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Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Mathematical theorem
infinity, and we recover Bernstein's theorem. Absolutely and completely monotonic functions and sequences Choquet's theorem Krein–Milman theorem Bernstein, Serge
Bernstein's theorem on monotone functions
Bernstein's_theorem_on_monotone_functions
Mathematics concept
Absolutely and completely monotonic functions and sequences Levin, Vladimir (1 March 1999). "Abstract Cyclical Monotonicity and Monge Solutions for the
Cyclical_monotonicity
inputs. Strict monotonic function: preserves the given order. Real function: a function whose domain is real. Complex function: a function whose domain
List_of_types_of_functions
Description of continuous random distribution
{\displaystyle g:\mathbb {R} \to \mathbb {R} } be a monotonic function, then the resulting density function is f Y ( y ) = f X ( g − 1 ( y ) ) | d d y ( g
Probability_density_function
Mathematical function, inverse of an exponential function
determined by reverse look up in the same table, since the logarithm is a monotonic function. The product and quotient of two positive numbers c and d were routinely
Logarithm
Nonparametric measure of rank correlation
well the relationship between two variables can be described using a monotonic function. The Spearman correlation between two variables is equal to the Pearson
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Topics referred to by the same term
object Monotonic function, a property of a mathematical function Monotonicity of entailment, a property of some logical systems Monotonically increasing
Monotone
Function that preserves distinctness
mathematical functions Injective metric space – Type of metric space Monotonic function – Order-preserving mathematical function Univalent function – Mathematical
Injective_function
Finite or infinite ordered list of elements
sequence. This is a special case of the more general notion of a monotonic function. The terms nondecreasing and nonincreasing are often used in place
Sequence
Theorem in order and lattice theory
be a complete lattice and let f : L → L be an order-preserving (monotonic) function with respect to ≤. Then the set of fixed points of f in L forms a
Knaster–Tarski_theorem
Functions of an angle
trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define
Trigonometric_functions
Mathematical function with convex lower level sets
both concave and quasiconvex. Any monotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and
Quasiconvex_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
has a positive derivative at each point. In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the
Sigmoid_function
Method for bounding the errors of numerical computations
[x]_{n}\right)\in [\mathbb {R} ]^{n}.} Interval functions beyond the four basic operators may also be defined. For monotonic functions in one variable, the range of values
Interval_arithmetic
Length of a line segment
since it allows convex analysis to be used. Since squaring is a monotonic function of non-negative values, minimizing squared distance is equivalent
Euclidean_distance
Association of one output to each input
value zero for x = 1, is a differentiable function called the natural logarithm. A real function f is monotonic in an interval if the sign of f ( x ) −
Function_(mathematics)
Technique for the generative modeling of a continuous probability distribution
increasing monotonic function σ {\displaystyle \sigma } of type R → ( 0 , 1 ) {\displaystyle \mathbb {R} \to (0,1)} , such as the sigmoid function. In that
Diffusion_model
Tendency of matter to change volume in response to a change in temperature
moving, the higher that body's temperature is. Specifically, it is a monotonic function of the average molecular kinetic energy of a substance. As the energy
Thermal_expansion
Monotone maps have countable discontinuities
MR 2356043. Gelbaum, Bernard R.; Olmsted, John M. H. (1964), "18: A monotonic function whose points of discontinuity form an arbitrary countable (possibly
Discontinuities of monotone functions
Discontinuities_of_monotone_functions
Examples of non-monotonic queries are aggregation queries, or queries with set difference. Identifying whether a query is monotonic can be crucial for
Monotonic_query
Product of a number by itself
square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on the
Square_(algebra)
Probability problem
x^{6}(1-x)^{4}d\mu (x)\geq 0.} Absolutely and completely monotonic functions and sequences Total monotonicity Hausdorff, F. "Summationsmethoden und Momentfolgen
Hausdorff_moment_problem
Mathematical property excluding equality
and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive
Strict
Growth of quantities at rate proportional to the current amount
of exponential topics Logarithmic growth Logistic function Malthusian growth model Monotonic function Power law Menger sponge Moore's law Quadratic growth
Exponential_growth
privacy that a person achieves. Third, privacy is described as a non-monotonic function. More privacy is not necessarily better. A person seeks an optimal
Privacy_regulation_theory
Test for the injectivity of a function
more than once, the function fails the horizontal line test and is not injective. Vertical line test Inverse function Monotonic function Stewart, James (2003)
Horizontal_line_test
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Mathematical relation making a non-equal comparison
and >) and — in the case of applying a function — monotonic functions are limited to strictly monotonic functions. The relations ≤ and ≥ are each other's
Inequality_(mathematics)
Statistical function that converts a probability to a standard normal score
model Multinomial probit Q–Q plot Continuous function Monotonic function Quantile function Sigmoid function Rankit analysis, also developed by Chester Bliss
Probit
Paradigm for the design, analysis, and scoring of tests
normally distributed, discrimination in the 2PL model is approximately a monotonic function of the point-biserial correlation. In particular: a i ≅ ρ i t 1 −
Item_response_theory
Function that is continuous everywhere but differentiable nowhere
differentiable continuous function proof of existence using Banach's contraction principle. Nowhere monotonic continuous function proof of existence using
Weierstrass_function
Approach to static program analysis
sound approximation of the semantics of computer programs, based on monotonic functions over ordered sets, especially lattices. It can be viewed as a partial
Abstract_interpretation
job-scheduling, the makespan-minimization social-choice function is not strongly-monotonic nor weakly-monotonic. Indeed, it cannot be implemented by a truthful
Monotonicity (mechanism design)
Monotonicity_(mechanism_design)
Mathematical theorem, used in calculus
in the previous argument. On the other hand, even though general monotonic functions are differentiable almost everywhere, the proof of the general formula
Integral_of_inverse_functions
Numerical measure of a statistical relationship between variables
well the relationship between two variables can be described by a monotonic function. The Kendall tau rank correlation coefficient is a measure of the
Correlation_coefficient
Everywhere except a set of measure zero
{\displaystyle f(x)=0} almost everywhere. If f : [a, b] → R is a monotonic function, then f is differentiable almost everywhere. If f : R → R is Lebesgue
Almost_everywhere
Mathematical function
F(x+1)=F(x)+{\frac {1}{x}}} that is monotonic on R+ and satisfies F(1) = −γ. This fact follows immediately from the uniqueness of the Γ function given its recurrence
Digamma_function
Quantity in information theory
self-information is a strictly decreasing monotonic function of the probability, or sometimes called an "antitonic" function. While standard probabilities are
Information_content
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Continuous function that is not absolutely continuous
while the function seems like a constant one that cannot grow, it does indeed monotonically grow. It is also called the Cantor ternary function, the Lebesgue
Cantor_function
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Polynomial function of degree 3
cubic function is strictly monotonic. See the figure for an example of the case Δ0 > 0. The inflection point of a function is where that function changes
Cubic_function
Fundamental trigonometric functions
derivative test, according to which the monotonicity of a function can be defined as the inequality of function's first derivative greater or less than
Sine_and_cosine
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Function with unusual fractal properties
ISBN 978-3-540-44141-0, Zbl 1014.11015 Salem, Raphaël (1943), "On some singular monotonic functions which are strictly increasing" (PDF), Transactions of the American
Minkowski's question-mark function
Minkowski's_question-mark_function
Theory of cognition
learned to any stimulus will generalize to any other is an invariant [monotonic function] of the distance between them". Using experimental evidence from both
Universal law of generalization
Universal_law_of_generalization
Model of future interest rates
Hull–White (as today's value of a swaption in the Hull–White model is a monotonic function of today's short rate). Thus knowing how to price caps is also sufficient
Hull–White_model
S-shaped curve
rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth
Logistic_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 is partitioned
Piecewise_function
Economics term
preferences which are weakly monotonic but not strongly monotonic are those represented by a Leontief utility function. If an agent has monotone preferences
Monotone_preferences
Meromorphic function
{\displaystyle m=0} , the digamma function, ψ ( x ) = ψ ( 0 ) ( x ) {\displaystyle \psi (x)=\psi ^{(0)}(x)} , is strictly monotonic increasing and strictly concave
Polygamma_function
Computer science algorithm
{\displaystyle FFD(S,C)} is not a monotonic function of C. Similarly, F F D ( S , C ) {\displaystyle FFD(S,C)} is not a monotonic function of the sizes of items in
First-fit-decreasing bin packing
First-fit-decreasing_bin_packing
Statistical function that defines the quantiles of a probability distribution
probability distribution's quantile function is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_function
Mathematical function that outputs real values
two L2 functions belongs to L1. Other contexts where real-valued functions and their special properties are used include monotonic functions (on ordered
Real-valued_function
Method of estimating the parameters of a statistical model, given observations
{\mathcal {L}}_{n}(\theta \,;\mathbf {y} )~.} Since the logarithm is a monotonic function, the maximum of ℓ ( θ ; y ) {\displaystyle \;\ell (\theta \,;\mathbf
Maximum_likelihood_estimation
Test for infinite series of monotonous terms for convergence
integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis
Integral_test_for_convergence
Mathematical function, denoted exp(x) or e^x
x} . Since the exponential function equals its derivative, this implies that the exponential function is monotonically increasing. Extension of exponentiation
Exponential_function
Form of error in digital signals; spurious signals near sharp transitions
formally, when an input signal which is monotonic on an interval has output response which is not monotonic. This occurs most severely when the impulse
Ringing_artifacts
Methane molecules that contain two or more rare isotopes
internal isotopic equilibrium are predicted and verified to vary as monotonic functions of temperature of equilibration as follows: Δ 18 = − 0.0117 ( 10
Methane_clumped_isotopes
Negative of a convex function
differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that
Concave_function
Economic formula of productivity
normalized to one for a utility function because normalization is a monotonic transformation of the original utility function. Thus, let us define λ = ∑ i
Cobb–Douglas production function
Cobb–Douglas_production_function
Measurable set whose measure is zero
strictly monotonic function, so consider g ( x ) = f ( x ) + x . {\displaystyle g(x)=f(x)+x.} Since g {\displaystyle g} is strictly monotonic and continuous
Null_set
Mathematical activation function in data analysis
S2CID 9959057. Misra, Diganta (2019). "Mish: A Self Regularized Non-Monotonic Neural Activation Function". arXiv:1908.08681 [cs.LG]. Hendrycks, Dan; Gimpel, Kevin
Swish_function
Type of statistical analysis
coefficient: measures statistical dependence between two variables using a monotonic function. Squared ranks test: tests equality of variances in two or more samples
Nonparametric_statistics
Marks added to letters in Greek
different orthographies: el-monoton for monotonic Greek. el-polyton for polytonic Greek. While the tónos of monotonic orthography looks similar to the oxeîa
Greek_diacritics
Statistics models class
_{p=1}^{n}\phi _{p}(x_{p})\right)} . where Φ {\displaystyle \Phi } is a smooth monotonic function. Writing g {\displaystyle g} for the inverse of Φ {\displaystyle \Phi
Generalized_additive_model
Causal relationships between points in a manifold
oriented if the parameter change is required to be a strictly increasing monotonic function. Smooth regular curves (or paths) in M {\displaystyle M} can be classified
Causal_structure
Property of graphs that depends only on abstract structure
hereditary, monotone, or minor-closed if the function formalizing the invariant forms a monotonic function from the corresponding partial order on graphs
Graph_property
Energy contained within a system
U. It expresses the entropy representation. Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides
Internal_energy
Form of artificial neural network
neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order
Hopfield_network
Characteristic of an optical system
frequency from 1 to 0 (at the diffraction limit); however, the function is often not monotonic. On the other hand, when also the pattern translation is important
Optical_transfer_function
Type of electronic component
11_3-11_5 Large positive numbers means loss is large. The loss is a monotonic function of the impedance ratio. Higher ratios require higher loss. This is
Attenuator_(electronics)
Derivation of the laws of probability theory
the function g {\displaystyle g} is an associative binary operation. Additionally, Cox postulates the function g {\displaystyle g} to be monotonic. All
Cox's_theorem
Impedance matching circuit element
resistors. Large positive numbers means loss is large. The loss is a monotonic function to the impedance ratio. Higher ratios require higher loss. Speaker
L_pad
Physical law for entropy and heat
for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is
Second_law_of_thermodynamics
Measure of temperature relative to absolute zero
ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our g ( T ) {\displaystyle
Thermodynamic_temperature
Bound on probability of a random variable being far from its mean
R {\displaystyle f,g:[a,b]\to \mathbb {R} } are two monotonic functions of the same monotonicity, then 1 b − a ∫ a b f ( x ) g ( x ) d x ≥ [ 1 b − a ∫
Chebyshev's_inequality
Type of activation function
is the softplus function. Mish was obtained by experimenting with functions similar to Swish (SiLU, see above). It is non-monotonic (has a "bump") like
Rectified_linear_unit
Generalization of means
)=x~.} Monotonicity: M f {\displaystyle \ M_{f}\ } is monotonic in each of its arguments (since f {\displaystyle \ f\ } is monotonic). Continuity:
Quasi-arithmetic_mean
Abstraction of linear independence of vectors
( E ) {\displaystyle r(A)\leq r(B)\leq r(E)} . That is, rank is a monotonic function. These properties can be used as one of the alternative definitions
Matroid
result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null
Fubini's theorem on differentiation
Fubini's_theorem_on_differentiation
Neural networks
neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order
Modern_Hopfield_network
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f:U\to \mathbb {R} }
Harmonic_function
Logical connective AND
concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction (AND-ing over an empty set of
Logical_conjunction
Function uniquely mapping two numbers into a single number
{\displaystyle \pi (x,y):={\frac {x^{2}+x+2xy+3y+y^{2}}{2}}} . It is also strictly monotonic with respect to each argument, that is, for all k 1 , k 1 ′ , k 2 , k
Pairing_function
Property of a planar simple closed curve
increasing monotonic function relating the parameter of one curve to the parameter of the other. When there is a decreasing continuous function relating
Curve_orientation
Concept in economics and decision theory
monotonic) transformation. This means that the ordinal preference induced by these functions is the same (although they are two different functions)
Utility
Mapping function
In mathematics, an additive set function is a function μ \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets
Sigma-additive_set_function
Italian economist (1930–2023)
production function allowed getting three noteworthy conclusions: A rate of interest determined by the marginal productivity of capital. A monotonic inverse
Luigi_Pasinetti
Signal analysis tool
The sifting process finally stops when the residue, rn, becomes a monotonic function from which no more IMF can be extracted. From the above equations
Hilbert–Huang_transform
Counterintuitive mathematical object
Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic continuous
Pathological_(mathematics)
Geometry of the location of polynomial roots
{\frac {h(x)}{x^{k}}}} (it converges rapidly, as the derivative is a monotonic function). One can increase the number of existing R k {\displaystyle R_{k}}
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Group with a compatible partial order
partially ordered groups if it is both a group homomorphism and a monotonic function. The partially ordered groups, together with this notion of morphism
Partially_ordered_group
examines a function's monotonic properties (where the function is increasing or decreasing) focusing on a particular point in its domain. If the function "switches"
Glossary_of_calculus
Type of numerical analysis
such as the linearity imposed by linear regression, as long as the function is monotonic increasing. Another application is nonmetric multidimensional scaling
Isotonic_regression
Mathematical function conceived as a crude model
Non-monotonic, unbounded, and oscillating activation functions with multiple zeros that outperform sigmoidal and ReLU-like activation functions on many
Artificial_neuron
Theorems on the convergence of bounded monotonic sequences
number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing.
Monotone_convergence_theorem
MONOTONIC FUNCTION
MONOTONIC FUNCTION
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Biblical
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Male
Egyptian
, the son of the functionary Heknofre.
Male
Egyptian
, Functionary of the Interior.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Egyptian
, a great functionary.
MONOTONIC FUNCTION
MONOTONIC FUNCTION
Girl/Female
Assamese, Hindu, Indian, Malayalam, Marathi, Sanskrit
Excited
Boy/Male
Hindu, Indian, Tamil
Honourable Man
Surname or Lastname
English
English : nickname for a cheerful or lively person, from Middle English bragge ‘lively’, ‘cheerful’, ‘active’, also ‘brave’, ‘proud’, ‘arrogant’ (of unknown origin).
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Crystal Clear; Lord Krishna
Boy/Male
Hindu, Indian, Tamil
Pure White
Girl/Female
Hindu, Indian
Advantage
Girl/Female
Indian, Telugu
Light; Smile
Boy/Male
Hindu
Blessing, Prayer
Male
English
Pet form of English Dustin, probably DUSTY means "Thor's stone."
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Tamil, Telugu
Grown; Awakened; Shining
MONOTONIC FUNCTION
MONOTONIC FUNCTION
MONOTONIC FUNCTION
MONOTONIC FUNCTION
MONOTONIC FUNCTION
n.
The utterance of successive syllables, words, or sentences, on one unvaried key or line of pitch.
n.
One who talks in the same strain or on the same subject until weariness is produced.
a.
Alt. of Monotonical
n.
A single unvaried tone or sound.
a.
Alt. of Monotypic
a.
Of or pertaining to monogenesis.
adv.
Consisting of, or containing, one atom; as, the molecule of mercury is monatomic.
a.
Having but one type; containing but one representative; as, a monotypic genus, which contains but one species.
n.
A departure from the monotone, or reciting note, in chanting.
n.
A frequent recurrence of the same tone or sound, producing a dull uniformity; absence of variety, as in speaking or singing.
a.
For one voice; monophonic.
adv.
Having the equivalence or replacing power of an atom of hydrogen; univalent; as, the methyl radical is monatomic.
a.
Single-voiced; having but one part; as, a monophonic composition; -- opposed to polyphonic.
n.
Any irksome sameness, or want of variety.
n.
Of or pertaining to mountains; consisting of mountains.
a.
Monogenic.
a.
Producing only one kind of germs, or young; developing only in one way.
n.
Hence, want of variety; tedious monotony.
a.
Of, pertaining to, or uttered in, a monotone; monotonous.
a.
Uttered in one unvarying tone; continued with dull uniformity; characterized by monotony; without change or variety; wearisome.