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20,600 | Systemic disease at various sites of the body can result from oral microbes entering the blood bypassing periodontal pockets and oral membranes.Persistent proper oral hygiene is the primary method for preventing oral and systemic disease. It reduces the density of biofilm and overgrowth of potential pathogenic bacteria... | Human microbiome | 0.810781 |
20,601 | This can be done using machine learning approaches that can reach an accuracy at genus-level of about 80%. Other popular analysis packages provide support for taxonomic classification using exact matches to reference databases and should provide greater specificity, but poor sensitivity. Unclassified microorganism shou... | Human microbiome | 0.810781 |
20,602 | Primer affinity varies among all DNA sequences, which may result in biases during the amplification reaction; indeed, low-abundance samples are susceptible to overamplification errors, since the other contaminating microorganisms result to be over-represented in case of increasing the PCR cycles. Therefore, the optimiz... | Human microbiome | 0.810781 |
20,603 | As of 2014, it was reported in popular media and in the scientific literature that there are about 10 times as many microbial cells in the human body as there are human cells; this figure was based on estimates that the human microbiome includes around 100 trillion bacterial cells and that an adult human typically has ... | Human microbiome | 0.810781 |
20,604 | Although cancer is generally a disease of host genetics and environmental factors, microorganisms are implicated in some 20% of human cancers. Particularly for potential factors in colon cancer, bacterial density is one million times higher than in the small intestine, and approximately 12-fold more cancers occur in th... | Human microbiome | 0.810781 |
20,605 | Bacteria of the same species found throughout the mouth are of multiple subtypes, preferring to inhabit distinctly different locations in the mouth. Even the enterotypes in the human gut, previously thought to be well understood, are from a broad spectrum of communities with blurred taxon boundaries.It is estimated tha... | Human microbiome | 0.810781 |
20,606 | X represents a random variable a triangle center Ẋ represents the rate of change of quantity X x represents a realized value of a random variable an unknown variable, most often (but not always) from the set of real numbers, while a complex unknown would rather be called z, and an integer by a letter like m from the mi... | List of mathematical uses of Latin letters | 0.810778 |
20,607 | I represents: the closed unit interval, which contains all real numbers from 0 to 1, inclusive the identity matrix the Irradiance the moment of inertia intensity in physics, typically the vector field I Luminous intensity, typically Iv the incenter of a triangle the electric current ionization energy, denoted I I repre... | List of mathematical uses of Latin letters | 0.810778 |
20,608 | H represents: a Hilbert space the unit henry of magnetic inductance the homology and cohomology functor the enthalpy the (Shannon) entropy of information the orthocenter of a triangle a partial sum of the harmonic series Auxiliary magnetic field, denoted H {\displaystyle {\boldsymbol {H}}} Hamiltonian in quantum mechan... | List of mathematical uses of Latin letters | 0.810778 |
20,609 | E represents: the digit "14" in hexadecimal and other positional numeral systems with a radix of 15 or greater an exponent in decimal numbers. For example, 1.2E3 is 1.2×103 or 1200 the set of edges in a graph or matroid the unit prefix exa (1018) energy in physics electric field denoted E {\displaystyle {\textbf {E}}} ... | List of mathematical uses of Latin letters | 0.810778 |
20,610 | A represents: the first point of a triangle the digit "10" in hexadecimal and other positional numeral systems with a radix of 11 or greater the unit ampere for electric current in physics the area of a figure the mass number or nucleon number of an element in chemistry the Helmholtz free energy of a closed thermodynam... | List of mathematical uses of Latin letters | 0.810778 |
20,611 | S represents a sum the unit siemens of electric conductance the unit sphere (with superscript denoting dimension) the scattering matrix entropy action in joule-seconds Apparent power in electrical engineering Area Spin operator Sulfur Symmetric group s represents: an arclength a path length the displacement in mechanic... | List of mathematical uses of Latin letters | 0.810778 |
20,612 | R represents: the Ricci tensor the circumradius of a cyclic polygon such as a triangle an arbitrary relation Riemann curvature tensor Electrical resistance Molar gas constant R {\displaystyle \mathbb {R} } represents the set of real numbers and various algebraic structures built upon the set of real numbers, such as R ... | List of mathematical uses of Latin letters | 0.810778 |
20,613 | Q represents: heat energy: 6 electroweak charge, denoted QW Reactive power in electrical engineering Volumetric flow rate Q {\displaystyle \mathbb {Q} } represents the rational numbers q represents: the deceleration parameter in cosmology electric charge of a particle a generalized coordinate Quark | List of mathematical uses of Latin letters | 0.810778 |
20,614 | L represents: length, used often in quantum mechanics as the size of an infinite square well angular momentum the unit of volume the litre the radiance the space of all integrable real (or complex) functions the space of linear maps, as in L(E,F) or L(E) = End(E) the likelihood function a formal language the lag operat... | List of mathematical uses of Latin letters | 0.810778 |
20,615 | In polymer chemistry, a structural unit is a building block of a polymer chain. It is the result of a monomer which has been polymerized into a long chain. There may be more than one structural unit in the repeat unit. When different monomers are polymerized, a copolymer is formed. It is a routine way of developing new... | Structural unit | 0.810777 |
20,616 | In particle physics, a system of natural units in which the speed of light in vacuum c and the reduced Planck constant ħ are dimensionless and equal to unity is widely used: c = ħ = 1. In these units, both distances and times are expressed in inverse energy units (while energy and mass are expressed in the same units, ... | Mega electronvolt | 0.810777 |
20,617 | The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. In 1900, David Hilbert posed a question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number th... | Transcendental numbers | 0.810775 |
20,618 | Then, since ei π = −1 is algebraic (see Euler's identity), i π must be transcendental. But since i is algebraic, π therefore must be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. | Transcendental numbers | 0.810775 |
20,619 | Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers. Cantor's work established the ubiquity of transcendental numbers. In 1882, Ferdinand von Lindemann published t... | Transcendental numbers | 0.810775 |
20,620 | Liouville showed that all Liouville numbers are transcendental.The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was e, by Charles Hermite in 1873. In 1874, Georg Cantor proved that the algebraic numbers are countable a... | Transcendental numbers | 0.810775 |
20,621 | = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers are called Liouville numbers, named in his honour. | Transcendental numbers | 0.810775 |
20,622 | The name "transcendental" comes from the Latin trānscendere 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin x is not an algebraic function of x . Euler, in the 18th century, was probably the first person to define transcendental nu... | Transcendental numbers | 0.810775 |
20,623 | The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: Assume, for purpose of finding a contradiction, that e is algebraic. ... | Transcendental numbers | 0.810775 |
20,624 | The paperfolding constant (also named as "Gaussian Liouville number"). Constructed irrational numbers which are not simply normal in any base. For β > 1 ∑ k = 0 ∞ 10 − ⌊ β k ⌋ ; {\displaystyle \sum _{k=0}^{\infty }10^{-\left\lfloor \beta ^{k}\right\rfloor };} where β ↦ ⌊ β ⌋ {\displaystyle \beta \mapsto \lfloor \beta \... | Transcendental numbers | 0.810775 |
20,625 | The aforementioned Liouville constant for any algebraic b ∈ (0, 1). The sum of reciprocals of exponential factorials. The Prouhet–Thue–Morse constant and the related rabbit constant. | Transcendental numbers | 0.810775 |
20,626 | Any number of the form ∑ n = 0 ∞ E n ( β r n ) F n ( β r n ) {\displaystyle \sum _{n=0}^{\infty }{\frac {E_{n}(\beta ^{r^{n}})}{F_{n}(\beta ^{r^{n}})}}} (where E n ( z ) {\displaystyle E_{n}(z)} , F n ( z ) {\displaystyle F_{n}(z)} are polynomials in variables n {\displaystyle n} and z {\displaystyle z} , β {\displayst... | Transcendental numbers | 0.810775 |
20,627 | The lemniscatic values of theta function ∑ n = − ∞ ∞ q n 2 {\displaystyle \sum _{n=-\infty }^{\infty }q^{n^{2}}} (under the same conditions for q {\displaystyle {q}} ) are also transcendental. j(q) where q ∈ C {\displaystyle {q}\in \mathbb {C} } is algebraic but not imaginary quadratic (i.e, the exceptional set of this... | Transcendental numbers | 0.810775 |
20,628 | The so-called Fredholm constants, such as ∑ n = 0 ∞ 10 − 2 n = 0. 1 1 0 1 000 1 0000000 1 … {\displaystyle \sum _{n=0}^{\infty }10^{-2^{n}}=0. {\textbf {1}}{\textbf {1}}0{\textbf {1}}000{\textbf {1}}0000000{\textbf {1}}\ldots } which also holds by replacing 10 with any algebraic number b > 1. arctan ( x ) π {\display... | Transcendental numbers | 0.810775 |
20,629 | In general, all numbers of the form π + β 1 ln ( a 1 ) + ⋯ + β n ln ( a n ) {\displaystyle \pi +\beta _{1}\ln(a_{1})+\cdots +\beta _{n}\ln(a_{n})} are transcendental, where β j {\displaystyle \beta _{j}} are algebraic for all 1 ≤ j ≤ n {\displaystyle 1\leq j\leq n} and a j {\displaystyle a_{j}} are non-zero algebra... | Transcendental numbers | 0.810775 |
20,630 | √xs, the square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem) Γ ( 1 3 ) {\displaystyle \operatorname {\Gamma } \left({\tfrac {1}{3}}\right)\ } , Γ ( 1 4 ) {\displaystyle \operatorname {\Gamma } \left({\tfrac {1}{4}}\right)\ } , and Γ ( 1 6 ) {\displayst... | Transcendental numbers | 0.810775 |
20,631 | arcsin a, arccos a, arctan a, arccsc a, arcsec a, arccot a and their hyperbolic counterparts, for any algebraic number a where a ∉ { 0 , 1 } {\displaystyle a\notin \{0,1\}} (by the Lindemann–Weierstrass theorem). The Bessel function of the first kind Jν(x), its first derivative, and the quotient J ν ′ ( x ) J ν ( x ) {... | Transcendental numbers | 0.810775 |
20,632 | The fixed point of the cosine function (also referred to as the Dottie number d) – the unique real solution to the equation cos x = x, where x is in radians (by the Lindemann–Weierstrass theorem). ln a if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theo... | Transcendental numbers | 0.810775 |
20,633 | Numbers proven to be transcendental: ea if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem). π (by the Lindemann–Weierstrass theorem). eπ, Gelfond's constant, as well as e−π/2 = ii (by the Gelfond–Schneider theorem). ab where a is algebraic but not 0 or 1, and b is irrational algebraic (by the Gelfond–... | Transcendental numbers | 0.810775 |
20,634 | Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers. Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion... | Transcendental numbers | 0.810775 |
20,635 | If (a + b) and a b were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic. But this is a contradiction, and thus it must be the case that at least one of... | Transcendental numbers | 0.810775 |
20,636 | For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as 5 π {\displaystyle 5\pi } , π − 3 2 {\displaystyle {\tfrac {\pi -3}{\sqrt {2}}}} , ( π − 3 ) 8 {\displaystyle ({\sqrt {\pi }}-{\sqrt {3}})^{8}} , and π 5 + 7 4 {\displaystyle {\sqrt{\pi ^{5}+7}}} are transcendental as... | Transcendental numbers | 0.810775 |
20,637 | No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals. Applying any non-constant single-variable algeb... | Transcendental numbers | 0.810775 |
20,638 | However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable. | Transcendental numbers | 0.810775 |
20,639 | A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since a rational number is the root of an integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials wi... | Transcendental numbers | 0.810775 |
20,640 | The terms Baxter operator algebra and Baxter algebra are also used. Let R {\displaystyle R} be a Rota–Baxter of weight λ {\displaystyle \lambda } . Then − λ I d − R {\displaystyle -\lambda Id-R} is also a Rota–Baxter operator of weight λ {\displaystyle \lambda } . Further, for μ {\displaystyle \mu } in k, μ R {\display... | Rota–Baxter algebra | 0.810775 |
20,641 | Let k be a commutative ring and let λ {\displaystyle \lambda } be given. A linear operator R on a k-algebra A is called a Rota–Baxter operator of weight λ {\displaystyle \lambda } if it satisfies the Rota–Baxter relation of weight λ {\displaystyle \lambda }: R ( x ) R ( y ) = R ( R ( x ) y ) + R ( x R ( y ) ) + λ R ( x... | Rota–Baxter algebra | 0.810775 |
20,642 | {\displaystyle I(f)(x)=\int _{0}^{x}f(t)dt\;.} Let G(x) = I(g)(x) and F(x) = I(f)(x). Then the formula for integration for parts can be written in terms of these variables as F ( x ) G ( x ) = ∫ 0 x f ( t ) G ( t ) d t + ∫ 0 x F ( t ) g ( t ) d t . {\displaystyle F(x)G(x)=\int _{0}^{x}f(t)G(t)dt+\int _{0}^{x}F(t)g(t)dt... | Rota–Baxter algebra | 0.810775 |
20,643 | Integration by parts Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let C ( R ) {\displaystyle C(R)} be the algebra of continuous functions from the real line to the real line. Let: f ( x ) ∈ C ( R ) {\displaystyle f(x)\in C(R)} be a continuous function. Define integration as the Rota–Baxter o... | Rota–Baxter algebra | 0.810775 |
20,644 | In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gi... | Rota–Baxter algebra | 0.810775 |
20,645 | However, this trend is not shown in the bond energies because fluorine is singular due to its small size, low polarisability, and inability to show hypervalence. As another difference, chlorine has a significant chemistry in positive oxidation states while fluorine does not. Chlorination often leads to higher oxidation... | Chlorine atom | 0.810774 |
20,646 | Like the other carbon–halogen bonds, the C–Cl bond is a common functional group that forms part of core organic chemistry. Formally, compounds with this functional group may be considered organic derivatives of the chloride anion. Due to the difference of electronegativity between chlorine (3.16) and carbon (2.55), the... | Chlorine atom | 0.810774 |
20,647 | The class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class nu... | List of number fields with class number one | 0.810772 |
20,648 | The principal axis theorem shows that for any (possibly reducible) quadric, a suitable change of Cartesian coordinates or, equivalently, a Euclidean transformation allows putting the equation of the quadric into a unique simple form on which the class of the quadric is immediately visible. This form is called the norma... | Quadric hypersurface | 0.81077 |
20,649 | Range of motion (or ROM) is the linear or angular distance that a moving object may normally travel while properly attached to another. In biomechanics and strength training, ROM refers to the angular distance and direction a joint can move between the flexed position and the extended position. The act of attempting to... | Range of motion | 0.810763 |
20,650 | Rather than characterize mathematics by deductive logic, intuitionism views mathematics as primarily about the construction of ideas in the mind: The only possible foundation of mathematics must be sought in this construction under the obligation carefully to watch which constructions intuition allows and which not. L.... | Definitions of mathematics | 0.810763 |
20,651 | Most contemporary reference works define mathematics by summarizing its main topics and methods: The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra. Oxfo... | Definitions of mathematics | 0.810763 |
20,652 | In his classification of the sciences, he further distinguished between arithmetic, which studies discrete quantities, and geometry that studies continuous quantities. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studyin... | Definitions of mathematics | 0.810763 |
20,653 | In the 19th century, as mathematics branched out into abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.Three leading types of definition of mathematics today are calle... | Definitions of mathematics | 0.810763 |
20,654 | This also raises the question of the place of humans in a radically automated society. AGI could also help to make rational decisions, and to anticipate and prevent disasters. It could also help to reap the benefits of potentially catastrophic technologies such as nanotechnology or climate engineering, while avoiding t... | Strong artificial intelligence | 0.810757 |
20,655 | Other aspects of the human mind besides intelligence are relevant to the concept of AGI or "strong AI", and these play a major role in science fiction and the ethics of artificial intelligence: consciousness: To have subjective experience. Thomas Nagel explains that it "feels like" something to be conscious. If we are ... | Strong artificial intelligence | 0.810757 |
20,656 | Hans Moravec wrote in 1988: I am confident that this bottom-up route to artificial intelligence will one day meet the traditional top-down route more than half way, ready to provide the real-world competence and the commonsense knowledge that has been so frustratingly elusive in reasoning programs. Fully intelligent ma... | Strong artificial intelligence | 0.810757 |
20,657 | In the 1990s and early 21st century, mainstream AI achieved commercial success and academic respectability by focusing on specific sub-problems where AI can produce verifiable results and commercial applications, such as artificial neural networks and statistical machine learning. These "applied AI" systems are now use... | Strong artificial intelligence | 0.810757 |
20,658 | While linear groups are a vast class of examples, among all infinite groups they are distinguished by many remarkable properties. Finitely generated linear groups have the following properties: They are residually finite; Burnside's theorem: a torsion group of finite exponent which is linear over a field of characteris... | Linear group | 0.810757 |
20,659 | A finite group G of order n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of G on the group ring K by left (or right) multiplication is linear and faithful. The finite groups of Lie type (classical groups over finite... | Linear group | 0.810757 |
20,660 | Once a group has been established to be linear it is interesting to try to find "optimal" faithful linear representations for it, for example of the lowest possible dimension, or even to try to classify all its linear representations (including those which are not faithful). These questions are the object of representa... | Linear group | 0.810757 |
20,661 | In some cases the fundamental group of a manifold can be shown to be linear by using representations coming from a geometric structure. For example, all closed surfaces of genus at least 2 are hyperbolic Riemann surfaces. Via the uniformization theorem this gives rise to a representation of its fundamental group in the... | Linear group | 0.810757 |
20,662 | The geographic areas analyzed in machine learning are defined by both administrative boundaries and top-down division into grids of regular shapes such as rectangles, for example. Both types are represented as polygons and, like points, can be assigned different demographic, transportation, or economic features. A poly... | Spatial embedding | 0.810753 |
20,663 | A single point of interest (POI) can be assigned multiple features that can be used in machine learning. These could be demographic, transportation, meteorological, or economic data, for example. When embedding single points, it is common to consider the entire set of available points as nodes in a graph. | Spatial embedding | 0.810753 |
20,664 | Spatial embedding is one of feature learning techniques used in spatial analysis where points, lines, polygons or other spatial data types. representing geographic locations are mapped to vectors of real numbers. Conceptually it involves a mathematical embedding from a space with many dimensions per geographic object t... | Spatial embedding | 0.810753 |
20,665 | Satellites and aircraft collect digital spatial data acquired from remotely sensed images which can be used in machine learning. They are sometimes hard to analyse using basic image analysis methods and convolutional neural networks can be used to acquire an embedding of images bound to a given geographical object or a... | Spatial embedding | 0.810753 |
20,666 | To motivate the general definitions, a definition is first formulated, in the case of finite groups and their representations on finite-dimensional vector spaces. If G is a finite group and U a representation of G on a finite-dimensional complex vector space H. The action of G on elements of H induces an action of G on... | System of imprimitivity | 0.810751 |
20,667 | Let G be a finite group and consider the space H of complex-valued functions on G. The left regular representation of G on H is defined by ( h ) = ψ ( g − 1 h ) . {\displaystyle (h)=\psi (g^{-1}h).} Now H can be considered as the algebraic direct sum of the one-dimensional spaces Wx, for x ∈ G, where W x = { ψ ∈ H: ψ ... | System of imprimitivity | 0.810751 |
20,668 | In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.It is related to the quadratic assignment problem in the same way as the linear bot... | Quadratic bottleneck assignment problem | 0.810751 |
20,669 | Dirhenium decacarbonyl is the most common entry to organorhenium chemistry. Its reduction with sodium amalgam gives Na with rhenium in the formal oxidation state −1. Dirhenium decacarbonyl can be oxidised with bromine to bromopentacarbonylrhenium(I): Re2(CO)10 + Br2 → 2 Re(CO)5BrReduction of this pentacarbonyl with zin... | Group 7 element | 0.810751 |
20,670 | The chemistry of organometallic compounds of Mn(II) are unusual among the transition metals due to the high ionic character of the Mn(II)-C bond. The reactivity of organomanganese compounds can be compared to that of organomagnesium and organozinc compounds. The electronegativity of Mn (1.55) is comparable to that of M... | Group 7 element | 0.810751 |
20,671 | Technetium-99m ("m" indicates that this is a metastable nuclear isomer) is used in radioactive isotope medical tests. For example, Technetium-99m is a radioactive tracer that medical imaging equipment tracks in the human body. It is well suited to the role because it emits readily detectable 140 keV gamma rays, and its... | Group 7 element | 0.810751 |
20,672 | A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness.Statistical randomness does not necessarily imply "true" randomness, i.e., objective unpredictability. Ps... | Local randomness | 0.810751 |
20,673 | A simple proof of the existence of a centerpoint may be obtained using Helly's theorem. Suppose there are n points, and consider the family of closed half-spaces that contain more than dn/(d + 1) of the points. Fewer than n/(d + 1) points are excluded from any one of these halfspaces, so the intersection of any subset ... | Centerpoint (geometry) | 0.81075 |
20,674 | In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two ro... | Centerpoint (geometry) | 0.81075 |
20,675 | Research has reported the application and usefulness of Markov chains in a wide range of topics such as physics, chemistry, biology, medicine, music, game theory and sports. | Equilibrium distribution | 0.810748 |
20,676 | The Australia Bioinformatics Resource (EMBL-ABR) (formerly the Bioinformatics Resource Australia - EMBL (BRAEMBL)) was a significant initiative under the associate membership to EMBL. Since 2019, all activities carried out under EMBL-ABR have rolled over into the Bioplatforms Australia (NCRIS-funded) Australian BioComm... | Australia Bioinformatics Resource | 0.810744 |
20,677 | Optical character recognition (OCR) – targets typewritten text, one glyph or character at a time. Optical word recognition – targets typewritten text, one word at a time (for languages that use a space as a word divider). Usually just called "OCR". Intelligent character recognition (ICR) – also targets handwritten prin... | Text recognition | 0.810743 |
20,678 | Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle. Some illusory visual proofs, such as the missing square puzzle, ... | Proof techniques | 0.810743 |
20,679 | A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems. In the probabilistic method, one seeks an object having a given property, starting with a large se... | Proof techniques | 0.810743 |
20,680 | Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four col... | Proof techniques | 0.810743 |
20,681 | In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because... | Proof techniques | 0.810743 |
20,682 | This does not specify which candidates have the property, but the probability could not be positive without at least one. A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work on the Collatz conjecture shows how far plausibility is from genui... | Proof techniques | 0.810743 |
20,683 | Proofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than full certainty. Inductive logic should not be confused with mathematical induction. Bayesian analysis uses Bayes' theorem ... | Proof techniques | 0.810743 |
20,684 | Prior to analysis, raw data (or unprocessed data) is typically cleaned: Outliers are removed and obvious instrument or data entry errors are corrected. Data can be seen as the smallest units of factual information that can be used as a basis for calculation, reasoning, or discussion. Data can range from abstract ideas ... | Data | 0.810741 |
20,685 | This usage is common in natural sciences, life sciences, social sciences, software development and computer science, and grew in popularity in the 20th and 21st centuries. Some style guides do not recognize the different meanings of the term, and simply recommend the form that best suits the target audience of the guid... | Data | 0.810741 |
20,686 | This usage is common in everyday language and in technical and scientific fields such as software development and computer science. One example of this usage is the term "big data". When used more specifically to refer to the processing and analysis of sets of data, the term retains its plural form. | Data | 0.810741 |
20,687 | Using traditional data analysis methods and computing, working with such large (and growing) datasets is difficult, even impossible. (Theoretically speaking, infinite data would yield infinite information, which would render extracting insights or intelligence impossible.) In response, the relatively new field of data ... | Data | 0.810741 |
20,688 | In common usage and statistics, data (US: ; UK: ) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a colle... | Data | 0.810741 |
20,689 | An important field in computer science, technology, and library science is the longevity of data. Scientific research generates huge amounts of data, especially in genomics and astronomy, but also in the medical sciences, e.g. in medical imaging. In the past, scientific data has been published in papers and books, stor... | Data | 0.810741 |
20,690 | 'Strong gravity' is a non-mainstream theoretical approach to particle confinement having both a cosmological scale and a particle scale gravity. In the 1960s, it was taken up as an alternative to the then young QCD theory by several theorists, including Abdus Salam, who showed that the particle level gravity approach c... | Strong gravity | 0.810741 |
20,691 | The second time derivative is of interest in physics, as it is found in equations of motion for classical mechanical systems. The second time derivative of a vector field in cylindrical coordinates is given by: To understand this expression, A is substituted for P, where P is the vector (ρ, φ, z). This means that A = P... | Vector fields in cylindrical and spherical coordinates | 0.810739 |
20,692 | Note: This page uses common physics notation for spherical coordinates, in which θ {\displaystyle \theta } is the angle between the z axis and the radius vector connecting the origin to the point in question, while ϕ {\displaystyle \phi } is the angle between the projection of the radius vector onto the x-y plane and t... | Vector fields in cylindrical and spherical coordinates | 0.810739 |
20,693 | In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional d... | First-order differential equation | 0.810737 |
20,694 | In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several l... | Polynomial method in combinatorics | 0.810736 |
20,695 | A variation of the polynomial method, often called polynomial partitioning, was introduced by Guth and Katz in their solution to the Erdős distinct distances problem. Polynomial partitioning involves using polynomials to divide the underlying space into regions and arguing about the geometric structure of the partition... | Polynomial method in combinatorics | 0.810736 |
20,696 | Many uses of the polynomial method follow the same high-level approach. The approach is as follows: Embed some combinatorial problem into a vector space. Capture the hypotheses of the problem by constructing a polynomial of low-degree that is zero on a certain set After constructing the polynomial, argue about its alge... | Polynomial method in combinatorics | 0.810736 |
20,697 | Consider one mole of gas composed of non-interacting point particles that satisfy the ideal gas law:(see any standard Physical Chemistry text, op. cit.) p = R T V m . | Van der Waals equation of state | 0.810733 |
20,698 | Such behaviour is qualitatively (though perhaps not quantitatively) predicted by the Van der Waals equation of state.However, the values of physical quantities as predicted with the Van der Waals equation of state "are in very poor agreement with experiment", so the model's utility is limited to qualitative rather than... | Van der Waals equation of state | 0.810733 |
20,699 | Any product i n i m {\displaystyle i_{n}i_{m}} of two distinct multicomplex units behaves as the j {\displaystyle j} of the split-complex numbers, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane. With respect to subalgebra C k {\displaystyle \mathbb {C} _{k}} , k = 0,... | Multicomplex number | 0.810731 |
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