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82,300 | That concatenation commutes and associates may then be demonstrated from the: Initial AC.D=CD.A and the consequence AA=A (Byrne 1946). This result holds for all lattices, because AA=A is an easy consequence of the absorption law, which holds for all lattices; Initials AC.D=AD.C and J0. Since J0 holds only for lattices ... | Laws of Form | 0.784233 |
82,301 | T13 in LoF generalizes C2 as follows. Any primary algebra (or sentential logic) formula B can be viewed as an ordered tree with branches. Then: T13: A subformula A can be copied at will into any depth of B greater than that of A, as long as A and its copy are in the same branch of B. Also, given multiple instances of A... | Laws of Form | 0.784233 |
82,302 | (T3 in LoF); Starting from an initial marked or unmarked state, "complicating" an expression by a finite number of repeated application of A1 and A2 cannot yield an expression whose simplification differs from the initial state. (T4 in LoF).Thus the relation of logical equivalence partitions all primary arithmetic expr... | Laws of Form | 0.784233 |
82,303 | This suggests that every primary arithmetic expression can, by repeated application of A1 and A2, be simplified to one of two states: the marked or the unmarked state. This is indeed the case, and the result is the expression's "simplification". The two fundamental metatheorems of the primary arithmetic state that: Eve... | Laws of Form | 0.784233 |
82,304 | Note that A=B is not required.R2 is employed very frequently in primary algebra demonstrations (see below), almost always silently. These rules are routinely invoked in logic and most of mathematics, nearly always unconsciously. The primary algebra consists of equations, i.e., pairs of formulae linked by an infix opera... | Laws of Form | 0.784233 |
82,305 | The sign '=' may link two logically equivalent expressions; the result is an equation. By "logically equivalent" is meant that the two expressions have the same simplification. Logical equivalence is an equivalence relation over the set of primary algebra formulas, governed by the rules R1 and R2. Let "C" and "D" be fo... | Laws of Form | 0.784233 |
82,306 | The following calculation of Leibniz's nontrivial Praeclarum Theorema exemplifies the demonstrative power of the primary algebra. Let C1 be A | ¯ | ¯ {\displaystyle {\overline {{\overline {A|}}{\Big |}}}} =A, C2 be A A B | ¯ = A B | ¯ {\displaystyle A\ {\overline {A\ B|}}=A\ {\overline {B|}}} , C3 be | ¯ A = | ¯ {\disp... | Laws of Form | 0.784233 |
82,307 | Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a primary algebra formula. Letters so employed in mathematics and logic are called variables. A primary algebra variable indicates a location where one can wr... | Laws of Form | 0.784233 |
82,308 | Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. LoF describes three distinct logical systems: The "primary arithmetic" (described in Chapter 4 of LoF), whose models include Boolean arithmetic; The "primary algebra" (Chapter... | Laws of Form | 0.784233 |
82,309 | The extension of the MPS system with the so called MOIS Executor, became the Smart-1 automation system. It permitted to operate 70% of the passes unmanned towards the end of the mission and allowed the validation of the first operational "spacecraft automation system" at ESA.The mission achieved all its objectives: get... | SMART 1 | 0.784232 |
82,310 | A linear function of a matrix M is a linear combination of its elements (with given coefficients), M ↦ tr(AM) where A is the matrix of the coefficients; see Trace (linear algebra)#Inner product. A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the ort... | Lyapunov condition | 0.784232 |
82,311 | By the limit of the exponential function ( e x = lim n → ∞ ( 1 + x n ) n {\textstyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}} ), the characteristic function of Z n {\displaystyle Z_{n}} equals All of the higher order terms vanish in the limit n → ∞ {\textstyle n\to \infty } . The right hand side equ... | Lyapunov condition | 0.784232 |
82,312 | Consider the random variable where in the last step we defined the new random variables Y i = X i − μ σ {\textstyle Y_{i}={\frac {X_{i}-\mu }{\sigma }}} , each with zero mean and unit variance ( var ( Y ) = 1 {\textstyle \operatorname {var} (Y)=1} ). The characteristic function of Z n {\textstyle Z_{n}} is given by w... | Lyapunov condition | 0.784232 |
82,313 | The central limit theorem has a proof using characteristic functions. It is similar to the proof of the (weak) law of large numbers. Assume { X 1 , … , X n , … } {\textstyle \{X_{1},\ldots ,X_{n},\ldots \}} are independent and identically distributed random variables, each with mean μ {\textstyle \mu } and finite varia... | Lyapunov condition | 0.784232 |
82,314 | The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (Berstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937. The first published complete proof of the GCLT was in 1937 by Paul Lévy in French. An English language version of the complete proof of the... | Lyapunov condition | 0.784232 |
82,315 | Let Sn be the sum of n random variables. Many central limit theorems provide conditions such that Sn/√Var(Sn) converges in distribution to N(0,1) (the normal distribution with mean 0, variance 1) as n → ∞. In some cases, it is possible to find a constant σ2 and function f(n) such that Sn/(σ√n⋅f(n)) converges in distrib... | Lyapunov condition | 0.784232 |
82,316 | For this reason, in public key cryptography, values for a are chosen to have large prime factors of similar size so that they cannot be factored by any publicly known method in a useful time period on any available computer system or computer cluster such as supercomputers and computer grids. The largest cryptography-g... | Trial division | 0.784232 |
82,317 | The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, whi... | Lyapunov condition | 0.784232 |
82,318 | Proofs that use characteristic functions can be extended to cases where each individual X i {\textstyle \mathbf {X} _{i}} is a random vector in R k {\textstyle \mathbb {R} ^{k}} , with mean vector μ = E {\textstyle {\boldsymbol {\mu }}=\operatorname {E} } and covariance matrix Σ {\textstyle \mathbf {\Sigma } } (amon... | Lyapunov condition | 0.784232 |
82,319 | In its common form, the random variables must be independent and identically distributed (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions. The earliest ver... | Lyapunov condition | 0.784232 |
82,320 | Then the limit as n → ∞ {\displaystyle n\to \infty } of the distribution of X ¯ n − μ σ X ¯ n , {\displaystyle {\frac {{\bar {X}}_{n}-\mu }{\sigma _{{\bar {X}}_{n}}}},} where σ X ¯ n = σ n , {\displaystyle \sigma _{{\bar {X}}_{n}}={\frac {\sigma }{\sqrt {n}}},} is the standard normal distribution.In other words, suppos... | Lyapunov condition | 0.784232 |
82,321 | Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. An elementary form of the theorem states the following. Let X 1 , X 2 , … ,... | Lyapunov condition | 0.784232 |
82,322 | In probability theory, the central limit theorem (CLT) establishes that, in many situations, for independent and identically distributed random variables, the sampling distribution of the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally d... | Lyapunov condition | 0.784232 |
82,323 | In gastronomy, "in situ" refers to the art of cooking with the different resources that are available at the site of the event. Here a person is not going to the restaurant, but the restaurant comes to the person's home. | In situ | 0.784231 |
82,324 | In experimental physics in situ typically refers to a method of data collection or manipulation of a sample without exposure to an external environment. For example, the Si(111) 7x7 surface reconstruction is visible using a scanning tunneling microscope when it is prepared and analyzed in situ. | In situ | 0.784231 |
82,325 | A fraction of the globular star clusters in our galaxy, as well as those in other massive galaxies, might have formed in situ. The rest might have been accreted from now-defunct dwarf galaxies. In astronomy, in situ also refers to in situ planet formation, in which planets are hypothesized to have formed at the orbital... | In situ | 0.784231 |
82,326 | In electrochemistry, the phrase in situ refers to performing electrochemical experiments under operating conditions of the electrochemical cell, i.e., under potential control. This is opposed to doing ex situ experiments that are performed under the absence of potential control. Potential control preserves the electroc... | In situ | 0.784231 |
82,327 | Another use of the term in-situ that appears in Computer Science focuses primarily on the use of technology and user interfaces to provide continuous access to situationally relevant information in various locations and contexts. Examples include athletes viewing biometric data on smartwatches to improve their performa... | In situ | 0.784231 |
82,328 | In computer science an in situ operation is one that occurs without interrupting the normal state of a system. For example, a file backup may be restored over a running system, without needing to take the system down to perform the restore. In the context of a database, a restore would allow the database system to cont... | In situ | 0.784231 |
82,329 | Examples of the former include the Corey-Chaykovsky reagent and adrenochrome. In biomedical engineering, protein nanogels made by the in situ polymerization method provide a versatile platform for storage and release of therapeutic proteins. It has tremendous applications for cancer treatment, vaccination, diagnosis, r... | In situ | 0.784231 |
82,330 | In chemistry, in situ typically means "in the reaction mixture." There are numerous situations in which chemical intermediates are synthesized in situ in various processes. This may be done because the species is unstable, and cannot be isolated, or simply out of convenience. | In situ | 0.784231 |
82,331 | This would not be in vivo as the donor is sacrificed by experimentation, but it would not be the same as working with the cell alone (a common scenario for in vitro experiments). For instance, an example of biomedical engineering in situ involves the procedures to directly create an implant from a patient's own tissue ... | In situ | 0.784231 |
82,332 | In biology and biomedical engineering, in situ means to examine the phenomenon exactly in place where it occurs (i.e., without moving it to some special medium). In the case of observations or photographs of living animals, it means that the organism was observed (and photographed) in the wild, exactly as it was found ... | In situ | 0.784231 |
82,333 | In situ (; often not italicized in English) is a Latin phrase that translates literally to "on site" or "in position" and is the only way to accurately measure and bend cable tray prior to installation to ensure the tray bending radii does not exceed that specified by the manufacturer. It can mean "locally", "on site",... | In situ | 0.784231 |
82,334 | It features processor and sensor technology from ST Microelectronics and a NuttX real-time operating system. The APM 2.6 is an open source autopilot system. It allows the user to turn fixed rotary wing or multirotor vehicles, including cars and boats, into fully autonomous vehicles capable of performing programmed GPS ... | 3D Robotics | 0.78423 |
82,335 | 3DR is an American company situated in Berkeley, California, that makes enterprise drone software for construction, engineering, and mining firms, along with government agencies.Prior to 2016, the company designed and marketed commercial and recreational unmanned aerial vehicles. They produced consumer drones, ready-to... | 3D Robotics | 0.78423 |
82,336 | Later however, according to Richard Amasino, Lysenko inaccurately asserted that the vernalized state could be inherited, i.e. the offspring of a vernalized plant would behave as if they themselves had also been vernalized and would not require vernalization in order to flower quickly. Opposing this view and supporting ... | Vernalization | 0.78423 |
82,337 | As of July 2012, the MML included 1150 articles written by 241 authors. In aggregate, these contain more than 10,000 formal definitions of mathematical objects and about 52,000 theorems proved on these objects. More than 180 named mathematical facts have so benefited from formal codification. Some examples are the Hahn... | Mizar system | 0.784229 |
82,338 | All MML articles are available in PDF form as the papers of the Journal of Formalized Mathematics. The full text of the MML is distributed with the Mizar checker and can be freely downloaded from the Mizar website. In an ongoing recent project the library was also made available in an experimental wiki form that only a... | Mizar system | 0.784229 |
82,339 | For a proof to be admitted, all steps have to be justified either by elementary logical arguments or by citing previously verified proofs. This results in a higher level of rigor and detail than is customary in mathematical textbooks and publications. Thus, a typical Mizar article is about four times as long as an equi... | Mizar system | 0.784229 |
82,340 | The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in the proof of new theorems. The system is maintained and developed by the M... | Mizar system | 0.784229 |
82,341 | The Mizar Project was started around 1973 by Andrzej Trybulec as an attempt to reconstruct mathematical vernacular so it can be checked by a computer. Its current goal, apart from the continual development of the Mizar System, is the collaborative creation of a large library of formally verified proofs, covering most o... | Mizar system | 0.784229 |
82,342 | The Mizar Mathematical Library (MML) includes all theorems to which authors can refer in newly written articles. Once approved by the proof checker they are further evaluated in a process of peer-review for appropriate contribution and style. If accepted they are published in the associated Journal of Formalized Mathem... | Mizar system | 0.784229 |
82,343 | A plasma railgun is a linear accelerator which, like a projectile railgun, uses two long parallel electrodes to accelerate a "sliding short" armature. However, in a plasma railgun, the armature and ejected projectile consists of plasma, or hot, ionized, gas-like particles, instead of a solid slug of material. Scientifi... | Plasma railgun | 0.784226 |
82,344 | Controlled jets from plasma rail guns can have peak densities in the 1013 to 1016 particles/m3 range, and velocities from 5 to 200 km/s, depending on device design configuration and operating parameters, and the upper limits may be higher. Plasma rail guns are being evaluated for applications in magnetic confinement fu... | Plasma railgun | 0.784226 |
82,345 | As plasma railguns are an open area of research, the method of armature formation varies. However, techniques including exploding foils, gas cell burst disk injection, neutral gas injection via fast gas valve, and plasma capillary injection have been employed. After armature formation, the plasmoid is then accelerated ... | Plasma railgun | 0.784226 |
82,346 | Project A119, also known as A Study of Lunar Research Flights, was a top-secret plan developed in 1958 by the United States Air Force. The aim of the project was to detonate a nuclear bomb on the Moon, which would help in answering some of the mysteries in planetary astronomy and astrogeology. If the explosive device d... | Project A119 | 0.784226 |
82,347 | Sagan's involvement with the project was apparent from his application for an academic scholarship at the Miller Institute of the University of California, Berkeley, in 1959. In the application, Sagan gave details of the project research, which Davidson felt constituted a violation of national security. The leak consis... | Project A119 | 0.784226 |
82,348 | In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. | Great Picard theorem | 0.784224 |
82,349 | In the case where the values of f {\textstyle f} are missing a single point, this point is called a lacunary value of the function. Great Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle w} , then on any punctured neighborhood of w , f ( z ) {\textstyle w,... | Great Picard theorem | 0.784224 |
82,350 | If f {\textstyle f} omits two values, then the composition of f {\textstyle f} with the inverse of the modular function maps the plane into the unit disc which implies that f {\textstyle f} is constant by Liouville's theorem. This theorem is a significant strengthening of Liouville's theorem which states that the image... | Great Picard theorem | 0.784224 |
82,351 | Little Picard Theorem: If a function f: C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is entire and non-constant, then the set of values that f ( z ) {\textstyle f(z)} assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the... | Great Picard theorem | 0.784224 |
82,352 | In molecular biology, Small nucleolar RNA MBII-202 is a non-coding RNA (ncRNA) molecule which functions in the modification of other small nuclear RNAs (snRNAs). This type of modifying RNA is usually located in the nucleolus of the eukaryotic cell which is a major site of snRNA biogenesis. It is known as a small nucleo... | Small nucleolar RNA MBII-202 | 0.784224 |
82,353 | These had played a role in p-adic cohomology theories (precursors of the crystalline theory, introduced in various forms by Dwork, Monsky, Washnitzer, Lubkin and Katz) particularly in Dwork's work. Such differential equations can be formulated easily enough by means of the algebraic Koszul connections, but in the p-adi... | Crystalline cohomology | 0.784224 |
82,354 | If X is a scheme over S then the sheaf OX/S is defined by OX/S(T) = coordinate ring of T, where we write T as an abbreviation for an object U → T of Cris(X/S). A crystal on the site Cris(X/S) is a sheaf F of OX/S modules that is rigid in the following sense: for any map f between objects T, T′ of Cris(X/S), the natural... | Crystalline cohomology | 0.784224 |
82,355 | One idea for defining a Weil cohomology theory of a variety X over a field k of characteristic p is to 'lift' it to a variety X* over the ring of Witt vectors of k (that gives back X on reduction mod p), then take the de Rham cohomology of this lift. The problem is that it is not at all obvious that this cohomology is ... | Crystalline cohomology | 0.784224 |
82,356 | The gaming industry embraced AR technology. A number of games were developed for prepared indoor environments, such as AR air hockey, Titans of Space, collaborative combat against virtual enemies, and AR-enhanced pool table games.In 2010, Ogmento became the first AR gaming startup to receive VC Funding. The company wen... | Applications of augmented reality | 0.784223 |
82,357 | Augmented reality has been explored for many applications, from gaming and entertainment to medicine, education and business. Example application areas described below include archaeology, architecture, commerce and education. Some of the earliest cited examples include augmented reality used to support surgery by prov... | Applications of augmented reality | 0.784223 |
82,358 | Primary school children learn easily from interactive experiences. As an example, astronomical constellations and the movements of objects in the solar system were oriented in 3D and overlaid in the direction the device was held, and expanded with supplemental video information. Paper-based science book illustrations c... | Applications of augmented reality | 0.784223 |
82,359 | Surgical Endoscopy is a peer-reviewed medical journal published by Springer Science+Business Media. It is the official journal of the Society of American Gastrointestinal and Endoscopic Surgeons and the European Association for Endoscopic Surgery.Surgical Endoscopy covers the surgical aspects of interventional endoscop... | Surgical Endoscopy | 0.784222 |
82,360 | A further construction in A1-homotopy theory is the category SH(S), which is obtained from the above unstable category by forcing the smash product with Gm to become invertible. This process can be carried out either using model-categorical constructions using so-called Gm-spectra or alternatively using infinity-catego... | Motivic homotopy theory | 0.784221 |
82,361 | The setup, especially the Nisnevich topology, is chosen as to make algebraic K-theory representable by a spectrum, and in some aspects to make a proof of the Bloch-Kato conjecture possible. After the Morel-Voevodsky construction there have been several different approaches to A1 homotopy theory by using other model cat... | Motivic homotopy theory | 0.784221 |
82,362 | The Berliner Astronomisches Jahrbuch (abbrev. B.A.J.) is an astronomical ephemeris almanac and one of the longest publication series in astronomy. | Berliner Astronomisches Jahrbuch | 0.78422 |
82,363 | The scope of MV is broad. MV is related to, though distinct from, computer vision.3.Three dimensional art Three-dimensional art is observed in terms of its height, width and depth. It is not flat like two-dimensional art, which consists of paintings, drawings and photographs. | 3D stereo view | 0.784219 |
82,364 | Equivalent statement 4 – connection to elliptic curves: If a, b, c is a non-trivial solution to ap + bp = cp, p odd prime, then y2 = x(x − ap)(x + bp) (Frey curve) will be an elliptic curve.Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles ... | Fermat's conjecture | 0.784219 |
82,365 | There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem. In order to state them, we use the following notations: let N be the set of natural numbers 1, 2, 3, ..., let Z be the set of integers 0, ±1, ±2, ..., and let Q be the set of ra... | Fermat's conjecture | 0.784219 |
82,366 | The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case m = n = k. The Beal conjecture, also known as the Mauldin conj... | Fermat's conjecture | 0.784219 |
82,367 | The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad et al. (1999), and Breuil et al. (2001) who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem. Severa... | Fermat's conjecture | 0.784219 |
82,368 | In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution (a, b, c) for exponent p > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-Hellegouarch) y2 = x (x − ap)(x + bp)would have such unusu... | Fermat's conjecture | 0.784219 |
82,369 | See the history of ideal numbers.) Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) that conjecturally occur approximately 39% of the time; the only irregula... | Fermat's conjecture | 0.784219 |
82,370 | He succeeded in that task by developing the ideal numbers. (Note: It is often stated that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Lejeune Dirichlet told him his arg... | Fermat's conjecture | 0.784219 |
82,371 | In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation xp + yp = zp in complex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes,... | Fermat's conjecture | 0.784219 |
82,372 | The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration. Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents. Althoug... | Fermat's conjecture | 0.784219 |
82,373 | Therefore, Fermat's Last Theorem could be proved for all n if it could be proved for n = 4 and for all odd primes p. In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th cent... | Fermat's conjecture | 0.784219 |
82,374 | After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation an + bn = cn has no positive integer solutions (a, b, c) when n is an odd prime number. This follows because... | Fermat's conjecture | 0.784219 |
82,375 | Nothing I ever do again will mean as much." — Andrew Wiles, as quoted by Simon SinghOn 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain ... | Fermat's conjecture | 0.784219 |
82,376 | : 289, 296–297 However, without this part proved, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor, without success. By the end of 1993, rumours had spread that under scrutiny, ... | Fermat's conjecture | 0.784219 |
82,377 | Their conclusion at the time was that the techniques Wiles used seemed to work correctly. : 261–265 By mid-May 1993, Wiles was ready to tell his wife he thought he had solved the proof of Fermat's Last Theorem,: 265 and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 J... | Fermat's conjecture | 0.784219 |
82,378 | Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, Andrew Wiles, an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the... | Fermat's conjecture | 0.784219 |
82,379 | Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer. His proof is equivalent to demonstrating that the equation x 4 − y 4 = z 2 {\displaystyle x^{4}-y^{4}=z^{2}} has no pr... | Fermat's conjecture | 0.784219 |
82,380 | In mathematics, Lehrbuch der Topologie (German for "textbook of topology") is a book by Herbert Seifert and William Threlfall, first published in 1934 and published in an English translation in 1980. It was one of the earliest textbooks on algebraic topology, and was the standard reference on this topic for many years.... | Lehrbuch der Topologie | 0.784219 |
82,381 | In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added t... | Fermat's conjecture | 0.784219 |
82,382 | The modularity theorem—if proved for semi-stable elliptic curves—would mean that all semistable elliptic curves must be modular. Ribet's theorem showed that any solution to Fermat's equation for a prime number could be used to create a semistable elliptic curve that could not be modular; The only way that both of these... | Fermat's conjecture | 0.784219 |
82,383 | : 203–205, 223, 226 Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was plausible but did not go as far a... | Fermat's conjecture | 0.784219 |
82,384 | Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem, or at least to prove it for the types of elliptical curves that included Frey's equation (known as semistable elliptic curves). This was widely believed inaccessible to proof by contemp... | Fermat's conjecture | 0.784219 |
82,385 | By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture. In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to dis... | Fermat's conjecture | 0.784219 |
82,386 | His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897).Fermat's Last Theorem was also proved for the exponents n = 6, 10, and 14. Proofs for ... | Fermat's conjecture | 0.784219 |
82,387 | As a byproduct of this latter work, she proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem (namely, the case in which p {\displaystyle p} does not divide x y z {\displaystyle xyz} ) for every odd prime exponent less than 270 {\displaystyle 270} , and for all primes p {\displaystyle ... | Fermat's conjecture | 0.784219 |
82,388 | In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes θ {\displaystyle \theta } constructed from the prime exponent p {\displaystyle p} by the equation θ = 2 h p + 1 {\displaystyle \theta =2hp+1} , wher... | Fermat's conjecture | 0.784219 |
82,389 | Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function.According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p {\displaystyle p} is an odd prime number where ! ! {\displa... | Superfactorial | 0.784218 |
82,390 | In mathematics, and more specifically number theory, the superfactorial of a positive integer n {\displaystyle n} is the product of the first n {\displaystyle n} factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials. | Superfactorial | 0.784218 |
82,391 | The implication (1) ⟹ {\displaystyle \implies } (2) in the following theorem is known as the Robinson–Ursescu theorem. | Ursescu theorem | 0.784217 |
82,392 | i S := aint aff ( S − S ) S {\displaystyle {}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S} denotes the relative algebraic interior of S {\displaystyle S} (i.e. the algebraic interior of S {\displaystyle S} in aff ( S − S ) {\displaystyle \operatorname {aff} (S-S)} ). i b S := i S {\displaystyle {}^{i... | Ursescu theorem | 0.784217 |
82,393 | The following notation and notions are used, where R: X ⇉ Y {\displaystyle {\mathcal {R}}:X\rightrightarrows Y} is a set-valued function and S {\displaystyle S} is a non-empty subset of a topological vector space X {\displaystyle X}: the affine span of S {\displaystyle S} is denoted by aff S {\displaystyle \operatorn... | Ursescu theorem | 0.784217 |
82,394 | In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle. | Ursescu theorem | 0.784217 |
82,395 | The two districts of 2012 are Michigan and Mid-Atlantic Robotics (MAR): Week 1Kettering University FIRST Robotics District Competition - Flint, MI USA Gull Lake FIRST Robotics District Competition - Richland, MI Hatboro-Horsham FIRST Robotics District Competition - Horsham, PA USAWeek 2Traverse City FIRST Robotics Dist... | Rebound Rumble | 0.784215 |
82,396 | Rebound Rumble is the 2012 FIRST Robotics Competition game. It is styled similarly to basketball. | Rebound Rumble | 0.784215 |
82,397 | Compact oriented manifolds M and N satisfy σ ( M ⊔ N ) = σ ( M ) + σ ( N ) {\displaystyle \sigma (M\sqcup N)=\sigma (M)+\sigma (N)} by definition, and satisfy σ ( M × N ) = σ ( M ) σ ( N ) {\displaystyle \sigma (M\times N)=\sigma (M)\sigma (N)} by a Künneth formula.If M is an oriented boundary, then σ ( M ) = 0 {\displ... | Signature (topology) | 0.784215 |
82,398 | In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four. This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem. | Signature (topology) | 0.784215 |
82,399 | Ming Wu, a UC Berkeley Professor of electrical engineering and computer sciences invented the new optoelectronic tweezers. Wu transformed the optical energy from low powered light emitting diodes (LED) into electrical energy via a photoconductive surface. The idea is to allow the LED to switch on and off the photocondu... | Laser tweezer | 0.784214 |
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