homogeneous function

homogeneous function (plural homogeneous functions)

homogeneous polynomials
homogeneous polynomials

(mathematics) the ratio of two homogeneous polynomials, such that the sum of the exponents in a term of the numerator is equal to the sum of the exponents in a term of the denominator.


constant of integration

en-noun

constant of integration (plural constants of integration)

(calculus) An unspecified constant term added to a particular antiderivative to make it represent its whole family of antiderivatives, usually denoted with C.

constant d’integració

constant to be added at the indefinite integral


morphism

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1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. Čencov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50,
1° The composition of two morphisms is defined if and only if the final object of the first morphism is the initial object of the second. This composition is also a morphism, whose initial object is the initial object of the first morphism and whose final object is the final object of the second.

1992, Terrance Brown (translator), Gil Henriques, Chapter 13: Morphisms and Transformations in the Construction of Invariants, Terrance Brown (translator), Jean Piaget, Gil Henriques, Edgar Ascher (editors), Morphisms and Categories: Comparing and Transforming, page 198,
In certain extreme cases in mathematics, the synthesis of morphisms and of transformations is so intimate that one can speak of a veritable fusion. […] Essentially, categories are sets of morphisms organized into operatory systems.

The proof of toroidalization of morphisms of 3-folds to surfaces in [C3] breaks up into two parts: a reduction to prepared morphisms and then a proof of toroidalization of prepared morphisms from n-folds to [surfaces] in [CK].

2007 November, Steven Dale Cutkosky, “Toroidalization of Dominant Morphisms of 3-Folds”, in Memoirs of the, volume 190, number 890, American Mathematical Society, page 3:

(mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.


continuous function

continuous function (plural continuous functions)

mathematical analysis
mathematical analysis

(mathematical analysis) a function whose value at any point in its domain is equal its limit at the same point

անընդհատ ֆունկցիա

function whose value changes only slightly when its input changes slightly


linear transformation

linear transformation (plural linear transformations)

(linear algebra) A map between vector spaces which preserves the operations of vector addition and scalar multiplication.

lineêre transformasie

a map between vector spaces preserving vector addition and scalar multiplication


homomorphism

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Coordinate term: homology

A field homomorphism is a map from one field to another one which is additive, multiplicative, zero-preserving, and unit-preserving.

1954, Kuo-Tsai Chen, Iterated Integrals and Exponential Homomorphisms, Proceedings of the London Mathematical Society, Reprinted in 2001, Philippe Tondeur (editor), Collected Papers of K.-T. Chen, Birkhäuser, page 54,
This motivates a generalization, and exponential homomorphisms are now defined, in an algebraic fashion, from certain free products to formal power series rings with non-commutative indeterminates.

A homomorphism of presheaves h#x3A;A#x5C;rightarrowB is a collection of homomorphisms h#x5F;U#x3A;A(U)#x5C;rightarrowB(U) commuting with restrictions.

1997, Glen E. Bredon, Sheaf Theory, 2nd edition, Springer, page 8:

Definition 1.15. Let G and H be matrix Lie groups. A map #x5C;Phi from G to H is called a Lie group homomorphism if (1) #x5C;Phi is a group homomorphism and (2) #x5C;Phi is continuous.

2003, Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, page 17:

(algebra) A structure-preserving map between two algebraic structures of the same type, such as groups, rings, or vector spaces.

automorphism

structure-preserving map between algebraic structures


Bessel function

Bessel function (plural Bessel functions)

For Friedrich Bessel, German mathematician.

differential equation
differential equation

(mathematics) Any of a class of functions that are solutions to a particular form of differential equation (a Bessel equation) and are typically used to describe waves in a cylindrically symmetric system.


Green's theorem

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George Green (mathematician)
Green's theorem

Named after the mathematician George Green.

fundamental theorem of calculus
fundamental theorem of calculus

(calculus) A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or

where ∧ is the wedge product, or equivalently, as
∬_R∇·⃑Gdx,dy=∮_(∂R)⃑G∧d⃑l,

with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem.

(calculus) Letting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as


natural logarithm

natural logarithm (plural natural logarithms)

From New Latin logarithmus nātūrālis, coined in 1668 by Nicholas Mercator in his book Logarithmo-technica.

The natural logarithm can be used to integrate ¹⁄ₓ.

(mathematics) The logarithm in base e: either the function that given x returns y such that eʸ = x, or the value of y given by the function.


precalculus

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2016, Pam Goodner (lead writer/editor), Eureka Math Precalculus Study Guide, Wiley (Jossey-Bass), page ix,
As we wrote the Precalculus course, we wanted to create interest for older high school students by presenting topics in a context, something that students could relate to so that they could see the value in their work.

2017, Sheldon Axler, Precalculus: A Prelude to Calculus, Wiley, 3rd Edition, page xviii,
Similarly, many precalculus textbooks consider, for example, a colony of bacteria doubling in size every 3 hours, with the textbook then producing the formula e/3) for the growth factor after t hours.

2023, Mehdi Rahmani-Andebili, Precalculus, Springer, page v,
Moreover, in many universities and colleges, a precalculus course is mandatory for under-prepared students as the prerequisite course of Calculus 1.

(education, mathematics) A course, or set of courses, that includes (specifically) trigonometry and algebra at an appropriate level to prepare students to study calculus.


inverse Fourier transform

inverse Fourier transform (plural inverse Fourier transforms)

(mathematics) A mathematical operation that transforms a function for a discrete or continuous spectrum into a function for the amplitude with the given spectrum; an inverse transform of the Fourier transform.


differentiable

en-adj

differentiable (comparative more differentiable, superlative most differentiable)

more differentiable (comparative)
most differentiable (superlative)

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A function which is differentiable wherever it is continuous is said to possess ordinary continuity.

1896 August, W. Williams, “On the Convergency of Fourier's Series”, in The London, Edinburgh and Dublin Philosophical Magazine and .Journal of Science, volume 42, number 255, page 137:

(calculus, not comparable) Having a derivative, said of a function whose domain and codomain are manifolds.


step function

(mathematics) A function from the real line to a finite subset of the real line.


fast Fourier transform

en-noun

fast Fourier transform (plural fast Fourier transforms)

discrete Fourier transform
discrete Fourier transform

A member of a certain family of algorithms for efficiently computing the discrete Fourier transform of data.


monotonic function

monotonic function (plural monotonic functions)

independent variable
independent variable

(mathematics) A function that either never decreases or never increases as its independent variable increases.


monotone function

monotone function (plural monotone functions)

monotone Funktion

order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)

Where defined, the first derivative of a monotone function never changes sign, although it may be zero.

Section 1 contains Lebesgue's main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere.

2005, Anthony W. Knapp, Basic Real Analysis, Springer, page 357:

Monotone functions on #x5B;a,b#x5D; have nice properties. For example, they are integrable on #x5B;a,b#x5D; and have only a countable number of jump discontinuities. In this section, we shall also show that every monotone function is a function of bounded variation, and hence the class #x5C;mathit#x7B;BV#x7D;(#x5B;a,b#x5D;) contains the class of monotone functions on #x5B;a,b#x5D;.

2011, Saminathan Ponnusamy, Foundations of Mathematical Analysis, Springer, page 469:

We saw in the previous section that monotone functions have some nice properties. For example, a monotone function is continuous except possibly on a countable set. They are also closed under scalar multiplication, and the sum of two increasing functions is increasing. (Exercise (4) on page 103). However, the difference and product of two monotone functions are not necessarily monotone (Exercise (5) on page 103).

2013, Donald Yau, A First Course in Analysis, World Scientific, page 104:

(calculus) A function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x).

monotone Funktion

calculus: real function that either never decreases or never increases

nondecreasing function

function that never decreases as its independent variable increases

nonincreasing function

function that never increases as its independent variable increases


objective function

objective function (plural objective functions)

(mathematics) A function to be maximized or minimized in optimization theory.


Riemann zeta function

Riemann zeta function (usually uncountable, plural Riemann zeta functions)

Named after German mathematician Bernhard Riemann.

named-after

It is straightforward to show that the Riemann zeta function has zeros at the negative even integers and these are called the trivial zeros of the Riemann zeta function.

2004, Paula B. Cohen, “Interactions between number theory and operator algebras in the study of the Riemann zeta function (d'apres Bost-Connes and Connes)”, in David Chudnovsky, Gregory Chudnovsky, Melvyn Bernard Nathanson, editors, Number Theory: New York Seminar 2003, Springer,, page 87:

The Riemann zeta-function (which has no relation to the Weierstrass function of Chapter 8, and must not be confused with it) was originally of interest because of its connection with number theory. Since then it has served as the model for a proliferation of "zeta-functions" throughout mathematics. Some mention of the Riemann zeta-function, and treatment of the prime number theorem as an asymptotic result have become a topic treated by writers of introductory texts in complex variables.

2008, Sanford L. Segal, Nine Introductions in Complex Analysis, Revised edition, Elsevier (North-Holland), page 397:

2009, Arthur T. Benjamin, Ezra Brown (editors), Biscuits of Number Theory, Mathematical Association of America, page 195,
The Riemann zeta function is the function ζ(s)=∑ₙ₌₁ ᪲n⁻ˢ for s a complex number whose real part is greater than 1. […] The historical moments include Euler's proof that there are infinitely many primes, in which he proves
ζ(s)=∏ₚₚᵣᵢₘₑ(1-1/(pˢ))⁻¹
as well as Riemann's statement of his hypothesis and several others. Beineke and Hughes then define the moment of the modulus of the Riemann zeta function by
I_k(T)=1/T∫₀ ᪲|ζ(1/2+it)|²ᵏdt
and take us through the work of several mathematicians on properties of the second and fourth moments.

analytic continuation
analytic continuation

(number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series ζ(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.

Riemann-zeta-functie

analytic continuation of a function defined as the sum of a Dirichlet series


incompressible

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From French incompressible, from in- + compressible.

(calculus, of a vector field) Having a divergence equal to zero.


utility function

en-noun

utility function (plural utility functions)

(game theory) A mathematical function that assigns a real number to every element of the outcome space in a way that captures the agent's preferences over both simple and compound lotteries.


attractor

(mathematics, physics) A set of points or states to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed.


parameterization

parameterization (countable and uncountable, plural parameterizations)

(design, engineering) The representation of physical characteristics or properties by parameters rather than as sui generis instances; usually and especially:

(computing) Such parameterized representation in a computer model.

The representation of a curve or surface in terms of a parametric equation.


totient

From Latin totiēns (“so many times, so often”). Coined by English mathematician James Joseph Sylvester in 1879.

coin

(mathematics) The number of positive integers not greater than a specified integer that are relatively prime to it.


binary logarithm

en-noun

binary logarithm (plural binary logarithms)


cosinus

Learned borrowing from New Latin cosinus, abbreviation of complementi sinus. Doublet of cosine.

lbor

When I came to these very buildings to pass my examination I knew far better the names of all the plants in this garden than the theory of the cubic roots or the long formulæ of the sum of two cosinus.

1884, A[lbert] A[ugustin] Fauvel, Chinese Plants in Normandy, Hong Kong: […], page 4, column 1:

So, in the helicopteron, as the helix is at the same time a sustaining plane, it should be likened to a surface moving horizontally, and in which, consequenty, the resistance to motion will be to the lifting power as the sinus is to the cosinus of the angle formed by such plane with the horizon.

1884 November 29, “Aerial Navigation”, in Scientific American: A Weekly Journal of Practical Information, Art, Science, Mechanics, Chemistry, and Manufactures, volume LI, number 22, New York, N.Y.: Munn & Co., translation of original by Victor Tatin in La Nature, page 342, column 1:

And according to our choice of a symmetrical conjunction or opposition, all the cosinuses are reduced to 1, namely to coefficients build up solely by scalar Keplerian elements a, e.

1949, Contributions from the Astronomical Institute of the Charles University Prague, page 38:

The variations are described in terms of cycles of sinuses and cosinuses.

1996, Pentti Zetterberg, Matti Eronen, Markus Lindholm, “Construction of a 7500-Year Tree-Ring Record for Scots Pine (Pinus sylvestris, L.) in Northern Fennoscandia and its Application to Growth Variation and Palaeoclimatic Studies”, in Heinrich Spiecker, Kari Mielikäinen, Michael Köhl, Jens Peter Skovsgaard, editors, Growth Trends in European Forests (European Forest Institute Research Report; No. 5), Springer-Verlag Berlin Heidelberg, →ISBN, page 15:

Basically, the rotation of the matrix of the factor loadings L represents its post-multiplication, i.e. L* = LO by the rotation matrix O, which itself resembles one of the matrices included in the classical rotational Lie groups SO(m) (containing the specific m–fold combination of sinuses and cosinuses.

2007, Vladimir G. Ivancevic, Tijana T. Ivancevic, “Introduction: Human and Computational Mind”, in Computational Mind: A Complex Dynamics Perspective (Studies in Computational Intelligence; 60), Springer-Verlag Berlin Heidelberg, →ISBN, →LCCN, section 1 (Natural Intelligence and Human Mind), pages 60–61:


hypergeometric series

en-noun

hypergeometric series (plural hypergeometric series)

(mathematics) Any power series such that the ratio of the (k+1)-th and the k-th terms is a rational function of the natural integer k.


hypergeometric

hypergeometric series
hypergeometric series

(mathematics) Pertaining to mathematical entities of different kinds (series, functions, equations, ...) but strictly related to hypergeometric series


normal distribution

normal distribution (plural normal distributions)

φ_(μ,σ²)(x)=1/(σ√),e²)/(2σ²))=1/σφ((x-μ)/σ), x∈ℝ.

It begins with the normal distribution, which is the limiting form of the sample sum over an arbitrary population.

2008, James L. Johnson, Probability and Statistics for Computer Science, Wiley, page 437:

The normal distribution is the backbone of traditional statistics. We learn very early in our statistics training that the distribution of sample means, regardless of the shape of the parent distribution, approaches a normal distribution as the sample size increases.

2012, Larry Pace, Beginning R: An Introduction to Statistical Programming, Apress, page 103:

The development of the general theories of the normal distributions began with the work of de Moivre (1733, 1738) in his studies of approximations to certain binomial distributions for large positive integer n > 0.

2014, Mohammad Ahsanullah, B.M. Golam Kibria, Mohammad Shakil, Normal and Student's t Distributions and Their Applications, Springer (Atlantis Press), page 7:

probability distribution
probability distribution

probability density function
probability density function

(statistics) Any of a family of continuous probability distributions such that the probability density function is the Gaussian function


probability density

probability density (plural probability densities)

(probability theory) probability density function


dynamical system

dynamical system (plural dynamical systems)

(mathematics, physics) A system which is time-dependent.


Gaussian

List of things named after Carl Friedrich Gauss


sinusoid

From the French sinusoïde, equivalent to sinus + -oid.

Temperature during the week can be represented with a sinusoid.

(mathematics) A curve having the shape of a sine wave.


sinusoid

en-adj

sinusoid (comparative more sinusoid, superlative most sinusoid)

more sinusoid (comparative)
most sinusoid (superlative)


probit

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Blend of probability + unit. Coined by Chester Ittner Bliss in 1934: “These arbitrary probability units have been termed ‘probits’…”

The probit function, the inverse of the cumulative distribution function.


bounded function

en-noun

bounded function (plural bounded functions)

(mathematics) Any function whose values remain bounded by some constant.


Fourier series

Named after French mathematician and physicist Joseph Fourier (1768-1830), who pioneered research into such series and their application to problems of heat transfer and vibration.

1951 [Blackie & Son], R. C. H. Young (translator), Konrad Knopp, Theory and Application of Infinite Series, [1947, Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, 4th Edition], 1990, Dover, Unabridged unaltered reprint of 2nd English Edition, page 493,
As we have seen (pp. 369—370), the question of the necessary and sufficient conditions under which the Fourier series of an integrable function converges and represents the given function is one which presents very great difficulties.

2001, David Cruz-Uribe (translator), Javier Duoandikoetxea Zuazo, Fourier Analysis, [1995, Javier Duoandikoetxea Zuazo, Análisis de Fourier], American Mathematical Society, page 20,
There are excellent discussions of Fourier series and integrals in Katznelson [10] and Dym and McKean [4]. The book by R. E. Edwards [5] is an exhaustive study of Fourier series from a more modern perspective.

Unlike Taylor series, a Fourier series can describe functions that are not everywhere continuous and/or differentiable.

2006, K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, 3rd edition, Cambridge University Press, page 415:

(mathematics, mathematical analysis) Any series resulting from the decomposition of a periodic function into terms involving cosines and sines (or, equivalently, complex exponentials).

フーリエ級数

フーリエきゅうすう,fūrie kyūsū


arccotangent

(trigonometry) An inverse of the cotangent function. Symbol: arccot


sine ratio

right-angled triangle
right-angled triangle

(trigonometry) The ratio of the side of a right-angled triangle opposite to a specified angle to the hypotenuse; when expressed as a real number between 0 and 1, it defines the sine of the angle.


DFT

DFT (countable and uncountable, plural DFTs)

discrete Fourier transform
discrete Fourier transform#English

(mathematics) Initialism of discrete Fourier transform.


discrete Fourier transform

discrete Fourier transform (plural discrete Fourier transforms)

(mathematics) The process of converting a discrete time-based function into its frequency-based representation.


arccosecant

(trigonometry) function that is the compositional inverse of the cosecant function. Symbol: arccsc


arcsecant

(trigonometry) function that is the compositional inverse of the secant function. Symbol: arcsec


antidifferentiation

antidifferentiation (usually uncountable, plural antidifferentiations)

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(calculus) The process of finding the antiderivative.

integração

the process of finding the antiderivative


isosurface

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an isosurface representing a concentration of 10 grams per cubic meter

(computer graphics, geometry) A surface that represents points of a constant value within a volume of space


computable function

computable function (plural computable functions)

(computer science) A mathematical function that can be computed using an abstract computation machine such as Turing machine.


codifferentiation

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From co- + differentiation or codifferentiate + -ion.

In addition, a is called a co-exact form if the r-form a can be written as codifferentiation of a (r + l)-form β, ie, a = δβ.

1997, Bo-Yu Hou, Bo-Yuan Hou, Differential Geometry for Physicists, →ISBN, page 113:

(calculus) The act or process of obtaining the codifferential of a function, or a function which obtains the codifferential


additive function

additive function (plural additive functions)

(mathematics) A unary function that preserves the addition operation; a unary function f such that for any x and y: f(x + y) = f(x) + f(y).


multivariable

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DDE

delay differential equation
delay differential equation#English

(calculus) Initialism of delay differential equation.


delay differential equation

delay differential equation (plural delay differential equations)

(calculus) a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times


SDE

SDE (countable and uncountable, plural SDEs)

(calculus) Initialism of stochastic differential equation.


stochastic differential equation

stochastic differential equation (plural stochastic differential equations)

stochastic differential equations (plural)

(calculus) a type of differential equation in which one or more of the terms is a stochastic process resulting in a solution which is itself a stochastic process

стохастическое дифференциальное уравне́ние


strange attractor

strange attractor (plural strange attractors)

The term was coined by David Ruelle and Floris Takens.

(mathematics) An attractor that has a non-integer dimension or whose dynamics on it are chaotic.


signum function

signum function (plural signum functions)

(mathematics) The function that determines the sign of a real number, yielding −1 if negative, +1 if positive, or otherwise zero.


tractrix

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(geometry) A curve that satisfies the following property: that segment of the tangent line that lies between the point of tangency and a fixed line has length independent of the point of tangency chosen.


CDF

Initialism of cumulative distribution function.


polylogarithm

(mathematics) A function of complex numbers related to logarithms.

polilogaritmo

a function of complex numbers related to logarithms


fluxions

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The sheltered world of academia suited Clarence, who for the first time was among his intellectual peers. He excelled at geometry, algebra and fluxions.

2022, Hernan Diaz, Trust, Picador (2023), page 141:

differential calculus
differential calculus

(obsolete, mathematics) differential calculus.


fluxion

fluxion (countable and uncountable, plural fluxions)

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From Middle French fluxion, from Late Latin fluxiō, from Latin flūxus + -iō.

(obsolete, mathematics) The derivative of a function.


hypoelliptic

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hypoelliptic operator
hypoelliptic operator

(calculus) Related to or involving hypoelliptic operators.

гипоэллиптический

related to or involving hypoelliptic operators


hypoelliptic operator

hypoelliptic operator (plural hypoelliptic operators)

(calculus) A differential operator that preserves mathematical smoothness.


ar-

Shortened from area (because the function describes the area under a hyperbola), by analogy with arc-, the corresponding prefix for the circular trigonometric functions.

inverse hyperbolic function
inverse hyperbolic function#English

(trigonometry) Used to form the names of inverse hyperbolic functions, and the symbols for these functions.


logit

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Blend of logistic + unit, by analogy with probit, coined by Joseph Berkson in 1944: “I use this term [logit] for ln p/q following Bliss, who called the analogous function which is linear on x for the normal curve ‘probit.’”

operatorname logit(p)= log (p/(1-p))= log (p)- log (1-p).!,

(mathematics) the inverse of the "sigmoid" or "logistic" function used in mathematics, especially in statistics. The logit of a number p between 0 and 1 is given by the formula:


cologarithm

(mathematics) The logarithm of the reciprocal of a number (equal to the negative of the original logarithm)


catenoid

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(geometry) A three-dimensional surface formed by rotation of a catenary

caténoïde

Three-dimensional surface formed by rotation of a catenary


square wave

(calculus, engineering, signal processing) A function or waveform that alternates regularly and instantaneously between two levels.


pseudosphere

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(geometry) Any of various surfaces with constant negative Gaussian curvature.


calculist

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A mathematician who specializes in calculus.


trilogarithm

(mathematics) The form of polylogarithm having a base of three


dilogarithm

In two space-time dimensions, we arrive at a result in terms of elliptic dilogarithms.

2015, Luise Adams, Christian Bogner, Stefan Weinzierl, “The Elliptic Sunrise”, in arXiv:

(mathematics) The form of polylogarithm having a base of two


intermediate value theorem

greatest lower bound
greatest lower bound

(calculus) A theorem that states for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value.


mean value theorem

mean value theorem (plural mean value theorems)

for some ζ on the line segment between ̃x and ̌x.

1990, A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, page 51, In order to get a true bound for the range we may replace the Taylor series in (2) by the mean value theorem, which tells us that f(̃x)=f(̌x)+f'(ζ)(̃x-̌x)

In what follows, we will use the mean value theorem, another one of Lagrange's many contributions to numerical analysis.

2003, Sylvain Raynes, Ann Rutledge, The Analysis of Structured Securities, Oxford University Press, page 397:

The main existence theorems in calculus are the Intermediate Value Theorem, the Extreme Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

2007, Denise Szecsei, Calculus, The Career Press, page 10:

(calculus, uncountable) The theorem that for any real-valued function that is differentiable on an interval, there is a point in that interval where the derivative of the curve equals the slope of the straight line between the graphed function values at the interval's end points.

differentiaalilaskennan väliarvolause

theorem that for a differentiable function on an interval there is a point in the interval where the derivative equals the overall slope

Lagrange mean value theorem

theorem that a point exists where the derivative equals the overall slope


real function

(mathematics) A function whose range is a subset of the set of real numbers.


complex function

complex function (plural complex functions)

(mathematics) A function whose range is a subset of the set of complex numbers.

kompleksifunktio

a function whose range is a subset of the set of complex numbers

komplexe Funktion

with the additional constraint of the domain being a subset of the complex numbers


characteristic function

characteristic function (plural characteristic functions)

(mathematics, probability theory) A complex function completely defining the probability distribution of a real-valued random variable

karakteristinen funktio

a complex function completely defining a probability distribution


positive linear functional

positive linear functional (plural positive linear functionals)

(mathematical analysis) A kind of linear functional which yields a non-negative scalar when given a non-negative function as parameter.


linear functional

linear functional (plural linear functionals)

(mathematics) A linear mapping from a vector space to its field of scalars.


linear operator

linear operator (plural linear operators)

(mathematics, functional analysis) An operator L such that for functions f and g and scalar λ, L (f + g) = L f + L g and L λf = λ L f.


law of cosines

(trigonometry) A generalization of the Pythagorean theorem that relates the lengths of the sides of a triangle to the cosine of one of its angles.


fixpoint

It feels like I've stepped through the looking glass and am wandering in a topsy-turvy world where the fixpoints we have lived with for decades have gone. Not just moved... gone.

2020 June 3, Stefanie Foster, “Comment: The recovery starts here”, in Rail, page 3:


Airy function

(physics) A function, denoted by Ai(x), that is a solution to the Airy equation.


Taylor series

Named after English mathematician Brook Taylor, who formally introduced the series in 1715. The concept was formulated by Scottish mathematician James Gregory.

A series solution about an ordinary point of a differential equation is always a Taylor series having a nonvanishing radius of convergence. A series solution about a singular point does not have this form (except in rare cases). Instead, it may be either a convergent series not in Taylor series form (such as a Frobenius series) or it may be a divergent series.

1978, [McGraw-Hill], Carl M. Bender, Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, published 1999, page 324:

The usual procedure for deriving finite-difference equations consists of approximating the derivatives in the differential equation via a truncated Taylor series.

1980, Suhas Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis (CRC Press), page 28:

This function has its only singularity at x = 0, implying that the radius of convergence for the Taylor series around x = 1 is only unity.

1998, Kenneth L. Judd, Numerical Methods in Economics, The MIT Press, page 197:

(calculus) A power series representation of given infinitely differentiable function f whose terms are calculated from the function's arbitrary order derivatives at given reference point a; the series f(a)+(f'(a))/(1!)(x-a)+(f(a))/(2!)(x-a)²+(f'(a))/(3!)(x-a)³+⋯=∑ₙ₌₀∞(f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ.

Brook Taylor
James Gregory (mathematician)
Taylor series

泰勒展開式

power series of a function calculated from derivatives at a reference point


Green's function

Green's function (plural Green's functions)

Named after British mathematician George Green, who first developed the concept in the 1830s.

(calculus) a type of function used in the analysis of inhomogeneous differential equations.

Greense functie

function used to analyse differential equations


one-form

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Fourier sine series

Fourier sine series (plural Fourier sine series)

a Fourier series consisting solely of sine functions


Fourier cosine series

Fourier cosine series (plural Fourier cosine series)

a Fourier series consisting solely of cosine functions


error function

normally distributed
normal distribution

(mathematics, statistics, probability theory) A sigmoid function giving the probability that a measurement, under the influence of normally distributed errors with standard deviation, is within a certain distance from the mean value.


tractricoid

(geometry) A surface formed by revolving a tractrix about its asymptote.


law of sines

(trigonometry) A statement that relates the lengths of the sides of a triangle to the sines of its angles.


Ackermann function

(computing theory) One of the simplest and earliest examples of a total computable function that is not primitive recursive.


Dirac delta function

Dirac delta function (plural Dirac delta functions)

Named after English physicist Paul Dirac.

named-after

(mathematics) A function of one real argument, whose value is zero when the argument is nonzero, and whose integral is one over any interval that includes zero.


SOHCAHTOA

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(mnemonic, trigonometry) sine: opposite over hypotenuse, cosine: adjacent over hypotenuse, tangent: opposite over adjacent


triangle wave

From triangle + wave, named for its triangular shape when graphed.

(mathematics, engineering) A function or waveform that alternately ramps linearly upwards and then downwards.


sawtooth wave

From sawtooth + wave, named for its resemblance to the jagged teeth of a saw when graphed.

(mathematics, engineering) A function or waveform that repeatedly ramps upwards (usually linearly) and then sharply drops.

鋸齒波

function or waveform that ramps upwards and then sharply drops


linear form

In #x5C;mathbb#x7B;R#x7D;ⁿ, if vectors are represented as column vectors, then linear forms are represented as row vectors, and their action on vectors is given by the matrix product, with the row vector on the left and the column vector on the right.

1961 [Prentice-Hall], Richard A. Silverman (translator), Georgi E. Shilov, An Introduction to the Theory of Linear Spaces, 1974, Dover, page 66,
A more general linear form in the same space is the expression
f(x)=∑ₖ₊₁ⁿc_kζₖ
with arbitrary fixed coefficients c_1,c_2…,c_n.

In the theory of distributions, functions are replaced by linear forms on an auxiliary vector space, whose members are called test functions.

1998, F. G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, 2nd edition, Cambridge University Press, page 2:

In order to better understand the effect of reducing a K-vector subspace V#x5C;subseteqP#x5F;d modulo a generic linear form, we generalize Proposition 5.5.18 as follows.

2005, Martin Kreuzer, Lorenzo Robbiano, Computational Commutative Algebra 2, Springer, page 264:


antihyperbolic

inverse hyperbolic
inverse hyperbolic function


logistic growth

(ecology) A pattern of growth that can be represented by a logistic function.


indicator function

en-noun

indicator function (plural indicator functions)

A function which is equal to 1 for all points in its domain which belong to a given set, and is equal to 0 for all points in the domain which do not belong to that given set.

indikaattorifunktio

function equal to 1 on elements of a given set


Pythagorean trigonometric identity

Pythagorean trigonometric identity (plural Pythagorean trigonometric identities)

Pythagorean trigonometric identities (plural)

(trigonometry) The equality sin (x)²+ cos (x)²=1, holding for all real and complex x, equivalent to the application of the Pythagorean theorem to the unit circle.

jedynka trygonometryczna

statement that sum of squares of sine and cosine equals one


alysoid

Ancient Greek ἅλυσις (hálusis, “chain”).


operational calculus

(calculus) The study of calculus from the perspective of the theory of operators.


integral transform

integral transform (plural integral transforms)

(Tf)(u)=∫ limits _(t_1)K(t,u),f(t),dt

(mathematics) Any transform of the form:


Rolle's theorem

Named after French mathematician Michel Rolle (1652–1719), although his 1691 proof covered only the case of polynomial functions and did not use the methods of differential calculus.

(calculus) The theorem that any real-valued differentiable function that attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. In mathematical terms, if f:ℝ→ℝ is differentiable on (a,b) and f(a)=f(b) then ∃c∈(a,b):f'(c)=0.

teorema di Rolle

theorem that a differentiable function with points of equal value must have a point of zero slope between them


closed curve

(topology) A map from the circle, S¹, to a topological space.^((McLarty, p. 4))


theta function

(mathematics) Any of a group of special functions of several complex variables, important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms.


squeeze theorem

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(calculus) A theorem used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. Supposing that


Newton's method

(algebra, calculus) A method for finding successively better approximations to the roots (or zeroes) of a real-valued function.


chain rule

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The backward() function uses PyTorch's automatic differentiation package, torch.autograd, to differentiate and compute gradients of tensors based on the chain rule.

2021, Joe Papa, PyTorch Pocket Reference, page 48:

(calculus) A formula for computing the derivative of the functional composition of two or more functions.


pulse wave

A kind of non-sinusoidal waveform that is similar to a square wave but lacks the perfectly symmetrical shape.


Dirac delta

Named after English theoretical physicist Paul Dirac (1902–1984); the notation was chosen because the symbol behaves analogously to the Kronecker delta.

named-after

(mathematics) A function-like symbol denoted as δ, with the property ∫f(t)δ(x-t)dt=f(x), informally understood as a function with an infinite value at 0 and zero elsewhere, and formally as a distribution or a measure.


pmf

probability mass function
probability mass function#English

Initialism of probability mass function.


pdf

probability density function
probability density function#English

Initialism of probability density function.


cumulative distribution function

en-noun

cumulative distribution function (plural cumulative distribution functions)

cumulative distribution functions (plural)

A function which at each point t of the sample space has as its value the probability that a given random variable is less than (or equal) t. In symbols, F_X(t)=Pr(X<t).


K-function

mathematical analysis
mathematical analysis

(mathematical analysis) A generalization of the hyperfactorial function in mathematics to real and complex numbers.


hyperfactorial

(mathematics) The result of multiplying a given number of consecutive integers from 1 to the given number, each raised to its own power.

The result of multiplying a given number of consecutive integers from 1 to the given number, each raised to its own power.
For example, the hyperfactorial of 3 is 1¹ * 2² * 3³ = 1 * 4 * 27 = 108


beta function

Considering the equation of the binomial coefficient n choose k described in terms of the factorial, replacing n with p + q and k with p, turning the RHS upside-down and replacing factorials with the Gamma function, then replacing (p + q) choose p with Β(p, q) yields Euler's equation for the Beta function.

Euler integral of the first kind
Euler integral of the first kind

(mathematical analysis) A mathematical function that is also called the Euler integral of the first kind; it is a symmetric special function.

Betafunktion

function which generalizes the notion of a factorial


concave function

concave function (plural concave functions)

(computer science) The negative of the convex function.


set function

(computer science) A mathematical function whose input is a set (usually of real numbers or a set of points in the Euclidean or some measure space), and whose output is usually a number.


density function

density function (plural density functions)

(probability theory) Clipping of probability density function.


convex function

convex function (plural convex functions)

(computer science) A mathematical function where the line segment between any two points on the graph of the function lies above the graph in a vector space of at least two dimensions.


u-substitution

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(calculus) A method of integration in which a function of a variable is replaced with a new variable, conventionally called u.


stress function

stress function (plural stress functions)

(physics) A function that finds a more tractable method for the elastic equations that describe the deformation of an elastic body subject only to boundary surface forces.


unit function

multiplicative function
multiplicative function

(mathematics) a completely multiplicative function on the positive integers that is defined as equal to one for a special value and zero otherwise:


mass function

(astronomy, mathematics) A relationship that describes the mass distribution.


number-theoretic function

number-theoretic function (plural number-theoretic functions)

(mathematics) a real or complex valued function of n defined on the set of natural numbers that expresses some arithmetical property of n.


correlation function

correlation function (plural correlation functions)

(mathematics) function giving the statistical correlation between random variables at two different points in space or time


distribution function

distribution function (plural distribution functions)

(mathematics, physics) A function of seven variables of the form f(x,y,z,t;v_x,v_y,v_z) that gives the number of particles per unit volume in single-particle phase space.


partition function

partition function (plural partition functions)

(mathematics) generalization of the definition of a partition function in statistical mechanics


stream function

stream function (plural stream functions)

(mathematics) A function used to plot streamlines that represent the trajectories of particles in a steady flow.


distance function

distance function (plural distance functions)

(mathematics) a function used to calculate the distance between two points.


Hankel function

Hankel function (plural Hankel functions)

(mathematics) Either of two functions that are solutions to Bessel's equation.


zeta function

(mathematics) function of the complex variable s that analytically continues the sum of the infinite series ∑ₙ₌₁ ᪲1/(nˢ) that converges when the real part of s is greater than 1.


discriminant function

discriminant function (plural discriminant functions)

(mathematics) A function used to predict a categorical dependent variable by one or more continuous or binary independent variables.


Dirac comb

Named after English physicist Paul Dirac. The functions, when plotted visually, resemble the teeth of a comb.

named-after

tempered distribution
tempered distribution

Dirac delta function
Dirac delta function

(mathematics) A periodic tempered distribution constructed from Dirac delta functions for some given period. It allows both continuous and discrete phenomena to be represented in a single framework.


modular form

analytic function
analytic function#English

functional equation
functional equation#English

(mathematics) A complex analytic function satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.


Dickman function

(number theory) A function, denoted by ρ, used to estimate the proportion of smooth numbers up to a given bound.


quasicircle

We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space, which also enable us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in Douglas-Osborn space on quasicircles.

2016, Eric Schippers, Wolfgang Staubach, “Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks”, in arXiv:

(mathematics) A Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself.


digamma function

en-noun

digamma function (plural digamma functions)

(mathematics) The first of the polygamma functions, being the logarithmic derivative of the gamma function.


polygamma function

polygamma function (plural polygamma functions)

(mathematics) A meromorphic function on ℂ, defined as the (m+1)-th derivative (for a given order m) of the logarithm of the gamma function.


trigamma function

trigamma function (plural trigamma functions)

(mathematics) The second of the polygamma functions, after the digamma function.


circle inversion

circle inversion (plural circle inversions)

The first of these is obtained by the method of circle inversion first discovered by Peaucellier.

1875, A. B. Kempe, On Some New Limkages: Messenger of Mathematics, volume 4, page 121:

Nikolai Joukowski (1847-1921) was a Russian mathematician who did research in aerodynamics (James and James, 1992). He used circle inversion in the complex number plane to study airfoil shapes.

1997, Mathematical Association of America, MAA Notes, numbers 41-42, page 169:

As is common with nonlinear projections, it is possible to come up with many variants of circle inversions.[…]An obvious (but perhaps not very useful) extension of circle inversion is sphere inversion, where the spaces inside and outside a sphere are swapped.

2011, David Salomon, The Computer Graphics Manual, page 88:

(geometry, inversive geometry, for a given circle with centre O and radius r) A transformation that maps any point P ≠ O to the point P’ on the ray from O through P for which OP × OP’ = r², and which maps O to the point at infinity.


differential operator

differential operator (plural differential operators)

Define P(D) to be the differential operator given by P(D)#x3D;D²#x2B;5D#x2B;6, where D is the differentiation operator. Then P(D)(#x5C;sin(x))#x3D;-#x5C;sin(x)#x2B;5#x5C;cos(x)#x2B;6.

[…]these conditions appeared in the very first papers on ordinary differential operators.

2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328:

A differential operator P#x5C;left(D#x5C;right) is hypoelliptic if for any domain ⊂ #x5C;mathbb#x7B;R#x7D;ⁿ any solution u of the equation P#x5C;left(D#x5C;right)u#x3D;0 from the class #x5C;mathfrak#x7B;D#x7D;'#x5C;left(#x5C;Omega#x5C;right) is a function from C#x5C;infty(ω) for any open set ω ⊂⊂ #x5C;Omega.
A complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].

2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27:

Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.

1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, page 251:

(mathematics, mathematical analysis) An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives).


product rule

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(mathematics) A formula used to find the derivatives of products of two or more functions: (f·g)'=f'·g+f·g',!


chainette

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colog

(mathematics) Abbreviation of cologarithm.


Brillouin function

Brillouin function (plural Brillouin functions)

Named after the physicist Léon Brillouin (1889–1969).

statistical mechanics
statistical mechanics

A special function that appears when studying an idealized paramagnetic material in statistical mechanics.


smooth function

smooth function (plural smooth functions)

Any ANALYTIC FUNCTION is smooth. But a smooth function is not necessarily analytic.

2003, Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, →ISBN, page 419:

(mathematics) a function that has derivatives of all finite orders everywhere in its domain.


functionhood

He did not distill the essential idea of functionhood: that to each x in the domain there corresponds a unique y in the range.

1999, William Dunham, Euler: The Master of Us All, The Mathematical Association of America, published 1999, page 17:

(mathematics, logic, grammar) The state or quality of being a function.


initial value problem

initial value problem (plural initial value problems)

ordinary differential equation
ordinary differential equation

(calculus) The problem of solving an ordinary differential equation together with an initial condition (a set of specified values for the unknown function or its derivatives at some point).


exponential decay

exponential decay (countable and uncountable, plural exponential decays)

The service value of the capital stock, i.e. the product of its price and quantity component, is decomposed into a depreciation component and an interest component, similar to the decomposition which holds under exponential decay.

2017, E. Biørn, Taxation, Technology, and the User Cost of Capital, →ISBN, page 239:

A decrease where the rate of decrease is proportional to the amount remaining.


Maclaurin series

Maclaurin series (plural Maclaurin series)

Named after Scottish mathematician Colin Maclaurin (1698-1746), who made extensive use of the series.

Analytically there are, of course, two Jacobi series branching off the Maclaurin series, but they are geometrically and physically identical, and involve only an interchange of a and b.

1953, Raymond Lyttleton, The Stability of Rotating Liquid Masses, Paperback edition, Cambridge University Press, published 2013, page 42:

1995, Ralph P. Boas, Gerald L. Alexanderson (editor), Dale H. Mugler (editor), Lion Hunting and Other Mathematical Pursuits, Mathematical Association of America, page 88,
If the Maclaurin series of f and g converge for |z| < r and g(z) ≠ 0 for 0 ≤ |z| < r, then if the Maclaurin series for f is divided by the Maclaurin series for g by long division (as if the series were polynomials), the resulting series represents f / g for |z| < r.

It was almost as a by-product of this work that, in the first Turin memoir, he proved the convergence of the Maclaurin series of a function up to the singularity nearest to the origin (Section 7.5); it was in this context that he created what he called 'calculus of limits', later known as the method of majorants.

1997, Frank Smithies, Cauchy and the Creation of Complex Function Theory, Cambridge University Press, page 203:

(calculus) Any Taylor series that is centred at 0 (i.e., for which the origin is the reference point used to derive the series from its associated function); for a given infinitely differentiable complex function f, the power series f(0)+(f'(0))/(1!)x+(f(0))/(2!)x²+(f'(0))/(3!)x³+⋯=∑ₙ₌₀ ᪲(f⁽ⁿ⁾(0))/(n!),xⁿ.


law of tangents

(trigonometry) A statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.


special function

special function (plural special functions)

Any of various specific mathematical functions that have more or less established names and notations due to their importance.


Dirichlet eta function

Named after Johann Peter Gustav Lejeune Dirichlet (1805–1859), German mathematician.

Riemann zeta function
Riemann zeta function

(mathematics, mathematical analysis) The alternating sum of the Dirichlet series expansion of the Riemann zeta function:η(s)=∑ₙ₌₁ ᪲(-1)ⁿ⁻¹/nˢ=1/(1ˢ)-1/(2ˢ)+1/(3ˢ)-1/(4ˢ)+⋯.


logarithmic derivative

logarithmic derivative (plural logarithmic derivatives)

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From this Coulomb function, we can then calculate the logarithmic derivative of U²(r) at the eigenvalue for any value of r outside the core. If there exists a radius R lying in the Coulomb region and sufficiently small that the logarithmic derivative at R is a smooth function of the energy, we can obtain the logarithmic derivative for arbitrary energies within a reasonable range by interpolating it between the eigenvalues.

1955, Frank S. Ham, “The Quantum Defect Method”, in Frederick Seitz, David Turnbull, editors, Solid State Physics, Volume 1, Academic Press, page 138:

In economics, it is popular to consider the logarithmic derivative (#x5C;ln#x5C;varphi(t))'#x3D;#x5C;varphi'(t)#x2F;#x5C;varphi(t), as a convenient growth measure.

1994, Bjarne S. Jensen, The Dynamic Systems of Basic Economic Growth Models, Kluwer Academic, page 329:

In this section we consider the R-matrix method in which a solution is first found in an inner region 0 ≤ r ≤ a by expanding in a basis set yielding the logarithmic derivative of the wave function on the boundary r = a.

1995, Philip G. Burke, Charles J. Joachain, Theory of Electron-Atom Collisions: Part 1: Potential Scattering, Plenum Press, page 86:

(calculus, mathematical analysis) Given a real or complex function f, the ratio of the value of the derivative to the value of the function, (f')/f, regarded as a function.


Borel transform

Borel transform (plural Borel transforms)

(mathematics) A generalization of this in Nachbin's theorem.


sigmoid function

sigmoid function (plural sigmoid functions)

1995, Jun Han, Claudio Moraga, The Influence of the Sigmoid Function Parameters on the Speed of Backpropagation Learning, José Mira, Francisco Sandoval (editors), From Natural to Artificial Neural Computation: International Workshop on Artificial Neural Networks, Proceedings, Springer, LNCS 930, page 195,
[The] [s]igmoid function is the most commonly known function used in feed forward neural networks because of its nonlinearity and the computational simplicity of its derivative.

The first significant new insight from gamma spatial analysis emerged on re-examination of the sigmoid function representing bilateral saturation.

2012, Walter Freeman, Neurodynamics: An Exploration in Mesoscopic Brain Dynamics, Springer, page 241:

2016, Roumen Anguelov, Svetoslav Markov, Hausdorff Continuous Interval Functions and Applications, Marco Nehmeier, Jürgen Wolff von Gudenberg, Warwick Tucker (editors), Scientific Computing, Computer Arithmetic, and Validated Numerics: 16th International Symposium, SCAN 2014, Springer, LNCS 9553, page 10,
Sigmoid functions find multiple applications to neural networks and cell growth population models [14,20]. A sigmoid function on R with a range [a,b] is defined as a monotone function s(t): R →[a,b] such that lim _(t→-∞)s(t)=a, and lim _(t→∞)s(t)=b.

(mathematical analysis, statistics) Any of various real functions whose graph resembles an elongated letter "S"; specifically, the logistic function y=(eˣ)/(eˣ+1)=1/(1+e⁻ˣ).


Smarandache function

Named after Florentin Smarandache, who rediscovered the function in 1980.

(number theory) A function, denoted by S(n) for some positive integer n, that yields the smallest number s such that n divides the factorial s!. For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, so S(8) = 4.


witch of Agnesi

witch of Agnesi (plural witches of Agnesi)

Named after Italian mathematician Maria Gaetana Agnesi, who wrote about such curves, and from a mistranslation of Italian versiera, the term she used for them which means "rope", but which is spelled the same as versiera (“witch”).

(geometry) A cubic plane curve defined from two diametrically opposite points of a circle.


exterior derivative

exterior derivative (plural exterior derivatives)

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The exterior derivative of a “scalar”, i.e., a function f=f(x¹,x²,...,xⁿ) where the xⁱ’s are coordinates of ℝⁿ, is df=∂f/∂x¹dx¹+∂f/∂x²dx²+...+∂f/∂xⁿdxⁿ.

The exterior derivative of a k-blade f,dx∧dx∧...∧dx is df∧dx∧dx∧...∧dx.

The exterior derivative d may be though of as a differential operator del wedge: ∇∧, where ∇=∂/∂x¹dx¹+∂/∂x_2dx²+...+∂/∂xⁿdxⁿ. Then the square of the exterior derivative is d²=∇∧∇∧=(∇∧∇)∧=0∧=0 because the wedge product is alternating. (If u is a blade and f a scalar (function), then fu≡f∧u, so d(fu)=∇∧(fu)=∇∧(f∧u)=(∇∧f)∧u=df∧u.) Another way to show that d²=0 is that partial derivatives commute and wedge products of 1-forms anti-commute (so when d² is applied to a blade then the distributed parts end up canceling to zero.)

differential operator
differential operator

(calculus) A differential operator which acts on a differential k-form to yield a differential (k+1)-form, unless the k-form is a pseudoscalar, in which case it yields 0.


common logarithm

common logarithm (plural common logarithms)


Napierian logarithm

Napierian logarithm (plural Napierian logarithms)

(mathematics) The logarithm to the base e, named after John Napier.


Walsh function

Named after the American mathematician Joseph L. Walsh.

(mathematics) In harmonic analysis, any of a complete orthogonal set of functions that can be used to represent any discrete function.


Möbius function

Named after August Ferdinand Möbius (1790–1868), a German mathematician and theoretical astronomer who introduced it in 1832.

(combinatorics) A function from the natural numbers to the set {−1, 0, 1} which maps perfect squares to 0, prime numbers to −1, and is multiplicative.


separation constant

separation constant (plural separation constants)

The partial differential equation a sin ²(θ)Φ(ϕ)+Φ(ϕ) cos (θ)(∂Θ(θ))/(∂θ)+(∂²Φ)/(∂ϕ²)=0 can be rewritten a sin ²(θ)+ cos (θ)(∂Θ(θ))/(∂θ)=-1/(Φ(ϕ))(∂²Φ)/(∂ϕ²)≡K, where K is a separation constant that depends on neither θ nor ϕ. This then yields two ordinary differential equations.

separation of variables
separation of variables

(calculus) a constant that may be introduced upon separation of variables

Seperationskonstante

constant introduced after separation of variables


Baire function

(mathematics) A function obtained from a continuous function by transfinite iteration of the operation of forming pointwise limits of sequences of functions.


TREE

Kruskal's tree theorem
Kruskal's tree theorem

(mathematics) An extremely fast-growing function based on Kruskal's tree theorem.


d'Alembert operator

d'Alembert operator (plural d'Alembert operators)

Named after Jean le Rond d'Alembert (1717–1783), a French mathematician, mechanician, physicist, philosopher, and music theorist.

(physics) A differential operator which may be expressed as ∂_μ∂^μ=∑_(μ=0)³∂/∂x^μ∂/∂x_μ; it is the four-dimensional (Minkowski space) equivalent of the three-dimensional Laplace operator.

Jean le Rond d'Alembert
d'Alembert operator


Tanzalin method

integration by parts
integration by parts

(calculus) A method to simplify integration by parts that precalculates the necessary derivatives and antiderivatives in a table.


Gaussian function

Gaussian function (plural Gaussian functions)

(mathematics) A function of the form f(x)=a· exp (-((x-b)²)/(2c²)) for arbitrary real-number constants a, b and non-zero c; used in statistics, signal processing, etc.


artanh

/ˈɑː(ɹ)ˌtænʃ/

(trigonometry) The area hyperbolic tangent function; i.e., the inverse hyperbolic tangent function (or hyperbolic arctangent)


arcosh

/ˈɑː(ɹ)ˌkɒʃ/

(trigonometry) The area hyperbolic cosine function, i.e., the inverse hyperbolic cosine function.


arsinh

/ˈɑː(ɹ)ˌsaɪnʃ/

/ˈɑː(ɹ)ˌʃaɪn/

(trigonometry) The area hyperbolic sine function, i.e., the inverse hyperbolic sine function.


Barnes integral

Barnes integral (plural Barnes integrals)

They were introduced by Ernest William Barnes.

(mathematics) A contour integral involving a product of gamma functions, closely related to generalized hypergeometric series.


busy beaver function

busy beaver function (plural busy beaver functions)

(mathematics, computing theory) The mathematical function, denoted by Σ(n), that maps each positive integer n to the number of steps required for the busy beaver among machines with two colors and n states to terminate.


BSDE

(calculus) Initialism of backward stochastic differential equation.


coordinate map

Let (W#x5F;0,#x5C;psi) be a smooth coordinate chart for P, and let (E#x5F;1,#x5C;dots,E#x5F;#x7B;n-p#x7D;) be a local orthonormal frame for the normal bundle NP; by shrinking W#x5F;0 if necessary, we can assume that the coordinates and the local frame are defined on the same open subset W#x5F;0#x5C;subseteqP. Let #x5C;hat#x7B;W#x7D;#x5F;0#x3D;#x5C;psi(W#x5F;0)#x5C;subseteq#x5C;mathbb#x7B;R#x7D;ᵖ, and let NP#x5C;vert#x5F;#x7B;W#x5F;0#x7D; be the portion of the normal bundle over W#x5F;0. The coordinate map #x5C;psi and frame (E#x5F;j) yield a diffeomorphism #x5C;hat#x7B;W#x7D;#x5F;0#x5C;times#x5C;mathbb#x7B;R#x7D;#x7B;n-p#x7D;#x5C;rightarrowNP#x5C;vert#x5F;#x7B;W#x5F;0#x7D; defined by[…]

2019 January 2, John M. Lee, Introduction to Riemannian Manifolds (Graduate Texts in Mathematics), Springer, →ISBN, →OCLC, page 135:

differential geometry
differential geometry

coordinate chart
coordinate chart#English

(differential geometry) Synonym of coordinate chart.


coordinate chart

coordinate chart (plural coordinate charts)

A set S#x5C;subset#x5C;mathbb#x7B;R#x7D;³ is called a regular surface if each of its points has a neighborhood in S that is diffeomorphic to an open set in #x5C;mathbb#x7B;R#x7D;². That is, for every p#x5C;inS, there exists a neighborhood, V, of p in S, an open set U#x5C;subset#x5C;mathbb#x7B;R#x7D;², and a diffeomorphism #x5C;sigma#x3A;U#x5C;rightarrowV. Such a diffeomorphism #x5C;sigma is called a surface patch or a coordinate chart. A collection of surface patches that together cover all of the points of S is called an atlas for S.

2016 September 30, Kristopher Tapp, Differential Geometry of Curves and Surfaces (Undergraduate Texts in Mathematics), Springer, →ISBN, →OCLC, page 125:

(differential geometry) A homeomorphism from a local neighborhood around a point on a topological n-manifold in a Banach space to an n-dimensional Banach space (typically a subspace of the space the manifold is embedded in).


Weierstrass function

Weierstrass function (plural Weierstrass functions)

(mathematics) A real-valued function that is continuous everywhere but differentiable nowhere.


inverse tangent

inverse tangent (plural inverse tangents)


surjective function

surjective function (plural surjective functions)


SFT

sparse Fourier transform
w:sparse Fourier transform

(mathematics) Initialism of sparse Fourier transform.


STFT

short-time Fourier transform
w:short-time Fourier transform

(mathematics) Initialism of short-time Fourier transform.


coshine

Should a shine or coshine be wanted which is not in the table, advantage may be taken of the differential of cosh x being sinh x, and vice versa. Thus, sinh x and cosh x being known, the shine or coshine of x + y, y being small compared to x, may be calculated from these relationships—[…]

1931, Charles George Watson, Sags and Tensions in Overhead Lines, London: Isaac Pitman & Sons, →OCLC, page 135:

Considering equation (8) in which ϵ = −1 we have a similar situation to our previous analysis in which the hyperbolic functions replace the trigonometric ones. However, this leads to significant changes in the results, e.g., if in equation (18) the cosines are replaced by coshines the equation has no solutions since coshines are always greater than or equal to one.

1984, D. E. Hodgkinson, “Empty space-times of embedding class two”, in General Relativity and Gravitation, volume 16, number 6, Springer, →ISSN, page 574:

In (4) the function cos{h} denotes that either cosine or coshine functions are used depending upon whether the third charateristic^([sic]) root is real or imaginary.

1996, L. N. McCartney, “Stress transfer mechanics for angle-ply laminates”, in ECCM-7: Seventh European Conference on Composite Materials, volume 2, pages 236–237:

hyperbolic cosine
hyperbolic cosine#English

(uncommon) Synonym of hyperbolic cosine.


IVP

initial value problem
initial value problem#English

(calculus) Initialism of initial value problem.


second derivative

second derivative (plural second derivatives)

Coordinate term: first derivative

(calculus) The derivative of the derivative of a function.


antidifferentiate

en-verb

antidifferentiate (third-person singular simple present antidifferentiates, present participle antidifferentiating, simple past and past participle antidifferentiated)

antidifferentiates (present singular third-person)
antidifferentiating (participle present)
antidifferentiated (participle past)
antidifferentiated (past)

LL-Q1860 (eng)-Flame, not lame-antidifferentiate.wav

Back-formation from antidifferentiation, equivalent to anti- + differentiate.

If you are asked to antidifferentiate a rational function, use Partial Fractions.

(Can we date this quote?), Dave Rusin, “How to antidifferentiate”, in Department of Mathematics, University of Texas at Austin:

(mathematics, calculus) To find the antiderivative of a function.


versed cosine

trigonometric function
trigonometric function

(trigonometry) The trigonometric function 1 + cos(x).


time derivative

en-noun

time derivative (plural time derivatives)

(mathematics, calculus) A derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function.


rocket equation

The equation for how much delta-v a rocket produces: Δv = vₑ ln(m₀/m₁); where Δv is delta-v, vₑ is effective exhaust velocity, ln is the natural log function, m₀ is initial mass of the rocket, and m₁ is final mass of the rocket.


time integration

differential equation
differential equation

(mathematics, calculus) A numerical method for solving differential equations that describes the manner in which a system changes over time.


directional derivative

directional derivative (plural directional derivatives)

(calculus) A function which measures the rate of change of some other function at any point in a particular direction; formally, the directional derivative of a scalar function f( mathbf x)=f(x_1,x_2,…,x_n) along a vector mathbf v=(v_1,…,v_n) is the function ∇_(mathbf v)f defined by the limit ∇_(mathbf v)f( mathbf x)= lim _(h→0)(f( mathbf x+h mathbf v)-f( mathbf x))/h.


superconstructive

LL-Q1860 (eng)-Flame, not lame-superconstructive.wav

propositional calculus
propositional calculus

(mathematics, calculus) In which the constructive propositional calculus can be simulated or is reducible; at least as powerful for proofs as the constructive propositional calculus.


vector calculus

(calculus) The branch of mathematics that covers the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space.