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eb0866709c7a4ac1b68e72f7fd6ee6f807053f60 | the mathematical concept of a function dates from | the mathematical concept of a function dates from the 17th century in connection with the development of the calculus; for example, the slope d y / d x of a graph at a point was regarded as a function of the x-coordinate of the point. functions were not explicitly considered in antiquity, but some precursors of the con... | wikipedia |
a9582031a28074db499afd07e1574b28b7f90077 | the reason for the disappearance of the words | the reason for the disappearance of the words "propositional function" e.g., in suppes (1960), and halmos (1970), is explained by tarski (1946) together with further explanation of the terminology: | wikipedia |
edc264ab012e4b784ddddd99e7f2f9a4804f99f5 | suppes (1960) in axiomatic set theory, formally defines | suppes (1960) in axiomatic set theory, formally defines a relation (p. 57) as a set of pairs, and a function (p. 86) as a relation where no two pairs have the same first member. | wikipedia |
e42164330644988b3416173b94ed7b6240aa1eb4 | in 1954, bourbaki, on p. 76 in chapitre | in 1954, bourbaki, on p. 76 in chapitre ii of theorie des ensembles (theory of sets), gave a definition of a function as a triple f = (f, a, b). here f is a functional graph, meaning a set of pairs where no two pairs have the same first member. on p. 77 (op. cit.) bourbaki states (literal translation): "often we shall ... | wikipedia |
b4cac348fd795115a4995a157add2a6d4bf64f49 | kleene (1952) defines the words as follows: "in | kleene (1952) defines the words as follows: "in word languages, a proposition is expressed by a sentence. then a 'predicate' is expressed by an incomplete sentence or sentence skeleton containing an open place. for example, "___ is a man" expresses a predicate... the predicate is a propositional function of one variabl... | wikipedia |
5a0140004c83ccee2ff1348974aa062754f471de | while the reader of suppes (1960) axiomatic set | while the reader of suppes (1960) axiomatic set theory or halmos (1970) naive set theory observes the use of function-symbolism in the axiom of separation, e.g., φ(x) (in suppes) and s(x) (in halmos), they will see no mention of "proposition" or even "first order predicate calculus". in their place are " expressions of... | wikipedia |
4920ffbc0a1b97ab32cb9cde1b7599eaa9520b17 | both axiomatic and naive forms of zermelo's set | both axiomatic and naive forms of zermelo's set theory as modified by fraenkel (1922) and skolem (1922) define "function" as a relation, define a relation as a set of ordered pairs, and define an ordered pair as a set of two "dissymetric" sets. | wikipedia |
14a5f88f621049400b7f8a234fa91fdcb513c04c | "let e and f be two sets, which | "let e and f be two sets, which may or may not be distinct. a relation between a variable element x of e and a variable element y of f is called a functional relation in y if, for all x ∈ e, there exists a unique y ∈ f which is in the given relation with x.we give the name of function to the operation which in this way... | wikipedia |
8d99658ff37df062e542dac24c9147f4f89f6b80 | suppes observes that von neumann's axiomatization was modified | suppes observes that von neumann's axiomatization was modified by bernays "in order to remain nearer to the original zermelo system... he introduced two membership relations: one between sets, and one between sets and classes". then gödel further modified the theory: "his primitive notions are those of set, class and m... | wikipedia |
2cf6ccb9bea6a77dc82c4fb039806ff714bf1056 | at the outset he begins with i-objects and | at the outset he begins with i-objects and ii-objects, two objects a and b that are i-objects (first axiom), and two types of "operations" that assume ordering as a structural property obtained of the resulting objects and (x, y). the two "domains of objects" are called "arguments" (i-objects) and "functions" (ii-objec... | wikipedia |
8524bba34d1271dfea67a57ffd0717e768865017 | by 1925 abraham fraenkel (1922) and thoralf skolem | by 1925 abraham fraenkel (1922) and thoralf skolem (1922) had amended zermelo's set theory of 1908. but von neumann was not convinced that this axiomatization could not lead to the antinomies. so he proposed his own theory, his 1925 an axiomatization of set theory. it explicitly contains a "contemporary", set-theoretic... | wikipedia |
e9a238eccb903116755f5d29eb9b83b9978399e0 | according to willard quine, schönfinkel 1924 "provide for... | according to willard quine, schönfinkel 1924 "provide for... the whole sweep of abstract set theory. the crux of the matter is that schönfinkel lets functions stand as arguments. for schönfinkel, substantially as for frege, classes are special sorts of functions. they are propositional functions, functions whose values... | wikipedia |
ff655997cad71ce9ff547bf523c85880f4250297 | where exactly the general notion of "function" as | where exactly the general notion of "function" as a many-one correspondence derives from is unclear. russell in his 1920 introduction to mathematical philosophy states that "it should be observed that all mathematical functions result form one-many relations... functions in this sense are descriptive functions". a reas... | wikipedia |
3a358d940c327bb1a88455b91fcfa3ba1a4b0f85 | observe that while wiener "reduced" the relational *12.11 | observe that while wiener "reduced" the relational *12.11 form of the axiom of reducibility he did not reduce nor otherwise change the propositional-function form *12.1; indeed he declared this "essential to the treatment of identity, descriptions, classes and relations". | wikipedia |
0ce47ef283d64c95813f8c8c2f1249181662f4a7 | an attempt to solve the problem of the | an attempt to solve the problem of the antinomies led russell to propose his "doctrine of types" in an appendix b of his 1903 the principles of mathematics. in a few years he would refine this notion and propose in his 1908 the theory of types two axioms of reducibility, the purpose of which were to reduce (single-vari... | wikipedia |
ef4997d778c309e7df85b85da43365a8ad42ad95 | by 1910–1913 and principia mathematica russell had given | by 1910–1913 and principia mathematica russell had given up on the requirement for an intensional definition of a relation, stating that "mathematics is always concerned with extensions rather than intensions" and "relations, like classes, are to be taken in extension ". to demonstrate the notion of a relation in exten... | wikipedia |
e2aa368690a2a69d1094a35f081258dc90f79258 | the history of the notion of " ordered | the history of the notion of " ordered pair " is not clear. as noted above, frege (1879) proposed an intuitive ordering in his definition of a two-argument function Ψ(a, b). norbert wiener in his 1914 (see below) observes that his own treatment essentially "revert(s) to schröder's treatment of a relation as a class of ... | wikipedia |
dd9363adf1769fdb2857ab8873a711dc77cb2e6a | in this quote the reader may observe a | in this quote the reader may observe a shift in terminology: nowhere is mentioned the notion of "propositional function", but rather one sees the words "formula", "predicate calculus", "predicate", and "logical calculus." this shift in terminology is discussed more in the section that covers "function" in contemporary ... | wikipedia |
bc2439b0228fb32edbdbe47cde039aff37ed9a6a | as there is no universal set — sets | as there is no universal set — sets originate by way of axiom ii from elements of (non-set) domain b – "...this disposes of the russell antinomy so far as we are concerned". but zermelo's "definite criterion" is imprecise, and is fixed by weyl, fraenkel, skolem, and von neumann. | wikipedia |
0641ef35bf7b73b082f71341998b423865fbbb23 | the notion of "function" appears as zermelo's axiom | the notion of "function" appears as zermelo's axiom iii—the axiom of separation (axiom der aussonderung). this axiom constrains us to use a propositional function Φ(x) to "separate" a subset m from a previously formed set m: | wikipedia |
0fb16888c98b69aae0c058a158cb2ef9ff6a2e75 | in 1902 russell sent a letter to frege | in 1902 russell sent a letter to frege pointing out that frege's 1879 begriffsschrift allowed a function to be an argument of itself: "on the other hand, it may also be that the argument is determinate and the function indeterminate...." from this unconstrained situation russell was able to form a paradox: | wikipedia |
70e12758099aaa4683dca06be106bc7c46fe4066 | set theory began with the work of the | set theory began with the work of the logicians with the notion of "class" (modern "set") for example de morgan (1847), jevons (1880), venn (1881), frege (1879) and peano (1889). it was given a push by georg cantor 's attempt to define the infinite in set-theoretic treatment (1870–1890) and a subsequent discovery of an... | wikipedia |
c51018669d06c4d898e4fcf5190d1c3e94744c84 | recursion theory and computability: but the unexpected outcome | recursion theory and computability: but the unexpected outcome of hilbert's and his student bernays 's effort was failure; see gödel's incompleteness theorems of 1931. at about the same time, in an effort to solve hilbert's entscheidungsproblem, mathematicians set about to define what was meant by an "effectively calcu... | wikipedia |
79c03bb4187490a88215b48a048837015ba115e9 | hilbert then illustrates the three ways how the | hilbert then illustrates the three ways how the ε-function is to be used, firstly as the "for all" and "there exists" notions, secondly to represent the "object of which holds", and lastly how to cast it into the choice function. | wikipedia |
3efe6cd3a98aadc29d434325e92081f32232ef47 | david hilbert set himself the goal of "formalizing" | david hilbert set himself the goal of "formalizing" classical mathematics "as a formal axiomatic theory, and this theory shall be proved to be consistent, i.e., free from contradiction". in hilbert 1927 the foundations of mathematics he frames the notion of function in terms of the existence of an "object": | wikipedia |
c3d94dbd44a06c6c789a4ba9a4ff1ac27699289d | russell symbolizes the descriptive function as "the object | russell symbolizes the descriptive function as "the object standing in relation to y ": r'y = (ιx)(x r y). russell repeats that " r'y is a function of y, but not a propositional function; we shall call it a descriptive function. all the ordinary functions of mathematics are of this kind. thus in our notation "sin y " w... | wikipedia |
f456de18ae2fb347cb0ab7b23fc32eb6243f65e3 | the notion of a "many-one" functional relation": russell | the notion of a "many-one" functional relation": russell first discusses the notion of "identity", then defines a descriptive function (pages 30ff) as the unique value ιx that satisfies the (2-variable) propositional function (i.e., "relation") φŷ. | wikipedia |
5ef8731fc9c22bbe98fab3db08c687e954340026 | russell defines functions of propositions with arguments, and | russell defines functions of propositions with arguments, and truth-functions f (p). for example, suppose one were to form the "function of propositions with arguments" p: "not(p) and q " and assign its variables the values of p: "bob is hurt" and q: "this bird is hurt". (we are restricted to the logical linkages not, ... | wikipedia |
166b038f1bce6a80f67a41d9faab57175b8ececa | to continue the example: suppose (from outside the | to continue the example: suppose (from outside the mathematics/logic) one determines that the propositions "bob is hurt" has a truth value of "falsity", "this bird is hurt" has a truth value of "truth", "emily the rabbit is hurt" has an indeterminate truth value because "emily the rabbit" doesn't exist, and " y is hurt... | wikipedia |
3653e9669ad228a3d25d4fbbbc3f45be99103b3b | propositional functions: because his terminology is different from | propositional functions: because his terminology is different from the contemporary, the reader may be confused by russell's "propositional function". an example may help. russell writes a propositional function in its raw form, e.g., as φŷ: " ŷ is hurt". (observe the circumflex or "hat" over the variable y). for our e... | wikipedia |
bab1be92612f7151facc524bfecc19ae5e8dc3a4 | russell would carry his ideas forward in his | russell would carry his ideas forward in his 1908 mathematical logical as based on the theory of types and into his and whitehead's 1910–1913 principia mathematica. by the time of principia mathematica russell, like frege, considered the propositional function fundamental: "propositional functions are the fundamental k... | wikipedia |
54b816facdc13e4735ad5d077ce56d1d6ef9f744 | as expressed by russell "the process of transforming | as expressed by russell "the process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition... so long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as... | wikipedia |
89a4911a95826c1bb9a444357d39f5e5fce1bcad | for russell the bedeviling notion is that of | for russell the bedeviling notion is that of variable: "6. mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain variables. the notion of the variable is one of the most difficult with which logic has to deal. for the present, i openly wis... | wikipedia |
cefd680c88bd05f66ebb9877034aeb024eddc641 | while the influence of cantor and peano was | while the influence of cantor and peano was paramount, in appendix a "the logical and arithmetical doctrines of frege" of the principles of mathematics, russell arrives at a discussion of frege's notion of function, "...a point in which frege's work is very important, and requires careful examination". in response to h... | wikipedia |
9abba9fd0c14137d16150c69d755078b40895fa0 | peano defined the notion of "function" in a | peano defined the notion of "function" in a manner somewhat similar to frege, but without the precision. first peano defines the sign "k means class, or aggregate of objects", the objects of which satisfy three simple equality-conditions, a = a, (a = b) = (b = a), if ((a = b) and (b = c)) then (a = c). he then introduc... | wikipedia |
efcabd686fda3da5fa6e1cfd166027e8b3241ada | the one-argument function frege generalizes into the form | the one-argument function frege generalizes into the form Φ(a) where a is the argument and Φ() represents the function, whereas the two-argument function he symbolizes as Ψ(a, b) with a and b the arguments and Ψ(,) the function and cautions that "in general Ψ(a, b) differs from Ψ(b, a)". using his unique symbolism he t... | wikipedia |
c85d865ea448a5f19380ec8e644cbb92bf6f09a9 | frege calls the argument of the function "he | frege calls the argument of the function "he sign, regarded as replaceable by others that denotes the object standing in these relations". he notes that we could have derived the function as "hydrogen is lighter than...." as well, with an argument position on the right; the exact observation is made by peano (see more ... | wikipedia |
38cb01347a7aa909d9577fbd2cab2d8f69797008 | frege begins his discussion of "function" with an | frege begins his discussion of "function" with an example: begin with the expression "hydrogen is lighter than carbon dioxide". now remove the sign for hydrogen (i.e., the word "hydrogen") and replace it with the sign for oxygen (i.e., the word "oxygen"); this makes a second statement. do this again (using either state... | wikipedia |
7ce9fc405902f484e77d51d7249d67ad6dc2ae46 | at the outset frege abandons the traditional "concepts | at the outset frege abandons the traditional "concepts subject and predicate ", replacing them with argument and function respectively, which he believes "will stand the test of time. it is easy to see how regarding a content as a function of an argument leads to the formation of concepts. furthermore, the demonstratio... | wikipedia |
01322379219f190c07fefa6bd1fb70fd31735a12 | gottlob frege 's begriffsschrift (1879) preceded giuseppe peano | gottlob frege 's begriffsschrift (1879) preceded giuseppe peano (1889), but peano had no knowledge of frege 1879 until after he had published his 1889. both writers strongly influenced russell (1903). russell in turn influenced much of 20th-century mathematics and logic through his principia mathematica (1913) jointly ... | wikipedia |
38804998196fd5defa1ca922509e102e136bceba | in his 1881 symbolic logic venn was using | in his 1881 symbolic logic venn was using the words "logical function" and the contemporary symbolism (x = f (y), y = f (x), cf page xxi) plus the circle-diagrams historically associated with venn to describe "class relations", the notions "'quantifying' our predicate", "propositions in respect of their extension", "th... | wikipedia |
b2fe8f76fb01357daea830d6a6f06e98eec7141b | boole then used algebraic expressions to define both | boole then used algebraic expressions to define both algebraic and logical notions, e.g., 1 − x is logical not(x), xy is the logical and(x, y), x + y is the logical or(x, y), x (x + y) is xx + xy, and "the special law" xx = x = x. | wikipedia |
372ce6b93d5ad8d34102a57f4074edac5d8c98a4 | the second group of logicians, the set-theorists, emerged | the second group of logicians, the set-theorists, emerged with georg cantor 's "set theory" (1870–1890) but were driven forward partly as a result of russell's discovery of a paradox that could be derived from frege's conception of "function", but also as a reaction against russell's proposed solution. zermelo 's set-t... | wikipedia |
feed425d6970ccc306eb9a4ea77aa6f40340fd17 | eves observes "that logicians have endeavored to push | eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of sets, or classes, from a foundation in the logic of propositions and propositional functions". but by the late 19th century the logicians' research into the fo... | wikipedia |
5e23d386e96ca34998771874f9b9ba3e65fd45a7 | in his 1848 the nature of logic boole | in his 1848 the nature of logic boole asserts that "logic... is in a more especial sense the science of reasoning by signs", and he briefly discusses the notions of "belonging to" and "class": "an individual may possess a great variety of attributes and thus belonging to a great variety of different classes". like de m... | wikipedia |
a5af515579df67e801913595110d5105e398c923 | de morgan's 1847 "formal logic or, the calculus | de morgan's 1847 "formal logic or, the calculus of inference, necessary and probable" observes that " logical truth depends upon the structure of the statement, and not upon the particular matters spoken of"; he wastes no time (preface page i) abstracting: "in the form of the proposition, the copula is made as abstract... | wikipedia |
31cf74c4d4042a902e50976d7ca67994e3fd47c3 | logicians of this time were primarily involved with | logicians of this time were primarily involved with analyzing syllogisms (the 2000-year-old aristotelian forms and otherwise), or as augustus de morgan (1847) stated it: "the examination of that part of reasoning which depends upon the manner in which inferences are formed,and the investigation of general maxims and ru... | wikipedia |
f34f765835d006bc5092e020d2c71ddbcd1511f5 | hardy 1908, pp. 26–28 defined a function as | hardy 1908, pp. 26–28 defined a function as a relation between two variables x and y such that "to some values of x at any rate correspond values of y." he neither required the function to be defined for all values of x nor to associate each value of x to a single value of y. this broad definition of a function encompa... | wikipedia |
df2662c9302b76bcd92fd670e4bfe051215da4b5 | dieudonné, who was one of the founding members | dieudonné, who was one of the founding members of the bourbaki group, credits a precise and general modern definition of a function to dedekind in his work was sind und was sollen die zahlen, which appeared in 1888 but had already been drafted in 1878. dieudonné observes that instead of confining himself, as in previou... | wikipedia |
6974f88a31af7f646900f9c096024aeb97858c9c | because lobachevsky and dirichlet have been credited as | because lobachevsky and dirichlet have been credited as among the first to introduce the notion of an arbitrary correspondence, this notion is sometimes referred to as the dirichlet or lobachevsky-dirichlet definition of a function. a general version of this definition was later used by bourbaki (1939), and some in the... | wikipedia |
81143803a4e43e45f7922c1d5c801da3d0b32e1f | however, gardiner says"...it seems to me that lakatos | however, gardiner says"...it seems to me that lakatos goes too far, for example, when he asserts that 'there is ample evidence that had no idea of concept'." moreover, as noted above, dirichlet's paper does appear to include a definition along the lines of what is usually ascribed to him, even though (like lobachevsky)... | wikipedia |
35becd7c8f7686a8af817f15d39ff394cd59cf44 | during the 19th century, mathematicians started to formalize | during the 19th century, mathematicians started to formalize all the different branches of mathematics. one of the first to do so was cauchy; his somewhat imprecise results were later made completely rigorous by weierstrass, who advocated building calculus on arithmetic rather than on geometry, which favoured euler's d... | wikipedia |
4f6d97854800c10c70eb0502384d3acfae0c6a37 | in his théorie analytique de la chaleur, fourier | in his théorie analytique de la chaleur, fourier claimed that an arbitrary function could be represented by a fourier series. fourier had a general conception of a function, which included functions that were neither continuous nor defined by an analytical expression. related questions on the nature and representation ... | wikipedia |
b30d3b6a869dae52a6c6822fff7a26abea18fb1c | in the first volume of his fundamental text | in the first volume of his fundamental text introductio in analysin infinitorum, published in 1748, euler gave essentially the same definition of a function as his teacher bernoulli, as an expression or formula involving variables and constants e.g., x 2 + 3 x + 2 {\displaystyle {x^{2}+3x+2}}. euler's own definition re... | wikipedia |
0b3c532c5da92debaa32b8a47357ca9abae214db | the functions considered in those times are called | the functions considered in those times are called today differentiable functions. for this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input. such functions are the basis of calculus. | wikipedia |
02997d334bc762b448c4b910a3949a300912fc28 | the term "function" was literally introduced by gottfried | the term "function" was literally introduced by gottfried leibniz, in a 1673 letter, to describe a quantity related to points of a curve, such as a coordinate or curve's slope. johann bernoulli started calling expressions made of a single variable "functions." in 1698, he agreed with leibniz that any quantity formed "i... | wikipedia |
dc8c0068098b99105d16886219539525471ce20b | the development of analytical geometry around 1640 allowed | the development of analytical geometry around 1640 allowed mathematicians to go between geometric problems about curves and algebraic relations between "variable coordinates x and y." calculus was developed using the notion of variables, with their associated geometric meaning, which persisted well into the eighteenth ... | wikipedia |
07a56dd6a8063214b2bb64366f071868068b401f | according to dieudonné and ponte, the concept of | according to dieudonné and ponte, the concept of a function emerged in the 17th century as a result of the development of analytic geometry and the infinitesimal calculus. nevertheless, medvedev suggests that the implicit concept of a function is one with an ancient lineage. ponte also sees more explicit approaches to ... | wikipedia |
9aedf34458bca1a320007d437da906348df49461 | already in the 12th century, mathematician sharaf al-din | already in the 12th century, mathematician sharaf al-din al-tusi analyzed the equation x + d = b ⋅ x in the form x ⋅ (b – x) = d, stating that the left hand side must at least equal the value of d for the equation to have a solution. he then determined the maximum value of this expression. it is arguable that the isola... | wikipedia |
ce759b0b0c396ab94612163b5cb06f80e8d106f9 | mathematicians of the 18th century typically regarded a | mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. in the 19th century, the demands of the rigorous development of analysis by weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by cantor, eventually led t... | wikipedia |
b70af292b93ea41904b471e4018ba374c3fa3879 | the mathematical concept of a function dates from | the mathematical concept of a function dates from the 17th century in connection with the development of the calculus; for example, the slope d y / d x {\displaystyle \operatorname {d} \!y/\operatorname {d} \!x} of a graph at a point was regarded as a function of the x -coordinate of the point. functions were not expli... | wikipedia |
ec02f454d6751cd3c322e30015db72ae6dfc9052 | brodie dupont is a retired canadian professional ice | brodie dupont is a retired canadian professional ice hockey forward and current coach. he was most recently head coach of the uk elite ice hockey league (eihl) side cardiff devils. he was drafted in the 3rd round, 66th overall by the new york rangers in the 2005 nhl entry draft. | wikipedia |
cab429683ed3dd2c848f19187cc19beffc75426a | dupont was born in russell, manitoba, but grew | dupont was born in russell, manitoba, but grew up in st. lazare, manitoba. on july 6, 2013, brodie dupont married kayleen kelly. brody now resides with kayleen and his two children in the town of hampton, new brunswick, canada. | wikipedia |
d53807d31396dbd09a708d92b344db544c3a4e24 | however, in april 2023, dupont and assistant coach | however, in april 2023, dupont and assistant coach christian horn left cardiff after one full season in charge. dupont led cardiff to 4th in the league standings, and to the runners-up spot in the elite league play-offs. | wikipedia |
334f220ce832fd016e9d44060a366fea124efa2e | in april 2022, dupont - alongside cardiff assistant | in april 2022, dupont - alongside cardiff assistant coach neil francis - took interim charge until the end of the season following the departure of head coach jarrod skalde. dupont then led cardiff to the 2022 elite league play-off championship with a 6-3 win over belfast giants in the final. | wikipedia |
f48d345d24bbe87aa4e553191a731664a09bdbea | as a free agent in the following off-season, | as a free agent in the following off-season, dupont returned to the norfolk admirals, securing a one-year deal on september 21, 2017. regaining his role as team captain through the 2017–18 season, dupont led the club in scoring at a point-per game rate through 68 appearances. he also had a one-game pto stint with the s... | wikipedia |
fbd06262dbd6ee42e5503548b4095813308bbf7d | after parts of three seasons abroad with the | after parts of three seasons abroad with the german club, iserlohn roosters of the deutsche eishockey liga, dupont returned to north america in initially signing with echl club, the norfolk admirals. as captain of the club in the 2016–17 season, dupont was the leading point scorer with 41 in 40 games before he was sign... | wikipedia |
f2ae63d77f46f0f5aea6c92b3510ba16f4efd680 | after playing in the american hockey league (ahl) | after playing in the american hockey league (ahl) with the hartford wolfpack for more than three seasons, dupont was first called up to the new york rangers on january 19, 2011 and made his nhl debut on january 22, playing seven shifts during a 3–2 shoot-out win over the atlanta thrashers. | wikipedia |
9cfd68104cf2cff5a38a4e10cd315ff86d9e9aa2 | brodie dupont (born february 17, 1987) is a | brodie dupont (born february 17, 1987) is a retired canadian professional ice hockey forward and current coach. he was most recently head coach of the uk elite ice hockey league (eihl) side cardiff devils. he was drafted in the 3rd round, 66th overall by the new york rangers in the 2005 nhl entry draft. | wikipedia |
3e6728375ae565a07c9892139aab71517adc40b2 | in mathematical logic, a theory is categorical if | in mathematical logic, a theory is categorical if it has exactly one model. such a theory can be viewed as defining its model, uniquely characterizing the model's structure. in first-order logic, only theories with a finite model can be categorical. higher-order logic contains categorical theories with an infinite mode... | wikipedia |
fe8a15d49e7a587d88df91e3803dfbc31c738515 | any theory t categorical in some infinite cardinal | any theory t categorical in some infinite cardinal κ is very close to being complete. more precisely, the Łoś–vaught test states that if a satisfiable theory has no finite models and is categorical in some infinite cardinal κ at least equal to the cardinality of its language, then the theory is complete. the reason is ... | wikipedia |
a4fe013027dfaecf3f4862d3bc63b8bdc7ae91a1 | there are also examples of theories that are | there are also examples of theories that are categorical in ω but not categorical in uncountable cardinals. the simplest example is the theory of an equivalence relation with exactly two equivalence classes, both of which are infinite. another example is the theory of dense linear orders with no endpoints; cantor prove... | wikipedia |
97abb5bc1ba494cb4f4f6f9b4b38c657df6c920e | in other words, he observed that, in all | in other words, he observed that, in all the cases he could think of, κ -categoricity at any one uncountable cardinal implied κ -categoricity at all other uncountable cardinals. this observation spurred a great amount of research into the 1960s, eventually culminating in michael morley 's famous result that these are i... | wikipedia |
67017517c0107cf5f6ce9f35330e547949347f15 | oswald veblen in 1904 defined a theory to | oswald veblen in 1904 defined a theory to be categorical if all of its models are isomorphic. it follows from the definition above and the löwenheim–skolem theorem that any first-order theory with a model of infinite cardinality cannot be categorical. one is then immediately led to the more subtle notion of κ -categori... | wikipedia |
555f0b21f3ef52cd98dbdb138450792b2a59f4e2 | saharon shelah (1974) extended morley's theorem to uncountable | saharon shelah (1974) extended morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ. | wikipedia |
0aa47f4717fed262a5331889672809b26730d551 | in model theory, the notion of a categorical | in model theory, the notion of a categorical theory is refined with respect to cardinality. a theory is κ - categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. morley's categoricity theorem is a theorem of michael d. morley (1965) stating that if a first-order theory in a c... | wikipedia |
8de233f2866addad64ba9d0da95fea959624011b | in first-order logic, only theories with a finite | in first-order logic, only theories with a finite model can be categorical. higher-order logic contains categorical theories with an infinite model. for example, the second-order peano axioms are categorical, having a unique model whose domain is the set of natural numbers n. {\displaystyle \mathbb {n}.} | wikipedia |
a477f9e3dceba7bc218b01f4ea4f6c230d310687 | cadmium selenide is an inorganic compound with the | cadmium selenide is an inorganic compound with the formula cdse. it is a black to red-black solid that is classified as a ii-vi semiconductor of the n-type. it is a pigment, but applications are declining because of environmental concerns. | wikipedia |
379dd8724f59ec35f5805186a5ea7d8eb1b4ff47 | cadmium is a toxic heavy metal and appropriate | cadmium is a toxic heavy metal and appropriate precautions should be taken when handling it and its compounds. selenides are toxic in large amounts. cadmium selenide is a known carcinogen to humans and medical attention should be sought if swallowed, dust inhaled, or if contact with skin or eyes occurs. | wikipedia |
6d19284c3921b7b764db10a1e5508ff0d9c69b33 | cdse material is transparent to infra-red (ir) light | cdse material is transparent to infra-red (ir) light and has seen limited use in photoresistors and in windows for instruments utilizing ir light. the material is also highly luminescent. cdse is a component of the pigment cadmium orange. cdse can also serve as the n-type semiconductor layer in photovoltaic cells. | wikipedia |
2055b447313ae7497740f76333549cb76b0a25ae | the cdse ligand shell may contain both x | the cdse ligand shell may contain both x type ligands which form covalent bonds with the metal and l type ligands that form dative bonds. it has been shown that these ligands can undergo exchange with other ligands. examples of x type ligands that have been studied in the context of cdse nanocrystal surface chemistry a... | wikipedia |
9e5206ebfb788439ecb96878679d8b47fa1d7d94 | a prevailing belief is that trioctylphosphine oxide (topo) | a prevailing belief is that trioctylphosphine oxide (topo) or trioctylphosphine (top), a neutral ligand derived from a common precursor used in the synthesis of cdse dots, caps the surface of cdse quantum dots. however, results from recent studies challenge this model. using nmr, quantum dots have been shown to be nons... | wikipedia |
be3cd5f2d68bfc2e8337c2f0ff54a688e5f769ab | cdse quantum dots are usually composed of a | cdse quantum dots are usually composed of a cdse core and a ligand shell. ligands play important roles in the stability and solubility of the nanoparticles. during synthesis, ligands stabilize growth to prevent aggregation and precipitation of the nanocrystals. these capping ligands also affect the quantum dot's electr... | wikipedia |
94a95844d2e6030d4067f9817823a4bcea69cfab | cdse quantum dots have been implemented in a | cdse quantum dots have been implemented in a wide range of applications including solar cells, light emitting diodes, and biofluorescent tagging. cdse-based materials also have potential uses in biomedical imaging. human tissue is permeable to near infra-red light. by injecting appropriately prepared cdse nanoparticles... | wikipedia |
7aedba80fb5680b5b8ca0240576ab14d189fcfca | cdse-derived nanoparticles with sizes below 10 nm exhibit | cdse-derived nanoparticles with sizes below 10 nm exhibit a property known as quantum confinement. quantum confinement results when the electrons in a material are confined to a very small volume. quantum confinement is size dependent, meaning the properties of cdse nanoparticles are tunable based on their size. one ty... | wikipedia |
5e08ecbf20138e41190758084f362de0b7469c07 | high temperature pyrolysis synthesis is usually carried out | high temperature pyrolysis synthesis is usually carried out using an aerosol containing a mixture of volatile cadmium and selenium precursors. the precursor aerosol is then carried through a furnace with an inert gas, such as hydrogen, nitrogen, or argon. in the furnace the precursors react to form cdse as well as seve... | wikipedia |
b63fc7f4beadc7bfe06d4ae87b67968ed24cd980 | synthesis in structured environments refers to the production | synthesis in structured environments refers to the production of cadmium selenide in liquid crystal or surfactant solutions. the addition of surfactants to solutions often results in a phase change in the solution leading to a liquid crystallinity. a liquid crystal is similar to a solid crystal in that the solution has... | wikipedia |
09f8bd8367f9fe6459c26c0565b5ea4c2df9a709 | cadmium selenide may also be produced in the | cadmium selenide may also be produced in the form of nanoparticles. (see applications for explanation) several methods for the production of cdse nanoparticles have been developed: arrested precipitation in solution, synthesis in structured media, high temperature pyrolysis, sonochemical, and radiolytic methods are jus... | wikipedia |
a3800e12ab2bf30228da35d99d87d8c0c45690a3 | three crystalline forms of cdse are known which | three crystalline forms of cdse are known which follow the structures of: wurtzite (hexagonal), sphalerite (cubic) and rock-salt (cubic). the sphalerite cdse structure is unstable and converts to the wurtzite form upon moderate heating. the transition starts at about 130 °c, and at 700 °c it completes within a day. the... | wikipedia |
2b87a4b7ba1177e213b4ff017d3a8e5aa1e3ad25 | cadmium selenide is an inorganic compound with the | cadmium selenide is an inorganic compound with the formula cd se. it is a black to red-black solid that is classified as a ii-vi semiconductor of the n-type. it is a pigment, but applications are declining because of environmental concerns. | wikipedia |
8f883bcdb4bcf7b6902a7a62c8d479c55b78dcb0 | cantilever enhanced photoacoustic spectroscopy enables the detection of | cantilever enhanced photoacoustic spectroscopy enables the detection of small amount of trace gases which is vital in many applications. photoacoustic spectroscopy is one of the most sensitive optical detection schemes. it is based on detecting a gas specific acoustic wave generated that originates from the absorption ... | wikipedia |
4ba182261462714c174b7e41c4745594e4aff8f2 | the cantilever sensor is made out of single | the cantilever sensor is made out of single crystal soi-silicon with a specially developed dry-etching process that leads to a highly stable and robust component; this is why the sensor is practically totally immune to temperature and humidity variations. in addition, the sensor does not suffer from wearing. the sensor... | wikipedia |
09fb1a9339bb0b03aa437f0e1fa71e942bc38dfb | an extremely thin cantilever portion moves like a | an extremely thin cantilever portion moves like a flexible door due to the pressure variations in the surrounding gas. the displacement of the cantilever is measured with an accurate interferometric readout system. this way the "breathing effect" can be avoided. the so-called breathing effect occurs in capacitive measu... | wikipedia |
a499bfbe41719177b51e7dd2388f66fd6382772b | the novel mems cantilever approach detects pressure changes | the novel mems cantilever approach detects pressure changes in a photoacoustic cell. high sensitivity is achieved by using a cantilever pressure sensor that is over hundred times more sensitive compared to a membrane, which is conventionally used in photoacoustic spectroscopy. a laser-based readout interferometer is ab... | wikipedia |
8f883bcdb4bcf7b6902a7a62c8d479c55b78dcb0 | cantilever enhanced photoacoustic spectroscopy enables the detection of | cantilever enhanced photoacoustic spectroscopy enables the detection of small amount of trace gases which is vital in many applications. photoacoustic spectroscopy is one of the most sensitive optical detection schemes. it is based on detecting a gas specific acoustic wave generated that originates from the absorption ... | wikipedia |
19b31954f2a7bcbe0742a32138880dba538ff525 | the chebotarev theorem on roots of unity was | the chebotarev theorem on roots of unity was originally a conjecture made by ostrowski in the context of lacunary series. chebotarev was the first to prove it, in the 1930s. this proof involves tools from galois theory and pleased ostrowski, who made comments arguing that it "does meet the requirements of mathematical ... | wikipedia |
4f947b0a39c8b795e264037224e0481a1f1d298f | let Ω {\displaystyle \omega } be a matrix | let Ω {\displaystyle \omega } be a matrix with entries a i j = ω i j, 1 ≤ i, j ≤ n {\displaystyle a_{ij}=\omega ^{ij},1\leq i,j\leq n}, where ω = e 2 i π / n, n ∈ n {\displaystyle \omega =e^{2\mathrm {i} \pi /n},n\in \mathbb {n} }. if n {\displaystyle n} is prime then any minor of Ω {\displaystyle \omega } is non-zero. | wikipedia |
7491f0e597252ae0eba16740c5151d1f2d850e72 | chebotarev was the first to prove it, in | chebotarev was the first to prove it, in the 1930s. this proof involves tools from galois theory and pleased ostrowski, who made comments arguing that it " does meet the requirements of mathematical esthetics ". several proofs have been proposed since, and it has even been discovered independently by dieudonné. | wikipedia |
57816bee11b82d219146cdd0fe8dd93d58728571 | chalcogenidotetrelates are chemical compounds containing a group 14 | chalcogenidotetrelates are chemical compounds containing a group 14 element, known as a tetrel, and a group 16 element, known as a chalcogen. the group 14 elements are carbon, silicon, germanium, tin, lead and flerovium. flerovium compounds like this are unknown due to its short half-life. the group 16 elements are oxy... | wikipedia |
6a6a2766e9c9de3f341707d34e030f891bb211c7 | chalcogenidotetrelates may be produced by heating together the | chalcogenidotetrelates may be produced by heating together the chalcogen compounds of the desired ingredients. a high temperature flux of a molten salt may be used. or solutions in amines or organic solvents may crystallise at low temperatures. | wikipedia |
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