Symmetric groupIn abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .
Problem solvingProblem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles.
Lagrange's theorem (group theory)In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index , defined as the number of left cosets of in . Lagrange's theorem This variant holds even if is infinite, provided that , , and are interpreted as cardinal numbers.
Creative problem-solvingCreative problem-solving (CPS) is the mental process of searching for an original and previously unknown solution to a problem. To qualify, the solution must be novel and reached independently. The creative problem-solving process was originally developed by Alex Osborn and Sid Parnes. Creative problem solving (CPS) is a way of using creativity to develop new ideas and solutions to problems. The process is based on separating divergent and convergent thinking styles, so that one can focus one's mind on creating at the first stage, and then evaluating at the second stage.
Classification of finite simple groupsIn mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
Knapsack problemThe knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.
Binomial proportion confidence intervalIn statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes nS are known. There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution.
Organized crimeOrganized crime (or organised crime) is a category of transnational, national, or local groupings of highly centralized enterprises run by criminals to engage in illegal activity, most commonly for profit. While organized crime is generally thought of as a form of illegal business, some criminal organizations, such as terrorist groups, rebel forces, and separatists, are politically motivated. Many criminal organizations rely on fear or terror to achieve their goals or aims as well as to maintain control within the organization and may adopt tactics commonly used by authoritarian regimes to maintain power.
Approximation algorithmIn computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time.
Counterfeit moneyCounterfeit money is currency produced without the legal sanction of a state or government, usually in a deliberate attempt to imitate that currency and so as to deceive its recipient. Producing or using counterfeit money is a form of fraud or forgery, and is illegal. The business of counterfeiting money is nearly as old as money itself: plated copies (known as Fourrées) have been found of Lydian coins, which are thought to be among the first Western coins.
CreativityCreativity is a characteristic of someone or some process that forms something new and valuable. The created item may be intangible (such as an idea, a scientific theory, a musical composition, or a joke) or a physical object (such as an invention, a printed literary work, or a painting). Scholarly interest in creativity is found in a number of disciplines, primarily psychology, business studies, and cognitive science.
Travelling salesman problemThe travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.
Bayes' theoremIn probability theory and statistics, Bayes' theorem (beɪz or beɪzɪz ; alternatively Bayes' law or Bayes' rule), and occasionally Bayes's theorem, named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.
Bound stateIn quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. The potential may be external or it may be the result of the presence of another particle; in the latter case, one can equivalently define a bound state as a state representing two or more particles whose interaction energy exceeds the total energy of each separate particle. One consequence is that, given a potential vanishing at infinity, negative-energy states must be bound.
