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Type of mathematical function
piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. A piecewise linear
Piecewise_linear_function
Function defined by multiple sub-functions
mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain
Piecewise_function
Topics referred to by the same term
Piecewise linear may refer to: Piecewise linear curve, a connected sequence of line segments Piecewise linear function, a function whose domain can be
Piecewise_linear
Linear combination of indicator functions of real intervals
intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. A function f : R → R {\displaystyle f\colon
Step_function
Linear map or polynomial function of degree one
graph of the function must pass through the origin. Homogeneous function Nonlinear system Piecewise linear function Linear approximation Linear interpolation
Linear_function
Topological manifold with a piecewise linear structure on it
an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation
Piecewise_linear_manifold
Method of curve fitting
higher-dimensional piecewise linear functions (see second figure below). Many libraries and shading languages have a "lerp" helper-function (in GLSL known
Linear_interpolation
Numerical method for solving physical or engineering problems
chooses basis functions. We used piecewise linear basis functions in our discussion, but it is common to use piecewise polynomial basis functions. Separate
Finite_element_method
Tent function, often used in signal processing
x_{j+1}=1} . A linear B-spline is the same as a continuous piecewise linear function f ( x ) {\displaystyle f(x)} , and this general triangle function is useful
Triangular_function
Set of functions between two fixed sets
functions B M O ( Ω ) {\displaystyle BMO(\Omega )} , space of bounded mean oscillation. Also called John–Nirenberg space linear functions piecewise linear
Function_space
Mathematical function conceived as a crude model
they may also take the form of other nonlinear functions, piecewise linear functions, or step functions. They are also often monotonically increasing,
Artificial_neuron
to 1, but using simpler functions, especially piecewise linear functions or piecewise constant functions. These are preferred where speed of computation
Hard_sigmoid
Model in digital signal processing
impulse response) so that each input impulse results in the correct piecewise linear function in the output. First-order hold is the hypothetical filter or
First-order_hold
Connected series of line segments
be partitioned into exactly two monotone chains. The graphs of piecewise linear functions form monotone chains with respect to a horizontal line. Each segment
Polygonal_chain
Piece-wise division of objects
A piecewise-constant valuation is a kind of a function that represents the utility of an agent over a continuous resource, such as land. It occurs when
Piecewise-constant_valuation
Topics referred to by the same term
Examples of functions with such piecewise properties are: Piecewise constant function, also known as a step function Piecewise linear function Piecewise continuous
Piecewise_property
First known wavelet basis
k<2^{n}.} These functions sn,k are continuous, piecewise linear, supported by the interval In,k that also supports ψn,k. The function sn,k is equal to 1
Haar_wavelet
details. Let K and L be two GSCs. A function f : | K | → | L | {\displaystyle f:|K|\to |L|} is called piecewise-linear (PL) if there exist a subdivision
Simplicial_map
Indicator function of positive numbers
and represented the function as 1. Taking the convention that H(0) = 1, the Heaviside function may be defined as: A piecewise function: H ( x ) := { 1 ,
Heaviside_step_function
Mathematical function defined piecewise by polynomials
In mathematics, a spline is a function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial
Spline_(mathematics)
Type of mathematical function
elementary, such as piecewise-defined functions. More generally, in some modern treatments, elementary functions comprise the set of functions previously enumerated
Elementary_function
Type of activation function
(rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU (
Rectified_linear_unit
Statistical method of dividing data into equal-sized intervals for analysis
floor and ceiling functions). The first three are piecewise constant, changing abruptly at each data point, while the last six use linear interpolation between
Quantile
Exponent of a power of two
of approximately 0.44. It can be made more accurate by using a piecewise linear function of m {\displaystyle m} , or more crudely by adding a constant
Binary_logarithm
Function that is continuous everywhere but differentiable nowhere
the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. G. H. Hardy showed that the function of the above
Weierstrass_function
Array of numbers
It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen concerning a sufficiently fine grid
Matrix_(mathematics)
Non-sinusoidal waveform
the extreme case of an asymmetric triangle wave. The equivalent piecewise linear functions x ( t ) = t − ⌊ t ⌋ {\displaystyle x(t)=t-\lfloor t\rfloor } x
Sawtooth_wave
Loss function used in robust regression
The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by L δ ( a ) = {
Huber_loss
Graphical representation of the distribution of income or of wealth
{\displaystyle f(y_{j}):=\Pr(Y=y_{j})} the Lorenz curve is the continuous piecewise linear function connecting the points (Fi, Li), i = 0 to n, where F0 = 0, L0 =
Lorenz_curve
Generalized function whose value is zero everywhere except at zero
numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η1 to be a hat function. With this choice
Dirac_delta_function
Family of second-order differential equations
Liénard system with piecewise-linear functions can also contain homoclinic orbits. Let f and g be two continuously differentiable functions on R , {\displaystyle
Liénard_equation
Type of numerical analysis
interpolate linearly between the points ( x i , y ^ i ) {\displaystyle (x_{i},{\hat {y}}_{i})} , as illustrated in the figure, yielding a continuous piecewise linear
Isotonic_regression
Algorithm for generating pseudo-randomized numbers
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear
Linear_congruential_generator
of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line Linear function: First degree polynomial, graph
List of mathematical functions
List_of_mathematical_functions
Piecewise function that clamps its input to be non-negative
the ramp function. The ramp function (R(x) : R → R0+) may be defined analytically in several ways. Possible definitions are: A piecewise function: R ( x
Ramp_function
Mapping arbitrary data to fixed-size values
poorly designed hash functions can result in access times approaching linear in the number of items in the table. Hash functions can be designed to give
Hash_function
Doubling map on the unit interval
transformation can also be defined as the iterated function map of the piecewise linear function T ( x ) = { 2 x 0 ≤ x < 1 2 2 x − 1 1 2 ≤ x < 1. {\displaystyle
Dyadic_transformation
is completed. These knees in functions are defined by piecewise functions. Below is an exemplary piecewise linear function defining that as soon as x equals
Bulk_dispatch_lapse
Non-sinusoidal waveform
waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd
Triangle_wave
function (e.g. arcsine is the inverse of sine). Implicit function: defined implicitly by a relation between the argument(s) and the value. Piecewise function:
List_of_types_of_functions
coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in x and y variables. The polyhedral terrain is a generalization
Polyhedral_terrain
Algorithm for measuring similarity between temporal sequences
where the fluctuations in the time axis are modeled using a non-linear time-warping function. Considering any two speech patterns, we can get rid of their
Dynamic_time_warping
Simplicial continuation, or piecewise linear continuation (Allgower and Georg), is a one-parameter continuation method which is well suited to small to
Piecewise_linear_continuation
temperature stability and other non-ideal effects. Piecewise linear functions can approximate non-linear functions as a series of connected line segments. Gain
Operational amplifier applications
Operational_amplifier_applications
Operation in mathematical calculus
Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly
Integral
Category of piecewise-smooth manifolds
smooth manifolds and smooth functions between them) and PL (the category of piecewise linear manifolds and piecewise linear maps between them), and the
PDIFF
Type of artificial neural network architecture
low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. In some modern neural architectures
Kolmogorov–Arnold_Networks
Interpolation on functions of more than one variable
neighbor Triangulated irregular network-based linear interpolation (a type of piecewise linear function) n-simplex (e.g. tetrahedron) interpolation (see
Multivariate_interpolation
Any mathematical model describing semiconductor diodes
diode is called piecewise linear (PWL) modelling. In mathematics, this means taking a function and breaking it down into several linear segments. This
Diode_modelling
Mathematical method for approximating solutions to differential and integral equations
finite-dimensional subspace of piecewise linear functions (as in trapezoidal rule), or cubic functions, or other piecewise polynomial functions. In orthogonal collocation
Collocation_method
Electronic circuit that behaves chaotically
containing a nonlinear element such that f(x) was a 3-segment piecewise-linear function. The easy experimental implementation of the circuit, combined
Chua's_circuit
Topics referred to by the same term
terms remains constant throughout Linear spline, a function defined piecewise by polynomials of degree (at most) 1 Linear term, a component of a formula
Linear_(disambiguation)
Musical interval unit
increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946, allowing for a piecewise linear approximation. Thus, although
Cent_(music)
Electronic sound synthesis method
directly equivalent. The phase transforms are all assembled from piecewise linear functions under binary logic control and shows characteristic sharp knees
Phase_distortion_synthesis
Differential calculus on function spaces
choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes
Calculus_of_variations
Type of function in mathematics
nonempty open subset of C {\displaystyle \mathbb {C} } . Piecewise defined functions (functions given by different formulae in different regions) are typically
Analytic_function
Failure of convergence in interpolation
( x ) {\displaystyle \lim _{N\to \infty }P_{N}(x)} approaches the linear piecewise polynomials, i.e. connecting the interpolation points with straight
Runge's_phenomenon
Method for estimating new data within known data points
fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable
Interpolation
Arithmetic operation
function of the function exp a {\displaystyle \exp _{a}} . The proof is that the second through fourth conditions trivially imply that f is a linear function
Tetration
Integral transform and linear operator
be mitigated by use of a window function to taper h ~ [ n ] {\displaystyle {\tilde {h}}[n]} to zero. Method 2: Piecewise convolution. It is well known that
Hilbert_transform
Form of taxation in the Netherlands
both for income and for tax), the tax is a continuous, convex, piecewise linear function of income. For people above the pension age, the social security
Income_tax_in_the_Netherlands
real-valued functions defined on the unit interval, but equip X with the L1 norm and Y with the supremum norm. For n ∈ N, let fn be the continuous, piecewise linear
Continuous_embedding
Mathematical method in functional analysis
{\displaystyle {\mathcal {PC}}} denote the space of bounded, piecewise continuous functions on [ a , b ] {\displaystyle [a,b]} that are continuous from
Continuous_linear_extension
{\displaystyle \textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}} . 2. Piecewise definition; for example, | x | = { x if x ≥ 0 − x if x < 0 {\displaystyle
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Multivalued function in mathematics
W function by piecewise minimax rational function approximation with variable transformation". doi:10.13140/RG.2.2.30264.37128. "Lambert W Functions -
Lambert_W_function
Model of signal reconstruction in digital-to-analog (DAC) converters
is the piecewise-constant signal depicted in Figure 2. The equation above for the output of the ZOH can also be modeled as the output of a linear time-invariant
Zero-order_hold
Electrical circuit
until the first limits. The resulting curve is a piecewise linear function approximation of the log function. If limiting amplifiers that clip "softly" are
Log_amplifier
World spatial reference system co-rotating with the Earth in its diurnal motion in space
parametric fits are to model post-seismic deformation as opposed to the traditional approach using piecewise linear functions. ITRF2020 2015.0 9988 9989 9990
International Terrestrial Reference System and Frame
International_Terrestrial_Reference_System_and_Frame
Method in numerical analysis
continuation must be used (see below). Simplicial Continuation, or Piecewise Linear Continuation (Allgower and Georg) is based on three basic results.
Numerical_continuation
Standard model of the structure of Earth's magnetic field
functions is omitted in this formulation and is instead accounted for in the Gauss coefficients. The Gauss coefficients are modelled as a piecewise-linear
International Geomagnetic Reference Field
International_Geomagnetic_Reference_Field
Concept in statistical mathematics
Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable
Segmented_regression
Function used in signal processing
astronomy, inspired by the Planck distribution. It is defined as a piecewise function: w [ 0 ] = 0 , w [ n ] = ( 1 + exp ( ε N n − ε N ε N − n ) ) − 1
Window_function
Non-parametric regression technique
the piecewise linear graph shown for the simple MARS model in the previous section. One might assume that only piecewise linear functions can be formed
Multivariate adaptive regression spline
Multivariate_adaptive_regression_spline
Near-cylindrical polyhedron with large area
approximates a smooth function (often, the solution to a physical simulation problem in science or engineering) by a piecewise-linear function on a triangulation
Schwarz_lantern
prescribed exact solution consists of displacements that vary as piecewise linear functions in space (called a constant strain solution). The elements pass
Patch_test_(finite_elements)
the partition function P ( n 1 , n 2 ) {\displaystyle P(n_{1},n_{2})} can be defined piecewise with the help of two auxiliary functions. If n 1 ≤ n 2
Kostant_partition_function
Representation of mathematical space
a triangulable space. Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various applications
Triangulation_(topology)
smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for continuous functions between
Maps_of_manifolds
Graph in engineering
equation will change for each linear section of the influence line. Therefore, the complete equation is a piecewise linear function with a separate influence-line
Influence_line
System with self-optimizing transfer function
An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters
Adaptive_filter
Measure of air pollution
calculated using breakpoint concentrations below, and using same piecewise linear function to calculate intermediate values as the US AQI scale. and The
Air_quality_index
Mathematical model of the time dependence of a point in space
with just second-degree polynomials; the horseshoe map is piecewise linear. For non-linear autonomous ODEs it is possible under some conditions to develop
Dynamical_system
Mathematical rule
continuity is important. Without this assumption, the discontinuous piecewise linear function f : [ 0 , 3 ) → [ 0 , 3 ) {\displaystyle f:[0,3)\to [0,3)} defined
Sharkovskii's_theorem
Nearest integers from a number
\end{aligned}}} None of the functions discussed in this article are continuous, but all are piecewise linear: the functions ⌊ x ⌋ {\displaystyle \lfloor
Floor_and_ceiling_functions
Theorem in vector calculus
let c0, c1: [0, 1] → U be piecewise smooth loops. If there is a function H: [0, 1] × [0, 1] → U such that [TLH0] H is piecewise smooth, [TLH1] H(t, 0) =
Stokes'_theorem
Every graph has evenly many odd vertices
the motion. One proof of this involves approximating the function by a piecewise linear function with the same extreme points, parameterizing the position
Handshaking_lemma
Economical computational problem
is a piecewise linear function of xj. This assumption is common in computer science, as it allows a finite representation of the utility functions. Utilities
Market equilibrium computation
Market_equilibrium_computation
Spline function
a fundamental building block for all spline functions of that degree. A B-spline is defined as a piecewise polynomial of order n {\displaystyle n} , meaning
B-spline
Standard for HDTV image encoding and signal characteristics
analog-to-digital converters. Thus the standard opted for a piecewise function that combines a simple linear function V = a L {\displaystyle V=a\,L} for low light levels
Rec._709
Method of mathematical integration
functions, mainly piecewise continuous functions, including elementary functions, for example polynomials. However, the graphs of other functions, for example
Lebesgue_integral
Finite volume method in partial differential equations
use piecewise linear approximations of each cell, which results in a central difference scheme that is second-order accurate in space. The piecewise linear
MUSCL_scheme
Strong form of uniform continuity
metric on M or N. This structure is intermediate between that of a piecewise-linear manifold and a topological manifold: a PL structure gives rise to a
Lipschitz_continuity
(but piecewise continuous) nonlinearity (e.g., an amplifier with saturation, or an element with deadband effects) cascaded with a slow stable linear system
Describing_function
Software product
electronic circuit simulator with ideal and piecewise-linear components. The first generation of NL (non-linear) analog simulator was developed in the late
NL5_circuit_simulator
Degree of differentiability of a function or map
approximating function to data Spline – Mathematical function defined piecewise by polynomials Sobolev mapping Weisstein, Eric W. "Smooth Function". mathworld
Smoothness
Property of functions which is weaker than continuity
( y , ∞ ] {\displaystyle (y,\infty ]} . Consider the function f , {\displaystyle f,} piecewise defined by: f ( x ) = { − 1 if x < 0 , 1 if x ≥ 0 {\displaystyle
Semi-continuity
Set of functions used to represent the electronic wave function
finite element methods (FEMs), the wave function is represented as a linear combination of a set of piecewise polynomials. Lagrange interpolating polynomials
Basis_set_(chemistry)
Geometric model of the physical space
curve. For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as ∫ C f d s = ∫ a b f ( r ( t ) ) | r ′
Three-dimensional_space
Mathematical function with no sudden changes
discontinuities Coarse function Continuous function (set theory) Continuous stochastic process Normal function Open and closed maps Piecewise Squeeze mapping
Continuous_function
Mathematical concept of energy in physics
function in a finite-dimensional subspace of the true solution space. For example, one might let u h {\displaystyle u_{h}} be a continuous piecewise linear
Energetic_space
PIECEWISE LINEAR-FUNCTION
PIECEWISE LINEAR-FUNCTION
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Surname or Lastname
English
English : occupational name for a whitewasher, Middle English limer, lymer, an agent derivative of Old English līm ‘lime’.
Surname or Lastname
English
English : variant of Lingard.French : occupational name for a maker of or dealer in linen goods, from Old French linge ‘linen (goods)’ (see Linge 1).
Surname or Lastname
English (Cornish)
English (Cornish) : habitational name from a place named with Cornish lan ‘church’. In England this surname is now found chiefly in the southern counties of Wiltshire and Hampshire, and Berkshire; it has no doubt moved there from Cornwall.
Male
Greek
(ΑἰνÎας) Variant spelling of Greek AineÃas, AINEAS means "praiseworthy."
Boy/Male
Irish
Meaning “â€fair-haired,â€â€ the name has been popular since the sixth century when St. Finbar came to an area of Cork that was being tormented by a serpent. The people begged him to do something to help them. One night he went to where the serpent was sleeping and sprinkled it with holy water. The angry serpent tore and devoured the land until she slithered into the sea at Cork Harbor. The track she left behind filled with water and became the River Lee and that’s why St. Finbar is the patron saint of Cork. It is said that the sun didn’t set for two weeks after Finbar’s death.
Boy/Male
Hindu
The Sun
Surname or Lastname
English
English : metronymic from Line.
Female
English
Variant spelling of English Linsey, LINSAY means "Lincoln's wetlands."
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Boy/Male
Hindu
Lingam
Surname or Lastname
Swedish
Swedish : ornamental name from lind ‘lime tree’ + either the German suffix -er denoting an inhabitant, or the surname suffix -ér, derived from the Latin adjectival ending -er(i)us.English (mainly southeastern) : variant of Lind 2.German : habitational name from any of numerous places called Linden or Lindern, named with German Linden ‘lime trees’.
Boy/Male
Sikh
Love unending
Surname or Lastname
English
English : habitational name from Lingart, Lancashire, or Lingards Wood in Marsden, West Yorkshire, both named from Old English līn ‘flax’ + garðr ‘enclosure’.
Female
English
English name probably derived from Germanic lindi, LINDA means "serpent."Â In some cases, it may have been derived from the Spanish word for "pretty."
Surname or Lastname
English
English : variant of Lanier 1.Dutch : variant of Leonard.Jewish (western Ashkenazic) : name taken by someone who was good at chanting the Pentateuch at public worship in the synagogue or who regularly did so, from West Yiddish layner ‘reader’ (a derivative of West Yiddish laynen ‘to read’, which comes ultimately from Latin legere ‘to read’).Jewish (Ashkenazic) : occupational name for a flax grower or merchant, from German Lein ‘flax’ + agent suffix -er.
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Girl/Female
Irish
Eimear possessed the “Six Gifts of Womanhood†– “beauty, a gentle voice, sweet words, wisdom, needlework and chastity!†She was bethrothed to the warrior Cuchulainn (read the legend) when they were children and they loved each other very deeply. But Cuchulainn had “a wandering eye†and Eimear endured this, realizing “everything new is fair,†but when he made love to Fand, wife of the sea god Manannan, Eimear confronted the lovers. After seeing the strength of Fand’s love she offered to withdraw. Touched by this display of unselfishness, Fand left Cuchulainn and returned to the sea. When Cuchulainn died Eimear spoke movingly and lovingly at his graveside.
Surname or Lastname
English (Devon; of Cornish origin)
English (Devon; of Cornish origin) : topographic name for someone who lived by a menhir, i.e. a tall standing stone erected in prehistoric times (Cornish men ‘stone’ + hir ‘long’).
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
PIECEWISE LINEAR-FUNCTION
PIECEWISE LINEAR-FUNCTION
Boy/Male
English
Spear strong.
Girl/Female
Greek
Guileless.
Boy/Male
Arabic
Fluency; Eloquence
Boy/Male
Sikh
Religion of righteousness
Boy/Male
German
Resolute.
Girl/Female
Arabic, Muslim
Excellence of the Women
Girl/Female
Tamil
Phaalguni | பாலà¯à®•à¯à®¨à¯€
The day of the full Moon, The month of Phalguna
Boy/Male
British, English
Path
Boy/Male
Australian, German, Polish
Son of a Farmer; Son of Talmai
Boy/Male
Arabic, English, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Telugu
Fresh; Nectar
PIECEWISE LINEAR-FUNCTION
PIECEWISE LINEAR-FUNCTION
PIECEWISE LINEAR-FUNCTION
PIECEWISE LINEAR-FUNCTION
PIECEWISE LINEAR-FUNCTION
a.
Composed of lines; delineated; as, lineal designs.
a.
Linear.
prep. & adv.
Near.
n.
Made of linen; as, linen cloth; a linen stocking.
adv.
In a linear manner; with lines.
n.
A dealer in linen; a linen draper.
a.
Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.
a.
Of or pertaining to a line; consisting of lines; in a straight direction; lineal.
v. t.
To convert into vinegar; to make like vinegar; to render sour or sharp.
a.
Descending in a direct line from an ancestor; hereditary; derived from ancestors; -- opposed to collateral; as, a lineal descent or a lineal descendant.
n.
Alt. of Lingam
n.
One who adjusts things to a line or lines or brings them into line.
v. t.
To mark with a line or lines; to cover with lines; as, to line a copy book.
n.
A vessel belonging to a regular line of packets; also, a line-of-battle ship; a ship of the line.
n.
A lunar distance.
a.
Of, pertaining to, or included by, two lines; as, bilinear coordinates.
a.
Of a linear shape.
a.
In the direction of a line; of or pertaining to a line; measured on, or ascertained by, a line; linear; as, lineal magnitude.
a.
Formed by right lines; rectilineal; as, a right-lined angle.
n.
One who lines, as, a liner of shoes.