In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, they appear in electrostatics and continuum mechanics. Elliptic regularity implies. Steady-state solutions to hyperbolic and parabolic equations solve elliptic equations. A linear differential operator L of order m on a domain Ω in Rn given by L u = ∑ | α | ≤ m a α ∂ α u is called elliptic if for every x in Ω and every non-zero ξ in Rn, ∑ | α | = m a α ξ α ≠ 0, where ξ α = ξ 1 α 1 ⋯ ξ n α n. In many applications, this condition is not strong enough, instead a uniform ellipticity condition may be imposed for operators of degree m = 2k: k ∑ | α | = 2 k a α ξ α > C | ξ | 2 k, where C is a positive constant.
Note that ellipticity only depends on the highest-order terms. A nonlinear operator L = F is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator. Example 1 The negative of the Laplacian in Rd given by − Δ u = − ∑ i = 1 d ∂ i 2 u is a uniformly elliptic operator; the Laplace operator occurs in electrostatics. If ρ is the charge density within some region Ω, the potential Φ must satisfy the equation − Δ Φ = 4 π ρ. Example 2 Given a matrix-valued function A, symmetric and positive definite for every x, having components aij, the operator L u = − ∂ i + b j ∂ j u + c u is elliptic; this is the most general form of a second-order divergence form linear elliptic differential operator. The Laplace operator is obtained by taking A = I; these operators occur in electrostatics in polarized media. Example 3 For p a non-negative number, the p-Laplacian is a nonlinear elliptic operator defined by L = − ∑ i = 1 d ∂ i. A similar nonlinear operator occurs in glacier mechanics.
The Cauchy stress tensor of ice, according to Glen's flow law, is given by τ i j = B − 1
Maxima and minima
In mathematical analysis, the maxima and minima of a function, known collectively as extrema, are the largest and smallest value of the function, either within a given range or on the entire domain of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, for finding the maxima and minima of functions; as defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no maximum. A real-valued function f defined on a domain X has a global maximum point at x∗ if f ≥ f for all x in X. Similarly, the function has a global minimum point at x∗ if f ≤ f for all x in X; the value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function. Symbolically, this can be written as follows: x 0 ∈ X is a global maximum point of function f: X → R if f ≥ f.
For global minimum point. If the domain X is a metric space f is said to have a local maximum point at the point x∗ if there exists some ε > 0 such that f ≥ f for all x in X within distance ε of x∗. The function has a local minimum point at x∗ if f ≤ f for all x in X within distance ε of x∗. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: Let be a metric space and function f: X → R. X 0 ∈ X is a local maximum point of function f if such that d X < ε ⟹ f ≥ f. For a local minimum point. In both the global and local cases, the concept of a strict extremum can be defined. For example, x∗ is a strict global maximum point if, for all x in X with x ≠ x∗, we have f > f, x∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x∗ with x ≠ x∗, we have f > f. Note that a point is a strict global maximum point if and only if it is the unique global maximum point, for minimum points.
A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function. Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum either must be a local maximum in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum is to look at all the local maxima in the interior, look at the maxima of the points on the boundary, take the largest one; the most important, yet quite obvious, feature of continuous real-valued functions of a real variable is that they decrease before local minima and increase afterwards for maxima. A direct consequence of this is the Fermat's theorem, which states that local extrema must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability.
For any function, defined piecewise, one finds a maximum by finding the maximum of each piece separately, seeing which one is largest. The function x2 has a unique global minimum at x = 0; the function x3 has maxima. Although the first derivative is 0 at x = 0, this is an inflection point; the function x. The function x−x has a unique global maximum over the positive real numbers at x = 1/e; the function x3/3 − x has first derivative x2 − second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign o
A capacitor is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit; the capacitor was known as a condenser or condensator. The original name is still used in many languages, but not in English; the physical form and construction of practical capacitors vary and many capacitor types are in common use. Most capacitors contain at least two electrical conductors in the form of metallic plates or surfaces separated by a dielectric medium. A conductor may be sintered bead of metal, or an electrolyte; the nonconducting dielectric acts to increase the capacitor's charge capacity. Materials used as dielectrics include glass, plastic film, mica and oxide layers. Capacitors are used as parts of electrical circuits in many common electrical devices. Unlike a resistor, an ideal capacitor does not dissipate energy.
When two conductors experience a potential difference, for example, when a capacitor is attached across a battery, an electric field develops across the dielectric, causing a net positive charge to collect on one plate and net negative charge to collect on the other plate. No current flows through the dielectric. However, there is a flow of charge through the source circuit. If the condition is maintained sufficiently long, the current through the source circuit ceases. If a time-varying voltage is applied across the leads of the capacitor, the source experiences an ongoing current due to the charging and discharging cycles of the capacitor. Capacitance is defined as the ratio of the electric charge on each conductor to the potential difference between them; the unit of capacitance in the International System of Units is the farad, defined as one coulomb per volt. Capacitance values of typical capacitors for use in general electronics range from about 1 picofarad to about 1 millifarad; the capacitance of a capacitor is proportional to the surface area of the plates and inversely related to the gap between them.
In practice, the dielectric between the plates passes a small amount of leakage current. It has an electric field strength limit, known as the breakdown voltage; the conductors and leads introduce an undesired resistance. Capacitors are used in electronic circuits for blocking direct current while allowing alternating current to pass. In analog filter networks, they smooth the output of power supplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems, they stabilize power flow; the property of energy storage in capacitors was exploited as dynamic memory in early digital computers. In October 1745, Ewald Georg von Kleist of Pomerania, found that charge could be stored by connecting a high-voltage electrostatic generator by a wire to a volume of water in a hand-held glass jar. Von Kleist's hand and the water acted as conductors, the jar as a dielectric. Von Kleist found that touching the wire resulted in a powerful spark, much more painful than that obtained from an electrostatic machine.
The following year, the Dutch physicist Pieter van Musschenbroek invented a similar capacitor, named the Leyden jar, after the University of Leiden where he worked. He was impressed by the power of the shock he received, writing, "I would not take a second shock for the kingdom of France."Daniel Gralath was the first to combine several jars in parallel to increase the charge storage capacity. Benjamin Franklin investigated the Leyden jar and came to the conclusion that the charge was stored on the glass, not in the water as others had assumed, he adopted the term "battery", subsequently applied to clusters of electrochemical cells. Leyden jars were made by coating the inside and outside of jars with metal foil, leaving a space at the mouth to prevent arcing between the foils; the earliest unit of capacitance was the jar, equivalent to about 1.11 nanofarads. Leyden jars or more powerful devices employing flat glass plates alternating with foil conductors were used up until about 1900, when the invention of wireless created a demand for standard capacitors, the steady move to higher frequencies required capacitors with lower inductance.
More compact construction methods began to be used, such as a flexible dielectric sheet sandwiched between sheets of metal foil, rolled or folded into a small package. Early capacitors were known as condensers, a term, still used today in high power applications, such as automotive systems; the term was first used for this purpose by Alessandro Volta in 1782, with reference to the device's ability to store a higher density of electric charge than was possible with an isolated conductor. The term became deprecated because of the ambiguous meaning of steam condenser, with capacitor becoming the recommended term from 1926. Since the beginning of the study of electricity non conductive materials like glass, porcelain and mica have been used as insulators; these materials some decades were well-suited for further use as the dielectric for the first capacitors. Paper capacitors made by sandwiching a strip of impregnated paper between strips of metal, rolling the result into a cylinder were used in the late 19th century.
In classical mechanics, the gravitational potential at a location is equal to the work per unit mass that would be needed to move the object from a fixed reference location to the location of the object. It is analogous to the electric potential with mass playing the role of charge; the reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance. In mathematics, the gravitational potential is known as the Newtonian potential and is fundamental in the study of potential theory, it may be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies. The gravitational potential at a location is the gravitational potential energy at that location per unit mass: V = U m, where m is the mass of the object. Potential energy is equal to the work done by the gravitational field moving a body to its given position in space from infinity. If the body has a mass of 1 unit the potential energy to be assigned to that body is equal to the gravitational potential.
So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity. In some situations, the equations can be simplified by assuming a field, nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant. In that case, the difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height: Δ U ≈ m g Δ h; the potential V of a unit mass m at a distance x from a point mass of mass M can be defined as the work W that needs to be done by an external agent to bring the unit mass in from infinity to that point: V = W m = 1 m ∫ ∞ x F ⋅ d x = 1 m ∫ ∞ x G m M x 2 d x = − G M x, where G is the gravitational constant, F is the gravitational force. The potential has units of energy per e.g. J/kg in the MKS system. By convention, it is always negative where it is defined, as x tends to infinity, it approaches zero.
The gravitational field, thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object; because the potential has no angular components, its gradient is a = − G M x 3 x = − G M x 2 x ^, where x is a vector of length x pointing from the point mass toward the small body and x ^ is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law: | a | = G M x 2; the potential associated with a mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, if the point masses are located at the points x1... xn and have masses m1... mn the potential of the distribution at the point x is V = ∑ i = 1 n − G m i | x − x i |. If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R3 the potential is the convolution of −G/|r| with dm.
In good cases this equals the integral V = − ∫ R 3 G | x − r | d m, where |x − r| is the distance between the points x and r. If there is a function ρ representing the density of the distribution at r, so that dm= ρdv, where dv is the Euclidean volume element the gravitational potential is the volume integral V = − ∫ R 3 G | x −
In probability theory and related fields, a stochastic or random process is a mathematical object defined as a collection of random variables. The random variables were associated with or indexed by a set of numbers viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are used as mathematical models of systems and phenomena that appear to vary in a random manner, they have applications in many disciplines including sciences such as biology, ecology and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science and telecommunications. Furthermore random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes.
Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time; these two stochastic processes are considered the most important and central in the theory of stochastic processes, were discovered and independently, both before and after Bachelier and Erlang, in different settings and countries. The term random function is used to refer to a stochastic or random process, because a stochastic process can be interpreted as a random element in a function space; the terms stochastic process and random process are used interchangeably with no specific mathematical space for the set that indexes the random variables. But these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space the collection of random variables is called a random field instead.
The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, branching processes; the study of stochastic processes uses mathematical knowledge and techniques from probability, linear algebra, set theory, topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications. A stochastic or random process can be defined as a collection of random variables, indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.
The set used to index. The index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time; each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real n - dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, a single outcome of a stochastic process is called, among other names, a sample function or realization. A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the state space; when interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers the stochastic process is said to be in discrete time.
If the index set is some interval of the real line time is said to be continuous. The two types of stochastic processes are referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge due to the index set being uncountable. If the index set is the integers, or some subset of them the stochastic process can be called a random sequence. If the state space is the integers or natural numbers the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is n -dimensional Euclidean space the stochastic process is called a n -dimensional vector process or n -vector process; the word stochastic in English was used as an adjective with the definition "pertaining to conjecturing", stemming from a Greek word meaning "to aim at a mark, guess", the Oxford English Dictionary gives the year 16
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions simplifies the study of heat transfer. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis refers to the study of both operations.
The decomposition process. Its output, the Fourier transform, is given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, the general field is known as harmonic analysis; each transform used for analysis has a corresponding inverse transform that can be used for synthesis. Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, signal processing, digital image processing, probability theory, forensics, option pricing, numerical analysis, oceanography, optics, geometry, protein structure analysis, other areas; this wide applicability stems from many useful properties of the transforms: The transforms are linear operators and, with proper normalization, are unitary as well. The transforms are invertible; the exponential functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones.
Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency independently. By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers; the discrete version of the Fourier transform can be evaluated on computers using Fast Fourier Transform algorithms. In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum; the FT method is used to record the wavelength data. And by using a computer, these Fourier calculations are carried out, so that in a matter of seconds, a computer-operated FT-IR instrument can produce an infrared absorption pattern comparable to that of a prism instrument. Fourier transformation is useful as a compact representation of a signal.
For example, JPEG compression uses a variant of the Fourier transformation of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, weak components are eliminated so that the remaining components can be stored compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are inverse-transformed to produce an approximation of the original image; when processing signals, such as audio, radio waves, light waves, seismic waves, images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, reversing the transformation; some examples include: Equalization of audio recordings with a series of bandpass filters. Most the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, it produces a continuous function of frequency, known as a frequency distribution.
One function is transformed into another, the operation is reversible. When the domain of the input function is time, the domain of the output function is ordinary frequency, the transform of function s at frequen
In mathematics convolution is a mathematical operation on two functions to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to the process of computing it; some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f or g is reflected about the y-axis. For continuous functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, computer vision, natural language processing and signal processing and differential equations; the convolution can be defined for functions on Euclidean space, other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, in the design and implementation of finite impulse response filters in signal processing.
Computing the inverse of the convolution operation is known as deconvolution. The convolution of f and g is written f ∗ g, using an star, it is defined as the integral of the product of the two functions after one is shifted. As such, it is a particular kind of integral transform: An equivalent definition is: ≜ ∫ − ∞ ∞ f g d τ. While the symbol t is used above, it need not represent the time domain, but in that context, the convolution formula can be described as a weighted average of the function f at the moment t where the weighting is given by g shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function. For functions f, g supported on only [0, ∞), the integration limits can be truncated, resulting in: = ∫ 0 t f g d τ for f, g: [ 0, ∞ ) → R. For the multi-dimensional formulation of convolution, see domain of definition. A common engineering convention is: f ∗ g ≜ ∫ − ∞ ∞ f g d τ ⏟, which has to be interpreted to avoid confusion. For instance, f ∗ g is equivalent to.
Convolution describes the output of an important class of operations known as linear time-invariant. See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created; the existing ones are only modified. In other words, the output transform is the pointwise product of the input transform with a third transform. See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. An expression of the type: ∫ f ⋅ g d u is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.
Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, others. The term itself did not come into wide use until the 60s. Prior to that it was sometimes known as Faltung, composition product, superposition integral, Carson's integral, yet it appears as early as 1903. The o