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6,800 | Synthetic accessibility is a simple approach to network simulation whose goal is to predict which metabolic gene knockouts are lethal. The synthetic accessibility approach uses the topology of the metabolic network to calculate the sum of the minimum number of steps needed to traverse the metabolic network graph from t... | Metabolic pathway analysis | 0.823542 |
6,801 | Pathway Tools: A bioinformatics software package that assists in the construction of pathway/genome databases such as EcoCyc. Developed by Peter Karp and associates at the SRI International Bioinformatics Research Group, Pathway Tools has several components. Its PathoLogic module takes an annotated genome for an organi... | Metabolic pathway analysis | 0.823542 |
6,802 | PUBMED: This is an online library developed by the National Center for Biotechnology Information, which contains a massive collection of medical journals. Using the link provided by ENZYME, the search can be directed towards the organism of interest, thus recovering literature on the enzyme and its use inside of the or... | Metabolic pathway analysis | 0.823542 |
6,803 | In simplified terms, a reconstruction collects all of the relevant metabolic information of an organism and compiles it in a mathematical model. Validation and analysis of reconstructions can allow identification of key features of metabolism such as growth yield, resource distribution, network robustness, and gene ess... | Metabolic pathway analysis | 0.823542 |
6,804 | Ion channels are pore-forming membrane proteins that allow ions to pass through the channel pore. Their functions include establishing a resting membrane potential, shaping action potentials and other electrical signals by gating the flow of ions across the cell membrane, controlling the flow of ions across secretory a... | Ion current | 0.82354 |
6,805 | The National Institutes of Health has defined a biomarker as "a characteristic that is objectively measured and evaluated as an indicator of normal biological processes, pathogenic processes, or pharmacologic responses to a therapeutic intervention. "Understanding the proteome, the structure and function of each protei... | Call-map proteomics | 0.823533 |
6,806 | Applied mechanics, most notably mechanical engineering disciplines such as continuum mechanics, mechanism analysis, structural analysis, kinematics and dynamics play prominent roles in the study of biomechanics. Usually biological systems are much more complex than man-built systems. Numerical methods are hence applied... | Biomechanics | 0.823529 |
6,807 | The dynamic loading of joints considered as impact is discussed in detail by Emanuel Willert. It is also tied to the field of engineering, because it often uses traditional engineering sciences to analyze biological systems. Some simple applications of Newtonian mechanics and/or materials sciences can supply correct ap... | Biomechanics | 0.823529 |
6,808 | Studying the mechanical properties of this complex tissues improves the possibility to better understanding cardiovascular diseases and drastically improve personalized medicine. Vascular tissues are inhomogeneous with a strongly non linear behaviour. Generally this study involves complex geometry with intricate load c... | Biomechanics | 0.823529 |
6,809 | The application of biomechanical principles to plants, plant organs and cells has developed into the subfield of plant biomechanics. Application of biomechanics for plants ranges from studying the resilience of crops to environmental stress to development and morphogenesis at cell and tissue scale, overlapping with mec... | Biomechanics | 0.823529 |
6,810 | Comparative biomechanics is the application of biomechanics to non-human organisms, whether used to gain greater insights into humans (as in physical anthropology) or into the functions, ecology and adaptations of the organisms themselves. Common areas of investigation are Animal locomotion and feeding, as these have s... | Biomechanics | 0.823529 |
6,811 | In sports biomechanics, the laws of mechanics are applied to human movement in order to gain a greater understanding of athletic performance and to reduce sport injuries as well. It focuses on the application of the scientific principles of mechanical physics to understand movements of action of human bodies and sports... | Biomechanics | 0.823529 |
6,812 | Consequently, bones must also increase disproportionately in girth rather than mere size. This is because the bending strength of a tubular structure (such as a bone) is much more efficient relative to its weight. Mason suggests that this insight was one of the first grasps of the principles of biological optimization.... | Biomechanics | 0.823529 |
6,813 | On his deathbed, he published his work, On the Revolutions of the Heavenly Spheres. This work not only revolutionized science and physics, but also the development of mechanics and later bio-mechanics.Galileo Galilei, the father of mechanics and part time biomechanic was born 21 years after the death of Copernicus. Gal... | Biomechanics | 0.823529 |
6,814 | The next major biomechanic would not be around until 1452, with the birth of Leonardo da Vinci. Da Vinci was an artist and mechanic and engineer. He contributed to mechanics and military and civil engineering projects. He had a great understanding of science and mechanics and studied anatomy in a mechanics context. | Biomechanics | 0.823529 |
6,815 | Simon Singh's Fermat's Last Theorem is available in audio and there is also a Horizon television program. 3Blue1Brown, YouTube channel by Grant Sanderson. Mathologer, YouTube channel by Burkard Polster. Numberphile, YouTube channel by Brady Haran. BetterExplained, YouTube channel and website by Kalid Azad. | Popular mathematics | 0.823529 |
6,816 | Rucker, Rudy (1984), The Fourth Dimension: Toward a Geometry of Higher Reality; Houghton Mifflin Harcourt. | Popular mathematics | 0.823529 |
6,817 | Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. | Popular mathematics | 0.823529 |
6,818 | King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. Walker Books. ISBN 0-887-84201-1. | Popular mathematics | 0.823529 |
6,819 | In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was ne... | 15 and 290 theorems | 0.823517 |
6,820 | (This does not give an alternative proof of Lagrange's theorem, because Lagrange's theorem is used in the proof of the 15 theorem.) On the other hand, w 2 + 2 x 2 + 5 y 2 + 5 z 2 , {\displaystyle w^{2}+2x^{2}+5y^{2}+5z^{2},} is a positive definite quadratic form with integral matrix that takes as values all positive in... | 15 and 290 theorems | 0.823517 |
6,821 | Moreover, for each of these 9 numbers, there is such a quadratic form taking all other 8 positive integers except for this number as values. For example, the quadratic form w 2 + x 2 + y 2 + z 2 {\displaystyle w^{2}+x^{2}+y^{2}+z^{2}} is universal, because every positive integer can be written as a sum of 4 squares, by... | 15 and 290 theorems | 0.823517 |
6,822 | A positive integral quadratic form taking all positive integers as values is called universal. The 15 theorem says that a quadratic form with integer matrix is universal if it takes the numbers from 1 to 15 as values. A more precise version says that, if a positive definite quadratic form with integral matrix takes the... | 15 and 290 theorems | 0.823517 |
6,823 | To arrive at a proof, Euler analyses what the situation would look like if the theorem were true. To that end, suppose the yellow line in Figure 1 goes through the center of the sphere and is the axis of rotation we are looking for, and point O is one of the two intersection points of that axis with the sphere. Then he... | Euler rotation theorem | 0.823517 |
6,824 | Euler's original proof was made using spherical geometry and therefore whenever he speaks about triangles they must be understood as spherical triangles. | Euler rotation theorem | 0.823517 |
6,825 | It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product Aθ is the "generator" of the particular rotation, being the vector (x,y,z) associated with the matrix A. This shows that the rotation matrix and the axis–angle format are related by the exponential fun... | Euler rotation theorem | 0.823517 |
6,826 | Such a set of four numbers is called a quaternion. While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary nu... | Euler rotation theorem | 0.823517 |
6,827 | It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvecto... | Euler rotation theorem | 0.823517 |
6,828 | However, for many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements (and hence also translation lengths of hyperbolic elements acting in the upper half-plane, as well as systoles of Fuchsian subgroups) can be calculated by means of the reduced trace in the quaternion algebra, and the formu... | (2,3,7) triangle group | 0.823514 |
6,829 | Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotien... | (2,3,7) triangle group | 0.823514 |
6,830 | {\displaystyle (2-\eta )^{3}=7(\eta -1)^{2}.} we see that Q(η) is a totally real cubic extension of Q. The (2,3,7) hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i2 = j2 = η, ij = −ji. One ... | (2,3,7) triangle group | 0.823514 |
6,831 | The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center ±1. Let η = 2cos(2π/7). Then from the identity ( 2 − η ) 3 = 7 ( η − 1 ) 2 . | (2,3,7) triangle group | 0.823514 |
6,832 | In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84(g − 1), of its automorphism group. A note on terminology – the "(2,3,7) t... | (2,3,7) triangle group | 0.823514 |
6,833 | Derivatization-free approaches aim to infer drug-target interactions by observing changes in protein stability or drug chromatography upon binding. Computational techniques complement the chemoproteomic toolkit as parallel lines of evidence supporting potential drug-target pairs, and are used to generate structural mod... | Chemoproteomics | 0.823511 |
6,834 | There are several varieties of photoreactive groups, but they are fundamentally different from ABPP probes: while ABPP specifically labels nucleophilic amino acids in a target's active site, photoaffinity labeling is non-specific, and thus is applicable to labeling a wider range of targets. The identification tag will ... | Chemoproteomics | 0.823511 |
6,835 | Unlike ABPP, which results in protein labeling upon probe binding, photoaffinity labeling probes require activation by photolysis before covalent bonding to a protein occurs. The presence of a photoreactive group makes this possible. These probes are composed of three connected moieties: (1) a drug scaffold; (2) a phot... | Chemoproteomics | 0.823511 |
6,836 | The conclusion of the Human Genome Project was followed with hope for a new paradigm in treating disease. Many fatal and intractable diseases were able to be mapped to specific genes, providing a starting point to better understand the roles of their protein products in illness. Drug discovery has made use of animal kn... | Chemoproteomics | 0.823511 |
6,837 | The Journal of Bioinformatics and Computational Biology was founded in 2003 and is published by World Scientific. The journal covers analysis of cellular information, especially in the technical aspect. The managing editor is Limsoon Wong (National University of Singapore). | Journal of Bioinformatics and Computational Biology | 0.823506 |
6,838 | The theorem still holds for the space R ∞ {\displaystyle \mathbb {R} ^{\infty }\,} of infinite sequences. A measurable function f: Ω → R ∞ {\displaystyle f:\Omega \to \mathbb {R} ^{\infty }\,} may be thought of as an infinite sequence of random variables X 1 , X 2 , ⋯: Ω → R , {\displaystyle X_{1},X_{2},\dots :\Omega \... | Standard probability space | 0.823504 |
6,839 | See (Rokhlin 1952, Sect. 2.6 (p. 23) and 3.2), (Kechris 1995, Sect. 17.F), (Petersen 1983, Theorem 4.7 on page 17). | Standard probability space | 0.823504 |
6,840 | Every measure preserving map f: Ω 1 → Ω 2 {\displaystyle \textstyle f:\Omega _{1}\to \Omega _{2}} leads to a homomorphism F {\displaystyle \textstyle F} of measure algebras; basically, F ( B ) = f − 1 ( B ) {\displaystyle \textstyle F(B)=f^{-1}(B)} for B ∈ F 2 {\displaystyle \textstyle B\in {\mathcal {F}}_{2}} . It may... | Standard probability space | 0.823504 |
6,841 | "There is a coherent way to ignore the sets of measure 0 in a measure space" (Petersen 1983, page 15). Striving to get rid of null sets, mathematicians often use equivalence classes of measurable sets or functions. Equivalence classes of measurable subsets of a probability space form a normed complete Boolean algebra c... | Standard probability space | 0.823504 |
6,842 | 2.5 (p. 20)) and (de la Rue 1993, Theorem 3-5). See also (Haezendonck 1973, Proposition 9 (and Remark after it)). | Standard probability space | 0.823504 |
6,843 | Given two probability spaces ( Ω 1 , F 1 , P 1 ) {\displaystyle \textstyle (\Omega _{1},{\mathcal {F}}_{1},P_{1})} , ( Ω 2 , F 2 , P 2 ) {\displaystyle \textstyle (\Omega _{2},{\mathcal {F}}_{2},P_{2})} and a measure preserving map f: Ω 1 → Ω 2 {\displaystyle \textstyle f:\Omega _{1}\to \Omega _{2}} , the image f ( Ω 1... | Standard probability space | 0.823504 |
6,844 | For example, the Wiener measure turns the Polish space C [ 0 , ∞ ) {\displaystyle \textstyle C[0,\infty )} (of all continuous functions [ 0 , ∞ ) → R , {\displaystyle \textstyle [0,\infty )\to \mathbb {R} ,} endowed with the topology of local uniform convergence) into a standard probability space. Another example: for ... | Standard probability space | 0.823504 |
6,845 | 2.7 (p. 24)), (Haezendonck 1973, Example 1 (p. 248)), (de la Rue 1993, Theorem 2-3), and (Itô 1984, Theorem 2.4.1). | Standard probability space | 0.823504 |
6,846 | Every probability distribution on the space R n {\displaystyle \textstyle \mathbb {R} ^{n}} turns it into a standard probability space. (Here, a probability distribution means a probability measure defined initially on the Borel sigma-algebra and completed.) The same holds on every Polish space, see (Rokhlin 1952, Sect... | Standard probability space | 0.823504 |
6,847 | Definition. ( Ω , F , P ) {\displaystyle \textstyle (\Omega ,{\mathcal {F}},P)} is standard if there exists a topology τ {\displaystyle \textstyle \tau } on Ω {\displaystyle \textstyle \Omega } such that the topological space ( Ω , τ ) {\displaystyle \textstyle (\Omega ,\tau )} is metrizable; F {\displaystyle \textstyl... | Standard probability space | 0.823504 |
6,848 | 249) and Remark 2 (p. 250)), and (de la Rue 1993, Theorem 4-3). See also (Kechris 1995, Sect. | Standard probability space | 0.823504 |
6,849 | Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be effectively substituted for many of these in probability theory. The dimension of the unit interval is not an obstacle, as was clear already to Norbert Wiener. He constructed the ... | Standard probability space | 0.823504 |
6,850 | In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a... | Standard probability space | 0.823504 |
6,851 | The product of two standard probability spaces is a standard probability space. The same holds for the product of countably many spaces, see (Rokhlin 1952, Sect. 3.4), (Haezendonck 1973, Proposition 12), and (Itô 1984, Theorem 2.4.3). | Standard probability space | 0.823504 |
6,852 | The Van der Waals equation can also be expressed in terms of reduced properties: ( P r + 3 V r 2 ) ( V r − 1 3 ) = 8 3 T r {\displaystyle \left(P_{r}+{\frac {3}{V_{r}^{2}}}\right)\left(V_{r}-{\frac {1}{3}}\right)={\frac {8}{3}}T_{r}} The equation in reduced form is exactly the same for every gas, this is consistent wit... | Van der Waals constant | 0.823502 |
6,853 | Further chemistry and analysis is necessary, first to identify the "triage" compounds that do not provide series displaying suitable SAR and chemical characteristics associated with long-term potential for development, then to improve remaining hit series with regard to the desired primary activity, as well as secondar... | Medicinal Chemistry | 0.823493 |
6,854 | Discovery is the identification of novel active chemical compounds, often called "hits", which are typically found by assay of compounds for a desired biological activity. Initial hits can come from repurposing existing agents toward a new pathologic processes, and from observations of biologic effects of new or existi... | Medicinal Chemistry | 0.823493 |
6,855 | Other areas of study include: metallomics, genomics, proteomics, diagnostic agents (e.g., MRI: Gd, Mn; X-ray: Ba, I) and radiopharmaceuticals (e.g., 99mTc for diagnostics, 186Re for therapeutics). In particular, medicinal chemistry in its most common practice—focusing on small organic molecules—encompasses synthetic or... | Medicinal Chemistry | 0.823493 |
6,856 | Medicines can also be inorganic and organometallic compounds, commonly referred to as metallodrugs (e.g., platinum, lithium and gallium-based agents such as cisplatin, lithium carbonate and gallium nitrate, respectively). The discipline of Medicinal Inorganic Chemistry investigates the role of metals in medicine (metal... | Medicinal Chemistry | 0.823493 |
6,857 | Medicinal or pharmaceutical chemistry is a scientific discipline at the intersection of chemistry and pharmacy involved with designing and developing pharmaceutical drugs. Medicinal chemistry involves the identification, synthesis and development of new chemical entities suitable for therapeutic use. It also includes t... | Medicinal Chemistry | 0.823493 |
6,858 | The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields. | Additive Combinatorics | 0.82349 |
6,859 | The Breuillard–Green–Tao theorem, proved by Emmanuel Breuillard, Ben Green, and Terence Tao in 2011, gives a complete classification of approximate groups. This result can be seen as a nonabelian version of Freiman's theorem, and a generalization of Gromov's theorem on groups of polynomial growth. | Additive Combinatorics | 0.82349 |
6,860 | Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theor... | Additive Combinatorics | 0.82349 |
6,861 | As randomised Bayes rules always have nonrandomised alternatives, they are unnecessary in Bayesian statistics. However, in frequentist statistics, randomised rules are theoretically necessary under certain situations, and were thought to be useful in practice when they were first invented: Egon Pearson forecast that th... | Randomised decision rule | 0.823485 |
6,862 | In statistical decision theory, a randomised decision rule or mixed decision rule is a decision rule that associates probabilities with deterministic decision rules. In finite decision problems, randomised decision rules define a risk set which is the convex hull of the risk points of the nonrandomised decision rules. ... | Randomised decision rule | 0.823485 |
6,863 | Since abstraction is the only way to manufacture functions in the lambda calculus, something must replace it in the combinatory calculus. Instead of abstraction, combinatory calculus provides a limited set of primitive functions out of which other functions may be built. | Combinatory algebra | 0.823483 |
6,864 | The upshot of these historical contingencies was that until theoretical computer science began taking an interest in combinatory logic in the 1960s and 1970s, nearly all work on the subject was by Haskell Curry and his students, or by Robert Feys in Belgium. Curry and Feys (1958), and Curry et al. (1972) survey the ear... | Combinatory algebra | 0.823483 |
6,865 | The original inventor of combinatory logic, Moses Schönfinkel, published nothing on combinatory logic after his original 1924 paper. Haskell Curry rediscovered the combinators while working as an instructor at Princeton University in late 1927. In the late 1930s, Alonzo Church and his students at Princeton invented a r... | Combinatory algebra | 0.823483 |
6,866 | 350 BC – Aristotle attempted a comprehensive classification of animals. His written works include Historion Animalium, a general biology of animals, De Partibus Animalium, a comparative anatomy and physiology of animals, and De Generatione Animalium, on developmental biology. c. | Timeline of biology and organic chemistry | 0.823476 |
6,867 | This timeline of biology and organic chemistry captures significant events from before 1600 to the present. | Timeline of biology and organic chemistry | 0.823476 |
6,868 | 2002 – First virus produced 'from scratch', an artificial polio virus that paralyzes and kills mice. 2007 – Commercialization of Illumina Next generation Sequencing tools. This has become the most popular high-throughput sequencing system. 2012 – Use of CRISPR-Cas9 as a DNA-editing biotechnology tool. | Timeline of biology and organic chemistry | 0.823476 |
6,869 | Richard Dawkins called the theory "one of the great achievements of twentieth-century evolutionary biology." 1968 – Fred Sanger used radioactive phosphorus as a tracer to chromatographically decipher a 120 base long RNA sequence. 1969 – Dorothy Crowfoot Hodgkin deciphered the three-dimensional structure of insulin. | Timeline of biology and organic chemistry | 0.823476 |
6,870 | 1937 – Dorothy Crowfoot Hodgkin discovered the three-dimensional structure of cholesterol. 1937 – Hans Adolf Krebs discovered the tricarboxylic acid cycle. 1937 – In Genetics and the Origin of Species, Theodosius Dobzhansky applies the chromosome theory and population genetics to natural populations in the first mature... | Timeline of biology and organic chemistry | 0.823476 |
6,871 | 1900 – Hugo de Vries, Carl Correns and Erich von Tschermak independently rediscovered Mendel's paper on heredity. 1902 – Walter Sutton and Theodor Boveri, independently proposed that the chromosomes carry the hereditary information. 1905 – William Bateson coined the term "genetics" to describe the study of biological i... | Timeline of biology and organic chemistry | 0.823476 |
6,872 | Mendel's trailblazing foundation for the science of genetics went unnoticed, to his lasting disappointment. 1865 – Friedrich August Kekulé von Stradonitz realized that benzene is composed of carbon and hydrogen atoms in a hexagonal ring. 1869 – Friedrich Miescher discovered nucleic acids in the nuclei of cells. | Timeline of biology and organic chemistry | 0.823476 |
6,873 | 1801 – Jean-Baptiste Lamarck began the detailed study of invertebrate taxonomy. 1802 – The term biology in its modern sense was propounded independently by Gottfried Reinhold Treviranus (Biologie oder Philosophie der lebenden Natur) and Lamarck (Hydrogéologie). The word was coined in 1800 by Karl Friedrich Burdach. 180... | Timeline of biology and organic chemistry | 0.823476 |
6,874 | Molecular and Cellular Biology is a biweekly peer-reviewed scientific journal covering all aspects of molecular and cellular biology. It is published by the American Society for Microbiology and the editor-in-chief is Peter Tontonoz (University of California, Los Angeles). It was established in 1981. The h-index (1981-... | Molecular and Cellular Biology | 0.823474 |
6,875 | Consider the endofunctor F: Set → Set sending X to 1 + X, where 1 is the one-point (singleton) set, the terminal object in the category. An algebra for this endofunctor is a set X (called the carrier of the algebra) together with a function f: (1 + X) → X. Defining such a function amounts to defining a point x ∈ X {\di... | Initial algebra | 0.823471 |
6,876 | In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion. | Initial algebra | 0.823471 |
6,877 | Such a structure can be viewed as a chain of sets, X0 on which f′ is not defined, X1 which elements map into X0 by f′, X2 which elements map into X1 by f′, etc., and Xω containing the remaining elements of X. With this in view, the set N ∪ { ω } {\displaystyle \mathbf {N} \cup \{\omega \}} consisting of the set of natu... | Initial algebra | 0.823471 |
6,878 | Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for coinduction and corecursion. For example, using the same functor 1 + (−) as before, a coalgebra is defined as a set X together with a function f: X → (1 + X). Defining such a function amounts to... | Initial algebra | 0.823471 |
6,879 | Initial algebras are minimal (i.e., have no proper subalgebra). Final coalgebras are simple (i.e., have no proper quotients). | Initial algebra | 0.823471 |
6,880 | In general, any rotation can be specified completely by the three angular displacements with respect to the rectangular-coordinate axes x, y, and z. Any change in the position of the rigid body is thus completely described by three translational and three rotational coordinates. Any displacement of a rigid body may be ... | Rotation around a fixed axis | 0.823461 |
6,881 | The kinematics and dynamics of rotation around a fixed axis of a rigid body are mathematically much simpler than those for free rotation of a rigid body; they are entirely analogous to those of linear motion along a single fixed direction, which is not true for free rotation of a rigid body. The expressions for the kin... | Rotation around a fixed axis | 0.82346 |
6,882 | Rotation around a fixed axis or axial rotation is a special case of rotational motion around a axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbl... | Rotation around a fixed axis | 0.82346 |
6,883 | They calibrated solar models against the low-degree data finding two similarly good fits, one with low Y {\displaystyle Y} and a corresponding low neutrino production rate L ν {\displaystyle L_{\nu }} , the other with higher Y {\displaystyle Y} and L ν {\displaystyle L_{\nu }} ; earlier envelope calibrations against hi... | Helioseismology | 0.823459 |
6,884 | The basic idea in helioseismic holography is that the wavefield, e.g., the line-of-sight Doppler velocity observed at the solar surface, can be used to make an estimate of the wavefield at any location in the solar interior at any instant in time. In this sense, holography is much like seismic migration, a technique in... | Helioseismology | 0.823459 |
6,885 | In 1929, von Neumann published an article containing the axioms that would lead to NBG. This article was motivated by his concern about the consistency of the axiom of limitation of size. He stated that this axiom "does a lot, actually too much." Besides implying the axioms of separation and replacement, and the well-o... | NBG set theory | 0.823457 |
6,886 | Morse–Kelley set theory has an axiom schema of class comprehension that includes formulas whose quantifiers range over classes. MK is a stronger theory than NBG because MK proves the consistency of NBG, while Gödel's second incompleteness theorem implies that NBG cannot prove the consistency of NBG. For a discussion of... | NBG set theory | 0.823457 |
6,887 | Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see Pudlák 1998. | NBG set theory | 0.823457 |
6,888 | In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global ... | NBG set theory | 0.823457 |
6,889 | Gödel extended the class existence theorem to formulas ϕ {\displaystyle \phi } containing relations over classes (such as Y 1 ⊆ Y 2 {\displaystyle Y_{1}\subseteq Y_{2}} and the unary relation M ( Y 1 ) {\displaystyle M(Y_{1})} ), special classes (such as O r d {\displaystyle Ord} ), and operations (such as ( x 1 , x 2 ... | NBG set theory | 0.823457 |
6,890 | {\displaystyle B.} By specifying the 3‑tuples, these axioms also specify the n {\displaystyle n} -tuples for n ≥ 4 {\displaystyle n\geq 4} since: The axioms for handling tuples and the domain axiom imply the following lemma, which is used in the proof of the class existence theorem. One more axiom is needed to prove th... | NBG set theory | 0.823457 |
6,891 | Class existence axioms will be used to prove the class existence theorem: For every formula in n {\displaystyle n} free set variables that quantifies only over sets, there exists a class of n {\displaystyle n} -tuples that satisfy it. The following example starts with two classes that are functions and builds a composi... | NBG set theory | 0.823457 |
6,892 | For example, it is a theorem of NBG that the global axiom of choice implies that the proper class V can be well-ordered and that every proper class can be put into one-to-one correspondence with V.One consequence of conservative extension is that ZFC and NBG are equiconsistent. Proving this uses the principle of explos... | NBG set theory | 0.823457 |
6,893 | NBG is not logically equivalent to ZFC because its language is more expressive: it can make statements about classes, which cannot be made in ZFC. However, NBG and ZFC imply the same statements about sets. Therefore, NBG is a conservative extension of ZFC. NBG implies theorems that ZFC does not imply, but since NBG is ... | NBG set theory | 0.823457 |
6,894 | Roche Diagnostics, Genzyme, QIAGEN, Argenomics S.A. & other companies make tests to detect EGFR mutations, designed to help predict which lung cancer patients may respond best to some therapies, including gefitinib and erlotinib. The tests examine the genetics of tumors removed for biopsy for mutations that make them s... | Gefitinib | 0.823456 |
6,895 | Order: 246 ⋅ 320 ⋅ 59 ⋅ 76 ⋅ 112 ⋅ 133 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 41 ⋅ 47 ⋅ 59 ⋅ 71 = 808017424794512875886459904961710757005754368000000000Schur multiplier: Trivial. Outer automorphism group: Trivial. Other names: F1, M1, Monster group, Friendly giant, Fischer's monster. Remarks: Contains all but 6 of the other sporad... | List of finite simple groups | 0.823453 |
6,896 | Unlike most CAS applications, LiveMath uses a full GUI with high-quality graphical representations of the equations at every step, including input. LiveMath also allows the user to interact with the equation in the sheet; for instance, one can drag an instance of x {\displaystyle x} to the left hand side of the equatio... | LiveMath | 0.823451 |
6,897 | LiveMath is a computer algebra system available on a number of platforms including Mac OS, macOS (Carbon), Microsoft Windows, Linux (x86) and Solaris (SPARC). It is the latest release of a system that originally emerged as Theorist for the "classic" Mac in 1989, became MathView and MathPlus in 1997 after it was sold to... | LiveMath | 0.823451 |
6,898 | Most computer algebra systems also use this as the default input method. In BASIC notation, the formula is entered as it would be entered in BASIC, using the PRINT command – the PRINT command itself being optional. On pressing "ENTER" or "=", the result would be displayed. As with standard infix notation, typing mistak... | Natural textbook display | 0.823451 |
6,899 | Early models, such as the HP-35, used RPN entirely without any alternative methods. Later models can switch between RPN and another notation, such as the HP-12C Platinum which includes both RPN and immediate execution (with operations performed strictly in the order in which inputs are entered), the HP 33s with both RP... | Natural textbook display | 0.823451 |
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