map

→ Irish: mapa

Shortening or back-formation of Middle English mappemounde, mapemounde (“world map”), from Old French mapamonde, from Medieval Latin mappa mundī, compound of Latin mappa (“napkin, cloth”) and mundus (“world”). See mop for more on the first component. Doublet of mop, nape, and nappe.

Let f be a map from ℝ to ℝ

(mathematics) A function, especially a function satisfying a certain property (e.g. continuity, linearity, etc.; see Usage notes).


derivative

/dɪˈɹɪvətɪv/

From Middle French dérivatif, from Latin dērīvātus, perfect passive participle of dērīvō (“to derive”). Related to derive; by surface analysis, derive + -ative.

The derivative of x² is 2x; if f(x)#61;x², then f'(x)#61;2x

(calculus) One of the two fundamental objects of study in calculus (the other being integration), which quantifies the rate of change, tangency, and other qualities arising from the local behavior of a function.

(of a function of a single variable f(x)) The derived function of f(x): the function giving the instantaneous rate of change of f; equivalently, the function giving the slope of the line tangent to the graph of f. Written f'(x) or (df)/(dx) in Leibniz's notation, ̇f(x) in Newton's notation (the latter used particularly when the independent variable is time).


action

en-noun

action (countable and uncountable, plural actions)

→ Arabic: أَكْشِن (ʔakšin)

→ Bulgarian: екшън (ekšǎn)

→ German: Action

→ Irish: aicsean

→ Hebrew: אקשן (akshn)

→ Japanese: アクション (akushon)

→ Korean: 액션 (aeksyeon)

→ Russian: экшн (ekšn)

From Middle English accioun, accion, from Old French aucion, acciun, from Latin āctiō(n) (“act of doing or making”), from āctus + action suffix -iō(n), perfect passive participle of agere (“do, act”), ultimately from Proto-Indo-European *h₂éǵeti; see also act, active. By surface analysis, act + -ion.

der

(mathematics) A way in which each element of some algebraic structure transforms some other structure or set, in a way which respects the structure of the first. Formally, this may be seen as a morphism from the first structure into some structure of endomorphisms of the second; for example, a group action of a group G on a set S can be seen as a group homomorphism from G into the set of bijections on S (which form a group under function composition), while a module M over a ring R can be defined as an abelian group together with a ring homomorphism from R into the ring of group endomorphisms of M (which is also called the action of R on M).

The Action Sermon is quite simply, then, the eucharistic sermon.

2008, Duncan B. Forrester, Doug Gay, Worship and Liturgy in Context, scm Press, page 88:

(Christianity) A religious performance or solemn function, i.e. action sermon, a sacramental sermon in the Scots Presbyterian Church.


argument

en-noun

argument (countable and uncountable, plural arguments)

arguments (plural)
arguement (alternative obsolete)

From Middle English argument, from Anglo-Norman and Old French arguement, from Latin argumentum. The English word is analysable as argue + -ment. Doublet of argumentum.
Displaced native Old English racu and ġeflit.

/ˈɑːɡjʊmənt/

/ˈɑɹɡjʊmənt/

argument

fact or statement used to support a proposition; a reason

وَسِيط

variable that is being passed to a function

(countable, mathematics)
The independent variable of a function.


path

From Middle English path, peth, from Old English pæþ (“path, track”), from Proto-West Germanic *paþ, from Proto-Germanic *paþaz (“path”). The Proto-Germanic term is possibly borrowed from Iranian, from Proto-Iranian *pántaHh, from Proto-Indo-Iranian *pántaHs, from Proto-Indo-European *póntoh₁s, from the root *pent- (“to pass”), however this is disputed. Cognate with West Frisian paad, Dutch pad, German Pfad. Indo-Iranian cognates could be Avestan 𐬞𐬀𐬧𐬙𐬃 (paṇtā̊, “way”), Old Persian 𐎱𐎰 (p-θ /paθi/)), Sanskrit पन्था (panthā). See also English find). Doublet of panth.

lead someone down the garden path, lead someone up the garden path

"Permissive" working allows more than one train to be in a block section at one time but trains must be run at low speed in order to stop on sight behind the train in front. Such working is often authorised to allow freight trains to "bunch" together to await a path through a bottleneck instead of being strung out over several block sections, as would be necessary if absolute working were in force.

1962 October, “Talking of Trains: The collisions at Connington”, in Modern Railways, page 232:

... while the planned hourly fast 'Connect' service from Middlesbrough to Newcastle has been postponed indefinitely due to problems in finding paths for it on the East Coast main line.

2019 October, James Abbott, “Esk Valley revival: December 2019 changes”, in Modern Railways, page 78:

Echoing McNaughton's comments in 2009, it adds: "The WCML has exhausted its available train paths and no extra services could be run without further significant investment to enhance current infrastructure or build a new line.

2020 May 6, Philip Haigh, “Just one more stop on the long journey to HS2 fulfillment [sic]”, in Rail, page 65:

(rail transport) A slot available for allocation to a railway train over a given route in between other trains.


distribution

distribution (countable and uncountable, plural distributions)

distributions (plural)
distribucion (alternative obsolete)

From Old French, from Latin distributio, from distribuere 'to distribute', itself from dis- 'apart' + tribuere 'to' (from tribus).

probability distribution
probability distribution

(mathematics, statistics) A probability distribution; the set of relative likelihoods that a variable will have a value in a given interval.

تَوْزِيع

statistics: set of relative likelihoods — see also probability distribution


hyperbola

hyperbola (plural hyperbolas or hyperbolae or hyperbolæ)

hyperbolas (plural)
hyperbolae (plural)
hyperbolæ (plural)

From New Latin hyperbola, from Ancient Greek ὑπερβολή (huperbolḗ). Doublet of hyperbole.

(geometry) A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. The function y(x) = 1/x draws a hyperbola.


graph

Clipping of graphic formula. From 1878; verb from 1889.

1969 [MIT Press], Thomas Walsh, Randell Magee (translators), I. M. Gelfand, E. G. Glagoleva, E. E. Shnol, Functions and Graphs, 2002, Dover, page 19,
Let us take any point of the first graph, for example, x=1/2,y=4/5, that is, the point M_1(1/2,4/5).

Graph of a function
w:Graph of a function

(mathematics) A set of points constituting a graphical representation of a real function; (formally) a set of tuples (x_1,x_2,…,x_m,y)∈ R ᵐ⁺¹, where y=f(x_1,x_2,…,x_m) for a given function f: R ᵐ→ R . See also Graph of a function on Wikipedia.Wikipedia


function

From Middle French function, from Old French fonction, from Latin functiō (“performance, execution”), from functus, perfect participle of fungor (“to perform, execute, discharge”), from Proto-Indo-European *bʰewg- (“to enjoy”).

der

(mathematics) A relation in which each element of the domain is associated with exactly one element of the codomain.

دَالَّة

mathematics: a relation between a set of inputs and a set of permissible outputs


logarithm

/ˈlɑ.ɡəɹ.ɹɪ.ðəm/

/ˈlɑɡ.ə.ɹɪðm/

/ˈlɑɡ.əɹ.ɹɪðm/

From New Latin logarithmus, term coined by Scottish mathematician John Napier from Ancient Greek λόγος (lógos, “word, reckoning”) and ἀριθμός (arithmós, “number”); compare rational number, from analogous Latin.

For a currency which uses denominations of 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, etc., each jump in the base-10 logarithm from one denomination to the next higher is either 0.3010 or 0.3979.

(mathematics) For a number x, the exponent by which a given base number must be raised in order to obtain the power x. Written log _bx. For example, log ₁₀1000=3 because 10³=1000 and log ₂16=4 because 2⁴=16.

أَسِيس

The power to which a given base number must be raised in order to obtain a given number


sine

Borrowed from Latin sinus (“curve, bend; bosom”), a translation of Arabic جَيْب (jayb, “bosom”), a misidentification of the notation جيب (j-y-b), written without vowel diacritics, standing for Arabic جِيبَ (jība, “sine”), in turn from Sanskrit ज्या (jyā, “sine, chord, bowstring”) through the similar Sanskrit जीव (jīva, “sine, chord, life, existence”). Doublet of sinus.

(trigonometry, mathematics) In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.


operator

en-noun

operator (countable and uncountable, plural operators)

/ˈɒpəˌɹeɪtə/

/ˈɑpəˌɹeɪtɚ/

Borrowed from Latin operātor, from operor (“work, labour”). Equivalent to operate + -or.

(mathematics) A function or other mapping that carries values defined on a domain into another value or set of values in a defined range.


log

logs (plural)
logg (alternative obsolete)
logge (alternative obsolete)

To multiply two numbers, add their logs.


monotonic

From Ancient Greek μονότονος (monótonos, “monotone”) + -ικός (-ikós, “-ic”), equivalent to monotone + -ic.

suffix

independent variable
independent variable

(mathematics) Of a function: that either never decreases or never increases as its independent variable increases.

monótonu

mathematics: never decreasing or increasing as its independent variable increases


functional

en-adj

functional (comparative more functional, superlative most functional)

more functional (comparative)
most functional (superlative)

/ˈfʌŋkt͡ʃənəl/

(mathematics) Of or relating to a function or functions.

функционален

of or relating to a function or functions

(computing theory) Having semantics defined purely in terms of mathematical functions, without side-effects.

функционален

having semantics defined purely in terms of mathematical functions, without side-effects


functional

/ˈfʌŋkt͡ʃənəl/

An example of a functional is the definite integration of integrable real functions in a real interval.

(mathematics) A function that takes a function as its argument; more precisely, a function whose argument varies in a space of (real or complex valued) functions and whose value belongs to a monodimensional space.

функционал

a function that takes a function as its argument


FFT

fast Fourier transform
fast Fourier transform#English

(signal processing) Initialism of fast Fourier transform.


sigmoid

Borrowing from Ancient Greek σῑγμοειδής (sīgmoeidḗs, “of the shape of sigma”), from σῖγμᾰ (sîgmă, “sigma”) + -ο- (-o-) + -ειδής (-eidḗs, “-oid, -like”); Equivalent to sigma + -oid. Compare with French sigmoïde.

af

If we had the textbook from a hundred years in the future that contains all the simple ideas that actually work, that take so long to identify in practice — the ReLUs instead of sigmoids — for those of us who’ve been following AI for, like, more than the last couple of years and know what I’m taking about there.

2023 June 29th, “AI will kill all of us | Eliezer Yudkowsky interview” (16:12–16:28 from the start), in The David Pakman Show, spoken by Eliezer Yudkowsky, via YouTube:

sigmoid function
sigmoid function#English

(mathematics) Ellipsis of sigmoid function.


dependent variable

en-noun

dependent variable (plural dependent variables)

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

In the equation z = x² + y, z is the dependent variable.

(algebra) In an equation, the variable whose value depends on one or more variables in the equation.

(sciences) An outcome measured to see the effectiveness of the treatment.


independent variable

independent variable (plural independent variables)

In the equation z = x² + y, x and y are the independent variables.

(algebra) In an equation, any variable whose value is not dependent on any other in the equation.

(sciences) The variable that is changed or manipulated in a series of experiments.


bijection

PIE word
 *dwóh₁
From French bijection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

The present text has defined a set to be finite if and only if there exists a bijection onto a natural number, and infinite if and only if there does not exist any such bijection.

2002, Yves Nievergelt, Foundations of Logic and Mathematics, page 214:

Note in particular that a function is a bijection if and only if it's both an injection and a surjection.

2007, C. J. Date, Logic and Databases: The Roots of Relational Theory, page 167:

The basic idea is that two sets A and B have the same cardinality if there is a bijection from A to B. Since the domain and range of the bijection is not relevant here, we often refer to a bijection from A to B as a bijection between the sets, or a one-to-one correspondence between the elements of the sets.

2013, William F. Basener, Topology and Its Applications, unnumbered page:

(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.

Nicolas Bourbaki
bijection
Éléments de mathématique

биекция

function that is both a surjection and an injection


surjection

/sɜː(ɹ)ˈd͡ʒɛk.ʃən/

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

From French surjection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.
Ultimately borrowed from Latin superiectiō (“a throwing over or on; (fig.) an exaggeration, a hyperbole”).

bor

In some special cases, however, the number of surjections A#92;rightarrowB can be identified.

1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, Institute of Physics Publishing, page 220:

Let J#61;#92;cap#95;im#95;i be the (irredundant) primary decomposition of J. We associate to the pair (J,#92;omega) the element #92;textstyle#92;sum#95;i(m#95;i,#92;omega#95;i)#92;inG, where #92;omega#95;i is the equivalence class of surjections from L#47;m#95;iL#92;oplus(A#47;m#95;i)#123;n-1#125; to m#95;i#47;m#95;i² induced by #92;omega.

1999, M. Pavaman Murthy, “A survey of obstruction theory for projective modules of top rank”, in Tsit-Yuen Lam, Andy R. Magid, editors, Algebra, K-theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday, American Mathematical Society, page 168:

In Banach space theory, a mapping u#58;E#92;rightarrowF (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isometric isomorphism. Moreover, by the classical open mapping theorem, u is a surjection iff the associated mapping from E#47;#92;text#123;ker#125;(u) to F is an isomorphism.

2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43:

(set theory) A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function f:X→Y for which for every y∈Y, there is at least one x∈X such that f(x)=y.

Nicolas Bourbaki
Surjective function
Éléments de mathématique


transform

en-verb

transform (third-person singular simple present transforms, present participle transforming, simple past and past participle transformed)

transforms (present singular third-person)
transforming (participle present)
transformed (participle past)
transformed (past)
xform (alternative)

From Middle English transformen, from Old French transformer, from Latin transformo, transformare, from trans (“across”, preposition) + forma (“form”).

Be ye transformed by the renewing of your mind.

1611, The Holy Bible, […] (King James Version), London: […] Robert Barker, […], →OCLC, Romans 12:2:

(transitive) To change the nature, condition or function of; to change in nature, disposition, heart, character, etc.; to convert.

превръщам

(transitive, mathematics) To subject to a transformation; to change into another form without altering the value.

превръщам

in mathematics: subject to a transformation


transform

mathematical analysis
mathematical analysis

(mathematical analysis) An operation (often an integration) that converts one function into another.

(geology, seismology) A transform fault.


transformation

en-noun

transformation (countable and uncountable, plural transformations)

Inherited from Middle English transformacioun, from Middle French, from Ecclesiastical Latin trānsfōrmātiō.
Morphologically transform + -ation

inh+

(mathematics) The replacement of the variables in an algebraic expression by their values in terms of another set of variables; a mapping of one space onto another or onto itself; a function that changes the position or direction of the axes of a coordinate system.


asymptote

LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-asymptote.wav

circa 1650, from Ancient Greek ἀσύμπτωτη (asúmptōtē), the feminine of Apollonius Pergaeus' (circa 200 BC) Ancient Greek adjective ἀσύμπτωτος (asúmptōtos, “not falling together”), from ἀ- (a-, “not”) + συν- (sun-, “together”) + πτωτός (ptōtós, “fallen”).

af

The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F₁) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore.

1974, Stanisław Lem, translated by Michael Kandel, The Cyberiad:

(mathematical analysis) A straight line which a curve approaches arbitrarily closely as it goes to infinity. The limit of the curve; its tangent "at infinity".

asíntota

a straight line which a curve approaches arbitrarily closely

Language, in relation to thought, must ever be regarded as an asymptote.

1860, Frederic William Farrar, An Essay on the Origin of Language, page 117:

(by extension, figuratively) Anything which comes near to but never meets something else.


asymptote

en-verb

asymptote (third-person singular simple present asymptotes, present participle asymptoting, simple past and past participle asymptoted)

asymptotes (present singular third-person)
asymptoting (participle present)
asymptoted (participle past)
asymptoted (past)

2006: Neil deGrasse Tyson, The Perimeter of Ignorance
As you become more scientific, yes, the religiosity drops off, but it asymptotes.

(mathematical analysis) To approach, but never quite touch, a straight line, as something goes to infinity.


Euler's totient function

Named after Swiss mathematician Leonhard Euler (1707–1783).

named-after

(number theory) The function that calculates the total number of integers below a given integer that are coprime to it.


integral

Borrowed from Middle French integral, from Medieval Latin integrālis, from Latin integer (“entire”); see integer.

The integral of #92;frac#123;1#125;#123;x#125; on #91;#92;frac#123;1#125;#123;2#125;,1#93; is #92;ln(2), but the integral of the same function on (0,1#93; diverges. In notation, #92;int#95;#92;frac#123;1#125;#123;2#125;¹#92;frac#123;1#125;#123;x#125;dx#61;#92;ln(2), but #92;int#95;0¹#92;frac#123;1#125;#123;x#125;dx#61;#92;infty.

Stokes' theorem relates the integral over a surface of the curl of a vector field to a line integral around the boundary of that surface: #92;int#95;S#92;nabla#92;times#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;S#125;#61;#92;int#95;#123;#92;partialS#125;#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;r#125;,.

(mathematics) A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.

The integral of x² is #92;frac#123;x³#125;#123;3#125; plus a constant.

(mathematics) An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;

(mathematics, historical or obsolete) The fluent of a given fluxion in Newtonian calculus.


eigenfunction

For many quantum-mechanical problems it is important to investigate the change of eigenvalues and eigenfunctions with the continuous change of one or more parameters. The case in which one knows the eigenvalues and eigenfunctions for two special values of the parameters, and is interested in the region in between is particularly interesting.

2000, John von Neumann, E. Wigner, “On the Behaviour of Eigenvalues in Adiabatic Processes”, in Hinne Hettema, transl., edited by Hinne Hettema, Quantum Chemistry: Classic Scientific Papers, World Scientific, published 1929, page 25:

Problems that require an investigation of eigenvalues and eigenfunctions arise in connection with numerous topics in mechanics, the theory of vibrations and stability, hydrodynamics, elasticity, acoustics, electrodynamics, quantum mechanics, etc.

2005, Leonid D. Akulenko, Sergei V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications, CRC Press (Chapman & Hall), page 1:

(mathematics) A function ϕ such that, for a given linear operator D, Dϕ=λϕ for some scalar λ (called an eigenvalue).


primitive

primitives (plural)
primative (alternative obsolete)

From Middle English primitif, from Old French primitif, from Latin prīmitīvus (“first or earliest of its kind”), from prīmus (“first”); see prime. Doublet of primitivo.

der

(mathematics) A function whose derivative is a given function; an antiderivative.


transition function

en-noun

transition function (plural transition functions)

LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-transition function.wav

If (U#95;1,#92;psi#95;1) and (U#95;2,#92;psi#95;2) are two complex charts, then the transition function is #92;psi#95;#123;12#125;#58;#92;psi#95;1(U#95;1#92;capU#95;2)#92;rightarrow#92;psi#95;2(U#95;1#92;capU#95;2), given by #92;psi#95;#123;12#125;#61;#92;psi#95;2#92;circ#92;psi#95;1#123;-1#125;. We say M is a complex manifold if it has an atlas of complex charts (U,#92;psi), such that all the transition functions are holomorphic, as maps from #92;mathbb#123;C#125;ᵐ to itself.

2007, Dominic D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry (Oxford graduate texts in mathematics), Oxford University Press, →ISBN, →LCCN, page 76:

differential geometry
differential geometry

(differential geometry) A homeomorphism that bijects between the subsets of the images of two overlapping coordinate charts that are shared in the preimage: For (U_1,φ₁) and (U_2,φ₂) coordinate charts with U_1∩U_2≠∅, the transition functions φ₁₂=φ₂∘φ₁⁻¹ and its inverse φ₂₁=φ₁∘φ₂⁻¹ may be constructed.

transitiokuvaus

homeomorphism between subsets of images of overlapping coordinate charts


factorial

The expression #123;#125;#95;nC#95;r means the number of combinations of n things taken r at a time. It is also written #92;binomnr, and is equal to#92;frac#123;n(n-1)(n-2)(n-3)#92;cdots#92;cdots(n-r#43;1)#125;#123;r#33;#125;or n#33;#47;r#33;(n-r)#33;, in which r#33; is read "factorial r" and denotes the product of all the integral numbers from 1 to r inclusive.

1916 July, M. Mott-Smith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, →DOI, page 428:

This playing around with consecutive integers reminded one of the students of factorials, and he asked about products of integers, which I said could be expressed similarly as#92;prod#95;#123;k#61;1#125;ⁿk#61;1#92;cdot 2#92;cdot 3#92;cdotsn.But what does this equal?
Well, of course we can say that it is equal to n factorial and write#92;prod#95;#123;k#61;1#125;ⁿk#61;n#33; and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!

2018 September, Colin Foster, “What is the formula for factorial?”, in Mathematics in School, volume 47, number 4, page 40:

(mathematics, combinatorics) The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.


norm

From Latin norma (“a carpenter's square, a rule, a pattern, a precept”). Doublet of norma.

норма

math: function that maps vectors to non-negative reals

(mathematics) A function which satisfies a particular set of formal conditions, created to generalize the notion of the length of a vector. Formally, a real-valued function on a vector space, generally denoted v↦|v| or v↦‖v‖, that satisfies the following properties:

given a scalar k, ‖kv‖=|k|·‖v‖, where |k| is the absolute value of k;

given two vectors v,w, ‖v+w‖<‖v‖+‖w‖ (the triangle inequality).


bell curve

→ Turkish: çan eğrisi (calque)

Holonym: graph

All this is enhanced by well-chosen illustrations: the bell curve of Galton and similar frequency-curves,[…]

(probability theory) The bell-shaped curve that is typical of the normal distribution.

гаусова крива

bell shaped curve typical of the normal distribution

Gaußkurve

strictly the PDF of a normal distribution


parabola

en-noun

parabola (plural parabolas or parabolae or parabolæ)

parabolas (plural)
parabolae (plural)
parabolæ (plural)

Borrowed from New Latin parabola, from Ancient Greek παραβολή (parabolḗ), from παραβάλλω (parabállō, “I set side by side”), from παρά (pará, “beside”) + βάλλω (bállō, “I throw”). Doublet of parable, parole, and palaver.

(geometry) The conic section formed by the intersection of a cone with a plane parallel to a tangent plane to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix).


PDF

PDF (countable and uncountable, plural PDFs)

Translingual: .pdf

probability density function
probability density function

(probability theory) Alternative letter-case form of pdf (“probability density function”)


wavefunction

wavefunctions (plural)
wave function (alternative)

The interpretation of the wavefunction was one of the thorny issues facing the early pioneers of quantum theory.

2011, Brian Cox, Jeff Forshaw, The Quantum World, Allen Lane, published 2011, page 41:

(physics) A mathematical function that describes the propagation of the quantum mechanical wave associated with a particle (or system of particles), related to the probability of finding the particle in a particular region of space.


Hilbert space

Hilbert space (countable and uncountable, plural Hilbert spaces)

Named after German mathematician David Hilbert (1862–1943).

named-after

(functional analysis) A generalized Euclidean space in which mathematical functions take the place of points; crucial to the understanding of quantum mechanics and other applications.


atomic orbital

(quantum mechanics, physical chemistry) A mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus.


exponential

There were two deaths on Monday, four on Tuesday, and eight on Wednesday. The rate of increase seems exponential.

... Students apply the definition of slope to various representations of growth functions to discover differences between exponential and constant rate of growth.

2013, Yvelyne Germain-McCarthy, Bringing the NCTM Standards to Life: Best Practices, High School, page 101:

(mathematics) Characterised by a rate of change that is proportional to the value of the varying quantity, or, equivalently, by a doubling or halving over successive fixed intervals of time or other parameter, described as exponential growth or exponential decay.


exponential growth

en-noun

exponential growth (usually uncountable, plural exponential growths)

(mathematics) The growth in the value of a quantity in which the rate of growth is proportional to the instantaneous value of the quantity; for example, when the value has doubled, the rate of increase will also have doubled. The rate may be positive or negative.


hypergeometric distribution

hypergeometric distribution (plural hypergeometric distributions)

The hypergeometric distribution differs from the binomial distribution in that the random sample of n items is selected from a finite population of N items. With the hypergeometric distribution, there is no replacement. If N is large in respect to n (N>10n), the binomial distribution is a good approximation to the hypergeometric distribution.

2002, Bryan Dodson, Dennis Nolan, Reliability Engineering Handbook, QA Publishing, page 57:

2013, Alexandr A. Borovkov, unspecified translator, Probability Theory, [2009, Alexandr A. Borovkov, Teoriya Veroyatnostei, Knizhnyi dom Librokom, 5th Edition], Springer, page 1,
Sampling without replacement from a large population is considered, and convergence of the emerging hypergeometric distributions to the binomial one is formally proved.

Knowing that the asymptotic form approaches that of a hypergeometric distribution is not all that helpful since this is a complicated function.[…]To demonstrate analytically what the figure suggests graphically, it is better to work directly with the probability function (2.1.7) than with a moment generating function or a characteristic function, neither of which is easily calculable or useful in the case of a hypergeometric distribution.

2014, Mark P. Silverman, A Certain Uncertainty: Nature's Random Ways, Cambridge University Press, page 99:

(probability theory, statistics) A discrete probability distribution that describes the probability of k "successes" in a sequence of n draws without replacement from a finite population.


waveform

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

(mathematics) The shape of a wave function represented by a graph showing some dependent variable as function of an independent variable.


transcendental function

transcendental function (plural transcendental functions)

An analytic function is transcendental (i.e., is a transcendental function) if and only if its Riemann surface is noncompact.

algebraically independent
algebraically independent

(mathematical analysis) Any function that is algebraically independent of its variable(s); a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials.


Fourier transform

Fourier transform (plural Fourier transforms)

Fourier transforms (plural)
FT (alternative initialism)

Named after French mathematician and physicist Jean Baptiste Joseph Fourier, who initiated the study of what is now harmonic analysis.

Fourier transforms are not limited to acting on functions of time, but the domain of the original function is commonly called the time domain.

The Fourier transform of a function of time is a complex function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency.

Since a separate integration is needed to give each point of the transformed function, the process would become extremely tedious if it were to be attempted manually and many ingenious devices have been invented for preforming Fourier transforms mechanically, electrically, acoustically and optically.

2002, J. F. James, A Student's Guide to Fourier Transforms, 2nd edition, Cambridge University Press, page 116:

2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 1,
The trigonometric sums of 𝒢^F are thus (up to a scalar) the Fourier transforms of the characteristic functions of the G^F!!-orbits of 𝒢^F.

2012, David Brandwood, Fourier Transforms in Radar and Signal Processing, Artech House, 2nd Edition, page 1,
The Fourier transform is a valuable theoretical technique, used widely in fields such as applied mathematics, statistics, physics, and engineering.

electrical engineering
electrical engineering

(mathematical analysis, harmonic analysis, physics, electrical engineering) A particular integral transform that when applied to a function of time (such as a signal), converts the function to one that plots the original function's frequency composition; the resultant function of such a conversion.

Fourier transform
Jean Baptiste Joseph Fourier

transformada de Fourier

mathematics: a type of integral transform


Laplace distribution

exponential distribution
exponential distribution

(statistics) A probability distribution consisting of two exponential distributions, one positive and one negative.


probability distribution

probability distribution (plural probability distributions)

(statistics) A distribution of all possible values of a random variable together with an indication of their probabilities.


exponential distribution

exponential distribution (plural exponential distributions)

The exponential distribution is a basic distribution for constructing a number of other distributions. For example, the gamma distribution is obtained from the distribution of the sum of random variables where each variable follows an exponential distribution.

1998, Vijay Singh, Entropy-Based Parameter Estimation in Hydrology, Kluwer Academic, page 49:

In addition, PH distributions provide a simple framework to demonstrate how one can extend many simple results on exponential distributions to more complex models without losing computational tractability.

1999, G. Latouche, V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, page 33:

The exponential distribution was introduced briefly in Chapter 3 in connection with the Poisson discrete probability distribution function. There it was shown that if the number of failures in a given time period t was Poisson distributed, then the time to failure is distributed as an exponential distribution.

2012, Kenneth S. Stephens, Reliability Data Analysis with Excel and Minitab, ASQ Quality Press, page 145:

(statistics) Any of a class of continuous probability distributions used to model the time between events that occur independently at a constant average rate.


Laplace transform

Laplace transform (plural Laplace transforms)

Named after French polymath Pierre-Simon Laplace.

named-after

F(s)=∫₀ ᪲f(t)e⁻ˢᵗ,dt.

(mathematics) an integral transform of positive real function f(t) to a complex function F(s); given by:


Laplace operator

en-noun

Laplace operator (plural Laplace operators)

1975, Various translators, V. Ja. Sikjrjavyĭ, A Quasidifferentiation Operator and Boundary Value Problems Connected With It, V. I. Averbvh, M. S. Birman, A. A. Blahin (editors), Transactions of the Moscow Mathematical Society for the Year 1972, Volume 27, [ТРУДЫ МОСКОВКОГО МАТЕМАҬИЧЕСКОГО ОБЩЕСТВА ТОМ 27 (1972)], American Mathematical Society, page 202,
The first notion of a Laplace operator for functionals on a Hilbert space was introduced by Levy in [l], and the idea was developed further in [2]. Levy's results depended on the posthumous work of Gateaux [3] in which the Dirichlet problem in Hilbert space was considered (without any concise definition of the Laplace operator).

Nevertheless, a systematic analysis of heat semigroups for Laplace operators on metric graphs is still missing.[…]Below we will prove an upper bound on the number of negative eigenvalues (Theorem 3.7) and a lower bound on the spectrum of Laplace operators (Theorem 3.10).

2006, Vadim Kostrykin, Robert Schrader, “Laplacians on Metric Graphs: Eigenvalues, Resolvents, and Semigroups”, in Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, Peter Kuchment, editors, Quantum Graphs and Their Applications: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, American Mathematical Society, page 201:

There are lots of approaches to get rational approximation of fractional order Laplace operator.[…]The rational function approximation gives direction to realize this fractional order Laplace operator in circuit impedance and admittance forms. This section gives insight into simple method of approximating the fractional Laplace operator.

2008, Shantanu Das, Functional Fractional Calculus for System Identification and Controls, Springer, page 153:

differential operator
differential operator

(mathematics, physics) A differential operator,denoted ∆ and defined on ℝⁿ as Δ=∑ᵢ₌₁ⁿ(∂²)/(∂x_i²), used in the modeling of wave propagation, heat flow and many other applications.

оператор на Лаплас

mathematics, physics: differential operator used in modeling of wave propagation, heat flow and many other applications


isomorphism

isomorphism (countable and uncountable, plural isomorphisms)

Isomorphism on Encyclopedia of Mathematics

Natural Isomorphism on Wolfram MathWorld

(group algebra) A bijection f such that both f and its inverse f⁻¹ are homomorphisms, that is, structure-preserving mappings.


logarytm

logarytm (nominative singular)
logarytmy (nominative plural)
logarytmu (genitive singular)
logarytmów (genitive plural)
logarytmowi (dative singular)
logarytmom (dative plural)
logarytm (accusative singular)
logarytmy (accusative plural)
logarytmem (instrumental singular)
logarytmami (instrumental plural)
logarytmie (locative singular)
logarytmach (locative plural)
logarytmie (singular vocative)
logarytmy (plural vocative)

Internationalism; compare English logarithm, French logarithme, German Logarithmus, ultimately from New Latin logarithmus, from Ancient Greek λόγος (lógos) + ἀριθμός (arithmós).


absolute value

The absolute value |x| of a real number x is √, which is equal to x if x is non-negative, and -x if x is negative.

principal square root
principal square root

(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √. Denoted by vertical bars |,|.


mapping

mapping (countable and uncountable, plural mappings)

(mathematics) A function that maps every element of a given mathematical structure (eg: a set) to a unique element of another structure; a correspondence.

zobrazení

function that maps every element of a given set to a unique element of another set


antilogarithm

If x is the logarithm of y, then y is the antilogarithm of x.

(mathematics) The number of which a given number is the logarithm (to a given base).


antilog

Shortening; see log (“logarithm”), equivalent to anti- + log.

Then the desired y path can be obtained by taking the antilog of #92;lny […]

2005, Alpha C. Chiang and Kevin Wainwright, Fundamental Methods of Mathematical Economics, fourth edition, McGraw-Hill, page 484:


gamma function

The function itself was initially defined as an integral (in modern representation, Γ(x)=∫₀ ᪲e⁻ᵗtˣ⁻¹dt) for positive real x by Swiss mathematician Leonhard Euler in 1730. The name derives from the notation, Γ(x), which was introduced by Adrien-Marie Legendre (1752—1833) (he referred to it, however, as the Eulerian integral of the second kind). Both Euler's integral and Legendre's notation shift the argument with respect to the factorial, so that for integer n>0, Γ(n) = (n−1)!. Carl Friedrich Gauss preferred π(x), with no shift, but Legendre's notation prevailed. Generalisation to non-integer negative and to complex numbers was achieved by analytic continuation.

Chapter 3 deals with the p-adic gamma function.

1987, Kit Ming Yeung, Applications of p-adic gamma function to congruences of binomial coefficients, University of California, San Diego, page 3:

2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman & Hall / CRC Press, page 2,
In particular, the exponential, circular, and hyperbolic functions are rational combinations of gamma functions.

We select one mathematical object, the gamma function, and show how it grew in concept and in content from the time of Euler to the recent mathematical treatise of Bourbaki, and how, in this growth, it partook of the general development of mathematics over the past two and a quarter centuries.

2007, Philip J. Davis, “Leonhard Euler's Integral: A Historical Profile of the Gamma Function”, in William Dunham, editor, The Genius of Euler: Reflections on His Life and Work, American Mathematical Society, page 167:

(mathematical analysis) A meromorphic function which generalizes the notion of factorial to complex numbers and has singularities at the nonpositive integers; any of certain generalizations or analogues of said function, such as extend the factorial to domains other than the complex numbers.

Adrien-Marie Legendre
American Mathematical Monthly
Carl Friedrich Gauss
Leonhard Euler

fonction gamma

function which generalizes the notion of a factorial

Euler integral of the second kind

function that extends the domain of the factorial


injection

injection (countable and uncountable, plural injections)

/ɪnˈd͡ʒɛk.ʃən/

Nicolas Bourbaki
Éléments de mathématique

Borrowed from Middle French injection, from Latin iniectio. The mathematical sense is from French injection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

(mathematics) A relation on sets (X,Y) that associates each element of Y with at most one element of X.


harmonic

harmonics (plural)
harmonick (alternative obsolete)

From Latin harmonicus, from Ancient Greek ἁρμονικός (harmonikós), from ἁρμονία (harmonía, “harmony”). By surface analysis, harmony + -ic.

(mathematics) One of a class of functions that enter into the development of the potential of a nearly spherical mass due to its attraction.


transfer function

transfer function (plural transfer functions)

(systems theory) a mathematical representation of the relation between the input and output of a linear time-invariant system


parametric equation

parametric equation (plural parametric equations)

(mathematics) A set of equations that defines the coordinates of the dependent variables (x, y and z) of a curve or surface in terms of one or more independent variables or parameters.

參數方程式 /参数方程式

equations that define coordinates of dependent variables of a curve or surface


parametric

[pæɹ.əˈmɛt͡ʃ.ɹɪk]

/ˌpɛɹ.əˈmɛt.ɹɪk/

[ˌpɛɹ.əˈmɛt͡ʃ.ɹɪk]


logistic curve

Calque of French courbe logistique, coined by Belgian mathematician Pierre François Verhulst in 1845. Verhulst did not explain his choice of the name logistique; see discussion at logistic.

coined

(mathematics, biology) An S-shaped curve that models the growth of many ecosystems.


logistic function

en-noun

logistic function (plural logistic functions)

Calque of French fonction logistique; see discussion at logistic.

exponential function
exponential function

(mathematics) A function, the result of the division of two exponential functions, that gives rise to the logistic curve.


Lagrangian function

Lagrangian function (plural Lagrangian functions)

From Joseph Louis Lagrange, French mathematician and astronomer.

Lagrange's equations
Lagrange's equations

(mathematics) A function of the generalized coordinates and velocities of a dynamic system from which Lagrange's equations may be derived.


Langevin function

Langevin function (plural Langevin functions)

(physics) a mathematical function that describes the behaviour of paramagnetic materials and the dielectric properties of insulators


particular integral

particular integral (plural particular integrals)

differential equation
differential equation

(mathematics) Any solution to a differential equation.


complementary function

complementary function (plural complementary functions)

(mathematics) For a nonhomogeneous linear differential equation a_ny⁽ⁿ⁾+a_n-1y⁽ⁿ⁻¹⁾+⋯+a_1y'+a_0y=g(x), the general solution of the associated homogenous linear differential equation a_ny⁽ⁿ⁾+a_n-1y⁽ⁿ⁻¹⁾+⋯+a_1y'+a_0y=0.

solution générale de l'équation homogène correspondante


Arrhenius equation

Arrhenius equation (plural Arrhenius equations)

(physical chemistry) An equation which approximates the dependence of the rate of any chemical reaction on the temperature.


fixed point

(mathematics) A value which is unchanged by a function or other mapping. Formally: a value x for which f(x)=x.


function space

(mathematics) Any metric space whose elements are functions


functor

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

In the category of categories, #92;mathbb#123;CAT#125;, the objects are categories and the morphisms are functors.

1991, Natalie Wadhwa (translator), Yu. A. Brudnyǐ, N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, Volume I, Elsevier (North-Holland), page 143,
Choosing for U the operation of closure, regularization or relative completion, we obtain from a given functor ℱ∈𝒥ℱ the functors
◌̅F: overrightarrow X→◌̅F( overrightarrow X),F⁰: overrightarrow X→F( overrightarrow X)⁰,Fᶜ: overrightarrow X→F( overrightarrow X)ᶜ.

Given a homotopical category X and a functor u#58;A#92;rightarrowB, a homotopical u-colimit (resp. u-limit) functor on X will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram functor Xᵘ#58;Xᴮ#92;rightarrowXᴬ (47.1).

2004, William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, American Mathematical Society, page 165:

2009, Benoit Fresse, Modules Over Operads and Functors, Springer, Lecture Notes in Mathematics: 1967, page 35,
In this chapter, we recall the definition of the category of Σ_*-objects and we review the relationship between Σ_*-objects and functors. In short, a Σ_*-object (in English words, a symmetric sequence of objects, or simply a symmetric object) is the coefficient sequence of a generalized symmetric functor S(M):X→S(M,X), defined by a formula of the form
S(M,X)=⨁ ᪲ᵣ₌₀(M(r)⊗X)_(Σᵣ).

(category theory) A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows (either covariantly or contravariantly), in such a way as to preserve morphism composition and identities.


algebraic function

algebraic function (plural algebraic functions)

Quite often, algebraic functions are algebraic expressions using a finite number of terms and involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Some algebraic functions, however, cannot be expressed by such finite expressions.

The value of an algebraic function at an algebraic number is always an algebraic number.

1989, Manuel Bronstein, An Algorithm for the Integration of Elementary Functions, James H. Davenport (editor), EUROCAL '87: European Conference on Computer Algebra, Proceedings, Springer, LNCS 378, page 491,
Trager (1984) recently gave a new algorithm for the indefinite integration of algebraic functions.

1992, A. A. Goldberg, V. A. Pyana, Uniqueness Theorems for Algebraic Functions, A. B. Sossinsky (translation editor), Boris Yakovlevich Levin (editor), Entire and Subharmonic Functions, American Mathematical Society, page 199,
It is known that the algebraic function w=w(z) enables one to determine the polynomial P(z,w) up to a constant factor.

1993, Goro Kato (translator), Kenkichi Iwasawa, Algebraic Functions, American Mathematical Society, page xiv,
Riemann provided a clear picture of many difficult topics. His most significant contribution, rather than his construction of a Riemann surface from an algebraic function, was to derive the existence of algebraic functions as he built his theory based on the concept of a Riemann surface.

(algebraic geometry) Informally, any function expressible using (only) the operations of addition, subtraction, multiplication, division and raising to a rational power; more precisely, any continuous function definable as the root of some polynomial equation.


alternating function

alternating function (plural alternating functions)

(mathematics) Any function in which the interchange of two independent variables changes the sign of the dependent variable.


boxcar function

boxcar function (plural boxcar functions)

Due to the graph resembling a rectangular boxcar on a railway line.

(calculus, geometry) Any function whose value is zero except for a finite part of its argument for which it has a constant non-zero value.


circular function

circular function (plural circular functions)

trigonometric function
trigonometric function#English

(trigonometry) Synonym of trigonometric function.


cofunction

Cosine and sine are each other's cofunctions.

trigonometric function
trigonometric function

(mathematics) The trigonometric function of the complement of the supplied angle.

(programming) A costatement that takes the form of a function and can accept parameters and return a value.


comparable function

comparable function (plural comparable functions)

(mathematics) Either of a pair of functions that have the same domain and that the values of one are equal to or greater than the other for all points in that domain.


composite function

composite function (plural composite functions)

(mathematics) A function of one or more independent variables, at least one of which is itself a function of one or more other independent variables; a function of function(s).


constant function

constant function (plural constant functions)

(mathematics) A function whose value is the same for all the elements of its domain.


cost function

Any mathematical relationship between an activity and its cost.


critical function

critical function (plural critical functions)

calculus of variations
calculus of variations

(mathematics) Any function, in the calculus of variations, that satisfies the Euler equations


definite integral

definite integral (plural definite integrals)

(mathematics) The integral of a function between an upper and lower limit


indefinite integral

indefinite integral (plural indefinite integrals)

(mathematics) A function whose derivative is a given function; an antiderivative.


cylinder function

cylinder function (plural cylinder functions)

(mathematics) Any function that is a solution of the Bessel equation


decreasing function

decreasing function (plural decreasing functions)

(mathematics) Any function of a real variable whose value decreases (or is constant) as the variable increases.


strictly decreasing function

strictly decreasing function (plural strictly decreasing functions)

(mathematics) Any function of a real variable whose value decreases as the variable increases.


elementary function

elementary function (plural elementary functions)

logarithmic function
logarithmic function

(mathematics) Any function that is composed of algebraic functions, trigonometric functions, exponential functions and/or logarithmic functions, combined using addition, subtraction, multiplication and/or division

elementary function (differential algebra)


exponential function

exponential function (plural exponential functions)

independent variable
independent variable

(mathematics) Any function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms


logarithmic function

logarithmic function (plural logarithmic functions)

(mathematics) Any function in which an independent variable appears in the form of a logarithm; they are the inverse functions of exponential functions


inverse function

inverse function (plural inverse functions)

Halving is the inverse function of doubling.

If an inverse function exists for a given function, then it is unique.

The inverse function of an inverse function is the original function.

In the context of linearization, we recall the reflective property of inverse functions; the ƒ curve contains the point (a,b) if and only if the ƒ⁻¹ curve contains the point (b,a).

1995, Nicholas M. Karayanakis, Advanced System Modelling and Simulation with Block Diagram Languages, CRC Press, page 217:

An example of another function that has an inverse function is f(x)#61;4x#43;5.
Its inverse is f#123;-1#125;(x)#61;#92;frac#123;x-5#125;#123;4#125;.

2014, Mary Jane Sterling, Trigonometry For Dummies, 2nd edition, Wiley, page 51:

In words, this formula says that the derivative of a function, f, with respect to x, is the reciprocal of the derivative of its inverse function with respect to f.

2014, Mark Ryan, Calculus For Dummies, Wiley, 2nd Edition, page 147, If f and g are inverse functions, then f'(x)=1/(g'(f(x)))

(mathematics) For a given function f, another function, denoted f⁻¹, that reverses the mapping action of f; (formally) given a function f:X→Y, a function g:Y→X such that, ∀x∈X,f(x)=y⟹g(y)=x.

funció inversa

function that reverses the mapping action of a given function


dissipation function

dissipation function (plural dissipation functions)

(physics) The function governing the rate at which mechanical energy is converted to heat during viscous flow.


elliptic function

elliptic function (plural elliptic functions)

(mathematics) Any function of a complex variable which is periodic in two directions.

椭圆函数

any function of a complex variable which is periodic in two directions


entire function

entire function (plural entire functions)

(mathematics) Any function of a complex variable that is holomorphic throughout the complex plane


even function

(mathematics) Any function whose value is unchanged if the independent variable changes sign i.e. f(x) = f(-x)


explicit function

explicit function (plural explicit functions)

(mathematics) Any function whose value may be directly calculated from the independent variable.


implicit function

implicit function (plural implicit functions)

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

An important class of implicit functions comprises those defined by equations of the form x-f(y)#61;0. Choosing y as the value means that the implicit function is the inverse of f. This type of implicit function is of particular interest when that inverse is not expressible as a closed-form expression, and thus is difficult to study directly.

Thus, in the equation,
#92;quad#92;quad#92;quaday²#43;bxy#43;cx²#43;d#61;0;
y is an implicit function of x. Implicit functions are generally connected with their variables by one or more equations. When these equations are solved the implicit equation becomes explicit.
[…]The differential of an implicit function may be found without first finding the function itself.

1871, William G. Peck, Practical Treatise on the Differential and Integral Calculus, A. S. Barnes & Company, page 50:

The process of evaluating the derivatives in the case of implicit functions is exemplified below.

1875, J. Minot Rice, W. Woolsey Johnson, The Elements of the Differential Calculus, Part 2, Washington: Government Printing Office, page 71:

In this section we discuss the analysis of implicit functions for addressing questions like this.

2002, Michael W. Klein, Mathematical Methods for Economics, Addison-Wesley, page 245:

(mathematical analysis, algebraic geometry) A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially where said equation is such that the value is not directly calculable from the other variables.

неявна функция

function defined as the solution to an implicit equation


function multiplier

function multiplier (plural function multipliers)

An electronic analog computing device that accepts as input two periodic functions and outputs their mathematical product


function table

(mathematics) A printed tabulation of selected values of a function corresponding to selected independent variables; e.g. a table of logarithms.


harmonic function

harmonic function (plural harmonic functions)

(mathematics) A function of two, three or n variables which is a solution to any of Laplace's equations


Heaviside unit function

Heaviside unit function (plural Heaviside unit functions)

(mathematics) The function whose value is zero if its independent variable is negative, and one otherwise.


hypergeometric function

hypergeometric function (plural hypergeometric functions)

hypergeometric equation
hypergeometric equation

hypergeometric series
hypergeometric series

(mathematics) Any function that is a solution of the hypergeometric equation; usually an expansion of an hypergeometric series


identity function

identity function (plural identity functions)

(mathematics) A function whose value is always the same as its independent variable, and for which the codomain equals the domain.

funció identitat

a function whose value is always the same as its independent variable


impulse function

impulse function (plural impulse functions)

(mathematics) Any generalized function which is defined by its behavior under integration.


incidence function

incidence function (plural incidence functions)

(mathematics) A function that assigns a pair of vertices to each edge of a graph.


increasing function

increasing function (plural increasing functions)

(mathematics) Any function of a real variable whose value increases (or is constant) as the variable increases.


strictly increasing function

strictly increasing function (plural strictly increasing functions)

(mathematics) Any function of a real variable whose value increases as the variable increases.


independent function

independent function (plural independent functions)

(mathematics) Any of a set of functions the value of which can not be deduced from that of all the others.


initial mass function

(astronomy) a relationship that specifies the mass distribution of a newly formed stellar population


integrable function

integrable function (plural integrable functions)

(mathematics) Any function that possesses a finite integral


integral function

integral function (plural integral functions)


linear function

linear function (plural linear functions)

(mathematics) Any function whose graph is a straight line: f(x)=ax+b

(mathematics) Any function whose value on the sum of two elements is the sum of the values of the function on the two elements and whose value on the product of a scalar times an element is the scalar times the value of the function on the element: f(ax+by)=af(x)+bf(y)


measurable function

measurable function (plural measurable functions)

mathematical analysis
mathematical analysis

(mathematical analysis) A function whose codomain is a topological space and domain is a measurable space, with the property that the inverse image of every open set in its codomain is a measurable set in its domain.


membership function

membership function (plural membership functions)

(mathematics) A generalization of the indicator function for a fuzzy set which assigns a truth value (0 or 1) to each element.


natural function

natural function (plural natural functions)

(mathematics) An ordinary trigonometric function


nuclear response function

nuclear response function (plural nuclear response functions)

(particle physics) A relationship that describes the probability of a particle that collides with another being scattered in any particular manner


odd function

(mathematics) Any function whose value changes sign if the independent variable changes sign i.e. f(-x) = -f(x)


orthogonal function

orthogonal function (plural orthogonal functions)

(functional analysis) Either of a pair of functions whose inner product is zero


orthonormal function

orthonormal function (plural orthonormal functions)

(functional analysis) An orthogonal function whose inner product with itself is unity


partial function

partial function (plural partial functions)

total function

1967 [John Wiley & Sons], Stephen Cole Kleene, Mathematical Logic, 2002, Dover, page 244,
The Church-Turing thesis applies to partial functions on the same grounds as to total functions (§ 41).

1991, Michel Bidoit, Hans-Jörg Kreowski, Pierre Lescanne, Fernando Orejas, Donald Sannella (editors), Algebraic System Specification and Development: A Survey and Annotated Bibliography, Springer, LNCS 501, page 15,
Nowadays it seems quite clear that if algebraic specifications are to be used as a powerful and realistic tool for the development of complex systems they should permit the specification of partial functions. […] There are essentially two ways of specifying partial functions.

2006, Paulo Oliva, Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis, Arnold Beckmann, Ulrich Berger, Benedikt Löwe, John V. Tucker (editors), Logical Approaches to Computational Barriers: 2nd Conference on Computability in Europe, Proceedings, Springer, LNCS 3988, page 432,
In the following σ and τ will denote finite partial functions from N to N , i.e. partial functions which are defined on a finite domain. A partial function which is everywhere undefined is denoted by ⟨⟩, whereas a partial function defined only at position k (with value n) is denoted by ⟨k,n⟩.

(mathematics) A function whose domain is a subset of the set on which it is formally defined; i.e., a function f: X→Y for which values f(x) are defined only for x ∈ W, where W ⊆ X.

hlutskilgreint fall

mathematics: a function whose domain is a subset of the set on which it is formally defined


penalty function

penalty function (plural penalty functions)

(mathematics) Any function that applies constraints to a maximum or minimum problem


periodic function

periodic function (plural periodic functions)

(mathematics) Any function whose value repeats after the regular addition of a period to its independent variable; i.e., a function ƒ such that, for some positive constant p, ƒ(x + p) = ƒ(x) for all x.


phase function

(astronomy) A function that describes the angular distribution of light reflected from a body when illuminated from a specific direction


point function

(mathematics) Any function whose values are points


probability density function

probability density function (plural probability density functions)

(probability theory) Any function whose integral over a set gives the probability that a random variable has a value in that set.


probability mass function

probability mass function (plural probability mass functions)

probability mass functions (plural)
pmf (alternative initialism)

(mathematics) A function that gives the relative probability that a discrete random variable is exactly equal to some value.


psi function

(mathematics) A function of a complex variable obtained by differentiating the logarithm of a gamma function; the digamma function


quadratic function

quadratic function (plural quadratic functions)

(mathematics) Any function whose value is the solution of a quadratic polynomial


radial distribution function

radial distribution function (plural radial distribution functions)

(mathematics, physics) A function which specifies the average density of atoms, molecules etc in three dimensions from a given point.


random function

random function (plural random functions)

(mathematics) A function chosen at random from a set of functions.


rational function

rational function (plural rational functions)

Our first problem is that of interpolation in prescribed points to a given function by a rational function whose poles are given.

1960, J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 3rd edition, American Mathematical Society, page 184:

By Theorem 2.3.2., we have that the right-hand side of this equation can be equal to a rational function only if that rational function is equal to zero.

1970, Ellis Horowitz, Algorithms for Symbolic Integration of Rational Functions, University of Wisconsin-Madison, page 24:

Let #92;mathcal#123;C#125; be the class of continuous maps of #92;mathbb#123;C#125;#95;#92;infty into itself and let #92;mathcal#123;R#125; be the subclass of rational functions.[…]Now #92;mathcal#123;R#125; is a closed subset of #92;mathcal#123;C#125;#95;#92;infty because if the rational functions R#95;n converge uniformly to R on the complex sphere, then R is analytic on the sphere and so it too is rational.

2000, Alan F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, Springer, page 45:

(mathematics, complex analysis, algebraic geometry) Any function expressible as the quotient of two (coprime) polynomials (and which thus has poles at a finite, discrete set of points which are the roots of the denominator).

rationaalifunktio

function expressible as the quotient of polynomials


recursive function

recursive function (plural recursive functions)

Computable function
Recursion (computer science)

(computing) Any function that uses recursion and can call itself until a certain condition is met.


regular function

regular function (plural regular functions)


sample function

sample function (plural sample functions)

(statistics) Any function used to obtain a set of samples from a given population


scalar function

scalar function (plural scalar functions)

(mathematics) Any function whose domain is a vector space and whose value is its scalar field


scattering function

scattering function (plural scattering functions)

(physics) A function which describes the intensity of scattered radiation in any given direction


simple function

simple function (plural simple functions)

(mathematical analysis) Any complex-valued measurable function whose range is finite.


single-valued function

single-valued function (plural single-valued functions)

(mathematics) Any function in which each point in the domain maps to just one point in the range


symmetric function

symmetric function (plural symmetric functions)

(mathematics) Any function, of multiple variables, whose value is independent of the order of the variables, i.e. f(a,b) = f(b,a).


truth function

truth functions (plural)
truth-function (alternative)

(mathematics, logic) A Boolean function whose value is interpreted as truth or falsity


Turing computable function

Turing computable function (plural Turing computable functions)

From Alan Turing, English mathematician, logician, and cryptographer.

(mathematics) Any function whose value may be computed using a Turing machine.

Turing-reiknanlegt fall

any function whose value may be computer using a Turing machine


user-defined function

user-defined function (plural user-defined functions)

(computing) A program or subroutine, provided by the user rather than being built into the system, that may return a value and may take input parameters.


spherical harmonics

(mathematics) The solutions to Laplace's equation using spherical coordinates


Boolean function

Boolean function (plural Boolean functions)

(algebra, logic, computing) Any function based on the operations AND, OR and NOT, and whose elements are from the domain of Boolean algebra

fonction booléenne

function based on AND, OR and NOT operations


antiderivative

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(calculus) A function whose derivative is a given function; an indefinite integral


regression

regression (countable and uncountable, plural regressions)

/ɹɪˈɡɹɛʃ.ən/

/ɹiːˈɡɹɛʃ.ən/

/ɹɪˈɡɹɛʃ.ən/

/ɹəˈɡɹɛʃ.ən/

/ɹəˈɡɹeʃ.ən/

/rɪˈɡrɛ.ʃən/

Learned borrowing from Latin regressio. Equivalent to regress + -ion.
The statistics sense comes from regression to the mean.

(statistics) An equation using specified and associated data for two or more variables such that one variable can be estimated from the remaining variable(s).


analytic function

analytic function (plural analytic functions)

There is a large and important literature concerned with the question of how to characterize precisely the boundary values of analytic functions. For example, when the analytic functions are of Hardy Class H#42; in a half plane, then their boundary values are attained in the L2 norm and, in fact, the analytic function can be reproduced by a Cauchy integral of the boundary value.

1966, E. J. Beltrami, M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions, Academic Press, page vii:

2000, Vladimir V. Mityushev, Sergei V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions: Theory and Applications, Chapman & Hall / CRC, page v,
Thus we have limited ourselves by boundary value problems for analytic functions, and some related problems. The constructive ideas are always in our mind, so, the basic goal of this book is to be useful for experts in analytic function theory and for nonspecialists in it, and even for non-mathematicians who apply these methods in their research.

2010, Emmanuel Fricain, Andreas Hartmann, Regularity on the Boundary in Spaces of Holomorphic Functions on the Unit Disk, Javad Mashreghi, Thomas Ransford, Kristian Seip (editors, Hilbert Spaces of Analytic Functions, American Mathematical Society, page 91,
We review some results on regularity on the boundary in spaces of analytic functions on the unit disk connected with backward shift invariant subspaces in Hᵖ.

(mathematical analysis) Any smooth (infinitely differentiable) function f, defined on an open set D⊆ℂ( textit or⊆ℝ), whose value in some neighbourhood of any given point x_0∈D is given by the Taylor series ∑ₙ₌₀ ᪲(f⁽ⁿ⁾(x_0))/(n!)(x-x_0)ⁿ.


Poisson distribution

Poisson distribution (plural Poisson distributions)

After French mathematician Siméon Denis Poisson (1781-1840), who introduced the distribution in his 1837 work Recherches sur la probabilité des jugements en matière criminelle et en matière civile ("Research on the Probability of Judgements in Criminal and Civil Matters"), although prior work had been done by Abraham de Moivre.

For the modeling of many phenomena there is a need for enlarging the family of Poisson distributions. One possibility of doing this is the use of mixed Poisson distributions (outlined above).

1993, Rolf-Dieter Reiss, A Course on Point Processes, Springer, page 62:

It is worth noting, therefore, that unlike the binomial distribution in which we are interested in observing both 'success' and 'failure', the application of the Poisson distribution is concerned only with occurrences rather than non-occurrences.

2000, Michael C. Fleming, Joseph G. Nellis, Principles of Applied Statistics: An Integrated Approach Using MINITAB and Excel, 2nd edition, Thomson, page 116:

A mixture of Poisson processes will have a spike count described by a mixture of Poisson distributions. A model allowing a mixture of a few Poisson distributions adds substantial flexibility, while keeping the calculations reasonably simple.

2007, “4: Order Statistics for Decoding of Spike Trains”, in Kenji Doya, Shin Ashii, Alexandre Pouget, Rajesh P. N. Rao, editors, Bayesian Brain: Probabilistic Approaches to Neural Coding, The MIT Press, page 83:

As indicated, problems involving the number of occurrences of independent, randomly occurring events per unit of space or time can often be modeled with the Poisson distribution.

2017, William Q. Meeker, Gerald J. Hahn, Luis A. Escobar, Statistical Intervals: A Guide for Practitioners and Researchers, 2nd edition, Wiley, page 128:

negative binomial distribution
negative binomial distribution

(statistics) Any of a class of discrete probability distributions that express the probability of a given number of events occurring in a fixed time interval, where the events occur independently and at a constant average rate; describable as a limit case of either binomial or negative binomial distributions.

Abraham de Moivre
Poisson distribution
Siméon Denis Poisson


primitive recursive

(mathematics) Of a function, capable of being constructed from the zero function, successor function, and projection functions, by a finite number of applications of composition and recursion.


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.


homomorphism

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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#58;A#92;rightarrowB is a collection of homomorphisms h#95;U#58;A(U)#92;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 #92;Phi from G to H is called a Lie group homomorphism if (1) #92;Phi is a group homomorphism and (2) #92;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


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) 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


Bessel function

Bessel function (plural Bessel functions)

For Friedrich Bessel, German mathematician.

(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.


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.


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.


step function

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


fast Fourier transform

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)

monotonic functions (plural)
monotone function (alternative)

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


monotone function

monotone function (plural monotone functions)

monotone functions (plural)
monotonic function (alternative)

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 #91;a,b#93; have nice properties. For example, they are integrable on #91;a,b#93; 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 #92;mathit#123;BV#125;(#91;a,b#93;) contains the class of monotone functions on #91;a,b#93;.

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).

монотонна функция

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)

Riemann zeta functions (plural)
Riemann zeta-function (alternative)
Riemann's zeta function (alternative)

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


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.


函數

/xän³⁵ ʂu⁵¹/

/xän³⁵ ʂuɻʷ⁵¹/

/ham²⁴⁻²² sɔ²¹/

/ham²⁴⁻²² sɔ⁴¹/

/ham¹³⁻²² sɔ²¹/

/ham²⁴⁻¹¹ sɔ¹¹/

/ham²³⁻³³ sɔ²¹/

/xän³⁵ ʂu⁵¹/

/xän³⁵ ʂuɻʷ⁵¹/

/ham²⁴⁻²² sɔ²¹/

/ham²⁴⁻²² sɔ⁴¹/

/ham¹³⁻²² sɔ²¹/

/ham²⁴⁻¹¹ sɔ¹¹/

/ham²³⁻³³ sɔ²¹/


modul

module (plural)
modul (accusative indefinite nominative singular)
modulul (accusative definite nominative singular)
module (accusative indefinite nominative plural)
modulele (accusative definite nominative plural)
modul (indefinite singular)
modulului (definite singular)
module (indefinite plural)
modulelor (definite plural)
modulule (singular vocative)
modulelor (plural vocative)


parameterization

parameterization (countable and uncountable, plural parameterizations)

parameterizations (plural)
parametrization (alternative)
parameterisation (alternative)
parametrisation (alternative)

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)


hypergeometric

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


hypergeometric series

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.


funkcja

funkcyjny (adjective)
funkcja (nominative singular)
funkcje (nominative plural)
funkcji (genitive singular)
funkcji (genitive plural)
funkcyj (archaic genitive plural)
funkcji (dative singular)
funkcjom (dative plural)
funkcję (accusative singular)
funkcje (accusative plural)
funkcją (instrumental singular)
funkcjami (instrumental plural)
funkcji (locative singular)
funkcjach (locative plural)
funkcjo (singular vocative)
funkcje (plural vocative)
funkcyja (alternative Middle Polish)
funkcya (alternative)

→ Kashubian: fónkcjô

Learned borrowing from Latin fūnctiō. First attested in 1656–1688.

ref

“FUNKCYJA”, in Elektroniczny Słownik Języka Polskiego XVII i XVIII Wieku [Electronic Dictionary of the Polish Language of the XVII and XVIII Century], 08.09.2008

(mathematics) function (a relation in which each element of the domain is associated with exactly one element of the codomain)


normal distribution

en-noun

normal distribution (plural normal distributions)

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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

Gaussians (plural)
Gaußian (alternative)

These conditions provide a framework which encompasses some notable approximation kernels including splines, so-called cardinal functions, and many radial basis functions such as the Gaussians and general multiquadrics.

2016, Keaton Hamm, Jeff Ledford, “Regular Families of Kernels for Nonlinear Approximation”, in arXiv:


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

bounded function (plural bounded functions)

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


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.


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.


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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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.


CDF

Initialism of cumulative distribution function.

КФР

initialism of cumulative distribution function


polylogarithm

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

polilogaritmo

a function of complex numbers related to logarithms


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)


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


trilogarithm

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


złożenie

złożenie (nominative singular)
złożenia (nominative plural)
złożenia (genitive singular)
złożeń (genitive plural)
złożeniu (dative singular)
złożeniom (dative plural)
złożenie (accusative singular)
złożenia (accusative plural)
złożeniem (instrumental singular)
złożeniami (instrumental plural)
złożeniu (locative singular)
złożeniach (locative plural)
złożenie (singular vocative)
złożenia (plural vocative)

(countable, mathematics) composition (applying a function to the result of another)


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

характеристична функция

a complex function completely defining a probability distribution


linear functional

linear functional (plural linear functionals)

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


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 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.


bijekcja

pl-noun

bijekcja f (related adjective bijekcyjny)

bijekcyjny (adjective)
bijekcja (nominative singular)
bijekcje (nominative plural)
bijekcji (genitive singular)
bijekcji (genitive plural)
bijekcyj (archaic genitive plural)
bijekcji (dative singular)
bijekcjom (dative plural)
bijekcję (accusative singular)
bijekcje (accusative plural)
bijekcją (instrumental singular)
bijekcjami (instrumental plural)
bijekcji (locative singular)
bijekcjach (locative plural)
bijekcjo (singular vocative)
bijekcje (plural vocative)

(set theory) bijection (one-to-one correspondence, a function which is both a surjection and an injection)


suriekcja

suriekcja f (related adjective suriekcyjny)

suriekcyjny (adjective)
suriekcja (nominative singular)
suriekcje (nominative plural)
suriekcji (genitive singular)
suriekcji (genitive plural)
suriekcyj (archaic genitive plural)
suriekcji (dative singular)
suriekcjom (dative plural)
suriekcję (accusative singular)
suriekcje (accusative plural)
suriekcją (instrumental singular)
suriekcjami (instrumental plural)
suriekcji (locative singular)
suriekcjach (locative plural)
suriekcjo (singular vocative)
suriekcje (plural vocative)

(set theory) alternative spelling of surjekcja


iniekcja

iniekcja f (related adjective iniekcyjny)

iniekcyjny (adjective)
iniekcja (nominative singular)
iniekcje (nominative plural)
iniekcji (genitive singular)
iniekcji (genitive plural)
iniekcyj (archaic genitive plural)
iniekcji (dative singular)
iniekcjom (dative plural)
iniekcję (accusative singular)
iniekcje (accusative plural)
iniekcją (instrumental singular)
iniekcjami (instrumental plural)
iniekcji (locative singular)
iniekcjach (locative plural)
iniekcjo (singular vocative)
iniekcje (plural vocative)

(set theory) injection (relation on sets (X,Y) that associates each element of Y with at most one element of X)


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.


Green's function

en-noun

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.

фунцкия на Грийн

function used to analyse differential equations


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.


Ackermann function

/ˈækə(ɹ)mən ˈfʌŋkʃən/

LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-Ackermann function.wav

(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)

Dirac delta functions (plural)
Dirac delta (alternative)
Dirac's delta function (alternative)
Dirac's delta (alternative)

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.


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 /锯齿波

function or waveform that ramps upwards and then sharply drops


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.


linear form

In #92;mathbb#123;R#125;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#92;subseteqP#95;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:


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.

характеристична функция

function equal to 1 on elements of a given set


funkcja podcałkowa

funkcja podcałkowa (nominative singular)
funkcje podcałkowe (nominative plural)
funkcji podcałkowej (genitive singular)
funkcji (genitive plural)
funkcyj  podcałkowych (archaic genitive plural)
funkcji podcałkowej (dative singular)
funkcjom podcałkowym (dative plural)
funkcję podcałkową (accusative singular)
funkcje podcałkowe (accusative plural)
funkcją podcałkową (instrumental singular)
funkcjami podcałkowymi (instrumental plural)
funkcji podcałkowej (locative singular)
funkcjach podcałkowych (locative plural)
funkcjo podcałkowa (singular vocative)
funkcje podcałkowe (plural vocative)


integral transform

integral transform (plural integral transforms)

(Tf)(u)=∫ limits _(t_1)K(t,u),f(t),dt

(mathematics) Any transform of the form:


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.


三角函數

ㄙㄢ ㄐㄧㄠˇ ㄏㄢˊ ㄕㄨˋ

ㄙㄢ ㄐㄧㄠˇ ㄏㄢˊ ㄕㄨˋ

саньцзяо ханьшу

/sän⁵⁵ t͡ɕi̯ɑʊ̯²¹⁴⁻²¹ xän³⁵ ʂu⁵¹/

ㄙㄢ ㄐㄧㄠˇㄦ ㄏㄢˊ ㄕㄨˋㄦ

саньцзяорханьшур

/sän⁵⁵ t͡ɕi̯aʊ̯ɻʷ²¹⁴⁻²¹ xän³⁵ ʂuɻʷ⁵¹/

/saːm⁵⁵ kɔːk̚³ haːm²¹ sou̯³³/

/sän⁵⁵ t͡ɕi̯ɑʊ̯²¹⁴⁻²¹ xän³⁵ ʂu⁵¹/

/sän⁵⁵ t͡ɕi̯aʊ̯ɻʷ²¹⁴⁻²¹ xän³⁵ ʂuɻʷ⁵¹/

/saːm⁵⁵ kɔːk̚³ haːm²¹ sou̯³³/


Newton's method

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


pulse wave

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