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Rayleigh–Faber–Krahn Inequality
In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eigenvalue of the Laplace operator on a bounded domain in \mathbb^n, n \ge 2. It states that the first Dirichlet eigenvalue is no less than the corresponding Dirichlet eigenvalue of a Euclidean ball having the same volume. Furthermore, the inequality is rigid in the sense that if the first Dirichlet eigenvalue is equal to that of the corresponding ball, then the domain must actually be a ball. In the case of n=2, the inequality essentially states that among all drums of equal area, the circular drum (uniquely) has the lowest voice. More generally, the Faber–Krahn inequality holds in any Riemannian manifold in which the isoperimetric inequality holds. In particular, according to Cartan–Hadamard conjecture, it should hold in all simply con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Geometry
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems. Inverse problems seek to identify features of the geometry from information about the eigenvalues of the Laplacian. One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator. This question is usually expressed as " Can one hear the shape of a drum?", the popula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Among many honors, he received the 1904 Nobel Prize in Physics "for his investigations of the densities of the most important gases and for his discovery of argon in connection with these studies." He served as president of the Royal Society from 1905 to 1908 and as chancellor of the University of Cambridge from 1908 to 1919. Rayleigh provided the first theoretical treatment of the elastic scattering of light by particles much smaller than the light's wavelength, a phenomenon now known as "Rayleigh scattering", which notably explains why the sky is blue. He studied and described transverse surface waves in solids, now known as "Rayleigh waves". He contributed extensively to fluid dynamics, with concepts such as the Rayleigh number (a dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgar Krahn
Edgar Krahn ( – 6 March 1961) was an Estonian mathematician. Krahn was born in Sootaga (now Laiuse, Jõgeva County), Governorate of Livonia, as a member of the Baltic German minority. He died in Rockville, Maryland, United States. Krahn studied at the University of Tartu and the University of Göttingen. He graduated at Tartu in 1918, received his doctoral degree at Göttingen in 1926, with Richard Courant as his advisor, and his habilitation took place at Tartu in 1928. He is co-author of the Rayleigh–Faber–Krahn inequality. Krahn worked in Estonia, Germany, the United Kingdom, and the United States in the following areas of pure and applied mathematics: * Differential geometry * Differential equations * ''Bausparmathematik'', which is distantly related to insurance mathematics * Probability theory * Gas dynamics * Elasticity theory In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its origi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Eigenvalue
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function ''u'' ≠ 0 and eigenvalue λ Here Δ is the Laplacian, which is given in ''xy''-coordinates by :\Delta u = \frac + \frac. The boundary value problem () is the Dirichlet problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator Δ appearing in () is often known as the Dirichlet Laplacian when it is considered as accepting only fu ... [...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] |
Rigidity (mathematics)
In mathematics, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c'' ∈ ''C'' is uniquely determined by less information about ''c'' than one would expect. The above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians. __FORCETOC__ Examples Some examples include: #Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. #Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem. #By the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. '' Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Hadamard Conjecture
In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926. Informally, the conjecture states that negative curvature allows regions with a given perimeter to hold more volume. This phenomenon manifests itself in nature through corrugations on coral reefs, or ripples on a petunia flower, which form some of the simplest examples of non-positively curved spaces. History The conjecture, in all dimensions, was first stated explicitly in 1976 by Thierry Aubin, and a few years later by Misha Gromov, Yuri Burago and Viktor Zalgaller. In dimension 2 this fact had already been established in 1926 by André Weil and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hearing The Shape Of A Drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" is the title of a 1966 article by Mark Kac in the '' American Mathematical Monthly'' which made the question famous, though this particular phrasing originates with Lipman Bers. Similar questions can be traced back all the way to physicist Arthur Schuster in 1882. For his paper, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968. The frequencies at which a drumhead can vibrate depend on its shape. The Helmholtz equation calculates the frequencies if the shape is known. These frequencies are the eigenvalues of the Laplacian in the space. A central question is whether the shape can be predicted if the frequencies are known; for example, whether a Reuleaux triangle can be recognized in this way. Kac admitted that he did not know whethe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Partial Differential Equations
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
