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c_kbhy7v95w6fi | 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 Burgers vector practically, one may draw ... |
c_cq9s99zrwut6 | 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, the Burgers vector and dislocation lin... |
c_kcxl9bwvix8g | 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 industry, since it is easy to pr... |
c_uby0a46r94vd | 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 test in the early 1900s due to th... |
c_yvb5so5ecui3 | 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 temperature, when creep is active. It is given by Z = ε ˙ exp ( Q ... |
c_81ra4xs7eoyy | 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. |
c_mxdf62uz4ut9 | 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 material flowing properly. Z has also been applied to some... |
c_eccr9w48ehk1 | 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) and face-centered cubic (FCC) materials, such a... |
c_1rrs180hhsvy | 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. |
c_66b6yencpmuk | 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 energy and contact angle, the surface energy of the soli... |
c_gxt7f0tsze3x | 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 (sol) that acts as the precurso... |
c_3inp81azfcg6 | 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 materials derived from single elements generally carry the -ene suffix in their nam... |
c_uewm8gstojj7 | 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. |
c_mdontgqe7ne2 | 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 exfoliation and bottom-up synthesis. The exfoliation methods include sonication, mech... |
c_vm8qazmssfgk | 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" or just "displacement energy". In a crystal, a separate threshold di... |
c_8mnybwlzoo22 | 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 average quantity of interest. Threshold displacement energies in typical s... |
c_hpxt4ryokzmi | 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 temperature. In metals, this decrease in yield strength is due... |
c_3zwbkrnlfhrv | 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 earliest discoveries such a phenomenon, and several o... |
c_waxo2j45td41 | 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 temperature of the yield strength peak is roughly 50% of th... |
c_jf2nvk1ajh6o | 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 different materials classes. The cra... |
c_fk6eaeg6453h | 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 movements and dislocation generation withi... |
c_bht1sely95f9 | 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, such as indium, however others can be stre... |
c_0m8faprm0esw | 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 ferromagnetism and ferrimagnetism, a manifestation of ordered m... |
c_w9o14dyomcd4 | 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 not apply for nonzero wavevector. The non-isotropic permittivity for nonzero wavevector leads to effects such as optical ... |
c_12nx9otvsb4i | 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 the schematic on the right. In a corrosive environment, the crack grows due to cyclic loading at a lower stress-intensity range; above th... |
c_ryyorxxhz4io | 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 are electrically attracted to each other). In this situation, there is a distinction between "op... |
c_feey2q68b3eq | 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 binding energy), and therefore the optical and electronic bandgap are essentially identical, an... |
c_r74aswhyrall | 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 same molecule, as in fullerenes. This Frenkel exci... |
c_n7o97qbc0obe | 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 partially-filled d-shells. While d-d transitions are i... |
c_nwumcr765b2o | 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. During this time, solutes (such as interstitial particles or substitutional impurities) diffuse a... |
c_2vrqnp68en38 | 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 manifestations are Lüders bands and the Portevin–Le Chatelier effect. However, the mechanism is known ... |
c_g7sifjefdyfg | 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 students think in the process of learning or problem solving makes it possible for teachers to help their stu... |
c_pcpy81di11ri | 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 mathematical models that include ultrametric pseudo-differential equations... |
c_r10n7omt7t8s | 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 equations as oscillatory integrals. |
c_jy30z1zkvejz | 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. |
c_ssgc38iexixj | 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 studied it in 1904, and named in th... |
c_92ucf7jhjfgv | 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 measurable space. The σ-algebras are a subset of... |
c_z0i5tpyypcuq | 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 notion of conditional density is not applicabl... |
c_1n5zj6tk9563 | 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 more useful example is the set of subsets... |
c_25ube1e2slk2 | 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 several variables. The "input" variables take real value... |
c_3boowh6klrtx | 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 set. |
c_z0siec50bpre | 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 metric. |
c_zh4ouc51rylv | 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 two parabolic curves x2 + 1 and x2... |
c_unryb77ey98h | 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 partial differential equations. Let u ∈ H 2 ( Ω )... |
c_eewcbmp9mkh8 | 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 )} where Ω ⊂ R 2 {\displaystyle \Omega \subset \m... |
c_psf2k03y94vi | 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 bound for the similar maximum of its derivative. Taking the k-th derivative of the theorem, max | z | ≤ 1 ( | P ( k ) ( z ) | ) ≤... |
c_eamlmfaz68yq | 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 first n partial sums of the series. This spec... |
c_d19gfyr2yyjp | 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 differentiable. It is a particular case of the Lagrange diff... |
c_rjhs7k6yz9e1 | 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, which is used for similar purposes and derived ... |
c_t3x6zj88hbr4 | 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. |
c_txjmrujqddki | 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 represented as the integral with respect to an appropriate Radon measure over the c... |
c_q2y1lwvxeng2 | 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 when the lower level set of a minimization problem is not compact, so that the Bolza... |
c_oz9d4vi2bgas | 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 minimization problems is not compact, so that the Bolzano–Weierstrass th... |
c_dyvm7d08ho59 | 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. A Fourier integral operator T {\displaystyle T} is given b... |
c_s63add43eqi1 | 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 answer when the integrand is replaced by ... |
c_9ckvffd75c39 | 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 , n = 1 ∞ {\textstyle \{a_{m,n}\}_{m=1,n=1... |
c_0dlwcj6anxxa | 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 by Jean Dieudonné.The theorem states: Let f: U → R 0 + {\displayst... |
c_cqynwwn4gggz | 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. |
c_27ozez4lvb8s | 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 has degree, m − 1⁄2, a half-integer, and argument, z, real and g... |
c_c2sspbmriwe1 | 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 gives a form of the Cauchy–Schwarz inequality. ... |
c_bksswb0h1cre | 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 ∈ [1, ∞). Hölder's inequality (in a slightly di... |
c_qsj7xttnjh0u | 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 complete vector lattices called "K-spaces" which are now referred ... |
c_5m7b3uau24ax | 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-symmetric matrix. Korn's theorem is a quantitati... |
c_a1yn0ar7gy9l | 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}\geq 0\right\},} to be dense in a weighted L2 ... |
c_2trdiav66fh8 | Lambert summation | Summary | Lambert_summation | In mathematical analysis, Lambert summation is a summability method for a class of divergent series. |
c_k951uhrjju6g | 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, for every pair of points on the graph of this ... |
c_afqh2ooya0qr | 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. Precisely, let B: c 0 × c 0 → C {\displaystyle B:c_{0}\times c... |
c_pohariczva0z | 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. |
c_f1hi546cwm5l | 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 characterized by a norm (technically ... |
c_fk2msctwtp51 | 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” convergence since it uses both the weak and strong to... |
c_7lbl95eoo576 | 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 higher-dimensional manifold to a one-dimens... |
c_w3ocvo6ryyq0 | 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. The conditions of the theorem can be re... |
c_d6r3xdd0ddei | 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 subsets of the plane that have nonzero ... |
c_bxalboy05fkt | 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. |
c_u9krlzlj22oc | 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 a surprise to many mathematicians and ki... |
c_7if1ykfbxx4s | 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..., cause this to be an injection rather than... |
c_a1blqtqa73d7 | 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 infinity of basis vectors). Informally, the ide... |
c_tq233r1211h2 | 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} ),} Another similar identity is a one which gives th... |
c_9jg0hvdgilnm | 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 differentiable form a set of Lebesgue measure zero. Different... |
c_m71f88gmogt5 | 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} } defined on a set Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} , if p ∈ R n {\d... |
c_bz73ee2tkahl | 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 ) {\displaystyle f(x,y)} be a differentiable function on an open rectangle Ω {\displaystyle \Omega } conta... |
c_jmd3ksciql6k | 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 ) . {\displaystyle {\begin{aligned}u\left(h,\,k\right)&=f\left(a+h,\,b+k\right)-f\left(a+h,\,b\rig... |
c_uacz48ehdpx4 | 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 h,\,b+\theta ^{\prime }k\right)&=\partial _{x}\partial _{y}f\left(a+\phi ^{\prime }h,\,b+\ph... |
c_9o2mb79t7zn7 | 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 _{x}f,\,\,\partial _{y}f} as t {\displaystyle t} tends to 0, with a similar statement for se... |
c_evcpvsoaz69z | 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 ) . {\displaystyle \int _{a}^{b}f^{\prime }(x)\,dx=f(b)-f(a).} Hence | f ( b ) − f ( a ) | ≤ ... |
c_9semhsxo22tk | 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 point c {\displaystyle c} in ( a , b ) {\displaystyle (a,b)} such that f ( b ) − f ( a ) b − ... |
c_r5b61lemb5cj | 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}\left|D_{1}D_{2}f(x_{0}+tr,y_{0}+ts)-D_{1}D_{2}f(x_{0},y_{0})\right|.} Thus Δ 1 t Δ 2 t f ( x ... |
c_un02waz6avk7 | 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 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}... |
c_pniensr7fpjc | 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 out of connections to the Fourier restriction pro... |
c_fsehp4treneu | 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. |
c_l08bl1uqrjot | 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 certain Sobolev space norm and an Orlicz space n... |
c_2ae5ylnf1tc2 | 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 {\displaystyle p>1} . Set A ( t ) = exp ( t n / ( n − m... |
c_mwnm3moikf6q | 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 } < ∞ } . {\displaystyle L_{A}(\Omega )=\left\{u\in... |
c_v0bkgoviys9w | 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 be approximated by linear combinations of translations of a gi... |
c_dtj6o8fs4p2h | 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. |
c_o7ow7trwjie0 | 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 ) {\displaystyle x=(x_{n})} , y = ( y ... |
c_2teeh2lwa914 | 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 Hahn–Banach theorem (analyst's app... |
c_d08fbltnl4l6 | 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 theorem asserts that there exists a constant cN depending only on the d... |
c_kbdtth82xqnc | 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 cover for E (every point y ∈ E belongs to at least one ball from ... |
c_feic4zhd7vc2 | 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. |
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