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24,500 | {\displaystyle {\begin{aligned}\operatorname {SU} (n)&\supset \operatorname {SO} (n),\\\operatorname {SU} (2n)&\supset \operatorname {Sp} (n).\end{aligned}}} Since the rank of SU(n) is n − 1 and of U(1) is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original ... | Special unitary group | 0.808176 |
24,501 | The representation theory of S U ( 3 ) {\displaystyle SU(3)} is well-understood. Descriptions of these representations, from the point of view of its complexified Lie algebra s l ( 3 ; C ) {\displaystyle {\mathfrak {sl}}(3;\mathbb {C} )} , may be found in the articles on Lie algebra representations or the Clebsch–Gorda... | Special unitary group | 0.808176 |
24,502 | The variability of the pole of a wave, as noted in studies of polarization, might view elliptical polarization as an exhibit of the elliptical shape of a wave with pole p ≠ ± i {\displaystyle ~p\neq \pm i~} . The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model, and the practice of... | Special unitary group | 0.808176 |
24,503 | {\displaystyle ~q\,q^{*}=w^{2}+x^{2}-y^{2}-z^{2}~.} Note that the 2-sheet hyperboloid { x i + y j + z k: x 2 − y 2 − z 2 = 1 } {\displaystyle \left\{xi+yj+zk:x^{2}-y^{2}-z^{2}=1\right\}} corresponds to the imaginary units in the algebra so that any point p on this hyperboloid can be used as a pole of a sinusoidal wave ... | Special unitary group | 0.808176 |
24,504 | The Weyl group of SU(n) is the symmetric group Sn, which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1). The Lie algebra of SU(n), denoted by s u ( n ) {\displaystyle {\mathfrak {su}}(n)} , can be identified with the set of traceless anti‑Hermitian n×n complex m... | Special unitary group | 0.808176 |
24,505 | Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).The center of SU(n) is isomorphic to the cyclic group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } , and is composed of the diagonal matrices ζ I for ζ an n‑th root of unity and I the n×n identity matrix. Its outer automorphism... | Special unitary group | 0.808176 |
24,506 | As noted above, the Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} of SU ( n ) {\displaystyle \operatorname {SU} (n)} consists of n × n {\displaystyle n\times n} skew-Hermitian matrices with trace zero.The complexification of the Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} is s l ( n ; C ) {\... | Special unitary group | 0.808176 |
24,507 | They also serve as unit vectors for the description of our 3 spatial dimensions in loop quantum gravity. They also correspond to the Pauli X, Y, and Z gates, which are standard generators for the single qubit gates, corresponding to 3d-rotations about the axes of the Bloch sphere. The Lie algebra serves to work out the... | Special unitary group | 0.808176 |
24,508 | {\displaystyle u_{3}=+i\ \sigma _{3}~.} This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons. | Special unitary group | 0.808176 |
24,509 | The Lie algebra of SU(2) consists of 2 × 2 {\displaystyle 2\times 2} skew-Hermitian matrices with trace zero. Explicitly, this means s u ( 2 ) = { ( i a − z ¯ z − i a ): a ∈ R , z ∈ C } . {\displaystyle {\mathfrak {su}}(2)=\left\{{\begin{pmatrix}i\ a&-{\overline {z}}\\z&-i\ a\end{pmatrix}}:\ a\in \mathbb {R} ,z\in \mat... | Special unitary group | 0.808176 |
24,510 | {\displaystyle \left\{T_{a},\,T_{b}\right\}~=~{\tfrac {1}{n}}\,\delta _{ab}\,I_{n}+\sum _{c=1}^{n^{2}-1}{d_{abc}\,T_{c}}~.} The factor of i {\displaystyle i} in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention. The conventional normalization condit... | Special unitary group | 0.808176 |
24,511 | In the physics literature, it is common to identify the Lie algebra with the space of trace-zero Hermitian (rather than the skew-Hermitian) matrices. That is to say, the physicists' Lie algebra differs by a factor of i {\displaystyle i} from the mathematicians'. With this convention, one can then choose generators Ta t... | Special unitary group | 0.808176 |
24,512 | In physics the special unitary group is used to represent bosonic symmetries. In theories of symmetry breaking it is important to be able to find the subgroups of the special unitary group. Subgroups of SU(n) that are important in GUT physics are, for p > 1, n − p > 1, SU ( n ) ⊃ SU ( p ) × SU ( n − p ) × U ( 1... | Special unitary group | 0.808176 |
24,513 | The groups SU(2n) are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n {\displaystyle n} qubits and thus 2 n {\displaystyle 2^{n}} basis states. (Alternatively, the more general unitary group U ( 2 n ) {\displaystyle U(2^{n})} can be used, since mu... | Special unitary group | 0.808176 |
24,514 | As a compact classical group, U(n) is the group that preserves the standard inner product on C n {\displaystyle \mathbb {C} ^{n}} . It is itself a subgroup of the general linear group, SU ( n ) ⊂ U ( n ) ⊂ GL ( n , C ) {\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n... | Special unitary group | 0.808176 |
24,515 | The Japanese Society for Bioinformatics (JSBi) is a Japanese research society on the subjects of bioinformatics and computational biology established in 1999. The society is an affiliated regional group of the International Society for Computational Biology (ISCB), and a member of Association of Asian Societies for Bio... | Japanese Society for Bioinformatics | 0.808173 |
24,516 | A neural substrate is a term used in neuroscience to indicate the part of the central nervous system (i.e., brain and spinal cord) that underlies a specific behavior, cognitive process, or psychological state. Neural is an adjective relating to "a nerve or the nervous system", while a substrate is an "underlying substa... | Neural substrate | 0.808172 |
24,517 | Most of the economic traits are controlled by genes with small but cumulative effects. Although the potential of DH populations in quantitative genetics has been understood for some time, it was the advent of molecular marker maps that provided the impetus for their use in identifying loci controlling quantitative trai... | Doubled haploidy | 0.808171 |
24,518 | Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group G of a certain arithmetic variety X, or some related geometric object, can help to restore X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi ... | Anabelian geometry | 0.808169 |
24,519 | The "anabelian question" has been formulated as How much information about the isomorphism class of the variety X is contained in the knowledge of the étale fundamental group A concrete example is the case of curves, which may be affine as well as projective. Suppose given a hyperbolic curve C, i.e., the complement of ... | Anabelian geometry | 0.808169 |
24,520 | Its main aim to construct algorithms which produce the curve, up to an isomorphism, from the étale fundamental group of such a curve. In particular, for the first time this theory produces a simultaneous functorial restoration of the ground number field and its completion, from the fundamental group of a large class of... | Anabelian geometry | 0.808169 |
24,521 | Shinichi Mochizuki introduced and developed the mono-anabelian geometry, an approach which restores, for a certain class of hyperbolic curves over number fields or some other fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute ... | Anabelian geometry | 0.808169 |
24,522 | The origin of combinatorial anabelian geometry is in some of such combinatorial ideas in Mochizuki's proofs of the Grothendieck conjecture. Some of the results of combinatorial anabelian geometry provide alternative proofs of partial cases of the Grothendieck conjecture without using p-adic Hodge theory. Combinatorial ... | Anabelian geometry | 0.808169 |
24,523 | Shinichi Mochizuki also introduced combinatorial anabelian geometry which deals with issues of hyperbolic curves and other related schemes over algebraically closed fields. The first results were published in Mochizuki's "A combinatorial version of the Grothendieck conjecture" (2007) and "On the combinatorial cuspidali... | Anabelian geometry | 0.808169 |
24,524 | Mass spectral interpretation is the method employed to identify the chemical formula, characteristic fragment patterns and possible fragment ions from the mass spectra. Mass spectra is a plot of relative abundance against mass-to-charge ratio. It is commonly used for the identification of organic compounds from electro... | Mass spectrum analysis | 0.808167 |
24,525 | An expression vector, otherwise known as an expression construct, is usually a plasmid or virus designed for gene expression in cells. The vector is used to introduce a specific gene into a target cell, and can commandeer the cell's mechanism for protein synthesis to produce the protein encoded by the gene. Expression ... | Expression vector | 0.808166 |
24,526 | Most protein pharmaceuticals are now produced through recombinant DNA technology using expression vectors. These peptide and protein pharmaceuticals may be hormones, vaccines, antibiotics, antibodies, and enzymes. The first human recombinant protein used for disease management, insulin, was introduced in 1982. Biotechn... | Expression vector | 0.808166 |
24,527 | Though well-quasi-ordering is an appealing notion, many important infinitary operations do not preserve well-quasi-orderedness. An example due to Richard Rado illustrates this. In a 1965 paper Crispin Nash-Williams formulated the stronger notion of better-quasi-ordering in order to prove that the class of trees of heig... | Better-quasi-ordering | 0.808161 |
24,528 | The relation < ∗ {\displaystyle <^{*}} is not the strict part of the relation ≤ ∗ {\displaystyle \leq ^{*}} . Theorem (Minimal Bad Array Lemma). Let Q {\displaystyle Q} be a quasi-order equipped with a partial ranking and suppose f {\displaystyle f} is a bad Q {\displaystyle Q} -array. Then there is a minimal bad Q {\d... | Better-quasi-ordering | 0.808161 |
24,529 | The Special Interest Group "Model-Driven Software Engineering" (SIG-MDSE) organizes the MDSE as an academic conference. This special interest group focus on model driven (or model based) and architecture centred software engineering techniques and tools, e.g. OMG's MDA (Model Driven Architecture). It is a platform for ... | SIGMDSE | 0.808161 |
24,530 | Simpson introduced an alternative definition of better-quasi-ordering in terms of Borel functions ω → Q {\displaystyle ^{\omega }\to Q} , where ω {\displaystyle ^{\omega }} , the set of infinite subsets of ω {\displaystyle \omega } , is given the usual product topology.Let Q {\displaystyle Q} be a quasi-ordering and ... | Better-quasi-ordering | 0.808161 |
24,531 | There is another definition of character grouppg 29 which uses U ( 1 ) = { z ∈ C ∗: | z | = 1 } {\displaystyle U(1)=\{z\in \mathbb {C} ^{*}:|z|=1\}} as the target instead of just C ∗ {\displaystyle \mathbb {C} ^{*}} . This is useful while studying complex tori because the character group of the lattice in a complex tor... | Character group | 0.808159 |
24,532 | The characters of irreducible representations are orthogonal.The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform. For locally ... | Character group | 0.808159 |
24,533 | Probe tips can be characterized for their shape, size, sharpness, bluntness, aspect ratio, radius of curvature, geometry and composition using many advanced instrumental techniques. For example, electron field emission measurement, scanning electron microscopy (SEM), transmission electron microscopy (TEM), scanning tun... | Probe tip | 0.808148 |
24,534 | It opened a new era for probing the nano-scale world of individual atoms and molecules as well as studying surface science, due to their unprecedented capability to characterize the mechanical, chemical, magnetic, and optical functionalities of various samples at nanometer-scale resolution in a vacuum, ambient, or flui... | Probe tip | 0.808148 |
24,535 | A probe tip is an instrument used in scanning probe microscopes (SPMs) to scan the surface of a sample and make nano-scale images of surfaces and structures. The probe tip is mounted on the end of a cantilever and can be as sharp as a single atom. In microscopy, probe tip geometry (length, width, shape, aspect ratio, a... | Probe tip | 0.808148 |
24,536 | Non-conductive nanoscale tips are widely used for AFM measurements. For non-conducting tip, surface forces acting on the tip/cantilever are responsible for deflection or attraction of tip. These attractive or repulsive forces are used for surface topology, chemical specifications, magnetic and electronic properties. | Probe tip | 0.808148 |
24,537 | In recognition of the impact cryo-EM has had on biochemistry, three scientists, Jacques Dubochet, Joachim Frank and Richard Henderson, were awarded the Nobel Prize in Chemistry "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution." | Cryo-Electron Microscopy | 0.808146 |
24,538 | For example, in the cubic system, all of the lengths are equal and all the angles are 90°, so only the a length needs to be given. This is the case of diamond, which has a = 3.57 Å = 357 pm at 300 K. Similarly, in hexagonal system, the a and b constants are equal, and the angles are 60°, 90°, and 90°, so the geometry i... | Lattice parameter | 0.808144 |
24,539 | The Stufe of a quadratically closed field is 1. The Stufe of an algebraic number field is ∞, 1, 2 or 4 (Siegel's theorem). Examples are Q, Q(√−1), Q(√−2) and Q(√−7). The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4. The Stufe of a local field of odd residue characteristic is equal to that of i... | Stufe (algebra) | 0.808141 |
24,540 | Symmetries may be broadly classified as global or local. A global symmetry is one that keeps a property invariant for a transformation that is applied simultaneously at all points of spacetime, whereas a local symmetry is one that keeps a property invariant when a possibly different symmetry transformation is applied a... | Symmetry in physics | 0.808141 |
24,541 | The 2006 Nobel Prize in Chemistry was awarded to Roger D. Kornberg for creating detailed molecular images of RNA polymerase during various stages of the transcription process.In most prokaryotes, a single RNA polymerase species transcribes all types of RNA. RNA polymerase "core" from E. coli consists of five subunits: ... | RNA Polymerase | 0.808138 |
24,542 | : 63: 196 The company was originally outfitted from the Cheiky's garage and was dedicated to the production of electronic teaching aids. The company's original name—Ohio Scientific Instruments, Inc.—reflected this initial purpose. The first products the company released included a calculator that also taught the basics... | Ohio Scientific | 0.808138 |
24,543 | Mike Cheiky had worked at the Solon-based Ohio Nuclear Company—makers of medical equipment—as director of engineering, while Charity Cheiky had been employed at Western Reserve Academy as a professor of math and computer science. : B3 Dreisbach meanwhile was a chairman and professor emeritus of Hiram College's chemistr... | Ohio Scientific | 0.808138 |
24,544 | The Challenger III was released alongside the Challenger II in 1977. It was a desktop computer featuring three microprocessors—a MOS 6502A, a Zilog Z80, and a Motorola 6800—on one board. This combination of processors allowed the computer to run virtually all software for microcomputers on the market at the time of its... | Ohio Scientific | 0.808138 |
24,545 | The C1P Series II was a redesign of Ohio Scientific's Challenger 1P with a vastly different case design made with a plastic shell, over top of the standard metal case. The 1P incorporates the DAC of the Challenger 4P to allow the computer to generate sound and music. The computer came stock with Microsoft 8K BASIC IN R... | Ohio Scientific | 0.808138 |
24,546 | Bayes' theorem in odds form is: O ( A 1: A 2 | B ) = O ( A 1: A 2 ) ⋅ Λ ( A 1: A 2 | B ) {\displaystyle O(A_{1}:A_{2}\vert B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\vert B)} where Λ ( A 1: A 2 | B ) = P ( B | A 1 ) P ( B | A 2 ) {\displaystyle \Lambda (A_{1}:A_{2}\vert B)={\frac {P(B\vert A_{1})}{P(B\vert A_{2})}}} i... | Bayes's Theorem | 0.808138 |
24,547 | A conditioned version of the Bayes' theorem results from the addition of a third event C {\displaystyle C} on which all probabilities are conditioned: P ( A | B ∩ C ) = P ( B | A ∩ C ) P ( A | C ) P ( B | C ) {\displaystyle P(A\vert B\cap C)={\frac {P(B\vert A\cap C)\,P(A\vert C)}{P(B\vert C)}}} | Bayes's Theorem | 0.808138 |
24,548 | In the Bayesian (or epistemological) interpretation, probability measures a "degree of belief". Bayes' theorem links the degree of belief in a proposition before and after accounting for evidence. For example, suppose it is believed with 50% certainty that a coin is twice as likely to land heads than tails. If the coin... | Bayes's Theorem | 0.808138 |
24,549 | Church and Turing independently demonstrated that Hilbert's Entscheidungsproblem (decision problem) was unsolvable, thus identifying the computational core of the incompleteness theorem. This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which, whe... | Turing completeness | 0.808137 |
24,550 | These rules were proved by Kurt Gödel in 1930 to be enough to produce every theorem. The actual notion of computation was isolated soon after, starting with Gödel's incompleteness theorem. This theorem showed that axiom systems were limited when reasoning about the computation that deduces their theorems. | Turing completeness | 0.808137 |
24,551 | All known laws of physics have consequences that are computable by a series of approximations on a digital computer. A hypothesis called digital physics states that this is no accident because the universe itself is computable on a universal Turing machine. This would imply that no computer more powerful than a univers... | Turing completeness | 0.808137 |
24,552 | All of these compute proper subsets of the total computable functions, since the full set of total computable functions is not computably enumerable. Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages, in contrast with Turing machines.... | Turing completeness | 0.808137 |
24,553 | This illustrates one reason why relatively powerful non-Turing-complete languages are rare: the more powerful the language is initially, the more complex are the tasks to which it is applied and the sooner its lack of completeness becomes perceived as a drawback, encouraging its extension until it is Turing-complete. T... | Turing completeness | 0.808137 |
24,554 | The computational systems (algebras, calculi) that are discussed as Turing-complete systems are those intended for studying theoretical computer science. They are intended to be as simple as possible, so that it would be easier to understand the limits of computation. Here are a few: Automata theory Formal grammar (lan... | Turing completeness | 0.808137 |
24,555 | In chemistry and materials science, molecular self-assembly is the process by which molecules adopt a defined arrangement without guidance or management from an outside source. There are two types of self-assembly: intermolecular and intramolecular. Commonly, the term molecular self-assembly refers to the former, while... | Molecular self-assembly | 0.808135 |
24,556 | DNA nanotechnology is an area of current research that uses the bottom-up, self-assembly approach for nanotechnological goals. DNA nanotechnology uses the unique molecular recognition properties of DNA and other nucleic acids to create self-assembling branched DNA complexes with useful properties. DNA is thus used as a... | Molecular self-assembly | 0.808135 |
24,557 | Molecular self-assembly is an important aspect of bottom-up approaches to nanotechnology. Using molecular self-assembly, the final (desired) structure is programmed in the shape and functional groups of the molecules. Self-assembly is referred to as a 'bottom-up' manufacturing technique in contrast to a 'top-down' tech... | Molecular self-assembly | 0.808135 |
24,558 | Molecular self-assembly is a key concept in supramolecular chemistry. This is because assembly of molecules in such systems is directed through non-covalent interactions (e.g., hydrogen bonding, metal coordination, hydrophobic forces, van der Waals forces, pi-stacking interactions, and/or electrostatic) as well as elec... | Molecular self-assembly | 0.808135 |
24,559 | In materials science and solid mechanics, biaxial tensile testing is a versatile technique to address the mechanical characterization of planar materials. It is a generalized form of tensile testing in which the material sample is simultaneously stressed along two perpendicular axes. Typical materials tested in biaxial... | Biaxial tensile testing | 0.808133 |
24,560 | This discovery, along with the publication by Friedrich Wöhler in 1828 of a paper on the chemical synthesis of urea, and is notable for being the first organic compound prepared from wholly inorganic precursors. This proved that the organic compounds and chemical reactions found in cells were no different in principle ... | Primary metabolism | 0.808132 |
24,561 | The history of the scientific study of metabolism spans several centuries and has moved from examining whole animals in early studies, to examining individual metabolic reactions in modern biochemistry. The first controlled experiments in human metabolism were published by Santorio Santorio in 1614 in his book Ars de s... | Primary metabolism | 0.808132 |
24,562 | Metabolism involves a vast array of chemical reactions, but most fall under a few basic types of reactions that involve the transfer of functional groups of atoms and their bonds within molecules. This common chemistry allows cells to use a small set of metabolic intermediates to carry chemical groups between different... | Primary metabolism | 0.808132 |
24,563 | Classicist Kurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally ... one can safely assume that it belongs to the very oldest period of Pythagorean mathematics." Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented. With contents known m... | The Pythagorean theorem | 0.808131 |
24,564 | 495 BC) starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath (1861–1940), no specific att... | The Pythagorean theorem | 0.808131 |
24,565 | 1800 BC, the Egyptian Middle Kingdom Berlin Papyrus 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet Plimpton 322, also written c. 1800 BC near Larsa, contains many entries closely related to Pythagorean triples.In India, the B... | The Pythagorean theorem | 0.808131 |
24,566 | There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period (20th... | The Pythagorean theorem | 0.808131 |
24,567 | The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram. This results in a larger square, with side a + b and area (a + b)2. The four triangles and the square side c must have the same area as the larger... | The Pythagorean theorem | 0.808131 |
24,568 | {\displaystyle {\frac {c}{b}}={\frac {b}{r}}\ .} Likewise, for the reflection of the other triangle, c a = a s . {\displaystyle {\frac {c}{a}}={\frac {a}{s}}\ .} Clearing fractions and adding these two relations: c s + c r = a 2 + b 2 , {\displaystyle cs+cr=a^{2}+b^{2}\ ,} the required result. The theorem remains valid... | The Pythagorean theorem | 0.808131 |
24,569 | As the angle θ approaches π/2, the base of the isosceles triangle narrows, and lengths r and s overlap less and less. When θ = π/2, ADB becomes a right triangle, r + s = c, and the original Pythagorean theorem is regained. One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. | The Pythagorean theorem | 0.808131 |
24,570 | Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an m-dimensional set of objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space is equal to the sum of the squares of the ... | The Pythagorean theorem | 0.808131 |
24,571 | For example, the starting center triangle can be replicated and used as a triangle C on its hypotenuse, and two similar right triangles (A and B ) constructed on the other two sides, formed by dividing the central triangle by its altitude. The sum of the areas of the two smaller triangles therefore is that of the third... | The Pythagorean theorem | 0.808131 |
24,572 | Thus, if similar figures with areas A, B and C are erected on sides with corresponding lengths a, b and c then: A a 2 = B b 2 = C c 2 , {\displaystyle {\frac {A}{a^{2}}}={\frac {B}{b^{2}}}={\frac {C}{c^{2}}}\,,} ⇒ A + B = a 2 c 2 C + b 2 c 2 C . {\displaystyle \Rightarrow A+B={\frac {a^{2}}{c^{2}}}C+{\frac {b^{2}}{c^{2... | The Pythagorean theorem | 0.808131 |
24,573 | The Pythagorean theorem generalizes beyond the areas of squares on the three sides to any similar figures. This was known by Hippocrates of Chios in the 5th century BC, and was included by Euclid in his Elements: If one erects similar figures (see Euclidean geometry) with corresponding sides on the sides of a right tri... | The Pythagorean theorem | 0.808131 |
24,574 | When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. The theorem can be generalized in various ways: to ... | The Pythagorean theorem | 0.808131 |
24,575 | The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. | The Pythagorean theorem | 0.808131 |
24,576 | In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other ... | The Pythagorean theorem | 0.808131 |
24,577 | The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral J {\displaystyle J} requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler–Lagran... | Variation calculus | 0.808131 |
24,578 | {\displaystyle B.} This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilb... | Variation calculus | 0.808131 |
24,579 | The result of this process is a diffraction pattern that consists of a summation or integration over the patterns generated during precession. While the geometry of this pattern matches the pattern associated with a normally incident beam, the intensities of the various reflections approximate those of the kinematical ... | Precession Electron Diffraction | 0.80813 |
24,580 | Precession electron diffraction is accomplished utilizing the standard instrument configuration of a modern TEM. The animation illustrates the geometry used to generate a PED pattern. Specifically, the beam tilt coils located pre-specimen are used to tilt the electron beam off of the optic axis so it is incident with t... | Precession Electron Diffraction | 0.80813 |
24,581 | PED possesses many advantageous attributes that make it well suited to investigating crystal structures via direct methods approaches: Quasi-kinematical diffraction patterns: While the underlying physics of the electron diffraction is still dynamical in nature, the conditions used to collect PED patterns minimize many ... | Precession Electron Diffraction | 0.808129 |
24,582 | First published in 1981 by Elsevier, Principles of Neural Science is an influential neuroscience textbook edited by Columbia University professors Eric R. Kandel, James H. Schwartz, and Thomas M. Jessell. The original edition was 468 pages; now on the sixth edition, the book has grown to 1646 pages. The second edition ... | Principles of Neural Science | 0.808127 |
24,583 | Principles of Neural Science is often assigned as a textbook for many undergraduate and graduate/medical neuroscience and neurobiology courses. The book attempts to at least introduce every aspect of our most modern understanding of the brain. The sixth edition is divided into sixty-four chapters, organized into nine p... | Principles of Neural Science | 0.808127 |
24,584 | Jessell became an editor of the book starting from the third edition. He was a professor of biochemistry and molecular biophysics at Columbia University, and an investigator at the Howard Hughes Medical Institute. Hudspeth is a professor of sensory neuroscience at Rockefeller University. He is also an investigator at t... | Principles of Neural Science | 0.808127 |
24,585 | Kandel was one of the recipients of the 2000 Nobel Prize in Physiology or Medicine. He is currently a professor of biochemistry, molecular biophysics, physiology, cellular biophysics, and psychiatry at Columbia University. He is a senior investigator at the Howard Hughes Medical Institute and a recipient of the Nationa... | Principles of Neural Science | 0.808127 |
24,586 | "Good genes" theory proposes that females select males seen to have genetic advantages that increase offspring quality. Increased viability of offspring provides compensation for any lower reproductive success that results from their being "picky". The good-gene hypothesis for polyandry proposes that when females encou... | Good genes hypothesis | 0.808127 |
24,587 | Sexual conflict refers to the conflicting goals of breeding males and females. It describes the diverging interests of males and females in optimizing their fitness. From the viewpoint of any one partner, the best outcome would be for the partner's mate to care for the young, thus freeing up his or her own resources (e... | Good genes hypothesis | 0.808127 |
24,588 | It is estimated that Jupiter radiates more energy through this mechanism than it receives from the Sun, but Saturn might not. Jupiter has been estimated to shrink at a rate of approximately 1 mm/year by this process, corresponding to an internal flux of 7.485 W/m2.The mechanism was originally proposed by Kelvin and Hel... | Kelvin-Helmholtz mechanism | 0.808124 |
24,589 | A major breakthrough in convex optimization was the introduction of interior-point methods. These methods were developed in a series of papers and became of true interest in the context of LMI problems in the work of Yurii Nesterov and Arkadi Nemirovski. | Linear matrix inequality | 0.808124 |
24,590 | In August 1982 Intel sued NEC for copyright infringement over the microcode implementation. NEC prevailed by demonstrating via cleanroom software engineering that the similarities in the implementation of microcode on its V20 and V30 processors was the result of the restrictions demanded by the architecture, rather tha... | Intel Microcode | 0.808122 |
24,591 | A carbon–nitrogen bond is a covalent bond between carbon and nitrogen and is one of the most abundant bonds in organic chemistry and biochemistry.Nitrogen has five valence electrons and in simple amines it is trivalent, with the two remaining electrons forming a lone pair. Through that pair, nitrogen can form an additi... | Carbon–nitrogen bond | 0.80812 |
24,592 | In molecular biology (specifically protein biosynthesis), a stop codon (or termination codon) is a codon (nucleotide triplet within messenger RNA) that signals the termination of the translation process of the current protein. Most codons in messenger RNA correspond to the addition of an amino acid to a growing polypep... | Stop codon | 0.80812 |
24,593 | In the framework of relativistic quantum mechanics, where α {\displaystyle \alpha } is the fine-structure constant. Here the small corrections to the relativistic result g = 2 come from the quantum field theory calculations of the anomalous magnetic dipole moment. The electron g-factor is known to twelve decimal places... | Gyromagnetic ratio | 0.808119 |
24,594 | In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the radian per second per tesla (rad⋅s−1⋅T−1) or, equivalently, the... | Gyromagnetic ratio | 0.808119 |
24,595 | In nuclear reactions, the fraction of mass that may be removed as light or heat, i.e. binding energy, is often a much larger fraction of the system mass. It may thus be measured directly as a mass difference between rest masses of reactants and (cooled) products. This is because nuclear forces are comparatively stronge... | Binding energies | 0.808118 |
24,596 | The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli. It took him... | Law of large numbers | 0.808118 |
24,597 | By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as All X1, X2, ... have the same characteristic function, so we will simply denote this φX. Among the basic properties of characteristic functions there are if X and Y are independent. Th... | Law of large numbers | 0.808118 |
24,598 | He named this his "Golden Theorem" but it became generally known as "Bernoulli's theorem". This should not be confused with Bernoulli's principle, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1837, S. D. Poisson further described it under the name "la loi des grands nombres" ("the law of large numbers"). | Law of large numbers | 0.808118 |
24,599 | Many of their important properties can only be rationalized by considering them to be porous media. The concept of porous media is used in many areas of applied science and engineering: filtration, mechanics (acoustics, geomechanics, soil mechanics, rock mechanics), engineering (petroleum engineering, bioremediation, c... | Porous media | 0.808115 |
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