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A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called
Singular_solution
Solution to the Einstein field equations
Schwarzschild solution with 0 ≤ r < rs, which contains the singularity at r = 0, is completely separated from the outer patch by the singularity at r = rs
Schwarzschild_metric
Differential equation containing derivatives with respect to only one variable
conditions'. A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution. In the
Ordinary differential equation
Ordinary_differential_equation
Point on a curve not given by a smooth embedding of a parameter
geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends
Singular_point_of_a_curve
Square matrix without an inverse
However, in case of singular matrix the result is often infinitely many solutions. In mechanical and robotic systems, singular Jacobian matrices indicate
Singular_matrix
Point where a mathematical object behaves irregularly
(mathematics) Hyperbolic growth Movable singularity Pathological (mathematics) Regular singularity Singular solution "Singularities, Zeros, and Poles". mathfaculty
Singularity_(mathematics)
x^{4}{\frac {d^{2}\psi }{dx^{2}}}=e^{-\psi }} The equation has a singular solution given by e − ψ s = 2 x 2 , or − ψ s = 2 ln x + ln 2 {\displaystyle
Emden–Chandrasekhar_equation
Matrix decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a scaling, followed
Singular_value_decomposition
Condition in which spacetime itself breaks down
In theoretical physics, a gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted
Gravitational_singularity
General relativity model near spacetime singularities
of the universe near the initial gravitational singularity, described by an anisotropic, chaotic solution of the Einstein field equation of gravitation
BKL_singularity
Type of ordinary differential equation
one solution y ( x ) {\displaystyle y(x)} , the so-called singular solution, whose graph is the envelope of the graphs of the general solutions. The
Clairaut's_equation
Ordinary differential equation
named after the mathematician George Chrystal, who discussed the singular solution of this equation in 1896. The equation reads as ( d y d x ) 2 + A
Chrystal's_equation
Mathematical theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold
Singularity_theory
Gender-neutral English pronoun
Singular they is an epicene (gender-neutral) third-person pronoun in English. It typically occurs with an indeterminate antecedent, to refer to an unknown
Singular_they
discovered a solution to the Einstein field equations containing singularities, which would later become known as the Schwarzschild solution. This solution was
History_of_black_hole_physics
Concept in differential equation mathematics
a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where
Regular_singular_point
Compact astronomical body
independently gave the same solution. At a certain radius from the centre of the mass, the Schwarzschild solution became singular, meaning that some of the
Black_hole
Exact solution for the Einstein field equations
be a black hole and will instead be a naked singularity. Although the Kerr solution appears to be singular at the roots of Δ = 0 {\displaystyle \Delta
Kerr_metric
2026 scientific priority controversy
Thomas Hou presented numerical evidence that solutions to Euler equations could reach a blowup singularity in finite time, tilting public opinion about
Navier–Stokes priority controversy
Navier–Stokes_priority_controversy
Concept in mathematics
satisfied; in algebraic equations, the possible number of solutions is decreased. Singular perturbation theory is a rich and ongoing area of exploration
Singular_perturbation
French mathematician (1869–1951)
the given system in such a way that any singular solution of the original system becomes a general solution of the new system. Although Cartan showed
Élie_Cartan
2017 video game
collapsing infrastructure. They did not want the player to solve singular-solution puzzles, such as simply finding a key for a locked door, but instead
Prey_(2017_video_game)
Riemann–Roch theorem Riemann surface Road curve Sato–Tate conjecture secant Singular solution Sinuosity Slope Space curve Spinode Square wheel Subtangent Tacnode
List_of_curves_topics
Time period of seeming infinite density just after the Big Bang
be singularity in the past. Lev Landau formalized the question in 1959. Vladimir Belinski, Isaak Khalatnikov, and Evgeny Lifshitz studied solution of
Initial_singularity
Solution of Einstein field equations
"electrovacuum" solution of Einstein's equations. The solution contains a singularity in the shape of a ring. The multipole structure of the solution suggests
Kerr–Newman_metric
Airflow above and below the speed of sound
Hideo Yoshihara, along with some input from Busemann, later used a singular solution of Tricomi's equations to analytically solve the behavior of transonic
Transonic
American mathematician
particularly singularity formation and the long-term behavior of solutions to the incompressible Euler equations. Elgindi established finite-time singularity formation
Tarek_Elgindi
Method in numerical analysis
curves cross is a singular point. In general solution components Γ {\displaystyle \Gamma } are branched curves. The branch points are singular points. Finding
Numerical_continuation
the curves are singular solutions. A solution graph that contains a singular solution curve becomes indeterminate beyond the singular point. Multiple
Monge_equation
Key results in general relativity on gravitational singularities
relativity". A singularity in solutions of the Einstein field equations is one of three things: Spacelike singularities: The singularity lies in the future
Penrose–Hawking singularity theorems
Penrose–Hawking_singularity_theorems
Phenomenon within general relativity
curvature singularity at the Cauchy horizon known as the mass-inflation singularity, the Cauchy horizon singularity, the infalling singularity, or the "fat
Mass_inflation
that if α ∈ R n {\displaystyle \alpha \in \mathbb {R} ^{n}} is a non-singular solution of the system f 1 ( x 1 , … , x n , e x 1 , … , e x n ) = … = f n
Tarski's exponential function problem
Tarski's_exponential_function_problem
Hypothetical particle with one magnetic pole
monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the
Magnetic_monopole
To find the minimal surface with a given boundary
1 {\displaystyle k=n-1} , singularities occur only for n ≥ 8 {\displaystyle n\geq 8} . An example of such singular solution of the Plateau problem is
Plateau's_problem
the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition, also known as convexity
Legendre–Clebsch_condition
Methods of mathematical approximation
comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature
Perturbation_theory
Four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations
From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed
Gravitational_instanton
Partial differential equation
Brendle and Richard Schoen. Following the possibility that the singularities of solutions of the Ricci flow could identify the topological data predicted
Ricci_flow
Hypothetical phenomenon
In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. When there exists at least one causal
Naked_singularity
the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS)
Singular_boundary_method
Solution of a confluent hypergeometric equation
with a regular singular point at z = 0 and an irregular singular point at z = ∞. It has two (usually) linearly independent solutions M(a, b, z) and U(a
Confluent hypergeometric function
Confluent_hypergeometric_function
Millennium Prize Problem
OpenAI claimed to have found a solution to the Navier–Stokes equations with a smooth external force that develops a singularity in finite time in three-dimensional
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Russian mathematician (born 1966)
Massachusetts: Int. Press: 275–379. Hamilton, Richard S. (1999). "Non-singular solutions of the Ricci flow on three-manifolds". Comm. Anal. Geom. 7 (4): 695–729
Grigori_Perelman
Conjecture in physics
structure of gravitational singularities in the context of general relativity. Singularities that arise in the solutions of Einstein's equations are
Cosmic_censorship_hypothesis
Line of fortifications in Malta
for the defence of Malta in the latter half of the 19th century. A singular solution which exploited the defensive advantages of geography and technology
Victoria_Lines
Mathematics award
understanding of singular solutions to the fundamental equations of fluids dynamics, including their construction of smooth self-similar solutions for the compressible
Clay_Research_Award
Matrix consisting of linearly independent solutions to a linear differential equation
{\displaystyle t} , any solution to Ψ ˙ ( t ) = A ( t ) Ψ ( t ) {\displaystyle {\dot {\Psi }}(t)=A(t)\Psi (t)} which is a non-singular matrix for any single
Fundamental matrix (linear differential equation)
Fundamental_matrix_(linear_differential_equation)
Exact solution in general relativity
In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations that corresponds to the gravitational
Reissner–Nordström_metric
Major unsolved problem in transcendental number theory
states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomials; this is, non-obviously, a consequence
Schanuel's_conjecture
Black hole which possesses angular momentum
Black hole bomb Black hole spin parameter Black hole spin-flip BKL singularity – solution representing interior geometry of black holes formed by gravitational
Rotating_black_hole
Theorem in geometric topology
eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar soliton solution, which looked like a strand sticking
Poincaré_conjecture
Hypothetical object of spacetime
general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy, matter, light
White_hole
Most widely known generalized inverse of a matrix
"best fit" (least squares) approximate solution to a system of linear equations that lacks an exact solution (see below under § Applications). Another
Moore–Penrose_inverse
Cameroonian philosopher (1934–2018)
furthers the debate by assessment and analysis without offering any singular solution, resulting in added discourse. Boulaga worked as an academic in various
Fabien_Eboussi_Boulaga
Large threat to the health and safety of many people
humanitarian crisis (or sometimes humanitarian disaster) is defined as a singular event or a series of events that are threatening in terms of health, safety
Humanitarian_crisis
series solutions. A non-ordinary point is called a singularity. At a singularity the maximal number of linearly independent power series solutions may be
Fuchsian_theory
1007/s10958-005-0350-5. S2CID 122382003. Bryson, Arthur E. Jr.; Ho, Yu-Chi (1969). "Singular Solutions of Optimization and Control Problems". Applied Optimal Control. Waltham:
Singular_control
American mathematician (b. 1949)
Fefferman, C.; Majda, A. J. (1996), "Geometric constraints on potentially singular solutions for the 3-D Euler equations", Communications in Partial Differential
Charles_Fefferman
Generalization of the hypergeometric differential equation
three regular singular points, 0, 1 and ∞ {\displaystyle \infty } . That equation admits two linearly independent solutions; near a singularity z s {\displaystyle
Riemann's differential equation
Riemann's_differential_equation
theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in
Movable_singularity
Concept in algebraic geometry
trivial solutions, such as taking X′ to be the subvariety of non-singular points of X. More generally, it is often useful to resolve the singularities of a
Resolution_of_singularities
Theodoros I. (2011-10-26). "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Recurrence plot Limit cycle Initial value problem Clairaut's equation Singular solution Poincaré–Bendixson theorem Riccati equations Functional differential
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it
Ancient_solution
American mathematician (1943–2024)
continuing work to uncover the geometric and analytic properties of singularities of the Ricci flow equation and related systems of differential equations
Richard_S._Hamilton
Computer vision algorithm
{e} \|=1} . The solution is to choose e {\displaystyle \mathbf {e} } as the left singular vector corresponding to the smallest singular value of Y {\displaystyle
Eight-point_algorithm
Indian physicist (1915–1997)
directions until one encounters a singularity or U(x,y,z) vanishes. It was later shown by Hartle and Hawking that these solutions can be "glued" together to
Sudhansu_Datta_Majumdar
2014 video game
being "so obtuse that you'll be spending quite a while divining the singular solution, or so easy that the concept feels completely wasted", but concluded
Oddworld:_New_'n'_Tasty!
Generalized function whose value is zero everywhere except at zero
hyperbolic and so may have more singular solutions. As a result, the approximate delta functions that arise as fundamental solutions of the associated Cauchy
Dirac_delta_function
completely arbitrary. In practice, regarding endgame methods for computing singular solutions using homotopy continuation, the target time being 0 can significantly
Numerical_algebraic_geometry
Concept from linear programming
basis the matrix A B {\displaystyle A_{B}} is non-singular, so the constraint has a unique solution: x B = A B − 1 ⋅ b {\displaystyle \mathbf {x_{B}}
Basic_feasible_solution
Hence, both limits exist and x = 0 is a regular singular point. Therefore, we assume the solution takes the form y = ∑ r = 0 ∞ a r x r + c {\displaystyle
Frobenius solution to the hypergeometric equation
Frobenius_solution_to_the_hypergeometric_equation
Partial differential equation with nonlinear terms
mathematics. The basic questions about singularities (their formation, propagation, and removal, and regularity of solutions) are the same as for linear PDE
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Method for approximating eigenvalues
exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all
Rayleigh–Ritz_method
singularities theorem (also called the Duistermaat–Hörmander theorem) is theorem which characterizes the wavefront set of the distributional solution
Propagation of singularities theorem
Propagation_of_singularities_theorem
Topic in systems theory
In systems theory, a singularity refers to a critical condition in which relatively small changes, perturbations, or events may produce disproportionately
Singularity_(systems_theory)
English gardener, and botanist (1630–1715)
2016) p.33. Noted by R Stungo, "Recording the Aloes at Chelsea. A Singular Solution to a Difficult Problem" Notes and Records of the Royal Society of
Mary Somerset, Duchess of Beaufort (gardener)
Mary_Somerset,_Duchess_of_Beaufort_(gardener)
British mathematician (1869–1950)
work on this topic but later, advised by Scott, she began to work on singular solutions of differential equations. Although she had earned the equivalent
Isabel_Maddison
Optical phenomenon
phenomena is known as singular optics. The concept of "optical vortices" was first described by Coullet et al. in 1989, based on solutions of the Maxwell-Bloch
Optical_vortex
Solutions of Legendre's differential equation
x−(n+3)+(n+1)(n+2)(n+3)(n+4)2⋅4(2n+3)(2n+5)x−(n+5)+⋯) This solution is necessarily singular when x=±1. The Legendre functions of the second kind can also
Legendre_function
American business consultancy company
Singularity Education Group (using the public names Singularity Group, Singularity University or SingularityU) is an American company that offers executive
Singularity_Group
Dimensionless astrophysics equation
the origin the numerical methods fail due to the singularity of the equation. In general, solutions are found by numerical integration. Many standard
Lane–Emden_equation
Physics problem related to laws of motion and gravity
two-body problem, the three-body problem has no general closed-form analytic solution. The differential equations that govern the motions of three gravitating
Three-body_problem
of Algeciras, Cádiz, in whose port is located the type locality of this singular species." Another species, found in Playa del Carmen, Mexico (where Paco
List of organisms named after famous people (born 1925–1949)
List_of_organisms_named_after_famous_people_(born_1925–1949)
Function for Heun's differential equation
\beta ,\gamma ,\delta ;z)} is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun function
Heun_function
Method of solving differential equations
reduces short-wavelength error) by a global correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem
Multigrid_method
{1+bt}}\left(1-{2V \over {K}}\right)\end{aligned}}} Solving for the singular solution V ∗ {\displaystyle V^{*}} , we find that: V ∗ = K 2 [ 1 − δ a ( 1
Optimal_rotation_age
Regularization technique for ill-posed problems
I} , this least-squares solution can be analyzed in a special way using the singular-value decomposition. Given the singular value decomposition A = U
Ridge_regression
Metric for black holes in general relativity
In general relativity, the de Sitter–Schwarzschild solution describes a black hole in a causal patch of de Sitter space. It is the positive-curvature case
De Sitter–Schwarzschild metric
De_Sitter–Schwarzschild_metric
Model of the early universe
Sunbeam Mixmaster, a brand of Sunbeam Products electric kitchen mixer) is a solution to Einstein field equations of general relativity studied by Charles Misner
Mixmaster_universe
Function defined by a hypergeometric series
It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can
Hypergeometric_function
Einstein field equation solution
named after Mael Melvin who rediscovered it in 1964. The Melvin magnetic solution is used in astrophysical models as a background for more complicated spacetimes
Melvin_metric
Application of game theory to evolving populations in biology
fitness landscape. (See hawk dove graph above as an example of this.) A singular solution: often several ESS conditions can exist in a competitive situation
Evolutionary_game_theory
Radius of the event horizon of a Schwarzschild black hole
mass M (see Schwarzschild metric). The solution contained terms of the form: 11−rsr. These terms have singularities at r=0 and r=rs. The parameter rs is
Schwarzschild_radius
German-American physicist
two-component fields with mass and charge, the prediction of particlelike singular solutions in nonlinear field theory, extension of the formalism of covariant
Herbert_Jehle
Physical theorem
conjecture is a theorem about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4. The theorem was proven
Painlevé_conjecture
Equations with an unknown function under an integral sign
used are all proper integrals. Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper
Integral_equation
Curve external to a family of curves in geometry
study of ordinary differential equations (ODEs), and in particular singular solutions of ODEs. Consider, for example, the one-parameter family of tangent
Envelope_(mathematics)
Type of unphysical field in quantum field theory which provides mathematical consistency
the requirement that quantum field theories yield unambiguous, non-singular solutions. This is not possible in the path integral formulation when a gauge
Faddeev–Popov_ghost
Matrix with a multiplicative inverse
In linear algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is
Invertible_matrix
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SINGULAR SOLUTION
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SINGULAR SOLUTION
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