text stringlengths 559 401k | source stringlengths 13 121 |
|---|---|
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to it... | Wikipedia/Weak_topology |
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant un... | Wikipedia/Spectrum_of_a_C*-algebra |
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical forma... | Wikipedia/Mathematical_formulation_of_quantum_mechanics |
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topologi... | Wikipedia/Lebesgue-measurable_function |
In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author).
In linear algebra, it is synonymous with a linear form, which is a linear mapping from a vector space
V
{\displaystyle V... | Wikipedia/Functional_(mathematics) |
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
A blog post by T. Tao lists several closed graph theorems throughout mathematics.
== Gra... | Wikipedia/Closed_graph_theorem |
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to t... | Wikipedia/Group_algebra_of_a_locally_compact_group |
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space
X
{\displaystyle X}
is a real-valued function with only some of the properties of a seminorm. Unlike se... | Wikipedia/Sublinear_function |
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory o... | Wikipedia/Spectral_theory_of_ordinary_differential_equations |
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of v... | Wikipedia/Functional_calculus |
Functional linguistics is an approach to the study of language characterized by taking systematically into account the speaker's and the hearer's side, and the communicative needs of the speaker and of the given language community.: 5–6 Linguistic functionalism spawned in the 1920s to 1930s from Ferdinand de Saussure'... | Wikipedia/Functional_analysis_(linguistics) |
In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a go... | Wikipedia/Tomita–Takesaki_theory |
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k ... | Wikipedia/Linear_functional |
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation lik... | Wikipedia/Partial_differential_equations |
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the b... | Wikipedia/Mahler's_conjecture |
The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are overlapped so that the last half of one block coincide... | Wikipedia/Modified_discrete_cosine_transform |
Rate–distortion theory is a major branch of information theory which provides the theoretical foundations for lossy data compression; it addresses the problem of determining the minimal number of bits per symbol, as measured by the rate R, that should be communicated over a channel, so that the source (input signal) ca... | Wikipedia/Rate–distortion_theory |
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are... | Wikipedia/Molecular_dynamics |
The move-to-front (MTF) transform is an encoding of data (typically a stream of bytes) designed to improve the performance of entropy encoding techniques of compression. When efficiently implemented, it is fast enough that its benefits usually justify including it as an extra step in data compression algorithm.
This a... | Wikipedia/Move-to-front_transform |
Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, game theory, systems theory, decision theory and theoretical linguistics. Whereas ... | Wikipedia/Formal_science |
In mathematics and empirical science, quantification (or quantitation) is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method.
== Natural science ==
Some measure of the undisputed general importance ... | Wikipedia/Quantification_(science) |
In telecommunications, average bitrate (ABR) refers to the average amount of data transferred per unit of time, usually measured per second, commonly for digital music or video. An MP3 file, for example, that has an average bit rate of 128 kbit/s transfers, on average, 128,000 bits every second. It can have higher bitr... | Wikipedia/Average_bitrate |
Because the mathematical expressions for information theory developed by Claude Shannon and Ralph Hartley in the 1940s are similar to the mathematics of statistical thermodynamics worked out by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, in which the concept of entropy is central, Shannon was persuaded to emplo... | Wikipedia/Entropy_in_thermodynamics_and_information_theory |
In cryptography, a brute-force attack or exhaustive key search is a cryptanalytic attack that consists of an attacker submitting many possible keys or passwords with the hope of eventually guessing correctly. This strategy can theoretically be used to break any form of encryption that is not information-theoretically s... | Wikipedia/Brute_force_attack |
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable
Y
{\displaystyle Y}
given that the value of another random variable
X
{\displaystyle X}
is known... | Wikipedia/Conditional_entropy |
In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared by Shannon's source coding theorem, which states that any lossless data compression method must have an expected code length greater than or equal to the entropy of t... | Wikipedia/Entropy_coding |
In signal processing, a lapped transform is a type of linear discrete block transformation where the basis functions of the transformation overlap the block boundaries, yet the number of coefficients overall resulting from a series of overlapping block transforms remains the same as if a non-overlapping block transform... | Wikipedia/Lapped_transform |
In information technology and computer science, a system is described as stateful if it is designed to remember preceding events or user interactions; the remembered information is called the state of the system.
The set of states a system can occupy is known as its state space. In a discrete system, the state space is... | Wikipedia/State_(computer_science) |
In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.
== Definition ==
The joint Shannon entropy (in bits) of two discrete random variables
X
{\displaystyle X}
and
Y
{\displaystyle Y}... | Wikipedia/Joint_entropy |
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.
== Definition ==
A function
... | Wikipedia/Wavelet_transform |
842, 8-4-2, or EFT is a data compression algorithm. It is a variation on Lempel–Ziv compression with a limited dictionary length. With typical data, 842 gives 80 to 90 percent of the compression of LZ77 with much faster throughput and less memory use. Hardware implementations also provide minimal use of energy and mini... | Wikipedia/842_(compression_algorithm) |
Frame rate, most commonly expressed in frame/s, frames per second or FPS, is typically the frequency (rate) at which consecutive images (frames) are captured or displayed. This definition applies to film and video cameras, computer animation, and motion capture systems. In these contexts, frame rate may be used interch... | Wikipedia/Frame_rate |
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of thermodynamics, which convey a quantitative description using measurable macrosc... | Wikipedia/Thermodynamics |
Constructor theory is a proposal for a new mode of explanation in fundamental physics in the language of ergodic theory, developed by physicists David Deutsch and Chiara Marletto, at the University of Oxford, since 2012. Constructor theory expresses physical laws exclusively in terms of which physical transformations, ... | Wikipedia/Constructor_theory |
A cryptographically secure pseudorandom number generator (CSPRNG) or cryptographic pseudorandom number generator (CPRNG) is a pseudorandom number generator (PRNG) with properties that make it suitable for use in cryptography. It is also referred to as a cryptographic random number generator (CRNG).
== Background ==
... | Wikipedia/Cryptographically_secure_pseudorandom_number_generator |
In computer networking, linear network coding is a program in which intermediate nodes transmit data from source nodes to sink nodes by means of linear combinations.
Linear network coding may be used to improve a network's throughput, efficiency, and scalability, as well as reducing attacks and eavesdropping. The node... | Wikipedia/Network_coding |
Sequitur (or Nevill-Manning–Witten algorithm) is a recursive algorithm developed by Craig Nevill-Manning and Ian H. Witten in 1997 that infers a hierarchical structure (context-free grammar) from a sequence of discrete symbols. The algorithm operates in linear space and time. It can be used in data compression software... | Wikipedia/Sequitur_algorithm |
A key in cryptography is a piece of information, usually a string of numbers or letters that are stored in a file, which, when processed through a cryptographic algorithm, can encode or decode cryptographic data. Based on the used method, the key can be different sizes and varieties, but in all cases, the strength of t... | Wikipedia/Key_(cryptography) |
In telecommunications and computing, bit rate (bitrate or as a variable R) is the number of bits that are conveyed or processed per unit of time.
The bit rate is expressed in the unit bit per second (symbol: bit/s), often in conjunction with an SI prefix such as kilo (1 kbit/s = 1,000 bit/s), mega (1 Mbit/s = 1,000 kbi... | Wikipedia/Information_rate |
Bayesian inference ( BAY-zee-ən or BAY-zhən) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian inference uses a prior distribution to estimate posterior prob... | Wikipedia/Bayesian_inference |
The Burrows–Wheeler transform (BWT) rearranges a character string into runs of similar characters, in a manner that can be reversed to recover the original string. Since compression techniques such as move-to-front transform and run-length encoding are more effective when such runs are present, the BWT can be used as a... | Wikipedia/Burrows–Wheeler_transform |
In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted
D
KL
(
P
∥
Q
)
{\displaystyle D_{\text{KL}}(P\parallel Q)}
, ... | Wikipedia/Relative_entropy |
Algebraic code-excited linear prediction (ACELP) is a speech coding algorithm in which a limited set of pulses is distributed as excitation to a linear prediction filter. It is a linear predictive coding (LPC) algorithm that is based on the code-excited linear prediction (CELP) method and has an algebraic structure. AC... | Wikipedia/Algebraic_code-excited_linear_prediction |
Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy (a measure of average surprisal) of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this f... | Wikipedia/Differential_entropy |
Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between the two keys. The keys, in practice, represent a shared secret between two... | Wikipedia/Symmetric-key_algorithm |
In information theory, the binary entropy function, denoted
H
(
p
)
{\displaystyle \operatorname {H} (p)}
or
H
b
(
p
)
... | Wikipedia/Binary_entropy_function |
Transform coding is a type of data compression for "natural" data like audio signals or photographic images. The transformation is typically lossless (perfectly reversible) on its own but is used to enable better (more targeted) quantization, which then results in a lower quality copy of the original input (lossy compr... | Wikipedia/Transform_coding |
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to app... | Wikipedia/Estimation_theory |
Information processing theory is the approach to the study of cognitive development evolved out of the American experimental tradition in psychology. Developmental psychologists who adopt the information processing perspective account for mental development in terms of maturational changes in basic components of a chi... | Wikipedia/Information_processing_theory |
LZ4 is a lossless data compression algorithm that is focused on compression and decompression speed. It belongs to the LZ77 family of byte-oriented compression schemes.
== Features ==
The LZ4 algorithm aims to provide a good trade-off between speed and compression ratio. Typically, it has a smaller (i.e., worse) comp... | Wikipedia/LZ4_(compression_algorithm) |
Embedded zerotrees of wavelet transforms (EZW) is a lossy image compression algorithm. At low bit rates, i.e. high compression ratios, most of the coefficients produced by a subband transform (such as the wavelet transform) will be zero, or very close to zero. This occurs because "real world" images tend to contain mos... | Wikipedia/Embedded_zerotrees_of_wavelet_transforms |
In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy.
It is proportional to the expectation of the q-logarithm of a distribution.
== History ==
The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identi... | Wikipedia/Tsallis_entropy |
In information theory, the cross-entropy between two probability distributions
p
{\displaystyle p}
and
q
{\displaystyle q}
, over the same underlying set of events, measures the average number of bits needed to identify an event draw... | Wikipedia/Cross-entropy |
Communication theory is a proposed description of communication phenomena, the relationships among them, a storyline describing these relationships, and an argument for these three elements. Communication theory provides a way of talking about and analyzing key events, processes, and commitments that together form comm... | Wikipedia/Communication_theory |
Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (called noise, consisting of background stimuli and random activity of the ... | Wikipedia/Detection_theory |
Active networking is a communication pattern that allows packets flowing through a telecommunications network to dynamically modify the operation of the network.
Active network architecture is composed of execution environments (similar to a unix shell that can execute active packets), a node operating system capable o... | Wikipedia/Active_networking |
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger popul... | Wikipedia/Statistical_inference |
Asymmetric numeral systems (ANS) is a family of entropy encoding methods introduced by Jarosław (Jarek) Duda from Jagiellonian University, used in data compression since 2014 due to improved performance compared to previous methods. ANS combines the compression ratio of arithmetic coding (which uses a nearly accurate p... | Wikipedia/Asymmetric_numeral_systems |
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathemati... | Wikipedia/Bernoulli_trial |
In telecommunications and computing, bit rate (bitrate or as a variable R) is the number of bits that are conveyed or processed per unit of time.
The bit rate is expressed in the unit bit per second (symbol: bit/s), often in conjunction with an SI prefix such as kilo (1 kbit/s = 1,000 bit/s), mega (1 Mbit/s = 1,000 kbi... | Wikipedia/Bit_rate |
Integrated information theory (IIT) proposes a mathematical model for the consciousness of a system. It comprises a framework ultimately intended to explain why some physical systems (such as human brains) are conscious, and to be capable of providing a concrete inference about whether any physical system is conscious,... | Wikipedia/Integrated_information_theory |
Constant bitrate (CBR) is a term used in telecommunications, relating to the quality of service. Compare with variable bitrate.
When referring to codecs, constant bit rate encoding means that the rate at which a codec's output data should be consumed is constant. CBR is useful for streaming multimedia content on limite... | Wikipedia/Constant_bitrate |
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources n... | Wikipedia/Algorithmic_complexity_theory |
A remote control, also known colloquially as a remote or clicker, is an electronic device used to operate another device from a distance, usually wirelessly. In consumer electronics, a remote control can be used to operate devices such as a television set, DVD player or other digital home media appliance. A remote cont... | Wikipedia/Remote_control |
Image resolution is the level of detail of an image. The term applies to digital images, film images, and other types of images. "Higher resolution" means more image detail.
Image resolution can be measured in various ways. Resolution quantifies how close lines can be to each other and still be visibly resolved. Resolu... | Wikipedia/Image_resolution |
In cryptography, encryption (more specifically, encoding) is the process of transforming information in a way that, ideally, only authorized parties can decode. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Despite its goal, encr... | Wikipedia/Cryptographic_algorithm |
This article discusses how information theory (a branch of mathematics studying the transmission, processing and storage of information) is related to measure theory (a branch of mathematics related to integration and probability).
== Measures in information theory ==
Many of the concepts in information theory have s... | Wikipedia/Information_theory_and_measure_theory |
Thermal physics is the combined study of thermodynamics, statistical mechanics, and kinetic theory of gases. This umbrella-subject is typically designed for physics students and functions to provide a general introduction to each of three core heat-related subjects. Other authors, however, define thermal physics loos... | Wikipedia/Thermal_physics |
In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real ... | Wikipedia/Discrete_sine_transform |
The free energy principle is a mathematical principle of information physics. Its application to fMRI brain imaging data as a theoretical framework suggests that the brain reduces surprise or uncertainty by making predictions based on internal models and uses sensory input to update its models so as to improve the accu... | Wikipedia/Free_energy_principle |
Cognitive neuroscience is the scientific field that is concerned with the study of the biological processes and aspects that underlie cognition, with a specific focus on the neural connections in the brain which are involved in mental processes. It addresses the questions of how cognitive activities are affected or con... | Wikipedia/Cognitive_neuroscience |
Quantum information science is a field that combines the principles of quantum mechanics with information theory to study the processing, analysis, and transmission of information. It covers both theoretical and experimental aspects of quantum physics, including the limits of what can be achieved with quantum informati... | Wikipedia/Quantum_information_science |
In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematical model for digital logic circuits. Circuits are defined by the gates the... | Wikipedia/Circuit_(computer_science) |
In probability and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete probability density function. The probability mass... | Wikipedia/Probability_mass_function |
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location... | Wikipedia/Discrete_wavelet_transform |
Variable bitrate (VBR) is a term used in telecommunications and computing that relates to the bitrate used in sound or video encoding. As opposed to constant bitrate (CBR), VBR files vary the amount of output data per time segment. VBR allows a higher bitrate (and therefore more storage space) to be allocated to the ... | Wikipedia/Variable_bitrate |
In cryptology, a code is a method used to encrypt a message that operates at the level of meaning; that is, words or phrases are converted into something else. A code might transform "change" into "CVGDK" or "cocktail lounge". The U.S. National Security Agency defined a code as "A substitution cryptosystem in which the... | Wikipedia/Code_(cryptography) |
In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s.
It is used in inductive inference theory and analyses of algorithms. In his general theory ... | Wikipedia/Algorithmic_probability |
Information field theory (IFT) is a Bayesian statistical field theory relating to signal reconstruction, cosmography, and other related areas. IFT summarizes the information available on a physical field using Bayesian probabilities. It uses computational techniques developed for quantum field theory and statistical fi... | Wikipedia/Information_field_theory |
An A-law algorithm is a standard companding algorithm, used in European 8-bit PCM digital communications systems to optimize, i.e. modify, the dynamic range of an analog signal for digitizing. It is one of the two companding algorithms in the G.711 standard from ITU-T, the other being the similar μ-law, used in North A... | Wikipedia/A-law_algorithm |
The search for extraterrestrial intelligence (usually shortened as SETI) is an expression that refers to the diverse efforts and scientific projects intended to detect extraterrestrial signals, or any evidence of intelligent life beyond Earth.
Researchers use methods such as monitoring electromagnetic radiation, searc... | Wikipedia/Search_for_extraterrestrial_intelligence |
Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as strings or any other data structure. In other words, it is shown within algori... | Wikipedia/Algorithmic_information_theory |
The min-entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome. The various Rényi entropies are all equal for a uniform d... | Wikipedia/Min-entropy |
Adaptive differential pulse-code modulation (ADPCM) is a variant of differential pulse-code modulation (DPCM) that varies the size of the quantization step, to allow further reduction of the required data bandwidth for a given signal-to-noise ratio.
Typically, the adaptation to signal statistics in ADPCM consists simpl... | Wikipedia/Adaptive_differential_pulse-code_modulation |
Ultra was the designation adopted by British military intelligence in June 1941 for wartime signals intelligence obtained by breaking high-level encrypted enemy radio and teleprinter communications at the Government Code and Cypher School (GC&CS) at Bletchley Park. Ultra eventually became the standard designation among... | Wikipedia/Ultra_(cryptography) |
Differential pulse-code modulation (DPCM) is a signal encoder that uses the baseline of pulse-code modulation (PCM) but adds some functionalities based on the prediction of the samples of the signal. The input can be an analog signal or a digital signal.
If the input is a continuous-time analog signal, it needs to be s... | Wikipedia/Differential_pulse-code_modulation |
The display resolution or display modes of a digital television, computer monitor, or other display device is the number of distinct pixels in each dimension that can be displayed. It can be an ambiguous term especially as the displayed resolution is controlled by different factors in cathode-ray tube (CRT) displays, f... | Wikipedia/Display_resolution |
The term "information algebra" refers to mathematical techniques of information processing. Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources... | Wikipedia/Information_algebra |
Statistical inference might be thought of as gambling theory applied to the world around us. The myriad applications for logarithmic information measures tell us precisely how to take the best guess in the face of partial information. In that sense, information theory might be considered a formal expression of the the... | Wikipedia/Gambling_and_information_theory |
In philosophy of mind, the computational theory of mind (CTM), also known as computationalism, is a family of views that hold that the human mind is an information processing system and that cognition and consciousness together are a form of computation. It is closely related to functionalism, a broader theory that def... | Wikipedia/Computational_theory_of_mind |
In the mathematical theory of probability, the entropy rate or source information rate is a function assigning an entropy to a stochastic process.
For a strongly stationary process, the conditional entropy for latest random variable eventually tend towards this rate value.
== Definition ==
A process
... | Wikipedia/Entropy_rate |
The μ-law algorithm (sometimes written mu-law, often abbreviated as u-law) is a companding algorithm, primarily used in 8-bit PCM digital telecommunications systems in North America and Japan. It is one of the two companding algorithms in the G.711 standard from ITU-T, the other being the similar A-law. A-law is used i... | Wikipedia/Μ-law_algorithm |
In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for i... | Wikipedia/Rényi_entropy |
In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces... | Wikipedia/Affine_function |
In mathematics, a Petersson algebra is a composition algebra over a field constructed from an order-3 automorphism of a Hurwitz algebra. They were first constructed by Petersson (1969).
== Construction ==
Suppose that C is a Hurwitz algebra and φ is an order 3 automorphism. Define the new product of x and y to be φ(x... | Wikipedia/Petersson_algebra |
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F,
A
... | Wikipedia/Quaternion_algebra |
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9.
In some cases when superscripts a... | Wikipedia/Square_(algebra) |
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.
... | Wikipedia/Pauli_algebra |
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.
In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only fo... | Wikipedia/Split_algebra |
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homo... | Wikipedia/Hurwitz's_theorem_(composition_algebras) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.