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Ingeometry,close-packing of equalspheresis a dense arrangement of congruent spheres in an infinite, regular arrangement (orlattice).Carl Friedrich Gaussproved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by alatticepacking is
The samepacking dens... | https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres |
Apollonian sphere packingis the three-dimensional equivalent of theApollonian gasket. The principle of construction is very similar: with any four spheres that arecotangentto each other, it is then possible to construct two more spheres that are cotangent to four of them, resulting in an infinitesphere packing.
Thefra... | https://en.wikipedia.org/wiki/Apollonian_sphere_packing |
In mathematics, the theory offinite sphere packingconcerns the question of how a finite number of equally-sizedspherescan be most efficiently packed. The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid byLászló Fejes Tóth.
The sim... | https://en.wikipedia.org/wiki/Finite_sphere_packing |
Inmathematics, theHermite constant, named afterCharles Hermite, determines how long a shortest element of alatticeinEuclidean spacecan be.
The constantγnfor integersn> 0 is defined as follows. For a latticeLin Euclidean spaceRnwith unit covolume, i.e. vol(Rn/L) = 1, letλ1(L) denote the least length of a nonzero eleme... | https://en.wikipedia.org/wiki/Hermite_constant |
Ingeometry, theinscribed sphereorinsphereof aconvex polyhedronis aspherethat is contained within the polyhedron andtangentto each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and isdualto thedual polyhedron'scircumsphere.
The radius of the sphere inscribed in a po... | https://en.wikipedia.org/wiki/Inscribed_sphere |
Ingeometry, thekissing numberof amathematical spaceis defined as the greatest number of non-overlapping unitspheresthat can be arranged in that space such that they each touch a common unit sphere. For a givensphere packing(arrangement of spheres) in a given space, a kissing number can also be defined for each individ... | https://en.wikipedia.org/wiki/Kissing_number |
Inmathematicsandcomputer science, in the field ofcoding theory, theHamming boundis a limit on the parameters of an arbitraryblock code: it is also known as thesphere-packing boundor thevolume boundfrom an interpretation in terms ofpacking ballsin theHamming metricinto thespaceof all possible words. It gives an importa... | https://en.wikipedia.org/wiki/Sphere-packing_bound |
Random close packing(RCP) of spheres is an empirical parameter used to characterize the maximumvolume fractionofsolidobjects obtained when they are packed randomly. For example, when a solid container is filled withgrain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to ... | https://en.wikipedia.org/wiki/Random_close_pack |
Sphere packing in a cylinderis a three-dimensionalpacking problemwith the objective of packing a given number of identicalspheresinside acylinderof specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are calledcolumnar structures.
Thes... | https://en.wikipedia.org/wiki/Cylinder_sphere_packing |
Sphere packing in a sphereis a three-dimensionalpacking problemwith the objective of packing a given number of equal spheres inside aunit sphere. It is the three-dimensional equivalent of thecircle packing in a circleproblem in two dimensions. | https://en.wikipedia.org/wiki/Sphere_packing_in_a_sphere |
Inmathematics,computer science,telecommunication,information theory, andsearching theory,error-correcting codes with feedbackareerror correcting codesdesigned to work in the presence of feedback from the receiver to the sender.[1]
Alice (the sender) wishes to send a valuexto Bob (the receiver). The communication chann... | https://en.wikipedia.org/wiki/Error-correcting_codes_with_feedback |
IEEE 802.11n-2009, or802.11n, is a wireless-networking standard that uses multiple antennas to increase data rates. TheWi-Fi Alliancehas also retroactively labelled the technology for the standard asWi-Fi 4.[4][5]It standardized support formultiple-input multiple-output(MIMO),frame aggregation, and security improvement... | https://en.wikipedia.org/wiki/IEEE_802.11n-2009#Data_rates |
IEEE 802.11ac-2013or802.11acis awireless networkingstandard in theIEEE 802.11set of protocols (which is part of theWi-Finetworking family), providing high-throughputwireless local area networks(WLANs) on the5 GHz band.[d]The standard has been retroactively labelled asWi-Fi 5byWi-Fi Alliance.[4][5]
The specification ha... | https://en.wikipedia.org/wiki/IEEE_802.11ac#Data_rates_and_speed |
Wi-Fi 6, orIEEE 802.11ax, is anIEEEstandard from theWi-Fi Alliance, for wireless networks (WLANs). It operates in the 2.4 GHz and 5 GHz bands,[4]with an extended version,Wi-Fi 6E, that adds the 6 GHz band.[5]It is an upgrade from Wi-Fi 5 (802.11ac), with improvements for better performance in crowded places. Wi-Fi 6 co... | https://en.wikipedia.org/wiki/IEEE_802.11ax#Rate_set |
Hierarchical modulation, also calledlayered modulation, is one of thesignal processingtechniques formultiplexingandmodulatingmultiple data streams into one single symbol stream, where base-layer symbols and enhancement-layer symbols are synchronously overlaid before transmission.
Hierarchical modulation is particularl... | https://en.wikipedia.org/wiki/Hierarchical_modulation |
This is a list of well-knowndata structures. For a wider list of terms, seelist of terms relating to algorithms and data structures. For a comparison ofrunning timesfor a subset of this list seecomparison of data structures.
Some properties of abstract data types:
"Ordered" means that the elements of the data type ha... | https://en.wikipedia.org/wiki/List_of_data_structures |
The followingoutlineis provided as an overview of, and topical guide to, machine learning:
Machine learning(ML) is a subfield ofartificial intelligencewithincomputer sciencethat evolved from the study ofpattern recognitionandcomputational learning theory.[1]In 1959,Arthur Samueldefined machine learning as a "field of ... | https://en.wikipedia.org/wiki/List_of_machine_learning_algorithms |
Pathfindingorpathingis the search, by a computer application, for the shortest route between two points. It is a more practical variant onsolving mazes. This field of research is based heavily onDijkstra's algorithmfor finding the shortest path on aweighted graph.
Pathfinding is closely related to theshortest path pro... | https://en.wikipedia.org/wiki/List_of_pathfinding_algorithms |
This is alist ofalgorithmgeneral topics. | https://en.wikipedia.org/wiki/List_of_algorithm_general_topics |
The NISTDictionary of Algorithms and Data Structures[1]is a reference work maintained by the U.S.National Institute of Standards and Technology. It defines a large number ofterms relating to algorithms and data structures. For algorithms and data structures not necessarily mentioned here, seelist of algorithmsandlist o... | https://en.wikipedia.org/wiki/List_of_terms_relating_to_algorithms_and_data_structures |
Collective intelligenceCollective actionSelf-organized criticalityHerd mentalityPhase transitionAgent-based modellingSynchronizationAnt colony optimizationParticle swarm optimizationSwarm behaviour
Social network analysisSmall-world networksCentralityMotifsGraph theoryScalingRobustnessSystems biologyDynamic networks
... | https://en.wikipedia.org/wiki/Heuristic |
Indigital logic, adon't-care term[1][2](abbreviatedDC, historically also known asredundancies,[2]irrelevancies,[2]optional entries,[3][4]invalid combinations,[5][4][6]vacuous combinations,[7][4]forbidden combinations,[8][2]unused statesorlogical remainders[9]) for a function is an input-sequence (a series of bits) for ... | https://en.wikipedia.org/wiki/Don%27t_care_alarm |
Indigital logic, adon't-care term[1][2](abbreviatedDC, historically also known asredundancies,[2]irrelevancies,[2]optional entries,[3][4]invalid combinations,[5][4][6]vacuous combinations,[7][4]forbidden combinations,[8][2]unused statesorlogical remainders[9]) for a function is an input-sequence (a series of bits) for ... | https://en.wikipedia.org/wiki/Forbidden_input |
Random number generatorsare important in many kinds of technical applications, includingphysics,engineeringormathematicalcomputer studies (e.g.,Monte Carlosimulations),cryptographyandgambling(ongame servers).
This list includes many common types, regardless of quality or applicability to a given use case.
The followi... | https://en.wikipedia.org/wiki/List_of_random_number_generators |
Alinear congruential generator(LCG) is analgorithmthat yields a sequence of pseudo-randomized numbers calculated with a discontinuouspiecewise linear equation. The method represents one of the oldest and best-knownpseudorandom number generatoralgorithms. The theory behind them is relatively easy to understand, and they... | https://en.wikipedia.org/wiki/Linear_congruential_generator |
An approach to nonlinear congruential methods ofgenerating uniform pseudorandom numbersin the interval [0,1) is theInversive congruential generatorwith prime modulus. A generalization for arbitrary composite modulim=p1,…pr{\displaystyle m=p_{1},\dots p_{r}}with arbitrary distinctprimesp1,…,pr≥5{\displaystyle p_{1},\dot... | https://en.wikipedia.org/wiki/Generalized_inversive_congruential_pseudorandom_numbers |
In 1997,Moni NaorandOmer Reingolddescribed efficient constructions for variouscryptographic primitivesin private key as well aspublic-key cryptography. Their result is the construction of an efficientpseudorandom function. Letpandlbeprime numberswithl|p−1. Select an elementg∈Fp∗{\displaystyle {\mathbb {F} _{p}}^{*}}ofm... | https://en.wikipedia.org/wiki/Naor-Reingold_Pseudorandom_Function |
TheReeds–Sloane algorithm, named afterJames ReedsandNeil Sloane, is an extension of theBerlekamp–Massey algorithm, an algorithm for finding the shortestlinear-feedback shift register(LFSR) for a given outputsequence, for use on sequences that take their values from theintegers modn.
This cryptography-related article i... | https://en.wikipedia.org/wiki/Reeds%E2%80%93Sloane_algorithm |
Inmathematicsandelectronics engineering, abinary Golay codeis a type of linearerror-correcting codeused indigital communications. The binary Golay code, along with theternary Golay code, has a particularly deep and interesting connection to the theory offinite sporadic groupsin mathematics.[1]These codes are named in h... | https://en.wikipedia.org/wiki/Binary_Golay_code |
Eugene August Prange(July 30, 1917 – February 12, 2006)[1][2]was an Americancoding theorist, a researcher at theAir Force Cambridge Research Laboratory(AFCRL) inMassachusettswho "introduced many of the early fundamental ideas of algebraic coding theory"[3]and was the first to investigatecyclic codesin 1957.[4][5]WithAn... | https://en.wikipedia.org/wiki/Eugene_Prange |
Reed–Muller codesareerror-correcting codesthat are used in wireless communications applications, particularly in deep-space communication.[1]Moreover, the proposed5G standard[2]relies on the closely relatedpolar codes[3]for error correction in the control channel. Due to their favorable theoretical and mathematical pro... | https://en.wikipedia.org/wiki/Reed%E2%80%93Muller_code |
Incoding theory, theternary Golay codesare two closely relatederror-correcting codes.
The code generally known simply as theternary Golay codeis an[11,6,5]3{\displaystyle [11,6,5]_{3}}-code, that is, it is alinear codeover aternaryalphabet; therelative distanceof the code is as large as it possibly can be for a ternary... | https://en.wikipedia.org/wiki/Ternary_Golay_code |
Incoding theory, theSingleton bound, named after the American mathematician Richard Collom Singleton (1928–2007), is a relatively crude upper bound on the size of an arbitraryblock codeC{\displaystyle C}with block lengthn{\displaystyle n}, sizeM{\displaystyle M}and minimum distanced{\displaystyle d}. It is also known a... | https://en.wikipedia.org/wiki/Singleton_bound |
Incoding theory, anerasure codeis aforward error correction(FEC) code under the assumption of bit erasures (rather than bit errors), which transforms a message ofksymbols into a longer message (code word) withnsymbols such that the original message can be recovered from a subset of thensymbols. The fractionr=k/nis call... | https://en.wikipedia.org/wiki/Erasure_code |
Quantum error correction(QEC) is a set of techniques used inquantum computingto protectquantum informationfrom errors due todecoherenceand otherquantum noise. Quantum error correction is theorised as essential to achievefault tolerant quantum computingthat can reduce the effects of noise on stored quantum information, ... | https://en.wikipedia.org/wiki/Quantum_error_correction |
Ininformation theory, asoft-decision decoderis a kind ofdecoding method– a class ofalgorithmused to decode data that has been encoded with anerror correcting code. Whereas ahard-decision decoderoperates on data that take on a fixed set of possible values (typically 0 or 1 in a binary code), the inputs to a soft-decisi... | https://en.wikipedia.org/wiki/Soft-decision_decoder |
Gödel's incompleteness theoremsare twotheoremsofmathematical logicthat are concerned with the limits of provability in formal axiomatic theories. These results, published byKurt Gödelin 1931, are important both in mathematical logic and in thephilosophy of mathematics. The theorems are widely, but not universally, inte... | https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems |
Inalgorithmic information theory(a subfield ofcomputer scienceandmathematics), theKolmogorov complexityof an object, such as a piece of text, is the length of a shortestcomputer program(in a predeterminedprogramming language) that produces the object as output. It is a measure of thecomputationalresources needed to spe... | https://en.wikipedia.org/wiki/Kolmogorov_complexity |
Ininformation theory,data compression,source coding,[1]orbit-rate reductionis the process of encodinginformationusing fewerbitsthan the original representation.[2]Any particular compression is eitherlossyorlossless. Lossless compression reduces bits by identifying and eliminatingstatistical redundancy. No information i... | https://en.wikipedia.org/wiki/Source_coding |
Elias δ codeorElias delta codeis auniversal codeencoding the positive integers developed byPeter Elias.[1]: 200
To code a numberX≥ 1:
An equivalent way to express the same process:
To represent a numberx{\displaystyle x}, Elias delta (δ) uses⌊log2(x)⌋+2⌊log2(⌊log2(x)⌋+1)⌋+1{\displaystyle \lfloor \log _{2}(x)\rflo... | https://en.wikipedia.org/wiki/Elias_delta_coding |
Equifinalityis the principle that inopen systemsa given end state can be reached by many potential means. The term and concept is due to the GermanHans Driesch, the developmental biologist, later applied by the AustrianLudwig von Bertalanffy, the founder of generalsystems theory, and byWilliam T. Powers, the founder of... | https://en.wikipedia.org/wiki/Equifinality |
Ergodic theoryis a branch ofmathematicsthat studiesstatisticalproperties of deterministicdynamical systems; it is the study ofergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior oftime averagesof various functions along trajectories of dynamical systems. T... | https://en.wikipedia.org/wiki/Ergodic_theory |
Incomputer programming,machine codeiscomputer codeconsisting ofmachine languageinstructions, which are used to control a computer'scentral processing unit(CPU). For conventionalbinary computers, machine code is the binary[nb 1]representation of a computer program that is actually read and interpreted by the computer. A... | https://en.wikipedia.org/wiki/Overlapping_instructions |
Incoding theory, especially intelecommunications, aself-synchronizing codeis auniquely decodable codein which thesymbolstream formed by a portion of onecode word, or by the overlapped portion of any two adjacent code words, is not a valid code word.[1]Put another way, a set of strings (called "code words") over an alph... | https://en.wikipedia.org/wiki/Self-synchronizing_code |
Aninstruction set architecture(ISA) is an abstract model of acomputer, also referred to ascomputer architecture. A realization of an ISA is called animplementation. An ISA permits multiple implementations that may vary inperformance, physical size, and monetary cost (among other things); because the ISA serves as thein... | https://en.wikipedia.org/wiki/Comparison_of_instruction_set_architectures |
Acompressed instruction set, or simplycompressed instructions, are a variation on amicroprocessor'sinstruction set architecture(ISA) that allows instructions to be represented in a more compact format. In most real-world examples, compressed instructions are 16 bits long in a processor that would otherwise use 32-bit i... | https://en.wikipedia.org/wiki/Compressed_instruction_set |
Incomputer scienceandcomputer engineering,computer architectureis a description of the structure of acomputersystem made from component parts.[1]It can sometimes be a high-level description that ignores details of the implementation.[2]At a more detailed level, the description may include theinstruction set architectur... | https://en.wikipedia.org/wiki/Computer_architecture |
[1]Incomputing, anemulatorishardwareorsoftwarethat enables onecomputer system(called thehost) to behave like another computer system (called theguest). An emulator typically enables the host system to run software or use peripheral devices designed for the guest system.
Emulation refers to the ability of acomputer prog... | https://en.wikipedia.org/wiki/Emulator |
Incomputercentral processing units,micro-operations(also known asmicro-opsorμops, historically also asmicro-actions[2]) are detailed low-level instructions used in some designs to implement complex machine instructions (sometimes termedmacro-instructionsin this context).[3]: 8–9
Usually, micro-operations perform basic... | https://en.wikipedia.org/wiki/Micro-operation |
No instruction set computing(NISC) is a computing architecture and compiler technology for designing highly efficient custom processors and hardware accelerators by allowing a compiler to have low-level control of hardware resources.
NISC is a statically scheduled horizontal nanocoded architecture (SSHNA). The term "s... | https://en.wikipedia.org/wiki/No_instruction_set_computing |
Asimulationis an imitative representation of a process or system that could exist in the real world.[1][2][3]In this broad sense, simulation can often be used interchangeably withmodel.[2]Sometimes a clear distinction between the two terms is made, in which simulations require the use of models; the model represents th... | https://en.wikipedia.org/wiki/Simulation |
Incomputer science,register transfer language(RTL) is a kind ofintermediate representation(IR) that is very close toassembly language, such as that which is used in acompiler. It is used to describe data flow at theregister-transfer levelof anarchitecture.[1]Academic papers and textbooks often use a form of RTL as an a... | https://en.wikipedia.org/wiki/Register_transfer_language |
Inmathematics, theKrylov–Bogolyubov theorem(also known as theexistence of invariant measures theorem) may refer to either of the two related fundamentaltheoremswithin the theory ofdynamical systems. The theorems guarantee the existence ofinvariant measuresfor certain "nice" maps defined on "nice" spaces and were named ... | https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_theorem |
Inmathematicsandphysics, thePoincaré recurrence theoremstates that certaindynamical systemswill, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state.
ThePoincaré recurrence timeis the ... | https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem |
Operating signalsare a type ofbrevity codeused in operational communication among radio and telegraph operators. For example: | https://en.wikipedia.org/wiki/Operating_signals |
SINPO, anacronymforSignal, Interference, Noise, Propagation, and Overall, is a Signal Reporting Code used to describe the quality of broadcast andradiotelegraphtransmissions. SINPFEMO, an acronym forSignal, Interference, Noise, Propagation, frequency of Fading, dEpth, Modulation, and Overallis used to describe the qual... | https://en.wikipedia.org/wiki/SINPO_code |
Telegram style,telegraph style,telegraphic style, ortelegraphese[1]is a clipped way of writing which abbreviates words and packs information into the smallest possible number of words or characters. It originated in thetelegraphage when telecommunication consisted only of short messages transmitted by hand over the tel... | https://en.wikipedia.org/wiki/Telegraphese |
Hypertext Transfer Protocol(HTTP) response status codes are issued by a server in response to aclient's requestmade to the server. It includes codes from IETFRequest for Comments(RFCs), other specifications, and some additional codes used in some common applications of the HTTP. The first digit of the status code speci... | https://en.wikipedia.org/wiki/List_of_HTTP_status_codes |
Policeunits in theUnited Statestend to use atactical designator(ortactical callsign) consisting of a letter of the policeradio alphabetfollowed by one or two numbers. For example, "Mary One" might identify the head of a city'shomicidedivision. Police and fire department radio systems are assigned officialcallsigns, how... | https://en.wikipedia.org/wiki/Tactical_designator |
The Medical Priority Dispatch System(MPDS), sometimes referred to as theAdvanced Medical Priority Dispatch System(AMPDS) is a unified system used to dispatch appropriate aid to medical emergencies including systematized caller interrogation and pre-arrival instructions. Priority Dispatch Corporation is licensed to desi... | https://en.wikipedia.org/wiki/Advanced_Medical_Priority_Dispatch_System |
Emergency service response codesare predefined systems used byemergency servicesto describe the priority and response assigned tocalls for service. Response codes vary from country to country, jurisdiction to jurisdiction, and even agency to agency, with different methods used to categorize responses to reported events... | https://en.wikipedia.org/wiki/Emergency_service_response_codes |
CB slangis the distinctiveanti-language,argot, orcantwhich developed among users ofCitizens Band radio(CB), especiallytruck driversin the United States during the 1970s and early 1980s,[1]when it was an important part of theculture of the trucking industry.
The slang itself is not only cyclical, but also geographical.... | https://en.wikipedia.org/wiki/List_of_CB_slang |
Alist of international common and basictechnical standards, which have been established worldwide and are related by their frequent and widespread use. These standards are conventionally accepted asbest practiceand used globally byindustryandorganizations.
In circumstances and situations there are certain methods and ... | https://en.wikipedia.org/wiki/List_of_international_common_standards |
The Medical Priority Dispatch System(MPDS), sometimes referred to as theAdvanced Medical Priority Dispatch System(AMPDS) is a unified system used to dispatch appropriate aid to medical emergencies including systematized caller interrogation and pre-arrival instructions. Priority Dispatch Corporation is licensed to desi... | https://en.wikipedia.org/wiki/Medical_Priority_Dispatch_System |
The Password Gameis a 2023puzzlebrowser gamedeveloped by Neal Agarwal, where the player creates apasswordthat follows increasingly unusual and complicated rules. Based on Agarwal's experience withpassword policies,[1]the game was developed in two months, releasing on June 27, 2023. The gamewent viraland was recognized ... | https://en.wikipedia.org/wiki/The_Password_Game |
Code wordmay refer to: | https://en.wikipedia.org/wiki/Code_word_(disambiguation) |
Infunctional analysisand related areas ofmathematicsanabsorbing setin avector spaceis asetS{\displaystyle S}which can be "inflated" or "scaled up" to eventually always include any given point of the vector space.
Alternative terms areradialorabsorbent set.
Everyneighborhood of the originin everytopological vector spa... | https://en.wikipedia.org/wiki/Absorbing_set |
Annihilator(s)may refer to: | https://en.wikipedia.org/wiki/Annihilator_(disambiguation) |
Inmathematics, theannihilatorof asubsetSof amoduleover aringis theidealformed by the elements of the ring that give always zero when multiplied by each element ofS.
Over anintegral domain, a module that has a nonzero annihilator is atorsion module, and afinitely generatedtorsion module has a nonzero annihilator.
The ... | https://en.wikipedia.org/wiki/Annihilator_(ring_theory) |
Inring theory, a branch ofmathematics, anidempotent elementor simplyidempotentof aringis an elementasuch thata2=a.[1][a]That is, the element isidempotentunder the ring's multiplication.Inductivelythen, one can also conclude thata=a2=a3=a4= ... =anfor any positiveintegern. For example, an idempotent element of amatrix r... | https://en.wikipedia.org/wiki/Idempotent_(ring_theory) |
Inmathematics, anull semigroup(also called azero semigroup) is asemigroupwith anabsorbing element, calledzero, in which the product of any two elements is zero.[1]If every element of a semigroup is aleft zerothen the semigroup is called aleft zero semigroup; aright zero semigroupis defined analogously.[2]
According to... | https://en.wikipedia.org/wiki/Null_semigroup |
Inmathematics, theabsolute valueormodulusof areal numberx{\displaystyle x},denoted|x|{\displaystyle |x|},is thenon-negativevalueofx{\displaystyle x}without regard to itssign. Namely,|x|=x{\displaystyle |x|=x}ifx{\displaystyle x}is apositive number, and|x|=−x{\displaystyle |x|=-x}ifx{\displaystyle x}isnegative(in which ... | https://en.wikipedia.org/wiki/Absolute_value |
Inmathematics, theinverse functionof afunctionf(also called theinverseoff) is afunctionthat undoes the operation off. The inverse offexistsif and only iffisbijective, and if it exists, is denoted byf−1.{\displaystyle f^{-1}.}
For a functionf:X→Y{\displaystyle f\colon X\to Y}, its inversef−1:Y→X{\displaystyle f^{-1}\co... | https://en.wikipedia.org/wiki/Inverse_function |
Inmathematics, areflection(also spelledreflexion)[1]is amappingfrom aEuclidean spaceto itself that is anisometrywith ahyperplaneas the set offixed points; this set is called theaxis(in dimension 2) orplane(in dimension 3) of reflection. The image of a figure by a reflection is itsmirror imagein the axis or plane of ref... | https://en.wikipedia.org/wiki/Reflection_(mathematics) |
Inmathematics,reflection symmetry,line symmetry,mirror symmetry, ormirror-image symmetryissymmetrywith respect to areflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In2-dimensional space, there is a line/axis of symmetry, in3-dimensional space, there is aplaneo... | https://en.wikipedia.org/wiki/Reflection_symmetry |
Inmathematics, ablock matrix pseudoinverseis a formula for thepseudoinverseof apartitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters insignal processing, which are based on theleast squaresmethod.
Consider a column-wise partitioned matrix:
If the above matrix is full ... | https://en.wikipedia.org/wiki/Block_matrix_pseudoinverse |
Inmathematics, aregular semigroupis asemigroupSin which every element isregular, i.e., for each elementainSthere exists an elementxinSsuch thataxa=a.[1]Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study viaGreen's relations.[2]
Regular semigroups... | https://en.wikipedia.org/wiki/Regular_semigroup |
Inaccounting,financeandeconomics, anaccounting identityis an equality that must be true regardless of the value of its variables, or a statement that by definition (or construction) must be true.[1][2]Where an accounting identity applies, any deviation from numerical equality signifies an error in formulation, calculat... | https://en.wikipedia.org/wiki/Accounting_identity |
This article listsmathematical identities, that is,identically true relationsholding inmathematics. | https://en.wikipedia.org/wiki/List_of_mathematical_identities |
Inmathematics, alawis aformulathat is always true within a given context.[1]Laws describe arelationship, between two or moreexpressionsorterms(which may containvariables), usually usingequalityorinequality,[2]or between formulas themselves, for instance, inmathematical logic. For example, the formulaa2≥0{\displaystyle ... | https://en.wikipedia.org/wiki/Law_(mathematics) |
Inlinear algebra, theidentity matrixof sizen{\displaystyle n}is then×n{\displaystyle n\times n}square matrixwithoneson themain diagonalandzeroselsewhere. It has unique properties, for example when the identity matrix represents ageometric transformation, the object remains unchanged by the transformation. In other cont... | https://en.wikipedia.org/wiki/Identity_matrix |
Inmathematics, ifA{\displaystyle A}is asubsetofB,{\displaystyle B,}then theinclusion mapis thefunctionι{\displaystyle \iota }that sends each elementx{\displaystyle x}ofA{\displaystyle A}tox,{\displaystyle x,}treated as an element ofB:{\displaystyle B:}ι:A→B,ι(x)=x.{\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x... | https://en.wikipedia.org/wiki/Inclusion_map |
Inmathematics, anindicator functionor acharacteristic functionof asubsetof asetis afunctionthat maps elements of the subset to one, and all other elements to zero. That is, ifAis a subset of some setX, then the indicator function ofAis the function1A{\displaystyle \mathbf {1} _{A}}defined by1A(x)=1{\displaystyle \mathb... | https://en.wikipedia.org/wiki/Indicator_function |
In mathematics, theexterior algebraorGrassmann algebraof avector spaceV{\displaystyle V}is anassociative algebrathat containsV,{\displaystyle V,}which has a product, calledexterior productorwedge productand denoted with∧{\displaystyle \wedge }, such thatv∧v=0{\displaystyle v\wedge v=0}for every vectorv{\displaystyle v}... | https://en.wikipedia.org/wiki/Exterior_algebra |
Inalgebra, agraded-commutative ring(also called askew-commutative ring) is agraded ringthat is commutative in the graded sense; that is,homogeneous elementsx,ysatisfy
where |x| and |y| denote the degrees ofxandy.
Acommutative (non-graded) ring, with trivial grading, is a basic example. For a nontrivial example, anext... | https://en.wikipedia.org/wiki/Graded-commutative_ring |
Inmathematics, anoperationis afunctionfrom asetto itself. For example, an operation onreal numberswill take in real numbers and return a real number. An operation can take zero or more input values (also called "operands" or "arguments") to a well-defined output value. The number of operands is thearityof the operation... | https://en.wikipedia.org/wiki/Operation_(mathematics) |
Symmetryoccurs not only ingeometry, but also in other branches of mathematics. Symmetry is a type ofinvariance: the property that a mathematical object remains unchanged under a set ofoperationsortransformations.[1]
Given a structured objectXof any sort, asymmetryis amappingof the object onto itself which preserves th... | https://en.wikipedia.org/wiki/Symmetry_in_mathematics |
Inphysics,canonical quantizationis a procedure forquantizingaclassical theory, while attempting to preserve the formal structure, such assymmetries, of the classical theory to the greatest extent possible.
Historically, this was not quiteWerner Heisenberg's route to obtainingquantum mechanics, butPaul Diracintroduced ... | https://en.wikipedia.org/wiki/Canonical_quantization |
InmathematicsandphysicsCCR algebras(aftercanonical commutation relations) andCAR algebras(after canonical anticommutation relations) arise from thequantum mechanicalstudy ofbosonsandfermions, respectively. They play a prominent role inquantum statistical mechanics[1]andquantum field theory.
LetV{\displaystyle V}be are... | https://en.wikipedia.org/wiki/CCR_and_CAR_algebras |
Conformastatic spacetimesrefer to a special class of static solutions toEinstein's equationingeneral relativity.
Theline elementfor the conformastatic class of solutions in Weyl's canonical coordinates reads[1][2][3][4][5][6](1)ds2=−e2Ψ(ρ,ϕ,z)dt2+e−2Ψ(ρ,ϕ,z)(dρ2+dz2+ρ2dϕ2),{\displaystyle (1)\qquad ds^{2}=-e^{2\Psi (\r... | https://en.wikipedia.org/wiki/Conformastatic_spacetimes |
Indifferential geometry, theLie derivative(/liː/LEE), named afterSophus LiebyWładysław Ślebodziński,[1][2]evaluates the change of atensor field(including scalar functions,vector fieldsandone-forms), along theflowdefined by another vector field. This change is coordinate invariant and therefore the Lie derivative is def... | https://en.wikipedia.org/wiki/Lie_derivative |
Inphysics, theMoyal bracketis the suitably normalized antisymmetrization of the phase-spacestar product.
The Moyal bracket was developed in about 1940 byJosé Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute withPaul Dirac.[1][2]In the meantime this idea was independently i... | https://en.wikipedia.org/wiki/Moyal_bracket |
Inmathematicsand intheoretical physics, theStone–von Neumann theoremrefers to any one of a number of different formulations of theuniquenessof thecanonical commutation relationsbetweenpositionandmomentumoperators. It is named afterMarshall StoneandJohn von Neumann.[1][2][3][4]
Inquantum mechanics, physicalobservablesa... | https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem |
Inmathematics,multipliers and centralizersare algebraic objects in the study ofBanach spaces. They are used, for example, in generalizations of theBanach–Stone theorem.
Let (X, ‖·‖) be a Banach space over a fieldK(either therealorcomplex numbers), and let Ext(X) be the set ofextreme pointsof theclosed unit ballof thec... | https://en.wikipedia.org/wiki/Multipliers_and_centralizers_(Banach_spaces) |
Inmathematics, agroup actionof a groupG{\displaystyle G}on asetS{\displaystyle S}is agroup homomorphismfromG{\displaystyle G}to some group (underfunction composition) of functions fromS{\displaystyle S}to itself. It is said thatG{\displaystyle G}actsonS{\displaystyle S}.
Many sets oftransformationsform agroupunderfunc... | https://en.wikipedia.org/wiki/Stabilizer_subgroup |
Mathematical diagrams, such aschartsandgraphs, are mainly designed to convey mathematical relationships—for example, comparisons over time.[1]
Acomplex numbercan be visually represented as a pair of numbers forming a vector on a diagram called anArgand diagramThecomplex planeis sometimes called theArgand planebecause ... | https://en.wikipedia.org/wiki/Mathematical_diagram |
Inmathematics,anticommutativityis a specific property of some non-commutativemathematicaloperations. Swapping the position oftwo argumentsof an antisymmetric operation yields a result which is theinverseof the result with unswapped arguments. The notioninverserefers to agroup structureon the operation'scodomain, possib... | https://en.wikipedia.org/wiki/Anticommutativity |
Inabstract algebra, the termassociatoris used in different ways as a measure of thenon-associativityof analgebraic structure. Associators are commonly studied astriple systems.
For anon-associative ringoralgebraR, theassociatoris themultilinear map[⋅,⋅,⋅]:R×R×R→R{\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\... | https://en.wikipedia.org/wiki/Associator |
Inmathematics, theBaker–Campbell–Hausdorff formulagives the value ofZ{\displaystyle Z}that solves the equationeXeY=eZ{\displaystyle e^{X}e^{Y}=e^{Z}}for possiblynoncommutativeXandYin theLie algebraof aLie group. There are various ways of writing the formula, but all ultimately yield an expression forZ{\displaystyle Z}i... | https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula |
Inmathematics, thePincherle derivative[1]T′{\displaystyle T'}of alinear operatorT:K[x]→K[x]{\displaystyle T:\mathbb {K} [x]\to \mathbb {K} [x]}on thevector spaceofpolynomialsin the variablexover afieldK{\displaystyle \mathbb {K} }is thecommutatorofT{\displaystyle T}with the multiplication byxin thealgebra of endomorphi... | https://en.wikipedia.org/wiki/Pincherle_derivative |
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