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79,800 | For functions of more than one variable, the theorem states that if f is a continuously differentiable function from an open subset A {\displaystyle A} of R n {\displaystyle \mathbb {R} ^{n}} into R n {\displaystyle \mathbb {R} ^{n}} , and the derivative f ′ ( a ) {\displaystyle f'(a)} is invertible at a point a (that ... | Inverse function theorem | 0.785208 |
79,801 | For functions of a single variable, the theorem states that if f {\displaystyle f} is a continuously differentiable function with nonzero derivative at the point a {\displaystyle a} ; then f {\displaystyle f} is injective (or bijective onto the image) in a neighborhood of a {\displaystyle a} , the inverse is continuous... | Inverse function theorem | 0.785208 |
79,802 | In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the invers... | Inverse function theorem | 0.785208 |
79,803 | Sanitation and hygiene are important to prevent typhoid. It can spread only in environments where human feces can come into contact with food or drinking water. Careful food preparation and washing of hands are crucial to prevent typhoid. Industrialization contributed greatly to the elimination of typhoid fever, as it ... | Typhoid fever | 0.785208 |
79,804 | Citing earlier work on the dissolution of potassium in 16-crown-4, he realized that the cyclic polyethers represented a new class of complexing agents that were capable of binding alkali metal cations. He proceeded to report systematic studies of the synthesis and binding properties of crown ethers in a seminal series ... | Crown ether | 0.785205 |
79,805 | In organic chemistry, crown ethers are cyclic chemical compounds that consist of a ring containing several ether groups (R−O−R’). The most common crown ethers are cyclic oligomers of ethylene oxide, the repeating unit being ethyleneoxy, i.e., −CH2CH2O−. Important members of this series are the tetramer (n = 4), the pen... | Crown ether | 0.785205 |
79,806 | Officially the Center for the Advancement of Natural Discoveries using Light Emission, more commonly CANDLE Synchrotron Research Institute, is a project and a research center-institute in Yerevan, Armenia. CANDLE is a project of 3 gigaelectronvolts energy, third generation synchrotron light source for fundamental, indu... | CANDLE Synchrotron Research Institute | 0.785204 |
79,807 | In the mathematical field of graph theory, the Biggs–Smith graph is a 3-regular graph with 102 vertices and 153 edges.It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3-vertex-connected graph and a 3-edge-connected graph. All the cubic distance-regular graphs are known. The B... | Biggs-Smith graph | 0.785204 |
79,808 | Similar research indicates that non-alcoholic fatty liver diseases are also accompanied by a low respiratory quotient value, and the non protein respiratory quotient value was a good indication of disease severity.Recently the respiratory quotient is also used from aquatic scientists to illuminate its environmental app... | Respiratory quotient | 0.785204 |
79,809 | The Journal of Commercial Biotechnology is a peer-reviewed academic journal covering the business of biotechnology. Topics covered include biotechnology management, policy, finance, law, and regulations. The Journal of Commercial Biotechnology was founded by Henry Stewart Publications in 1992 as the Journal of Biotechn... | Journal of Commercial Biotechnology | 0.785203 |
79,810 | The journal was later published by Palgrave MacMillan and was purchased by thinkBiotech in 2010. It is published quarterly. An external Editorial Board assists with peer review. The current editor of the Journal of Commercial Biotechnology is Yali Friedman, who has served since 2007. | Journal of Commercial Biotechnology | 0.785203 |
79,811 | The third and most widely used definition is similar to Halmos's definition, modified so that the Baire sets form a σ-algebra rather than just a σ-ring. A subset of a locally compact Hausdorff topological space is called a Baire set if it is a member of the smallest σ–algebra containing all compact Gδ sets. In other wo... | Baire set | 0.785203 |
79,812 | Halmos (1950, page 220) defined Baire sets of a locally compact Hausdorff space to be the elements of the σ-ring generated by the compact Gδ sets. This definition is no longer used much, as σ-rings are somewhat out of fashion. When the space is σ-compact, this definition is equivalent to the next definition. One reason... | Baire set | 0.785203 |
79,813 | Thus, the three definitions are non-equivalent on an uncountable discrete space. For non-Hausdorff spaces the definitions of Baire sets in terms of continuous functions need not be equivalent to definitions involving Gδ compact sets. For example, if X is an infinite countable set whose closed sets are the finite sets a... | Baire set | 0.785203 |
79,814 | Baire sets coincide with Borel sets in Euclidean spaces. For every compact Hausdorff space, every finite Baire measure (that is, a measure on the σ-algebra of all Baire sets) is regular.For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel measure.The Kolmogorov extensi... | Baire set | 0.785203 |
79,815 | Kunihiko Kodaira defined what we call Baire sets (although he confusingly calls them "Borel sets") of certain topological spaces to be the sets whose characteristic function is a Baire function (the smallest class of functions containing all continuous real-valued functions and closed under pointwise limits of sequence... | Baire set | 0.785203 |
79,816 | Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on... | Baire set | 0.785203 |
79,817 | In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the s... | Baire set | 0.785203 |
79,818 | In molecular biology, Small nucleolar RNA F1/F2/snoR5a refers to a group of related non-coding RNA (ncRNA) molecules which function in the biogenesis of other small nuclear RNAs (snRNAs). These small nucleolar RNAs (snoRNAs) are modifying RNAs and usually located in the nucleolus of the eukaryotic cell which is a major... | Small nucleolar RNA F1/F2/snoR5a | 0.785201 |
79,819 | In molecular biology, the protein domain Stirrup is a domain, found only in found in the domain, archaea. The Stirrup protein domain is found in prokaryotic protein ribonucleotide reductases. It obtains its name due to its resemblance to an old fashioned Japanese stirrup. Stirrip has a molecular mass of 9 kDa and is fo... | Stirrup protein domain | 0.785199 |
79,820 | That is, one forms the direct sum of the Leech lattice VOA with the twisted module, and takes the fixed points under an induced involution. Frenkel, Lepowsky, and Meurman conjectured in 1988 that V ♮ {\displaystyle V^{\natural }} is the unique holomorphic vertex operator algebra with central charge 24, and partition fu... | Vertex algebras | 0.785199 |
79,821 | The monster vertex algebra V ♮ {\displaystyle V^{\natural }} (also called the "moonshine module") is the key to Borcherds's proof of the Monstrous moonshine conjectures. It was constructed by Frenkel, Lepowsky, and Meurman in 1988. It is notable because its character is the j-invariant with no constant term, j ( τ ) − ... | Vertex algebras | 0.785199 |
79,822 | There are both Ramond and Neveu–Schwarz N=2 superconformal algebras, depending on whether the indexing on the G fields is integral or half-integral. However, the U(1) current gives rise to a one-parameter family of isomorphic superconformal algebras interpolating between Ramond and Neveu–Schwartz, and this deformation ... | Vertex algebras | 0.785199 |
79,823 | These algebras have unitary discrete series representations at central charge c ^ = 2 3 c = 1 − 8 m ( m + 2 ) m ≥ 3 {\displaystyle {\hat {c}}={\frac {2}{3}}c=1-{\frac {8}{m(m+2)}}\quad m\geq 3} and unitary representations for all c greater than 3/2, with lowest weight h only constrained by h≥ 0 for Neveu–Schwarz and h ... | Vertex algebras | 0.785199 |
79,824 | The Virasoro algebra has some supersymmetric extensions that naturally appear in superconformal field theory and superstring theory. The N=1, 2, and 4 superconformal algebras are of particular importance. Infinitesimal holomorphic superconformal transformations of a supercurve (with one even local coordinate z and N od... | Vertex algebras | 0.785199 |
79,825 | Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism. For an automorphism σ of order N, the action has the form V ⊗ M → M((z1/N)), with the following monodromy condition: if u ∈ V satisfies σ u = exp(2πik/N)u, then un = 0 unless n satisfies n+k/N ∈ Z (there is some disagre... | Vertex algebras | 0.785199 |
79,826 | In the general case of rotation of an unsymmetric body, which has different values of the moment of inertia about the three principal axes, the rotational motion can be quite complex unless the body is rotating around a principal axis. As described in the tennis racket theorem, rotation of an object around its first or... | Poinsot's ellipsoid | 0.785198 |
79,827 | Various results have been established towards the full conjecture by Ben Moonen, Yves André, Andrei Yafaev, Bas Edixhoven, Laurent Clozel, Bruno Klingler and Emmanuel Ullmo, among others. Some of these results were conditional upon the generalized Riemann hypothesis (GRH) being true. In fact, the proof of the full conj... | André–Oort conjecture | 0.785197 |
79,828 | In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a theorem (proven in several different ways). The conjecture concerns itself with a characterization of the Zariski closure ... | André–Oort conjecture | 0.785197 |
79,829 | In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi con... | Exact solutions of nonlinear partial differential equations | 0.785196 |
79,830 | The Geneva Rules are the rules established by the International Chemistry Committee in 1892. These rules were the beginning of international cooperation for organic chemistry nomenclature. They were decided upon by a group of 34 of leading chemists from 9 different European nations. Their goal was to provide rules for ... | Geneva Rules | 0.785195 |
79,831 | Let E be a vector space over a field k equipped with a linear mapping d: E → E ∧ E {\displaystyle d\colon E\to E\wedge E} from E to the exterior product of E with itself. It is possible to extend d uniquely to a graded derivation (this means that, for any a, b ∈ E which are homogeneous elements, d ( a ∧ b ) = ( d a ) ∧... | Lie coalgebra | 0.785194 |
79,832 | In mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely. | Lie coalgebra | 0.785194 |
79,833 | In this context, the Jacobi identity corresponds to the cocycle condition. More explicitly, let E be a Lie coalgebra over a field of characteristic neither 2 nor 3. The dual space E* carries the structure of a bracket defined by α() = dα(x∧y), for all α ∈ E and x,y ∈ E*.We show that this endows E* with a Lie bracket. | Lie coalgebra | 0.785194 |
79,834 | Just as the exterior algebra (and tensor algebra) of vector fields on a manifold form a Lie algebra (over the base field K), the de Rham complex of differential forms on a manifold form a Lie coalgebra (over the base field K). Further, there is a pairing between vector fields and differential forms. However, the situat... | Lie coalgebra | 0.785194 |
79,835 | So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in... | Ferrers diagram | 0.785192 |
79,836 | In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, ... | Ferrers diagram | 0.785192 |
79,837 | Most theorems on integration and differentiation of scalar functions can be generalized to vector-valued functions, often using essentially the same proofs. Perhaps the most important exception is that absolutely continuous functions need not equal the integrals of their (a.e.) derivatives (unless, for example, X {\dis... | Infinite-dimensional-vector function | 0.785191 |
79,838 | The distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a t-test. The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distri... | Hotelling's T-squared distribution | 0.78519 |
79,839 | In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural genera... | Hotelling's T-squared distribution | 0.78519 |
79,840 | In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl:The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1.In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval [0, 1... | Weyl sequence | 0.785188 |
79,841 | In philosophical logic, metaphysics, and the philosophy of language, the problem of absolute generality is the problem of referring to absolutely everything. Historically, philosophers have assumed that some of their statements are absolutely general, referring to truly everything. In recent years, logicians working in... | Absolute generality | 0.785187 |
79,842 | In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is... | Primary ideal | 0.785185 |
79,843 | In statistics, the quartile coefficient of dispersion is a descriptive statistic which measures dispersion and is used to make comparisons within and between data sets. Since it is based on quantile information, it is less sensitive to outliers than measures such as the coefficient of variation. As such, it is one of s... | Quartile coefficient of dispersion | 0.785184 |
79,844 | In the mathematical field of graph theory, the Livingstone graph is a distance-transitive graph with 266 vertices and 1463 edges. Its intersection array is {11,10,6,1;1,1,5,11}. It is the largest distance-transitive graph with degree 11. | Livingstone graph | 0.785179 |
79,845 | Polymers formed via topochemical polymerisation are generally crystalline. In many cases, the monomer to polymer transition occurs with the retention of crystallinity. Often one can determine the crystal structure of such polymers and the mechanism of polymerisation via single crystal X-ray diffraction. Since the polym... | Semi-crystalline polymer | 0.785179 |
79,846 | In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps. | Gluing scheme | 0.785177 |
79,847 | The category of schemes admits both a finite fiber product and a finite pushout; they both are constructed by gluing affine schemes. For affine schemes, fiber products and pushouts correspond to tensor products and fiber squares of algebras. | Gluing scheme | 0.785177 |
79,848 | It was the first 3-axis stabilized spacecraft in geostationary orbit. It was also the first to use experimentally with some success electric propulsion in geostationary orbit. It also carried several particle physics experiments, including the first heavy ion detector in geostationary orbit. | ATS-6 | 0.785176 |
79,849 | Several particle physics experiments were on board ATS-6. The most significant measured low energy protons (from 25 keV to 3.6 MeV), as well as detected heavy ions (up to 6 MeV). This latter experiment allowed to detect the first heavy ions (Z > 6) with an energy E > 4 MeV, in geostationary orbit. | ATS-6 | 0.785176 |
79,850 | The final audio file is generated, including the synthetic simulation audio in a waveform format, creating speech audio in the voice of many speakers, even those not in training. The first breakthrough in this regard was introduced by WaveNet, a neural network for generating raw audio waveforms capable of emulating the... | Voice cloning | 0.785172 |
79,851 | Replay-based deepfakes are malicious works that aim to reproduce a recording of the interlocutor's voice.There are two types: far-field detection and cut-and-paste detection. In far-field detection, a microphone recording of the victim is played as a test segment on a hands-free phone. On the other hand, cut-and-paste ... | Voice cloning | 0.785171 |
79,852 | The European Conference on Computer Vision (ECCV) is a biennial research conference with the proceedings published by Springer Science+Business Media. Similar to ICCV in scope and quality, it is held those years which ICCV is not. It is considered to be one of the top conferences in computer vision, alongside CVPR and ... | European Conference on Computer Vision | 0.785171 |
79,853 | In mathematics – and in particular the study of games on the unit square – Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem. It states that a particular class of games has a mixed value, provided that at least one of the players has a strategy that is restricted to absolutely continuous dist... | Parthasarathy's theorem | 0.78517 |
79,854 | It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance"... | Autocovariance function | 0.785169 |
79,855 | Autocovariance can be used to calculate turbulent diffusivity. Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations. Reynolds decomposition is used to define the velocity fluctuations u ′ ( x , t ) {\displayst... | Autocovariance function | 0.785169 |
79,856 | In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question. | Autocovariance function | 0.785169 |
79,857 | The Pixel Imaging Mass Spectrometry camera (PImMS) is an ultrafast imaging sensor designed for time-of-flight particle imaging. It was invented by professors of chemistry at the University of Oxford, Mark Brouard and Claire Vallance., Renato Turchetta from IMASENIC (formerly at the STFC Rutherford Appleton Laboratory),... | Pixel Imaging Mass Spectrometry camera | 0.785169 |
79,858 | These cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization. Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its dir... | Quillen–Suslin theorem | 0.785168 |
79,859 | Jean-Pierre Serre, in his 1955 paper Faisceaux algébriques cohérents, remarked that the corresponding question was not known for algebraic vector bundles: "It is not known whether there exist projective A-modules of finite type which are not free." Here A {\displaystyle A} is a polynomial ring over a field, that is, A ... | Quillen–Suslin theorem | 0.785168 |
79,860 | Geometrically, finitely generated projective modules over the ring R {\displaystyle R} correspond to vector bundles over affine space A R n {\displaystyle \mathbb {A} _{R}^{n}} , where free modules correspond to trivial vector bundles. This correspondence (from modules to (algebraic) vector bundles) is given by the 'g... | Quillen–Suslin theorem | 0.785168 |
79,861 | A generalization relating projective modules over regular Noetherian rings A and their polynomial rings is known as the Bass–Quillen conjecture. Note that although G L n {\displaystyle GL_{n}} -bundles on affine space are all trivial, this is not true for G-bundles where G is a general reductive algebraic group. | Quillen–Suslin theorem | 0.785168 |
79,862 | The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space. The theor... | Quillen–Suslin theorem | 0.785168 |
79,863 | Another potentially important effect that needs to be better characterized for the post-perovskite phase transition is the influence of other chemical components that are known to be present to some degree in Earth's lowermost mantle. The phase transition pressure (characterized by a two-phase loop in this system), was... | Post-perovskite | 0.785167 |
79,864 | Post-perovskite phase is stable above 120 GPa at 2500 K, and exhibits a positive Clapeyron slope such that the transformation pressure increases with temperature. Because these conditions correspond to a depth of about 2600 km and the D" seismic discontinuity occurs at similar depths, the perovskite to post-perovskite ... | Post-perovskite | 0.785167 |
79,865 | The name "post-perovskite" derives from silicate perovskite, the stable phase of MgSiO3 throughout most of Earth's mantle, which has the perovskite structure. The prefix "post-" refers to the fact that it occurs after perovskite structured MgSiO3 as pressure increases (and historically, the progression of high pressure... | Post-perovskite | 0.785167 |
79,866 | In plasma physics, a drift wave is a type of collective excitation that is driven by a pressure gradient within a magnetised plasma, which can be destabilised by differences between ion and electron motion (then known as drift-wave instability or drift instability). The drift wave typically propagates across the pressu... | Drift waves | 0.785166 |
79,867 | In the presence of spin–orbit interaction, total angular momentum should take part in. From the three valence band, l=1 and s=1/2 state generate six state of | j , m j ⟩ {\displaystyle \left|j,m_{j}\right\rangle } as | 3 2 , ± 3 2 ⟩ , | 3 2 , ± 1 2 ⟩ , | 1 2 , ± 1 2 ⟩ {\displaystyle \left|{\frac {3}{2}},\pm {\frac {3}{... | Luttinger parameter | 0.785166 |
79,868 | Metascape implements a CAME analysis workflow: Conversion: Convert gene identifiers from popular types (such as Symbol, RefSeq, Ensembl, UniProt, UCSC) into human Entrez gene IDs and vice versa. Annotation: Extract from dozens of function-relevant gene annotations, including protein families, transmembrane/secreted pre... | Metascape | 0.785165 |
79,869 | In the "OMICs" age, it is important to gain biological insights into a list of genes. Although a number of bioinformatics sources exist for this purpose, such as DAVID, they are not all free, easy to use, and well maintained. To analyze multiple lists of genes originated from orthogonal but complementary "OMICs" studie... | Metascape | 0.785165 |
79,870 | Most alkali metal halides crystallize with the face-centered cubic lattices. In this structure both the metals and halides feature octahedral coordination geometry, in which each ion has a coordination number of six. Caesium chloride, bromide, and iodide crystallize in a body-centered cubic lattice that accommodates co... | Alkali halide | 0.785163 |
79,871 | The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces: Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of μ (or equivalently its sets of measure 0) is unique... | Self-adjoint operators | 0.785163 |
79,872 | As noted above, the spectral theorem applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric operator that has an orthonormal basis of eigenvectors. (This operator is actually "essentially self-adjoint.") | Self-adjoint operators | 0.785163 |
79,873 | {\displaystyle |\langle Ax,y\rangle |=|\langle x,Ay\rangle |\leq \|x\|\cdot \|Ay\|.} The equality A = A ∗ | Dom ( A ) {\displaystyle A=A^{*}|_{\operatorname {Dom} (A)}} holds due to the equality ⟨ x , A ∗ y ⟩ = ⟨ A x , y ⟩ = ⟨ x , A y ⟩ , {\displaystyle \langle x,A^{*}y\rangle =\langle Ax,y\rangle =\langle x,Ay\rangl... | Self-adjoint operators | 0.785163 |
79,874 | A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis {ei}i ∈ I consisting of eigenvectors for A. Example. The Hamiltonian for the harmonic oscillator has a quadratic potential V, that is − Δ + | x | 2 . {\displaystyle -\Delta +|x|^{2}.} This Hamiltonian has pure point spectru... | Self-adjoint operators | 0.785163 |
79,875 | Other versions of the spectral theorem can be found in the spectral theorem article linked to above. The spectral theorem for unbounded self-adjoint operators can be proved by reduction to the spectral theorem for unitary (hence bounded) operators. This reduction uses the Cayley transform for self-adjoint operators whi... | Self-adjoint operators | 0.785163 |
79,876 | Partially defined operators A, B on Hilbert spaces H, K are unitarily equivalent if and only if there is a unitary transformation U: H → K such that U maps dom A bijectively onto dom B, B U ξ = U A ξ , ∀ ξ ∈ dom A . {\displaystyle BU\xi =UA\xi ,\qquad \forall \xi \in \operatorname {dom} A.} A multiplication operator ... | Self-adjoint operators | 0.785163 |
79,877 | It has been customary to introduce the following notation E T ( λ ) = 1 ( − ∞ , λ ] ( T ) {\displaystyle \operatorname {E} _{T}(\lambda )=\mathbf {1} _{(-\infty ,\lambda ]}(T)} where 1 ( − ∞ , λ ] {\displaystyle \mathbf {1} _{(-\infty ,\lambda ]}} is the characteristic function (indicator function)of the interval ( −... | Self-adjoint operators | 0.785163 |
79,878 | Physicists would then go on to say that these "eigenvectors" are orthonormal in a continuous sense, where the usual Kronecker delta δ i , j {\displaystyle \delta _{i,j}} is replaced by a Dirac delta function δ ( p − p ′ ) {\displaystyle \delta \left(p-p'\right)} . Although these statements may seem disconcerting to mat... | Self-adjoint operators | 0.785163 |
79,879 | In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an orthonormal basis in the cl... | Self-adjoint operators | 0.785163 |
79,880 | chemists recommended that element 106, which is placed under tungsten (74), be called 'seaborgium.' I was looking forward to the day when chemical investigators will refer to such compounds as seaborgous chloride, seaborgic nitrate, and perhaps, sodium seaborgate.This is the greatest honor ever bestowed upon me—even be... | Organoseaborgium chemistry | 0.78516 |
79,881 | Very few properties of seaborgium or its compounds have been measured; this is due to its extremely limited and expensive production and the fact that seaborgium (and its parents) decays very quickly. A few singular chemistry-related properties have been measured, but properties of seaborgium metal remain unknown and o... | Organoseaborgium chemistry | 0.78516 |
79,882 | In a manner similar to the formation of the oxychloride, the results of the experiment indicated the formation of seaborgium oxide hydroxide, a reaction well known among the lighter group 6 homologues as well as the pseudohomologue uranium. 2 Sg + 3 O2 → 2 SgO3 SgO3 + H2O → SgO2(OH)2Predictions on the aqueous chemistry... | Organoseaborgium chemistry | 0.78516 |
79,883 | Experimental chemical investigation of seaborgium has been hampered due to the need to produce it one atom at a time, its short half-life, and the resulting necessary harshness of the experimental conditions. The isotope 265Sg and its isomer 265mSg are advantageous for radiochemistry: they are produced in the 248Cm(22N... | Organoseaborgium chemistry | 0.78516 |
79,884 | The chemical properties of seaborgium are characterized only partly, but they compare well with the chemistry of the other group 6 elements. In 1974, a few atoms of seaborgium were produced in laboratories in the Soviet Union and in the United States. The priority of the discovery and therefore the naming of the elemen... | Organoseaborgium chemistry | 0.78516 |
79,885 | It is also radioactive; the most stable known isotope, 269Sg, has a half-life of approximately 14 minutes.In the periodic table of the elements, it is a d-block transactinide element. It is a member of the 7th period and belongs to the group 6 elements as the fourth member of the 6d series of transition metals. Chemist... | Organoseaborgium chemistry | 0.78516 |
79,886 | In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole. Emergence plays a central role in theories of integrative levels and of complex systems. For instance, the phe... | Emergence | 0.785158 |
79,887 | The trend to implement software programs into the civil engineering industry began as educational concerns for the future as civil engineering prepared to enter the twenty-first century. Today, these concerns and trends are centered on the continuing education unit which have become required as part of maintaining the ... | Civil engineering software | 0.785158 |
79,888 | Hydraulic engineering as a sub-discipline of civil engineering is concerned with the flow and conveyance of fluids, principally water and sewage. One feature of these systems is the extensive use of gravity as the motive force to cause the movement of the fluids. This area of civil engineering is intimately related to ... | Civil engineering software | 0.785158 |
79,889 | Different software packages rely on different formulas and theories as the basis for these calculations. Consulting engineers also take advantage of the insight software can provide as far as crossing services are concerned. A road design may have to accommodate the presence of underground pipes for example. Civil Desi... | Civil engineering software | 0.785158 |
79,890 | Many software programs are available for each discipline of civil engineering. Most civil engineers practice in specialized subsets of civil engineering, such as geotechnical engineering, structural engineering, transportation engineering, hydraulic engineering, environmental engineering, project and construction manag... | Civil engineering software | 0.785158 |
79,891 | In statistics, the Sobel test is a method of testing the significance of a mediation effect. The test is based on the work of Michael E. Sobel, a statistics professor at Columbia University in New York, NY, and is an application of the delta method. In mediation, the relationship between the independent variable and th... | Sobel test | 0.785156 |
79,892 | A few mathematicians of Late Antiquity wrote commentaries on the Almagest, including Pappus of Alexandria as well as Theon of Alexandria and his daughter Hypatia. Ptolemaic astronomy became standard in medieval western European and Islamic astronomy until it was displaced by Maraghan, heliocentric and Tychonic systems ... | Hellenistic astronomy | 0.785153 |
79,893 | in The Crime of Claudius Ptolemy, he argued that Ptolemy faked his observations and falsely claimed the catalogue of Hipparchus as his own work. Newton's theories have not been adopted by most historians of astronomy. Claudius Ptolemy of Alexandria performed a deep examination of the shape and motion of the Earth and c... | Hellenistic astronomy | 0.785153 |
79,894 | The Almagest is one of the most influential books in the history of Western astronomy. In this book, Ptolemy explained how to predict the behavior of the planets, as Hipparchus could not, with the introduction of a new mathematical tool, the equant. The Almagest gave a comprehensive treatment of astronomy, incorporatin... | Hellenistic astronomy | 0.785153 |
79,895 | Since all his own works are lost, our knowledge of Eudoxus is obtained from secondary sources. Aratus's poem on astronomy is based on a work of Eudoxus, and possibly also Theodosius of Bithynia's Spherics. They give us an indication of his work in spherical astronomy as well as planetary motions. | Hellenistic astronomy | 0.785153 |
79,896 | By tilting the axes of the spheres, and by assigning each a different period of revolution, he was able to approximate the celestial "appearances." Thus, he was the first to attempt a mathematical description of the motions of the planets. A general idea of the content of On Speeds, his book on the planets, can be glea... | Hellenistic astronomy | 0.785153 |
79,897 | 347 BCE), to develop a system of Greek astronomy. According to a modern historian of science, David Lindberg: In their work we find (1) a shift from stellar to planetary concerns, (2) the creation of a geometrical model, the "two-sphere model," for the representation of stellar and planetary phenomena, and (3) the esta... | Hellenistic astronomy | 0.785153 |
79,898 | In classical Greece, astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions. This tradition began with the Pythagoreans, who placed astronomy among the four mathematical arts (along with arithmetic, geometry, and music). The study o... | Hellenistic astronomy | 0.785153 |
79,899 | In addition to the authors named in the article, the following list of people who worked on mathematical astronomy or cosmology may be of interest. | Hellenistic astronomy | 0.785153 |
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