In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of , its subsequence has a low discrepancy.
In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory.
In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, Analytic Combinatorics, while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions.
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided dice rolled n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function.
In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In statistics, an empirical distribution function is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.
Discrepancy of hypergraphs is an area of discrepancy theory that studies the discrepancy of general set systems.
Amplitude amplification is a technique in quantum computing which generalizes the idea behind Grover's search algorithm, and gives rise to a family of quantum algorithms. It was discovered by Gilles Brassard and Peter Høyer in 1997, and independently rediscovered by Lov Grover in 1998.
A positional game is a kind of a combinatorial game for two players. It is described by:
A Maker-Breaker game is a kind of positional game. Like most positional games, it is described by its set of positions/points/elements and its family of winning-sets. It is played by two players, called Maker and Breaker, who alternately take previously untaken elements.
In mathematics, an algebraic number field is an extension field of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .
The set balancing problem in mathematics is the problem of dividing a set to two subsets that have roughly the same characteristics. It arises naturally in design of experiments.
A Waiter-Client game is a kind of positional game. Like most positional games, it is described by its set of positions/points/elements, and its family of winning-sets. It is played by two players, called Waiter and Client. Each round, Waiter picks two elements, Client chooses one element and Waiter gets the other element.
A biased positional game is a variant of a positional game. Like most positional games, it is described by a set of positions/points/elements and a family of subsets, which are usually called the winning-sets. It is played by two players who take turns picking elements until all elements are taken. While in the standard game each player picks one element per turn, in the biased game each player takes a different number of elements.
In computer science, a parallel external memory (PEM) model is a cache-aware, external-memory abstract machine. It is the parallel-computing analogy to the single-processor external memory (EM) model. In a similar way, it is the cache-aware analogy to the parallel random-access machine (PRAM). The PEM model consists of a number of processors, together with their respective private caches and a shared main memory.
