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Absorbing Markov Chain
In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this article concentrates on the discrete-time discrete-state-space case. Formal definition A Markov chain is an absorbing chain if # there is at least one absorbing state and # it is possible to go from any state to at least one absorbing state in a finite number of steps. In an absorbing Markov chain, a state that is not absorbing is called transient. Canonical form Let an absorbing Markov chain with transition matrix ''P'' have ''t'' transient states and ''r'' absorbing states. Unlike a typical transition matrix, the rows of ''P'' represent sources, while columns represent destinations. Then : P = \begin Q & R\\ \mathbf & I_r \end, where ''Q'' is a ''t' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain For String Generation Example
Markov ( Bulgarian, ), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics * Ivana Markova (1938–2024), Czechoslovak-British emeritus professor of psychology at the University of Stirling *John Markoff (sociologist) (born 1942), American professor of sociology and history at the University of Pittsburgh * Konstantin Markov (1905–1980), Soviet geomorphologist and quaternary geologist Mathematics, science, and technology * Alexander V. Markov (born 1965), Russian biologist *Andrey Markov (1856–1922), Russian mathematician * Andrey Markov Jr. (1903–1979), Russian mathematician and son of Andrey Markov *Elena Vladimirovna Markova (1923–2023), Soviet and Russian cyberneticist, Doctor of Technical Sciences, gulag convict and memoirist. *John Markoff (born 1949), American journalist of computer industry and technology *Moisey Markov (1908–1994), Russian physicist * Vladimir Andreevich Markov (1871–1897), Russia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing Set (random Dynamical Systems)
In the mathematical theory of random dynamical systems, an absorbing set is a subset of the phase space that exhibits a capturing property. It acts like a gravitational center, with the property that all trajectories of the system eventually enter and remain within that set. Absorbing sets ultimately contain the transformed images of any initially bounded set as the system evolves over time. As with many concepts related to random dynamical systems, it is defined in the pullback sense, which means they are understood through their long-term behavior. Absorbing sets are a key concept in the study of the long-term behavior of dynamical systems, particularly in the context of dissipative systems, as they provide a bound on the possible future states of the system. The existence and properties of absorbing sets are fundamental to establishing the existence of global attractors and understanding the asymptotic behavior of solutions. Definition Consider a random dynamical system ''φ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Phase-type Distribution
The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of an absorbing Markov chain with one absorbing state. Each of the states of the Markov chain represents one of the phases. It has continuous time equivalent in the phase-type distribution. Definition A terminating Markov chain is a Markov chain where all states are transient, except one which is absorbing. Reordering the states, the transition probability matrix of a terminating Markov chain with m transient states is : =\left begin&\mathbf^0\\\mathbf^\mathsf&1\end\right where is a m\times m matrix, \mathbf^0 and \mathbf are column vectors with m entries, and \mathbf^0+\mathbf=\mathbf. The transition matrix is charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snakes And Ladders
Snakes and ladders is a board game for two or more Player (game), players regarded today as a worldwide classic. The game Traditional games of India, originated in ancient India as ''Moksha Patam'', and was brought to the United Kingdom in the 1890s. It is played on a game board with numbered, gridded squares. A number of "ladders" and "snakes" are pictured on the board, each connecting two specific board squares. The object of the game is to navigate one's game piece, according to Dice, die rolls, from the start (bottom square) to the finish (top square), helped by climbing ladders but hindered by falling down snakes. The game is a simple race based on sheer luck, and it is popular with young children. The historic version had its roots in morality lessons, on which a player's progression up the board represented a life journey complicated by virtues (ladders) and vices (snakes). The game is also sold under other names, such as the morality themed ''Chutes and Ladders'', whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Of Winning Snakes And Ladders By Turns
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty String
In formal language theory, the empty string, or empty word, is the unique String (computer science), string of length zero. Formal theory Formally, a string is a finite, ordered sequence of character (symbol), characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with ε or sometimes Λ or λ. The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string. The empty string has several properties: * , ε, = 0. Its string (computer science)#Formal theory, string length is zero. * ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the concatenation operation. The set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Coin
In probability theory and statistics, a sequence of Independence (probability theory), independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a Crown (British coin), crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000. In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the met ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Product (matrices)
In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur product) is a binary operation that takes in two Matrix (mathematics), matrices of the same dimensions and returns a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the Matrix multiplication, matrix product. It is attributed to, and named after, either French mathematician Jacques Hadamard or German mathematician Issai Schur. The Hadamard product is associative and Distributive property, distributive. Unlike the matrix product, it is also commutative. Definition For two matrices and of the same dimension , the Hadamard product A \odot B (sometimes A \circ B) is a matrix of the same dimension as the operands, with elements given by :(A \odot B)_ = (A)_ (B)_. For matrices of different dimensions ( and , where or ), the Hadamard product is undefined. An example of the Hadamard product for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 4 \end\right/math>. An identity matrix of any size, or any multiple of it is a diagonal matrix called a ''scalar matrix'', for example, \left begin 0.5 & 0 \\ 0 & 0.5 \end\right/math>. In geometry, a diagonal matrix may be used as a '' scaling matrix'', since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with columns and rows is diagonal if \forall i,j \in \, i \ne j \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
