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Lions–Lax–Milgram Theorem
In mathematics, the Lions–Lax–Milgram theorem (or simply Lions's theorem) is a result in functional analysis with applications in the study of partial differential equations. It is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear function can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Jacques-Louis Lions, Peter Lax and Arthur Milgram. Statement of the theorem Let ''H'' be a Hilbert space and ''V'' a normed space. Let ''B'' : ''H'' × ''V'' → R be a continuous function, continuous, bilinear function. Then the following are equivalent: * (coercive function, coercivity) for some constant ''c'' > 0, ::\inf_ \sup_ , B(h, v) , \geq c; * (existence of a "weak inverse") for each continuous linear functional ''f'' ∈ ''V''∗, there is an element ''h'' & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coercive Function
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use. Coercive vector fields A vector field ''f'' : R''n'' → R''n'' is called coercive if :\frac \to + \infty \mbox \, x \, \to + \infty, where "\cdot" denotes the usual dot product and \, x\, denotes the usual Euclidean norm of the vector ''x''. A coercive vector field is in particular norm-coercive since \, f(x)\, \geq (f(x) \cdot x) / \, x \, for x \in \mathbb^n \setminus \ , by Cauchy–Schwarz inequality. However a norm-coercive mapping ''f'' : R''n'' → R''n'' is not necessarily a coercive vector field. For instance the rotation ''f'' : R''2'' → R''2'', ''f(x) = (-x2, x1)'' by 90° is a norm-coercive mapping which fails to be a coercive vector field since f(x) \cdot x = 0 for every x \in \mathbb^2. Coercive operators and forms A self-adjoint operator A:H\to H, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Babuška–Lax–Milgram Theorem
In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška, Peter Lax and Arthur Milgram. Background In the modern, functional-analytic approach to the study of partial differential equations, one does not attempt to solve a given partial differential equation directly, but by using the structure of the vector space of possible solutions, e.g. a Sobolev space ''W'' ''k'',''p''. Abstractly, consider two real normed spaces ''U'' and ''V'' with their continuous dual spaces ''U''∗ and ''V''∗ respectively. In many applications, ''U'' is the space of possible solutions; given some partial differential operator Λ : ''U'' → ''V''∗ and a specified element ''f'' ∈ ''V''∗, the objecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iceberg
An iceberg is a piece of freshwater ice more than 15 m long that has broken off a glacier or an ice shelf and is floating freely in open (salt) water. Smaller chunks of floating glacially-derived ice are called "growlers" or "bergy bits". The sinking of the ''Titanic'' in 1912 led to the formation of the International Ice Patrol in 1914. Much of an iceberg is below the surface, which led to the expression "tip of the iceberg" to illustrate a small part of a larger unseen issue. Icebergs are considered a serious maritime hazard. Icebergs vary considerably in size and shape. Icebergs that calve from glaciers in Greenland are often irregularly shaped while Antarctic ice shelves often produce large tabular (table top) icebergs. The largest iceberg in recent history (2000), named B-15, measured nearly 300 km × 40 km. The largest iceberg on record was an Antarctic tabular iceberg of over [] sighted west of Scott Island, in the South Pacific Ocean, by the USS Glacier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evaporate
Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidity affects rate of evaporation of water. When the molecules of the liquid collide, they transfer energy to each other based on how they collide. When a molecule near the surface absorbs enough energy to overcome the vapor pressure, it will escape and enter the surrounding air as a gas. When evaporation occurs, the energy removed from the vaporized liquid will reduce the temperature of the liquid, resulting in evaporative cooling. On average, only a fraction of the molecules in a liquid have enough heat energy to escape from the liquid. The evaporation will continue until an equilibrium is reached when the evaporation of the liquid is equal to its condensation. In an enclosed environment, a liquid will evaporate until the surrounding air is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Ice Cap
A polar ice cap or polar cap is a high-latitude region of a planet, dwarf planet, or natural satellite that is covered in ice. There are no requirements with respect to size or composition for a body of ice to be termed a polar ice cap, nor any geological requirement for it to be over land, but only that it must be a body of solid phase matter in the polar region. This causes the term "polar ice cap" to be something of a misnomer, as the term ice cap itself is applied more narrowly to bodies that are over land, and cover less than 50,000 km2: larger bodies are referred to as ice sheets. The composition of the ice will vary. For example, Earth's polar caps are mainly water ice, whereas Mars's polar ice caps are a mixture of solid carbon dioxide and water ice. Polar ice caps form because high-latitude regions receive less energy in the form of solar radiation from the Sun than equatorial regions, resulting in lower surface temperatures. Earth's polar caps have changed d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galerkin Approximation
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used: * Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions.A. Ern, J.L. Guermond, ''Theory and practice of finite elements'', Springer, 2004, * Bubnov–Galerkin method (after Ivan Bubnov) does not require the bilinear fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based a work on special relativity on two postulates: * The laws of physics are invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that densi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuously Embedded
In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are continuous embedding theorems. Definition Let ''X'' and ''Y'' be two normed vector spaces, with norms , , ·, , ''X'' and , , ·, , ''Y'' respectively, such that ''X'' ⊆ ''Y''. If the inclusion map (identity function) :i : X \hookrightarrow Y : x \mapsto x is continuous, i.e. if there exists a constant ''C'' > 0 such that :\, x \, _Y \leq C \, x \, _X for every ''x'' in ''X'', then ''X'' is said to be continuously embedded in ''Y''. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "''X'' ↪ ''Y''" means "''X'' and ''Y'' are normed spaces with ''X'' continuously embedded in ''Y''". This is a consist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Linear Functional
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous linear operators Characterizations of continuity Suppose that F : X \to Y is a linear operator between two topological vector spaces (TVSs). The following are equivalent: F is continuous. F is continuous at some point x \in X. F is continuous at the origin in X. if Y is locally convex then this list may be extended to include: for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that q \circ F \leq p. if X and Y are both Hausdorff locally convex spaces then this list may be extended to include: F is weakly continuous and its transpose ^t F : Y^ \to X^ maps equicontinuous subsets of Y^ to equicontinuous subsets of X^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
