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Modes of vibration in mathematics
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
Dirichlet_eigenvalue
Theorem in Riemannian geometry
geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami
Cheng's eigenvalue comparison theorem
Cheng's_eigenvalue_comparison_theorem
Mathematical measure of a function's variability
tools for obtaining extremal solutions. Dirichlet's principle – Concept in potential theory Dirichlet eigenvalue – Modes of vibration in mathematics Total
Dirichlet_energy
Spectral Geometry Phenomenon
G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eigenvalue of the Laplace operator on a bounded domain in R n {\displaystyle
Rayleigh–Faber–Krahn inequality
Rayleigh–Faber–Krahn_inequality
theory) Dirichlet eigenvalue Dirichlet's ellipsoidal problem Dirichlet eta function (number theory) Dirichlet form Dirichlet function (topology) Dirichlet hyperbola
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Mathematical problem in spectral theory
domain D in the plane. Denote by λn the Dirichlet eigenvalues for D: that is, the eigenvalues of the Dirichlet problem for the Laplacian: { Δ u + λ u =
Hearing_the_shape_of_a_drum
Number, approximately 3.14
variational form of the Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any
Pi
Theorem in analysis
about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on metric balls in Euclidean space: the first Dirichlet eigenvalue of the Laplace−Beltrami
Wirtinger's inequality for functions
Wirtinger's_inequality_for_functions
Description in spectral theory
proved that the number, N ( λ ) {\displaystyle N(\lambda )} , of Dirichlet eigenvalues (counting their multiplicities) less than or equal to λ {\displaystyle
Weyl_law
used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Mathematical method
conditions to boundary conditions independent of the eigenvalue parameter – one of the Dirichlet, Neumann or Robin conditions. On the other hand, it also
Darboux_transformation
of Dirichlet, Neumann, and Periodic boundary conditions using Kronecker sums of discrete 1D Laplacians. The code also provides the exact eigenvalues and
Kronecker sum of discrete Laplacians
Kronecker_sum_of_discrete_Laplacians
Conjecture on zeros of the zeta function
this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation ( 1 − 2 2 s ) ζ ( s ) = η ( s ) = ∑
Riemann_hypothesis
Construct for Hermitian matrices
principle Min-max theorem Rayleigh's quotient in vibrations analysis Dirichlet eigenvalue Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and
Rayleigh_quotient
Type of vector space in math
itself? The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental
Hilbert_space
Mathematical functions and constants
So we consider only the first n of these values to be the n eigenvalues of the Dirichlet - Neumann problem. λ k = − 4 h 2 sin 2 ( π ( k − 0.5 ) 2 n
Eigenvalues and eigenvectors of the second derivative
Eigenvalues_and_eigenvectors_of_the_second_derivative
Canadian mathematician
first "Steklov eigenvalue" of a compact Riemannian manifold-with-boundary. This is defined as the minimal nonzero eigenvalue of the "Dirichlet to Neumann"
Ailana_Fraser
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains. Green's theorem for a bounded region Ω in
Neumann–Poincaré_operator
Field in mathematics
space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator. This question
Spectral_geometry
Second-order partial differential equation
relative to the new coordinates and Γ denotes its Christoffel symbols. The Dirichlet problem for Laplace's equation consists of finding a solution φ on some
Laplace's_equation
the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns
Stochastic processes and boundary value problems
Stochastic_processes_and_boundary_value_problems
Summability method in physics
sometimes be understood as eigenvalues of the heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging
Zeta_function_regularization
Differential calculus on function spaces
Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area
Calculus_of_variations
Shimura (1973). It has the property that the eigenvalue of a Hecke operator Tn2 on F is equal to the eigenvalue of Tn on f. Let f {\displaystyle f} be a holomorphic
Shimura_correspondence
Chinese-American mathematician (born 1949)
usually Dirichlet or Neumann conditions. Paul Yang and Yau showed that in the case of a closed two-dimensional manifold, the first eigenvalue is bounded
Shing-Tung_Yau
Spectrum of eigenvalues
have different Dirac spectra. Can you hear the shape of a drum? Dirichlet eigenvalue Spectral asymmetry Angle-resolved photoemission spectroscopy Bär
Dirac_spectrum
Differential operator
compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are
Eta_invariant
Square (0,1) matrix
if j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving
Redheffer_matrix
It states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization
List_of_Chinese_discoveries
Infinite series summing alternating 1 and -1 terms
limits of the Dirichlet, Fejér, and Poisson kernels, respectively. Multiplying the terms of Grandi's series by 1/nz yields the Dirichlet series η ( z )
Grandi's_series
Class of ordinary differential equations
Such values λ {\displaystyle \lambda } are called the eigenvalues of the problem. For each eigenvalue λ {\displaystyle \lambda } , to find the corresponding
Sturm–Liouville_theory
symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues Convergent matrix — square matrix whose successive powers approach the
List of numerical analysis topics
List_of_numerical_analysis_topics
Mathematical operation
homogeneous Dirichlet boundary conditions (i.e., v ( 0 ) = v ( L ) = 0 {\displaystyle v(0)=v(L)=0} where v is the eigenvector), the eigenvalues are λ j =
Second_derivative
{\displaystyle \exp({\text{length of }}p)} (equivalently, the square of the bigger eigenvalue of p). For any hyperbolic surface of finite area there is an associated
Selberg_zeta_function
Extended objects found in string theory
theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after
D-brane
Russian-American mathematician
1090/S0002-9947-99-02186-8 with Thomas Kappeler: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM journal on mathematical analysis
Boris_Mityagin
Generalization of beta distribution
distribution of the largest eigenvalue is well approximated by a transform of the Tracy–Widom distribution. Matrix variate Dirichlet distribution (Potters &
Matrix variate beta distribution
Matrix_variate_beta_distribution
Visual representation used in non-linear control system analysis
the coefficients of the right hand side written in matrix form using eigenvalues λ, given by the determinant: det ( [ A B C D ] − λ I ) = 0 {\displaystyle
Phase_plane
Swedish mathematician
contributed to the development of the theory of Bergman spaces and spaces of Dirichlet series. After 2010, Hedenmalm became interested in complex normal random
Hakan_Hedenmalm
dimension of, and is tangent to, the eigenspace with an associated eigenvalue (or eigenvalue pair) that has the smallest real part in magnitude. This generalizes
Slow_manifold
Hong Kong mathematician
differential geometry and partial differential equations, including Cheng's eigenvalue comparison theorem, Cheng's maximal diameter theorem, and a number of
Shiu-Yuen_Cheng
Fundamental solution to the heat equation, given boundary values
domain, consider the Dirichlet problem in a connected domain (or manifold with boundary) U. Let λn be the eigenvalues for the Dirichlet problem of the Laplacian
Heat_kernel
Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced
Minakshisundaram–Pleijel zeta function
Minakshisundaram–Pleijel_zeta_function
Continuous-time linear system with only negative real parts
time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real
Exponential_stability
equation in the domain (0 < x < L) for boundary conditions of type 1 (Dirichlet) at both boundaries x = 0 and x = L. Here X denotes the Cartesian coordinate
Green's_function_number
Identity obeyed by many special functions related to the gamma function
gamma functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg formula. Formally similar duplication
Multiplication_theorem
Method of solution to differential equations
0 or Poisson's equation ∇2φ(x) = −ρ(x), subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for φ(x) everywhere
Green's_function
Linear operator acting on modular forms
possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p
Hecke_operator
Mathematical algorithms
algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball (see Rayleigh–Faber–Krahn inequality
Symmetrization_methods
thought of as the steady state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a
Elliptic boundary value problem
Elliptic_boundary_value_problem
Mathematical function often applied to matrices
differentiable functions u , v {\displaystyle u,v} satisfying homogeneous Dirichlet conditions u ( 0 ) = u ( 1 ) = 0 {\displaystyle u(0)=u(1)=0} , with the
Logarithmic_norm
Axiomatic definition of a class of L-functions
axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties
Selberg_class
System of complete and orthogonal polynomials
of Sturm–Liouville theory. We rewrite the differential equation as an eigenvalue problem, d d x ( ( 1 − x 2 ) d d x P ( x ) ) = − λ P ( x ) , {\displaystyle
Legendre_polynomials
Israeli mathematician
Roditty-Gershon, Edva; Rudnick, Zeév (2013). "Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory". Geometric
Zeev_Rudnick
Differential operator in mathematics
same existence and uniqueness theory is applicable, and semi-elliptic Dirichlet problems can be solved using the methods of stochastic analysis. A second-order
Semi-elliptic_operator
{\left(\mathbf {r} \right)}=E\psi {\left(\mathbf {r} \right)},} which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation
List of quantum-mechanical systems with analytical solutions
List_of_quantum-mechanical_systems_with_analytical_solutions
Unsolved problem in mathematics
L-function, nowadays called the Ramanujan L-function. It can be defined as a Dirichlet series for Ramanujan tau function: L ( s , τ ) = ∑ n = 1 ∞ τ ( n ) n s
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
American mathematician (born 1946)
Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math
Joel_Spruck
Type of Dirichlet series associated to number field extensions
In mathematics, Artin L-functions are a type of Dirichlet series defined for finite extensions of number fields, encoding informations about linear representations
Artin_L-function
Function in discrete mathematics
linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Various researchers have proposed different choices
Discrete_Fourier_transform
Differential operator in mathematics
Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms. For spaces with additional structure
Laplace_operator
Yury A.; Vinogradov, Sergey S. (5 January 2009). "Regularization of the Dirichlet Problem for Laplace's Equation: Surfaces of Revolution". Electromagnetics
Analytical_regularization
Italian mathematician
of the points effect. She is also known for her contributions to the Dirichlet problem for pluriharmonic functions on the unit sphere of C n . {\displaystyle
Maria_Adelaide_Sneider
Technique for solving differential equations
only on x and the left hand side only on t. Thus: and −λ here is the eigenvalue for both differential operators, and T(t) and X(x) are corresponding eigenfunctions
Separation_of_variables
Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient
Dimensional_reduction
German mathematician (1804–1851)
Unabridged (Online). n.d. Koenigsberger 1904. Pierpont 1906, pp. 261–262. Dirichlet 1855, pp. 193–217. James 2002, pp. 69–74. "Carl Jacobi - Biography". Maths
Carl_Gustav_Jacob_Jacobi
Plot of a dynamical system's trajectories in phase space
phase portrait behavior of a system of ODEs can be determined by the eigenvalues or the trace and determinant (trace = λ1 + λ2, determinant = λ1λ2) of
Phase_portrait
Type of differential equation
there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. The theory of elliptic, parabolic
Partial_differential_equation
Mathematical conjecture
Ali Erhan Özlük, proved the pair correlation conjecture for some of Dirichlet's L-functions (A. E. Ozluk (1982)). The connection with random unitary
Montgomery's pair correlation conjecture
Montgomery's_pair_correlation_conjecture
Matrix of partial derivatives of a vector-valued function
Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Singular perturbation problem dealing with confinement of Brownian particles
( x ) {\displaystyle p_{\varepsilon }(x,0)=\rho _{0}(x)\,} and mixed Dirichlet–Neumann boundary conditions ( t > 0 {\displaystyle t>0} ) p ε ( x , t
Narrow_escape_problem
Japanese mathematician (1930–2019)
Mathematical Society. ISBN 978-0-8218-3573-9. Shimura, Goro (2007). Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Hardcover ed
Goro_Shimura
Type of graph in mathematics and physics
single graph edge and the eigenvalues are n 2 π 2 L e 2 {\displaystyle {\frac {n^{2}\pi ^{2}}{L_{e}^{2}}}} . The Dirichlet conditions don't allow interaction
Quantum_graph
Matrix used in finite element analysis
\nabla \varphi _{j}\,dx\right)u_{j}.} as a consequence of the homogenous Dirichlet boundary conditions. The stiffness matrix is the n-element square matrix
Stiffness_matrix
; Spruck, J. (1985), "The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian" (PDF), Acta
Hessian_equation
Armenian scientist and mathematician (1958–2018)
Karapetyan G.A., Convergence of Galerkin approximations to the solution of the Dirichlet problem // DAN SSSR, vol. 264, No.2, 1982, pp. 291–294. Karapetyan G.A
Garnik_A._Karapetyan
American mathematician
Hirschman's 1955 research program. In 1964 Hirschman published Extreme eigenvalues of Toeplitz forms associated with Jacobi polynomials, showing that for
Isidore_Isaac_Hirschman_Jr.
Model-based clustering in statistics
components, G {\displaystyle G} , is infinite, using a Dirichlet process prior, yielding a Dirichlet process mixture model for clustering. An advantage of
Model-based_clustering
Equation used in quantum scattering problems
1/A denotes the inverse of A. However E − H0 is singular since E is an eigenvalue of H0. As is described below, this singularity is eliminated in two distinct
Lippmann–Schwinger_equation
Concept in mathematical physics
critical line 1 2 + i s {\textstyle {\frac {1}{2}}+is} ; also they use the Dirichlet generating function ζ ′ ( s ) ζ ( s ) = − ∑ n = 1 ∞ Λ ( n ) e − s l n
Wu–Sprung_potential
Periodic distribution ("function") of "point-mass" Dirac delta sampling
function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel: Ш T ( t ) = 1 T ∑ n = − ∞ ∞ e i 2 π n t / T . {\displaystyle
Dirac_comb
the analogous generalized eigenvalue problem for MAX and MIN matrices. They determined the MAX–MIN generalized eigenvalues for every n ≥ 1 {\displaystyle
GCD_matrix
Model for representing text documents
document frequency, latent semantic indexing, random projections and latent Dirichlet allocation. Weka. Weka is a popular data mining package for Java including
Vector_space_model
Semi-classical method for computing quantum eigenvalues
Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from
Einstein–Brillouin–Keller method
Einstein–Brillouin–Keller_method
Mathematical theorem
zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated
Selberg_trace_formula
Nonlinear second-order partial differential equation of special kind
x {\displaystyle x} if Q x ( ξ ) {\displaystyle Q_{x}(\xi )} if all eigenvalues are of the same sign, hyperbolic at x {\displaystyle x} if Q x ( ξ )
Monge–Ampère_equation
Linear operators with a common spectrum
spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectral
Isospectral
Special function in the physical sciences
}\!\cos \left({\dfrac {t^{3}}{3}}+xt\right)\,dt} , which converges by Dirichlet's test. The Airy equation y ″ − x y = 0 {\displaystyle y''-xy=0} has two
Airy_function
Norwegian mathematician (1917–2007)
discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series". Journal of the Indian Mathematical Society. New Series. 20 (1–3):
Atle_Selberg
Part of spectral theory
operator whenever λ is not an eigenvalue of D and hence that the Fredholm determinant det I − μ(D − λ)−1 is defined. The Dirichlet boundary conditions imply
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
Australian and American mathematician (born 1975)
matrices and their eigenvalues. Wigner studied the case of hermitian and symmetric matrices, proving a "semicircle law" for their eigenvalues. In 2010, Tao
Terence_Tao
of the solvability of the Dirichlet problem for the Laplacian to make useful choices for u. Applications include eigenvalue estimates in spectral geometry
Reilly_formula
Concept in mathematical analysis
given fixed volume, the ball has the smallest first eigenvalue for the Laplacian with Dirichlet boundary conditions. The proof goes by restating the
Pólya–Szegő_inequality
Branch of ordinary differential equations
( t ) {\displaystyle x(t)} are determined by the eigenvalues of R {\displaystyle R} . The eigenvalues of e T B {\displaystyle e^{TB}} are called the characteristic
Floquet_theory
Mathematical concept
parameter. The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then gets extra factors
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Model from mathematical physics
\Phi } is canonical. If we take a simple quantum Hamiltonian H to have eigenvalues proportional to log p, that is, H | p ⟩ = E p | p ⟩ {\displaystyle H|p\rangle
Primon_gas
Type of differential equation
even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues in any vertical strip of the complex plane. This
Delay_differential_equation
Linear operator equal to its own adjoint
with eigenvalues − i β {\displaystyle -i\beta } , and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary
Self-adjoint_operator
Theorem in combinatorics on words
f^{\omega }(b)} , and α > 1 {\displaystyle \alpha >1} is the dominant eigenvalue of the matrix of morphism f {\displaystyle f} , namely, the matrix M (
Cobham's_theorem
a rank-one symmetric space Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE
Male
Slavic
(Влатко) Pet form of Slavic Vladimir, VLATKO means "peaceful ruler."
Girl/Female
Tamil
Camerina | காமேரீநா
Surname or Lastname
English
English : habitational name for someone from Debenham in Suffolk, probably named from the genitive case of the Old English river name DÄ“ope ‘deep one’ + Old English hÄm ‘homestead’.
Boy/Male
Tamil
Worship
Girl/Female
German, Hebrew, Swedish
Pledged to God; God's Promise; God is My Oath
Girl/Female
Hindu
Name of Lord Murugan, Goddess Saraswati (Goddess of education
Boy/Male
Arabic, Muslim, Sindhi
Praise; Loving
Surname or Lastname
English
English : habitational name from Gowlands in Moor Monkton, West Yorkshire.
Boy/Male
Indian, Punjabi, Sikh
Wealth of Virtues
Boy/Male
Arabic, Armenian, British, English, French, Gaelic, German, Hindu, Indian, Irish, Muslim, Romanian, Sanskrit
Fair; Handsome; Both a Diminutive of Albert; Noble; Rock; Comely; To Soothe; Bearer of the Light; Scorpion; The Zodiac Sign Scorpio
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE
DIRICHLET EIGENVALUE