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c_cqynwwn4gggz | 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. | Haar's Tauberian theorem | [
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c_jf2nvk1ajh6o | 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 crack tip plasticity is important in toughening of metals and long-chain polymers. Ceramics have limited crack tip plasticity and primarily rely on different toughening mechanisms. | Toughening | [
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c_oz9d4vi2bgas | 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 theorem cannot be applied. The principle relies on the completeness of the metric space.The principle has been shown to be equivalent to completeness of metric spaces. In proof theory, it is equivalent to Π11CA0 over RCA0, i.e. relatively strong. It also leads to a quick proof of the Caristi fixed point theorem. | Ekeland's variational principle | [
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c_hpxt4ryokzmi | 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 to the thermal activation of dislocation motion, resulting in easier plastic deformation at higher temperatures.In some cases, a yield strength anomaly refers to a decrease in the ductility of a material with increasing temperature, which is also opposite the trend in the majority of materials. Anomalies in ductility can be more clear, as an anomalous effect on yield strength can be obscured by its typical decrease with temperature. | Yield strength anomaly | [
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c_waxo2j45td41 | 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 the absolute melting temperature. | Yield strength anomaly | [
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c_3zwbkrnlfhrv | 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 other ordered intermetallic alloys demonstrate this effect. Precipitation-hardened superalloys exhibit a yield strength anomaly over a considerable temperature range. For these materials, the yield strength shows little variation between room temperature and several hundred degrees Celsius. | Yield strength anomaly | [
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c_b580hxt3vsmg | 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} }{=}}\ {\frac {\tau _{xy}}{\gamma _{xy}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}} where τ x y = F / A {\displaystyle \tau _{xy}=F/A\,} = shear stress F {\displaystyle F} is the force which acts A {\displaystyle A} is the area on which the force acts γ x y {\displaystyle \gamma _{xy}} = shear strain. In engineering := Δ x / l = tan θ {\displaystyle :=\Delta x/l=\tan \theta } , elsewhere := θ {\displaystyle :=\theta } Δ x {\displaystyle \Delta x} is the transverse displacement l {\displaystyle l} is the initial length of the area.The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional form is M1L−1T−2, replacing force by mass times acceleration. | Modulus of rigidity | [
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c_oicgefilxbsc | 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 atoms per length). Close-packed planes are known as slip or glide planes. | Slip (materials science) | [
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c_h6lqfwoyftsb | 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 lattice glide along each other, changing the material's geometry. A critical resolved shear stress is required to initiate a slip. | Slip (materials science) | [
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c_92ucf7jhjfgv | 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 the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. | Sigma field | [
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c_1n5zj6tk9563 | 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 of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy). | Sigma field | [
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c_z0i5tpyypcuq | 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 applicable. If X = { a , b , c , d } {\displaystyle X=\{a,b,c,d\}} one possible σ-algebra on X {\displaystyle X} is Σ = { ∅ , { a , b } , { c , d } , { a , b , c , d } } , {\displaystyle \Sigma =\{\varnothing ,\{a,b\},\{c,d\},\{a,b,c,d\}\},} where ∅ {\displaystyle \varnothing } is the empty set. | Sigma field | [
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c_feic4zhd7vc2 | 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. | Contraction semigroup | [
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c_a1yn0ar7gy9l | 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 space on the real line. It was discovered by Mark Krein in the 1940s. A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem. | Krein's condition | [
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c_mwnm3moikf6q | {\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 M_{f}(\Omega ):\|u\|_{A,\Omega }=\inf\{k>0:\int _{\Omega }A\left({\frac {|u(x)|}{k}}\right)~dx\leq 1\}<\infty \right\}.} The space L A ( Ω ) {\displaystyle L_{A}(\Omega )} is an example of an Orlicz space. | Trudinger's theorem | [
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c_l08bl1uqrjot | 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 norm of a function. | Trudinger's theorem | [
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c_2ae5ylnf1tc2 | 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 ) ) − 1. | Trudinger's theorem | [
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c_mdontgqe7ne2 | 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, mechanical, hydrothermal, electrochemical, laser-assisted, and microwave-assisted exfoliation. | Two dimensional (2D) nanomaterials | [
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c_uewm8gstojj7 | 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. | Two dimensional (2D) nanomaterials | [
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c_3inp81azfcg6 | 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 names, e.g. graphene. Single-layer materials that are compounds of two or more elements have -ane or -ide suffixes. | Two dimensional (2D) nanomaterials | [
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c_eewcbmp9mkh8 | {\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 \mathbb {R} ^{2}} . Then Agmon's inequality in 2D states that there exists a constant C {\displaystyle C} such that ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ L 2 ( Ω ) 1 / 2 ‖ u ‖ H 2 ( Ω ) 1 / 2 . {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2}.} For the n {\displaystyle n} -dimensional case, choose s 1 {\displaystyle s_{1}} and s 2 {\displaystyle s_{2}} such that s 1 < n 2 < s 2 {\displaystyle s_{1}<{\tfrac {n}{2}} | Agmon's inequality | [
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c_unryb77ey98h | 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 ) ∩ H 0 1 ( Ω ) {\displaystyle u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} where Ω ⊂ R 3 {\displaystyle \Omega \subset \mathbb {R} ^{3}} . Then Agmon's inequalities in 3D state that there exists a constant C {\displaystyle C} such that ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) 1 / 2 ‖ u ‖ H 2 ( Ω ) 1 / 2 , {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2},} and ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ L 2 ( Ω ) 1 / 4 ‖ u ‖ H 2 ( Ω ) 3 / 4 . | Agmon's inequality | [
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c_dtj6o8fs4p2h | In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938. | Zorich's theorem | [
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] |
c_pniensr7fpjc | 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 problem. | Strichartz estimate | [
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c_n7o97qbc0obe | 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 in principle forbidden by symmetry, they become weakly-allowed in a crystal when the symmetry is broken by structural relaxations or other effects. Absorption of a photon resonant with a d-d transition leads to the creation of an electron-hole pair on a single atomic site, which can be treated as a Frenkel exciton. | Exciton | [
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c_r74aswhyrall | 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 exciton, named after Yakov Frenkel, has a typical binding energy on the order of 0.1 to 1 eV. | Exciton | [
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c_yvb5so5ecui3 | 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 / R T ) {\displaystyle Z={\dot {\varepsilon }}\exp(Q/RT)} where ε ˙ {\textstyle {\dot {\varepsilon }}} is the strain rate, Q is the activation energy, R is the gas constant, and T is the temperature. The Zener–Hollomon parameter is also known as the temperature compensated strain rate, since the two are inversely proportional in the definition. | Zener–Hollomon parameter | [
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c_mxdf62uz4ut9 | 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 metals over a large range of strain rates and temperatures and shown comparable microstructures at the end-of-processing, as long as Z remained similar. This is because the relative activity of various deformation mechanisms is typically inversely proportional to temperature or strain-rate, such that decreasing strain rate or increasing temperature will increase Z and promote plastic deformation. | Zener–Hollomon parameter | [
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] |
c_81ra4xs7eoyy | It is named after Clarence Zener and John Herbert Hollomon, Jr. who established the formula based on the stress-strain behavior in steel. | Zener–Hollomon parameter | [
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c_g7sifjefdyfg | 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 students overcome conceptual difficulties and, in turn, improve learning. In that sense, formative assessment is diagnostic. To employ formative assessment in the classrooms, a teacher has to make sure that each student participates in the learning process by expressing their ideas; there is a trustful environment in which students can provide each other with feedback; s/he (the teacher) provides students with feedback; and the instruction is modified according to students' needs. In math classes, thought revealing activities such as model-eliciting activities (MEAs) and generative activities provide good opportunities for covering these aspects of formative assessment. | Formative evaluation | [
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c_eccr9w48ehk1 | 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 as iron or nickel, with small impurity atoms, such as boron, carbon, or nitrogen. As these interstitial atoms distort the lattice slightly, there will be an associated residual stress field surrounding the interstitial. This stress field can be relaxed by the interstitial atom diffusing towards a dislocation, which contains a small gap at its core (as it is a more open structure), see Figure 1. | Cottrell atmosphere | [
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c_1rrs180hhsvy | 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. | Cottrell atmosphere | [
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c_kcxl9bwvix8g | 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 prepare and conduct and results can be obtained quickly and cheaply. | Charpy test | [
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c_uby0a46r94vd | 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 the technical contributions and standardization efforts by Charpy. | Charpy test | [
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c_v0bkgoviys9w | 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 given function.Informally, if the Fourier transform of a function f {\displaystyle f} vanishes on a certain set Z {\displaystyle Z} , the Fourier transform of any linear combination of translations of f {\displaystyle f} also vanishes on Z {\displaystyle Z} . Therefore, the linear combinations of translations of f {\displaystyle f} cannot approximate a function whose Fourier transform does not vanish on Z {\displaystyle Z} . Wiener's theorems make this precise, stating that linear combinations of translations of f {\displaystyle f} are dense if and only if the zero set of the Fourier transform of f {\displaystyle f} is empty (in the case of L 1 {\displaystyle L^{1}} ) or of Lebesgue measure zero (in the case of L 2 {\displaystyle L^{2}} ). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L 1 {\displaystyle L^{1}} group ring L 1 ( R ) {\displaystyle L^{1}(\mathbb {R} )} of the group R {\displaystyle \mathbb {R} } of real numbers is the dual group of R {\displaystyle \mathbb {R} } . A similar result is true when R {\displaystyle \mathbb {R} } is replaced by any locally compact abelian group. | Wiener's Tauberian theorem | [
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c_t3x6zj88hbr4 | In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous. | Dini continuity | [
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c_5m7b3uau24ax | 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 quantitative version of this statement, which intuitively says that if the gradient of a vector field is on average not far from the space of skew-symmetric matrices, then the gradient must not be far from a particular skew-symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity. In (linear) elasticity theory, the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate in linear elasticity theory. | Korn's inequality | [
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c_bksswb0h1cre | 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 different form) was first found by Leonard James Rogers (1888). Inspired by Rogers' work, Hölder (1889) gave another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named for work of Johan Jensen building on Hölder's work. | Hölder's inequality | [
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c_c2sspbmriwe1 | 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. Hölder's inequality holds even if ‖fg‖1 is infinite, the right-hand side also being infinite in that case. | Hölder's inequality | [
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c_fk2msctwtp51 | 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 topologies on a topological vector space X. In finite dimensional spaces, Mosco convergence coincides with epi-convergence, while in infinite-dimensional ones, Mosco convergence is strictly stronger property. Mosco convergence is named after Italian mathematician Umberto Mosco, a current Harold J. Gay professor of mathematics at Worcester Polytechnic Institute. | Mosco convergence | [
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c_k951uhrjju6g | 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 function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity). For instance, every function that is defined on an interval and has bounded first derivative is Lipschitz continuous.In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem.We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line: Continuously differentiable ⊂ Lipschitz continuous ⊂ α {\displaystyle \alpha } -Hölder continuous,where 0 < α ≤ 1 {\displaystyle 0<\alpha \leq 1} . We also have Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous. | Lipschitz function | [
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c_9ckvffd75c39 | 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}^{\infty }} is a doubly-indexed sequence of real numbers, and if ∑ ( m , n ) ∈ N × N a m , n {\textstyle \sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n}} is absolutely convergent, then ∑ ( m , n ) ∈ N × N a m , n = ∑ m = 1 ∞ ∑ n = 1 ∞ a m , n = ∑ n = 1 ∞ ∑ m = 1 ∞ a m , n {\displaystyle \sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }a_{m,n}=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }a_{m,n}} Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series. In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series. | Fubini's Theorem | [
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c_s63add43eqi1 | 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 its absolute value. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. | Fubini's Theorem | [
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c_uacz48ehdpx4 | {\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+\phi k\right).\end{aligned}}} Letting h , k {\displaystyle h,\,k} tend to zero in the last equality, the continuity assumptions on ∂ y ∂ x f {\displaystyle \partial _{y}\partial _{x}f} and ∂ x ∂ y f {\displaystyle \partial _{x}\partial _{y}f} now imply that ∂ 2 ∂ x ∂ y f ( a , b ) = ∂ 2 ∂ y ∂ x f ( a , b ) . {\displaystyle {\frac {\partial ^{2}}{\partial x\partial y}}f\left(a,\,b\right)={\frac {\partial ^{2}}{\partial y\partial x}}f\left(a,\,b\right).} This account is a straightforward classical method found in many text books, for example in Burkill, Apostol and Rudin.Although the derivation above is elementary, the approach can also be viewed from a more conceptual perspective so that the result becomes more apparent. | Equality of mixed partials | [
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c_jmd3ksciql6k | {\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\right),\\v\left(h,\,k\right)&=f\left(a+h,\,b+k\right)-f\left(a,\,b+k\right),\\w\left(h,\,k\right)&=f\left(a+h,\,b+k\right)-f\left(a+h,\,b\right)-f\left(a,\,b+k\right)+f\left(a,\,b\right).\end{aligned}}} These functions are defined for | h | , | k | < ε {\displaystyle \left|h\right|,\,\left|k\right|<\varepsilon } , where ε > 0 {\displaystyle \varepsilon >0} and × {\displaystyle \left\times \left} is contained in Ω . | Equality of mixed partials | [
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c_9semhsxo22tk | 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 − a = f ′ ( c ) . {\displaystyle {f(b)-f(a) \over b-a}=f^{\prime }(c).} For vector-valued functions with V {\displaystyle V} a finite-dimensional normed space, there is no analogue of the equality above, indeed it fails. | Equality of mixed partials | [
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c_nfv299d68m3p | In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f: U → R m {\displaystyle f\colon U\to \mathbb {R} ^{m}} is a convex function, then f {\displaystyle f} has a second derivative almost everywhere. In this context, having a second derivative at a point means having a second-order Taylor expansion at that point with a local error smaller than any quadratic. The result is closely related to Rademacher's theorem. | Alexandrov theorem | [
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c_lle53npqq5gd | In general, there may be many equilibria. Arrow (1972) and Debreu (1983) were separately awarded the Nobel Prize in Economics for their development of the model. McKenzie however was not awarded. | Arrow–Debreu model | [
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c_s82u7aynzbj9 | In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.The model is central to the theory of general (economic) equilibrium and it is often used as a general reference for other microeconomic models. It was proposed by Kenneth Arrow, Gérard Debreu in 1954, and Lionel W. McKenzie independently in 1954, with later improvements in 1959.The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. | Arrow–Debreu model | [
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c_0mblbcgchezj | In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one-sided) Laplace transform of ƒ(t). If f {\displaystyle f} is bounded on ( 0 , ∞ ) {\displaystyle (0,\infty )} (or if just f ( t ) = O ( e c t ) {\displaystyle f(t)=O(e^{ct})} ) and lim t → 0 + f ( t ) {\displaystyle \lim _{t\to 0^{+}}f(t)} exists then the initial value theorem says lim t → 0 f ( t ) = lim s → ∞ s F ( s ) . {\displaystyle \lim _{t\,\to \,0}f(t)=\lim _{s\to \infty }{sF(s)}.} | Initial value theorem | [
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c_lwsqgqy8nsfb | The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. : 226 In 1930, Jovan Karamata gave a new and much simpler proof. : 226 | Karamata's tauberian theorem | [
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c_bwerdz5lw6ba | In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equivalence ∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0} then there is also an asymptotic equivalence ∑ k = 0 n a k ∼ n {\displaystyle \sum _{k=0}^{n}a_{k}\sim n} as n → ∞ {\displaystyle n\to \infty } . The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform. | Karamata's tauberian theorem | [
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c_c86zit2s654k | In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970. | Wavefront set | [
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c_9h6j1b2sawhb | In mathematical analysis, the Whitney covering lemma, or Whitney decomposition, asserts the existence of a certain type of partition of an open set in a Euclidean space. Originally it was employed in the proof of Hassler Whitney's extension theorem. The lemma was subsequently applied to prove generalizations of the Calderón–Zygmund decomposition. Roughly speaking, the lemma states that it is possible to decompose an open set by cubes each of whose diameters is proportional, within certain bounds, to its distance from the boundary of the open set. | Whitney covering lemma | [
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c_ftnpdwzl09u4 | {\displaystyle {\sqrt {n}}\ell (Q_{j})\leq \mathrm {dist} (Q_{j},\Omega ^{c})\leq 4{\sqrt {n}}\ell (Q_{j}).} If the boundaries of two cubes Q j {\displaystyle Q_{j}} and Q k {\displaystyle Q_{k}} touch then 1 4 ≤ ℓ ( Q j ) ℓ ( Q k ) ≤ 4. {\displaystyle {\frac {1}{4}}\leq {\frac {\ell (Q_{j})}{\ell (Q_{k})}}\leq 4.} For a given Q j {\displaystyle Q_{j}} there exist at most 12 n Q k {\displaystyle 12^{n}Q_{k}} 's that touch it.Where ℓ ( Q ) {\displaystyle \ell (Q)} denotes the length of a cube Q {\displaystyle Q} . | Whitney covering lemma | [
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c_kvhr6u4gx5qt | More precisely: Whitney Covering Lemma (Grafakos 2008, Appendix J) Let Ω {\displaystyle \Omega } be an open non-empty proper subset of R n {\displaystyle \mathbb {R} ^{n}} . Then there exists a family of closed cubes { Q j } j {\displaystyle \{Q_{j}\}_{j}} such that ∪ j Q j = Ω {\displaystyle \cup _{j}Q_{j}=\Omega } and the Q j {\displaystyle Q_{j}} 's have disjoint interiors. n ℓ ( Q j ) ≤ d i s t ( Q j , Ω c ) ≤ 4 n ℓ ( Q j ) . | Whitney covering lemma | [
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c_uargji9dm1ae | In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S {\displaystyle S} be a measure space, let 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } and let f {\displaystyle f} and g {\displaystyle g} be elements of L p ( S ) . {\displaystyle L^{p}(S).} Then f + g {\displaystyle f+g} is in L p ( S ) , {\displaystyle L^{p}(S),} and we have the triangle inequality with equality for 1 < p < ∞ {\displaystyle 1 | Minkowski inequality | [
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c_p06adjfs6tc8 | In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. | Lagrange–Bürmann formula | [
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c_44ff4kz09ovs | In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. | Haar integral | [
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c_wyuqdqiuiw8w | Earlier analytic work with these models has examined the distortionary effects of taxes, tariffs, and other policies, along with functional incidence questions. More recent applied models, including those discussed here, provide numerical estimates of efficiency and distributional effects within the same framework. | Applied general equilibrium | [
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c_lwrrej6yijxf | Scarf never built an AGE model, but hinted that “these novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environment” (Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price vector could be solved for any changes in policies (or exogenous shocks), giving the equilibrium ‘adjustments’ needed for the prices. This method was first used by Shoven and Whalley (1972 and 1973), and then was developed through the 1970s by Scarf’s students and others.Most contemporary applied general equilibrium models are numerical analogs of traditional two-sector general equilibrium models popularized by James Meade, Harry Johnson, Arnold Harberger, and others in the 1950s and 1960s. | Applied general equilibrium | [
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c_low86vcaqmj5 | Scarf's fixed-point method was a break-through in the mathematics of computation generally, and specifically in optimization and computational economics. Later researchers continued to develop iterative methods for computing fixed-points, both for topological models like Scarf's and for models described by functions with continuous second derivatives or convexity or both. Of course, "global Newton methods" for essentially convex and smooth functions and path-following methods for diffeomorphisms converged faster than did robust algorithms for continuous functions, when the smooth methods are applicable. | Applied general equilibrium | [
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] |
c_ub11upedklwe | In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory. | Generalized Fourier series | [
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c_m82za17bp2ta | In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". | Asymptotic theory | [
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c_292in3v00cno | This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less than or equal to x. Then the theorem states that Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. | Asymptotic theory | [
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c_u4ia7o5wmg8v | In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover any two-dimensional region, distinguishing them from space-filling curves. Osgood curves are named after William Fogg Osgood. | Osgood curve | [
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c_23dpnyu2x98v | In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. | Measure zero | [
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c_hrzud5fhbjje | Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = ( X , Σ , μ ) {\displaystyle M=(X,\Sigma ,\mu )} a null set is a set S ∈ Σ {\displaystyle S\in \Sigma } such that μ ( S ) = 0. {\displaystyle \mu (S)=0.} | Measure zero | [
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c_ht0qqj2v63cb | In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C ∞ {\displaystyle C^{\infty }} function). | Differentiability class | [
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c_86vmrmh0psr7 | In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K. If K is 0, then the function is conformal. | Geometric function theory | [
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] |
c_p3g1n4jus90h | In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő. | Szegő limit theorems | [
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c_9sfkypp2vtrv | In mathematical analysis, a thin set is a subset of n-dimensional complex space Cn with the property that each point has a neighbourhood on which some non-zero holomorphic function vanishes. Since the set on which a holomorphic function vanishes is closed and has empty interior (by the Identity theorem), a thin set is nowhere dense, and the closure of a thin set is also thin. The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets. | Thin set (analysis) | [
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c_khzni0hyuoxu | In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes. Informally, they provide sharp bounds on a measure from above and from below in terms of its first moments. | Chebyshev–Markov–Stieltjes inequalities | [
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c_ay269pz2lgye | In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. | Alternating series test | [
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c_4t2139uerihn | In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent variable, in the limit as the changes approach zero in magnitude. For an elasticity coefficient r {\displaystyle r} (which can take on any real value), the function's general form is given by f ( x ) = k x r , {\displaystyle f(x)={kx^{r}},} where k {\displaystyle k} and r {\displaystyle r} are constants. The elasticity is by definition elasticity = d f ( x ) d x x f ( x ) = d ln f ( x ) d ln x , {\displaystyle {\text{elasticity}}={\frac {df(x)}{dx}}{\frac {x}{f(x)}}={\frac {d{\text{ln}}f(x)}{d{\text{ln}}x}},} which for this function simply equals r. | Isoelastic function | [
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c_39slakibjw8g | {\displaystyle X.} The space C ( X ) {\displaystyle {\mathcal {C}}(X)} is a Banach algebra with respect to this norm. (Rudin 1973, §11.3) | Continuous functions on a compact Hausdorff space | [
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c_ff60itbs94n5 | In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted by C ( X ) , {\displaystyle {\mathcal {C}}(X),} is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X . | Continuous functions on a compact Hausdorff space | [
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c_bdwrt468123b | In mathematical analysis, the Foias constant is a real number named after Ciprian Foias. It is defined in the following way: for every real number x1 > 0, there is a sequence defined by the recurrence relation x n + 1 = ( 1 + 1 x n ) n {\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}} for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x1 = α then the sequence diverges to infinity. For all other values of x1, the sequence is divergent as well, but it has two accumulation points: 1 and infinity. Numerically, it is α = 1.187452351126501 … {\displaystyle \alpha =1.187452351126501\ldots } .No closed form for the constant is known. When x1 = α then the growth rate of the sequence (xn) is given by the limit lim n → ∞ x n log n n = 1 , {\displaystyle \lim _{n\to \infty }x_{n}{\frac {\log n}{n}}=1,} where "log" denotes the natural logarithm.The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences. | Foias constant | [
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c_we71p1ov834n | In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. | Continuous-time signal | [
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c_ds0wskt2db5j | In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaystyle x_{0}} if, roughly speaking, the function values for arguments near x 0 {\displaystyle x_{0}} are not much higher (respectively, lower) than f ( x 0 ) . {\displaystyle f\left(x_{0}\right).} A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x 0 {\displaystyle x_{0}} to f ( x 0 ) + c {\displaystyle f\left(x_{0}\right)+c} for some c > 0 {\displaystyle c>0} , then the result is upper semicontinuous; if we decrease its value to f ( x 0 ) − c {\displaystyle f\left(x_{0}\right)-c} then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. | Upper semi-continuous | [
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c_sd3h4gygewrr | In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0 by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,\mathrm {d} t} as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex.A treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the AMS in a collection of Artin's writings.The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.The theorem admits a far-reaching generalization to a wide variety of functions (that have convexity or concavity properties of any order). | Bohr–Mollerup theorem | [
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c_97loffc8uqrj | In the calculus of relations, the semicolon is used in infix notation for the composition of relations: A ; B = { ( x , z ): ∃ y x A y ∧ y B z } . {\displaystyle A;B\ =\ \{(x,z):\exists y\ \ xAy\ \land \ yBz\}~.} The ; Humphrey point is sometimes used as the "decimal point" in duodecimal numbers: 54;612 equals 64.510. | Semi colon | [
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c_w4sqaow8w3gp | In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If π ( x ) = 1 Π ( x ) = 1 Γ ( x + 1 ) {\displaystyle \pi (x)={\frac {1}{\Pi (x)}}={\frac {1}{\Gamma (x+1)}}} is the entire function defined by means of the reciprocal gamma function, then the Bessel–Clifford function is defined by the series C n ( z ) = ∑ k = 0 ∞ π ( k + n ) z k k ! {\displaystyle {\mathcal {C}}_{n}(z)=\sum _{k=0}^{\infty }\pi (k+n){\frac {z^{k}}{k!}}} The ratio of successive terms is z/k(n + k), which for all values of z and n tends to zero with increasing k. By the ratio test, this series converges absolutely for all z and n, and uniformly for all regions with bounded |z|, and hence the Bessel–Clifford function is an entire function of the two complex variables n and z. | Bessel–Clifford function | [
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c_3zzwxh5ffd2p | In mathematical analysis, the concept of a mean-periodic function is a generalization of the concept of a periodic function introduced in 1935 by Jean Delsarte. Further results were made by Laurent Schwartz. | Mean-periodic function | [
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c_nyj1sgp3hn89 | The set of tempered distributions forms a vector subspace of the space of distributions D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support. Use of analytic test functions leads to Sato's theory of hyperfunctions. | Spaces of test functions and distributions | [
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c_mp6iow7lt3zv | There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If U = R n {\displaystyle U=\mathbb {R} ^{n}} then the use of Schwartz functions as test functions gives rise to a certain subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} whose elements are called tempered distributions. These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. | Spaces of test functions and distributions | [
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c_zu6x6i8ok53a | In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} that have compact support. The space of all test functions, denoted by C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} is endowed with a certain topology, called the canonical LF-topology, that makes C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} into a complete Hausdorff locally convex TVS. The strong dual space of C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} is called the space of distributions on U {\displaystyle U} and is denoted by D ′ ( U ) := ( C c ∞ ( U ) ) b ′ , {\displaystyle {\mathcal {D}}^{\prime }(U):=\left(C_{c}^{\infty }(U)\right)_{b}^{\prime },} where the " b {\displaystyle b} " subscript indicates that the continuous dual space of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} denoted by ( C c ∞ ( U ) ) ′ , {\displaystyle \left(C_{c}^{\infty }(U)\right)^{\prime },} is endowed with the strong dual topology. | Spaces of test functions and distributions | [
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c_0p92msopfslb | In mathematical analysis, nested intervals provide one method of axiomatically introducing the real numbers as the completion of the rational numbers, being a necessity for discussing the concepts of continuity and differentiability. Historically, Isaac Newton's and Gottfried Wilhelm Leibniz's discovery of differential and integral calculus from the late 1600s has posed a huge challenge for mathematicians trying to prove their methods rigorously; despite their success in physics, engineering and other sciences. The axiomatic description of nested intervals (or an equivalent axiom) has become an important foundation for the modern understanding of calculus. In the context of this article, R {\displaystyle \mathbb {R} } in conjunction with + {\displaystyle +} and ⋅ {\displaystyle \cdot } is an Archimedean ordered field, meaning the axioms of order and the Archimedean property hold. | Nested sequence of closed intervals | [
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c_kq72707aex7d | Let Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} be the exterior or the interior of a bounded domain with regular boundary, or R 2 {\displaystyle \mathbb {R} ^{2}} itself. Then the Brezis–Gallouët inequality states that there exists a real C {\displaystyle C} only depending on Ω {\displaystyle \Omega } such that, for all u ∈ H 2 ( Ω ) {\displaystyle u\in H^{2}(\Omega )} which is not a.e. equal to 0, ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) ( 1 + ( log ( 1 + ‖ u ‖ H 2 ( Ω ) ‖ u ‖ H 1 ( Ω ) ) ) 1 / 2 ) . | Brezis–Gallouët inequality | [
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c_w027eor13ec2 | In mathematical analysis, the Brezis–Gallouët inequality, named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provides an explicit bound, which depends only logarithmically on the second derivatives. It is useful in the study of partial differential equations. | Brezis–Gallouët inequality | [
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c_0uoen8kunmnh | In mathematical analysis, the characteristic variety of a microdifferential operator P is an algebraic variety that is the zero set of the principal symbol of P in the cotangent bundle. It is invariant under a quantized contact transformation. The notion is also defined more generally in commutative algebra. A basic theorem says a characteristic variety is involutive. | Characteristic variety | [
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c_5ra1mmj6ygs6 | In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval , then it takes on any given value between f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} at some point within the interval. This has two important corollaries: If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). The image of a continuous function over an interval is itself an interval. | Intermediate Value Theorem | [
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c_ihdm0005xs38 | In mathematical analysis, epi-convergence is a type of convergence for real-valued and extended real-valued functions. Epi-convergence is important because it is the appropriate notion of convergence with which to approximate minimization problems in the field of mathematical optimization. The symmetric notion of hypo-convergence is appropriate for maximization problems. Mosco convergence is a generalization of epi-convergence to infinite dimensional spaces. | Epi-convergence | [
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c_1r5r4mmylhxi | In mathematical complex analysis, universal Teichmüller space T(1) is a Teichmüller space containing the Teichmüller space T(G) of every Fuchsian group G. It was introduced by Bers (1965) as the set of boundary values of quasiconformal maps of the upper half-plane that fix 0, 1, and ∞. | Universal Teichmüller space | [
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c_h3rvm1ertmi1 | In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with ( a ; q ) 0 = 1. {\displaystyle (a;q)_{0}=1.} It is a q-analog of the Pochhammer symbol ( x ) n = x ( x + 1 ) … ( x + n − 1 ) {\displaystyle (x)_{n}=x(x+1)\dots (x+n-1)} , in the sense that The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product: This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms. | Q-Pochhammer symbol | [
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c_l4ypjjqbm0vh | In mathematical complex analysis, Schottky's theorem, introduced by Schottky (1904) is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1, the value of |f(z)| can be bounded in terms of z and f(0). Schottky's original theorem did not give an explicit bound for f. Ostrowski (1931, 1933) gave some weak explicit bounds. Ahlfors (1938, theorem B) gave a strong explicit bound, showing that if f is holomorphic in the open unit disk and does not take the values 0 or 1, then log | f ( z ) | ≤ 1 + | z | 1 − | z | ( 7 + max ( 0 , log | f ( 0 ) | ) ) {\displaystyle \log |f(z)|\leq {\frac {1+|z|}{1-|z|}}(7+\max(0,\log |f(0)|))} .Several authors, such as Jenkins (1955), have given variations of Ahlfors's bound with better constants: in particular Hempel (1980) gave some bounds whose constants are in some sense the best possible. | Schottky theorem | [
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c_nvojqkvjva7a | In mathematical analysis, the Young's inequality for integral operators, is a bound on the L p → L q {\displaystyle L^{p}\to L^{q}} operator norm of an integral operator in terms of L r {\displaystyle L^{r}} norms of the kernel itself. | Young's inequality for integral operators | [
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c_w8rc3xari60y | In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point. For example, the Lotka–Volterra predator–prey model is d x d t = x ( α − β y ) d y d t = − y ( γ − δ x ) , {\displaystyle {\begin{array}{rcl}{\dfrac {dx}{dt}}&=&x(\alpha -\beta y)\\{\dfrac {dy}{dt}}&=&-y(\gamma -\delta x),\end{array}}} where x(t) denotes the number of prey, y(t) the number of predators, and α, β, γ and δ are constants. | Community matrix | [
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c_ihoq9flspnif | By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point (x*, y*), which has the form = A , {\displaystyle {\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},} where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix A {\displaystyle \mathbf {A} } evaluated at the equilibrium point (x*, y*) is called the community matrix. By the stable manifold theorem, if one or both eigenvalues of A {\displaystyle \mathbf {A} } have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable. | Community matrix | [
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c_vac81ky92r4q | In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces. | Function (mathematics) | [
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Subsets and Splits