text stringlengths 1 2.7k | source stringlengths 21 174 |
|---|---|
Two moves in: At this point, the requirement that the graph be proper comes into effect, as a red area must be made which does not touch the existing one: Once the third region is coloured: Note that areas only count as touching if they share edges, not if they only share vertices, so this move is legal. The game conti... | Wikipedia - Col (game) - Example game |
In the following generation, John Locke sought to answer Filmer, creating a rationale for a balanced constitution in which the monarch had a part to play, but not an overwhelming part. Since Filmer's views essentially require that the Stuart family be uniquely descended from the patriarchs of the Bible, and even in the... | Wikipedia - Proprietary right - John Locke (17th century) |
"To this end" (to achieve the previously specified goal), he wrote, "it is that men give up all their natural power to the society they enter into, and the community put the Legislative power into such hands as they think fit, with this trust, that they shall be governed by declared laws, or else their peace, quiet, an... | Wikipedia - Proprietary right - John Locke (17th century) |
Whereas by supposing they have given themselves up to the absolute arbitrary power and will of a legislator, they have disarmed themselves, and armed him to make a prey of them when he pleases..." Both "persons" and "estates" are to be protected from the arbitrary power of any magistrate, including legislative power an... | Wikipedia - Proprietary right - John Locke (17th century) |
In the following generation, leaders of the Musar movement included Zissel's student Nosson Tzvi Finkel of Slabodka, and Rabbi Yosef Yozel Horwitz of Novaradok. The schools founded by these two men became the largest and most influential schools of Musar. The Slobodka school founded by Finkel became especially influent... | Wikipedia - Mussar movement - Third generation |
In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color. The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger ... | Wikipedia - Inverse hyperbolic sine - Graphical representation |
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram). | Wikipedia - Triangle identities - Triangle identities |
In the following identities, δ {\displaystyle \delta } is a derivation of a differential ring R . {\displaystyle R. }: 76 If r ∈ R {\displaystyle r\in R} and c {\displaystyle c} is a constant in R {\displaystyle R} (that is, δ c = 0 {\displaystyle \delta c=0} ), then If r ∈ R {\displaystyle r\in R} and u {\displaystyle... | Wikipedia - Derivation algebra - Basic formulas |
In the following it will be assumed that the Zak transform is as given in Definition 2. 1. Linearity Let a and b be any real or complex numbers. Then Z ( t , w ) = a Z ( t , w ) + b Z ( t , w ) {\displaystyle Z(t,w)=aZ(t,w)+bZ(t,w)} 2. | Wikipedia - Zak transform - Properties of the Zak transform |
Periodicity Z ( t , w + 1 ) = Z ( t , w ) {\displaystyle Z(t,w+1)=Z(t,w)} 3. Quasi-periodicity Z ( t + 1 , w ) = e 2 π w i Z ( t , w ) {\displaystyle Z(t+1,w)=e^{2\pi wi}Z(t,w)} 4. Conjugation Z ( t , w ) = Z ¯ ( t , − w ) {\displaystyle Z(t,w)={\overline {Z}}(t,-w)} 5. Symmetry If f(t) is even then Z ( t , w ) ... | Wikipedia - Zak transform - Properties of the Zak transform |
In the following jurisdictions, Creative Commons is either currently working to establish affiliate teams, or a team is in place but the drafting process has not begun yet: Iceland Kenya Lebanon Lithuania Macau Qatar Russia Rwanda Slovakia Syria Tanzania Uganda United Arab Emirates Venezuela | Wikipedia - Creative Commons jurisdiction ports - Jurisdictions for which development is planned |
In the following languages, reflexes separated by slashes mean: Albanian: Tosk Albanian / Gheg Albanian Armenian: Classical Armenian / Eastern Armenian / Western Armenian English: Old English / Modern English German: Old High German / New High German Irish: Old Irish / Modern Irish Ossetic: Iron / Digor Persian: Old Pe... | Wikipedia - Proto-Indo-European numerals - Reflexes of the cardinal numbers |
In the following list all envelopes can be defined as functors. 1. The completion X ▾ {\displaystyle X^{\blacktriangledown }} of a locally convex topological vector space X {\displaystyle X} is an envelope of X {\displaystyle X} in the category LCS {\displaystyle {\text{LCS}}} of all locally convex spaces with respect ... | Wikipedia - Envelope (category theory) - Examples |
Obviously, X ▾ {\displaystyle X^{\blacktriangledown }} is the inverse limit of the quotient Banach spaces X / U {\displaystyle X/U} (defined above): X ▾ = lim 0 ← U X / U . {\displaystyle X^{\blacktriangledown }=\lim _{0\gets U}X/U.} 2. | Wikipedia - Envelope (category theory) - Examples |
The Stone–Čech compactification β: X → β X {\displaystyle \beta :X\to \beta X} of a Tikhonov topological space X {\displaystyle X} is an envelope of X {\displaystyle X} in the category Tikh {\displaystyle {\text{Tikh}}} of all Tikhonov spaces in the class Com {\displaystyle {\text{Com}}} of compact spaces with respect ... | Wikipedia - Envelope (category theory) - Examples |
The Arens-Michael envelope A AM {\displaystyle A^{\text{AM}}} of a locally convex topological algebra A {\displaystyle A} with a separately continuous multiplication is an envelope of A {\displaystyle A} in the category TopAlg {\displaystyle {\text{TopAlg}}} of all (locally convex) topological algebras (with separately... | Wikipedia - Envelope (category theory) - Examples |
4. The holomorphic envelope Env O A {\displaystyle {\text{Env}}_{\cal {O}}A} of a stereotype algebra A {\displaystyle A} is an envelope of A {\displaystyle A} in the category SteAlg {\displaystyle {\text{SteAlg}}} of all stereotype algebras in the class DEpi {\displaystyle {\text{DEpi}}} of all dense epimorphisms in St... | Wikipedia - Envelope (category theory) - Examples |
5. The smooth envelope Env E A {\displaystyle {\text{Env}}_{\cal {E}}A} of a stereotype algebra A {\displaystyle A} is an envelope of A {\displaystyle A} in the category InvSteAlg {\displaystyle {\text{InvSteAlg}}} of all involutive stereotype algebras in the class DEpi {\displaystyle {\text{DEpi}}} of all dense epimor... | Wikipedia - Envelope (category theory) - Examples |
6. The continuous envelope Env C A {\displaystyle {\text{Env}}_{\cal {C}}A} of a stereotype algebra A {\displaystyle A} is an envelope of A {\displaystyle A} in the category InvSteAlg {\displaystyle {\text{InvSteAlg}}} of all involutive stereotype algebras in the class DEpi {\displaystyle {\text{DEpi}}} of all dense ep... | Wikipedia - Envelope (category theory) - Examples |
In the following list of Archons, years where the name of the archon is unknown are identified as such. Years listed as "anarchy" mean that there was literally "no archon". There are various conflicting reconstructions of lists; sources for this list are given at the end. Note that the term of an archon covered two of ... | Wikipedia - Eponymous archon - List of archons of Athens |
In the following list the "(substituting X)" refers to all of the materials preceding said parenthesis. Gallium arsenide n-type: tellurium, sulfur (substituting As); tin, silicon, germanium (substituting Ga) p-type: beryllium, zinc, chromium (substituting Ga); silicon, germanium, carbon (substituting As) Gallium phosph... | Wikipedia - Doped silicon - Other semiconductors |
In the following list the dates in parentheses give the year of merger with the present-day sort code holder, or its subsidiary. | Wikipedia - Sort code - England and Wales |
In the following list, the dates in parentheses are the earliest approved use of the drug. | Wikipedia - Anti-seizure medication - Drugs |
In the following list, ⟨ɐ⟩ is assumed to be unrounded. The rounded variant is transcribed as ⟨ɐ̹⟩. Some instances of the latter may actually be fully open. | Wikipedia - Near-open central vowel - Occurrence |
In the following listing, the distinction between languages and dialects follows Ethnologue. In the Northern groupKistane (Soddo) Dialects: Soddo, Dobi (Gogot)In the Eastern groupSilte (Silte, formerly) Wolane (Welene) Zay (Zway)In the Western groupSebat Bet Gurage Dialects: Chaha, Ezha, Gumer, Geta, Muher, Mesqan, Ino... | Wikipedia - Gurage languages - Languages |
In the following lists, the designation letter ("Des." column) is a reference assigned to the chronometer while it was on board ship and was used to identify the instrument in logged tables of chronometer readings during the voyage. It was a temporary designation with no relevance once the ship's mission was over. The ... | Wikipedia - List of chronometers on HMS Beagle - Nomenclature |
When referring to chronometers, the maker's name is frequently abbreviated; for instance, Earnshaw's chronometer no. 509 is known simply as E509. The "type" of all chronometers listed here is either box or pocket. A boxed chronometer is mounted on gimbals attached to its box. | Wikipedia - List of chronometers on HMS Beagle - Nomenclature |
A pocket chronometer is in the style of a pocketwatch. "Winding" refers to the number of days that a chronometer kept going before needing rewinding. However, they were all wound at precisely the same time every day, except for the eight-day chronometers which were wound weekly. | Wikipedia - List of chronometers on HMS Beagle - Nomenclature |
In the following n is a natural number, in sequence: n = 1 , 2 , 3 , 4 , … {\displaystyle \ n=1,\ 2,\ 3,\ 4,\ \ldots \ } 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , … , 1 n , … {\displaystyle 1,{\tfrac {\ 1\ }{2}},\ {\tfrac {\ 1\ }{3}},\ {\tfrac {\ 1\ }{4}},\ {\tfrac {\ 1\ }{5}},\ {\tfrac {\ 1\ }{6}},\ \ldots \ ,\ {\tfrac {\ 1\ ... | Wikipedia - Harmonic progression (mathematics) - Examples |
In the following one must remember that these models are abstract models with two fundamental differences from anything physically real: unbounded numbers of registers each with unbounded capacities. The problem appears most dramatically when one tries to use a counter-machine model to build a RASP that is Turing equiv... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
As noted below, Minsky (1967) hints at the problem for a RASP but doesn't offer a solution. Elgot and Robinson (1964) proved that their RASP model P0 – it has no indirection capability – cannot compute all "recursive sequential functions" (ones that have parameters of arbitrary length) if it does not have the capabilit... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
395). On the other hand their RASP model P'0 equipped with an "index register" (indirect addressing) can compute all the "partial recursive sequential functions" (the mu recursive functions) (p. | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
397-398).Cook and Reckhow (1973) say it most succinctly: The indirect instructions are necessary in order for a fixed program to access an unbounded number of registers as the inputs vary." (p. 73)Unbounded capacities of registers versus bounded capacities of state-machine instructions: The so-called finite state part ... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
So how does a state machine move an arbitrarily large constant directly into a register, e.g. MOVE (k, r) (Move constant k to register r)? If huge constants are necessary they must either start out in the registers themselves or be created by the state machine using a finite number of instructions e.g. multiply and add... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
This trick is mentioned by Kleene (1952) p. 223. The problem arises when the number to be created exhausts the number of instructions available to the finite state machine; there is always a bigger constant than the number of instructions available to the finite state machine.Unbounded numbers of registers versus bound... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
In particular, this problem arises when we attempt to build a so-called RASP, a "universal machine" (see more at Universal Turing machine) that uses its finite-state machine to interpret a "program of instructions" located in its registers – i.e. we are building what is nowadays called a computer with the von Neumann a... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
One approach would be to modify the program-instructions (the ones stored in the registers) so that they contain more than one command. But this too can be exhausted unless an instruction is of (potentially) unbounded size. So why not use just one "über-instruction" – one really really big number – that contains all th... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
This is how Minsky solves the problem, but the Gödel numbering he uses represents a great inconvenience to the model, and the result is nothing at all like our intuitive notion of a "stored program computer". Elgot and Robinson (1964) come to a similar conclusion with respect to a RASP that is "finitely determined". In... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
396). In the context of a more computer-like model using his RPT (repeat) instruction Minsky (1967) tantalizes us with a solution to the problem (cf p. 214, p. | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
259) but offers no firm resolution. He asserts: "In general a RPT operation could not be an instruction in the finite-state part of the machine ... this might exhaust any particular amount of storage allowed in the finite part of the computer . RPT operations require infinite registers of their own." | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
(p. 214).He offers us a bounded RPT that together with CLR (r) and INC (r) can compute any primitive recursive function, and he offers the unbounded RPT quoted above that as playing the role of μ operator; it together with CLR (r) and INC (r) can compute the mu recursive functions. | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
But he does not discuss "indirection" or the RAM model per se. From the references in Hartmanis (1971) it appears that Cook (in his lecture notes while at UC Berkeley, 1970) has firmed up the notion of indirect addressing. This becomes clearer in the paper of Cook and Reckhow (1973) – Cook is Reckhow's Master's thesis ... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
Hartmanis' model – quite similar to Melzak's (1961) model – uses two and three-register adds and subtracts and two parameter copies; Cook and Reckhow's model reduce the number of parameters (registers called out in the program instructions) to one call-out by use of an accumulator "AC". The solution in a nutshell: Desi... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
In this sense the solution represents the unbounded μ operator that can, if necessary, hunt ad infinitum along the unbounded string of registers until it finds what it is looking for. The pointer register is exactly like any other register with one exception: under the circumstances called "indirect addressing" it prov... | Wikipedia - Random-access machine - Why the need for an indirect operation: Two major problems with the counter-machine model |
In the following piece of Java code, the Java keyword synchronized makes the method thread-safe: In the C programming language, each thread has its own stack. However, a static variable is not kept on the stack; all threads share simultaneous access to it. If multiple threads overlap while running the same function, it... | Wikipedia - Thread safety - Examples |
One common way to avoid this is to use another shared variable as a "lock" or "mutex" (from mutual exclusion). In the following piece of C code, the function is thread-safe, but not reentrant: In the above, increment_counter can be called by different threads without any problem since a mutex is used to synchronize all... | Wikipedia - Thread safety - Examples |
In the following prefixes, a final vowel is normally dropped before a root that begins with a vowel, with the exceptions of bi-, which is bis- before a vowel, and of the other monosyllables, du-, di-, dvi-, tri-, which are invariable. The cardinal series are derived from cardinal numbers, such as the English one, two, ... | Wikipedia - Numerical prefix - Table of number prefixes in English |
The ordinal series are based on ordinal numbers such as the English first, second, third (for numbers higher than 2, the ordinal forms are also used for fractions; only the fraction 1⁄2 has special forms). For the hundreds, there are competing forms: those in -gent-, from the original Latin, and those in -cent-, derive... | Wikipedia - Numerical prefix - Table of number prefixes in English |
In the following properties, which result almost immediately from the definition of an inner product, x, y and z are arbitrary vectors, and a and b are arbitrary scalars. ⟨ 0 , x ⟩ = ⟨ x , 0 ⟩ = 0. {\displaystyle \langle \mathbf {0} ,x\rangle =\langle x,\mathbf {0} \rangle =0.} ⟨ x , x ⟩ {\displaystyle \langle x,x\rang... | Wikipedia - Orthogonal vector - Basic properties |
⟨ x , x ⟩ = 0 {\displaystyle \langle x,x\rangle =0} if and only if x = 0 . {\displaystyle x=\mathbf {0} .} | Wikipedia - Orthogonal vector - Basic properties |
⟨ x , a y + b z ⟩ = a ¯ ⟨ x , y ⟩ + b ¯ ⟨ x , z ⟩ . {\displaystyle \langle x,ay+bz\rangle ={\overline {a}}\langle x,y\rangle +{\overline {b}}\langle x,z\rangle .} This implies that an inner product is a sesquilinear form. | Wikipedia - Orthogonal vector - Basic properties |
⟨ x + y , x + y ⟩ = ⟨ x , x ⟩ + 2 Re ( ⟨ x , y ⟩ ) + ⟨ y , y ⟩ , {\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\operatorname {Re} (\langle x,y\rangle )+\langle y,y\rangle ,} where Re {\displaystyle \operatorname {Re} } denotes the real part of its argument.Over R {\displaystyle \mathbb {R} } , conjugate... | Wikipedia - Orthogonal vector - Basic properties |
In the following pseudocode algorithm, dist is an array that contains the current distances from the source to other vertices, i.e. dist is the current distance from the source to the vertex u. The prev array contains pointers to previous-hop nodes on the shortest path from source to the given vertex (equivalently, it ... | Wikipedia - Dijkstra's algorithm - Pseudocode |
Now we can read the shortest path from source to target by reverse iteration: 1 S ← empty sequence 2 u ← target 3 if prev is defined or u = source: // Do something only if the vertex is reachable 4 while u is defined: // Construct the shortest path with a stack S 5 insert u at the beginning of S // Push the vertex onto... | Wikipedia - Dijkstra's algorithm - Pseudocode |
Then instead of storing only a single node in each entry of prev we would store all nodes satisfying the relaxation condition. For example, if both r and source connect to target and both of them lie on different shortest paths through target (because the edge cost is the same in both cases), then we would add both r a... | Wikipedia - Dijkstra's algorithm - Pseudocode |
In the following query, four different types of data are retrieved from a single table (status) and for a single user ("me"): This query can run by querying the Facebook graph endpoint /fql with the parameters set to q= | Wikipedia - Facebook Query Language - Example |
In the following reaction network, A is irreversibly converted to an intermediate B, which irreversibly reacts to form C. A → δ b , α b / A ϕ a b B → δ c , α c / B ϕ b c C {\displaystyle {\ce {A->B->C}}} The pools of A, B, and C have delta values defined as δA, δB, and δC respectively. These values are related to the r... | Wikipedia - Stable isotope composition of amino acids - Linear irreversible network |
For A → B, α b / A ≡ δ b + 1 δ A + 1 {\displaystyle \alpha _{b/A}\equiv {\frac {\delta _{b}+1}{\delta _{A}+1}}} Rearranging for δb gives δ b = α b / A δ A + ϵ b / A {\displaystyle \delta _{b}=\alpha _{b/A}\delta _{A}+\epsilon _{b/A}} in which ϵ b / A ≡ α b / A − 1 {\displaystyle \epsilon _{b/A}\equiv \alpha _{b/A}-1} .... | Wikipedia - Stable isotope composition of amino acids - Linear irreversible network |
If the isotope effect is small, as is typical for C and N, α b / A ≈ 1 {\displaystyle \alpha _{b/A}\approx 1} and δ b ≈ δ A + ϵ b / A {\displaystyle \delta _{b}\approx \delta _{A}+\epsilon _{b/A}} . From this, we can seen that the product produced from A will be depleted by roughly ϵ b / A {\displaystyle \epsilon _{b/A... | Wikipedia - Stable isotope composition of amino acids - Linear irreversible network |
In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector defines the axis of rotation, e ^ r {\displaystyle \scriptstyle \mathbf {\hat {e}} _{r}... | Wikipedia - Linear-rotational analogs - Kinematics |
In the following rules, φ ( β / α ) {\displaystyle \varphi (\beta /\alpha )} is exactly like φ {\displaystyle \varphi } except for having the term β {\displaystyle \beta } wherever φ {\displaystyle \varphi } has the free variable α {\displaystyle \alpha } . Universal Generalization (or Universal Introduction) φ ( β / α... | Wikipedia - List of rules of inference - Rules of classical predicate calculus |
Existential Generalization (or Existential Introduction) φ ( β / α ) _ {\displaystyle {\underline {\varphi (\beta /\alpha )}}} ∃ α φ {\displaystyle \exists \alpha \,\varphi } Restriction: No free occurrence of α {\displaystyle \alpha } in φ {\displaystyle \varphi } falls within the scope of a quantifier quantifying a v... | Wikipedia - List of rules of inference - Rules of classical predicate calculus |
In the following scenario a web browser is developed using defense in depth - the browser developers receive security training the codebase is checked automatically using security analysis tools the browser is regularly audited by an internal security team ... is occasionally audited by an external security team ... is... | Wikipedia - Defense in depth (computing) - Example |
In the following section it is shown that for 455 spheres the sausage packing is non-optimal, and that there instead exists a special cluster packing that occupies a smaller volume. The volume of a convex hull of a sausage packing with n {\displaystyle n} spheres of radius r {\displaystyle r} is calculable with element... | Wikipedia - Finite sphere packing - Example of the sausage packing being non-optimal |
In the following section, these symbols are used: Length of the cable, ℓ . {\displaystyle \ \ell ~.} Outside diameter of inner conductor, d . {\displaystyle \ d~.} | Wikipedia - Coax cable - Physical parameters |
Inside diameter of the shield, D . {\displaystyle \ D~.} Dielectric constant of the insulator, ϵ . | Wikipedia - Coax cable - Physical parameters |
{\displaystyle \ \epsilon ~.} The dielectric constant is often quoted as the relative dielectric constant ϵ r {\displaystyle \ \epsilon _{\mathsf {r}}\ } referred to the dielectric constant of free space ϵ o: {\displaystyle \ \epsilon _{\mathsf {o}}\ :} ϵ = ϵ r ϵ o . {\displaystyle \ \epsilon =\epsilon _{\mathsf {r}}\ ... | Wikipedia - Coax cable - Physical parameters |
When the insulator is a mixture of different dielectric materials (e.g., polyethylene foam is a mixture of polyethylene and air), then the term effective dielectric constant ϵ e f f {\displaystyle \ \epsilon _{\mathsf {eff}}\ } is often used. Magnetic permeability of the insulator, μ . {\displaystyle \ \mu ~.} | Wikipedia - Coax cable - Physical parameters |
Permeability is often quoted as the relative permeability μ r {\displaystyle \ \mu _{\mathsf {r}}\ } referred to the permeability of free space μ o: {\displaystyle \ \mu _{\mathsf {o}}\ :} μ = μ r μ o . {\displaystyle \ \mu =\mu _{\mathsf {r}}\ \mu _{\mathsf {o}}~.} The relative permeability will almost always be 1. | Wikipedia - Coax cable - Physical parameters |
In the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems.The follow... | Wikipedia - Integral equations - Converting IVP to integral equations |
In the following sections on sib-crossing and similar topics, a number of "averaging rules" are useful. These derive from path analysis. The rules show that any co-ancestry coefficient can be obtained as the average of cross-over co-ancestries between appropriate grand-parental and parental combinations. Thus, referrin... | Wikipedia - Quantitative genetics - Cross-multiplication rules |
In a similar fashion, cross-multiplier 2 states that fPC = (1/2) —while cross-multiplier 3 states that fPD = (1/2) . Returning to the first multiplier, it can now be seen also to be fPQ = (1/2) , which, after substituting multipliers 2 and 3, resumes its original form. In much of the following, the grand-parental gene... | Wikipedia - Quantitative genetics - Cross-multiplication rules |
In the following sections, (x,y) = xT y denotes the dot product of vectors. To solve a linear system Ax = b, BiCGSTAB starts with an initial guess x0 and proceeds as follows: r0 = b − Ax0 Choose an arbitrary vector r̂0 such that (r̂0, r0) ≠ 0, e.g., r̂0 = r0 ρ0 = (r̂0, r0) p0 = r0 For i = 1, 2, 3, … v = Api−1 α = ρi−1/... | Wikipedia - Biconjugate gradient stabilized method - Unpreconditioned BiCGSTAB |
In the following sentence with antithesis, two topics, "the land journey" and "the journey by sea", are contrasted. Because they are contrasted, the topics are more emphatic than the focus: breve terrā iter eō, brevis nāvigātiō ab Naupactō est. "The journey there by land is short, and the sea journey from Naupactus is ... | Wikipedia - Latin word order - Emphatic topics |
"I defended the republic as a young man, I shall not desert her as an old one. "Devine and Stephens refer to such emphatic topics used contrastively as "co-focus".Another example is the following, where the phrase "this place" (namely the city of Laurentum which was mentioned in the previous sentence) is in the emphati... | Wikipedia - Latin word order - Emphatic topics |
In the following sentences, verbs are used without a direct object: "Rivers flow." "I sneezed." "My dog ran." "Water evaporates when it's hot." | Wikipedia - Intransitive verb - Examples |
"You've grown since I last saw you!" "I wonder how old I will be when I die. "The following sentences contain transitive verbs (they entail one or more objects): "We watched a movie last night." | Wikipedia - Intransitive verb - Examples |
"She's making promises." "When I said that, my sister smacked me." "Santa gave me a present." | Wikipedia - Intransitive verb - Examples |
"He continuously clicked his pen and it was incredibly annoying to me. "Some verbs, called ambitransitive verbs, may entail objects but do not always require one. Such a verb may be used as intransitive in one sentence, and as transitive in another. In general, intransitive verbs often involve weather terms, involuntar... | Wikipedia - Intransitive verb - Examples |
In the following series λ {\displaystyle \lambda } is the difference of the longitude of an arbitrary point and the longitude of the chosen central meridian: λ {\displaystyle \lambda } is in radians and is positive east of the central meridian. The W coefficients are functions of ϕ {\displaystyle \phi } listed below. T... | Wikipedia - Transverse Mercator: Redfearn series - Direct series |
In the following simple stack implementation in Java, each element popped from the stack becomes semantic garbage once there are no outside references to it: This is because elements still contains a reference to the object, but the object will never be accessed again through this reference, because elements is private... | Wikipedia - Garbage (computer science) - Example |
In the following situations, the overarching purpose of stereotyping is for people to put their collective self (their in-group membership) in a positive light: when stereotypes are used for explaining social events when stereotypes are used for justifying activities of one's own group (ingroup) to another group (outgr... | Wikipedia - Stereotype - Social categorization |
In the following soft clustering example, the reference vector Y {\displaystyle Y\,} contains sample categories and the joint probability p ( X , Y ) {\displaystyle p(X,Y)\,} is assumed known. A soft cluster c k {\displaystyle c_{k}\,} is defined by its probability distribution over the data samples x i: p ( c k | x i ... | Wikipedia - Information bottleneck method - Clusters |
Line 2: Second matrix-valued set of conditional probabilities. By definition p ( y i | c k ) = ∑ j p ( y i | x j ) p ( x j | c k ) = ∑ j p ( y i | x j ) p ( x j , c k ) / p ( c k ) = ∑ j p ( y i | x j ) p ( c k | x j ) p ( x j ) / p ( c k ) {\displaystyle {\begin{aligned}p(y_{i}|c_{k})&=\sum _{j}p(y_{i}|x_{j})p(x_{j}|c... | Wikipedia - Information bottleneck method - Clusters |
Further inputs to the algorithm are the marginal sample distribution p ( x ) {\displaystyle p(x)\,} which has already been determined by the dominant eigenvector of P {\displaystyle P\,} and the matrix valued Kullback–Leibler divergence function D i , j K L = D K L ) {\displaystyle D_{i,j}^{KL}=D^{KL}{\Big }{\Big )}} ... | Wikipedia - Information bottleneck method - Clusters |
In the following some Elliptic Integral Singular Values are derived: | Wikipedia - Jacobi theta functions - Exemplary elliptic integrals |
In the following statements, the notation ' s → 0 {\displaystyle s\to 0} ' means that s {\displaystyle s} approaches 0, whereas ' s ↓ 0 {\displaystyle s\downarrow 0} ' means that s {\displaystyle s} approaches 0 through the positive numbers. | Wikipedia - Final value theorem - Deducing limt → ∞ f(t) |
In the following subsections we discuss versions of Itô's lemma for different types of stochastic processes. | Wikipedia - Itō's lemma - Mathematical formulation of Itô's lemma |
In the following subsections we will first compute the variation of the Einstein–Hilbert term and then the variation of the boundary term, and show that their sum results in δ S T O T A L = δ S E H + δ S G H Y = 1 16 π ∫ M G α β δ g α β − g d 4 x {\displaystyle \delta S_{TOTAL}=\delta S_{EH}+\delta S_{GHY}={\frac {1}{1... | Wikipedia - Gibbons–Hawking–York boundary term - On proving the main result |
In the following subsections, a selection of significant tests (by no means exhaustive) is listed, representative of the testing effort in each nuclear country. | Wikipedia - List of nuclear weapons tests - Known tests |
In the following summations, n P k {\displaystyle {}_{n}P_{k}} is the number of k-permutations of n. ∑ i = 0 n i P k ( n i ) = n P k ( 2 n − k ) {\displaystyle \sum _{i=0}^{n}{}_{i}P_{k}{n \choose i}={}_{n}P_{k}(2^{n-k})} ∑ i = 1 n i + k P k + 1 = ∑ i = 1 n ∏ j = 0 k ( i + j ) = ( n + k + 1 ) ! ( n − 1 ) ! ( k + 2 ) {\... | Wikipedia - Summation symbol - Involving permutation numbers |
(k+2)}}} ∑ i = 0 n i ! ⋅ ( n i ) = ∑ i = 0 n n P i = ⌊ n ! ⋅ e ⌋ , n ∈ Z + {\displaystyle \sum _{i=0}^{n}i!\cdot {n \choose i}=\sum _{i=0}^{n}{}_{n}P_{i}=\lfloor n!\cdot e\rfloor ,\quad n\in \mathbb {Z} ^{+}} , where and ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } denotes the floor function. | Wikipedia - Summation symbol - Involving permutation numbers |
In the following table N {\displaystyle N} is the number of points (charges) in a configuration, U Thom {\displaystyle U_{\textrm {Thom}}} is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and r i {\displaystyle r_{i}} are the positions of the charges. Most symmetr... | Wikipedia - Thomson Problem - Configurations of smallest known energy |
Thus, v i {\displaystyle v_{i}} is the number of vertices where the given number of edges meet, e {\displaystyle e} is the total number of edges, f 3 {\displaystyle f_{3}} is the number of triangular faces, f 4 {\displaystyle f_{4}} is the number of quadrilateral faces, and θ 1 {\displaystyle \theta _{1}} is the smalle... | Wikipedia - Thomson Problem - Configurations of smallest known energy |
In the following table and notes, "support" refers to official Microsoft support, as described above. | Wikipedia - Windows Virtual PC - Table of supported operating systems |
In the following table are ranked the 30 countries by the largest national net wealth from 2000 to 2022 according to UBS and Credit Suisse S.A. (August 2023). | Wikipedia - National wealth - Countries ranking through time |
In the following table are the decimal values of the Kaktovik digits up to three places to the left and to the right of the units' place. | Wikipedia - Kaktovik numerals - Values |
In the following table contains a list of codices and tractates in the Nag Hammadi library as given by Aleksandr Leonovich Khosroev. Abbreviations are from The Coptic Gnostic Library. The so-called "Codex XIII" is not a codex, but rather the text of Trimorphic Protennoia, written on "eight leaves removed from a thirtee... | Wikipedia - Nag Hammadi library - List of codices and tractates |
In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length. The list applies to the most general cases, and excludes named subsets. Note 1: The most general trapezoids and isosceles trape... | Wikipedia - Irregular quadrilateral - Properties of the diagonals in quadrilaterals |
In the following table men's giant slalom World Cup podiums from the World Cup first edition in 1967. | Wikipedia - Giant slalom - Men's World Cup podiums |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.