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wiki_200_chunk_0 | Lamellar structure | In materials science, lamellar structures or microstructures are composed of fine, alternating layers of different materials in the form of lamellae. They are often observed in cases where a phase transition front moves quickly, leaving behind two solid products, as in rapid cooling of eutectic (such as solder) or eute... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_201_chunk_0 | Lamellar structure | A deeper eutectic or more rapid cooling will result in finer lamellae; as the size of an individual lamellum approaches zero, the system will instead retain its high-temperature structure. Two common cases of this include cooling a liquid to form an amorphous solid, and cooling eutectoid austenite to form martensite. I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_202_chunk_0 | Liquefaction | In materials science, liquefaction is a process that generates a liquid from a solid or a gas or that generates a non-liquid phase which behaves in accordance with fluid dynamics. It occurs both naturally and artificially. As an example of the latter, a "major commercial application of liquefaction is the liquefaction ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_203_chunk_0 | Material failure theory | In materials science, material failure is the loss of load carrying capacity of a material unit. This definition introduces to the fact that material failure can be examined in different scales, from microscopic, to macroscopic. In structural problems, where the structural response may be beyond the initiation of nonli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_204_chunk_0 | Metallic elements | In materials science, metallurgy, and engineering, a refractory metal is a metal that is extraordinarily resistant to heat and wear. Which metals belong to this category varies; the most common definition includes niobium, molybdenum, tantalum, tungsten, and rhenium. They all have melting points above 2000 °C, and a hi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_205_chunk_0 | Misorientation | In materials science, misorientation is the difference in crystallographic orientation between two crystallites in a polycrystalline material. In crystalline materials, the orientation of a crystallite is defined by a transformation from a sample reference frame (i.e. defined by the direction of a rolling or extrusion ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_206_chunk_0 | Paracrystallinity | In materials science, paracrystalline materials are defined as having short- and medium-range ordering in their lattice (similar to the liquid crystal phases) but lacking crystal-like long-range ordering at least in one direction. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_207_chunk_0 | Permeance | In materials science, permeance is the degree to which a material transmits another substance. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_208_chunk_0 | Polymorphism (crystallography) | In materials science, polymorphism describes the existence of a solid material in more than one form or crystal structure. Polymorphism is a form of isomerism. Any crystalline material can exhibit the phenomenon. Allotropy refers to polymorphism for chemical elements. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_209_chunk_0 | Quenching | In materials science, quenching is the rapid cooling of a workpiece in water, oil, polymer, air, or other fluids to obtain certain material properties. A type of heat treating, quenching prevents undesired low-temperature processes, such as phase transformations, from occurring. It does this by reducing the window of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_210_chunk_0 | Quenching | In steel alloyed with metals such as nickel and manganese, the eutectoid temperature becomes much lower, but the kinetic barriers to phase transformation remain the same. This allows quenching to start at a lower temperature, making the process much easier. High-speed steel also has added tungsten, which serves to rais... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_211_chunk_0 | Radar absorbent material | In materials science, radiation-absorbent material (RAM) is a material which has been specially designed and shaped to absorb incident RF radiation (also known as non-ionising radiation), as effectively as possible, from as many incident directions as possible. The more effective the RAM, the lower the resulting level ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_212_chunk_0 | Recrystallization temperature | In materials science, recrystallization is a process by which deformed grains are replaced by a new set of defect-free grains that nucleate and grow until the original grains have been entirely consumed. Recrystallization is usually accompanied by a reduction in the strength and hardness of a material and a simultaneou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_213_chunk_0 | Reinforcement (composite) | In materials science, reinforcement is a constituent of a composite material which increases the composite's stiffness and tensile strength. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_214_chunk_0 | Segregation in materials | In materials science, segregation is the enrichment of atoms, ions, or molecules at a microscopic region in a materials system. While the terms segregation and adsorption are essentially synonymous, in practice, segregation is often used to describe the partitioning of molecular constituents to defects from solid solut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_215_chunk_0 | Modulus of rigidity | In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: G = d e f τ x y γ x y = F / A Δ x / l = F l A Δ x {\displaystyle G\ {\stackrel {\mathrm {def} }{=}}\ {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_216_chunk_0 | Slip (materials science) | In materials science, slip is the large displacement of one part of a crystal relative to another part along crystallographic planes and directions. Slip occurs by the passage of dislocations on close/packed planes, which are planes containing the greatest number of atoms per area and in close-packed directions (most a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_217_chunk_0 | Slip (materials science) | A slip system describes the set of symmetrically identical slip planes and associated family of slip directions for which dislocation motion can easily occur and lead to plastic deformation. The magnitude and direction of slip are represented by the Burgers vector, b. An external force makes parts of the crystal lattic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_218_chunk_0 | Stress relaxation | In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing some amount of plastic strain. This should not be confused with creep, which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_219_chunk_0 | Stress relaxation | Thus, relaxation has the same effect as cold springing, except it occurs over a longer period of time. The amount of relaxation which takes place is a function of time, temperature and stress level, thus the actual effect it has on the system is not precisely known, but can be bounded. Stress relaxation describes how p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_220_chunk_0 | Stress relaxation | Because they are viscoelastic, polymers behave in a nonlinear, non-Hookean fashion. This nonlinearity is described by both stress relaxation and a phenomenon known as creep, which describes how polymers strain under constant stress. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_221_chunk_0 | Stress relaxation | Experimentally, stress relaxation is determined by step strain experiments, i.e. by applying a sudden one-time strain and measuring the build-up and subsequent relaxation of stress in the material (see figure), in either extensional or shear rheology. Viscoelastic materials have the properties of both viscous and elast... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_222_chunk_0 | Stress relaxation | Although the Maxwell model is good at predicting stress relaxation, it is fairly poor at predicting creep. On the other hand, the Voigt model is good at predicting creep but rather poor at predicting stress relaxation (see viscoelasticity). The extracellular matrix and most tissues are stress relaxing, and the kinetics... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_223_chunk_0 | Superplastic deformation | In materials science, superplasticity is a state in which solid crystalline material is deformed well beyond its usual breaking point, usually over about 400% during tensile deformation. Such a state is usually achieved at high homologous temperature. Examples of superplastic materials are some fine-grained metals and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_224_chunk_0 | Burgers vector | In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as b, that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice. The vector's magnitude and direction is best understood when the dislocation-bearin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_225_chunk_0 | Burgers vector | This dislocation will have the effect of deforming, not only the perfect crystal structure, but the rectangle as well. The said rectangle could have one of its sides disjoined from the perpendicular side, severing the connection of the length and width line segments of the rectangle at one of the rectangle's corners, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_226_chunk_0 | 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 Burgers vector practically, one may draw ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_227_chunk_0 | 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, the Burgers vector and dislocation lin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_228_chunk_0 | 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 industry, since it is easy to pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_229_chunk_0 | 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 temperature, when creep is active. It is given by Z = ε ˙ exp ( Q ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_230_chunk_0 | 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 |
wiki_231_chunk_0 | 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 material flowing properly. Z has also been applied to some... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_232_chunk_0 | 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) and face-centered cubic (FCC) materials, such a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_233_chunk_0 | 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 energy and contact angle, the surface energy of the soli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_234_chunk_0 | Sol–gel process | 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 (sol) that acts as the precurso... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_235_chunk_0 | 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 materials derived from single elements generally carry the -ene suffix in their nam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_236_chunk_0 | 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 exfoliation and bottom-up synthesis. The exfoliation methods include sonication, mech... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_237_chunk_0 | 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" or just "displacement energy". In a crystal, a separate threshold di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_238_chunk_0 | 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 average quantity of interest. Threshold displacement energies in typical s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_239_chunk_0 | 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 temperature. In metals, this decrease in yield strength is due... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_240_chunk_0 | 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 different materials classes. The cra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_241_chunk_0 | Cold pressing | 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 movements and dislocation generation withi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_242_chunk_0 | Antiferromagnetic interaction | 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 ferromagnetism and ferrimagnetism, a manifestation of ordered m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_243_chunk_0 | Spatial dispersion | 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 not apply for nonzero wavevector. The non-isotropic permittivity for nonzero wavevector leads to effects such as optical ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_244_chunk_0 | 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 the schematic on the right. In a corrosive environment, the crack grows due to cyclic loading at a lower stress-intensity range; above th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_245_chunk_0 | Narrow gap | 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 are electrically attracted to each other). In this situation, there is a distinction between "op... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_246_chunk_0 | Narrow gap | 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 binding energy), and therefore the optical and electronic bandgap are essentially identical, an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_247_chunk_0 | 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 same molecule, as in fullerenes. This Frenkel exci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_248_chunk_0 | Dynamic strain aging | 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. During this time, solutes (such as interstitial particles or substitutional impurities) diffuse a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_249_chunk_0 | Dynamic strain aging | 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 manifestations are Lüders bands and the Portevin–Le Chatelier effect. However, the mechanism is known ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_250_chunk_0 | 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 equations as oscillatory integrals. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_251_chunk_0 | 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 studied it in 1904, and named in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_252_chunk_0 | Sigma field | 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 measurable space. The σ-algebras are a subset of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_253_chunk_0 | Sigma field | 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 notion of conditional density is not applicabl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_254_chunk_0 | Sigma field | 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 more useful example is the set of subsets... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_255_chunk_0 | 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 several variables. The "input" variables take real value... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_256_chunk_0 | 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 set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_257_chunk_0 | 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 partial differential equations. Let u ∈ H 2 ( Ω )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_258_chunk_0 | 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 )} where Ω ⊂ R 2 {\displaystyle \Omega \subset \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_259_chunk_0 | Bernstein's theorem (polynomials) | 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 bound for the similar maximum of its derivative. Taking the k-th derivative of the theorem, max | z | ≤ 1 ( | P ( k ) ( z ) | ) ≤... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_260_chunk_0 | Cesàro 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 first n partial sums of the series. This spec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_261_chunk_0 | 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 differentiable. It is a particular case of the Lagrange diff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_262_chunk_0 | 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, which is used for similar purposes and derived ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_263_chunk_0 | 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 |
wiki_264_chunk_0 | 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 represented as the integral with respect to an appropriate Radon measure over the c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_265_chunk_0 | Ivar Ekeland | 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 when the lower level set of a minimization problem is not compact, so that the Bolza... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_266_chunk_0 | 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 minimization problems is not compact, so that the Bolzano–Weierstrass th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_267_chunk_0 | 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. A Fourier integral operator T {\displaystyle T} is given b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_268_chunk_0 | 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 answer when the integrand is replaced by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_269_chunk_0 | 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 , n = 1 ∞ {\textstyle \{a_{m,n}\}_{m=1,n=1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_270_chunk_0 | 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 by Jean Dieudonné.The theorem states: Let f: U → R 0 + {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_271_chunk_0 | 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 |
wiki_272_chunk_0 | Heine's Reciprocal Square Root 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 has degree, m − 1⁄2, a half-integer, and argument, z, real and g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_273_chunk_0 | 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 gives a form of the Cauchy–Schwarz inequality. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_274_chunk_0 | 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 ∈ [1, ∞). Hölder's inequality (in a slightly di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_275_chunk_0 | Leonid Kantorovich | 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 complete vector lattices called "K-spaces" which are now referred ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_276_chunk_0 | 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-symmetric matrix. Korn's theorem is a quantitati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_277_chunk_0 | 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 |
wiki_278_chunk_0 | Lipschitz function | 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, for every pair of points on the graph of this ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_279_chunk_0 | 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. Precisely, let B: c 0 × c 0 → C {\displaystyle B:c_{0}\times c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_280_chunk_0 | 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 |
wiki_281_chunk_0 | 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 characterized by a norm (technically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_282_chunk_0 | 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” convergence since it uses both the weak and strong to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_283_chunk_0 | 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 higher-dimensional manifold to a one-dimens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_284_chunk_0 | 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. The conditions of the theorem can be re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_285_chunk_0 | 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 subsets of the plane that have nonzero ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_286_chunk_0 | 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 |
wiki_287_chunk_0 | 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 a surprise to many mathematicians and ki... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_288_chunk_0 | 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..., cause this to be an injection rather than... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_289_chunk_0 | 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 infinity of basis vectors). Informally, the ide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_290_chunk_0 | 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} ),} Another similar identity is a one which gives th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_291_chunk_0 | 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 differentiable form a set of Lebesgue measure zero. Different... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_292_chunk_0 | Equality of mixed partials | 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} } defined on a set Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} , if p ∈ R n {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_293_chunk_0 | Equality of mixed partials | {\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 ) {\displaystyle f(x,y)} be a differentiable function on an open rectangle Ω {\displaystyle \Omega } conta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_294_chunk_0 | Equality of mixed partials | 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 _{x}f,\,\,\partial _{y}f} as t {\displaystyle t} tends to 0, with a similar statement for se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_295_chunk_0 | Equality of mixed partials | 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 ) . {\displaystyle \int _{a}^{b}f^{\prime }(x)\,dx=f(b)-f(a).} Hence | f ( b ) − f ( a ) | ≤ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_296_chunk_0 | Equality of mixed partials | 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 point c {\displaystyle c} in ( a , b ) {\displaystyle (a,b)} such that f ( b ) − f ( a ) b − ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_297_chunk_0 | Equality of mixed partials | 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 Δ 2 t f ( x 0 , y 0 ) = D 1 D 2 f ( x 0 + t θ , y 0 + t θ ′ ) {\displaystyle \Delta _{1}^{t}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_298_chunk_0 | 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 |
wiki_299_chunk_0 | 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 certain Sobolev space norm and an Orlicz space n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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