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design management | In market and brand focused companies, design management focuses mainly on brand design management, including corporate brand management and product brand management. Focusing on the brand as the core for design decisions results in a strong focus on the brand experience, customer touch points, reliability, recognition... | wikipedia |
design management | Corporate design management implements, develops, and maintains the corporate identity, or brand. This type of brand management is strongly anchored in the organization to control and influence corporate design activities. The design program plays the role of a quality program within many fields of the organization to ... | wikipedia |
design management | It is strongly linked to strategy, corporate culture, product development, marketing, organizational structure, and technological development. Achieving a consistent corporate brand requires the involvement of designers and a widespread design awareness among employees. A creative culture, knowledge sharing processes, ... | wikipedia |
design management | Product brand design managementThe main focus of product brand management lies on the single product or product family. Product design management is linked to research and development, marketing, and brand management, and is present in the fast-moving consumer goods (FMCG) industry. It is responsible for the visual exp... | wikipedia |
marketing engineering | In marketing engineering methods and models can be classified in several categories: | wikipedia |
observational techniques | In marketing research, the most frequently used types of observational techniques are: Personal observation observing products in use to detect usage patterns and problems observing license plates in store parking lots determining the socio-economic status of shoppers determining the level of package scrutiny determini... | wikipedia |
product model | In marketing, a product is an object, or system, or service made available for consumer use as of the consumer demand; it is anything that can be offered to a market to satisfy the desire or need of a customer. In retailing, products are often referred to as merchandise, and in manufacturing, products are bought as raw... | wikipedia |
product model | In project management, products are the formal definition of the project deliverables that make up or contribute to delivering the objectives of the project. A related concept is that of a sub-product, a secondary but useful result of a production process. Dangerous products, particularly physical ones, that cause inju... | wikipedia |
brand development | In marketing, brand management begins with an analysis on how a brand is currently perceived in the market, proceeds to planning how the brand should be perceived if it is to achieve its objectives and continues with ensuring that the brand is perceived as planned and secures its objectives. Developing a good relations... | wikipedia |
linear discriminant analysis | In marketing, discriminant analysis was once often used to determine the factors which distinguish different types of customers and/or products on the basis of surveys or other forms of collected data. Logistic regression or other methods are now more commonly used. The use of discriminant analysis in marketing can be ... | wikipedia |
linear discriminant analysis | Survey questions ask the respondent to rate a product from one to five (or 1 to 7, or 1 to 10) on a range of attributes chosen by the researcher. Anywhere from five to twenty attributes are chosen. They could include things like: ease of use, weight, accuracy, durability, colourfulness, price, or size. | wikipedia |
linear discriminant analysis | The attributes chosen will vary depending on the product being studied. The same question is asked about all the products in the study. The data for multiple products is codified and input into a statistical program such as R, SPSS or SAS. | wikipedia |
linear discriminant analysis | (This step is the same as in Factor analysis). Estimate the Discriminant Function Coefficients and determine the statistical significance and validity—Choose the appropriate discriminant analysis method. The direct method involves estimating the discriminant function so that all the predictors are assessed simultaneous... | wikipedia |
linear discriminant analysis | The stepwise method enters the predictors sequentially. The two-group method should be used when the dependent variable has two categories or states. The multiple discriminant method is used when the dependent variable has three or more categorical states. | wikipedia |
linear discriminant analysis | Use Wilks's Lambda to test for significance in SPSS or F stat in SAS. The most common method used to test validity is to split the sample into an estimation or analysis sample, and a validation or holdout sample. The estimation sample is used in constructing the discriminant function. | wikipedia |
linear discriminant analysis | The validation sample is used to construct a classification matrix which contains the number of correctly classified and incorrectly classified cases. The percentage of correctly classified cases is called the hit ratio. Plot the results on a two dimensional map, define the dimensions, and interpret the results. | wikipedia |
linear discriminant analysis | The statistical program (or a related module) will map the results. The map will plot each product (usually in two-dimensional space). | wikipedia |
linear discriminant analysis | The distance of products to each other indicate either how different they are. The dimensions must be labelled by the researcher. This requires subjective judgement and is often very challenging. See perceptual mapping. | wikipedia |
bundled software | In marketing, product bundling is offering several products or services for sale as one combined product or service package. It is a common feature in many imperfectly competitive product and service markets. Industries engaged in the practice include telecommunications services, financial services, health care, inform... | wikipedia |
bundled software | A software bundle might include a word processor, spreadsheet, and presentation program into a single office suite. The cable television industry often bundles many TV and movie channels into a single tier or package. The fast food industry combines separate food items into a "meal deal" or "value meal". | wikipedia |
bundled software | A bundle of products may be called a package deal; in recorded music or video games, a compilation or box set; or in publishing, an anthology. Most firms are multi-product or multi-service companies faced with the decision whether to sell products or services separately at individual prices or whether combinations of p... | wikipedia |
émery topology | In martingale theory, Émery topology is a topology on the space of semimartingales. The topology is used in financial mathematics. The class of stochastic integrals with general predictable integrands coincides with the closure of the set of all simple integrals.The topology was introduced in 1979 by the french mathema... | wikipedia |
computer media | In mass communication, digital media is any communication media that operate in conjunction with various encoded machine-readable data formats. Digital content can be created, viewed, distributed, modified, listened to, and preserved on a digital electronics device, including digital data storage media (in contrast to ... | wikipedia |
proteinogenic amino acids | In mass spectrometry of peptides and proteins, knowledge of the masses of the residues is useful. The mass of the peptide or protein is the sum of the residue masses plus the mass of water (Monoisotopic mass = 18.01056 Da; average mass = 18.0153 Da). The residue masses are calculated from the tabulated chemical formula... | wikipedia |
direct analysis in real time | In mass spectrometry, direct analysis in real time (DART) is an ion source that produces electronically or vibronically excited-state species from gases such as helium, argon, or nitrogen that ionize atmospheric molecules or dopant molecules. The ions generated from atmospheric or dopant molecules undergo ion-molecule ... | wikipedia |
direct analysis in real time | This ionization can occur for species desorbed directly from surfaces such as bank notes, tablets, bodily fluids (blood, saliva and urine), polymers, glass, plant leaves, fruits & vegetables, clothing, and living organisms. DART is applied for rapid analysis of a wide variety of samples at atmospheric pressure and in t... | wikipedia |
fragmentation pattern | In mass spectrometry, fragmentation is the dissociation of energetically unstable molecular ions formed from passing the molecules mass spectrum. These reactions are well documented over the decades and fragmentation patterns are useful to determine the molar weight and structural information of unknown molecules. Frag... | wikipedia |
single-cell analysis | In mass spectroscopy based proteomics there are three major steps needed for peptide identification: sample preparation, separation of peptides, and identification of peptides. Several groups have focused on oocytes or very early cleavage-stage cells since these cells are unusually large and provide enough material for... | wikipedia |
single-cell analysis | The sensitivity and consistency of these methods have been further improved by prioritization, and massively parallel sample preparation in nanoliter size droplets. Another direction for single-cell protein analysis is based on a scalable framework of multiplexed data-independent acquisition (plexDIA) enables time savi... | wikipedia |
single-cell analysis | This step gives order to the peptides before quantification using tandem mass-spectroscopy (MS/MS). The major difference between quantification methods is some use labels on the peptides such as tandem mass tags (TMT) or dimethyl labels which are used to identify which cell a certain protein came from (proteins coming ... | wikipedia |
golden record (informatics) | In master data management (MDM), the golden copy refers to the master data (master version) of the reference data which works as an authoritative source for the "truth" for all applications in a given IT landscape. | wikipedia |
glutaraldehyde | In material science glutaraldehyde application areas range from polymers to metals and biomaterials. Glutaraldehyde is commonly used as fixing agent before characterization of biomaterials for microscopy. Glutaraldehyde is a powerful crosslinking agent for many polymers containing primary amine groups.. Glutaraldehdye ... | wikipedia |
resilience (materials science) | In material science, resilience is the ability of a material to absorb energy when it is deformed elastically, and release that energy upon unloading. Proof resilience is defined as the maximum energy that can be absorbed up to the elastic limit, without creating a permanent distortion. The modulus of resilience is def... | wikipedia |
resilience (materials science) | It can be calculated by integrating the stress–strain curve from zero to the elastic limit. In uniaxial tension, under the assumptions of linear elasticity, U r = σ y 2 2 E = σ y ε y 2 {\displaystyle U_{r}={\frac {\sigma _{y}^{2}}{2E}}={\frac {\sigma _{y}\varepsilon _{y}}{2}}} where Ur is the modulus of resilience, σy ... | wikipedia |
abc analysis | In materials management, ABC analysis is an inventory categorisation technique. ABC analysis divides an inventory into three categories—"A items" with very tight control and accurate records, "B items" with less tightly controlled and good records, and "C items" with the simplest controls possible and minimal records. ... | wikipedia |
abc analysis | Thus, the inventory is grouped into three categories (A, B, and C) in order of their estimated importance. 'A' items are very important for an organization. | wikipedia |
abc analysis | Because of the high value of these 'A' items, frequent value analysis is required. In addition to that, an organization needs to choose an appropriate order pattern (e.g. 'just-in-time') to avoid excess capacity. 'B' items are important, but of course less important than 'A' items and more important than 'C' items. The... | wikipedia |
functionally graded material | In materials science Functionally Graded Materials (FGMs) may be characterized by the variation in composition and structure gradually over volume, resulting in corresponding changes in the properties of the material. The materials can be designed for specific function and applications. Various approaches based on the ... | wikipedia |
yield (engineering) | In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied stress is removed. Once the yie... | wikipedia |
yield (engineering) | The yield strength is often used to determine the maximum allowable load in a mechanical component, since it represents the upper limit to forces that can be applied without producing permanent deformation. In some materials, such as aluminium, there is a gradual onset of non-linear behavior, and no precise yield point... | wikipedia |
yield (engineering) | Yielding is a gradual failure mode which is normally not catastrophic, unlike ultimate failure. In solid mechanics, the yield point can be specified in terms of the three-dimensional principal stresses ( σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} ) with a yield surface or a yield criterion. A v... | wikipedia |
functionally graded element | In materials science and mathematics, functionally graded elements are elements used in finite element analysis. They can be used to describe a functionally graded material. | wikipedia |
work hardened | 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 ... | wikipedia |
work hardened | 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. | wikipedia |
work hardened | 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, ... | wikipedia |
sandwich structured composite | In materials science, a sandwich-structured composite is a special class of composite materials that is fabricated by attaching two thin-but-stiff skins to a lightweight but thick core. The core material is normally low strength, but its higher thickness provides the sandwich composite with high bending stiffness with ... | wikipedia |
sandwich structured composite | Sometimes, the honeycomb structure is filled with other foams for added strength. Open- and closed-cell metal foam can also be used as core materials. Laminates of glass or carbon fiber-reinforced thermoplastics or mainly thermoset polymers (unsaturated polyesters, epoxies...) are widely used as skin materials. Sheet m... | wikipedia |
advanced composite materials (engineering) | 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 m... | wikipedia |
advanced composite materials (engineering) | 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 com... | wikipedia |
advanced composite materials (engineering) | 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 valu... | wikipedia |
advanced composite materials (engineering) | Even more specifically ACMs are very attractive for aircraft and aerospace structural parts. ACMs have been developing for NASA's Advanced Space Transportation Program, armor protection for Army aviation and the Federal Aviation Administration of the USA, and high-temperature shafting for the Comanche helicopter. Addit... | wikipedia |
intrinsic properties | 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. Extri... | wikipedia |
asperity (material science) | In materials science, asperity, defined as "unevenness of surface, roughness, ruggedness" (from the Latin asper—"rough"), has implications (for example) in physics and seismology. Smooth surfaces, even those polished to a mirror finish, are not truly smooth on a microscopic scale. They are rough, with sharp, rough or r... | wikipedia |
asperity (material science) | The fractal dimension of these structures has been correlated with the contact mechanics exhibited at an interface in terms of friction and contact stiffness. When two macroscopically smooth surfaces come into contact, initially they only touch at a few of these asperity points. These cover only a very small portion of... | wikipedia |
asperity (material science) | Friction and wear originate at these points, and thus understanding their behavior becomes important when studying materials in contact. When the surfaces are subjected to a compressive load, the asperities deform through elastic and plastic modes, increasing the contact area between the two surfaces until the contact ... | wikipedia |
dispersion (materials science) | 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 s... | wikipedia |
lamellar structure | In materials science, lamellar structures or microstructures are composed of fine, alternating layers of different materials in the form of lamellae. They are often observed in cases where a phase transition front moves quickly, leaving behind two solid products, as in rapid cooling of eutectic (such as solder) or eute... | wikipedia |
lamellar structure | A deeper eutectic or more rapid cooling will result in finer lamellae; as the size of an individual lamellum approaches zero, the system will instead retain its high-temperature structure. Two common cases of this include cooling a liquid to form an amorphous solid, and cooling eutectoid austenite to form martensite. I... | wikipedia |
material failure theory | In materials science, material failure is the loss of load carrying capacity of a material unit. This definition introduces to the fact that material failure can be examined in different scales, from microscopic, to macroscopic. In structural problems, where the structural response may be beyond the initiation of nonli... | wikipedia |
permeance | In materials science, permeance is the degree to which a material transmits another substance. | wikipedia |
quenching | In materials science, quenching is the rapid cooling of a workpiece in water, oil, polymer, air, or other fluids to obtain certain material properties. A type of heat treating, quenching prevents undesired low-temperature processes, such as phase transformations, from occurring. It does this by reducing the window of t... | wikipedia |
quenching | In steel alloyed with metals such as nickel and manganese, the eutectoid temperature becomes much lower, but the kinetic barriers to phase transformation remain the same. This allows quenching to start at a lower temperature, making the process much easier. High-speed steel also has added tungsten, which serves to rais... | wikipedia |
slip (materials science) | In materials science, slip is the large displacement of one part of a crystal relative to another part along crystallographic planes and directions. Slip occurs by the passage of dislocations on close/packed planes, which are planes containing the greatest number of atoms per area and in close-packed directions (most a... | wikipedia |
slip (materials science) | A slip system describes the set of symmetrically identical slip planes and associated family of slip directions for which dislocation motion can easily occur and lead to plastic deformation. The magnitude and direction of slip are represented by the Burgers vector, b. An external force makes parts of the crystal lattic... | wikipedia |
zener–hollomon parameter | In materials science, the Zener–Hollomon parameter, typically denoted as Z, is used to relate changes in temperature or strain-rate to the stress-strain behavior of a material. It has been most extensively applied to the forming of steels at increased temperature, when creep is active. It is given by Z = ε ˙ exp ( Q ... | wikipedia |
zener–hollomon parameter | It is named after Clarence Zener and John Herbert Hollomon, Jr. who established the formula based on the stress-strain behavior in steel. | wikipedia |
zener–hollomon parameter | When plastically deforming a material, the flow stress depends heavily on both the strain-rate and temperature. During forming processes, Z may help determine appropriate changes in strain-rate or temperature when the other variable is altered, in order to keep material flowing properly. Z has also been applied to some... | wikipedia |
threshold displacement energy | In materials science, the threshold displacement energy (Td) is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. It is also known as "displacement threshold energy" or just "displacement energy". In a crystal, a separate threshold di... | wikipedia |
threshold displacement energy | Then one should distinguish between the minimum (Td,min) and average (Td,ave) over all lattice directions' threshold displacement energies. In amorphous solids, it may be possible to define an effective displacement energy to describe some other average quantity of interest. Threshold displacement energies in typical s... | wikipedia |
formative evaluation | In math education, it is important for teachers to see how their students approach the problems and how much mathematical knowledge and at what level students use when solving the problems. That is, knowing how students think in the process of learning or problem solving makes it possible for teachers to help their stu... | wikipedia |
sigma field | In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair ( X , Σ ) {\displaystyle (X,\Sigma )} is called a measurable space. The σ-algebras are a subset of... | wikipedia |
sigma field | Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic, particularly when the statistic is a function or a random process and the notion of conditional density is not applicabl... | wikipedia |
sigma field | In general, a finite algebra is always a σ-algebra. If { A 1 , A 2 , A 3 , … } , {\displaystyle \{A_{1},A_{2},A_{3},\ldots \},} is a countable partition of X {\displaystyle X} then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra. A more useful example is the set of subsets... | wikipedia |
multi-variable function | In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real value... | wikipedia |
bernstein's theorem (polynomials) | In mathematical analysis, Bernstein's inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative. Taking the k-th derivative of the theorem, max | z | ≤ 1 ( | P ( k ) ( z ) | ) ≤... | wikipedia |
clairaut's equation | In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form y ( x ) = x d y d x + f ( d y d x ) {\displaystyle y(x)=x{\frac {dy}{dx}}+f\left({\frac {dy}{dx}}\right)} where f {\displaystyle f} is continuously differentiable. It is a particular case of the Lagrange diff... | wikipedia |
ivar ekeland | In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist a nearly optimal solution to a class of optimization problems.Ekeland's variational principle can be used when the lower level set of a minimization problem is not compact, so that the Bolza... | wikipedia |
fubini's theorem | In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by ... | wikipedia |
fubini's theorem | Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. A related theorem is often called Fubini's theorem for infinite series, which states that if { a m , n } m = 1 , n = 1 ∞ {\textstyle \{a_{m,n}\}_{m=1,n=1... | wikipedia |
glaeser's continuity theorem | In mathematical analysis, Glaeser's continuity theorem is a characterization of the continuity of the derivative of the square roots of functions of class C 2 {\displaystyle C^{2}} . It was introduced in 1963 by Georges Glaeser, and was later simplified by Jean Dieudonné.The theorem states: Let f: U → R 0 + {\displayst... | wikipedia |
haar's tauberian theorem | In mathematical analysis, Haar's Tauberian theorem named after Alfréd Haar, relates the asymptotic behaviour of a continuous function to properties of its Laplace transform. It is related to the integral formulation of the Hardy–Littlewood Tauberian theorem. | wikipedia |
leonid kantorovich | In mathematical analysis, Kantorovich had important results in functional analysis, approximation theory, and operator theory. In particular, Kantorovich formulated some fundamental results in the theory of normed vector lattices, especially in Dedekind complete vector lattices called "K-spaces" which are now referred ... | wikipedia |
lipschitz function | In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this ... | wikipedia |
netto's theorem | In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after Eugen Netto.The case for maps from a higher-dimensional manifold to a one-dimens... | wikipedia |
netto's theorem | The faults were later recognized and corrected.An important special case of this theorem concerns the non-existence of continuous bijections from one-dimensional spaces, such as the real line or unit interval, to two-dimensional spaces, such as the Euclidean plane or unit square. The conditions of the theorem can be re... | wikipedia |
netto's theorem | Examples include the Peano curve and Hilbert curve. Neither of these examples has any self-crossings, but by Netto's theorem there are many points of the square that are covered multiple times by these curves. Osgood curves are continuous bijections from one-dimensional spaces to subsets of the plane that have nonzero ... | wikipedia |
netto's theorem | They form Jordan curves in the plane. However, by Netto's theorem, they cannot cover the entire plane, unit square, or any other two-dimensional region. | wikipedia |
netto's theorem | If one relaxes the requirement of continuity, then all smooth manifolds of bounded dimension have equal cardinality, the cardinality of the continuum. Therefore, there exist discontinuous bijections between any two of them, as Georg Cantor showed in 1878. Cantor's result came as a surprise to many mathematicians and ki... | wikipedia |
netto's theorem | A near-bijection from the unit square to the unit interval can be obtained by interleaving the digits of the decimal representations of the Cartesian coordinates of points in the square. The ambiguities of decimal, exemplified by the two decimal representations of 1 = 0.999..., cause this to be an injection rather than... | wikipedia |
rademacher's theorem | In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f: U → Rm is Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentiable form a set of Lebesgue measure zero. Different... | wikipedia |
equality of mixed partials | In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) named after Alexis Clairaut and Hermann Schwarz, states that for a function f: Ω → R {\displaystyle f\colon \Omega \to \mathbb {R} } defined on a set Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} , if p ∈ R n {\d... | wikipedia |
equality of mixed partials | {\displaystyle f.} An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case). Let f ( x , y ) {\displaystyle f(x,y)} be a differentiable function on an open rectangle Ω {\displaystyle \Omega } conta... | wikipedia |
equality of mixed partials | {\displaystyle \Omega .} Define u ( h , k ) = f ( a + h , b + k ) − f ( a + h , b ) , v ( h , k ) = f ( a + h , b + k ) − f ( a , b + k ) , w ( h , k ) = f ( a + h , b + k ) − f ( a + h , b ) − f ( a , b + k ) + f ( a , b ) . {\displaystyle {\begin{aligned}u\left(h,\,k\right)&=f\left(a+h,\,b+k\right)-f\left(a+h,\,b\rig... | wikipedia |
equality of mixed partials | {\displaystyle {\begin{aligned}hk\,\partial _{y}\partial _{x}f\left(a+\theta h,\,b+\theta ^{\prime }k\right)&=hk\,\partial _{x}\partial _{y}f\left(a+\phi ^{\prime }h,\,b+\phi k\right),\\\partial _{y}\partial _{x}f\left(a+\theta h,\,b+\theta ^{\prime }k\right)&=\partial _{x}\partial _{y}f\left(a+\phi ^{\prime }h,\,b+\ph... | wikipedia |
equality of mixed partials | Indeed the difference operators Δ x t , Δ y t {\displaystyle \Delta _{x}^{t},\,\,\Delta _{y}^{t}} commute and Δ x t f , Δ y t f {\displaystyle \Delta _{x}^{t}f,\,\,\Delta _{y}^{t}f} tend to ∂ x f , ∂ y f {\displaystyle \partial _{x}f,\,\,\partial _{y}f} as t {\displaystyle t} tends to 0, with a similar statement for se... | wikipedia |
equality of mixed partials | By the fundamental theorem of calculus for C 1 {\displaystyle C^{1}} functions f {\displaystyle f} on an open interval I {\displaystyle I} with ( a , b ) ⊂ I {\displaystyle (a,b)\subset I} ∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) . {\displaystyle \int _{a}^{b}f^{\prime }(x)\,dx=f(b)-f(a).} Hence | f ( b ) − f ( a ) | ≤ ... | wikipedia |
equality of mixed partials | Recall that the elementary discussion on maxima or minima for real-valued functions implies that if f {\displaystyle f} is continuous on {\displaystyle } and differentiable on ( a , b ) {\displaystyle (a,b)} , then there is a point c {\displaystyle c} in ( a , b ) {\displaystyle (a,b)} such that f ( b ) − f ( a ) b − ... | wikipedia |
equality of mixed partials | {\displaystyle \left|\Delta _{1}^{t}\Delta _{2}^{t}f(x_{0},y_{0})-D_{1}D_{2}f(x_{0},y_{0})\right|\leq \sup _{0\leq s\leq 1}\left|\Delta _{1}^{t}D_{2}f(x_{0},y_{0}+ts)-D_{1}D_{2}f(x_{0},y_{0})\right|\leq \sup _{0\leq r,s\leq 1}\left|D_{1}D_{2}f(x_{0}+tr,y_{0}+ts)-D_{1}D_{2}f(x_{0},y_{0})\right|.} Thus Δ 1 t Δ 2 t f ( x ... | wikipedia |
equality of mixed partials | Hence, since the difference operators commute, so do the partial differential operators D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} , as claimed.Remark. By two applications of the classical mean value theorem, Δ 1 t Δ 2 t f ( x 0 , y 0 ) = D 1 D 2 f ( x 0 + t θ , y 0 + t θ ′ ) {\displaystyle \Delta _{1}^{t}... | wikipedia |
tannery's theorem | In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery. | wikipedia |
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