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In the following C code, the value of zero is used to indicate a null pointer: | Wikipedia - Tagged pointer - Example 1 |
In the following Diophantine equations, w, x, y, and z are the unknowns and the other letters are given constants: | Wikipedia - Diophantine equations - Examples |
In the following English sentence, no word indicates the quoted speech. John said, "Wow," That is indicated only typographically. In Sinhala, on the other hand, here is the equivalent sentence: John Wow kiyalaa kivvaa It has an overt indication of quoted speech after the quoted string Wow, the quotative kiyalaa. | Wikipedia - Quotative - Sinhala |
In the following Haskell example, the guards occur between each pair of "|" and "=": This is similar to the respective mathematical notation: f ( x ) = { 1 if x > 0 0 otherwise {\displaystyle f(x)=\left\{{\begin{matrix}1&{\mbox{if }}x>0\\0&{\mbox{otherwise}}\end{matrix}}\right.} In this case the guards are in the "if" ... | Wikipedia - Guard (computer science) - Mathematics |
In the following IN BCSMs, bold Detection Points and Points In Call are also present in the CAMEL Ph1 subset. TODO: Expand to Ph2,3,4 | Wikipedia - Basic call state model - State machine description |
In the following ITS markup example, the elements and attributes with the its prefix are part of the ITS namespace. The its:rules element lists the different rules to apply to this file. There is one its:translateRule rule that indicates that any content inside the head element should not be translated. The its:transla... | Wikipedia - Internationalization Tag Set - Translate data category |
In the following ITS markup example, the its:locNote element specifies that any node corresponding to the XPath expression "//msg/data" has an associated note. The location of that note is expressed by the locNotePointer attribute, which holds a relative XPath expression pointing to the node where the note is, here =".... | Wikipedia - Internationalization Tag Set - Localization Note data category |
In the following Java example, the Client class contains a Service member variable initialized in the constructor. The client directly constructs and controls which service it uses, creating a hard-coded dependency. | Wikipedia - Contexts and Dependency Injection - Without dependency injection |
In the following PHP program, the value of $_GET is used to build a message to be shown to the user. This opens the possibility for XSS attacks that incorporate their payload into the HTTP GET parameter name. As a security filter against XSS attacks, this program sanitizes the value it reads from $_GET via the htmlenti... | Wikipedia - Double encoding - XSS |
This filter prevents payloads such as and its URI-encoded form %3Cscript%3Ealert%281%29%3C%2Fscript%3E. However, %253Cscript%253Ealert%25281%2529%253C%252Fscript%253E, which is the double-URI-encoded form of , will bypass this filter. When double-URI-encoded payload %253Cscript%253Ealert%25281%2529%253C%252Fscript%253E... | Wikipedia - Double encoding - XSS |
In the following PHP program, the value of $_GET is used to build the path of the file to be sent to the user. This opens the possibility for directory traversal attacks that incorporate their payload into the HTTP GET parameter file. As a security filter against directory traversal attacks, this program searches the v... | Wikipedia - Double encoding - Directory traversal |
However, after this filter, the program URI-decodes the data that it has read from $_GET, which makes it vulnerable to double URI-encoding attacks. This filter prevents payloads such as ../../../../etc/passwd and its URI-encoded form %2E%2E%2F%2E%2E%2F%2E%2E%2F%2E%2E%2Fetc%2Fpasswd. However, %252E%252E%252F%252E%252E%2... | Wikipedia - Double encoding - Directory traversal |
In the following a brief description of applicable rules organized by source. | Wikipedia - IT risk - IT risk laws and regulations |
In the following animation, three layers are moving leftward at different speeds. Their speeds decrease from front to back and correspond to increases in relative distance from the viewer. The ground layer is moving 8 times as fast as the vegetation layer. The vegetation layer is moving two times as fast as the cloud l... | Wikipedia - Parallax scrolling - Example |
In the following case, one Proposer achieves acceptance of value V1 by one Acceptor before failing. A new Proposer prepares the Acceptors that never accepted V1, allowing it to propose V2. Then V2 is accepted by all Acceptors, including the one that initially accepted V1. Proposer Acceptor Learner | | | | | | | X------... | Wikipedia - Paxos (computer science) - Basic Paxos where an Acceptor accepts Two Different Values |
(1,V1) | | X------------>|->| Accepted(1,V1) ! | | | | | | !! | Wikipedia - Paxos (computer science) - Basic Paxos where an Acceptor accepts Two Different Values |
FAIL !! | | | | | | X--------->|->| | | Prepare(2) |<---------X--X | | Promise(2,{null,null}) X------>|->|->| | | Accept! (2,V2) |<------X--X--X------>|->| Accepted(2,V2) | | | | | | | Wikipedia - Paxos (computer science) - Basic Paxos where an Acceptor accepts Two Different Values |
In the following case, one Proposer achieves acceptance of value V1 of one Acceptor before failing. A new Proposer prepares the Acceptors that never accepted V1, allowing it to propose V2. This Proposer is able to get one Acceptor to accept V2 before failing. A new Proposer finds a majority that includes the Acceptor t... | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
The Proposer manages to get two Acceptors to accept it before failing. At this point, three Acceptors have accepted V1, but not for the same identifier. Finally, a new Proposer prepares the majority that has not seen the largest accepted identifier. | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
The value associated with the largest identifier in that majority is V2, so it must propose it. This Proposer then gets all Acceptors to accept V2, achieving consensus. Proposer Acceptor Learner | | | | | | | | | | | X--------------->|->|->|->|->| | | Prepare(1) |<---------------X--X--X--X--X | | Promise(1,{null,null,n... | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
(1,V1) | | | | X------------------>|->| Accepted(1,V1) ! | | | | | | | | | | !! FAIL !! | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
| | | | | | | | | | X--------------->|->|->|->| | | Prepare(2) |<---------------X--X--X--X | | Promise(2,{null,null,null,null}) X--------------->| | | | | | Accept! (2,V2) | | | | X--------------->|->| Accepted(2,V2) ! | | | | | | | | | !! | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
FAIL !! | | | | | | | | | X--------->|---->|->|->| | | Prepare(3) |<---------X-----X--X--X | | Promise(3,{V1,null,null,null}) X--------------->|->| | | | Accept! (3,V1) | | | | X--X--------->|->| Accepted(3,V1) ! | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
| | | | | | | | !! FAIL !! | | | | | | | | X------>|->|------->| | | Prepare(4) |<------X--X--|--|--X | | Promise(4,{V1(1),V2(2),null}) X------>|->|->|->|->| | | Accept! (4,V2) | X--X--X--X--X------>|->| Accepted(4,V2) | Wikipedia - Paxos (computer science) - Basic Paxos where a multi-identifier majority is insufficient |
In the following case, one Proposer achieves acceptance of value V1 of two Acceptors before failing. A new Proposer may start another round, but it is now impossible for that proposer to prepare a majority that doesn't include at least one Acceptor that has accepted V1. As such, even though the Proposer doesn't see the... | Wikipedia - Paxos (computer science) - Basic Paxos where new Proposers cannot change an existing consensus |
Proposer Acceptor Learner | | | | | | | X--------->|->|->| | | Prepare(1) |<---------X--X--X | | Promise(1,{null,null,null}) x--------->|->| | | | Accept! (1,V1) | | X--X--------->|->| Accepted(1,V1) ! | | | | | | !! | Wikipedia - Paxos (computer science) - Basic Paxos where new Proposers cannot change an existing consensus |
FAIL !! | | | | | | X--------->|->| | | Prepare(2) |<---------X--X | | Promise(2,{V1,null}) X------>|->|->| | | Accept! (2,V1) |<------X--X--X------>|->| Accepted(2,V1) | | | | | | | Wikipedia - Paxos (computer science) - Basic Paxos where new Proposers cannot change an existing consensus |
In the following case, one of the (redundant) Learners fails, but the Basic Paxos protocol still succeeds. Client Proposer Acceptor Learner | | | | | | | X-------->| | | | | | Request | X--------->|->|->| | | Prepare(1) | |<---------X--X--X | | Promise(1,{Va,Vb,Vc}) | X--------->|->|->| | | Accept! (1,V) | |<---------X... | Wikipedia - Paxos (computer science) - Basic Paxos when a redundant learner fails |
In the following cases, assume that the retail price, p {\displaystyle p} , is $7 per unit and the purchase price is c {\displaystyle c} , is $5 per unit. This gives a critical fractile of p − c p = 7 − 5 7 = 2 7 {\displaystyle {\frac {p-c}{p}}={\frac {7-5}{7}}={\frac {2}{7}}} | Wikipedia - Newsvendor model - Numerical examples |
In the following cases, every finite-dimensional central division algebra over a field K is K itself, so that the Brauer group Br(K) is trivial: K is an algebraically closed field. K is a finite field (Wedderburn's theorem). Equivalently, every finite division ring is commutative. | Wikipedia - Brauer group - Examples |
K is the function field of an algebraic curve over an algebraically closed field (Tsen's theorem). More generally, the Brauer group vanishes for any C1 field. K is an algebraic extension of Q containing all roots of unity. | Wikipedia - Brauer group - Examples |
The Brauer group Br(R) of the field R of real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras with center R: R itself and the quaternion algebra H. Since H ⊗ H ≅ M(4, R), the class of H has order two in the Brauer group. Let K be a non-Archimedean local field, meaning ... | Wikipedia - Brauer group - Examples |
In the following cases, the suspect arrested for the offense subsequently confessed and was sentenced without going to trial. Case results: Dismissals In the following cases, the suspect arrested for the offense was either released or found not guilty in a court of law. Many factors may result in such an outcome, rangi... | Wikipedia - Parabon NanoLabs - Case results: Confessions and guilty pleas |
In the following cities high levels of non-revenue water have been substantially reduced: Dolphin Coast (iLembe), South Africa, 30% in 1999 to 16% in 2003 by the private utility Siza Water Company; Istanbul, Turkey, from more than 50% prior to 1994 to 34% in 2000 by the public utility ISKI; Jamshedpur, India, from an e... | Wikipedia - Non-revenue water - Successful programs |
80 million liters per day in 2008 to 650 million liters per day in 2014 in cooperation with the private utility Miya.These successes were achieved by both public and private utilities, in every continent, in emerging countries as well as very poor countries, in large cities and smaller towns. All required a long-term c... | Wikipedia - Non-revenue water - Successful programs |
In the following code sample, two optimizations can be applied. Although the calculation x = y + z and x * x is loop-invariant, precautions must be taken before moving the code outside the loop. It is possible that the loop condition is false (for example, if n holds a negative value), and in such case, the loop body s... | Wikipedia - Loop-invariant code motion - Example |
Evaluating the loop condition can have side effects, so an additional evaluation by the if construct should be compensated by replacing the while loop with a do {} while. If the code used do {} while in the first place, the whole guarding process is not needed, as the loop body is guaranteed to execute at least once. T... | Wikipedia - Loop-invariant code motion - Example |
In the following code, neither f nor g functions is reentrant. In the above, f() depends on a non-constant global variable v; thus, if f() is interrupted during execution by an ISR which modifies v, then reentry into f() will return the wrong value of v. The value of v and, therefore, the return value of f, cannot be p... | Wikipedia - Reentrancy (computing) - Further examples |
These slightly altered versions are reentrant: In the following, the function is thread-safe, but not (necessarily) reentrant: In the above, function() can be called by different threads without any problem. But, if the function is used in a reentrant interrupt handler and a second interrupt arises inside the function,... | Wikipedia - Reentrancy (computing) - Further examples |
In the following code: a = b * c + g; d = b * c * e; it may be worth transforming the code to: tmp = b * c; a = tmp + g; d = tmp * e; if the cost of storing and retrieving tmp is less than the cost of calculating b * c an extra time. | Wikipedia - Common subexpression - Example |
In the following compilation, M is the length of the longest pattern, m their total length, n the length of the searchable text, o the number of occurrences. | Wikipedia - Text search - Algorithms using a finite set of patterns |
In the following compilation, m is the length of the pattern, n the length of the searchable text, and k = |Σ| is the size of the alphabet. 1.^ Asymptotic times are expressed using O, Ω, and Θ notation. 2.^ Used to implement the memmem and strstr search functions in the glibc and musl C standard libraries. 3.^ Can be e... | Wikipedia - String-searching algorithm - Single-pattern algorithms |
In the following context of mathematical logic, the turnstile symbol ⊢ {\displaystyle \vdash } means "provable from". That is, a ⊢ b {\displaystyle a\vdash b} reads: b is provable from a (in some specified formal system). | Wikipedia - Self consistent - Notation |
In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve): ∮ ∂ S A ⋅ d ℓ = ∬ S ( ∇ × A ) ⋅ d S {\displaystyle \oint _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}\ =\ \iint _{S}\left(\nabla \times \mathbf {A} \right)\cdot d\mathbf {S} }... | Wikipedia - List of vector calculus identities - Curve–surface integrals |
In the following decade, the US accumulated significant data on personnel protection, decontamination, and immunization; and, in the offensive program, on the potential for mosquitoes to be used as biological vectors. A new Department of Defense (DoD) Biological and Chemical Defense Planning Board was created in 1960 t... | Wikipedia - United States biological defense program - 1960s |
The U.S. Army Chemical Corps was given the responsibility to conduct biological warfare research for all of the services. In 1962, the responsibility for the testing of promising biological warfare agents was given to a separate Testing and Evaluation Command (TEC). | Wikipedia - United States biological defense program - 1960s |
Depending on the particular program, different test centers were used, such as the Deseret Test Center at Fort Douglas, Utah, the headquarters for the new biological and chemical warfare testing organization. In response to increasing concerns over public safety and the environment, the TEC implemented a complex system... | Wikipedia - United States biological defense program - 1960s |
During the last 10 years of the offensive research and development program (1959–69), many scientific advances were made that proved that biological warfare was clearly feasible, although dependent on careful planning, especially with regard to meteorological conditions. Large-scale fermentation, purification, concentr... | Wikipedia - United States biological defense program - 1960s |
Arnold G. Wedum, M.D., Ph.D., a civilian scientist who was Director of Industrial Health and Safety at Fort Detrick, was the leader in the development of containment facilities.During the 1960s, the US program underwent a philosophical change, and attention was now directed more towards biological agents that could inc... | Wikipedia - United States biological defense program - 1960s |
Research initiatives also included new therapy and prophylaxis. Pathogens studied included the agents causing anthrax, glanders, brucellosis, melioidosis, plague, psittacosis, Venezuelan equine encephalitis, Q fever, coccidioidomycosis, and a variety of plant and animal pathogensParticular attention was directed at che... | Wikipedia - United States biological defense program - 1960s |
Although the development of sensitive biological warfare agent detectors was at a standstill, two systems were, nonetheless, investigated. The first was a monitor that detected increases in the number of particles sized 1 to 5 µm in diameter, based on the assumption that a biological agent attack would include airborne... | Wikipedia - United States biological defense program - 1960s |
Both systems lacked enough specificity and sensitivity to be of any practical use.But in 1966, a research effort directed at detecting the presence of adenosine triphosphate (a chemical found only in living organisms) was begun. By using a fluorescent material found in fireflies, preliminary studies indicated that it w... | Wikipedia - United States biological defense program - 1960s |
The US Army also experimented with and developed highly effective barrier protective measures against both chemical and biological agents. Special impervious tents and personal protective equipment were developed, including individual gas masks even for military dogs.During the late 1960s, funding for the biological wa... | Wikipedia - United States biological defense program - 1960s |
Although the offensive program had been stopped in 1969, both offensive and defensive programs continued to be defended. John S. Foster, Jr, Director of Defense Research and Engineering, responded to a query by Congressman Richard D. McCarthy: It is the policy of the U.S. to develop and maintain a defensive chemical-bi... | Wikipedia - United States biological defense program - 1960s |
On 25 November 1969, President Richard Nixon visited Fort Detrick to announce a new policy on biological warfare. In two National Security Memoranda, the U.S. | Wikipedia - United States biological defense program - 1960s |
government renounced all development, production, and stockpiling of biological weapons and declared its intent to maintain only small research quantities of biological agents, such as are necessary for the development of vaccines, drugs, and diagnostics. Ground was broken in 1967 for the construction of a new, modern ... | Wikipedia - United States biological defense program - 1960s |
With the disestablishment of the biological warfare laboratories, the name of the U.S. Army Medical Unit, which was to have been housed in the new laboratories, was formally changed to U.S. Army Medical Research Institute of Infectious Diseases (USAMRIID) in 1969. The institute's new mission was stated in General Order... | Wikipedia - United States biological defense program - 1960s |
In the following definition, parentheses around tensor indices are notation for symmetrization. For example: T ( α β γ ) = 1 6 ( T α β γ + T α γ β + T β α γ + T β γ α + T γ α β + T γ β α ) {\displaystyle T_{(\alpha \beta \gamma )}={\frac {1}{6}}(T_{\alpha \beta \gamma }+T_{\alpha \gamma \beta }+T_{\beta \alpha \gamma }... | Wikipedia - Killing tensor - Definition and properties |
In the following definitions, the hypergraph is denoted by H=(V,E). H is called a k-uniform hypergraph if every edge in E consists of exactly k vertices. P {\displaystyle P} is a hypergraph packing if it is a subset of edges in H such that there is no pair of distinct edges with a common vertex. H {\displaystyle H} is ... | Wikipedia - Packing in a hypergraph - Definition and terminology |
In the following definitions, x T {\displaystyle \mathbf {x} ^{\textsf {T}}} is the transpose of x {\displaystyle \mathbf {x} } , x ∗ {\displaystyle \mathbf {x} ^{*}} is the conjugate transpose of x {\displaystyle \mathbf {x} } and 0 {\displaystyle \mathbf {0} } denotes the n-dimensional zero-vector. | Wikipedia - Positive-definite matrices - Definitions |
In the following derivation of the Einstein radius, we will assume that all of mass M of the lensing galaxy L is concentrated in the center of the galaxy. For a point mass the deflection can be calculated and is one of the classical tests of general relativity. For small angles α1 the total deflection by a point mass M... | Wikipedia - Einstein radius - Derivation |
In terms of the Einstein angle, the lens equation for a point mass becomes θ 1 = θ S + θ E 2 θ 1 {\displaystyle \theta _{1}=\theta _{\rm {S}}+{\frac {\theta _{E}^{2}}{\theta _{1}}}} Substituting for the constants gives θ E = ( M 10 11.09 M ⨀ ) 1 / 2 ( D L D S / D L S G p c ) − 1 / 2 a r c s e c {\displaystyle \theta _{... | Wikipedia - Einstein radius - Derivation |
For a Gravitational microlensing event (with masses of order 1 M☉) search for at galactic distances (say D ~ 3 kpc), the typical Einstein radius would be of order milli-arcseconds. Consequently, separate images in microlensing events are impossible to observe with current techniques. | Wikipedia - Einstein radius - Derivation |
Likewise, for the lower ray of light reaching the observer from below the lens, we have θ 2 D S = − θ S D S + α 2 D L S {\displaystyle \theta _{2}\;D_{\rm {S}}=-\;\theta _{\rm {S}}\;D_{\rm {S}}+\alpha _{2}\;D_{\rm {LS}}} and θ 2 + θ S = 4 G c 2 M θ 2 D L S D S D L {\displaystyle \theta _{2}+\theta _{\rm {S}}={\frac {4G... | Wikipedia - Einstein radius - Derivation |
In the following description, call the 3 containers: (A) Top: basin (B) Middle: water supply (C) Bottom: air supplyAnd three pipes: P1 (on the left in the picture) from a hole in the bottom of basin (A) to the bottom of air supply container (C) P2 (on the right in the picture) from the top of the air supply container (... | Wikipedia - Heron's fountain - Construction |
B and C, however, must be airtight and resistant to atmospheric pressure. Plastic bottles suffice, but glass containers work better. Balloons do not work because they cannot hold pressure without deforming. | Wikipedia - Heron's fountain - Construction |
The fountain works in the following way: The energy for moving the water ultimately comes from the water in A descending into C. This means the water in B can rise into A only as much as it falls from A to C. Water falling from A down to C through pipe P1 builds up pressure in the bottom container; this pressure is pro... | Wikipedia - Heron's fountain - Construction |
In the following description, the terminology used will be particular to the Windows NT operating system, but the concepts are applicable to other virtual memory operating systems.When a new application on a 32-bit OS is executed, the process has a 4 GiB VAS: each one of the memory addresses (from 0 to 232 − 1) in that... | Wikipedia - Virtual address - Example |
Addresses in the process VAS are mapped to bytes in the exe file. The OS manages the mapping: 0 4 GiB VAS |---vvv----------------------------------------| mapping ||| file bytes app The v's are values from bytes in the mapped file. Then, required DLL files are mapped (this includes custom libraries as well as system on... | Wikipedia - Virtual address - Example |
However, the only way the process can use or set '-' values in its VAS is to ask the OS to map them to bytes from a file. A common way to use VAS memory in this way is to map it to the page file. The page file is a single file, but multiple distinct sets of contiguous bytes can be mapped into a VAS: 0 4 GiB VAS |---vvv... | Wikipedia - Virtual address - Example |
The other 2 GiB are used by the operating system. On later 32-bit editions of Microsoft Windows, it is possible to extend the user-mode virtual address space to 3 GiB while only 1 GiB is left for kernel-mode virtual address space by marking the programs as IMAGE_FILE_LARGE_ADDRESS_AWARE and enabling the /3GB switch in ... | Wikipedia - Virtual address - Example |
In the following descriptions, α, β, and γ represent any string of terminals/nonterminals (including the empty string), X and Y represent single nonterminals, and a represents a terminal symbol. Earley's algorithm is a top-down dynamic programming algorithm. In the following, we use Earley's dot notation: given a produ... | Wikipedia - Earley parser - The algorithm |
Input position 0 is the position prior to input. Input position n is the position after accepting the nth token. (Informally, input positions can be thought of as locations at token boundaries.) | Wikipedia - Earley parser - The algorithm |
For every input position, the parser generates a state set. Each state is a tuple (X → α • β, i), consisting of the production currently being matched (X → α β) the current position in that production (visually represented by the dot •) the position i in the input at which the matching of this production began: the ori... | Wikipedia - Earley parser - The algorithm |
The state set at input position k is called S(k). The parser is seeded with S(0) consisting of only the top-level rule. The parser then repeatedly executes three operations: prediction, scanning, and completion. | Wikipedia - Earley parser - The algorithm |
Prediction: For every state in S(k) of the form (X → α • Y β, j) (where j is the origin position as above), add (Y → • γ, k) to S(k) for every production in the grammar with Y on the left-hand side (Y → γ). Scanning: If a is the next symbol in the input stream, for every state in S(k) of the form (X → α • a β, j), add ... | Wikipedia - Earley parser - The algorithm |
These three operations are repeated until no new states can be added to the set. The set is generally implemented as a queue of states to process, with the operation to be performed depending on what kind of state it is. The algorithm accepts if (X → γ •, 0) ends up in S(n), where (X → γ) is the top level-rule and n th... | Wikipedia - Earley parser - The algorithm |
In the following diagram, one of the Acceptors in the Quorum fails, so the Quorum size becomes 2. In this case, the Basic Paxos protocol still succeeds. Client Proposer Acceptor Learner | | | | | | | X-------->| | | | | | Request | X--------->|->|->| | | Prepare(1) | | | | ! | Wikipedia - Paxos algorithm - Basic Paxos when an Acceptor fails |
| | !! FAIL !! | |<---------X--X | | Promise(1,{Va, Vb, null}) | X--------->|->| | | Accept! (1,V) | |<---------X--X--------->|->| Accepted(1,V) |<---------------------------------X--X Response | | | | | | | Wikipedia - Paxos algorithm - Basic Paxos when an Acceptor fails |
In the following diagram, only one instance (or "execution") of the basic Paxos protocol, with an initial Leader (a Proposer), is shown. Note that a Multi-Paxos consists of several instances of the basic Paxos protocol. Client Proposer Acceptor Learner | | | | | | | --- First Request --- X-------->| | | | | | Request |... | Wikipedia - Paxos (computer science) - Multi-Paxos without failures |
In the following discussion and positions, assume that the side with the pawn is White. White will be attempting to win and Black will be attempting to draw. Ranks are counted from that player's side of the board. | Wikipedia - Rook and pawn versus rook endgame - Terminology |
Thus, "the rook's third rank" would be the third rank counting from that player's side of the board. The ranks for the white pieces correspond to the ranks in algebraic notation, whereas the ranks for the black pieces are reversed. In these positions with one pawn, the pawn's file divides the board into a short side an... | Wikipedia - Rook and pawn versus rook endgame - Terminology |
A pawn is referred to by the file on which it stands: a rook pawn is on the a- or h-file, a knight pawn is on the b- or g-file, a bishop pawn is on the c- or f-file. A central pawn is a queen pawn or a king pawn, on the d- or e-file. When designating a position as a win or a draw, optimal play by both sides is assumed. | Wikipedia - Rook and pawn versus rook endgame - Terminology |
In the following discussion we will always assume that the representing matrix H {\displaystyle {\mathfrak {H}}} is normalized such that det H = a d − b c = 1 {\displaystyle \det {\mathfrak {H}}=ad-bc=1} . Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and l... | Wikipedia - Möbius group - Classification |
The four types can be distinguished by looking at the trace tr H = a + d {\displaystyle \operatorname {tr} {\mathfrak {H}}=a+d} . The trace is invariant under conjugation, that is, and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing ... | Wikipedia - Möbius group - Classification |
In the following discussion, g {\displaystyle g} represents the acceleration due to gravity, ω {\displaystyle \omega } represents the angular speed of the liquid's rotation, in radians per second, m {\displaystyle m} is the mass of an infinitesimal parcel of liquid material on the surface of the liquid, r {\displaystyl... | Wikipedia - Liquid mirror telescopes - Explanation of the equilibrium |
The green arrow shows the buoyancy force exerted on the parcel by the bulk of the liquid. Since, in equilibrium, the liquid cannot exert a force parallel with its surface, the green arrow must be perpendicular to the surface. The short blue arrow shows the net force on the parcel. | Wikipedia - Liquid mirror telescopes - Explanation of the equilibrium |
It is the vector sum of the forces of weight and buoyancy, and acts horizontally toward the axis of rotation. (It must be horizontal, since the parcel has no vertical acceleration.) It is the centripetal force that constantly accelerates the parcel toward the axis, keeping it in circular motion as the liquid rotates. | Wikipedia - Liquid mirror telescopes - Explanation of the equilibrium |
The buoyancy force (green arrow) has a vertical component, which must equal the weight m g {\displaystyle mg} of the parcel (red arrow), and the horizontal component of the buoyancy force must equal the centripetal force m ω 2 r {\displaystyle m\omega ^{2}r} (blue arrow). Therefore, the green arrow is tilted from the v... | Wikipedia - Liquid mirror telescopes - Explanation of the equilibrium |
Since the green arrow is perpendicular to the surface of the liquid, the slope of the surface must be the same quotient of the forces: d h d r = m ω 2 r m g . {\displaystyle {\frac {dh}{dr}}={\frac {m\omega ^{2}r}{mg}}.} Cancelling the m {\displaystyle m} on both sides, integrating, and setting h = 0 {\displaystyle h=0... | Wikipedia - Liquid mirror telescopes - Explanation of the equilibrium |
In the following discussion, in accordance with a preceding observation, the term "Middle Mongol" is used merely as a cover term for texts written in any of three scripts, Uighur Mongolian script (UM), Chinese (SM), or Arabic (AM). The case system of Middle Mongol has remained mostly intact down to the present, althoug... | Wikipedia - Mongolic language - Nominal system |
There have been three different case suffixes in the dative-locative-directive domain that are grouped in different ways: -a as locative and -dur, -da as dative or -da and -a as dative and -dur as locative, in both cases with some functional overlapping. As -dur seems to be grammaticalized from dotur-a 'within', thus i... | Wikipedia - Mongolic language - Nominal system |
Of these, -da was lost, -dur was first reduced to -du and then to -d and -a only survived in a few frozen environments. Finally, the directive of modern Mongolian, -ruu, has been innovated from uruγu 'downwards'. Social gender agreement was abandoned. | Wikipedia - Mongolic language - Nominal system |
In the following discussion, let K(s) be the complexity of the string s. It is not hard to see that the minimal description of a string cannot be too much larger than the string itself — the program GenerateString2 above that outputs s is a fixed amount larger than s. Theorem: There is a constant c such that ∀s. K(s) ≤... | Wikipedia - Kolmogorov Complexity - Basic results |
In the following discussion, the following conventions are adopted. K {\displaystyle \mathbb {K} } will denote one of the fields of real or complex numbers, denoted R {\displaystyle \mathbb {R} } , C {\displaystyle \mathbb {C} } , respectively. The vector space of m × n {\displaystyle m\times n} matrices over K {\displ... | Wikipedia - Moore–Penrose pseudoinverse - Notation |
If K = R {\displaystyle \mathbb {K} =\mathbb {R} } , then A ∗ = A T {\displaystyle A^{*}=A^{\textsf {T}}} . For A ∈ K m × n {\displaystyle A\in \mathbb {K} ^{m\times n}} , ran ( A ) {\displaystyle \operatorname {ran} (A)} (standing for "range") denotes the column space (image) of A {\displaystyle A} (the space spanne... | Wikipedia - Moore–Penrose pseudoinverse - Notation |
In the following discussion, the term "antioxidant" refers mainly to non-nutrient compounds in foods, such as polyphenols, which have antioxidant capacity in vitro and so provide an artificial index of antioxidant strength – the ORAC measurement. Other than for dietary antioxidant vitamins – vitamin A, vitamin C and vi... | Wikipedia - List of antioxidants in food - Regulatory guidance |
In the following discussion, the term "antioxidant" refers mainly to non-nutrient compounds in foods, such as polyphenols, which have antioxidant capacity in vitro, so provide an artificial index of antioxidant strength—the ORAC measurement. Other than for dietary antioxidant vitamins—vitamin A, vitamin C and vitamin E... | Wikipedia - Oxygen Radical Absorbance Capacity - Regulatory guidance |
In the following discussion, we define volumetric flow rate V̇ (i.e. volume of fluid flowing per time) as V ˙ = π r 2 v {\displaystyle {\dot {V}}=\pi r^{2}v} where r = radius of the pipe (for a pipe of circular section, the internal radius of the pipe). v = mean velocity of fluid flowing through the pipe. A = cross sec... | Wikipedia - Friction loss - Calculating volumetric flow |
{\displaystyle {\frac {\Delta p}{L}}.} When the pressure is expressed in terms of the equivalent height of a column of that fluid, as is common with water, the friction loss is expressed as S, the "head loss" per length of pipe, a dimensionless quantity also known as the hydraulic slope. | Wikipedia - Friction loss - Calculating volumetric flow |
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