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84,500 | The Chinese remainder theorem can be generalized to any ring, by using coprime ideals (also called comaximal ideals). Two ideals I and J are coprime if there are elements i ∈ I {\displaystyle i\in I} and j ∈ J {\displaystyle j\in J} such that i + j = 1. {\displaystyle i+j=1.} This relation plays the role of Bézout's id... | Linear congruence theorem | 0.783293 |
84,501 | Suppose that x and y are both solutions to all the congruences. As x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N also divides x − y, and thus x and y are congruent modulo N. If x and y are supposed to be non-negative... | Linear congruence theorem | 0.783293 |
84,502 | This may be much faster than the direct computation if N and the number of operations are large. This is widely used, under the name multi-modular computation, for linear algebra over the integers or the rational numbers. The theorem can also be restated in the language of combinatorics as the fact that the infinite ar... | Linear congruence theorem | 0.783293 |
84,503 | New biotechnology firms often lack the complementary assets to commercialize their innovations and thus form collaborative partnerships with large incumbent firms who do possess the necessary complementary assets such as manufacturing capabilities, marketing channels, brand name, etc. (Rothaermel, 2001)RC Cola was the ... | Complementary assets | 0.783292 |
84,504 | The Science Museum of the University of Coimbra (Museu da Ciência da Universidade de Coimbra) gathers the historical scientific collections of several units of the University of Coimbra, in Coimbra, Portugal. It includes the collection of scientific instruments from the 18th and 19th century of the Physics Museum, the ... | Science Museum of the University of Coimbra | 0.783291 |
84,505 | In molecular biology, pertactin (PRN) is a highly immunogenic virulence factor of Bordetella pertussis, the bacterium that causes pertussis. Specifically, it is an outer membrane protein that promotes adhesion to tracheal epithelial cells. PRN is purified from Bordetella pertussis and is used for the vaccine production... | Pertactin | 0.78329 |
84,506 | The De Donder–Weyl theory has been a part of the calculus of variations since the 1930s and initially it found very few applications in physics. Recently it was applied in theoretical physics in the context of quantum field theory and quantum gravity.In 1970, Jedrzej Śniatycki, the author of Geometric quantization and ... | De Donder–Weyl theory | 0.78329 |
84,507 | In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theo... | De Donder–Weyl theory | 0.78329 |
84,508 | The Artin reciprocity law says that for a global field K {\displaystyle K} , C K ^ = A K × / K × ^ ≃ Gal ( K ab / K ) {\displaystyle {\widehat {C_{K}}}={\widehat {\mathbf {A} _{K}^{\times }/K^{\times }}}\ \simeq \ {\text{Gal}}(K^{\text{ab}}/K)} where K a b {\displaystyle K^{ab}} is the maximal abelian algebraic extensi... | Ring of finite adeles | 0.783288 |
84,509 | Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if one place (or more) is omitted, the property of discreteness of K {\displaystyle K} is turned into a denseness of K . {\displaystyle K.} | Ring of finite adeles | 0.783288 |
84,510 | {\displaystyle \forall n\in \{1,\ldots ,N\}:\quad |\alpha _{n}-\xi |_{n}<\varepsilon .} Strong Approximation Theorem. | Ring of finite adeles | 0.783288 |
84,511 | Weak Approximation Theorem. Let | ⋅ | 1 , … , | ⋅ | N , {\displaystyle |\cdot |_{1},\ldots ,|\cdot |_{N},} be inequivalent valuations of K . {\displaystyle K.} Let K n {\displaystyle K_{n}} be the completion of K {\displaystyle K} with respect to | ⋅ | n . | Ring of finite adeles | 0.783288 |
84,512 | Theorem. Let K {\displaystyle K} be a number field. There exists a finite set of places S , {\displaystyle S,} such that: I K = ( I K , S × ∏ v ∉ S O v × ) K × = ( ∏ v ∈ S K v × × ∏ v ∉ S O v × ) K × . {\displaystyle I_{K}=\left(I_{K,S}\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }=\left(\prod _{v\in S}K_{v... | Ring of finite adeles | 0.783288 |
84,513 | {\displaystyle K.} Let E ⋆ {\displaystyle E^{\star }} and A E ⋆ {\displaystyle \mathbb {A} _{E}^{\star }} be the algebraic duals of E {\displaystyle E} and A E . {\displaystyle \mathbb {A} _{E}.} | Ring of finite adeles | 0.783288 |
84,514 | {\displaystyle K_{v}.} The idea has been illustrated by showing R {\displaystyle \mathbb {R} } is self-dual. Define: { e ∞: R → T e ∞ ( t ) := exp ( 2 π i t ) {\displaystyle {\begin{cases}e_{\infty }:\mathbb {R} \to \mathbb {T} \\e_{\infty }(t):=\exp(2\pi it)\end{cases}}} Then the following map is an isomorphism whic... | Ring of finite adeles | 0.783288 |
84,515 | {\displaystyle G.} One can show that G ^ {\displaystyle {\widehat {G}}} is also a locally compact abelian group. Theorem. | Ring of finite adeles | 0.783288 |
84,516 | Equivalently G ^ {\displaystyle {\widehat {G}}} is the set of all continuous group homomorphisms from G {\displaystyle G} to T := { z ∈ C: | z | = 1 } . {\displaystyle \mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}.} Equip G ^ {\displaystyle {\widehat {G}}} with the topology of uniform convergence on compact subsets of G . | Ring of finite adeles | 0.783288 |
84,517 | {\displaystyle K,\mathbb {A} _{K}=\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.} Remark. Using A Q ⊗ Q K ≅ A Q ⊕ ⋯ ⊕ A Q , {\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },} where there are {\displaystyle } summand... | Ring of finite adeles | 0.783288 |
84,518 | Define A Z := Z ^ × R . {\displaystyle \mathbb {A} _{\mathbb {Z} }:={\widehat {\mathbb {Z} }}\times \mathbb {R} .} This results in an algebraic isomorphism A Q ≅ A Z ⊗ Z Q . | Ring of finite adeles | 0.783288 |
84,519 | This follows from the Chinese Remainder Theorem. Lemma. A Q , fin = Z ^ ⊗ Z Q . | Ring of finite adeles | 0.783288 |
84,520 | Let A {\displaystyle A} be a finite-dimensional algebra over K . {\displaystyle K.} Since A A × {\displaystyle \mathbb {A} _{A}^{\times }} is not a topological group with the subset-topology in general, equip A A × {\displaystyle \mathbb {A} _{A}^{\times }} with the topology similar to I K {\displaystyle I_{K}} above a... | Ring of finite adeles | 0.783288 |
84,521 | {\displaystyle \forall \alpha ,\beta \in \mathbb {A} _{K}{\text{ and }}\forall a,b\in A:\qquad (\alpha \otimes _{K}a)\cdot (\beta \otimes _{K}b):=(\alpha \beta )\otimes _{K}(ab).} As a consequence, A A {\displaystyle \mathbb {A} _{A}} is an algebra with a unit over A K . {\displaystyle \mathbb {A} _{K}.} | Ring of finite adeles | 0.783288 |
84,522 | Let A {\displaystyle A} be a finite-dimensional algebra over K . {\displaystyle K.} In particular, A {\displaystyle A} is a finite-dimensional vector-space over K . {\displaystyle K.} | Ring of finite adeles | 0.783288 |
84,523 | Inside class field theory one can distinguish special class field theory and general class field theory. Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used ... | Global class field theory | 0.783287 |
84,524 | The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of C F . {\displaystyle C_{F}.} A standard method for developing global class field theory since the 1930s was to construct local class field t... | Global class field theory | 0.783287 |
84,525 | In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.Hilbert is credited as one of pioneers of the notion of a class field. However, this notion wa... | Global class field theory | 0.783287 |
84,526 | Stress-induced leakage current (SILC) is an increase in the gate leakage current of a MOSFET, used in semiconductor physics. It occurs due to defects created in the gate oxide during electrical stressing. SILC is perhaps the largest factor inhibiting device miniaturization. Increased leakage is a common failure mode of... | Stress-induced leakage current | 0.783286 |
84,527 | The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".The modern words "sine" and "cosine" are derived from the Latin word sinus via mistranslation from Arabic (see Sine and cosine#Etymology). Particularly Fibonacci's sinus rectus arcus proved influential in establish... | History of trigonometry | 0.783286 |
84,528 | Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculati... | History of trigonometry | 0.783286 |
84,529 | In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry ... | History of trigonometry | 0.783286 |
84,530 | In 1342, Levi ben Gershon, known as Gersonides, wrote On Sines, Chords and Arcs, in particular proving the sine law for plane triangles and giving five-figure sine tables.A simplified trigonometric table, the "toleta de marteloio", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate... | History of trigonometry | 0.783286 |
84,531 | In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It becam... | History of trigonometry | 0.783286 |
84,532 | Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic as... | History of trigonometry | 0.783286 |
84,533 | {\displaystyle \sin ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos(x)}{2}}.} Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although de... | History of trigonometry | 0.783286 |
84,534 | A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the ... | History of trigonometry | 0.783286 |
84,535 | Ptolemy's table of chords gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees n in the corresponding arc of the circle, for n ranging from 1/2 to 180 by increments of 1/2. The thirteen books of the Almagest are the most influential and significant trigonometric work of all ant... | History of trigonometry | 0.783286 |
84,536 | 90 – ca. 168 AD) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. | History of trigonometry | 0.783286 |
84,537 | And Book III contains the "theorem of Menelaus". He further gave his famous "rule of six quantities".Later, Claudius Ptolemy (ca. | History of trigonometry | 0.783286 |
84,538 | He established a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles. Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater ... | History of trigonometry | 0.783286 |
84,539 | In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts. It is due to the Babylonian sexagesimal numeral system t... | History of trigonometry | 0.783286 |
84,540 | 260 BC), since he measured an angle in terms of a fraction of a quadrant. It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day tha... | History of trigonometry | 0.783286 |
84,541 | And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles. To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the statement that, in modern notation, sin α/sin β < α/β < tan α/tan β whenever 0° < β < α < 90°, now ... | History of trigonometry | 0.783286 |
84,542 | Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a g... | History of trigonometry | 0.783286 |
84,543 | 1680–1620 BC), contains the following problem related to trigonometry: "If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?" Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other word... | History of trigonometry | 0.783286 |
84,544 | There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table. The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. The Rhind Mathematical Papyrus, written by the Egyptian ... | History of trigonometry | 0.783286 |
84,545 | The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.The Babylonian astronomers kept detailed records on the ... | History of trigonometry | 0.783286 |
84,546 | Later in the 7th century, Brahmagupta redeveloped the formula 1 − sin 2 ( x ) = cos 2 ( x ) = sin 2 ( π 2 − x ) {\displaystyle \ 1-\sin ^{2}(x)=\cos ^{2}(x)=\sin ^{2}\left({\frac {\pi }{2}}-x\right)} (also derived earlier, as mentioned above) and the Brahmagupta interpolation formula for computing sine values.Ano... | History of trigonometry | 0.783286 |
84,547 | Some of the early and very significant developments of trigonometry were in India. Influential works from the 4th–5th century AD, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chor... | History of trigonometry | 0.783286 |
84,548 | Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a form suitable for triangulation. In France, the law of cosines is still referred to as the theorem o... | History of trigonometry | 0.783286 |
84,549 | In the late 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherica... | History of trigonometry | 0.783286 |
84,550 | {\displaystyle {\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.} Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity: cos a cos b = cos ( a + b ) + co... | History of trigonometry | 0.783286 |
84,551 | He also developed the following trigonometric formula: sin ( 2 x ) = 2 sin ( x ) cos ( x ) {\displaystyle \ \sin(2x)=2\sin(x)\cos(x)} (a special case of Ptolemy's angle-addition formula; see above)In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine minutes, and th... | History of trigonometry | 0.783286 |
84,552 | To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, ... | History of trigonometry | 0.783286 |
84,553 | In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applyi... | History of trigonometry | 0.783286 |
84,554 | Previous works were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due t... | History of trigonometry | 0.783286 |
84,555 | The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E ( Q ) / 2 E ( Q ) {\displaystyle E(\mathbb {Q} )/2E(\mat... | Mordell–Weil theorem | 0.783286 |
84,556 | In mathematics, the Mordell–Weil theorem states that for an abelian variety A {\displaystyle A} over a number field K {\displaystyle K} , the group A ( K ) {\displaystyle A(K)} of K-rational points of A {\displaystyle A} is a finitely-generated abelian group, called the Mordell–Weil group. The case with A {\displaystyl... | Mordell–Weil theorem | 0.783286 |
84,557 | Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as: T f = − {\displaystyle T_{f}=-} ,where T P / Y = Ω P / Y ∨ {\displaystyle T_{P/Y}=\Omega _{P/Y}^{\vee }} is the relative tangent sheaf of p {\displaystyle p} (which is locally free since p {\displaystyle p} is smo... | Virtual tangent bundle | 0.783285 |
84,558 | In plasma physics, the Lundquist number (denoted by S {\displaystyle S} ) is a dimensionless ratio which compares the timescale of an Alfvén wave crossing to the timescale of resistive diffusion. It is a special case of the magnetic Reynolds number when the Alfvén velocity is the typical velocity scale of the system, a... | Lundquist number | 0.783283 |
84,559 | The Oliver E. Buckley Condensed Matter Prize is an annual award given by the American Physical Society "to recognize and encourage outstanding theoretical or experimental contributions to condensed matter physics." It was endowed by AT&T Bell Laboratories as a means of recognizing outstanding scientific work. The prize... | Oliver E. Buckley Condensed Matter Prize | 0.783283 |
84,560 | A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L1 ... | Taxicab geometry | 0.78328 |
84,561 | The geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In R 2 {\displaystyle \mathbb {R} ^{2}} , the taxicab distance between two points ( x 1 , y 1 ) {... | Taxicab geometry | 0.78328 |
84,562 | In solving an underdetermined system of linear equations, the regularization term for the parameter vector is expressed in terms of the ℓ 1 {\displaystyle \ell _{1}} norm (taxicab geometry) of the vector. This approach appears in the signal recovery framework called compressed sensing. | Taxicab geometry | 0.78328 |
84,563 | This stain was originally discovered by ICI in the 1940s as a member of the competitive dye industry for the purpose of industrial dying. It was used for some time for staining textiles, leather products and inks. ICI sold thousands of tons of alcian blue and filed multiple patents regarding its manufacturing process t... | Alcian blue | 0.783279 |
84,564 | Oxford online dictionary mentions that it was a trademark and also specifies "1940s: Alcian perhaps from (phth)al(o)cyan'(ine) with a phonetic respelling".This hypothesis is consistent with the name of Alcian green, which is a tetraphenyl-phthalocyanine with copper.However Prof. J. E. Scott who had cracked the chemistr... | Alcian blue | 0.783279 |
84,565 | While the popularity of Alcian blue expanded exponentially, the difficulty involved in its production due to environmentally hazardous intermediate steps made its availability difficult and ICI stopped producing it by 1973. Many of the alternate sources sold similar looking color products with unreliable staining. Prof... | Alcian blue | 0.783279 |
84,566 | Alcian blue can be used to quantitate acidic glycans both in microspectrophotometric quantitation in solution or for staining glycoproteins in polyacrylamide gels or on western blots. Biochemists had used it to assay acid polysaccharides in urine since the 1960s for diagnosis of diseases like mucopolysaccharidosis but ... | Alcian blue | 0.783279 |
84,567 | Thus we have found an involution with the desired properties and completed the proof of the Lindström-Gessel-Viennot lemma. Remark. Arguments similar to the one above appear in several sources, with variations regarding the choice of which tails to switch. A version with j smallest (unequal to i) rather than largest ap... | Lindström–Gessel–Viennot lemma | 0.783276 |
84,568 | {\displaystyle s_{(3,2,2,1)}={\begin{vmatrix}h_{3}&h_{4}&h_{5}&h_{6}\\h_{1}&h_{2}&h_{3}&h_{4}\\1&h_{1}&h_{2}&h_{3}\\0&0&1&h_{1}\end{vmatrix}}.} To prove the equivalence, given any partition λ as above, one considers the r starting points a i = ( r + 1 − i , 1 ) {\displaystyle a_{i}=(r+1-i,1)} and the r ending points b ... | Lindström–Gessel–Viennot lemma | 0.783276 |
84,569 | In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): ∀ i ≠ j x i j ≥ 0. {\displaystyle \forall _{i\neq j}\,x_{ij}\geq 0.} It is named after the American economist Lloyd Metzler. Metzler matrices appear in stability analysis of time delaye... | Metzler matrix | 0.783275 |
84,570 | In the late 2010s, a trend of SoCs implementing communications subsystems in terms of a network-like topology instead of bus-based protocols has emerged. A trend towards more processor cores on SoCs has caused on-chip communication efficiency to become one of the key factors in determining the overall system performanc... | System on a Chip | 0.783275 |
84,571 | In mathematics, the Chung–Fuchs theorem, named after Chung Kai-lai and Wolfgang Heinrich Johannes Fuchs, states that for a particle undergoing a random walk in m-dimensions, it is certain to come back infinitely often to any neighborhood of the origin on a one-dimensional line (m = 1) or two-dimensional plane (m = 2), ... | Chung–Fuchs theorem | 0.783274 |
84,572 | Inhibitors of ALS are used as herbicides that slowly starve affected plants of these amino acids, which eventually leads to inhibition of DNA synthesis. They affect grasses and dicots alike. They are not a chemistry class but rather a mechanism of action class with diverse chemistries. The ALS inhibitor family includes... | Acetolactate synthase | 0.783274 |
84,573 | It has been called the "fundamental theorem of Boolean algebra". Besides its theoretical importance, it paved the way for binary decision diagrams (BDDs), satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits. In such engineering contexts (especia... | Boole's expansion theorem | 0.783273 |
84,574 | Binary decision diagrams follow from systematic use of this theorem Any Boolean function can be implemented directly in a switching circuit using a hierarchy of basic multiplexer by repeated application of this theorem. | Boole's expansion theorem | 0.783273 |
84,575 | A more explicit way of stating the theorem is: f ( X 1 , X 2 , … , X n ) = X 1 ⋅ f ( 1 , X 2 , … , X n ) + X 1 ′ ⋅ f ( 0 , X 2 , … , X n ) {\displaystyle f(X_{1},X_{2},\dots ,X_{n})=X_{1}\cdot f(1,X_{2},\dots ,X_{n})+X_{1}'\cdot f(0,X_{2},\dots ,X_{n})} | Boole's expansion theorem | 0.783273 |
84,576 | Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: F = x ⋅ F x + x ′ ⋅ F x ′ {\displaystyle F=x\cdot F_{x}+x'\cdot F_{x'}} , where F {\displaystyle F} is any Boolean function, x {\displaystyle x} is a variable, x ′ {\displaystyle x'} is the complement of x {\display... | Boole's expansion theorem | 0.783273 |
84,577 | In semiconductor physics, an acceptor is a dopant atom that when substituted into a semiconductor lattice forms a p-type region. When silicon (Si), having four valence electrons, is doped with elements from group III of the periodic table, such as boron (B) and aluminium (Al), both having three valence electrons, a p-t... | Acceptor (semiconductors) | 0.783273 |
84,578 | Like vancomycin, telavancin inhibits bacterial cell wall synthesis by binding to the D-Ala-D-Ala terminus of the peptidoglycan in the growing cell wall (see Pharmacology and chemistry of vancomycin). In addition, it disrupts bacterial membranes by depolarization. | Telavancin | 0.78327 |
84,579 | The Yale Interdisciplinary Center for Bioethics, or YICB, is an academic research center based primarily in the study of biomedical ethics.It is partnered with the Hastings Center to sponsor visiting Yale/Hastings Bioethics scholars. It also hosts the Sherwin Nuland international Summer Bioethics Institute (SBI). It is... | Yale Interdisciplinary Center for Bioethics | 0.783269 |
84,580 | In 1986, V.A. Okhonin (Institute of Biophysics, USSR Academy of Sciences, Siberian Branch, Krasnoyarsk) had patented the STED idea. This patent was unknown to Hell and Wichmann in 1994. | STED microscope | 0.783267 |
84,581 | Stimulated emission depletion (STED) microscopy is one of the techniques that make up super-resolution microscopy. It creates super-resolution images by the selective deactivation of fluorophores, minimizing the area of illumination at the focal point, and thus enhancing the achievable resolution for a given system. It... | STED microscope | 0.783267 |
84,582 | The IEEE International Conference on Robotics and Automation (ICRA) is an annual academic conference covering advances in robotics. It is one of the premier conferences in its field, alongside the International Conference on Intelligent Robots and Systems (IROS), with an 'A' rating from the Australian Ranking of ICT Co... | International Conference on Robotics and Automation | 0.783265 |
84,583 | 2010 VZ98 orbits the Sun at a distance of 34.3–266.6 AU once every 1846 years (674,262 days; semi-major axis of 150.5 AU). Its orbit has a high eccentricity of 0.77 and an inclination of 5° with respect to the ecliptic. Small number statistics suggest that this body may be trapped in a 3:2 orbital resonance with an uns... | (445473) 2010 VZ98 | 0.783265 |
84,584 | Other estimates indicate about 73,000 new cases occur per year.Obstetric fistulae were very common throughout the world, but since the late 19th century, the rise of gynecology developed safe practices for childbirth, including giving birth at local hospitals rather than at home, which dramatically reduced rates of obs... | Obstetrical fistulae | 0.783264 |
84,585 | Molecular imprinted polymers pose this wide range of capabilities in extraction through highly specific micro-cavity binding sites. Due to the specific binding site created in a MIP this technique is showing promise in analytical chemistry as a useful method for solid phase extraction. The capability for MIPs to be a c... | Molecularly imprinted polymer | 0.783261 |
84,586 | Niche areas for application of MIPs are in sensors and separation. Despite the current good health of molecular imprinting in general, one difficulty which appears to remain to this day is the commercialization of molecularly imprinted polymers. Despite this, many patents (1035 patents, up to October 2018, according to... | Molecularly imprinted polymer | 0.783261 |
84,587 | In polymer chemistry, anionic addition polymerization is a form of chain-growth polymerization or addition polymerization that involves the polymerization of monomers initiated with anions. The type of reaction has many manifestations, but traditionally vinyl monomers are used. Often anionic polymerization involves liv... | Anionic addition polymerization | 0.783261 |
84,588 | Martian Time-Slip is a 1964 science fiction novel by American writer Philip K. Dick. The novel uses the common science fiction concept of a human colony on Mars. However, it also includes the themes of mental illness, the physics of time and the dangers of centralized authority. The novel was first published under the ... | Martian Time-Slip | 0.78326 |
84,589 | The Food Safety and Inspection Service (FSIS) of the United States Department of Agriculture (USDA) provides a test method for analyzing cyromazine and melamine in animal tissues in its Chemistry Laboratory Guidebook which "contains test methods used by FSIS Laboratories to support the Agency's inspection program, ensu... | Cyromazine | 0.783259 |
84,590 | The Sea Ice Physics and Ecosystem eXperiment (SIPEX) was an Australian contribution to the International Polar Year (IPY). The expedition studied the sea ice in the Antarctica zone in September and October 2007, investigating links between the sea ice and the Southern Ocean ecosystems. Taking part were 45 scientists fr... | Sea Ice Physics and Ecosystem eXperiment | 0.783259 |
84,591 | Mechanical properties are important in structural and building materials as well as textile fabrics. In modern materials science, fracture mechanics is an important tool in improving the mechanical performance of materials and components. It applies the physics of stress and strain, in particular the theories of elasti... | Ceramic material | 0.783258 |
84,592 | The Wigner fusion research groups are involved in magnetically confined nuclear fusion experiments around the world. Wigner fusion consists of research groups from four different research institutes and universities, 3 if which are located in the Department of Plasma Physics at the Wigner Research Centre for Physics, o... | Wigner fusion | 0.783257 |
84,593 | The Clinical Pharmacogenetics Implementation Consortium (CPIC) is an international consortium including members of NIH Pharmacogenomics Research Network (PGRN), PharmGKB staff, and experts in PGx and medicine, who are committed to facilitating the use of pharmacogenetic tests to improve patient care. | Clinical Pharmacogenetics Implementation Consortium | 0.783257 |
84,594 | Part B covers various topics in the field of optical physics, such as guided waves, laser spectroscopy, nonlinear optics, quantum optics, lasers, organic and polymer materials for optics, and ultrafast phenomena. The editor-in-chief is Kurt Busch (Humboldt University of Berlin, Germany). | Journal of the Optical Society of America | 0.783255 |
84,595 | Children are disproportionately affected by environmental hazards. WHO found that children under the age of five are more prone to diseases from environmental factors than the rest of the total population.WHO has supported the United Nation's Millennium Development Goals to provide sustainable access to safe drinking w... | Environmental health policy | 0.783254 |
84,596 | Mining in the country had also resulted in water pollution that affects human and animal health. Coal mining has been linked to environmental and health issues, leading the government to close all mining pits by 2018. Germany has been a leading actor in dealing with carbon emissions related to climate change through po... | Environmental health policy | 0.783254 |
84,597 | The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings C ( X ) {\displaystyle C(X)} (where X {\displaystyle X} is realcompact) together with ring homomorphisms as maps. For example one can reconstruct X {\displaystyle X} from C ( X ) {\displaystyle C(X)} when X {\displaystyle X} is... | Completely regular space | 0.783247 |
84,598 | Let ρ be the initial topology on X {\displaystyle X} induced by C τ ( X ) {\displaystyle C_{\tau }(X)} or, equivalently, the topology generated by the basis of cozero sets in ( X , τ ) . {\displaystyle (X,\tau ).} Then ρ will be the finest completely regular topology on X {\displaystyle X} that is coarser than τ . | Completely regular space | 0.783247 |
84,599 | {\displaystyle C_{b}(X).} A space X {\displaystyle X} is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in X {\displaystyle X} (i.e. the zero sets form a basis for the closed sets of X {\displaystyle X} ). A space X {\displaystyle X} is completely regular ... | Completely regular space | 0.783247 |
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