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Operational definition-Examples-Temperature The thermodynamic definition of temperature, due to Nicolas Léonard Sadi Carnot, refers to heat "flowing" between "infinite reservoirs". This is all highly abstract and unsuited for the day-to-day world of science and trade. In order to make the idea concrete, temperature is ... | milkshake721/2.1M-wiki-STEM |
Operational definition-Examples-For day-to-day use, the International Temperature Scale of 1990 (ITS) is used, defining temperature in terms of characteristics of the several specific sensor types required to cover the full range. One such is the electrical resistance of a thermistor, with specified construction, calib... | milkshake721/2.1M-wiki-STEM |
Operational definition-Examples-Mechanical hardness Unlike temperature and electric current, there is no abstract physical concept of the hardness of a material. It is a slightly vague, subjective idea, somewhat like the idea of intelligence. In fact, it leads to three more specific ideas: Scratch hardness measured on ... | milkshake721/2.1M-wiki-STEM |
Operational definition-Examples-The constellation Virgo The constellation Virgo is a specific constellation of stars in the sky, hence the process of forming Virgo cannot be an operational definition, since it is historical and not repeatable. Nevertheless, the process whereby we locate Virgo in the sky is repeatable, ... | milkshake721/2.1M-wiki-STEM |
Operational definition-Examples-Academic discipline New academic disciplines appear in response to interdisciplinary activity at universities. An academic suggested that a subject matter area becomes a discipline when there are more than a dozen university departments using the same name for roughly the same subject ma... | milkshake721/2.1M-wiki-STEM |
Fractography-Fractography-Fractography is the study of the fracture surfaces of materials. Fractographic methods are routinely used to determine the cause of failure in engineering structures, especially in product failure and the practice of forensic engineering or failure analysis. In material science research, fract... | milkshake721/2.1M-wiki-STEM |
Fractography-Fractography-One of the aims of fractographic examination is to determine the cause of failure by studying the characteristics of a fractured surface. Different types of crack growth (e.g. fatigue, stress corrosion cracking, hydrogen embrittlement) produce characteristic features on the surface, which can ... | milkshake721/2.1M-wiki-STEM |
Fractography-Usage-Fractography is a widely used technique in forensic engineering, forensic materials engineering and fracture mechanics to understand the causes of failures and also to verify theoretical failure predictions with real life failures. It is of use in forensic science for analysing broken products which ... | milkshake721/2.1M-wiki-STEM |
Fractography-Usage-Fractography is used also in materials research, since fracture properties can correlate with other properties and with structure of materials. | milkshake721/2.1M-wiki-STEM |
Fractography-Feature identification-Origin An important aim of fractography is to establish and examine the origin of cracking, as examination at the origin may reveal the cause of crack initiation. Initial fractographic examination is commonly carried out on a macro scale utilising low power optical microscopy and obl... | milkshake721/2.1M-wiki-STEM |
Fractography-Feature identification-Common features that may cause crack initiation are inclusions, voids or empty holes in the material, contamination, and stress concentrations. | milkshake721/2.1M-wiki-STEM |
Fractography-Feature identification-Fatigue crack growth The image of a broken crankshaft shows the component failed from a surface defect near the bulb at lower centre. The semi-circular marks near the origin indicate a crack growing up into the bulk material by process known as fatigue. The crankshaft also shows hach... | milkshake721/2.1M-wiki-STEM |
Fractography-Microscopy-Microscopes can be used to determine the initiation point and the mechanism that caused crack growth. The information can be obtained from images of the fracture surface known as fractographs and used in constructing diagrams. A schematic fracture surface map can be used to isolate and identify ... | milkshake721/2.1M-wiki-STEM |
Fractography-Microscopy-USB Microscopy USB microscopes are especially useful for examining fracture surface features since they are small enough to be hand-held. A variety of camera sizes and resolution are available commercially at low cost. The camera cable plugs into the computer via a USB plug and most such devices... | milkshake721/2.1M-wiki-STEM |
Fractography-Microscopy-Scanning electron microscopy In many cases, fractography requires examination at a finer scale, which is usually carried out in a scanning electron microscope or SEM. The resolution is much higher than the optical microscope, although samples are examined in a partial vacuum and colour is absent... | milkshake721/2.1M-wiki-STEM |
Fractography-Example-Breast implant A cusp is formed where brittle cracks meet, as shown on the picture of a failed catheter (Cp). The cusp was formed by brittle failure of the catheter on a breast implant in silicone rubber. The origin of the cracks is at the shoulder at the left-hand side. Identifying such features w... | milkshake721/2.1M-wiki-STEM |
Fractography-Example-In the case of the failed breast implant catheter, the crack path was very simple, but the cause more subtle. Further scanning electron microscopy showed numerous microcracks between the bag and the catheter, indicating that the adhesive bond between the two components had failed prematurely, perha... | milkshake721/2.1M-wiki-STEM |
Fractography-Example-Maritime Patrol Aircraft A non-critical crack occurred in the fastener hole of a lower wing plank. The plank was made from a 3.2 mm thick AA7075-T6 aluminium alloy. The time of the detection of the crack and the aircraft's counting g-meter allowed investigators to find out the load on the aircraft ... | milkshake721/2.1M-wiki-STEM |
Retirement home-Retirement home-A retirement home – sometimes called an old people's home, old folks' home, or old age home, although old people's home can also refer to a nursing home – is a multi-residence housing facility intended for the elderly. Typically, each person or couple in the home has an apartment-style r... | milkshake721/2.1M-wiki-STEM |
Retirement home-Retirement home-Retirement homes offer meal-making and some personal care services. Assisted living facilities, memory care facilities and nursing homes can all be referred to as retirement homes. The cost of living in a retirement home varies from $25,000 to $100,000 per year, although it can exceed th... | milkshake721/2.1M-wiki-STEM |
Retirement home-United States-Proper design is integral to the experience within retirement homes, especially for those experiencing dementia. Wayfinding and spatial orientation become difficult for residents with dementia, causing confusion, agitation and a general decline in the physical and mental wellbeing. | milkshake721/2.1M-wiki-STEM |
Retirement home-United States-Signage Those living with dementia often display difficulty with distinguishing relevance of information within signage. This phenomenon can be attributed to a combination of fixative behaviors as well as a tendency towards non discriminatory reading. Therefore in creating appropriate sign... | milkshake721/2.1M-wiki-STEM |
Retirement home-United States-Architectural cues Another relevant method of wayfinding is the presence of architectural cues within North American senior retirement homes. This method is most often considered during the design of new senior care centers, however there are still multiple items that can easily be impleme... | milkshake721/2.1M-wiki-STEM |
Retirement home-United States-Assistance in navigation through reducing a need for complicated cognitive mapping is an asset that can be achieved in multiple ways within care centers. Visual landmarks existing in both architectural and interior design helps provide differentiation between spaces. Burton notes “resident... | milkshake721/2.1M-wiki-STEM |
Calciferous sandstone-Calciferous sandstone-Calciferous sandstone is a geological term relating to strata at the base of the Carboniferous formation, below the entire sequence of coal measures. This term may be unique to the UK.
Typically this part of the geological sequence, as in the Touch Hills and Fintry Hills to t... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Virasoro algebra-In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Definition-The Virasoro algebra is spanned by generators Ln for n ∈ ℤ and the central charge c.
These generators satisfy [c,Ln]=0 and The factor of 12 is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-Highest weight representations A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector v such that Ln>0v=0,L0v=hv, where the number h is called the conformal dimension or conformal weight of v .A highest weight represen... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-More precisely, a highest weight representation is spanned by L0 -eigenstates of the type L−n1L−n2⋯L−nkv with 0<n1≤n2≤⋯nk and k≥0 , whose levels are N=∑i=1kni . Any state whose level is not zero is called a descendant state of v For any pair of complex numbers h and c, the V... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-A Verma module is irreducible if and only if it has no singular vectors.
Singular vectors A singular vector or null vector of a highest weight representation is a state that is both descendent and primary.
A sufficient condition for the Verma module Vc,h to have a singular vector... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-In particular, h1,1(c)=0 , and the reducible Verma module Vc,0 has a singular vector L−1v at the level N=1 . Then h2,1(c)=−12−34b2 , and the corresponding reducible Verma module has a singular vector (L−12+b2L−2)v at the level N=2 This condition for the existence of a singu... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-The representation is called unitary if that Hermitian form is positive definite. Since any singular vector has zero norm, all unitary highest weight representations are irreducible. | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-The Gram determinant of a basis of the level N is given by the Kac determinant formula, AN∏1≤r,s≤N(h−hr,s(c))p(N−rs), where the function p(N) is the partition function, and AN is a positive constant that does not depend on h or c . The Kac determinant formula was stated by V... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-The irreducible highest weight representation with values h and c is unitary if and only if either c ≥ 1 and h ≥ 0, or 10 25 28 ,…} and h is one of the values h=hr,s(c)=((m+1)r−ms)2−14m(m+1) for r = 1, 2, 3, ..., m − 1 and s = 1, 2, 3, ..., r.
Daniel Friedan, Zongan Qiu, and Ste... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-For any c∈C and for r,s∈N∗ , the Verma module Vc,hr,s is reducible due to the existence of a singular vector at level rs . This singular vector generates a submodule, which is isomorphic to the Verma module Vc,hr,s+rs . The quotient of Vc,hr,s by this submodule is irreducibl... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-Let c=cp,p′ with 2≤p<p′ and p,p′ coprime, and 1≤r≤p−1 and 1≤s≤p′−1 . (Then (r,s) is in the Kac table of the corresponding minimal model). The Verma module Vc,hr,s has infinitely many singular vectors, and is therefore reducible with infinitely many submodules. This Verma mo... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Representation theory-This expression is an infinite sum because the submodules Vc,hr,s+rs and Vc,hr,s+(p−r)(p′−s) have a nontrivial intersection, which is itself a complicated submodule. | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Applications-Conformal field theory In two dimensions, the algebra of local conformal transformations is made of two copies of the Witt algebra.
It follows that the symmetry algebra of two-dimensional conformal field theory is the Virasoro algebra. Technically, the conformal bootstrap approach to two-d... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Applications-String theory Since the Virasoro algebra comprises the generators of the conformal group of the worldsheet, the stress tensor in string theory obeys the commutation relations of (two copies of) the Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphis... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-Generalizations-Super Virasoro algebras There are two supersymmetric N = 1 extensions of the Virasoro algebra, called the Neveu–Schwarz algebra and the Ramond algebra. Their theory is similar to that of the Virasoro algebra, now involving Grassmann numbers. There are further extensions of these algebra... | milkshake721/2.1M-wiki-STEM |
Virasoro algebra-History-The Witt algebra (the Virasoro algebra without the central extension) was discovered by É. Cartan (1909). Its analogues over finite fields were studied by E. Witt in about the 1930s. The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic p >... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar fricative-The voiceless alveolar fricatives are a type of fricative consonant pronounced with the tip or blade of the tongue against the alveolar ridge (gum line) just behind the teeth. This refers to a class of sounds, not a single sound. There are at least six types wit... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar fricative-The voiceless denti-alveolar sibilant [s̄] (an ad hoc notation), also called apico-dental, has a weaker lisping sound like English th in thin. It occurs in Spanish dialects in southern Spain (eastern Andalusia). | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar fricative-The voiceless alveolar retracted sibilant [s̠], and the subform apico-alveolar [s̺], or called grave, has a weak hushing sound reminiscent of retroflex fricatives. It is used in the languages of northern Iberia, like Asturleonese, Basque, Castilian Spanish (excl... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar fricative-The voiceless alveolar non-sibilant fricative [θ̠] or [θ͇], using the alveolar diacritic from the Extended IPA, is similar to the th in English thin. It occurs in Icelandic as well as an intervocalic and word-final allophone of English /t/ in dialects such as Hi... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar fricative-The voiceless alveolar lateral fricative [ɬ] sounds like a voiceless, strongly articulated version of English l (somewhat like what the English cluster **hl would sound like) and is written as ll in Welsh.The first three types are sibilants, meaning that they ar... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar sibilant-The voiceless alveolar sibilant is a common consonant sound in vocal languages. It is the sound in English words such as sea and pass, and is represented in the International Phonetic Alphabet with ⟨s⟩. It has a characteristic high-pitched, highly perceptible his... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar sibilant-The voiceless alveolar sibilant [s] is one of the most common sounds cross-linguistically. If a language has fricatives, it will most likely have [s]. However, some languages have a related sibilant sound, such as [ʃ], but no [s]. In addition, sibilants are absen... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-The voiceless alveolar retracted sibilant (commonly termed the voiceless apico-alveolar sibilant) is a fricative that is articulated with the tongue in a hollow shape, usually with the tip of the tongue (apex) against the alveolar ridge. It is a sibilant so... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Occurrence in Europe Modern In Romance languages, it occurs as the normal voiceless alveolar sibilant in Astur-Leonese, Castilian Spanish, Catalan, Galician, northern European Portuguese, and some Occitan dialects. It also occurs in Basque and Mirandese, wh... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Amongst Germanic languages, it occurs in Dutch (and closely related Low German), Icelandic, many dialects in Scandinavia, and working-class Glaswegian English.
It also occurs in Modern Greek (with a laminal articulation), as well as the Baltic languages. | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-There is no single IPA symbol used for this sound. The symbol ⟨s̺⟩ is often used, with a diacritic indicating an apical pronunciation. However, that is potentially problematic in that not all alveolar retracted sibilants are apical (see below), and not all ... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Medieval In medieval times, it occurred in a wider area, including the Romance languages spoken in most or all of France and Iberia (Old Spanish, Galician-Portuguese, Catalan, French, etc.), as well as in the Old and Middle High German of central and southe... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Description of the retracted sibilant Often, to speakers of languages or dialects that do not have the sound, it is said to have a "whistling" quality, and to sound similar to palato-alveolar ʃ. For this reason, when borrowed into such languages or represen... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-One of the clearest descriptions of this sound is from Obaid: "There is a Castilian s, which is a voiceless, concave, apicoalveolar fricative: The tip of the tongue turned upward forms a narrow opening against the alveoli of the upper incisors. It resembles... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-In French and most dialects of Portuguese, the four alveolar sibilants have merged into non-retracted [s] and [z], while in European Portuguese, most other Old World Portuguese variants and some recently European-influenced dialects of Brazil all instances ... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-In the remaining dialects of Portuguese, found in northern Portugal, they merged into the retracted [s̺] [z̺], or, as in Mirandese (which is, however, not a Portuguese dialect, but belongs to Asturian-Leonese), conserved the medieval distinction. | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-In German, [s̺] was early on voiced to [z̺] in prevocalic position. This sound was then fronted to [z], but did not merge with any other sound (except that it was later re-devoiced in some southern dialects). In pre-consonantal and final position, [s̺] merg... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Loss-causing events Those languages in which the sound occurs typically did not have a phonological process from which either [s] or [ʃ] appeared, two similar sounds with which ⟨s̺⟩ was eventually confused. In general, older European languages only had a si... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-In Romance languages, [s] was reached from -ti-, -ci-, -ce- ([ti], [ki], [ke]) clusters that eventually became [ts], [tsi], [tse] and later [s], [si], [se] (as in Latin fortia "force", civitas "city", centum "hundred"), while [ʃ] was reached: From a [sk] or... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-from a deaffricated [tʃ] in Northern France and southern-central Portugal, as in French chat "cat", Portuguese achar "find".In High German, [s] was reached through a [t] > [ts] > [s] process, as in German Wasser compared to English water. In English, the sa... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Exceptions Standard Modern Greek, which has apical [s̺], lacked both processes.
The Germanic-speaking regions that did not have either phenomenon have normally preserved the apical [s̺], that is, Icelandic, Dutch and many Scandinavian lects. It is also foun... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-Reach in ancient times Because of the widespread medieval distribution, it has been speculated that retracted [s̺] was the normal pronunciation in spoken Latin. Certain borrowings suggest that it was not far off from the sh-sound [ʃ], e.g. Aramaic Jeshua > ... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-For the same reasons, it can be speculated that retracted [s̺] was the pronunciation of Proto-Germanic s. Its presence in many branches of Indo-European and its presence particularly in the more conservative languages inside each branch (e.g. Icelandic, Spa... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless apico-alveolar sibilant-[ʃ], but not [s], was developed in Italian. However, where Spanish and Catalan have apical [s̺], Italian uses the same laminal [s] that occurs in standard forms of English: evidence, it could be argued, that S was not pronounced apically in Latin. But Neapo... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless lamino-dental sibilant-A voiceless laminal dental or dentialveolar sibilant contrasts with a voiceless apical alveolar or post-alveolar sibilant in Basque and several languages of California, including Luiseño of the Uto-Aztecan family and Kumeyaay of the Yuman family. | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Comparison between English and Spanish-The term "voiceless alveolar sibilant" is potentially ambiguous in that it can refer to at least two different sounds. Various languages of northern Iberia (e.g., Astur-Leonese, Catalan, Basque, Galician, Portuguese and Spanish) have a so-called "voice... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Comparison between English and Spanish-There is no general agreement about what actual feature distinguishes these sounds. Spanish phoneticians normally describe the difference as apical (for the northern Iberian sound) vs. laminal (for the more common sound), but Ladefoged and Maddieson cl... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Comparison between English and Spanish-Some authors have instead suggested that the difference lies in tongue shape. Adams describes the northern Iberian sibilant as "retracted". Ladefoged and Maddieson appear to characterize the more common hissing variant as grooved, and some phoneticians... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Features-Features of the voiceless alveolar sibilant: Its manner of articulation is sibilant fricative, which means it is generally produced by channeling air flow along a groove in the back of the tongue up to the place of articulation, at which point it is focused against the sharp edge o... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Examples-Dentalized laminal alveolar Non-retracted alveolar Retracted alveolar Variable | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar non-sibilant fricative-The voiceless alveolar non-sibilant fricative (also known as a "slit" fricative) is a consonantal sound. As the International Phonetic Alphabet does not have separate symbols for the alveolar consonants (the same symbol is used for all coronal place... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar non-sibilant fricative-Some scholars also posit the voiceless alveolar approximant distinct from the fricative. The approximant may be represented in the IPA as ⟨ɹ̥⟩. | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar non-sibilant fricative-Few languages also have the voiceless alveolar tapped fricative, which is simply a very brief apical alveolar non-sibilant fricative, with the tongue making the gesture for a tapped stop but not making full contact. This can be indicated in the IPA ... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-Voiceless alveolar non-sibilant fricative-Flapped fricatives are theoretically possible but are not attested.
Features Its manner of articulation is fricative, which means it is produced by constricting air flow through a narrow channel at the place of articulation, causing turbulence. Howe... | milkshake721/2.1M-wiki-STEM |
Voiceless alveolar fricative-In popular culture-The voiceless alveolar fricative was used by Capt Jean-Luc Picard in the TNG episode The Big Goodbye in a complicated greeting to representatives of an alien race. | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Equivalence of categories-In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathem... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Equivalence of categories-If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Equivalence of categories-An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Definition-Formally, given two categories C and D, an equivalence of categories consists of a functor F : C → D, a functor G : D → C, and two natural isomorphisms ε: FG→ID and η : IC→GF. Here FG: D→D and GF: C→C denote the respective compositions of F and G, and IC: C→C and ID: D→D denote the ... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Definition-One often does not specify all the above data. For instance, we say that the categories C and D are equivalent (respectively dually equivalent) if there exists an equivalence (respectively duality) between them. Furthermore, we say that F "is" an equivalence of categories if an inve... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Alternative characterizations-A functor F : C → D yields an equivalence of categories if and only if it is simultaneously: full, i.e. for any two objects c1 and c2 of C, the map HomC(c1,c2) → HomD(Fc1,Fc2) induced by F is surjective; faithful, i.e. for any two objects c1 and c2 of C, the map H... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Alternative characterizations-Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories. (Unfortunately this conflicts with terminology from homotopy type theory.) There is also a close relation to the concept of adjoint functors F⊣G , wher... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Alternative characterizations-When adjoint functors F⊣G are not both full and faithful, then we may view their adjointness relation as expressing a "weaker form of equivalence" of categories. Assuming that the natural transformations for the adjunctions are given, all of these formulations al... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-Consider the category C having a single object c and a single morphism 1c , and the category D with two objects d1 , d2 and four morphisms: two identity morphisms 1d1 , 1d2 and two isomorphisms α:d1→d2 and β:d2→d1 . The categories C and D are equivalent; we can (for exampl... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-Consider a category C with one object c , and two morphisms 1c,f:c→c . Let 1c be the identity morphism on c and set f∘f=1 . Of course, C is equivalent to itself, which can be shown by taking 1c in place of the required natural isomorphisms between the functor IC and itself. H... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-Consider the category C of finite-dimensional real vector spaces, and the category D=Mat(R) of all real matrices (the latter category is explained in the article on additive categories). Then C and D are equivalent: The functor G:D→C which maps the object An of D to the vector ... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings. The functor G associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. Its adjoint F associates to ev... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-In lattice theory, there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces. Probably the most well-known theorem of this kind is Stone's representation theorem for Boolean algebras, which is a special ... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Examples-In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.
For two rings R and S, the product category R-Mod×S-Mod is equivalent to (R×S)-Mod.
Any category is equivalent to its skeleton. | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Properties-As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true: the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an init... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Properties-the functor H : I → C has limit (or colimit) l if and only if the functor FH : I → D has limit (or colimit) Fl. This can be applied to equalizers, products and coproducts among others. Applying it to kernels and cokernels, we see that the equivalence F is an exact functor.
C is a ca... | milkshake721/2.1M-wiki-STEM |
Equivalence of categories-Properties-If F : C → D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category), then D may be turned into a preadditive category (or additive category, or abelian category) in such a way that F becomes an additive functor. On the other h... | milkshake721/2.1M-wiki-STEM |
Market engineering-Market engineering-Market engineering comprises the structured, systematic and theoretically founded procedure of analyzing, designing, introducing and also quality assuring of markets as well as their legal framework regarding simultaneously their market mechanisms and trading rules, systems, platfo... | milkshake721/2.1M-wiki-STEM |
Market engineering-Framework-The Market Engineering Framework takes into account the various aspects of market engineering. The Market Engineering Framework diagram depicts the building blocks that form the framework. The framework operates in the economic and legal environment in which it is situated. This effects the... | milkshake721/2.1M-wiki-STEM |
Market engineering-Process-The market engineering approach is structured by means of a process as depicted in Market Engineering Process diagram. Taking into account the objectives of the new electronic markets, the requirements of the new electronic market are deduced in the first stage. Subsequently, the new market i... | milkshake721/2.1M-wiki-STEM |
Market engineering-Process-Environmental analysis The objective of the environmental analysis is to formalize the strategies and objectives of a new electronic market. The stage comprises two phases: the environment definition and the requirement analysis. The goal of the environment definition is to characterize the e... | milkshake721/2.1M-wiki-STEM |
Market engineering-Process-Design The second stage of the process comprises the design of the market with simultaneous consideration of the transaction object, the market microstructure, the IT infrastructure, and the business model. Supported by different tools and methodologies such as mechanism and parametric design... | milkshake721/2.1M-wiki-STEM |
Market engineering-Process-Evaluation Having designed the market, it is tested upon its technical and economic properties. The evaluation stage comprises functionality tests of a software prototype to ensure its correctness, economic tests to measure the outcome performance of the market, and an assessment of the busin... | milkshake721/2.1M-wiki-STEM |
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