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Formula in Lie group theory
group, the exponential map reduces to the matrix exponential. The exponential map, denoted exp:g → G, is analytic and has as such a derivative d/dtexp(X(t)):Tg
Derivative of the exponential map
Derivative_of_the_exponential_map
Map from a Lie algebra to its Lie group
In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to
Exponential_map_(Lie_theory)
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Matrix operation generalizing exponentiation of scalar numbers
differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group
Matrix_exponential
Characterizations of the exponential function Compound interest C0-semigroup De Moivre's formula Derivative of the exponential map Doléans-Dade exponential Double
List_of_exponential_topics
Instantaneous rate of change (mathematics)
the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a
Derivative
Type of derivative in differential geometry
} . Covariant derivative Connection (mathematics) Frölicher–Nijenhuis bracket Geodesic Killing field Derivative of the exponential map Trautman, A. (2008)
Lie_derivative
Formula in Lie theory
Matrix exponential Logarithm of a matrix Lie product formula (Trotter product formula) Lie group–Lie algebra correspondence Derivative of the exponential map
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
Mathematical term
needed] to the Lie derivative LXY = [X,Y] of vector fields on the group G considered as a manifold. Further see the derivative of the exponential map. Let g
Adjoint_representation
Matrix representing a Euclidean rotation
the Cayley map specifies the same rotation matrix through the map exp(2 artanh A). For a detailed derivation, see Derivative of the exponential map.
Rotation_matrix
Algebraic generalization of the derivative
differential Hasse derivative p-derivation Wirtinger derivatives Derivative of the exponential map Bourbaki, Nicolas (1989), Algebra I, Elements of mathematics
Derivation (differential algebra)
Derivation_(differential_algebra)
Fundamental construction of differential calculus
mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical
Generalizations of the derivative
Generalizations_of_the_derivative
Mathematical operation on invertible matrices
logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar
Logarithm_of_a_matrix
Exponential representation for differential equations
physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the product integral solution of a first-order
Magnus_expansion
Family of probability distributions related to the normal distribution
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special
Exponential_family
Concept in topology
fixed point theorem. Mostow (1955) showed that the derivative of the exponential map at any point of G/K satisfies |d exp X| ≥ |X|. This implies that
Maximal_compact_subgroup
Branch of mathematics studying functions of a complex variable
infinitely differentiable, whereas the existence of the nth derivative need not imply the existence of the (n + 1)th derivative for real functions. Furthermore
Complex_analysis
Complex-differentiable (mathematical) function
coordinate space C n {\displaystyle \mathbb {C} ^{n}} . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies
Holomorphic_function
Generalisation of the exponential integral to non-commutative algebras
exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras. It is a kind of
Ordered_exponential
Parameterization of a rotation into a unit vector and angle
them. Here the unit vector is denoted ω instead of e. The exponential map effects a transformation from the axis-angle representation of rotations to
Axis–angle_representation
German mathematician
Mathematische Annalen, Bd. 55, 1902 Baker–Campbell–Hausdorff formula Derivative of the exponential map K3 surface Owens, Frederick William (1912). "Review: Grundlagen
Friedrich_Schur
Mathematical function that preserves angles
. However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. The Riemann mapping
Conformal_map
Complex exponential in terms of sine and cosine
determined in the course of the proof. From any of the definitions of the exponential function it can be shown that the derivative of eix is ieix. Therefore
Euler's_formula
Strong form of uniform continuity
bounded first derivative is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf
Lipschitz_continuity
S-shaped curve
-\infty } is 0, and the limit as x {\displaystyle x} tends to + ∞ {\displaystyle +\infty } is L {\displaystyle L} . The exponential function with negated
Logistic_function
Logarithm of a complex number
are all mapped to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard
Complex_logarithm
Quadratic polynomial
alternate planes". aleph0.clarku.edu. "Exponential Map, Mu-Ency at MROB". mrob.com. Trees of visible components in the Mandelbrot set by Virpi Kauko, FUNDAMENTA
Complex_quadratic_polynomial
Functional square root of an exponential
In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f {\displaystyle f} such that f
Half-exponential_function
Special coordinate system in differential geometry
neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection
Normal_coordinates
Simple polynomial map exhibiting chaotic behavior
Schwarzian derivative of a map f (of class C3 ) is In fact, when calculating the Schwarzian derivative of the logistic map, the result is where the Schwarzian
Logistic_map
Numerical integration method
singularities or infinite derivatives exist at one or both endpoints. The method uses hyperbolic functions in the change of variables x = tanh ( 1 2
Tanh-sinh_quadrature
American-Canadian mathematician (1930-1995)
"The derivative of the exponential map". Proc. Amer. Math. Soc. 112 (3): 909–911. doi:10.1090/s0002-9939-1991-1086328-8. MR 1086328. Carl Herz at the Mathematics
Carl_S._Herz
implies the first stratum V 1 {\displaystyle V_{1}} generates the whole Lie algebra g {\displaystyle {\mathfrak {g}}} . The exponential map is a diffeomorphism
Carnot_group
Assignment of vector fields to manifolds
of the inverse function theorem to maps between manifolds. Coordinate-induced basis Cotangent space Differential geometry of curves Exponential map Vector
Tangent_space
Arithmetic operation
using the exponential identity if x is rational, and the continuity of the exponential function otherwise. The limit that defines the exponential function
Exponentiation
Rules for computing derivatives of functions
This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all
Differentiation_rules
Discrete analog of a derivative
expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such
Finite_difference
half-plane model Poincaré metric Angle of parallelism Prime geodesic Geodesic flow Exponential map (Lie theory) Exponential map (Riemannian geometry) Injectivity
List of differential geometry topics
List_of_differential_geometry_topics
Arithmetic operation
software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.[citation needed]
Tetration
Mathematical function, inverse of an exponential function
Moreover, as the derivative of f(x) evaluates to ln(b) bx by the properties of the exponential function, the chain rule implies that the derivative of logb x
Logarithm
Generalization of the inverse function theorem
contrast to the Banach space case, in which the invertibility of the derivative at a point is sufficient for a map to be locally invertible, the Nash–Moser
Nash–Moser_theorem
Hyperbolic analogues of trigonometric functions
with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are
Hyperbolic_functions
Logarithm to the base of the mathematical constant e
more. The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function
Natural_logarithm
Type of activation function
function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters
Rectified_linear_unit
Group that is also a differentiable manifold with group operations that are smooth
compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for
Lie_group
derivative . For v t = φ ˙ t ∘ φ t − 1 , t ∈ [ 0 , 1 ] {\displaystyle v_{t}={\dot {\varphi }}_{t}\circ \varphi _{t}^{-1},t\in [0,1]} , the inverse of
Bayesian model of computational anatomy
Bayesian_model_of_computational_anatomy
Structurally stable features in coupled oscillation dynamical systems.
driven rotator. The simplest mathematical model that exhibits mode-locking is the circle map, which attempts to capture the motion of the spinning disks
Arnold_tongue
Polynomial function of degree at most one
by a unique linear function. The derivative f ′ ( c ) {\displaystyle f\,'(c)} is the slope of this linear function, and the approximation is: f ( x ) ≈
Linear_function_(calculus)
Mathematical concept
considering the number α in F as the constant function fα that maps every x in R to α. This is a field since F is, and since the derivative of every function
Hardy_field
Mathematical concept
{dy}{dx}}} denotes the derivative of y. Functional equation. The exponential function e x {\displaystyle e^{x}} is the unique function f with the multiplicative
Characterizations of the exponential function
Characterizations_of_the_exponential_function
Differential equation that is linear with respect to the unknown function
differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0 ( x ) y + a 1 ( x ) y ′ + a 2 (
Linear_differential_equation
Generalization of the exponential function
one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient
C0-semigroup
Function that is holomorphic on the whole complex plane
function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums
Entire_function
Degree of differentiability of a function or map
k} such that a function has all derivatives up to order k {\displaystyle k} , and such that all of these derivatives are continuous. One says that such
Smoothness
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over
Softmax_function
Generalization of the standard Boltzmann–Gibbs entropy
Generalizations of the exponential family using the q-exponential function have been widely explored in statistical physics and information geometry. In the machine
Tsallis_entropy
Mathematics of smooth surfaces
If the radius δ is taken small enough, a slight sharpening of the Gauss lemma shows that the image U of the disc ‖v‖ < δ under the exponential map is
Differential geometry of surfaces
Differential_geometry_of_surfaces
Rate of separation of infinitesimally close trajectories
mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the exponential rate of separation
Lyapunov_exponent
Concepts in mathematics
by X. The exponential map is smooth. For a fixed X, the map t ↦ exp(tX) is the one-parameter subgroup of G generated by X. The exponential map restricts
Vector_flow
Second derivative Inflection point – found using second derivatives Directional derivative, Total derivative, Partial derivative Linearity of differentiation
List_of_real_analysis_topics
Free online crowdsourced encyclopedia
After an early period of exponential growth, the growth rate of the English Wikipedia in terms of the numbers of new articles and of editors appears to have
Wikipedia
Bijective holomorphic function with a holomorphic inverse
have nonzero derivative everywhere. Other authors define a conformal map as one with nonzero derivative, but without requiring that the map be injective
Biholomorphism
Diffeomorphism that has a hyperbolic structure on the tangent bundle
= 2 J . {\displaystyle [J,X]=X,\qquad [J,Y]=-Y,\qquad [X,Y]=2J.} The exponential maps g t = exp ( t J ) = ( e t / 2 0 0 e − t / 2 ) h t ∗ = exp ( t
Anosov_diffeomorphism
Association of one output to each input
defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that
Function_(mathematics)
Function used in computer graphics
indeed, a geodesic. In the tangent space at any point on a quaternion slerp curve, the inverse of the exponential map transforms the curve into a line segment
Spherical linear interpolation
Spherical_linear_interpolation
Nonlinear differential operator used to study conformal mappings
the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the
Schwarzian_derivative
Type of mathematical function
functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions
Elementary_function
Function that maps matrices to matrices
function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which
Analytic_function_of_a_matrix
Mathematical concept
condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in
Maurer–Cartan_form
Sum of elements on the main diagonal
respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). The trace of an n × n square matrix A is
Trace_(linear_algebra)
Type of function in mathematics
coefficients of the series, or equivalently by derivatives of the function evaluated at the center of the series expansion. In other words, an analytic
Analytic_function
Formula for the derivative of a matrix determinant
expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If A is a differentiable map from the real
Jacobi's_formula
Set of functions between two fixed sets
function space of paths of the process (functions of time); In category theory, the function space is called an exponential object or map object. It appears
Function_space
Algorithm to smooth data points
sub-sets, to give estimates of the smoothed signal, (or derivatives of the smoothed signal) at the central point of each sub-set. The method, based on established
Savitzky–Golay_filter
Doubling map on the unit interval
The dyadic transformation (also known as the dyadic map, bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e
Dyadic_transformation
Fundamental trigonometric functions
the higher-order derivatives. As mentioned in § Continuity and differentiation, the derivative of sine is cosine and the derivative of cosine is the negative
Sine_and_cosine
efficient since it involves computing the GCD of polynomials of a degree which is exponential in the degree of the input polynomial. However, g = gcd (
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Algorithms and methods of plotting the Mandelbrot set on a computing device
fact gives a formula for the uniformizing map of the complement of M {\displaystyle M} (and the derivative of this map). By the Koebe quarter theorem, one
Plotting algorithms for the Mandelbrot set
Plotting_algorithms_for_the_Mandelbrot_set
Group of rotations in 3 dimensions
of the exponential map. The exponential map provides a diffeomorphism between a neighborhood of the origin in the 𝖘𝖔(3) and a neighborhood of the identity
3D_rotation_group
Economic theory
production theory. The concept of diminishing returns can be explained by considering other theories, such as the concept of exponential growth. It is commonly
Diminishing_returns
In mathematics, invariant of square matrices
\textstyle {\frac {dA}{dx}}} denotes the derivative of the matrix, that is the matrix of the derivatives of the entries of A {\displaystyle A} . In particular
Determinant
Real function with secant line between points above the graph itself
of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a
Convex_function
representations. See also List of types of functions Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) Algebraic
List of mathematical functions
List_of_mathematical_functions
Exponential map Exponential map (Lie theory), Exponential map (Riemannian geometry) Finsler metric A generalization of Riemannian manifolds where the
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Homotopic map of a graph
In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy
Train_track_map
Function that preserves distinctness
redefined so that its domain is the non-negative real numbers [0, +∞), then g {\displaystyle g} is injective. The exponential function exp : R → R {\displaystyle
Injective_function
Exterior algebraic map taking tensors from p forms to n-p forms
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed
Hodge_star_operator
Operation in calculus
function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates
Integral
Function acting on the space of physical states in physics
is the derivative. The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations
Operator_(physics)
Calculus of functions generalization
coordinate-free way). Derivatives of such maps at a point are then vectors or linear maps, not real numbers. Let f : X → Y {\displaystyle f:X\to Y} be a map from an
Calculus_on_Euclidean_space
Polynomial interpolation using derivative values
derivatives have the same values at m (fewer than n) given points as the given function and its first few derivatives at those points. The number of pieces
Hermite_interpolation
combinations of the others. Matrix exponential — defined by the exponential series. Matrix representation of conic sections Pseudoinverse — a generalization of the
List_of_named_matrices
Method of estimating the parameters of a statistical model
Bayesian statistics, the maximum a posteriori (MAP) estimate of an unknown quantity is the mode of the posterior density. The MAP can be used to obtain
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Construct allowing differentiation of tangent vector fields of manifolds
definition of an exponential map associated to the affine connection. In particular, when M is a (pseudo-)Riemannian manifold and ∇ is the Levi-Civita
Affine_connection
Type of feedforward neural network
activations produce near-zero derivatives; these small values multiply across layers, causing error signals to diminish exponentially. ReLU activations address
Multilayer_perceptron
Family of probability distributions
about the same topic in The Annals of Statistics. The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion
Tweedie_distribution
Mathematical concept
^{\prime }} . The Gauss map is in fact much more than ergodic: it is exponentially mixing, but the proof is not elementary. The Gauss map, over the Gauss measure
Gauss–Kuzmin–Wirsing_operator
Linear mathematical operator which translates a function
functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian
Shift_operator
equation containing derivatives of non-integer order. Such systems are said to have fractional dynamics. Derivatives and integrals of fractional orders
Fractional-order_system
then the chain rule expresses the derivative of their composition f ∘ g (the function which maps x to f(g(x)) ) in terms of the derivatives of f and
Glossary_of_calculus
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DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
DERIVATIVE OF-THE-EXPONENTIAL-MAP
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