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Ergodic Sequence
In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties. Definition Let A = \ be an infinite, strictly increasing sequence of positive integers. Then, given an integer ''q'', this sequence is said to be ergodic mod ''q'' if, for all integers 1\leq k \leq q, one has :\lim_ \frac = \frac where :N(A,t) = \mbox \ and card is the count (the number of elements) of a set, so that N(A,t) is the number of elements in the sequence ''A'' that are less than or equal to ''t'', and :N(A,t,k,q) = \mbox \ so N(A,t,k,q) is the number of elements in the sequence ''A'', less than ''t'', that are equivalent to ''k'' modulo ''q''. That is, a sequence is an ergodic sequence if it becomes uniformly distributed mod ''q'' as the sequence is taken to infinity. An equivalent definition is that the sum :\lim_ \frac \sum_ \exp \frac = 0 vanish for every integer ''k'' with k \mod q \ne 0. If a sequence is ergodic for all ''q'', then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "a negligible amount of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only exception). * Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Sequence
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables ''X''''i'' are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable ''X''''i'' in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme. The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of checking whether a coin is fair. Definition A Bernoulli process is a fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Process
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables ''X''''i'' are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable ''X''''i'' in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme. The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of checking whether a coin is fair. Definition A Bernoulli process is a fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A sample space, \Omega, which is the set of all possible outcomes. # An event space, which is a set of events \mathcal, an event being a set of outcomes in the sample space. # A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be \. For the event space, we could simply use the set of all subsets of the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Process
In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime. Conversely, a process that is not in ergodic regime is said to be in non-ergodic regime. Specific definitions One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process X(t) has constant mean :\mu_X= E (t)/math>, and autocovariance :r_X(\tau) = E X(t)-\mu_X) (X(t+\tau)-\mu_X)/math>, that depends only on the lag \tau and not on time t. The properties \mu_X and r_X(\tau) are ''ensemble averages'' (calculated over all possible sample functions X), not time averages. The process X(t) is said to be mean-ergodicPapoulis, p.428 or mean-square ergodic in the first momentPorat, p.14 if the time average estima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was published in the Bell System Technical Journal. The paper laid the groundwork for later development of information c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
