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S = h f L = 1 ρ g Δ p L . {\displaystyle S={\frac {h_{f}}{L}}={\frac {1}{\rho \mathrm {g} }}{\frac {\Delta p}{L}}.} where ρ = density of the fluid, (SI kg / m3) g = the local acceleration due to gravity; | Wikipedia - Friction loss - Calculating volumetric flow |
In the following discussions, we consider steady-state flow. The speed cannot thus change with time, and the component forces producing tangential acceleration need to sum up to zero. In other words, active and resistive forces must balance out in the forward direction in order that D V / D t = 0 {\displaystyle DV/Dt=0... | Wikipedia - Cyclostrophic balance - Steady-state assumption |
Moreover, trajectories and streamlines coincide in steady-state conditions, and the pairs of adjectives tangential/normal and streamwise/cross-stream become interchangeable. An atmospheric flow in which the tangential acceleration is not negligible is called allisobaric. The velocity direction can still change in space... | Wikipedia - Cyclostrophic balance - Steady-state assumption |
In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points q − p = ∂ P {\displaystyle \mathbf {q} -\mathbf {p} =\partial P} and integration along P is from p {\displaystyle \mathbf {p} } to q {\displaystyle \mathbf {q} }: ψ | ∂ P = ψ ( q ) − ψ ( p ) = ∫ P ∇ ψ ⋅ d ℓ {\disp... | Wikipedia - Vector calculus identities - Endpoint-curve integrals |
In the following equations f refers to the fundamental frequency, and 2f refers to the second-order distortion component frequencies. OSOIdBm = ISOIdBm+GdBPout,f,dBm = Pin,f,dBm - ISOIdBm + OSOIdBmPout,2f,dBm = 2Pin,f,dBm - 2ISOIdBm + OSOIdBmPin,2f,dBm = 2Pin,f,dBm - ISOIdBmPout,2f,dBm = 2Pout,f,dBm - OSOIdBm △ S O , d... | Wikipedia - Second-order intercept point - Useful second order equations |
In the following equations, all subscripts correspond to the different steps in the Fickett–Jacobs cycle as shown in Figure 2. In addition, a representation of the work done by the system and the external work applied on the system is shown is Figure 1. Initially, the work done to the system to begin a cycling detonati... | Wikipedia - Fickett–Jacobs cycle - Mathematical interpretation of the cycle's total work |
The fact that the mass of the explosive is M = ρ L A {\displaystyle M=\rho LA} , where ρ is the explosive's density, the equation above becomes W i = − P i u p ρ U C J {\displaystyle W_{i}=-{\frac {P_{i}u_{p}}{\rho U_{CJ}}}} The work done by the system (detonation) per unit mass of explosive is W 01 = 1 2 u p 2 {\displ... | Wikipedia - Fickett–Jacobs cycle - Mathematical interpretation of the cycle's total work |
In the following equations, s {\displaystyle s} denotes the sagitta (the depth or height of the arc), r {\displaystyle r} equals the radius of the circle, and l {\displaystyle l} the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides o... | Wikipedia - Sagitta (geometry) - Formulas |
This may be rearranged to give any of the other three: s = r − r 2 − 1 4 l 2 , l = 2 2 r s − s 2 , r = s 2 + 1 4 l 2 2 s = s 2 + l 2 8 s . {\displaystyle {\begin{aligned}s&=r-{\sqrt {r^{2}-{\tfrac {1}{4}}l^{2}}},\\l&=2{\sqrt {2rs-s^{2}}},\\r&={\frac {s^{2}+{\tfrac {1}{4}}l^{2}}{2s}}={\frac {s}{2}}+{\frac {l^{2}}{8s}}.\... | Wikipedia - Sagitta (geometry) - Formulas |
In the following example 45 voters rank 5 candidates. The pairwise preferences have to be computed first. For example, when comparing A and B pairwise, there are 5+5+3+7=20 voters who prefer A to B, and 8+2+7+8=25 voters who prefer B to A. So d = 20 {\displaystyle d=20} and d = 25 {\displaystyle d=25} . The full set ... | Wikipedia - Schulze method - Example |
There is no undisputed winner by only looking at the pairwise differences here. Now the strongest paths have to be identified. To help visualize the strongest paths, the set of pairwise preferences is depicted in the diagram on the right in the form of a directed graph. | Wikipedia - Schulze method - Example |
An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d > d (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red ... | Wikipedia - Schulze method - Example |
The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined. Now the output of the Schulze method can be determined. For example, when comparing A and B, since ( 28 ... | Wikipedia - Schulze method - Example |
In the following example expressed in C, a program has two variables which are adjacent in memory: an 8-byte-long string buffer, A, and a two-byte big-endian integer, B. Initially, A contains nothing but zero bytes, and B contains the number 1979. Now, the program attempts to store the null-terminated string "excessive... | Wikipedia - Buffer overflow - Example |
By failing to check the length of the string, it also overwrites the value of B: B's value has now been inadvertently replaced by a number formed from part of the character string. In this example "e" followed by a zero byte would become 25856. Writing data past the end of allocated memory can sometimes be detected by ... | Wikipedia - Buffer overflow - Example |
In the following example from Kuuk Thaayorre, the specific borrowed noun tin.meat 'tinned meat' is preceded by its generic classifier minh 'meat.' In the next example, the same classifier minh stands in on its own for a generic crocodile (punc), another member of the minh class: Classifiers and specific nouns in Kuuk T... | Wikipedia - Numeral classifier - Kuuk Thaayorre |
In the following example green edges are being evaluated by the algorithm and red edges have been deleted. | Wikipedia - Reverse-delete algorithm - Example |
In the following example green edges are used to form a MBST and dashed red areas indicate super vertices formed during the algorithm steps. | Wikipedia - Minimum bottleneck spanning tree - Example |
In the following example of conversion from non-standard to standard form, the notation 1 is used to represent the signed digit -1. 211.01φ is not a standard base-φ numeral, since it contains a "11" and additionally a "2" and a "1" = −1, which are not "0" or "1". To put a numeral in standard form, we may use the follow... | Wikipedia - Golden ratio base - Writing golden ratio base numbers in standard form |
The substitutions may be applied in any order we like, as the result will be the same. Below, the substitutions applied to the number on the previous line are on the right, the resulting number on the left. 211.0 1 _ 0 ϕ = 211 . | Wikipedia - Golden ratio base - Writing golden ratio base numbers in standard form |
1 _ 01 ϕ 0 1 _ 0 → 1 _ 01 = 210 .011 ϕ 1 1 _ 0 → 001 = 1011 .011 ϕ 200 → 1001 = 1100 .100 ϕ 011 → 100 = 10000 .1 ϕ 011 → 100 {\displaystyle {\begin{aligned}211.0{\underline {1}}0_{\phi }=211&. {\underline {1}}01_{\phi }&0{\underline {1}}0\rightarrow {\underline {1}}01\\=210&.011_{\phi }&1{\underline {1}}0\rightarrow 00... | Wikipedia - Golden ratio base - Writing golden ratio base numbers in standard form |
(The exception to this is when the first digit is negative one and the next two digits are one, like 1111.001=1.001.) This can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then marking it as negative. For example, use a minus sign, or some other signific... | Wikipedia - Golden ratio base - Writing golden ratio base numbers in standard form |
In the following example sentences, independent clauses are underlined, and conjunctions are in bold. Single independent clauses: I have enough money to buy an ice cream cone. My favourite flavour is chocolate. Let's go to the shop.Multiple independent clauses: I have enough money to buy an ice cream cone; my favourite... | Wikipedia - Matrix clause - Examples |
In the following example the function name foo and the variable name bar are both metasyntactic variables. Lines beginning with // are comments. | Wikipedia - Metasyntactic variable - C |
In the following example the result of the subtraction has fewer digits than x {\displaystyle x}: 123410 - 123401 Using the first method the sum of the nines' complement of x {\displaystyle x} and y {\displaystyle y} is 876589 + 123401 ———————— 999990 The nines' complement of 999990 is 000009. Removing the leading ... | Wikipedia - 9s complement - Magnitude of numbers |
In the following example there are seven tasks, labeled A through G. Some tasks can be done concurrently (A and B) while others cannot be done until their predecessor task is complete (C cannot begin until A is complete). Additionally, each task has three time estimates: the optimistic time estimate (o), the most likel... | Wikipedia - PERT chart - Example |
In the following example, "car" (車, kuruma) is the subject, and it is marked as the topic. The が that would normally be there to mark the subject has been replaced by は. The topic normally goes at the beginning of the clause. In the following example, "television" (テレビ, terebi) is the direct object, and it is marked as... | Wikipedia - Topic marker - Examples |
As before, the topic goes at the beginning of the clause. In this third example, "today" (今日, kyou) is used adverbially, and it is marked as the topic. Normally there would be no particle marking it as an adverb, and so は is simply added without replacing any particle. The subject, which is omitted, is assumed to be "I... | Wikipedia - Topic marker - Examples |
In the following example, "school" (Korean: 학교; Hanja: 學校; RR: hakkyo) is the subject, and it is marked as the topic. | Wikipedia - Topic marker - Example |
In the following example, Fc is the canonical cover of F. Given the following, we can find the canonical cover: R = (A, B, C, G, H, I) F = {A→BC, B→C, A→B, AB→C} {A→BC, B→C, A→B, AB→C} {A → BC, B →C, AB → C} {A → BC, B → C} {A → B, B →C}Fc = {A → B, B →C} | Wikipedia - Canonical cover - Canonical cover example |
In the following example, Tupaq is the subject, and it is marked as the topic. The topic normally goes at the beginning of the clause. | Wikipedia - Topic marker - Example |
In the following example, a Mediator object controls the values of several Storage objects, forcing the user code to access the stored values through the mediator. When a storage object wants to emit an event indicating that its value has changed, it also goes back to the mediator object (via the method notifyObservers... | Wikipedia - Mediator pattern - Java |
In the following example, also Dyalog, the first line assigns some HTML code to a variable txt and then uses an APL expression to remove all the HTML tags (explanation): | Wikipedia - Apl (command) - HTML tags removal |
In the following example, both the service provider and the identity provider use an HTTP POST binding. Initially, the service provider responds to a request from the user agent with a document containing an XHTML form: The value of the SAMLRequest parameter is the base64-encoding of a element, which is transmitted to ... | Wikipedia - SAML 2.0 - HTTP POST Binding |
In the following example, elemental aluminium reduces the oxide of another metal, in this common example iron oxide, because aluminium forms stronger and more stable bonds with oxygen than iron: Fe2O3 + 2 Al → 2 Fe + Al2O3The products are aluminium oxide, elemental iron, and a large amount of heat. The reactants are co... | Wikipedia - Thermite reaction - Chemical reactions |
In the following example, for each of ten "subjects" ( N {\displaystyle N} ) fourteen raters ( n {\displaystyle n} ), sampled from a larger group, assign a total of five categories ( k {\displaystyle k} ). The categories are presented in the columns, while the subjects are presented in the rows. Each cell lists the num... | Wikipedia - Fleiss' kappa - Worked example |
In the following example, graph G and graph H are homeomorphic. If G′ is the graph created by subdivision of the outer edges of G and H′ is the graph created by subdivision of the inner edge of H, then G′ and H′ have a similar graph drawing: Therefore, there exists an isomorphism between G' and H', meaning G and H are ... | Wikipedia - Subdivision (graph theory) - Example |
In the following example, graph H is a minor of graph G: H. G. The following diagram illustrates this. First construct a subgraph of G by deleting the dashed edges (and the resulting isolated vertex), and then contract the gray edge (merging the two vertices it connects): | Wikipedia - Topological minor - Example |
In the following example, statement b explicitly negates statement a: Statements can also be mutually exclusive, without explicitly negating each other as in the following example: | Wikipedia - Correlative-based fallacies - Examples |
In the following example, the four bytes represent a KLV set where the key is one byte, the length field is one byte (or possibly BER - you cannot tell from the example), and the value is two bytes: a zero and a three. In your application you would have previously agreed to a) use one-byte keys and b) use one-byte leng... | Wikipedia - KLV - Example |
In the following example, the lexical entry is associated with a lemma clergyman and two inflected forms clergyman and clergymen. The language coding is set for the whole lexical resource. The language value is set for the whole lexicon as shown in the following UML instance diagram. The elements Lexical Resource, Glob... | Wikipedia - Lexical Markup Framework - Example |
They are specified within the LMF document. On the contrary, languageCoding, language, partOfSpeech, commonNoun, writtenForm, grammaticalNumber, singular, plural are data categories that are taken from the Data Category Registry. These marks adorn the structure. | Wikipedia - Lexical Markup Framework - Example |
The values ISO 639-3, clergyman, clergymen are plain character strings. The value eng is taken from the list of languages as defined by ISO 639-3. With some additional information like dtdVersion and feat, the same data can be expressed by the following XML fragment: This example is rather simple, while LMF can represe... | Wikipedia - Lexical Markup Framework - Example |
In the following example, the set of PBEs is strictly smaller than the set of SPEs and BNEs. It is a variant of the above gift-game, with the following change to the receiver's utility: If the sender is a friend, then the receiver's utility is 1 (if they accept) or 0 (if they reject). If the sender is an enemy, then th... | Wikipedia - Perfect Bayesian equilibrium - Gift game 2 |
Let's look at the following potential pooling equilibria: The sender's strategy is: always give. The receiver's beliefs are not updated: they still believe in the a-priori probability, that the sender is a friend with probability p {\displaystyle p} and an enemy with probability 1 − p {\displaystyle 1-p} . Their payoff... | Wikipedia - Perfect Bayesian equilibrium - Gift game 2 |
This is a PBE - it is a best-response for both sender and receiver. The sender's strategy is: never give. Suppose the receiver's beliefs when receiving a gift is that the sender is a friend with probability q {\displaystyle q} , where q {\displaystyle q} is any number in {\displaystyle } . | Wikipedia - Perfect Bayesian equilibrium - Gift game 2 |
Regardless of q {\displaystyle q} , the receiver's optimal strategy is: accept. This is NOT a PBE, since the sender can improve their payoff from 0 to 1 by giving a gift. The sender's strategy is: never give, and the receiver's strategy is: reject. | Wikipedia - Perfect Bayesian equilibrium - Gift game 2 |
This is NOT a PBE, since for any belief of the receiver, rejecting is not a best-response.Note that option 3 is a Nash equilibrium! If we ignore beliefs, then rejecting can be considered a best-response for the receiver, since it does not affect their payoff (since there is no gift anyway). Moreover, option 3 is even a... | Wikipedia - Perfect Bayesian equilibrium - Gift game 2 |
Such implausible equilibria might arise also in games with complete information, but they may be eliminated by applying subgame perfect Nash equilibrium. However, Bayesian games often contain non-singleton information sets and since subgames must contain complete information sets, sometimes there is only one subgame—th... | Wikipedia - Perfect Bayesian equilibrium - Gift game 2 |
In the following example, the text "Paul Schuster was born in Dresden" on a website will be annotated, connecting a person with their place of birth. The following HTML fragment shows how a small graph is being described, in RDFa-syntax using a schema.org vocabulary and a Wikidata ID: The example defines the following ... | Wikipedia - Data Web - Example |
The triples result in the graph shown in the given figure. One of the advantages of using Uniform Resource Identifiers (URIs) is that they can be dereferenced using the HTTP protocol. According to the so-called Linked Open Data principles, such a dereferenced URI should result in a document that offers further data abo... | Wikipedia - Data Web - Example |
In this example, all URIs, both for edges and nodes (e.g. http://schema.org/Person, http://schema.org/birthPlace, http://www.wikidata.org/entity/Q1731) can be dereferenced and will result in further RDF graphs, describing the URI, e.g. that Dresden is a city in Germany, or that a person, in the sense of that URI, can b... | Wikipedia - Data Web - Example |
In the following example, the total vote is 100,000. There are 10 seats. The number at each cell in the "pink" table denotes the number of votes divided by the corresponding divisor d ( k ) {\displaystyle d(k)} . For example, for D'Hondt's method, in the row k = 0 {\displaystyle k=0} , the numbers are just the parties'... | Wikipedia - Highest averages method - Comparative example |
In the row k = 1 {\displaystyle k=1} , the numbers are the votes divided by 2. For the Saint-Lague method, in the row k = 1 {\displaystyle k=1} , the numbers are the votes divided by 3 (the second element in the sequence of divisors), and so on. | Wikipedia - Highest averages method - Comparative example |
As can be seen in the example, D'Hondt, Sainte-Laguë and Huntington-Hill allow different strategies by parties looking to maximize their seat allocation. D'Hondt and Huntington–Hill can favor the merging of parties, while Sainte-Laguë favors parties with one seat. In these examples, under D'Hondt and Huntington–Hill th... | Wikipedia - Highest averages method - Comparative example |
In the following example, the vector p does not escape into g, so it can be allocated on the stack and then removed from the stack before calling g. If, however, we had then either p would need to be allocated on the heap or (if g is known to the compiler when f is compiled, and behaves well) allocated on the stack in ... | Wikipedia - Escape analysis - Examples (Scheme) |
In the following example, there is a common Swedish surname Åström written in the two alternative methods, the first one with a precomposed Å (U+00C5) and ö (U+00F6), and the second one using a decomposed base letter A (U+0041) with a combining ring above (U+030A) and an o (U+006F) with a combining diaeresis (U+0308). ... | Wikipedia - Precomposed character - Comparing precomposed and decomposed characters |
To overcome the problems, some applications may simply attempt to replace the decomposed characters with the equivalent precomposed characters. With an incomplete font, however, precomposed characters may also be problematic – especially if they are more exotic, as in the following example (showing the reconstructed Pr... | Wikipedia - Precomposed character - Comparing precomposed and decomposed characters |
In the following example, we are retrieving invoices and sending them to a collection agency if they are overdue, notices have been sent, and they are not already with the collection agency.This example is meant to show the end result of how the logic is 'chained' together. This usage example assumes a previously defin... | Wikipedia - Specification pattern - Example of use |
In the following example, we use the keys JANEAUSTEN and AEROPLANES to encrypt the following plaintext: "Transposition ciphers scramble letters like puzzle pieces to create an indecipherable arrangement." The colors show how the letters are scrambled in each transposition step. While a single step only causes a minor r... | Wikipedia - Permutation cipher - Visual demonstration of double transposition |
In the following examples F {\displaystyle F} is a field, and C , R , Q {\displaystyle \mathbb {C} ,\mathbb {R} ,\mathbb {Q} } are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F. | Wikipedia - Galois groups - Examples |
In the following examples indentation and formatting are critical for parsing the code: expressions are terminated by the end of the line, lists of expressions need to be on the same level of indentation. This feature, named the off-side rule, is also found in other languages such as Haskell and Python. Communication b... | Wikipedia - Occam programming language - Overview |
while another one inputs data with ?. Input and output cannot proceed until the other end is ready to accept or offer data. (In the not proceeding case it is often said that the process blocks on the channel. | Wikipedia - Occam programming language - Overview |
However, the program will neither spin nor poll; thus terms like wait, hang or yield may also convey the behaviour; also in the context that it will not block other independent processes from running.) Examples (c is a variable): keyboard ? | Wikipedia - Occam programming language - Overview |
c screen ! c SEQ introduces a list of expressions that are evaluated sequentially. This is not implicit as it is in most other programming languages. | Wikipedia - Occam programming language - Overview |
Example: SEQ x := x + 1 y := x * x PAR begins a list of expressions that may be evaluated concurrently. Example: PAR p() q() ALT specifies a list of guarded commands. The guards are a combination of a boolean condition and an input expression, both optional. | Wikipedia - Occam programming language - Overview |
Each guard for which the condition is true and the input channel is ready is successful. One of the successful alternatives is selected for execution. Example: ALT count1 < 100 & c1 ? | Wikipedia - Occam programming language - Overview |
data SEQ count1 := count1 + 1 merged ! data count2 < 100 & c2 ? data SEQ count2 := count2 + 1 merged ! | Wikipedia - Occam programming language - Overview |
data status ? request SEQ out ! count1 out ! | Wikipedia - Occam programming language - Overview |
count2 This will read data from channels c1 or c2 (whichever is ready) and pass it into a merged channel. If countN reaches 100, reads from the corresponding channel will be disabled. A request on the status channel is answered by outputting the counts to out. | Wikipedia - Occam programming language - Overview |
In the following examples the values shown above each figure should be m, n1, n2 and n3. A GNU Octave program for generating these figures | Wikipedia - Superformula - 2D plots |
In the following examples you can observe a normal step sequence and a double step sequence. The used rotors are (from left to right) I, II, III, with turnovers on Q, E and V. It is the right rotor's behavior we observe here (turnover V). Normal sequence:AAU — normal step of right rotor AAV — right rotor (III) goes in ... | Wikipedia - Wheels of the Enigma - Normalized Enigma sequences |
In the following examples, a Gaulish toponym was replaced by the name of the local tribe, according to a process well known in the later Roman Empire: Bayeux: rooted in the civitas named Bajocassensis; which was formerly known as Augustodurum, meaning "forum dedicated to Augustus". Evreux: rooted in the civitas named E... | Wikipedia - Norman toponymy - AD era |
Lillebonne: derived from Juliobona, where Julio meant "dedicated to Julius", and bona, from the Old Celtic, meant "foundation" or "spring". (Also see Ratisbona or Vindobona).Some of these would disappear later, replaced by Normanic names; thus Coriovallum became Cherbourg and Caracotinum became Harfleur. Such changes i... | Wikipedia - Norman toponymy - AD era |
In the following examples, a common-value auction is modeled as a Bayesian game. We try to find a Bayesian Nash equilibrium (BNE), which is a function from the information held by a player, to the bid of that player. We focus on a symmetric BNE (SBNE), in which all bidders use the same function. | Wikipedia - Common value auction - Examples |
In the following examples, computed values are in bold, while Register numbers are not.For example, to write the value 3 to register 1, (which already contains a 6), and then add 7 to register 1 and store the result in register 2, i.e.: i0: R1 = 6 i1: R1 = 3 i2: R2 = R1 + 7 = 10 Following execution, register 2 should c... | Wikipedia - Pipeline flush - Examples |
Forwarding (described below) helps correct such errors by depending on the fact that the output of i1 (which is 3) can be used by subsequent instructions before the value 3 is committed to/stored in Register 1. Forwarding applied to the example means that there is no wait to commit/store the output of i1 in Register 1 ... | Wikipedia - Pipeline flush - Examples |
The effect is that i2 uses the correct (the more recent) value of Register 1: the commit/store was made immediately and not pipelined. With forwarding enabled, the Instruction Decode/Execution (ID/EX) stage of the pipeline now has two inputs: the value read from the register specified (in this example, the value 6 from... | Wikipedia - Pipeline flush - Examples |
In the following examples, one or more of the assumptions of a strictly idealised population are relaxed, while other assumptions are retained. The variance effective population size of the more relaxed population model is then calculated with respect to the strict model. | Wikipedia - Effective population - Theoretical examples |
In the following examples, the active and passive voice are illustrated with pairs of sentences using the same transitive verb. | Wikipedia - Active Voice - Examples |
In the following examples, the natural numbers refer to the set of positive integers. The equation x = ( y 1 + 1 ) ( y 2 + 1 ) {\displaystyle x=(y_{1}+1)(y_{2}+1)} is an example of a Diophantine equation with a parameter x and unknowns y1 and y2. The equation has a solution in y1 and y2 precisely when x can be expresse... | Wikipedia - Matiyasevich's theorem - Examples |
Other examples of Diophantine definitions are as follows: The equation x = y 1 2 + y 2 2 {\displaystyle x=y_{1}^{2}+y_{2}^{2}} with parameter x and unknowns y1, y2 only has solutions in N {\displaystyle \mathbb {N} } when x is a sum of two perfect squares. The Diophantine set of the equation is {2, 5, 8, 10, 13, 17, 18... | Wikipedia - Matiyasevich's theorem - Examples |
This is a Pell equation, meaning it only has solutions in N {\displaystyle \mathbb {N} } when x is not a perfect square. The Diophantine set is {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, ...}. The equation x 1 + y = x 2 {\displaystyle x_{1}+y=x_{2}} is a Diophantine equation with two parameters x1, x2 and an unknown y, which d... | Wikipedia - Matiyasevich's theorem - Examples |
In the following examples, the set X is the real interval , and Σ {\displaystyle \Sigma } is the Borel sigma-algebra on X. μ {\displaystyle \mu } is the length measure on X. ν {\displaystyle \nu } assigns to each subset Y of X, twice the length of Y. Then, d ν d μ = 2 {\textstyle {\frac {d\nu }{d\mu }}=2} . μ {\display... | Wikipedia - Radon−Nikodym derivative - Examples |
In the following examples, the use of the distributive law on the set of real numbers R {\displaystyle \mathbb {R} } is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a field, which ensur... | Wikipedia - Distributive law - Real numbers |
Written multiplication is also based on the distributive law. Second example (with variables) Third example (with two sums) Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out. Fourth exampleHere the distributive law is applied the other way around compared to the p... | Wikipedia - Distributive law - Real numbers |
In the following examples, there are two commodities, 1 and 2. 1. The utility function has the Cobb–Douglas form: u ( x 1 , x 2 ) = x 1 α x 2 β . {\displaystyle u(x_{1},x_{2})=x_{1}^{\alpha }x_{2}^{\beta }.} | Wikipedia - Marshallian demand function - Examples |
The constrained optimization leads to the Marshallian demand function: x ∗ ( p 1 , p 2 , I ) = ( α I ( α + β ) p 1 , β I ( α + β ) p 2 ) . {\displaystyle x^{*}(p_{1},p_{2},I)=\left({\frac {\alpha I}{(\alpha +\beta )p_{1}}},{\frac {\beta I}{(\alpha +\beta )p_{2}}}\right).} 2. | Wikipedia - Marshallian demand function - Examples |
The utility function is a CES utility function: u ( x 1 , x 2 ) = 1 δ . {\displaystyle u(x_{1},x_{2})=\left^{\frac {1}{\delta }}.} Then x ∗ ( p 1 , p 2 , I ) = ( I p 1 ϵ − 1 p 1 ϵ + p 2 ϵ , I p 2 ϵ − 1 p 1 ϵ + p 2 ϵ ) , with ϵ = δ δ − 1 . | Wikipedia - Marshallian demand function - Examples |
{\displaystyle x^{*}(p_{1},p_{2},I)=\left({\frac {Ip_{1}^{\epsilon -1}}{p_{1}^{\epsilon }+p_{2}^{\epsilon }}},{\frac {Ip_{2}^{\epsilon -1}}{p_{1}^{\epsilon }+p_{2}^{\epsilon }}}\right),\quad {\text{with}}\quad \epsilon ={\frac {\delta }{\delta -1}}.} In both cases, the preferences are strictly convex, the demand is uni... | Wikipedia - Marshallian demand function - Examples |
The utility function has the linear form: u ( x 1 , x 2 ) = x 1 + x 2 . {\displaystyle u(x_{1},x_{2})=x_{1}+x_{2}.} The utility function is only weakly convex, and indeed the demand is not unique: when p 1 = p 2 {\displaystyle p_{1}=p_{2}} , the consumer may divide his income in arbitrary ratios between product types 1... | Wikipedia - Marshallian demand function - Examples |
4. The utility function exhibits a non-diminishing marginal rate of substitution: u ( x 1 , x 2 ) = ( x 1 α + x 2 α ) , with α > 1. {\displaystyle u(x_{1},x_{2})=(x_{1}^{\alpha }+x_{2}^{\alpha }),\quad {\text{with}}\quad \alpha >1.} The utility function is not convex, and indeed the demand is not continuous: when p 1 <... | Wikipedia - Marshallian demand function - Examples |
In the following examples: let F, G, H be predicate letters; let a, b, c be individual constants; let x, y, z be variables. | Wikipedia - Atomic sentence - Assumptions |
In the following figure, the graph C is a covering graph of the graph H. The covering map f from C to H is indicated with the colours. For example, both blue vertices of C are mapped to the blue vertex of H. The map f is a surjection: each vertex of H has a preimage in C. Furthermore, f maps bijectively each neighbourh... | Wikipedia - Covering graph - Examples |
In the following figure, the minimization problem on the left side of the equation is illustrated. One seeks to vary x such that the vertical distance between the convex and concave curves at x is as small as possible. The position of the vertical line in the figure is the (approximate) optimum. The next figure illustr... | Wikipedia - Fenchel duality - One-dimensional illustration |
Tangents are drawn to each of the two curves such that both tangents have the same slope p. The problem is to adjust p in such a way that the two tangents are as far away from each other as possible (more precisely, such that the points where they intersect the y-axis are as far from each other as possible). Imagine th... | Wikipedia - Fenchel duality - One-dimensional illustration |
In the following formulas Gi denotes a group contribution. Gi are counted for every single available group. If a group is present multiple times, each occurrence is counted separately. | Wikipedia - Joback method - Formulas |
In the following formulas, erf is the error function and Ei is the exponential integral. | Wikipedia - List of integrals of exponential functions - Integrals involving the error function |
In the following game, the first of the two players is using red, and the second is using blue. The last move in each image is shown brighter than the other areas. The starting graph: The first player may colour any of the areas to begin. However, the region around the outside of the graph is not included as an area fo... | Wikipedia - Col (game) - Example game |
After the first move: The second player now colours a white cell. As no areas are currently blue, any white cell is allowed. | Wikipedia - Col (game) - Example game |
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