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c_tiiyrzhekdxb | In materials science, critical resolved shear stress (CRSS) is the component of shear stress, resolved in the direction of slip, necessary to initiate slip in a grain. Resolved shear stress (RSS) is the shear component of an applied tensile or compressive stress resolved along a slip plane that is other than perpendicular or parallel to the stress axis. The RSS is related to the applied stress by a geometrical factor, m, typically the Schmid factor: τ RSS = σ app m = σ app ( cos ϕ cos λ ) {\displaystyle \tau _{\text{RSS}}=\sigma _{\text{app}}m=\sigma _{\text{app}}(\cos \phi \cos \lambda )} where σapp is the magnitude of the applied tensile stress, Φ is the angle between the normal of the slip plane and the direction of the applied force, and λ is the angle between the slip direction and the direction of the applied force. The Schmid factor is most applicable to FCC single-crystal metals, but for polycrystal metals the Taylor factor has been shown to be more accurate. | Critical resolved shear stress | [
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c_5c1eqay1lkk8 | In materials science, dispersion is the fraction of atoms of a material exposed to the surface. In general, D = NS/N, where D is the dispersion, NS is the number of surface atoms and NT is the total number of atoms of the material. It is an important concept in heterogeneous catalysis, since only atoms exposed to the surface can affect catalytic surface reactions. Dispersion increases with decreasing crystallite size and approaches unity at a crystallite diameter of about 0.1 nm. | Dispersion (materials science) | [
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c_drx3y18rtp2d | These classifications are polymer matrix composites (PMCs), ceramic matrix composites (CMCs), and metal matrix composites (MMCs). Also, materials within these categories are often called "advanced" if they combine the properties of high (axial, longitudinal) strength values and high (axial, longitudinal) stiffness values, with low weight, corrosion resistance, and in some cases special electrical properties. Advanced composite materials have broad, proven applications, in the aircraft, aerospace, and sports equipment sectors. | Advanced composite materials (engineering) | [
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c_w5sfv8mxe48t | Advanced composites exhibit desirable physical and chemical properties that include light weight coupled with high stiffness (elasticity), and strength along the direction of the reinforcing fiber, dimensional stability, temperature and chemical resistance, flex performance, and relatively easy processing. Advanced composites are replacing metal components in many uses, particularly in the aerospace industry. Composites are classified according to their matrix phase. | Advanced composite materials (engineering) | [
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c_40u7zaxd657w | In materials science, advanced composite materials (ACMs) are materials that are generally characterized by unusually high strength fibres with unusually high stiffness, or modulus of elasticity characteristics, compared to other materials, while bound together by weaker matrices. These are termed "advanced composite materials" in comparison to the composite materials commonly in use such as reinforced concrete, or even concrete itself. The high strength fibers are also low density while occupying a large fraction of the volume. | Advanced composite materials (engineering) | [
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c_e0xd77hfutfu | In materials science, an intrinsic property is independent of how much of a material is present and is independent of the form of the material, e.g., one large piece or a collection of small particles. Intrinsic properties are dependent mainly on the fundamental chemical composition and structure of the material. Extrinsic properties are differentiated as being dependent on the presence of avoidable chemical contaminants or structural defects.In biology, intrinsic effects originate from inside an organism or cell, such as an autoimmune disease or intrinsic immunity. In electronics and optics, intrinsic properties of devices (or systems of devices) are generally those that are free from the influence of various types of non-essential defects. Such defects may arise as a consequence of design imperfections, manufacturing errors, or operational extremes and can produce distinctive and often undesirable extrinsic properties. The identification, optimization, and control of both intrinsic and extrinsic properties are among the engineering tasks necessary to achieve the high performance and reliability of modern electrical and optical systems. | Intrinsic properties | [
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c_fergsvt73ubp | However, such substances are not generally considered friable because of the degree of difficulty involved in breaking the substance's chemical bonds through mechanical means. Some substances, such as polyurethane foams, show an increase in friability with exposure to ultraviolet radiation, as in sunlight. Friable is sometimes used metaphorically to describe "brittle" personalities who can be "rubbed" by seemingly-minor stimuli to produce extreme emotional responses. | Friability | [
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c_ycig73mvowdi | In materials science, friability ( FRY-ə-BIL-ə-tee), the condition of being friable, describes the tendency of a solid substance to break into smaller pieces under duress or contact, especially by rubbing. The opposite of friable is indurate. Substances that are designated hazardous, such as asbestos or crystalline silica, are often said to be friable if small particles are easily dislodged and become airborne, and hence respirable (able to enter human lungs), thereby posing a health hazard. Tougher substances, such as concrete, may also be mechanically ground down and reduced to finely divided mineral dust. | Friability | [
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c_vxrgt43ar50j | In materials science, intergranular corrosion (IGC), also known as intergranular attack (IGA), is a form of corrosion where the boundaries of crystallites of the material are more susceptible to corrosion than their insides. (Cf. transgranular corrosion.) | Intergranular corrosion | [
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c_jb1lnwf3ce1p | 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. | Metallic elements | [
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c_b9nwx65gpusa | In materials science, galfenol is the general term for an alloy of iron and gallium. The name was first given to iron-gallium alloys by United States Navy researchers in 1998 when they discovered that adding gallium to iron could amplify iron's magnetostrictive effect up to tenfold. Galfenol is of interest to sonar researchers because magnetostrictor materials are used to detect sound, and amplifying the magnetostrictive effect could lead to better sensitivity of sonar detectors. Galfenol is also proposed for vibrational energy harvesting, actuators for precision machine tools, active anti-vibration systems, and anti-clogging devices for sifting screens and spray nozzles. | Galfenol | [
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c_h5yjetldfh17 | These clumps have been described by Peter Gehring of the NIST Center for Neutron Research as "something like raisins within a cake". It has also been proposed that there is an intrinsic mechanism generating this enhanced magnetostriction, which has its origins in the electronic structure of the material as described by density functional theory. It is understood that the addition of gallium to pure iron alters the electronic structure and atomic arrangements in the material in such a way as to enhance the material's magnetoelastic constant. | Galfenol | [
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c_k9pycyecblu0 | Galfenol is machinable and can be produced in sheet and wire form.In 2009, scientists from Virginia Polytechnic Institute and State University, and National Institute of Standards and Technology (NIST) used neutron beams to determine the structure of galfenol. They determined that the addition of gallium changes the lattice structure of the iron atoms from regular cubic cells to one in which the faces of some of the cells become slightly rectangular. The elongated cells tend to clump together in the alloy, forming localized clumps within the material. | Galfenol | [
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c_hjs69pjlumlf | In materials science parlance, dislocations are defined as line defects in a material's crystal structure. The bonds surrounding the dislocation are already elastically strained by the defect compared to the bonds between the constituents of the regular crystal lattice. Therefore, these bonds break at relatively lower stresses, leading to plastic deformation. The strained bonds around a dislocation are characterized by lattice strain fields. | Work hardened | [
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c_xg9mse1blbjw | Specifically, the strain fields of dislocations obey similar laws of attraction and repulsion; in order to reduce overall strain, compressive strains are attracted to tensile strains, and vice versa. The visible (macroscopic) results of plastic deformation are the result of microscopic dislocation motion. For example, the stretching of a steel rod in a tensile tester is accommodated through dislocation motion on the atomic scale. | Work hardened | [
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c_pq9y7hbrjhkl | For example, there are compressively strained bonds directly next to an edge dislocation and tensilely strained bonds beyond the end of an edge dislocation. These form compressive strain fields and tensile strain fields, respectively. Strain fields are analogous to electric fields in certain ways. | Work hardened | [
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c_ox85u72ul6l2 | In materials science, effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. EMAs or EMTs are developed from averaging the multiple values of the constituents that directly make up the composite material. At the constituent level, the values of the materials vary and are inhomogeneous. Precise calculation of the many constituent values is nearly impossible. | Effective medium | [
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] |
c_70r19t2lkv6b | They both were derived in quasi-static approximation when the electric field inside a mixture particle may be considered as homogeneous. So, these formulae can not describe the particle size effect. Many attempts were undertaken to improve these formulae. | Effective medium | [
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c_77jiew9ogxh5 | However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters including the effective permittivity and permeability of the materials as a whole. In this sense, effective medium approximations are descriptions of a medium (composite material) based on the properties and the relative fractions of its components and are derived from calculations, and effective medium theory. There are two widely used formulae.Effective permittivity and permeability are averaged dielectric and magnetic characteristics of a microinhomogeneous medium. | Effective medium | [
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c_szgd2lmaqayk | This phenomenon is analogous to half of a piece of paper inserted into a stack of paper, where the defect in the stack is noticeable only at the edge of the half sheet. The theory describing the elastic fields of the defects was originally developed by Vito Volterra in 1907. In 1934, Egon Orowan, Michael Polanyi and G. I. Taylor, proposed that the low stresses observed to produce plastic deformation compared to theoretical predictions at the time could be explained in terms of the theory of dislocations. | Dislocation | [
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c_uxr3y093vmv1 | In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to slide over each other at low stress levels and is known as glide or slip. The crystalline order is restored on either side of a glide dislocation but the atoms on one side have moved by one position. The crystalline order is not fully restored with a partial dislocation. | Dislocation | [
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c_c33usal1ny8c | The two main types of mobile dislocations are edge and screw dislocations. Edge dislocations can be visualized as being caused by the termination of a plane of atoms in the middle of a crystal. In such a case, the surrounding planes are not straight, but instead bend around the edge of the terminating plane so that the crystal structure is perfectly ordered on either side. | Dislocation | [
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c_2ig0bptrnigt | In materials science, chemical force microscopy (CFM) is a variation of atomic force microscopy (AFM) which has become a versatile tool for characterization of materials surfaces. With AFM, structural morphology is probed using simple tapping or contact modes that utilize van der Waals interactions between tip and sample to maintain a constant probe deflection amplitude (constant force mode) or maintain height while measuring tip deflection (constant height mode). CFM, on the other hand, uses chemical interactions between functionalized probe tip and sample. Choice chemistry is typically gold-coated tip and surface with R−SH thiols attached, R being the functional groups of interest. | Friction force microscopy | [
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c_9548qs1o7og2 | CFM enables the ability to determine the chemical nature of surfaces, irrespective of their specific morphology, and facilitates studies of basic chemical bonding enthalpy and surface energy. Typically, CFM is limited by thermal vibrations within the cantilever holding the probe. This limits force measurement resolution to ~1 pN which is still very suitable considering weak COOH/CH3 interactions are ~20 pN per pair. Hydrophobicity is used as the primary example throughout this consideration of CFM, but certainly any type of bonding can be probed with this method. | Friction force microscopy | [
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c_vd0mxahbkk3a | Fast ion conductors are used primarily in solid oxide fuel cells. As solid electrolytes they allow the movement of ions without the need for a liquid or soft membrane separating the electrodes. The phenomenon relies on the hopping of ions through an otherwise rigid crystal structure. | Fast ion conductor | [
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c_ft528bu363si | In materials science, fast ion conductors are solid conductors with highly mobile ions. These materials are important in the area of solid state ionics, and are also known as solid electrolytes and superionic conductors. These materials are useful in batteries and various sensors. | Fast ion conductor | [
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c_momczbpchs2l | In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack suddenly becomes rapid and unlimited. A component's thickness affects the constraint conditions at the tip of a crack with thin components having plane stress conditions and thick components having plane strain conditions. Plane strain conditions give the lowest fracture toughness value which is a material property. The critical value of stress intensity factor in mode I loading measured under plane strain conditions is known as the plane strain fracture toughness, denoted K Ic {\displaystyle K_{\text{Ic}}} . | Fracture toughening mechanisms | [
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c_t4gk9poohonf | When a test fails to meet the thickness and other test requirements that are in place to ensure plane strain conditions, the fracture toughness value produced is given the designation K c {\displaystyle K_{\text{c}}} . Fracture toughness is a quantitative way of expressing a material's resistance to crack propagation and standard values for a given material are generally available. Slow self-sustaining crack propagation known as stress corrosion cracking, can occur in a corrosive environment above the threshold K Iscc {\displaystyle K_{\text{Iscc}}} and below K Ic {\displaystyle K_{\text{Ic}}} . Small increments of crack extension can also occur during fatigue crack growth, which after repeated loading cycles, can gradually grow a crack until final failure occurs by exceeding the fracture toughness. | Fracture toughening mechanisms | [
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c_a75gn71w5f7c | In materials science, bulk density, also called apparent density or volumetric density, is a property of powders, granules, and other "divided" solids, especially used in reference to mineral components (soil, gravel), chemical substances, pharmaceutical ingredients, foodstuff, or any other masses of corpuscular or particulate matter (particles). Bulk density is defined as the mass of the many particles of the material divided by the total volume they occupy. The total volume includes particle volume, inter-particle void volume, and internal pore volume.Bulk density is not an intrinsic property of a material; it can change depending on how the material is handled. For example, a powder poured into a cylinder will have a particular bulk density; if the cylinder is disturbed, the powder particles will move and usually settle closer together, resulting in a higher bulk density. For this reason, the bulk density of powders is usually reported both as "freely settled" (or "poured" density) and "tapped" density (where the tapped density refers to the bulk density of the powder after a specified compaction process, usually involving vibration of the container.) In contrast, particle density is an intrinsic property of the solid and does not include the volume for voids between particles. | Bulk density | [
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c_x5joemf2iwp0 | In materials science, ceramic matrix composites (CMCs) are a subgroup of composite materials and a subgroup of ceramics. They consist of ceramic fibers embedded in a ceramic matrix. The fibers and the matrix both can consist of any ceramic material, whereby carbon and carbon fibers can also be regarded as a ceramic material. | Ceramic Matrix Composite | [
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c_66b6yencpmuk | 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. | Sessile drop technique | [
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c_0dlwcj6anxxa | 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 == | Glaeser's continuity theorem | [
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c_bxalboy05fkt | 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. | Netto's theorem | [
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c_7if1ykfbxx4s | 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 == | Netto's theorem | [
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c_d6r3xdd0ddei | 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. | Netto's theorem | [
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c_wavpa0728pz8 | In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions. | Metric differential | [
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c_d08fbltnl4l6 | In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant cN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there are cN subcollections of balls A1 = {Bn1}, …, AcN = {BncN} contained in F such that each collection Ai consists of disjoint balls, and E ⊆ ⋃ i = 1 c N ⋃ B ∈ A i B . {\displaystyle E\subseteq \bigcup _{i=1}^{c_{N}}\bigcup _{B\in A_{i}}B.} | Besicovitch covering theorem | [
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c_kbdtth82xqnc | Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant). There exists a constant bN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most bN different balls from the subcover G.In other words, the function SG equal to the sum of the indicator functions of the balls in G is larger than 1E and bounded on RN by the constant bN, 1 E ≤ S G := ∑ B ∈ G 1 B ≤ b N . {\displaystyle \mathbf {1} _{E}\leq S_{\mathbf {G} }:=\sum _{B\in \mathbf {G} }\mathbf {1} _{B}\leq b_{N}.} | Besicovitch covering theorem | [
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c_pohariczva0z | {\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. | Littlewood's 4/3 inequality | [
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c_afqh2ooya0qr | 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 } . | Littlewood's 4/3 inequality | [
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c_rjhs7k6yz9e1 | 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. | Darboux's formula | [
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c_d19gfyr2yyjp | 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. | Clairaut's equation | [
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c_vm8qazmssfgk | 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. | Threshold displacement energy | [
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c_8mnybwlzoo22 | 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. | Threshold displacement energy | [
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c_r10n7omt7t8s | 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. | Oscillatory integral | [
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c_nwumcr765b2o | 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. | Dynamic strain aging | [
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c_2vrqnp68en38 | 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. | Dynamic strain aging | [
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c_rfp9djq2gl6t | In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used for the domain of a function, but in general, functions may be defined on sets that are not topological spaces. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term domain, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset. | Bounded domain | [
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c_o7ow7trwjie0 | In mathematical analysis, a Banach limit is a continuous linear functional ϕ: ℓ ∞ → C {\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space ℓ ∞ {\displaystyle \ell ^{\infty }} of all bounded complex-valued sequences such that for all sequences x = ( x n ) {\displaystyle x=(x_{n})} , y = ( y n ) {\displaystyle y=(y_{n})} in ℓ ∞ {\displaystyle \ell ^{\infty }} , and complex numbers α {\displaystyle \alpha }: ϕ ( α x + y ) = α ϕ ( x ) + ϕ ( y ) {\displaystyle \phi (\alpha x+y)=\alpha \phi (x)+\phi (y)} (linearity); if x n ≥ 0 {\displaystyle x_{n}\geq 0} for all n ∈ N {\displaystyle n\in \mathbb {N} } , then ϕ ( x ) ≥ 0 {\displaystyle \phi (x)\geq 0} (positivity); ϕ ( x ) = ϕ ( S x ) {\displaystyle \phi (x)=\phi (Sx)} , where S {\displaystyle S} is the shift operator defined by ( S x ) n = x n + 1 {\displaystyle (Sx)_{n}=x_{n+1}} (shift-invariance); if x {\displaystyle x} is a convergent sequence, then ϕ ( x ) = lim x {\displaystyle \phi (x)=\lim x} .Hence, ϕ {\displaystyle \phi } is an extension of the continuous functional lim: c → C {\displaystyle \lim :c\to \mathbb {C} } where c ⊂ ℓ ∞ {\displaystyle c\subset \ell ^{\infty }} is the complex vector space of all sequences which converge to a (usual) limit in C {\displaystyle \mathbb {C} } . In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case. | Banach limit | [
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c_2teeh2lwa914 | As a consequence of the above properties, a real-valued Banach limit also satisfies: lim inf n → ∞ x n ≤ ϕ ( x ) ≤ lim sup n → ∞ x n . {\displaystyle \liminf _{n\to \infty }x_{n}\leq \phi (x)\leq \limsup _{n\to \infty }x_{n}.} The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach), or using ultrafilters (this approach is more frequent in set-theoretical expositions). These proofs necessarily use the axiom of choice (so called non-effective proof). | Banach limit | [
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c_qsj7xttnjh0u | 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. | Leonid Kantorovich | [
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c_q1gxm1yb22n4 | As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous. | Equicontinuous linear maps | [
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c_zcmu3kyq6mo2 | In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. | Equicontinuous linear maps | [
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c_w9o14dyomcd4 | 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. | Spatial dispersion | [
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c_9jg0hvdgilnm | 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. | Rademacher's theorem | [
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c_8au9lylqhd7c | Segregation can occur in various materials classes. In polycrystalline solids, segregation occurs at defects, such as dislocations, grain boundaries, stacking faults, or the interface between two phases. In liquid solutions, chemical gradients exist near second phases and surfaces due to combinations of chemical and electrical effects. Segregation which occurs in well-equilibrated systems due to the instrinsic chemical properties of the system is termed equilibrium segregation. Segregation that occurs due to the processing history of the sample (but that would disappear at long times) is termed non-equilibrium segregation. | Segregation in materials | [
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c_18idd1opcosu | 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. | Segregation in materials | [
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c_partjb2f3dxj | In the case of several variables, a function f defined on an open subset Ω of R n {\displaystyle \mathbb {R} ^{n}} is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure. One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering. We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line: Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ continuous and bounded variation ⊆ differentiable almost everywhere | Function of bounded variation | [
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c_e23xy11ykvkr | In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed x-axis and to the y-axis. Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. | Function of bounded variation | [
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c_hcetedjy7667 | Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many. | Function of bounded variation | [
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c_f1hi546cwm5l | 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. | Lorentz space | [
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c_zh4ouc51rylv | Boundary is a distinct concept: for example, a circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R2 constrained by two parabolic curves x2 + 1 and x2 - 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded). | Bounded poset | [
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c_z0siec50bpre | In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric. | Bounded poset | [
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c_eamlmfaz68yq | 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. | Cesàro summation | [
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c_0m8faprm0esw | 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. | Antiferromagnetic interaction | [
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c_fsehp4treneu | 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. | Tannery's theorem | [
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c_3boowh6klrtx | 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. | Laakso space | [
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c_ryyorxxhz4io | 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. | Narrow gap | [
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c_feey2q68b3eq | 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. | Narrow gap | [
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c_d9xzebpplleo | Superplastically deformed material gets thinner in a very uniform manner, rather than forming a "neck" (a local narrowing) that leads to fracture. Also, the formation of microvoids, which is another cause of early fracture, is inhibited. Superplasticity must not be confused with superelasticity. | Superplastic deformation | [
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c_uxbfk3wstu91 | 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. | Superplastic deformation | [
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c_q2y1lwvxeng2 | 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. | Ivar Ekeland | [
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c_a1blqtqa73d7 | 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}.} | Parseval's formula | [
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c_tq233r1211h2 | 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 | Parseval's formula | [
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c_u7hiaqu7wczn | In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function is indeed Lebesgue-measurable, but in many situations, continuity is a too restrictive assumption. Carathéodory functions are more general than continuous functions, but still allow a composition with Lebesgue-measurable function to be measurable. Carathéodory functions play a significant role in calculus of variation, and it is named after the Greek mathematician Constantin Carathéodory. | Carathéodory function | [
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c_2trdiav66fh8 | In mathematical analysis, Lambert summation is a summability method for a class of divergent series. | Lambert summation | [
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] |
c_pcpy81di11ri | In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space. | Pseudo-differential operator | [
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c_jy30z1zkvejz | In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions. | Multidimensional transform | [
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c_oadm9fbx2agx | In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)} (where the ∗ {\displaystyle ^{*}} indicates the complex conjugate) for all x {\displaystyle x} in the domain of f {\displaystyle f} . In physics, this property is referred to as PT symmetry. This definition extends also to functions of two or more variables, e.g., in the case that f {\displaystyle f} is a function of two variables it is Hermitian if f ∗ ( x 1 , x 2 ) = f ( − x 1 , − x 2 ) {\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})} for all pairs ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} in the domain of f {\displaystyle f} . From this definition it follows immediately that: f {\displaystyle f} is a Hermitian function if and only if the real part of f {\displaystyle f} is an even function, the imaginary part of f {\displaystyle f} is an odd function. | Hermitian function | [
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] |
c_gxt7f0tsze3x | 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. | Sol–gel process | [
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c_txjmrujqddki | 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 == | Ehrenpreis's fundamental principle | [
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c_27ozez4lvb8s | 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 == | Heine's Reciprocal Square Root Identity | [
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c_zo5g9y5372q9 | 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. | Stress relaxation | [
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c_5bj4oc46vldj | 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 | Stress relaxation | [
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c_bknllzmzmyzn | 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. | Stress relaxation | [
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c_xredb32kysli | In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in the calculus of variations, especially models from material science, and the study of nonlinear partial differential equations, as well as in various optimization (or optimal control problems). They are named after Laurence Chisholm Young who invented them, already in 1937 in one dimension (curves) and later in higher dimensions in 1942. | Young measure | [
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c_dyvm7d08ho59 | 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}} . | Fourier integral operator | [
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c_4s95ktme8r12 | 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. | Burgers vector | [
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c_kbhy7v95w6fi | 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. | Burgers vector | [
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c_cq9s99zrwut6 | 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. | Burgers vector | [
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c_psf2k03y94vi | 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)|).} | Bernstein's theorem (polynomials) | [
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c_25ube1e2slk2 | 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}} . | Multi-variable function | [
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c_1e9o61ruheje | 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. | Recrystallization temperature | [
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c_ssgc38iexixj | 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. | Z-order curve | [
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c_8i8ndhegcru4 | In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. 2 {\displaystyle {\sqrt {2}}} is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below. | Complete metric space | [
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c_12nx9otvsb4i | 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. | Corrosion fatigue | [
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c_erkw62454c48 | In mathematical analysis, a modulus of continuity is a function ω: → used to measure quantitatively the uniform continuity of functions. So, a function f: I → R admits ω as a modulus of continuity if and only if | f ( x ) − f ( y ) | ≤ ω ( | x − y | ) , {\displaystyle |f(x)-f(y)|\leq \omega (|x-y|),} for all x and y in the domain of f. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families. For instance, the modulus ω(t) := kt describes the k-Lipschitz functions, the moduli ω(t) := ktα describe the Hölder continuity, the modulus ω(t) := kt(|log t|+1) describes the almost Lipschitz class, and so on. | Modulus of continuity | [
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c_fk6eaeg6453h | 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. | Cold pressing | [
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c_bht1sely95f9 | Many non-brittle metals with a reasonably high melting point as well as several polymers can be strengthened in this fashion. Alloys not amenable to heat treatment, including low-carbon steel, are often work-hardened. Some materials cannot be work-hardened at low temperatures, such as indium, however others can be strengthened only via work hardening, such as pure copper and aluminum. | Cold pressing | [
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