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(exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are
Dimension_function
Property of a mathematical space
mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur
Dimension
Notion in supervised machine learning
increasing function of its input, such as the sign function or the sigmoid function. This function is called the activation function. The VC dimension of a
Vapnik–Chervonenkis_dimension
Function valued in a vector space; typically a real or complex one
vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could
Vector-valued_function
Property of functions which is weaker than continuity
lower dimension if the perturbation is small enough. Another example of a similar character is that matrix rank is a lower semicontinuous function on the
Semi-continuity
Dimension of a subset of a metric space
the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some
Packing_dimension
Mathematical transform that expresses a function of time as a function of frequency
generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional "position space" to a function of 3-dimensional momentum
Fourier_transform
Characteristic of an optical system
typically only a one-dimensional, or sometimes a two-dimensional section is used, the three-dimensional optical transfer function can improve the understanding
Optical_transfer_function
Mathematical function with convex lower level sets
convexity: all convex functions are also quasiconvex, but not all quasiconvex functions are convex. For one-dimensional functions (functions on R), to check
Quasiconvex_function
Function that is continuous everywhere but differentiable nowhere
close, the function will not be monotone. The computation of the Hausdorff dimension D of the graph of the classical Weierstrass function was an open
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Computational statistics technique
region under the graph of its density function. Note that this property can be extended to N-dimension functions. The algorithm, which was used by John
Rejection_sampling
Whose values lie in an infinite-dimensional vector space
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of
Continuous_geometry
Mathematical function
under the Gaussian function is given by V=∫−∞∞∫−∞∞f(x,y)dxdy=2πAσXσY. In general, a two-dimensional elliptical Gaussian function is expressed as f(x
Gaussian_function
Chamber of the heart
measured in, e.g. the dimension of the longitudinal plane. Fractional shortening (FS) is the fraction of any diastolic dimension that is lost in systole
Ventricle_(heart)
Continuous function that is not absolutely continuous
showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, ( log
Cantor_function
Topics referred to by the same term
geometry, a synonym for dimension function; in control theory and dynamical systems, a synonym for Lyapunov candidate function; in gauge theory, a synonym
Gauge_function
Analysis of the dimensions of different physical quantities
engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical
Dimensional_analysis
Function for incompressible divergence-free flows in two dimensions
dynamics, two types of stream function (or streamfunction) are defined: The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange
Stream_function
Techniques to preserve differential privacy when releasing computational results
Appendix A in Dwork and Roth for a proof of this result). For high dimensional functions of the form f : D → R d {\displaystyle f\colon {\mathcal {D}}\to
Additive noise differential privacy mechanisms
Additive_noise_differential_privacy_mechanisms
Computing the fixed point of a function
interval is denoted by E:=[0,1], and the unit d-dimensional cube is denoted by Ed. A continuous function f is defined on Ed (from Ed to itself). Often,
Fixed-point_computation
Shape containing unit line segments in all directions
integrable function f, we define the Kakeya maximal function of f to be f∗δ(e)=supa∈𝐑n1m(Teδ(a))∫Teδ(a)|f(y)|dm(y), where m denotes the n-dimensional Lebesgue
Kakeya_set
Function used as a performance test problem for optimization algorithms
typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin as a 2-dimensional function and has been generalized by Rudolph
Rastrigin_function
Element of a basis for a function space
basis functions. In finite-dimensional vector spaces, this representation is purely algebraic and involves only finitely many basis functions, whereas
Basis_function
Real function with secant line between points above the graph itself
convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue
Convex_function
Smooth approximation of one-hot arg max
softmax function maps the Euclidean space ℝK to the boundary of the standard (K−1)-simplex, cutting the dimension by one (the range is a (K−1)-dimensional simplex
Softmax_function
Geometric model of the physical space
three-dimensional Euclidean space, which is the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces
Three-dimensional_space
Algebraic structure in linear algebra
are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that
Vector_space
Mathematical description of quantum state
This is a complex-valued function of two real variables x and t. For one spinless particle in one dimension, if the wave function is interpreted as a probability
Wave_function
Number of vectors in any basis of the vector space
is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there
Dimension_(vector_space)
Process of reducing the number of random variables under consideration
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the
Dimensionality_reduction
Tabletop role-playing web series
Dimension 20 is an actual play show produced by and broadcast on Dropout, and created and generally hosted by Brennan Lee Mulligan as the show's regular
Dimension_20
System for defining and representing engineering tolerances
or functions. The purpose of GD&T is to describe the engineering intent of parts and assemblies. GD&T can more accurately define the dimensional requirements
Geometric dimensioning and tolerancing
Geometric_dimensioning_and_tolerancing
Topological space that locally resembles Euclidean space
neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing
Manifold
Machine learning framework
deep learning architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial
Neural_operators
Bidirectional texture function (BTF) is a 6-dimensional function depending on planar texture coordinates (x,y) as well as on view and illumination spherical
Bidirectional texture function
Bidirectional_texture_function
Theorem in mathematics
calculations are equal: Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform
Projection-slice_theorem
Structure in data warehousing
techniques function. The type 5 technique builds on the type 4 mini-dimension by embedding a “current profile” mini-dimension key in the base dimension that's
Slowly_changing_dimension
Number specifying how a quantum operator changes under dilations
number called the scaling dimension of O {\displaystyle O} . This implies in particular that the two point correlation function ⟨ O ( x ) O ( 0 ) ⟩ {\displaystyle
Scaling_dimension
Mathematical function whose derivative exists
higher dimensional spaces above, and the absolute-value norm is used. However, a function f:ℂ→ℂ that is differentiable as a multi-variable function may not
Differentiable_function
Natarajan dimension characterizes the complexity of learning a set of functions, generalizing from the Vapnik–Chervonenkis dimension for boolean functions to
Natarajan_dimension
Invariant measure of fractal dimension
In mathematics, the Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician
Hausdorff_dimension
numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician Sergey
Sparse_grid
Difficulties arising when analyzing data with many aspects ("dimensions")
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional
Curse_of_dimensionality
Geometric space with four dimensions
Four-dimensional (4D) space is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible
Four-dimensional_space
Function of four real variables that defines how light is reflected at an opaque surface
Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF) is a 6-dimensional function, f r ( ω i , ω r , x ) {\displaystyle f_{\text{r}}(\omega
Bidirectional reflectance distribution function
Bidirectional_reflectance_distribution_function
generalization of the bisection method from one-dimensional functions to multi-dimensional functions. It is theoretically important as it attains the
Center-of-gravity_method
Construction for n-dimensional noise functions
Simplex noise is the result of an n-dimensional noise function comparable to Perlin noise ("classic" noise) but with fewer directional artifacts, in higher
Simplex_noise
Branch of mathematics studying functions of a complex variable
complex variables generalizes one-variable complex function theory to more than one complex dimension. While many of the techniques of a single complex
Complex_analysis
Conditions under which a chaotic system can be reconstructed by observation
generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other
Takens's_theorem
Least variables needed to represent data
intrinsic dimension of the two-variable function f is neither zero or one, it is two. In the literature, functions which are of intrinsic dimension zero,
Intrinsic_dimension
Association of one output to each input
y in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of
Function_(mathematics)
Infinitely detailed mathematical structure
arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales
Fractal
Mathematical function, in linear algebra
quotient space of the differentiable functions by the linear space of constant functions. If V and W are finite-dimensional vector spaces over a field F, of
Linear_map
Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Fundamental theorem in condensed matter physics
Particle in a one-dimensional lattice (periodic potential). Bloch's theorem—For electrons in a perfect crystal, there is a basis of wave functions with the following
Bloch's_theorem
Analytic function that does not satisfy a polynomial equation
operation to a dimension creates meaningless results. Complex function Function (mathematics) Generalized function List of special functions and eponyms
Transcendental_function
Real-valued number of spatial dimensions
Balay-Karperien, Audrey (2004). Defining Microglial Morphology: Form, Function, and Fractal Dimension. Charles Sturt University. p. 86. Retrieved 9 July 2013. Losa
Fractal_dimension
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
Kronecker delta function sometimes means the unit sample function δ[n] , which represents a special case of the 2-dimensional Kronecker delta function δij where
Kronecker_delta
Mathematical operation
cylindrical or spherical symmetry. Consider a function f(𝐫) of a d-dimensional vector r. Its d-dimensional Fourier transform is defined asF(𝐤)=∫ℝdf(𝐫)e−i𝐤⋅𝐫d𝐫
Hankel_transform
Software library for numerical integration
FORTRAN 77 library for numerical integration (quadrature) of one-dimensional functions. It was included in the SLATEC Common Mathematical Library and is
QUADPACK
Function of propagation delay and Doppler frequency
pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of propagation delay τ {\displaystyle \tau } and Doppler
Ambiguity_function
Fundamental object of geometry
As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves
Point_(geometry)
Topics referred to by the same term
needed to cover the set as a function of the box size Equilateral dimension of a metric space (also called the metric dimension), the maximum number of points
Metric_dimension
Size of a mathematical ball
expressed via a two-dimension recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also
Volume_of_an_n-ball
Type of gradient noise in computer graphics
Perlin noise is most commonly implemented as a two-, three- or four-dimensional function, but can be defined for any number of dimensions. An implementation
Perlin_noise
Function specifying the behavior of a component in an electronic or control system
be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical
Transfer_function
Measure of a mathematical object studied in the field of algebraic geometry
R=K[x1,…,xn]. Let A=R/I be the K-algebra of the polynomial functions over V. The dimension of V is any of the following integers. It does not change if
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Visual artifact that depicts or records perception
representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images
Image
Matrix of second derivatives
matrix takes Θ(n2) memory, which is infeasible for high-dimensional functions such as the loss functions of neural nets, conditional random fields, and other
Hessian_matrix
Generalized function whose value is zero everywhere except at zero
supported continuous function f. As a measure, the n-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable
Dirac_delta_function
Extension of the factorial function
the gamma function. The gamma function can also be used to calculate "volume" and "area" of n-dimensional hyperspheres. The gamma function's ability to
Gamma_function
Area of mathematics
finite-dimensional situation. This is explained in the dual space article. Also, the notion of derivative can be extended to arbitrary functions between
Functional_analysis
Special functions of several complex variables
functions in two dimensions are functions of two complex arguments. In one choice of parameter, for example, z encodes position on a two-dimensional lattice
Theta_function
Vector function in optics
point in a three-dimensional space. The mathematical space of all possible light rays is given by the five-dimensional plenoptic function (with three position
Light_field
Method in evaluating divergent integrals
1-sphere. Now we generalize this into dimension d. The volume of a d-sphere is 2πd/2Γ(d/2), where Γ is the gamma function. Now the integral becomes
Dimensional_regularization
Fourier transform, and is a two-dimensional function when it corresponds to the Fourier transform of one-dimensional functions. It is complex-valued and has
Fourier_operator
Permissible limit or limits of variation
tolerance is the permissible limit or limits of variation in: a physical dimension; a measured value or physical property of a material, manufactured object
Engineering_tolerance
Function describing equilibrium states of a system
thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a function relating several state variables
State_function
In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting
Effective_dimension
Theorem in mathematics
finite dimension, by replacing "derivative" with "Jacobian matrix" and "nonzero derivative" with "nonzero Jacobian determinant". If the function of the
Inverse_function_theorem
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Relation between genus, degree, and dimension of function spaces over surfaces
and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates
Riemann–Roch_theorem
Type of artificial neural network
first layer in this block is a 1×1 convolution for dimension reduction (e.g., to 1/2 of the input dimension); the second layer performs a 3×3 convolution;
Residual_neural_network
Type of vector space in math
and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract
Hilbert_space
Mathematical theorem
sides on dimension i. By the Poincaré-Miranda conditions, fi(x)<0 at the start of the path and fi(x)>0 at the end of the path, and the function is continuous
Steinhaus_chessboard_theorem
Mathematical function having a characteristic S-shaped curve or sigmoid curve
EpsilonDelta (2022-08-16). "Smooth Transition Function in One Dimension | Smooth Transition Function Series Part 1". 13:29/14:04 – via www.youtube.com
Sigmoid_function
to extremize. In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically
Palais–Smale compactness condition
Palais–Smale_compactness_condition
Linear map or polynomial function of degree one
a function f(x1,…,xk) of any finite number of variables, the general formula is f(x1,…,xk)=b+a1x1+⋯+akxk, and the graph is a hyperplane of dimension k
Linear_function
Methods of calculating definite integrals
multi-dimensional integrals. They may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods
Numerical_integration
function spaces, which are infinite-dimensional and within which most functions are "anonymous", with special functions picked out by properties such as
List of mathematical functions
List_of_mathematical_functions
Pictures encoded as binary data
for its intensity or gray level that is an output from its two-dimensional functions fed as input by its spatial coordinates denoted with x, y on the
Digital_image
Representation of a mathematical function
y,z) where f(x,y)=z. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface
Graph_of_a_function
Integration kernels for smoothing out sharp features
smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution
Mollifier
use progressively more complex models to estimate an unknown high-dimensional function as more data becomes available, with the aim of asymptotically reducing
Sieve_estimator
Function used as a performance test problem for optimization algorithms
of 2-dimensional Rosenbrock function optimization by adaptive coordinate descent from starting point x0=(−3,−4). The solution with the function value
Rosenbrock_function
Fractal curve
is an irrep-7 irrep-tile (see Rep-tile for discussion). The Hausdorff dimension of the Koch curve is d=ln4ln3≈1.26186. This is greater than that of
Koch_snowflake
Mathematical idealization of the surface of a body
may also be defined as the image, in some space of dimension at least 3, of a continuous function of two variables (some further conditions are required
Surface_(mathematics)
Analyzes the topology of a manifold by studying differentiable functions on that manifold
inequalities. An infinite dimensional analog of Morse homology in symplectic geometry is known as Floer homology. The notion of a Morse function can be generalized
Morse_theory
C*-algebra
origins in topological K-theory and serves as the range of a kind of "dimension function." For an AF algebra A, K0(A) can be defined as follows. Let Mn(A)
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
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DIMENSION FUNCTION
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