attained using a lottery. So in the simple example above, Alice and Batya will toss a fair coin and the winner will get the eastern room. History Random assignment is mentioned already in the Bible: a lottery was used to allocate the lands of Canaan among the Tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected Graph (discrete mathematics), graph is a set of Edge (graph theory), edges without common vertex (graph theory), vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a Flow network, network flow problem. Definitions Given a Graph (discrete mathematics), graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loop (graph theory), loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
