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Equation for function that computes iterated values
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form f ( h ( x ) ) = h ( x + 1 ) {\displaystyle f(h(x))=h(x+1)}
Abel_equation
Norwegian mathematician (1802–1829)
and Abel had studied all the latest mathematical literature in the university library. During that time, Abel started working on the quintic equation in
Niels_Henrik_Abel
Identity relating to differential equations
In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two
Abel's_identity
In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown
Abel equation of the first kind
Abel_equation_of_the_first_kind
Equations of degree 5 or higher cannot be solved by radicals
the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree
Abel–Ruffini_theorem
Equation whose unknown is a function
g(x+y)+g(x-y)=2[g(x)g(y)]} (d'Alembert's functional equation) f ( h ( x ) ) = h ( x + 1 ) {\displaystyle f(h(x))=h(x+1)} (Abel equation) f ( h ( x ) ) = c f ( x ) {\displaystyle
Functional_equation
Family of second-order differential equations
order differential equation: v d v d x + f ( x ) v + g ( x ) = 0 {\displaystyle v{dv \over dx}+f(x)v+g(x)=0} which is an Abel equation of the second kind
Liénard_equation
Equation for fixed point of functional composition
Schröder's equation to the older Abel equation, α(h(x)) = α(x) + 1. Similarly, the change of variables Ψ(x) = log(φ(x)) converts Schröder's equation to Böttcher's
Schröder's_equation
Polynomial equation, generally univariate
Tartaglia to equations of degree 3 and that of Lodovico Ferrari for equations of degree 4. Finally Niels Henrik Abel proved, in 1824, that equations of degree
Algebraic_equation
Curve for which the time to roll to the end is equal for all starting points
starting height, find an equation of the curve that yields this result. The tautochrone problem is a special case of Abel's mechanical problem when T
Tautochrone_curve
Result of repeatedly applying a mathematical function
this relation is called the translation functional equation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the nesting
Iterated_function
Polynomial equation of degree 3
(second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.) geometrically: using Omar Khayyam's
Cubic_equation
interpolation Abel–Plana formula Abel function Abel's integral equation Abel's identity Abel's inequality Abel's irreducibility theorem Abel–Jacobi map Abel–Jacobi
List of things named after Niels Henrik Abel
List_of_things_named_after_Niels_Henrik_Abel
Mathematical formula expressing equality
an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and
Equation
Functional equation characterizing associative binary operations
The associativity equation or associativity functional equation is the functional equation F ( F ( x , y ) , z ) = F ( x , F ( y , z ) ) {\displaystyle
Associativity_equation
Type of firearm propellant
Frederick Abel, worked to improve the properties of gunpowder during the late 19th century. This formed the basis for the Noble-Abel gas equation for internal
Gunpowder
of Schröder's equation, for his proof, Kneser had constructed the "superfunction" of the exponential map through the corresponding Abel function X {\displaystyle
Superfunction
Motion of particles in a fluid
Anosov flows are isomorphic to Bernoulli shifts. Abel equation Iterated function Schröder's equation Infinite compositions of analytic functions Irwin
Flow_(mathematics)
Norwegian international mathematics prize
The Abel Prize (/ˈɑːbəl/ AH-bəl; Norwegian: Abelprisen [ˈɑ̀ːbl̩ˌpriːsn̩]) is awarded annually by the King of Norway to one or more outstanding mathematicians
Abel_Prize
Function that, applied twice, gives another function
Wayback Machine.) Iterated function Function composition Abel equation Schröder's equation Flow (mathematics) Superfunction Fractional calculus Half-exponential
Functional_square_root
Expression in differential equations
mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution
Liouville's_formula
Type of differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives
Partial_differential_equation
Functional square root of an exponential
mathematical function Schröder's equation – Equation for fixed point of functional composition Abel equation – Equation for function that computes iterated
Half-exponential_function
Nonlinear second-order partial differential equation of special kind
geometry of surfaces. Luis Caffarelli earned the 2023 Abel Prize for his work on this equation. Given two independent variables x {\displaystyle x} and
Monge–Ampère_equation
17th-century conjecture proved by Andrew Wiles in 1994
Singh 2022, pp. 120–125. Singh 2022, p. 284. "The Abel Prize citation 2016". The Abel Prize. The Abel Prize Committee. March 2016. Archived from the original
Fermat's_Last_Theorem
Differential equation that is linear with respect to the unknown function
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written
Linear_differential_equation
Mathematical connection between field theory and group theory
roots, cube roots, etc)? The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot
Galois_theory
Type of functional equation (mathematics)
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions
Differential_equation
Argentine mathematician
differential equations and their applications. Caffarelli is a professor of mathematics at the University of Texas at Austin, and the winner of the 2023 Abel Prize
Luis_Caffarelli
North Carolina. Retrieved April 8, 2017. Stewart, A. D (1964). On the Abel equation in n-dimensions, n ≥ 2 (Ph.D. thesis). Austin, Tex.: University of Texas
List of African-American mathematicians
List_of_African-American_mathematicians
Polynomial function of degree 5
quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini
Quintic_function
Differential equation containing derivatives with respect to only one variable
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other
Ordinary differential equation
Ordinary_differential_equation
Polynomial function of degree 4
the highest degree such that every polynomial equation can be solved by radicals, according to the Abel–Ruffini theorem. Lodovico Ferrari is credited
Quartic_function
German mathematician (1804–1851)
fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Jacobi was born of Ashkenazi Jewish parentage
Carl_Gustav_Jacob_Jacobi
Study of polynomial equations
algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial
Theory_of_equations
American mathematician and Nobel Laureate (1928–2015)
Louis Nirenberg were awarded the Abel Prize for their contributions to the field of partial differential equations. As a graduate student in the Princeton
John_Forbes_Nash_Jr.
Polynomial equation of degree 4
mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is a x 4 + b x 3 +
Quartic_equation
Formulation of classical mechanics
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics
Hamilton–Jacobi_equation
Operator equation in the style of Fredholm theory
integral equation of the first kind can conveniently be transformed into a classical Abel integral equation. The Volterra integral equations were introduced
Volterra_integral_equation
Mathematical field
for instance formal solutions e ω {\displaystyle e_{\omega }} of the Abel equation e e ω ( x ) = e ω ( x + 1 ) {\displaystyle e^{e_{\omega }(x)}=e_{\omega
Transseries
British mathematician who proved Fermat's Last Theorem
known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Knight Commander
Andrew_Wiles
Dimensionless quantity in fluid dynamics
of the equation, and for practical purposes a root-finding algorithm must be used for a numerical solution (the equation is a septic equation in M2 and
Mach_number
Danish mathematician (1766–1825)
he proposed the equation x5 − 2x4 + 3x2 − 4x + 5 = 0. He ended the letter with the wish that .... the time and efforts that Mr. Abel in my eyes spends
Carl_Ferdinand_Degen
Differential equations involving stochastic processes
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution
Stochastic differential equation
Stochastic_differential_equation
Hardware description language and software
in 1983 by Data I/O Corporation, in Redmond, Washington. ABEL includes both concurrent equation and truth table logic formats as well as a sequential state
Advanced Boolean Expression Language
Advanced_Boolean_Expression_Language
Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)". International Journal of Mathematics
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
(1804–1851), a German mathematician. Abel–Jacobi theorem, a statement about the Jacobian variety of a curve Abel–Jacobi–Liouville identity Carathéodory–Jacobi–Lie
List of things named after Carl Gustav Jacob Jacobi
List_of_things_named_after_Carl_Gustav_Jacob_Jacobi
The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation: p n ( x ) = x ( x − a n ) n − 1 {\displaystyle
Abel_polynomials
Russian mathematician (born 1943)
doi:10.1007/BF02465190 Ilyashenko, Yu (2000). "Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions". Nonlinearity. 13 (4): 1337
Yulij_Ilyashenko
Class of ordinary differential equations
Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x ) y {\displaystyle
Sturm–Liouville_theory
Italian mathematician and philosopher (1765–1822)
chiefly for what is now known as the Abel–Ruffini theorem, Ruffini also made a major contribution to the theory of equations, developing the so-called theory
Paolo_Ruffini
Type of mathematical expression
degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel–Ruffini theorem asserts
Polynomial
Type of ordinary differential equation
In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form y ′ + P ( x ) y = Q ( x ) y n , {\displaystyle
Bernoulli differential equation
Bernoulli_differential_equation
Special function in the physical sciences
differential equation d 2 y d x 2 − x y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=0,} known as the Airy equation or the Stokes equation. Because the
Airy_function
Power series theorem in mathematics
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician
Abel's_theorem
Equations with an unknown function under an integral sign
integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be
Integral_equation
Branch of mathematics
methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them
Algebra
Partial differential equations with random force terms and coefficients
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary
Stochastic partial differential equation
Stochastic_partial_differential_equation
Infinite series with alternating signs
paradoxical equation: 1 − 2 + 3 − 4 + ⋯ = 1 4 . {\displaystyle 1-2+3-4+\cdots ={\frac {1}{4}}.} A rigorous explanation of this equation would not arrive
1_−_2_+_3_−_4_+_⋯
Analytic function in mathematics
a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime
Riemann_zeta_function
Mathematical question-answering engine
and Lev Alyshayev (CTO). At the onset, it could interpret a user-entered equation or symbolic problem and find the solution if it existed. Later, the ability
Symbolab
Mapping involving integration between function spaces
integral transform "maps" an equation from its original "domain" into another domain, in which manipulating and solving the equation may be much easier than
Integral_transform
Type of differential equation subject to a particular solution methodology
mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics
Exact_differential_equation
Approach to finding numerical solutions of ordinary differential equations
differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and
Euler_method
Mathematical formula involving a given set of operations
limiting their usefulness. In higher degrees, the Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals,
Closed-form_expression
Type of differential equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time
Delay_differential_equation
Type of ordinary differential equation
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written
Homogeneous differential equation
Homogeneous_differential_equation
Branch of mathematical analysis
Calculus and Fractional Differential Equations. New York: Wiley. pp. 1–2. ISBN 978-0-471-58884-9. Niels Henrik Abel (1823). "Oplösning af et Par Opgaver
Fractional_calculus
Branch of ordinary differential equations
branch of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form x ˙ = A ( t ) x , {\displaystyle
Floquet_theory
Solution in radicals of a polynomial equation
than the quadratic formula. The Abel–Ruffini theorem, and, more generally Galois theory, state that some quintic equations, such as x 5 − x + 1 = 0 , {\displaystyle
Solution_in_radicals
Family of solutions to related differential equations
(u-v)} for α > −1. Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions: A α (
Bessel_function
Determinant of the matrix of first derivatives of a set of functions
functions are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions themselves are
Wronskian
Concepts from linear algebra
certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that
Eigenvalues_and_eigenvectors
Canadian-American mathematician (1925–2020)
Caffarelli and Robert Kohn, for their article [CKN82] on the Navier–Stokes equations Abel Prize (2015) Nirenberg is especially known for his collaboration with
Louis_Nirenberg
Norwegian mathematician
had become interested in the work of Niels Henrik Abel, and especially in an unfinished work on equation theory that had been left behind. However, it was
Peter_Ludvig_Sylow
Method for solving continuous operator problems (such as differential equations)
methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear
Galerkin_method
Topics referred to by the same term
Abels may refer to: Abel's test, a mathematical test Abel's theorem, a mathematical theorem Abel's identity, a mathematical equation Abel's inequality
Abels
Technique for solving hyperbolic partial differential equations
partial differential equations. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODEs) along which
Method_of_characteristics
Numerical method for solving physical or engineering problems
method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas
Finite_element_method
Generalization of elliptic integrals
an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form ∫ z 0 z R ( x , w ) d x
Abelian_integral
German mathematician (born 1954)
proofs of the Mordell conjecture and several related conjectures. He won the Abel Prize in 2026 for these achievements. Gerd Faltings was born on 28 July 1954
Gerd_Faltings
Swedish mathematician
knew from the past work by Paolo Ruffini and Niels Henrik Abel that a general quintic equation can not be solved, this fact was not known to Bring, putting
Erland_Samuel_Bring
Number with a real and an imaginary part
imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} . Since no real number satisfies the above equation, i was called an imaginary
Complex_number
Type of ordinary differential equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form y ( x ) = x d y d x + f ( d y d x ) {\displaystyle
Clairaut's_equation
Class of numerical techniques
methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial
Finite_difference_method
Roots of multiple multivariate polynomials
closed-form expression of the solutions (in the case of a single equation, this is Abel–Ruffini theorem). The Barth surface, shown in the figure is the
System of polynomial equations
System_of_polynomial_equations
Mathematical expression using basic operations
of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just
Algebraic_expression
Existence and uniqueness of solutions to initial value problems
In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of sufficient (but not necessary) conditions
Picard–Lindelöf_theorem
Differential calculus on function spaces
maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to
Calculus_of_variations
Mathematical concept
undefined so that it follows the rules of a norm in a Euclidean domain. Abel–Ruffini theorem Fundamental theorem of algebra For simplicity, this is a
Degree_of_a_polynomial
Mathematical technique for simplification
Sixth-degree polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be
Change_of_variables
System of equations in mathematics
differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to
Differential-algebraic system of equations
Differential-algebraic_system_of_equations
Integration by parts version of Abel's method for summation by parts
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions
Abel's_summation_formula
Technique for solving differential equations
differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. A differential
Separation_of_variables
Cauchy–Euler equation Riccati equation Hill differential equation Gauss–Codazzi equations Chandrasekhar's white dwarf equation Lane-Emden equation Emden–Chandrasekhar
List of named differential equations
List_of_named_differential_equations
Special mathematical function
Gamma function and Zeta function. One can begin with equation for Bose integral, then use series equation. ∫ 0 ∞ x s e x − 1 d x = ∫ 0 ∞ x s 1 e x − 1 d x
Polylogarithm
Technique for solving differential equations
the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within
Integrating_factor
essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not considered as belonging
History_of_algebra
Partial differential equation with nonlinear terms
mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
French mathematician (1811–1832)
that is, that fifth and higher degree equations are not generally solvable by radicals. Although Niels Henrik Abel had already proved the impossibility
Évariste_Galois
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