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Burgers vector Summary Burgers_vector Specifically, the breadth of the opening defines the magnitude of the Burgers vector, and, when a set of fixed coordinates is introduced, an angle between the termini of the dislocated rectangle's length line segment and width line segment may be specified. When calculating the Bur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Burgers vector Summary Burgers_vector In most metallic materials, the magnitude of the Burgers vector for a dislocation is of a magnitude equal to the interatomic spacing of the material, since a single dislocation will offset the crystal lattice by one close-packed crystallographic spacing unit. In edge dislocations, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Charpy test Summary Charpy_test In materials science, the Charpy impact test, also known as the Charpy V-notch test, is a standardized high strain rate test which determines the amount of energy absorbed by a material during fracture. Absorbed energy is a measure of the material's notch toughness. It is widely used in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Charpy test Summary Charpy_test A disadvantage is that some results are only comparative. The test was pivotal in understanding the fracture problems of ships during World War II.The test was developed around 1900 by S. B. Russell (1898, American) and Georges Charpy (1901, French). The test became known as the Charpy t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zener–Hollomon parameter Summary Zener–Hollomon_parameter In materials science, the Zener–Hollomon parameter, typically denoted as Z, is used to relate changes in temperature or strain-rate to the stress-strain behavior of a material. It has been most extensively applied to the forming of steels at increased temperatur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zener–Hollomon parameter Summary Zener–Hollomon_parameter It is named after Clarence Zener and John Herbert Hollomon, Jr. who established the formula based on the stress-strain behavior in steel. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zener–Hollomon parameter Summary Zener–Hollomon_parameter When plastically deforming a material, the flow stress depends heavily on both the strain-rate and temperature. During forming processes, Z may help determine appropriate changes in strain-rate or temperature when the other variable is altered, in order to keep ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cottrell atmosphere Summary Cottrell_atmosphere In materials science, the concept of the Cottrell atmosphere was introduced by A. H. Cottrell and B. A. Bilby in 1949 to explain how dislocations are pinned in some metals by boron, carbon, or nitrogen interstitials. Cottrell atmospheres occur in body-centered cubic (BCC)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cottrell atmosphere Summary Cottrell_atmosphere Once the atom has diffused into the dislocation core the atom will stay. Typically only one interstitial atom is required per lattice plane of the dislocation. The collection of solute atoms around the dislocation core due to this process is the Cottrell atmosphere. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sessile drop technique Summary Sessile_drop_technique In materials science, the sessile drop technique is a method used for the characterization of solid surface energies, and in some cases, aspects of liquid surface energies. The main premise of the method is that by placing a droplet of liquid with a known surface en... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sol–gel process Summary Sol_gel In materials science, the sol–gel process is a method for producing solid materials from small molecules. The method is used for the fabrication of metal oxides, especially the oxides of silicon (Si) and titanium (Ti). The process involves conversion of monomers into a colloidal solution... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Two dimensional (2D) nanomaterials Summary Two_dimensional_(2D)_nanomaterials In materials science, the term single-layer materials or 2D materials refers to crystalline solids consisting of a single layer of atoms. These materials are promising for some applications but remain the focus of research. Single-layer mater... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Two dimensional (2D) nanomaterials Summary Two_dimensional_(2D)_nanomaterials 2D materials can generally be categorized as either 2D allotropes of various elements or as compounds (consisting of two or more covalently bonding elements). It is predicted that there are hundreds of stable single-layer materials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Two dimensional (2D) nanomaterials Summary Two_dimensional_(2D)_nanomaterials The atomic structure and calculated basic properties of these and many other potentially synthesisable single-layer materials, can be found in computational databases. 2D materials can be produced using mainly two approaches: top-down exfolia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Threshold displacement energy Summary Threshold_displacement_energy In materials science, the threshold displacement energy (Td) is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. It is also known as "displacement threshold energy" ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Threshold displacement energy Summary Threshold_displacement_energy Then one should distinguish between the minimum (Td,min) and average (Td,ave) over all lattice directions' threshold displacement energies. In amorphous solids, it may be possible to define an effective displacement energy to describe some other averag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Yield strength anomaly Summary Yield_strength_anomaly In materials science, the yield strength anomaly refers to materials wherein the yield strength (i.e., the stress necessary to initiate plastic yielding) increases with temperature. For the majority of materials, the yield strength decreases with increasing temperat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Yield strength anomaly Summary Yield_strength_anomaly In concert with yield strength or ductility anomalies, some materials demonstrate extrema in other temperature dependent properties, such as a minimum in ultrasonic damping, or a maximum in electrical conductivity.The yield strength anomaly in β-brass was one of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Yield strength anomaly Summary Yield_strength_anomaly Eventually, a maximum yield strength is reached. For even higher temperatures, the yield strength decreases and, eventually, drops to zero when reaching the melting temperature, where the solid material transforms into a liquid. For ordered intermetallics, the tempe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Toughening Summary Toughening In materials science, toughening refers to the process of making a material more resistant to the propagation of cracks. When a crack propagates, the associated irreversible work in different materials classes is different. Thus, the most effective toughening mechanisms differ among differ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cold pressing Summary Strain_hardening In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context. This strengthening occurs because of dislocation mov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cold pressing Summary Strain_hardening Many non-brittle metals with a reasonably high melting point as well as several polymers can be strengthened in this fashion. Alloys not amenable to heat treatment, including low-carbon steel, are often work-hardened. Some materials cannot be work-hardened at low temperatures, suc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Antiferromagnetic interaction Summary Antiferromagnetism In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. This is, like ferrom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spatial dispersion In isotropic media Spatial_dispersion > Spatial dispersion in electromagnetism > In isotropic media In materials that have no relevant crystalline structure, spatial dispersion can be important. Although symmetry demands that the permittivity is isotropic for zero wavevector, this restriction does no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Corrosion fatigue Stress-corrosion fatigue Corrosion_fatigue > Crack-propagation studies in corrosion fatigue > Stress-corrosion fatigue In materials where the maximum applied-stress-intensity factor exceeds the stress-corrosion cracking-threshold value, stress corrosion adds to crack-growth velocity. This is shown in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Narrow gap Optical versus electronic bandgap Bandgap_energy > Optical versus electronic bandgap In materials with a large exciton binding energy, it is possible for a photon to have just barely enough energy to create an exciton (bound electron–hole pair), but not enough energy to separate the electron and hole (which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Narrow gap Optical versus electronic bandgap Bandgap_energy > Optical versus electronic bandgap The optical bandgap is at lower energy than the transport gap. In almost all inorganic semiconductors, such as silicon, gallium arsenide, etc., there is very little interaction between electrons and holes (very small exciton... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Exciton Frenkel exciton Exciton > Frenkel exciton In materials with a relatively small dielectric constant, the Coulomb interaction between an electron and a hole may be strong and the excitons thus tend to be small, of the same order as the size of the unit cell. Molecular excitons may even be entirely located on the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Exciton Frenkel exciton Exciton > Frenkel exciton Frenkel excitons are typically found in alkali halide crystals and in organic molecular crystals composed of aromatic molecules, such as anthracene and tetracene. Another example of Frenkel exciton includes on-site d-d excitations in transition metal compounds with part... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dynamic strain aging Description of mechanism Dynamic_strain_aging > Description of mechanism In materials, the motion of dislocations is a discontinuous process. When dislocations meet obstacles during plastic deformation (such as particles or forest dislocations), they are temporarily arrested for a certain time. Dur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dynamic strain aging Description of mechanism Dynamic_strain_aging > Description of mechanism Eventually these dislocations will overcome the obstacles with sufficient stress and will quickly move to the next obstacle where they are stopped and the process can repeat. This process's most well-known macroscopic manifest... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formative evaluation In math education Formative_evaluation > Specific applications > In math education In math education, it is important for teachers to see how their students approach the problems and how much mathematical knowledge and at what level students use when solving the problems. That is, knowing how stude... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-differential operator Summary Pseudo-differential_operators In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Oscillatory integral Summary Oscillatory_integral In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multidimensional transform Summary Multidimensional_transform In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Z-order curve Summary Z-order_curve In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It is named in France after Henri Lebesgue, who... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sigma field Summary Join_(sigma_algebra) In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair ( X , Σ ) {\displaystyle (X,\Sigma )} is called a measura... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sigma field Summary Join_(sigma_algebra) Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic, particularly when the statistic is a function or a random process and the notio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sigma field Summary Join_(sigma_algebra) In general, a finite algebra is always a σ-algebra. If { A 1 , A 2 , A 3 , … } , {\displaystyle \{A_{1},A_{2},A_{3},\ldots \},} is a countable partition of X {\displaystyle X} then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra. A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multi-variable function Summary Multi-variable_function In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Laakso space Summary Laakso_space In mathematical analysis and metric geometry, Laakso spaces are a class of metric spaces which are fractal, in the sense that they have non-integer Hausdorff dimension, but that admit a notion of differential calculus. They are constructed as quotient spaces of × K where K is a Cantor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bounded poset Summary Bounded_set In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bounded poset Summary Bounded_set Boundary is a distinct concept: for example, a circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R2 constrained by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Agmon's inequality Summary Agmon's_inequality In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space L ∞ {\displaystyle L^{\infty }} and the Sobolev spaces H s {\displaystyle H^{s}} . It is useful in the study of par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Agmon's inequality Summary Agmon's_inequality {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/4}\|u\|_{H^{2}(\Omega )}^{3/4}.} In 2D, the first inequality still holds, but not the second: let u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) {\displaystyle u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bernstein's theorem (polynomials) Bernstein's inequality Bernstein's_inequality_(mathematical_analysis) > Bernstein's inequality In mathematical analysis, Bernstein's inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cesàro summation Summary Cesaro_summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the fir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Clairaut's equation Summary Clairaut's_equation In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form y ( x ) = x d y d x + f ( d y d x ) {\displaystyle y(x)=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)} where f {\displaystyle f} is continuously differentiab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Darboux's formula Summary Darboux's_formula In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux (1876) for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalization to the complex plane of the Euler–Maclaurin summation formula, whi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dini continuity Summary Dini_continuity In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ehrenpreis's fundamental principle Summary Ehrenpreis's_fundamental_principle In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states: Every solution of a system (in general, overdetermined) of homogeneous partial differential equations with constant coefficients can be repre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ivar Ekeland Variational principle Ivar_Ekeland > Research > Variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist a nearly optimal solution to a class of optimization problems.Ekeland's variational principle can be used w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ekeland's variational principle Summary Ekeland's_variational_principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems. Ekeland's principle can be used when the lower level set of a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier integral operator Summary Fourier_integral_operator In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fubini's Theorem Summary Fubini's_Theorem In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fubini's Theorem Summary Fubini's_Theorem Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. A related theorem is often called Fubini's theorem for infinite series, which states that if { a m , n } m = 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Glaeser's continuity theorem Summary Glaeser's_continuity_theorem In mathematical analysis, Glaeser's continuity theorem is a characterization of the continuity of the derivative of the square roots of functions of class C 2 {\displaystyle C^{2}} . It was introduced in 1963 by Georges Glaeser, and was later simplified ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Haar's Tauberian theorem Summary Haar's_Tauberian_theorem In mathematical analysis, Haar's Tauberian theorem named after Alfréd Haar, relates the asymptotic behaviour of a continuous function to properties of its Laplace transform. It is related to the integral formulation of the Hardy–Littlewood Tauberian theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Heine's Reciprocal Square Root Identity Summary Heine's_identity In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as where Q m − 1 2 {\displaystyle Q_{m-{\frac {1}{2}}}} is a Legendre function of the second kind, which... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hölder's inequality Summary Hölder's_inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces. The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hölder's inequality Summary Hölder's_inequality Conversely, if f is in Lp(μ) and g is in Lq(μ), then the pointwise product fg is in L1(μ). Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ), and also to establish that Lq(μ) is the dual space of Lp(μ) for p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leonid Kantorovich Mathematics Leonid_Kantorovich > Mathematics In mathematical analysis, Kantorovich had important results in functional analysis, approximation theory, and operator theory. In particular, Kantorovich formulated some fundamental results in the theory of normed vector lattices, especially in Dedekind co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Korn's inequality Summary Korn's_inequality In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Krein's condition Summary Krein's_condition In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums { ∑ k = 1 n a k exp ( i λ k x ) , a k ∈ C , λ k ≥ 0 } , {\displaystyle \left\{\sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lambert summation Summary Lambert_summation In mathematical analysis, Lambert summation is a summability method for a class of divergent series. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lipschitz function Summary Lipschitz_constant In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Littlewood's 4/3 inequality Summary Littlewood's_4/3_inequality In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood, is an inequality that holds for every complex-valued bilinear form defined on c 0 {\displaystyle c_{0}} , the Banach space of scalar sequences that converge to zero... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Littlewood's 4/3 inequality Summary Littlewood's_4/3_inequality {\displaystyle \|B\|=\sup\{|B(x_{1},x_{2})|:\|x_{i}\|_{\infty }\leq 1\}.} The exponent 4/3 is optimal, i.e., cannot be improved by a smaller exponent. It is also known that for real scalars the aforementioned constant is sharp. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lorentz space Summary Lorentz_space In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar L p {\displaystyle L^{p}} spaces. The Lorentz spaces are denoted by L p , q {\displaystyle L^{p,q}} . Like the L p {\displaystyle L^{p}} spaces, they are c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mosco convergence Summary Mosco_convergence In mathematical analysis, Mosco convergence is a notion of convergence for functionals that is used in nonlinear analysis and set-valued analysis. It is a particular case of Γ-convergence. Mosco convergence is sometimes phrased as “weak Γ-liminf and strong Γ-limsup” convergen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Netto's theorem Summary Netto's_theorem In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after Eugen Netto.The case for maps from a hig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Netto's theorem Summary Netto's_theorem The faults were later recognized and corrected.An important special case of this theorem concerns the non-existence of continuous bijections from one-dimensional spaces, such as the real line or unit interval, to two-dimensional spaces, such as the Euclidean plane or unit square.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Netto's theorem Summary Netto's_theorem Examples include the Peano curve and Hilbert curve. Neither of these examples has any self-crossings, but by Netto's theorem there are many points of the square that are covered multiple times by these curves. Osgood curves are continuous bijections from one-dimensional spaces to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Netto's theorem Summary Netto's_theorem They form Jordan curves in the plane. However, by Netto's theorem, they cannot cover the entire plane, unit square, or any other two-dimensional region. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Netto's theorem Summary Netto's_theorem If one relaxes the requirement of continuity, then all smooth manifolds of bounded dimension have equal cardinality, the cardinality of the continuum. Therefore, there exist discontinuous bijections between any two of them, as Georg Cantor showed in 1878. Cantor's result came as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Netto's theorem Summary Netto's_theorem A near-bijection from the unit square to the unit interval can be obtained by interleaving the digits of the decimal representations of the Cartesian coordinates of points in the square. The ambiguities of decimal, exemplified by the two decimal representations of 1 = 0.999..., c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parseval's formula Summary Parseval's_formula In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parseval's formula Summary Parseval's_formula A similar result is the Plancherel theorem, which asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself. In one-dimension, for f ∈ L 2 ( R ) , {\displaystyle f\in L^{2}(\mathbb {R} ),} An... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rademacher's theorem Summary Rademacher's_theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f: U → Rm is Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) named after Alexis Clairaut and Hermann Schwarz, states that for a function f: Ω → R {\displaystyle f\colon \Omega \to \mathbb {R}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem {\displaystyle f.} An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case). Let f ( x , y ) {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem {\displaystyle \Omega .} Define u ( h , k ) = f ( a + h , b + k ) − f ( a + h , b ) , v ( h , k ) = f ( a + h , b + k ) − f ( a , b + k ) , w ( h , k ) = f ( a + h , b + k ) − f ( a + h , b ) − f ( a , b + k ) + f ( a , b ) . {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem {\displaystyle {\begin{aligned}hk\,\partial _{y}\partial _{x}f\left(a+\theta h,\,b+\theta ^{\prime }k\right)&=hk\,\partial _{x}\partial _{y}f\left(a+\phi ^{\prime }h,\,b+\phi k\right),\\\partial _{y}\partial _{x}f\left(a+\theta ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem Indeed the difference operators Δ x t , Δ y t {\displaystyle \Delta _{x}^{t},\,\,\Delta _{y}^{t}} commute and Δ x t f , Δ y t f {\displaystyle \Delta _{x}^{t}f,\,\,\Delta _{y}^{t}f} tend to ∂ x f , ∂ y f {\displaystyle \partial ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem By the fundamental theorem of calculus for C 1 {\displaystyle C^{1}} functions f {\displaystyle f} on an open interval I {\displaystyle I} with ( a , b ) ⊂ I {\displaystyle (a,b)\subset I} ∫ a b f ′ ( x ) d x = f ( b ) − f ( a )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem Recall that the elementary discussion on maxima or minima for real-valued functions implies that if f {\displaystyle f} is continuous on {\displaystyle } and differentiable on ( a , b ) {\displaystyle (a,b)} , then there is a p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem {\displaystyle \left|\Delta _{1}^{t}\Delta _{2}^{t}f(x_{0},y_{0})-D_{1}D_{2}f(x_{0},y_{0})\right|\leq \sup _{0\leq s\leq 1}\left|\Delta _{1}^{t}D_{2}f(x_{0},y_{0}+ts)-D_{1}D_{2}f(x_{0},y_{0})\right|\leq \sup _{0\leq r,s\leq 1}\l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equality of mixed partials Schwarz's theorem Equality_of_mixed_partials > Schwarz's theorem Hence, since the difference operators commute, so do the partial differential operators D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} , as claimed.Remark. By two applications of the classical mean value theorem, Δ 1 t ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strichartz estimate Summary Strichartz_estimate In mathematical analysis, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tannery's theorem Summary Tannery's_theorem In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trudinger's theorem Summary Trudinger's_theorem In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trudinger's theorem Summary Trudinger's_theorem The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem: Let Ω {\displaystyle \Omega } be a bounded domain in R n {\displaystyle \mathbb {R} ^{n}} satisfying the cone condition. Let m p = n {\displaystyle mp=n} and p > 1 {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trudinger's theorem Summary Trudinger's_theorem {\displaystyle A(t)=\exp \left(t^{n/(n-m)}\right)-1.} Then there exists the embedding W m , p ( Ω ) ↪ L A ( Ω ) {\displaystyle W^{m,p}(\Omega )\hookrightarrow L_{A}(\Omega )} where L A ( Ω ) = { u ∈ M f ( Ω ): ‖ u ‖ A , Ω = inf { k > 0: ∫ Ω A ( | u ( x ) | k ) d x ≤ 1 } <... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Wiener's Tauberian theorem Summary Wiener's_Tauberian_theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L 1 {\displaystyle L^{1}} or L 2 {\displaystyle L^{2}} can ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zorich's theorem Summary Zorich's_theorem In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Banach limit Summary Banach_limit In mathematical analysis, a Banach limit is a continuous linear functional ϕ: ℓ ∞ → C {\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space ℓ ∞ {\displaystyle \ell ^{\infty }} of all bounded complex-valued sequences such that for all sequences x = ( x n ) {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Banach limit Summary Banach_limit As a consequence of the above properties, a real-valued Banach limit also satisfies: lim inf n → ∞ x n ≤ ϕ ( x ) ≤ lim sup n → ∞ x n . {\displaystyle \liminf _{n\to \infty }x_{n}\leq \phi (x)\leq \limsup _{n\to \infty }x_{n}.} The existence of Banach limits is usually proved using the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Besicovitch covering theorem Summary Besicovitch_covering_theorem In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Besicovitch covering theorem Summary Besicovitch_covering_theorem Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Contraction semigroup Summary Quasicontraction_semigroup In mathematical analysis, a C0-semigroup Γ(t), t ≥ 0, is called a quasicontraction semigroup if there is a constant ω such that ||Γ(t)|| ≤ exp(ωt) for all t ≥ 0. Γ(t) is called a contraction semigroup if ||Γ(t)|| ≤ 1 for all t ≥ 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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