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22-465: [REDACTED] Look up ST , St , st , .st , or s.t. in Wiktionary, the free dictionary. ST , St , or St. may refer to: Arts and entertainment [ edit ] Stanza , in poetry Suicidal Tendencies , an American heavy metal/hardcore punk band Star Trek , a science-fiction media franchise Summa Theologica ,

44-448: A , b ] {\displaystyle [a,b]} , one defines the integral ∫ a b f ( x ) d x {\textstyle \int _{a}^{b}f(x)\,dx} as the standard part of an infinite Riemann sum S ( f , a , b , Δ x ) {\displaystyle S(f,a,b,\Delta x)} when the value of Δ x {\displaystyle \Delta x}

66-594: A compendium of Catholic philosophy and theology by St. Thomas Aquinas St or St., abbreviation of "State", especially in the name of a college or university Businesses and organizations [ edit ] Transportation [ edit ] Germania (airline) (IATA airline designator ST) Maharashtra State Road Transport Corporation , abbreviated as State Transport Sound Transit , Central Puget Sound Regional Transit Authority, Washington state, US Springfield Terminal Railway (Vermont) (railroad reporting mark ST) Suffolk County Transit , or Suffolk Transit,

88-411: A least upper bound. Alternatively, the range of "st" is R ⊆ ∗ R {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} } , which is not internal; in fact every internal set in ∗ R {\displaystyle {}^{*}\mathbb {R} } that is a subset of R {\displaystyle \mathbb {R} }

110-505: A state of Germany Split, Croatia (vehicle plate code ST) Stoke-on-Trent postcode area , United Kingdom St or St., abbreviation of Saint St or St., abbreviation of Street St or St., abbreviation of Strait Language and typography [ edit ] Sesotho language (ISO 639-1 language code "st") ſt, or st, a typographic ligature Standard Theory in generative grammar Science and technology [ edit ] Computing [ edit ] ST connector ,

132-472: A steam tug Sine tempore (s.t.), Latin term indicating that a lecture will begin at the exact time; see Academic quarter (class timing) Striker (association football) , a position in association football ST, a type of London bus ST, a court-ordered pseudonym used during a British court case; see Sudiksha Thirumalesh case See also [ edit ] STST (disambiguation) STFC (disambiguation) for uses of ST F.C. Topics referred to by

154-481: A term used in non-standard analysis Physics [ edit ] Stanton number St, used in physics Strouhal number St, used in fluid mechanics String theory Units of measurement [ edit ] Stokes (unit) (St), a CGS unit of kinematic viscosity Stone (weight) (st.), a unit of mass used in the British Isles and other countries Medicine [ edit ] ST segment ,

176-636: A type of optical fiber connector Atari ST , a personal computer Prefix of hard disk drives made by Seagate Technology , e.g. ST-506 Internet Stream Protocol , an experimental Internet protocol Structured text , a high-level programming language that syntactically resembles Pascal and is designed for programmable logic controllers (PLC) .st , the Internet country code top-level domain (ccTLD) for São Tomé and Príncipe St (terminal emulator) , minimalist terminal emulator by suckless.org Mathematics [ edit ] Standard part function ,

198-476: Is a rigorous formalization of calculations with infinitesimals . The standard part of x is sometimes referred to as its shadow . Nonstandard analysis deals primarily with the pair R ⊆ ∗ R {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} } , where the hyperreals ∗ R {\displaystyle {}^{*}\mathbb {R} } are an ordered field extension of

220-406: Is an infinite index. Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen. A real function f {\displaystyle f} is continuous at a real point x {\displaystyle x} if and only if the composition st ∘ f {\displaystyle \operatorname {st} \circ f} is constant on

242-552: Is different from Wikidata All article disambiguation pages All disambiguation pages ST">ST Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.133 via cp1102 cp1102, Varnish XID 551919612 Upstream caches: cp1102 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 05:51:14 GMT Standard part function The standard part function

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264-436: Is expressed symbolically by writing The standard part of any infinitesimal is 0. Thus if N is an infinite hypernatural , then 1/ N is infinitesimal, and st(1/ N ) = 0. If a hyperreal u {\displaystyle u} is represented by a Cauchy sequence ⟨ u n : n ∈ N ⟩ {\displaystyle \langle u_{n}:n\in \mathbb {N} \rangle } in

286-515: Is necessarily finite . All the traditional notions of calculus can be expressed in terms of the standard part function, as follows. The standard part function is used to define the derivative of a function f . If f is a real function, and h is infinitesimal, and if f ′( x ) exists, then Alternatively, if y = f ( x ) {\displaystyle y=f(x)} , one takes an infinitesimal increment Δ x {\displaystyle \Delta x} , and computes

308-574: Is taken to be infinitesimal, exploiting a hyperfinite partition of the interval [ a , b ]. Given a sequence ( u n ) {\displaystyle (u_{n})} , its limit is defined by lim n → ∞ u n = st ⁡ ( u H ) {\textstyle \lim _{n\to \infty }u_{n}=\operatorname {st} (u_{H})} where H ∈ ∗ N ∖ N {\displaystyle H\in {}^{*}\mathbb {N} \setminus \mathbb {N} }

330-481: The ultrapower construction, then More generally, each finite u ∈ ∗ R {\displaystyle u\in {}^{*}\mathbb {R} } defines a Dedekind cut on the subset R ⊆ ∗ R {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} } (via the total order on ∗ R {\displaystyle {}^{\ast }\mathbb {R} } ) and

352-445: The bus system serving Suffolk County, New York Other businesses and organizations [ edit ] Statstjänstemannaförbundet , or Swedish Union of Civil Servants, a trade union STMicroelectronics , a worldwide manufacturer of semiconductors Geography [ edit ] São Tomé and Príncipe (ISO 3166-1 country code ST) .st , Internet country code top-level domain for São Tomé and Príncipe Saxony-Anhalt ,

374-470: The corresponding Δ y = f ( x + Δ x ) − f ( x ) {\displaystyle \Delta y=f(x+\Delta x)-f(x)} . One forms the ratio Δ y Δ x {\textstyle {\frac {\Delta y}{\Delta x}}} . The derivative is then defined as the standard part of the ratio: Given a function f {\displaystyle f} on [

396-453: The corresponding real number is the standard part of u . The standard part function "st" is not defined by an internal set . There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have

418-666: The part of an electrocardiogram connecting the QRS complex and the T wave Sulfotransferase , enzymes that catalyze the transfer of a sulfo group Heat-stable enterotoxin , secretory peptides produced by some bacterial strains, such as enterotoxigenic Escherichia coli Other uses [ edit ] -st, a suffix for an ordinal number , such as 1 or 21 Saint (St or St.), especially in Christianity Scheduled Tribes , in India Ship prefix for

440-420: The reals R {\displaystyle \mathbb {R} } , and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a monad , or halo ) of hyperreals infinitely close to it. The standard part function associates to a finite hyperreal x , the unique standard real number x 0 that is infinitely close to it. The relationship

462-401: The same term [REDACTED] This disambiguation page lists articles associated with the title ST . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=ST&oldid=1237068603 " Category : Disambiguation pages Hidden categories: Short description

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484-419: Was first defined by Abraham Robinson who used the notation ∘ x {\displaystyle {}^{\circ }x} for the standard part of a hyperreal x {\displaystyle x} (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis . The latter theory

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