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Lamellar structure
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In materials science, lamellar structures or microstructures are composed of fine, alternating layers of different materials in the form of lamellae. They are often observed in cases where a phase transition front moves quickly, leaving behind two solid products, as in rapid cooling of eutectic (such as solder) or eutectoid (such as pearlite) systems. Such conditions force phases of different composition to form but allow little time for diffusion to produce those phases' equilibrium compositions. Fine lamellae solve this problem by shortening the diffusion distance between phases, but their high surface energy makes them unstable and prone to break up when annealing allows diffusion to progress.
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Lamellar structure
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A deeper eutectic or more rapid cooling will result in finer lamellae; as the size of an individual lamellum approaches zero, the system will instead retain its high-temperature structure. Two common cases of this include cooling a liquid to form an amorphous solid, and cooling eutectoid austenite to form martensite. In biology, normal adult bones possess a lamellar structure which may be disrupted by some diseases. == References ==
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Liquefaction
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In materials science, liquefaction is a process that generates a liquid from a solid or a gas or that generates a non-liquid phase which behaves in accordance with fluid dynamics. It occurs both naturally and artificially. As an example of the latter, a "major commercial application of liquefaction is the liquefaction of air to allow separation of the constituents, such as oxygen, nitrogen, and the noble gases." Another is the conversion of solid coal into a liquid form usable as a substitute for liquid fuels.
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Material failure theory
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In materials science, material failure is the loss of load carrying capacity of a material unit. This definition introduces to the fact that material failure can be examined in different scales, from microscopic, to macroscopic. In structural problems, where the structural response may be beyond the initiation of nonlinear material behaviour, material failure is of profound importance for the determination of the integrity of the structure. On the other hand, due to the lack of globally accepted fracture criteria, the determination of the structure's damage, due to material failure, is still under intensive research.
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Metallic elements
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In materials science, metallurgy, and engineering, a refractory metal is a metal that is extraordinarily resistant to heat and wear. Which metals belong to this category varies; the most common definition includes niobium, molybdenum, tantalum, tungsten, and rhenium. They all have melting points above 2000 °C, and a high hardness at room temperature.
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Misorientation
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In materials science, misorientation is the difference in crystallographic orientation between two crystallites in a polycrystalline material. In crystalline materials, the orientation of a crystallite is defined by a transformation from a sample reference frame (i.e. defined by the direction of a rolling or extrusion process and two orthogonal directions) to the local reference frame of the crystalline lattice, as defined by the basis of the unit cell. In the same way, misorientation is the transformation necessary to move from one local crystal frame to some other crystal frame. That is, it is the distance in orientation space between two distinct orientations.
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Paracrystallinity
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In materials science, paracrystalline materials are defined as having short- and medium-range ordering in their lattice (similar to the liquid crystal phases) but lacking crystal-like long-range ordering at least in one direction.
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Permeance
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In materials science, permeance is the degree to which a material transmits another substance.
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Polymorphism (crystallography)
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In materials science, polymorphism describes the existence of a solid material in more than one form or crystal structure. Polymorphism is a form of isomerism. Any crystalline material can exhibit the phenomenon. Allotropy refers to polymorphism for chemical elements.
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Quenching
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In materials science, quenching is the rapid cooling of a workpiece in water, oil, polymer, air, or other fluids to obtain certain material properties. A type of heat treating, quenching prevents undesired low-temperature processes, such as phase transformations, from occurring. It does this by reducing the window of time during which these undesired reactions are both thermodynamically favorable, and kinetically accessible; for instance, quenching can reduce the crystal grain size of both metallic and plastic materials, increasing their hardness. In metallurgy, quenching is most commonly used to harden steel by inducing a martensite transformation, where the steel must be rapidly cooled through its eutectoid point, the temperature at which austenite becomes unstable.
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Quenching
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In steel alloyed with metals such as nickel and manganese, the eutectoid temperature becomes much lower, but the kinetic barriers to phase transformation remain the same. This allows quenching to start at a lower temperature, making the process much easier. High-speed steel also has added tungsten, which serves to raise kinetic barriers, which among other effects gives material properties (hardness and abrasion resistance) as though the workpiece had been cooled more rapidly than it really has. Even cooling such alloys slowly in air has most of the desired effects of quenching; high-speed steel weakens much less from heat cycling due to high-speed cutting.Extremely rapid cooling can prevent the formation of all crystal structures, resulting in amorphous metal or "metallic glass".
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Radar absorbent material
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In materials science, radiation-absorbent material (RAM) is a material which has been specially designed and shaped to absorb incident RF radiation (also known as non-ionising radiation), as effectively as possible, from as many incident directions as possible. The more effective the RAM, the lower the resulting level of reflected RF radiation. Many measurements in electromagnetic compatibility (EMC) and antenna radiation patterns require that spurious signals arising from the test setup, including reflections, are negligible to avoid the risk of causing measurement errors and ambiguities.
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Recrystallization temperature
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In materials science, recrystallization is a process by which deformed grains are replaced by a new set of defect-free grains that nucleate and grow until the original grains have been entirely consumed. Recrystallization is usually accompanied by a reduction in the strength and hardness of a material and a simultaneous increase in the ductility. Thus, the process may be introduced as a deliberate step in metals processing or may be an undesirable byproduct of another processing step. The most important industrial uses are softening of metals previously hardened or rendered brittle by cold work, and control of the grain structure in the final product. Recrystallization temperature is typically 0.3–0.4 times the melting point for pure metals and 0.5 times for alloys.
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Reinforcement (composite)
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In materials science, reinforcement is a constituent of a composite material which increases the composite's stiffness and tensile strength.
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Segregation in materials
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In materials science, segregation is the enrichment of atoms, ions, or molecules at a microscopic region in a materials system. While the terms segregation and adsorption are essentially synonymous, in practice, segregation is often used to describe the partitioning of molecular constituents to defects from solid solutions, whereas adsorption is generally used to describe such partitioning from liquids and gases to surfaces. The molecular-level segregation discussed in this article is distinct from other types of materials phenomena that are often called segregation, such as particle segregation in granular materials, and phase separation or precipitation, wherein molecules are segregated in to macroscopic regions of different compositions. Segregation has many practical consequences, ranging from the formation of soap bubbles, to microstructural engineering in materials science, to the stabilization of colloidal suspensions.
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Modulus of rigidity
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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.
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Slip (materials science)
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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.
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Slip (materials science)
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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.
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Stress relaxation
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In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing some amount of plastic strain. This should not be confused with creep, which is a constant state of stress with an increasing amount of strain. Since relaxation relieves the state of stress, it has the effect of also relieving the equipment reactions.
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Stress relaxation
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Thus, relaxation has the same effect as cold springing, except it occurs over a longer period of time. The amount of relaxation which takes place is a function of time, temperature and stress level, thus the actual effect it has on the system is not precisely known, but can be bounded. Stress relaxation describes how polymers relieve stress under constant strain.
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Stress relaxation
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Because they are viscoelastic, polymers behave in a nonlinear, non-Hookean fashion. This nonlinearity is described by both stress relaxation and a phenomenon known as creep, which describes how polymers strain under constant stress.
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Stress relaxation
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Experimentally, stress relaxation is determined by step strain experiments, i.e. by applying a sudden one-time strain and measuring the build-up and subsequent relaxation of stress in the material (see figure), in either extensional or shear rheology. Viscoelastic materials have the properties of both viscous and elastic materials and can be modeled by combining elements that represent these characteristics. One viscoelastic model, called the Maxwell model predicts behavior akin to a spring (elastic element) being in series with a dashpot (viscous element), while the Voigt model places these elements in parallel.
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Stress relaxation
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Although the Maxwell model is good at predicting stress relaxation, it is fairly poor at predicting creep. On the other hand, the Voigt model is good at predicting creep but rather poor at predicting stress relaxation (see viscoelasticity). The extracellular matrix and most tissues are stress relaxing, and the kinetics of stress relaxation have been recognized as an important mechanical cue that affects the migration, proliferation, and differentiation of embedded cells.Stress relaxation calculations can differ for different materials: To generalize, Obukhov uses power dependencies: σ ( t ) = σ 0 1 − {\displaystyle \sigma (t)={\frac {\sigma _{0}}{1-}}} where σ 0 {\displaystyle \sigma _{0}} is the maximum stress at the time the loading was removed (t*), and n is a material parameter. Vegener et al. use a power series to describe stress relaxation in polyamides: σ ( t ) = ∑ m , n A m n m ( ϵ 0 ′ ) n {\displaystyle \sigma (t)=\sum _{m,n}^{}{A_{mn}^{m}(\epsilon '_{0})^{n}}} To model stress relaxation in glass materials Dowvalter uses the following: σ ( t ) = 1 b ⋅ log 10 α ( t − t n ) + 1 10 α ( t − t n ) − 1 {\displaystyle \sigma (t)={\frac {1}{b}}\cdot \log {\frac {10^{\alpha }(t-t_{n})+1}{10^{\alpha }(t-t_{n})-1}}} where α {\displaystyle \alpha } is a material constant and b and t n {\displaystyle t_{n}} depend on processing conditions. The following non-material parameters all affect stress relaxation in polymers: Magnitude of initial loading Speed of loading Temperature (isothermal vs non-isothermal conditions) Loading medium Friction and wear Long-term storage
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Superplastic deformation
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In materials science, superplasticity is a state in which solid crystalline material is deformed well beyond its usual breaking point, usually over about 400% during tensile deformation. Such a state is usually achieved at high homologous temperature. Examples of superplastic materials are some fine-grained metals and ceramics. Other non-crystalline materials (amorphous) such as silica glass ("molten glass") and polymers also deform similarly, but are not called superplastic, because they are not crystalline; rather, their deformation is often described as Newtonian fluid.
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Burgers vector
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In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as b, that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice. The vector's magnitude and direction is best understood when the dislocation-bearing crystal structure is first visualized without the dislocation, that is, the perfect crystal structure. In this perfect crystal structure, a rectangle whose lengths and widths are integer multiples of a (the unit cell edge length) is drawn encompassing the site of the original dislocation's origin. Once this encompassing rectangle is drawn, the dislocation can be introduced.
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Burgers vector
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This dislocation will have the effect of deforming, not only the perfect crystal structure, but the rectangle as well. The said rectangle could have one of its sides disjoined from the perpendicular side, severing the connection of the length and width line segments of the rectangle at one of the rectangle's corners, and displacing each line segment from each other. What was once a rectangle before the dislocation was introduced is now an open geometric figure, whose opening defines the direction and magnitude of the Burgers vector.
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Burgers vector
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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 a rectangular counterclockwise circuit (Burgers circuit) from a starting point to enclose the dislocation (see the picture above). The Burgers vector will be the vector to complete the circuit, i.e., from the end to the start of the circuit.The direction of the vector depends on the plane of dislocation, which is usually on one of the closest-packed crystallographic planes.
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Burgers vector
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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 line are perpendicular to one another. In screw dislocations, they are parallel.The Burgers vector is significant in determining the yield strength of a material by affecting solute hardening, precipitation hardening and work hardening. The Burgers vector plays an important role in determining the direction of dislocation line.
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Charpy test
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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.
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Zener–Hollomon parameter
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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.
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Zener–Hollomon parameter
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It is named after Clarence Zener and John Herbert Hollomon, Jr. who established the formula based on the stress-strain behavior in steel.
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Zener–Hollomon parameter
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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.
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Cottrell atmosphere
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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.
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Sessile drop technique
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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 solid substrate can be calculated. The liquid used for such experiments is referred to as the probe liquid, and the use of several different probe liquids is required.
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Sol–gel process
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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 precursor for an integrated network (or gel) of either discrete particles or network polymers. Typical precursors are metal alkoxides. Sol-gel process is used to produce ceramic nanoparticles.
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Two dimensional (2D) nanomaterials
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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.
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Two dimensional (2D) nanomaterials
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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.
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Threshold displacement energy
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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 displacement energy exists for each crystallographic direction.
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Threshold displacement energy
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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 solids are of the order of 10-50 eV.
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Yield strength anomaly
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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.
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Toughening
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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.
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Cold pressing
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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 within the crystal structure of the material.
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Antiferromagnetic interaction
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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 magnetism. The phenomenon of antiferromagnetism was first introduced by Lev Landau in 1933.Generally, antiferromagnetic order may exist at sufficiently low temperatures, but vanishes at and above the Néel temperature – named after Louis Néel, who had first identified this type of magnetic ordering. Above the Néel temperature, the material is typically paramagnetic.
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Spatial dispersion
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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 activity in solutions of chiral molecules. In isotropic materials without optical activity, the permittivity tensor can be broken down to transverse and longitudinal components, referring to the response to electric fields either perpendicular or parallel to the wavevector.For frequencies nearby an absorption line (e.g., an exciton), spatial dispersion can play an important role.
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Corrosion fatigue
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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 the threshold stress intensity for stress corrosion cracking, additional crack growth (the red line) occurs due to SCC. The lower stress-intensity regions are not affected, and the threshold stress-intensity range for fatigue-crack propagation is unchanged in the corrosive environment. In the most-general case, corrosion-fatigue crack growth may exhibit both of the above effects; crack-growth behavior is represented in the schematic on the left.
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Narrow gap
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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 "optical band gap" and "electronic band gap" (or "transport gap"). The optical bandgap is the threshold for photons to be absorbed, while the transport gap is the threshold for creating an electron–hole pair that is not bound together.
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Narrow gap
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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, and the distinction between them is ignored. However, in some systems, including organic semiconductors and single-walled carbon nanotubes, the distinction may be significant.
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Exciton
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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.
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Dynamic strain aging
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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 around the pinned dislocations, further strengthening the obstacles' hold on the dislocations.
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Dynamic strain aging
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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 to affect materials without these physical observations.
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Oscillatory integral
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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.
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Z-order curve
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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 the United States after Guy Macdonald Morton, who first applied the order to file sequencing in 1966. The z-value of a point in multidimensions is simply calculated by interleaving the binary representations of its coordinate values. Once the data are sorted into this ordering, any one-dimensional data structure can be used, such as simple one dimensional arrays, binary search trees, B-trees, skip lists or (with low significant bits truncated) hash tables. The resulting ordering can equivalently be described as the order one would get from a depth-first traversal of a quadtree or octree.
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Sigma field
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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.
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Sigma field
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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.
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Sigma field
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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).
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Multi-variable function
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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 values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article. The domain of a function of n variables is the subset of R n {\displaystyle \mathbb {R} ^{n}} for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of R n {\displaystyle \mathbb {R} ^{n}} .
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Laakso space
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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.
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Agmon's inequality
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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 .
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Agmon's inequality
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{\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}}
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Bernstein's theorem (polynomials)
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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 ) | ) ≤ n ! ( n − k ) ! ⋅ max | z | ≤ 1 ( | P ( z ) | ) . {\displaystyle \max _{|z|\leq 1}(|P^{(k)}(z)|)\leq {\frac {n!}{(n-k)! }}\cdot \max _{|z|\leq 1}(|P(z)|).}
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Cesàro summation
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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 special case of a matrix summability method is named for the Italian analyst Ernesto Cesàro (1859–1906). The term summation can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the sum of that series is 1/2.
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Clairaut's equation
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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 differential equation. It is named after the French mathematician Alexis Clairaut, who introduced it in 1734.
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Darboux's formula
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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 in a similar manner (by repeated integration by parts of a particular choice of integrand). Darboux's formula can also be used to derive the Taylor series from calculus.
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Dini continuity
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In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.
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Ehrenpreis's fundamental principle
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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 complex “characteristic variety” of the system. == References ==
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Ivar Ekeland
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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 Bolzano–Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space.Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.Ekeland was associated with the University of Paris when he proposed this theorem.
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Ekeland's variational principle
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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.
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Fourier integral operator
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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 by: ( T f ) ( x ) = ∫ R n e 2 π i Φ ( x , ξ ) a ( x , ξ ) f ^ ( ξ ) d ξ {\displaystyle (Tf)(x)=\int _{\mathbb {R} ^{n}}e^{2\pi i\Phi (x,\xi )}a(x,\xi ){\hat {f}}(\xi )\,d\xi } where f ^ {\displaystyle {\hat {f}}} denotes the Fourier transform of f {\displaystyle f} , a ( x , ξ ) {\displaystyle a(x,\xi )} is a standard symbol which is compactly supported in x {\displaystyle x} and Φ {\displaystyle \Phi } is real valued and homogeneous of degree 1 {\displaystyle 1} in ξ {\displaystyle \xi } . It is also necessary to require that det ( ∂ 2 Φ ∂ x i ∂ ξ j ) ≠ 0 {\displaystyle \det \left({\frac {\partial ^{2}\Phi }{\partial x_{i}\,\partial \xi _{j}}}\right)\neq 0} on the support of a. Under these conditions, if a is of order zero, it is possible to show that T {\displaystyle T} defines a bounded operator from L 2 {\displaystyle L^{2}} to L 2 {\displaystyle L^{2}} .
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Fubini's Theorem
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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.
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Fubini's Theorem
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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.
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Glaeser's continuity theorem
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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 + {\displaystyle f\ :\ U\rightarrow \mathbb {R} _{0}^{+}} be a function of class C 2 {\displaystyle C^{2}} in an open set U contained in R n {\displaystyle \mathbb {R} ^{n}} , then f {\displaystyle {\sqrt {f}}} is of class C 1 {\displaystyle C^{1}} in U if and only if its partial derivatives of first and second order vanish in the zeros of f. == References ==
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Haar's Tauberian theorem
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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.
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Heine's Reciprocal Square Root Identity
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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 greater than one. This expression can be generalized for arbitrary half-integer powers as follows where Γ {\displaystyle \scriptstyle \,\Gamma } is the Gamma function. == References ==
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Hölder's inequality
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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.
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Hölder's inequality
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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.
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Leonid Kantorovich
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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 to as "Kantorovich spaces" in his honor. Kantorovich showed that functional analysis could be used in the analysis of iterative methods, obtaining the Kantorovich inequalities on the convergence rate of the gradient method and of Newton's method (see the Kantorovich theorem). Kantorovich considered infinite-dimensional optimization problems, such as the Kantorovich-Monge problem in transport theory. His analysis proposed the Kantorovich–Rubinstein metric, which is used in probability theory, in the theory of the weak convergence of probability measures.
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Korn's inequality
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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.
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Lambert summation
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In mathematical analysis, Lambert summation is a summability method for a class of divergent series.
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Lipschitz function
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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.
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Littlewood's 4/3 inequality
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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_{0}\to \mathbb {C} } or R {\displaystyle \mathbb {R} } be a bilinear form. Then the following holds: ( ∑ i , j = 1 ∞ | B ( e i , e j ) | 4 / 3 ) 3 / 4 ≤ 2 ‖ B ‖ , {\displaystyle \left(\sum _{i,j=1}^{\infty }|B(e_{i},e_{j})|^{4/3}\right)^{3/4}\leq {\sqrt {2}}\|B\|,} where ‖ B ‖ = sup { | B ( x 1 , x 2 ) |: ‖ x i ‖ ∞ ≤ 1 } .
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Littlewood's 4/3 inequality
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{\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.
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Lorentz space
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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 a quasinorm) that encodes information about the "size" of a function, just as the L p {\displaystyle L^{p}} norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the L p {\displaystyle L^{p}} norms, by exponentially rescaling the measure in both the range ( p {\displaystyle p} ) and the domain ( q {\displaystyle q} ). The Lorentz norms, like the L p {\displaystyle L^{p}} norms, are invariant under arbitrary rearrangements of the values of a function.
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Mosco convergence
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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.
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Netto's theorem
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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-dimensional manifold was proven by Jacob Lüroth in 1878, using the intermediate value theorem to show that no manifold containing a topological circle can be mapped continuously and bijectively to the real line. Both Netto in 1878, and Georg Cantor in 1879, gave faulty proofs of the general theorem.
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Netto's theorem
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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 relaxed in different ways to obtain interesting classes of functions from one-dimensional spaces to two-dimensional spaces: Space-filling curves are surjective continuous functions from one-dimensional spaces to two-dimensional spaces. They cover every point of the plane, or of a unit square, by the image of a line or unit interval.
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Netto's theorem
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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 area.
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Netto's theorem
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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.
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Netto's theorem
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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 kicked off the line of research leading to space-filling curves, Osgood curves, and Netto's theorem.
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Netto's theorem
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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 a bijection, but this issue can be repaired by using the Schröder–Bernstein theorem. == References ==
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Parseval's formula
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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 identity asserts that the sum of squares of the Fourier coefficients of a function is equal to the integral of the square of the function, where the Fourier coefficients c n {\displaystyle c_{n}} of f {\displaystyle f} are given by More formally, the result holds as stated provided f {\displaystyle f} is a square-integrable function or, more generally, in Lp space L 2 . {\displaystyle L^{2}.}
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Parseval's formula
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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 the integral of the fourth power of the function f ∈ L 4 {\displaystyle f\in L^{4}} in terms of its Fourier coefficients given f {\displaystyle f} has a finite-length discrete Fourier transform with M {\displaystyle M} number of coefficients c ∈ C {\displaystyle c\in \mathbb {C} } . if c ∈ R {\displaystyle c\in \mathbb {R} } the identity is simplified to
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Rademacher's theorem
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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. Differentiability here refers to infinitesimal approximability by a linear map, which in particular asserts the existence of the coordinate-wise partial derivatives.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Equality of mixed partials
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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 {\displaystyle \mathbf {p} \in \mathbb {R} ^{n}} is a point such that some neighborhood of p {\displaystyle \mathbf {p} } is contained in Ω {\displaystyle \Omega } and f {\displaystyle f} has continuous second partial derivatives on that neighborhood of p {\displaystyle \mathbf {p} } , then for all i and j in { 1 , 2 … , n } , {\displaystyle \{1,2\ldots ,\,n\},} ∂ 2 ∂ x i ∂ x j f ( p ) = ∂ 2 ∂ x j ∂ x i f ( p ) . {\displaystyle {\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}f(\mathbf {p} )={\frac {\partial ^{2}}{\partial x_{j}\,\partial x_{i}}}f(\mathbf {p} ).} The partial derivatives of this function commute at that point. One easy way to establish this theorem (in the case where n = 2 {\displaystyle n=2} , i = 1 {\displaystyle i=1} , and j = 2 {\displaystyle j=2} , which readily entails the result in general) is by applying Green's theorem to the gradient of f .
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Equality of mixed partials
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{\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 } containing a point ( a , b ) {\displaystyle (a,b)} and suppose that d f {\displaystyle df} is continuous with continuous ∂ x ∂ y f {\displaystyle \partial _{x}\partial _{y}f} and ∂ y ∂ x f {\displaystyle \partial _{y}\partial _{x}f} over Ω .
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Equality of mixed partials
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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 second order operators. Here, for z {\displaystyle z} a vector in the plane and u {\displaystyle u} a directional vector ( 1 0 ) {\displaystyle {\tbinom {1}{0}}} or ( 0 1 ) {\displaystyle {\tbinom {0}{1}}} , the difference operator is defined by Δ u t f ( z ) = f ( z + t u ) − f ( z ) t . {\displaystyle \Delta _{u}^{t}f(z)={f(z+tu)-f(z) \over t}.}
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Equality of mixed partials
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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 ) | ≤ ( b − a ) sup c ∈ ( a , b ) | f ′ ( c ) | {\displaystyle |f(b)-f(a)|\leq (b-a)\,\sup _{c\in (a,b)}|f^{\prime }(c)|} .This is a generalized version of the mean value theorem.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Equality of mixed partials
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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.
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Equality of mixed partials
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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}\Delta _{2}^{t}f(x_{0},y_{0})=D_{1}D_{2}f(x_{0}+t\theta ,y_{0}+t\theta ^{\prime })} for some θ {\displaystyle \theta } and θ ′ {\displaystyle \theta ^{\prime }} in ( 0 , 1 ) {\displaystyle (0,1)} . Thus the first elementary proof can be reinterpreted using difference operators. Conversely, instead of using the generalized mean value theorem in the second proof, the classical mean valued theorem could be used.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Tannery's theorem
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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.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Trudinger's theorem
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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.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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