Jacobian

From Jacobi + -an, after Carl Gustav Jacob Jacobi, a German mathematician of the 19th century.

As a demonstration, we first establish the correctness of the Jacobian matrices for kinetic models of hydrogen, methane, ethylene, and isopentanol oxidation, then demonstrate the performance achievable on CPUs and GPUs using pyJac via matrix evaluation timing comparisons..

2016, Kyle E. Niemeyer, Nicholas J. Curtis, Chih-Jen Sung, “pyJac: analytical Jacobian generator for chemical kinetics”, in arXiv:

(calculus) A Jacobian matrix or its associated operator.

In terms of per-bit security, elliptic curves and Jacobians of genus 2 curves appear to be roughly equivalent. However, when it comes to efficient and side-channel-aware implementations, we see a curious divergence.

2015, Ping Ngai Chung, Craig Costello, Benjamin Smith, “Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes”, in arXiv:

(calculus) The determinant of such a matrix.


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)#x3D;x², then f'(x)#x3D;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).

The derivative of f(x)#x3D;x³ at x#x3D;2 is 12.

The value of such a derived function for a given value of its independent variable: the rate of change of a function at a point in its domain.

(Of more general classes of functions) Any of several related generalizations of the derivative: the directional derivative, partial derivative, Fréchet derivative, functional derivative, etc.

differential operator
differential operator

(generally) The linear operator that maps functions to their derived functions, usually written D; the simplest differential operator.


map

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).


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 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

Pressing a piano key causes the action of the hammer on the string.

The way in which a mechanical device acts when used; especially a firearm.

pump-action shotgun

(firearms) The way in which cartridges are loaded, locked, and extracted from the mechanism.

(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).


argument

en-noun

argument (countable and uncountable, plural arguments)

From Middle English argument, from Anglo-Norman and Old French arguement, from Latin argūmentum (“argument (for a position); evidence, proof; point, theme; thesis, topic; plot (in theatre)”), from arguere + -mentum (suffix indicating the instrument, medium, or result of something). Arguere is the present active infinitive of arguō (“to argue, assert, declare; to make clear, prove, show; to accuse, charge with, reprove; to blame, censure; to denounce as false”), either ultimately from Proto-Indo-European *h₂erǵ- (“silver, white; glittering”), or from Hittite [script needed] (arkuwā(i)-, “to make a plea, state one’s case”). The English word is analysable as argue + -ment. Doublet of argumentum.
Displaced native Old English racu and ġeflit.

der

/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.

A parameter at a function call; an actual parameter, as opposed to a formal parameter.

[I]n vttering the ſtuffe ye receiued of the one, in declaring the order ye tooke with the other, ye ſhall neuer lacke, neither matter, nor maner, what to write, nor how to write in this kinde of Argument.

1570, Margaret Ascham, “To the Honorable Sir William Cecill Knight, Principall Secretarie to the Quenes Most Excellent Maiestie”, in Roger Ascham, edited by Margaret Ascham, The Scholemaster: Or Plaine and Perfite Way of Teaching Children, to Vnderstand, Write, and Speake, the Latin Tong, […], London: […] John Daye, […], →OCLC:

Belike this ſhow imports the argument of the play.

c. 1599–1602 (date written), William Shakespeare, The Tragicall Historie of Hamlet, Prince of Denmarke: […] (Second Quarto), London: […] I[ames] R[oberts] for N[icholas] L[ing] […], published 1604, →OCLC, [Act III, scene ii]:

This show is perhaps the subject of the play.

O know ſweet loue I alwaies write of you,
 And you and loue are ſtill my argument: […]

1609, William Shakespeare, “Sonnet 76”, in Shake-speares Sonnets. […], London: By G[eorge] Eld for T[homas] T[horpe] and are to be sold by William Aspley, →OCLC:

[N]earer view / Briſtl'd with upright beams innumerable / Of rigid Spears, and Helmets throng'd, and Shields / Various, with boaſtful Argument portraid, / The banded Powers of Satan haſting on / With furious expedition; […]

1667, John Milton, “Book VI”, in Paradise Lost. […], London: […] [Samuel Simmons], and are to be sold by Peter Parker […]; [a]nd by Robert Boulter […]; [a]nd Matthias Walker, […], →OCLC; republished as Paradise Lost in Ten Books: […], London: Basil Montagu Pickering […], 1873, →OCLC, lines 81–86:

The abstract, or argument of the piece, is shortly as follows.

1844, Francis Jeffrey, “Sardanapalus, a Tragedy. The Two Foscari, a Tragedy. Cain, a Mystery. By Lord Byron. 8vo. pp. 440. Murray, London: 1822. [book review]”, in Contributions to the Edinburgh Review. […], 2nd edition, volume II, London: […] [Andrew Spottiswoode] for Longman, Brown, Green, and Longmans, […], →OCLC, part III (Poetry), page 357:

The subject matter of an artistic representation, discourse, or writing; a theme or topic.


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 could be borrowed from Iranian, from Proto-Iranian *pántaHh, from Proto-Indo-Iranian *pántaHs, from Proto-Indo-European *póntoh₁s, from the root *pent- (“to pass”), but many go against it and it is far from being universally accepted. https://blog.oup.com/2015/11/path-word-origin-etymology/
Germanic cognates include 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.

(paganism) A Pagan tradition, for example witchcraft, Wicca, druidism, Heathenry.

Use the network path \\Marketing\Files to find the documents you need.

(computing) A human-readable specification for a location within a hierarchical or tree-like structure, such as a file system or as part of a URL.

camí

computing: a specification for a location within a hierarchical or tree-like structure

(graph theory) A sequence of vertices from one vertex to another using the arcs (edges). A path does not visit the same vertex more than once (unless it is a closed path, where only the first and the last vertex are the same).

ruta

graph theory: a sequence of vertices from one vertex to another

(topology) A continuous map f from the unit interval I=[0,1] to a topological space X.


path

en-verb

path (third-person singular simple present paths, present participle pathing, simple past and past participle pathed)

paths (present singular third-person)
pathing (participle present)
pathed (participle past)
pathed (past)

Next, you need to path to the location of the executable and run it from there.

(computing, intransitive) To navigate through a file system directory tree (to a desired file or folder).


distribution

en-noun

distribution (countable and uncountable, plural distributions)

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.


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

der

The Dirichlet kernel convolved with a function yields its Fourier series approximation.

(calculus) A function used to define an integral transform.


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


logarithm

/ˈlɑɡəɹ.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 power to which a given base number must be raised in order to obtain 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


cosine

From co- + sine, short for complementi sinus in Latin, meaning “sine of the complement”. Doublet of cosinus.

(trigonometry) In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse. Symbol: cos


calculus

en-noun

calculus (countable and uncountable, plural calculi or calculuses)

* Borrowed from Latin calculus (“a pebble or stone used as reckoning counters in abacus”), diminutive of calx (“limestone”) + -ulus.
* Mathematical topic is from differential calculus.

bor

Near-synonyms: analysis, mathematical analysis

Differential calculus
differential calculus

(uncountable, often definite, the calculus) Differential calculus and integral calculus considered as a single subject.

analise

differential calculus and integral calculus considered as a single subject


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”).

دَالَّة

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

Our hotel is hosting a function for all the warehouse workers on the night of April 18.

An official or social occasion.
(slang) A party.

An official or social occasion.
A party.

A relation where one thing is dependent on another for its existence, value, or significance.

фу́нкция

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

(computing) A routine that receives zero or more arguments and may return a result.

وَظِيفَة

computing: routine that returns a result


tangent

Borrowed from Latin tangentem, the accusative of tangēns (“touching”) (in the phrase līnea tangēns (“a touching line”)), the present participle of the verb tangō (“touch”, verb), from Proto-Italic *tangō, from Proto-Indo-European *teh₂g- (“to touch”).
Cognate with Old English þaccian (“to touch lightly, pat, stroke”). More at thack, thwack.

cog

The collision occurred on a two-mile stretch of tangent track.

(rail transport, of track) Straight; not horizontally curved.


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.


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.

der

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


monotonic

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

suffix

Of or using the Greek system of diacritics which discards the breathings and employs a single accent to indicate stress. It replaced polytonic system in 1982.

μονοτονικός

of or using the Greek system of diacritics which discards the breathings and employs a single accent to indicate stress

independent variable
independent variable

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

monoton

mathematics: never decreasing or increasing as its independent variable increases

The Same Spade talk-alike, super-cynical heroine of the piece is named Emma Victor, and there's no end to her rapid-fire, monotonic, street-smart patter.

1988 December 11, Georgia Cotrell, “A Lezbeen Sam Spade”, in Gay Community News, volume 16, number 22, page 8:


log

To multiply two numbers, add their logs.


FFT

fast Fourier transform
fast Fourier transform#English

(signal processing) Initialism of fast Fourier transform.


functional

en-adj

functional (comparative more functional, superlative most functional)

more functional (comparative)
most functional (superlative)

/ˈfʌŋkt͡ʃənəl/

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

funkcionální

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.

泛函

a function that takes a function as its argument

(mathematics, functional analysis) A scalar-valued linear function on a vector space.

(computing) An object encapsulating a function pointer (or equivalent).

関数オブジェクト

an object encapsulating a function pointer


sigmoid

Borrowing from Ancient Greek σῑγμοειδής (sīgmoeidḗs, “of the shape of sigma”), from σῖγμᾰ (sîgma, “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.


independent variable

en-noun

independent variable (plural independent variables)

Coordinate term: dependent variable

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.

Coordinate term: dependent variable

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


dependent variable

dependent variable (plural dependent variables)

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.


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

funció bijectiva

function that is both a surjection and an injection


surjection

/sɜː(ɹ)ˈd͡ʒɛk.ʃən/

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#x5C;rightarrowB can be identified.

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

Let J#x3D;#x5C;cap#x5F;im#x5F;i be the (irredundant) primary decomposition of J. We associate to the pair (J,#x5C;omega) the element #x5C;textstyle#x5C;sum#x5F;i(m#x5F;i,#x5C;omega#x5F;i)#x5C;inG, where #x5C;omega#x5F;i is the equivalence class of surjections from L#x2F;m#x5F;iL#x5C;oplus(A#x2F;m#x5F;i)#x7B;n-1#x7D; to m#x5F;i#x2F;m#x5F;i² induced by #x5C;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#x3A;E#x5C;rightarrowF (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from E#x2F;#x5C;text#x7B;ker#x7D;(u) to F is an isometric isomorphism. Moreover, by the classical open mapping theorem, u is a surjection iff the associated mapping from E#x2F;#x5C;text#x7B;ker#x7D;(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


transformation

en-noun

transformation (countable and uncountable, plural transformations)

Borrowed from Middle French, from Ecclesiastical Latin trānsfōrmātiō.
Morphologically transform + -ation

(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.


transform

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

(geology, seismology) A transform fault.


asymptote

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
mathematical analysis

(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".

asintota

a straight line which a curve approaches arbitrarily closely


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.

(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).


limit

German: Limit

From Middle English limit, from Old French limit, from Latin līmes (“a cross-path or balk between fields, hence a boundary, boundary line or wall, any path or road, border, limit”). Displaced native Old English ġemǣre. Doublet of limes.

der

the dateless limit of thy dear exile

1595 December 9 (first known performance), William Shakespeare, “The life and death of King Richard the Second”, in Mr. William Shakespeares Comedies, Histories, & Tragedies: Published According to the True Originall Copies (First Folio), London: […] Isaac Iaggard, and Ed[ward] Blount, published 1623, →OCLC, [Act I, scene iii]:

The limit of your lives is out.

c. 1593 (date written), William Shakespeare, “The Tragedy of Richard the Third: […]”, in Mr. William Shakespeares Comedies, Histories, & Tragedies […] (First Folio), London: […] Isaac Iaggard, and Ed[ward] Blount, published 1623, →OCLC, [Act III, scene iii]:

(obsolete) That which terminates a period of time; hence, the period itself; the full time or extent.


curl

From metathesis of Middle English crulle (“curled, curly”), from Middle Dutch crul, crulle (“curl”), ultimately from Proto-Germanic *kruzlǭ (“bent or crooked object, curl”), of unknown origin; possibly from Pre-Germanic *grus-, contracted from Proto-Indo-European *gurus- (“twist, curl”), same source as Persian گرس (gors, “braid of hair”).
Cognate with Saterland Frisian Krulle (“curl, lock”), West Frisian krul (“curl”), Dutch krul (“curl”), German Low German Krull (“curl”), dialectal German Krolle (“curl”), Danish krølle (“curl”), Norwegian krull (“curl”). Related also to Saterland Frisian Kruus (“curl”), German kraus (“frizzy, crumpled, curly”), Danish krus (“curl”), Swedish krusa (“to crimp, curl”). Compare also Gothic 𐌺𐍂𐌹𐌿𐍃𐍄𐌰𐌽 (kriustan, “to grind, crush, gnash”).

unk

unknown

cog

In 2D, when Q is a polygonal domain, the singularities of Type (2) disappear because ψ is the scalar curl of u and is such that its vectorial curl is zero.

The curl of the vector field #x5C;vec#x7B;F#x7D;(x,y,z) is the vector field #x5C;operatorname#x7B;curl#x7D;#x5C;,#x5C;vec#x7B;F#x7D;#x5C;equiv#x5C;vec#x7B;#x5C;nabla#x7D;#x5C;times#x5C;vec#x7B;F#x7D;#x3D;#x5C;left(#x5C;frac#x7B;#x5C;partialF#x5F;z#x7D;#x7B;#x5C;partialy#x7D;-#x5C;frac#x7B;#x5C;partialF#x5F;y#x7D;#x7B;#x5C;partialz#x7D;,#x5C;frac#x7B;#x5C;partialF#x5F;x#x7D;#x7B;#x5C;partialz#x7D;-#x5C;frac#x7B;#x5C;partialF#x5F;z#x7D;#x7B;#x5C;partialx#x7D;,#x5C;frac#x7B;#x5C;partialF#x5F;y#x7D;#x7B;#x5C;partialx#x7D;-#x5C;frac#x7B;#x5C;partialF#x5F;x#x7D;#x7B;#x5C;partialy#x7D;#x5C;right).

(calculus) The vector field denoting the rotationality of a given vector field.

rotace

vector field denoting rotationality of a given vector field

(calculus, proper noun) The vector operator, denoted rm curl; or ⃑∇×⃑(·), that generates this field.


cusp

From Latin cuspis (“a point, spear, pointed end”); first used in astrology. Doublet of cuspid and cuspis.


secant

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

From Latin secāns, present participle of secō (“to cut”).

(trigonometry) In a right triangle, the reciprocal of the cosine of an angle. Symbol: sec


differentiation

en-noun

differentiation (countable and uncountable, plural differentiations)

/ˌdɪf.əˌɹɛn.ʃiˈeɪ.ʃən/

From differentiate + -ion, from different + -iate, from differ + -ent, from Middle English differen, from Old French differer, from Latin differō (“carry apart, put off, defer; differ”), from dis- (“apart”) + ferō (“carry, bear”); cognate with Ancient Greek διαφέρω (diaphérō, “to differ”).

видоизменение

gradual formation or production of organs or parts by a process of evolution or development

The integration and differentiation of vital function on the one hand, and the preparation and composition of food-material on the other hand form — as we will become fully aware further on — the two great divisions in the subject-matter of the science of organization, divisions corresponding to the fundamental biplicity of all advanced organization, its animal and its vegetative life.

1878, Edmund Montgomery, “Monera, and the problem of life”, in The Popular Science Monthly, volume XIII, page 680:

(mathematics, calculus) The process of applying the derivative operator to a function; of calculating a function's derivative.


derivation

en-noun

derivation (countable and uncountable, plural derivations)

From Middle English derivacioun, borrowed from Middle French dérivation, from Latin dērīvātiō, dērīvātiōnem.
Morphologically derive + -ation

(mathematics, calculus) The process of application of the derivative operator to a function, yielding another function called the derived function of the first.


integration

integration (countable and uncountable, plural integrations)

/ˌɪn.tɪˈɡɹeɪ.ʃən/

/ˌɪn.təˈɡɹeɪ.ʃən/

/ˌɪn.təˈɡɹæɪ.ʃən/

From French intégration, from Latin integratio.
Morphologically integrate + -ion

(calculus) The operation of finding the integral of a function.


primitive

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.


differential calculus

(calculus) The calculus that deals with instantaneous rates of change.


sine wave

(trigonometry, signal processing) A periodically oscillating waveform as described by the sine function, possibly multiplied by some amplitude, shifted by some phase, and translated by an additive constant.


arccosine

(trigonometry) Any of several single-valued or multivalued functions that are inverses of the cosine function.


arcsine

/ˈɑːɹkˌsaɪn/

(trigonometry) Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin⁻¹


integral calculus

(calculus) The calculus that generalizes summation to find areas, masses, volumes, sums, and totals of quantities described by continuously varying functions.


transition function

en-noun

transition function (plural transition functions)

If (U#x5F;1,#x5C;psi#x5F;1) and (U#x5F;2,#x5C;psi#x5F;2) are two complex charts, then the transition function is #x5C;psi#x5F;#x7B;12#x7D;#x3A;#x5C;psi#x5F;1(U#x5F;1#x5C;capU#x5F;2)#x5C;rightarrow#x5C;psi#x5F;2(U#x5F;1#x5C;capU#x5F;2), given by #x5C;psi#x5F;#x7B;12#x7D;#x3D;#x5C;psi#x5F;2#x5C;circ#x5C;psi#x5F;1#x7B;-1#x7D;. We say M is a complex manifold if it has an atlas of complex charts (U,#x5C;psi), such that all the transition functions are holomorphic, as maps from #x5C;mathbb#x7B;C#x7D;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


derivable

[dɪˈɹaɪvəbəɫ]

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


factorial

The expression #x7B;#x7D;#x5F;nC#x5F;r means the number of combinations of n things taken r at a time. It is also written #x5C;binomnr, and is equal to#x5C;frac#x7B;n(n-1)(n-2)(n-3)#x5C;cdots#x5C;cdots(n-r#x2B;1)#x7D;#x7B;r#x21;#x7D;or n#x21;#x2F;r#x21;(n-r)#x21;, in which r#x21; 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#x5C;prod#x5F;#x7B;k#x3D;1#x7D;ⁿk#x3D;1#x5C;cdot 2#x5C;cdot 3#x5C;cdotsn.But what does this equal?
Well, of course we can say that it is equal to n factorial and write#x5C;prod#x5F;#x7B;k#x3D;1#x7D;ⁿk#x3D;n#x21; 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.

norm

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)

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.

Gaussova křivka

bell shaped curve typical of the normal distribution

Gaußkurve

strictly the PDF of a normal distribution


PDF

en-noun

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”)


arc-

From Latin arcus (“bow, arc”), as they represent an angle or arc. Compare etymology of arch.

arc-sine (British), arcsine (US)

arctan

inverse trigonometric function
inverse trigonometric function#English

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

inverse hyperbolic function
inverse hyperbolic function#English

(mathematics, sometimes proscribed) Alternative form of ar- used to form the names of inverse hyperbolic functions, and the symbols for these functions.

arkusz-

forms the names of inverse trigonometric and hyperbolic functions


parabola

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).


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.


wavefunction

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.


atomic orbital

The physical region or space where an electron can be calculated to be present, as defined by the particular mathematical form of the orbital.


exponential growth

en-noun

exponential growth (usually uncountable, plural exponential growths)

(by extension, proscribed) Very rapid growth.


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.


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.


catenary

From Late Latin catenaria, in turn from Latin catēna (“chain”). Attested since 1788.

/ˈkætəˌnɛɹi/

(geometry) The curve described by a flexible chain or a rope if it is supported at each end and is acted upon by no other forces than a uniform gravitational force due to its own weight and variations involving additional and non-uniform forces. It is described by the hyperbolic cosine function.


simple harmonic motion

simple harmonic motion (plural simple harmonic motions)

(mathematics, mechanics) oscillating motion (as of a pendulum) in which the acceleration of the oscillator has an equal magnitude but opposite direction to the displacement of it from the equilibrium position.


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.


integrate

en-verb

integrate (third-person singular simple present integrates, present participle integrating, simple past and past participle integrated)

integrates (present singular third-person)
integrating (participle present)
integrated (participle past)
integrated (past)

/ˈɪn.təˌɡɹeɪt/

/ˈɪn.tɪ.ɡɹæɪt/

Borrowed from Latin integrātus, perfect participle of integrō (“I make whole, I renew, I repair, I begin again”), from integer (“whole, fresh”); see integer, integral.

President Eisenhower had to call out the National Guard to integrate Little Rock Central High School.

I continued on the bus without him. It worked out fine. I went on to Tampa, Fla. That was the first time we integrated the buses. All the way down, sitting in the front row.

2020 July 18, Bernard Lafayette Jr., “The First Time John Lewis and I Integrated the Buses”, in New York Times:

To desegregate, as a school or neighborhood.

integrarsi

to desegregate, as a school or neighborhood


gradient

From Latin gradiēns, present participle of gradior (“to step, to walk”).

(calculus, of a function) The ratio of the rates of change of a dependent variable and an independent variable, the slope of a curve's tangent.

(calculus) A differential operator that maps each point of a scalar field to a vector pointed in the direction of the greatest rate of change of the scalar. Notation for a scalar field φ: ∇φ


transcendental function

en-noun

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)

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

Fouriertransformatie

mathematics: a type of integral transform


trigonometric function

trigonometric function (plural trigonometric functions)

(trigonometry) Any function of an angle expressible as the ratio of two of the sides of a right triangle that has that angle (e.g., sine), or that subtracts 1 from such a ratio or subtracts such a ratio from 1 (e.g., coversed sine).


cosecant

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

(trigonometry) In a right triangle, the reciprocal of the sine of an angle. Symbols: cosec, csc


cotangent

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

(trigonometry) In a right triangle, the reciprocal of the tangent of an angle. Symbols: cot, ctg, or ctn


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 operator

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:

(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.

Laplace'i operaator

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


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:


infinitesimal

en-adj

infinitesimal (comparative more infinitesimal, superlative most infinitesimal)

more infinitesimal (comparative)
most infinitesimal (superlative)

Learned borrowing from Latin *infinitesimus. Displaced earlier coordinate term fluxion.

lbor

Do you ever get the feeling that you are but an infinitesimal speck, swallowed by the vastness of the universe and beyond?

"You will conceive a bunch of grapes," said he, "which are covered by some infinitesimal but noxious bacillus."

1913, Arthur Conan Doyle, “(please specify the page)”, in The Poison Belt […], London; New York, N.Y.: Hodder and Stoughton, →OCLC:

Then you could say that the doorway exploded. But the particular verb doesn't do the action justice. Rather, it shattered into infinitesimal pieces.

2001, Eoin Colfer, Artemis Fowl, page 221:

Incalculably, exceedingly, or immeasurably minute; vanishingly small.


arctangent

(trigonometry) Any of several single-valued or multivalued functions that are inverses of the tangent function. Symbol: arctan, tan⁻¹


differential

[dɪfəˈɹənʃəɫ]

nonstandard analysis
nonstandard analysis

(calculus) A quantity representing an infinitesimal change in a variable, now only used as a heuristic aid except in nonstandard analysis but considered rigorous until the 20th century; a fluxion in Newtonian calculus, now usually written in Leibniz's notation as operatorname d!x.

If f(x)#x3D;x², the differential of f is the function #x5C;operatorname#x7B;d#x7D;#x5C;#x21;f(x,#x5C;Deltax)#x3D;f'(x)#x5C;Deltax#x3D;2x#x5C;Deltax.

(calculus, of a univariate differentiable function f(x)) A function giving the change in the linear approximation of f at a point x over a small interval Δx or operatorname d!x, the function being called the differential of f and denoted operatorname d!f(x,Δx), operatorname d!f(x), or simply operatorname d!f.

partial differential
partial differential

Any of several generalizations of this concept to functions of several variables or to higher orders: the partial differential, total differential, Gateaux differential, etc.


rapidity

en-noun

rapidity (countable and uncountable, plural rapidities)

From rapid + -ity, from French rapidité, from Latin rapiditas.

Although appreciating the rapidity and frequency of the Southern electric services I was now to use on short journeys, I became more than ever convinced that electric traction offers very little of interest to the dyed-in-the-wool railwayist.

1944 November and December, Talisman, “A Broadening Horizon”, in Railway Magazine, page 340:

speed, swiftness; the condition of being rapid

(physics) A measure of velocity relative to the speed of light; defined as artanh(v), where artanh is the hyperbolic arctangent, and v = speed (with c = speed of light = 1).

(physics) A measure of the velocity of a particle in a beam relative to the beam's axis


sinus

Learned borrowing from Latin sinus (“a bent surface, curve, hollow”). Doublet of sine.

lbor

(anatomy, zootomy) A pouch or cavity in a bone or other tissue, especially one in the bones of the face or skull connecting with the nasal cavities (the paranasal sinus).


infinitesimal calculus

(calculus) Differential calculus and integral calculus considered together as a single subject.


derived function

en-noun

derived function (plural derived functions)

(calculus) Of a function, another function, the value of which for any value of the independent variable is the instantaneous rate of change of the given function at that value of the independent variable.


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.


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 | |.


antilog

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

Then the desired y path can be obtained by taking the antilog of #x5C;lny […]

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


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).


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


differentiate

en-verb

differentiate (third-person singular simple present differentiates, present participle differentiating, simple past and past participle differentiated)

differentiates (present singular third-person)
differentiating (participle present)
differentiated (participle past)
differentiated (past)

equate

antonym(s) of “to show the distinction between things”

mix up

antonym(s) of “to perceive the difference between things”

From New Latin differentiātus, past participle of differentiō, from Latin differentia (“difference”); see difference.

Like most teaching, differentiating is a mix of art and science. When it works, "it's like a jazz rhythm," said Carol Tomlinson, an education professor at the University of Virginia and an expert on differentiation.

2012 May 16, Michael Alison Chandler, “Teaching for all levels — in one class”, in The Washington Post, archived from the original on 2023-02-11:

Moreover, children of STIP-teachers who showed many types of differentiation activities learnt more than children of STIP-teachers who differentiated less.

2017 August, Tessa H.S. Eysink, Manon Hulsbeek, Hannie Gijlers, “Supporting primary school teachers in differentiating in the regular classroom”, in Teaching and Teacher Education, volume 66, Amsterdam: Elsevier, →DOI, →ISSN, page 107116:

I use technology to differentiate in my Algebra classroom in many different ways. Sometimes, I put this technology in the hands of students and sometimes I use it myself to streamline a process.

2018 June 22, Larry Ferlazzo, quoting Sarah Shartzer, “Response: Ways to Use Tech to Differentiate Instruction”, in Education Week, archived from the original on 2022-05-19:

(education) To teach a lesson in multiple different ways in order to meet the needs of more or less advanced students.

differenciál

education: to teach a lesson in multiple different ways in order to meet the needs of more or less advanced students


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


inviscid

en-adj

inviscid (comparative more inviscid, superlative most inviscid)

more inviscid (comparative)
most inviscid (superlative)

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

(calculus, of a scalar field) having a gradient of zero


hyperbolic sine

en-noun

hyperbolic sine (plural hyperbolic sines)

(mathematics) A hyperbolic function that is the analogue of the sine function for hyperbolic spaces, taking in a hyperbolic angle as an argument and returning the y-coordinate for the corresponding point on the unit hyperbola. It is written in symbol sinh and can be represented as: sinh (x)= tfrac 1 2(eˣ-e⁻ˣ)


hyperbolic cosine

hyperbolic cosine (plural hyperbolic cosines)

(mathematics) A hyperbolic function that is written in symbol cosh and defined as the following: cosh (x)= tfrac 1 2(eˣ+e⁻ˣ).


hyperbolic function

hyperbolic function (plural hyperbolic functions)

trigonometric function
trigonometric function

(calculus) A set of functions that are analogs of the trigonometric functions for hyperbolic spaces.


hyperbolic tangent

hyperbolic tangent (plural hyperbolic tangents)

tanh (x)=(sinh (x))/(cosh (x))=(eˣ-e⁻ˣ)/(eˣ+e⁻ˣ).

(mathematics) A hyperbolic function that is written in symbol tanh and defined as the following:


hyperbolic

hyperbolic partial differential equation

The hyperbolic cosine of zero is one.

hyperbolic function
hyperbolic function#English

trigonometric function
trigonometric function#English

Indicates that the specified function is a hyperbolic function rather than a trigonometric function.


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

From Latin harmonicus, from Ancient Greek ἁρμονικός (harmonikós), from ἁρμονία (harmonía, “harmony”).

(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


turning point

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

(calculus) A maximum or minimum on a graph.


parametric equation

parametric equation (plural parametric equations)

independent variable
independent variable

(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

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.


Langevin function

Langevin function (plural Langevin functions)

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


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.


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) The general solution of a homogenous linear differential equation.


integrand

/ˈɪntɪɡɹænd/

(calculus) The function that is to be integrated

antidifferentiation: integral sign, ∫ integrand differential, dx


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.


arccos

arccos (1/2ϕ)=2π/5

[…]with the appropriate constraint angles measured in terms of the arccoses and arctans.

1973, The Soviet Economy in Regional Perspective, Praeger Publishers, page 3:


arctan

tan (arctan 2/2)=1/ϕ


div

(vector calculus) Divergence; a kind of differential operator.


differential equation

differential equation (plural differential equations)

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

First appears c. 1704 in the publications of Charles Hayes. Calque of Latin aequatio differentialis, which was coined by Leibniz in 1676.

(calculus) An equation involving the derivatives of a function.

Charles Hayes (mathematician)
Leibniz
differential equation


partial differential equation

partial differential equation (plural partial differential equations)

(calculus) A differential equation that involves the partial derivatives of a function of several variables.

partial differential equation
simple:partial differential equation

equació diferencial en derivades parcials

偏微分方程

piān wēifēn fāngchéng, piān wéifēn fāngchéng

équation différentielle aux dérivées partielles

equazione differenziale alle derivate parziali


funicular

en-adj

funicular (comparative more funicular, superlative most funicular)

more funicular (comparative)
most funicular (superlative)

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

From Latin fūniculus (“cord”), diminutive from fūnis (“cord”) + -culus.


arccosh

/ˈɑː(ɹ)kˌkɒʃ/

(mathematics) Alternative form of arcosh (“the inverse hyperbolic cosine function”)


arcsinh

/ˈɑː(ɹ)kˌʃaɪn/

(trigonometry) Alternative form of arsinh (“the inverse hyperbolic sine function”)


arctanh

/ˈɑː(ɹ)kˌtænʃ/

(trigonometry) Alternative form of artanh (“the inverse hyberbolic tangent function”)


fixed point

(computing) A fractional-number representation with a fixed number of digits after the decimal point. Compare floating point and integer.

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


fixed point

(computing) Pertaining to fixed point representations or operations.


hacoversed sine

en-noun

hacoversed sine (plural hacoversed sines)

(trigonometry) The trigonometric function equal to half the coversed sine. Abbreviation: hacoversin


versine

The simplest method of calculating the radius of a curve in situ is to measure the versine; in railway practice this is done by extending a tape 66 ft. (1 ch.) long in a straight line or chord between two points on the periphery of a curve, and then measuring the maximum distance of the rail from the chord at the centre of the 66 ft. The radius in chains is found by dividing the versine in inches into 99. Thus if a versine measures 3 in., the radius will be 33 ch.

1941 September, “The Why and the Wherefore: Curves”, in Railway Magazine, page 430:


coversed sine

From contraction of complement + versed sine.

The cosine of an arc is the sine of its complement.† In like manner, the coversed sine, cotangent, and cosecant of an arc, are respectively the versed sine, tangent, and secant of its complement.

1825, Jas Thomson, Elements of Plane and Spherical Trigonometry, page 2:

At the top of the page on the left-hand side is placed the number of degrees, and in the left-hand column each minute of the degree, opposite to which are arranged the numerical values of the sine, coversed sine, &c, of the corresponding angle in those columns, at the top of which those terms are placed.

1830, W. Hopkins, “Book II: Elements of Trigonometry”, in Geometry, Plane, Solid, and Spherical, page 46:

For the coversed sine; subtract the sine of the angle from 1. Thus, for the same angle we have the coversed sine,
1 — ·36650 = ·6335.

1838, William Grier, The Mechanic′s Pocket Dictionary, page 21:

Hence the cosines and versed sines of zenith distances are respectively equal to the sines and coversed sines of the corresponding angular elevations.

1871, Homersham Cox, The Law and Science of Ancient Lights, 2nd edition, page 109:

(trigonometry) The trigonometric function 1 − sin(x).


haversed sine

(trigonometry, rare) The trigonometric function defined as half the versed sine. Abbreviation: haversin


versed sine

From New Latin sinus versus (“flipped sine”); the sine function being sometimes called sinus rectus.

For the divisions, to be laid on this line, are the differences between the logarithm versed sine of 180°, and the logarithm versed sines of the successive arcs.

1753, John Robertson, “The Construction of the Logarithmic Lines on the Gunter′s Scale”, in Philosophical Transactions: 1750-1755, Great Britain: Royal Society, page 340:

But if the given arch exceed 90°, find the natural co-sine of its supplement, which add to 1000000, and the sum will be the natural versed sine required.

1805, John William Norie, A New and Complete Epitome of Practical Navigation, page xxii:

The Natural Sines in this table are fractions (radius being unit); and are particularly adapted to Dunthorne′s and other methods of clearing the moon′s distance from the effects of parallax and refraction, where natural sines or versed sines are used ;[…].

1806, John Garnett, Tables Requisite To Be Used with The Nautical Ephemeris, page 41:

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


havercosine

(trigonometry) the trigonometric function (1 + cos(x)) / 2, equal to half the vercosine


excosecant

(trigonometry) The trigonometric function csc(x) − 1. Abbreviation: excsc


exsecant

(trigonometry) The trigonometric function sec(x) − 1. Abbreviation: exsec


function space

(mathematics) Any metric space whose elements are functions


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.


antihyperbolic function

antihyperbolic function (plural antihyperbolic functions)

inverse hyperbolic function
inverse hyperbolic function


arc-hyperbolic function

arc-hyperbolic function (plural arc-hyperbolic functions)


inverse hyperbolic function

inverse hyperbolic function (plural inverse hyperbolic functions)


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.


boxcar function

boxcar function (plural boxcar functions)

The graph resembles 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.


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.


circular function

circular function (plural circular functions)

(trigonometry) Synonym of trigonometric function


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).


cofunction

Cosine and sine are each other's cofunctions.

(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.


constant function

constant function (plural constant functions)

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


inverse trigonometric function

inverse trigonometric function (plural inverse trigonometric functions)

inverse trigonometric functions (plural)

(mathematics) A function that is the inverse of an appropriate restriction of a trigonometric function.


functor

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

In the category of categories, #x5C;mathbb#x7B;CAT#x7D;, 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#x3A;A#x5C;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#x3A;Xᴮ#x5C;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.


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


cost function

Any mathematical relationship between an activity and its cost.


strictly decreasing function

strictly decreasing function (plural strictly decreasing functions)

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


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.


dissipation function

dissipation function (plural dissipation functions)

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


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


definite integral

definite integral (plural definite integrals)

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


explicit function

explicit function (plural explicit functions)

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


even function

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


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


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)


function multiplier

function multiplier (plural function multipliers)

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


indefinite integral

indefinite integral (plural indefinite integrals)

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


function table

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


exponential function

exponential function (plural exponential functions)

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


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


entire function

entire function (plural entire functions)

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


calculus of variations

the calculus of variations (uncountable)

The next advance was made by Cauchy in a memoir on the calculus of variations published in the third volume of his Exercices d'analyse et de Physique Mathématique, 1844, extending from page 50 to page 130.

1885, Lewis Buffett Carll, A Treatise on the Calculus of Variations, Macmillan and Co., page 550:

2004, Bernard Dacorogna, Introduction to the Calculus of Variations, 2nd Edition, Imperial College Press, page 2,
In the nineteenth century and in parallel to some of the work that was mentioned above, probably, the most celebrated problem of the calculus of variations emerged, namely the study of the Dirichlet integral; a problem of multiple integrals.

On the other hand, we can also consider that the calculus of variations includes all optimization problems where the unknown are functions[…].

2023, Filippo Santambrogio, A Course in the Calculus of Variations, Springer, page 57:

(calculus) The form of calculus that deals with the maxima and minima of definite integrals of functions of many variables.


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


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


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)#x3D;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,
#x5C;quad#x5C;quad#x5C;quaday²#x2B;bxy#x2B;cx²#x2B;d#x3D;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
mathematical analysis

(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


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)#x3D;4x#x2B;5.
Its inverse is f#x7B;-1#x7D;(x)#x3D;#x5C;frac#x7B;x-5#x7D;#x7B;4#x7D;.

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


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


integral function

integral function (plural integral functions)


integrable function

integrable function (plural integrable functions)

(mathematics) Any function that possesses a finite integral


linear function

linear function (plural linear functions)

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


natural function

natural function (plural natural functions)

(mathematics) An ordinary trigonometric function


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.


orthonormal function

orthonormal function (plural orthonormal functions)

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


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


penalty function

penalty function (plural penalty functions)

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


point function

(mathematics) Any function whose values are points


phase function

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


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


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


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.


probability mass function

probability mass function (plural probability mass functions)

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


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.


measurable function

measurable function (plural measurable functions)

(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.


psi function

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


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.


quadratic function

quadratic function (plural quadratic functions)

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


scattering function

scattering function (plural scattering functions)

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


regular function

regular function (plural regular functions)


scalar function

scalar function (plural scalar functions)

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


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).


random function

random function (plural random functions)

(mathematics, computing) A function whose formula is always the same, but whose initial value is chosen at random from a set of possible values (its domain).

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

probability distribution
probability distribution

(computing) A function that returns a random or pseudo-random value that is distributed according to a specific probability distribution associated with the function.


trigonometric cofunction

trigonometric cofunction (plural trigonometric cofunctions)

(mathematics) Either of a pair of trigonometric functions that have equal values when their arguments are complementary angles, e.g. sine and cosine.


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.


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


truth function

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


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


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.


sample function

sample function (plural sample functions)

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


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 #x5C;mathcal#x7B;C#x7D; be the class of continuous maps of #x5C;mathbb#x7B;C#x7D;#x5F;#x5C;infty into itself and let #x5C;mathcal#x7B;R#x7D; be the subclass of rational functions.[…]Now #x5C;mathcal#x7B;R#x7D; is a closed subset of #x5C;mathcal#x7B;C#x7D;#x5F;#x5C;infty because if the rational functions R#x5F;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


infinitesimal analysis

Of the new or infinitesimal analysis, we are to consider Sir Isaac Newton as the first inventor, Leibnitz, a German philosopher, as the second; ...

1816, John Playfair, Dissertation on the Progress of Mathematical and Physical Science since the Revival of Letters in Europe:


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

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

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


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#x2A; 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.


regression

regression (countable and uncountable, plural regressions)

/ɹɪˈɡɹɛʃ.ən/

/ɹiːˈɡɹɛʃ.ən/

/ɹɪˈɡɹɛʃ.ən/

/ɹəˈɡɹɛʃ.ən/

/ɹəˈɡɹeʃ.ən/

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).


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


divergence

divergence (countable and uncountable, plural divergences)

(calculus) the operator which maps a function F=(F₁, ... Fₙ) from a n-dimensional vector space to itself to the function ∑ᵢ₌₁ⁿ(∂F_i)/(∂x_i)


irrotational

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


ordinary differential equation

ordinary differential equation (plural ordinary differential equations)

ordinary differential equations (plural)

(calculus) An equation involving the derivatives of a function of only one independent variable.

ordinary differential equation
simple:ordinary differential equation

常微分方程

cháng wēifēn fāngchéng, piān wéifēn fāngchéng


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.