automorphism

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Since every linear automorphism of V fixes 0 our interest in the transitivity properties of GL(V) is confined to its action on V* = V - {0}. GL(V) is transitive on V* since any two elements of V* may be chosen as the initial members of two ordered bases; it is not in general 2-transitive because there is no linear automorphism taking an independent pair to a dependent pair.

1971, Norman Biggs, Finite Groups of Automorphisms: Course Given at the University of Southampton, Cambridge University Press, page 25:

In mathematics, automorphisms of algebraic structures play an important role. Automorphisms capture the symmetries inherent in the structures and many important results have been proved by analyzing the automorphism group of the structure.

2005, Maninda Agrawal, Nitin Saxena, “Automorpisms of Finite Rings and Applications to Complexity of Problems”, in Volker Diekert, Bruno Durand, editors, STACS 2005: 22nd Annual Symposium on Theoretical Aspects of Computer Science, Springer, LNCS 3404, page 1:

A tame automorphism is a product of elementary automorphisms, and a non-tame automorphism is called wild. The “tame automorphism problem” asks whether any automorphism is tame.

2014, Alexei Belov, Leonid Bokut, Louis Rowen, Jie-Tai Yu, “The Jacobian Conjecture, Together with Specht and Burnside-Type Problems”, in Ivan Cheltsov, Ciro Ciliberto, Hubert Flenner, James McKernan, Yuri G. Prokhorov, Mikhail Zaidenberg, editors, Automorphisms in Birational and Affine Geometry, Springer, page 274:

(algebra) An isomorphism of a mathematical object or system of objects onto itself.


covariance

The elements of such a correlation matrix do not have asymptotic variances and covariances of the form (1.2), even if S has a Wishart distribution.

2002, Karl G. Jöreskog, Dag Sörbom, PRELIS 2 User's Reference Guide, Scientific Software International, page 28:

Consequently, it can be shown that a covariance of two binary variables measures the extent to which the observed joint distribution of these variables differs from their expected joint distribution under the assumption that they are statistically independent.

1997, Michael Patrick Allen, Understanding Regression Analysis, Plenum Press, page 31:

The covariance of X and Y is the expected value of the product of two random variables, X − E(X) and Y − E(Y).[…]If two random variables tend to act like opposites, one is high when the other is low and vice versa, then the covariance will be negative. If two random variables tend to be high and low at the same time, then the covariance will be positive. In fact, the covariance measures the extent of a linear relationship between the two random variables.

2005, Steven J. Janke, Frederick Tinsley, Introduction to Linear Models and Statistical Inference, Wiley, page 133:

(statistics) A statistical measure defined as scriptstyle operatorname Cov(X,Y)= operatorname E((X-μ)(Y-ν)) given two real-valued random variables X and Y, with expected values scriptstyle E(X),=,μ and scriptstyle E(Y),=,ν.


category theory

As mentioned earlier set theory provides a general conceptual framework for mathematics. Now, since category theory, through the notion of topos, has succeeded in axiomatising set-theory, the outcome is an entirely new categorial foundation of mathematics! The category-theorists attitude that "function" rather than "set membership" can be seen as the fundamental mathematical concept has been entirely vindicated. The pre-eminent role of set theory in contemporary mathematics is suddenly challenged. A revolution has occurred in the history of mathematical ideas (albeit a peaceful one) that will undoubtedly influence the direction of the path to the future.

1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page 3:

(mathematics) A branch of mathematics which deals with spaces and maps between them in abstraction, taking similar theorems from various disparate more concrete branches of mathematics and unifying them.


abstract algebra

/ˈæbˌstɹækt ˈæl.d͡ʒɪ.bɹə/

/ˈæbˌstɹækt ˈæl.d͡ʒə.bɹə/

From 1860, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science

The operations are necessarily algebraic, because the relative magnitudes of the given quantities and the quantity sought for are unknown; and it is the essential principle of abstract algebra to furnish rules and symbols of operation which are proper for calculating independently of the knowledge of relative magnitudes.

1860, Prof. Challis, VIII: On the possibility of finding a Root, real or imaginary, of every Equation: The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, page 46:

This is neither the best nor the worst of the many abstract algebra texts written for advanced undergraduate courses. The author devotes 300 pages to a first course in abstract algebra and 200 pages to a (presumably) second course in linear algebra.

1973, “AAAS Science Books: A Quarterly Review, Volume 9”, in American Association for the Advancement of Science, page 29:

(mathematics) The branch of mathematics concerned with algebraic structures, such as groups, rings, and fields.

抽象代数学

ちゅうしょうだいすうがく


real analysis

(mathematics) The branch of mathematics dealing with the real numbers and related structures.

իրական անալիզ

mathematics: a branch dealing with the real numbers and related structures


discrete mathematics

(mathematics) Collectively, most discrete or computer science-related branches of mathematics, such as graph theory and combinatorics.


ring theory

The branch of mathematics dealing with the algebraic structure of rings.


field theory

en-noun

field theory (countable and uncountable, plural field theories)

The branch of mathematics dealing with the algebraic structure of fields.


Galois theory

Galois theory (usually uncountable, plural Galois theories)

/ˈɡælwɑ ˈθi.əɹi/

Standardly constructed calque of French théorie de Galois (which was used as a section heading in 1870, Camille Jordan, Traité des substitutions et des équations algébriques). First appeared in print in 1893, Bulletin of the New York Mathematical Society.
Ultimately, named after French mathematician Evariste Galois (1811-1832), who first developed the theory to explore how the roots of a given polynomial equation relate to each other.

In this chapter we present the Galois theory which may be described as the analysis of field extensions by means of automorphism groups.

1989, G. Karpilovsky, Topics in Field Theory, North-Holland, page 299:

The Galois theory of noncommutative rings is a natural outgrowth of the Galois theory of fields.

1989, Donald S. Passman, Infinite Crossed Products, Academic Press, page 297:

1992, Journal of Contemporary Mathematical Analysis, Volume 27, Allerton Press, page 4,
Though often our results are prompted by the classical or parallel Galois theories, their proofs are completely different and are based on the set-theoretical approach.

Classical Galois theory has developed a number of extensions and ramifications into more specific theories, which combine it with other areas of mathematics or restate its main problems in different situations. Three of the most important are geometric Galois theory, differential Galois theory, and Lie Galois theory, all of which have undergone very rapid development in recent years.

2003, “Introduction”, in Leila Schneps, editor, Galois Groups and Fundamental Groups, Cambridge University Press, page ix:

(algebra, field theory) The branch of mathematics dealing with Galois groups, Galois fields, and polynomial equations. It provides a link between field theory and group theory: it permits certain problems in the former to be reduced to the latter, which in some respects is simpler and better understood.


character theory

(mathematics) The branch of abstract algebra dealing with group characters.


coding theory

(mathematics) The branch of mathematics dealing with the art of transmitting messages reliably over a noisy channel by adding redundancy to the message.


Ramsey theory

Named after British mathematician and philosopher Frank P. Ramsey.

Ramsey theory can be loosely described as the study of structure which is preserved under finite decomposition.

1987, R. L. Graham, V. Rôdl, “Numbers in Ramsey Theory”, in C. Whitehead, editor, Surveys in Combinatorics 1987: Invited Papers for the Eleventh British Combinatorial Conference, Cambridge University Press, page 111:

1999, Randall McCutcheon, Elemental Methods in Ergodic Ramsey Theory, Springer, Lecture Notes in Mathematics: 1722, page 8,
We are now ready to offer a loose definition of Ramsey theory.
Ramsey theory is a collection of results which, given a finite coloring of some structure, guarantee the existence of certain monochromatic configurations or substructures.

In the 35 years since the lectures which form this book were given the area of Ramsey theory has continued to undergo tremendous growth, particularly in the last decade.

2015, Ron Graham, Steve Butler, Rudiments of Ramsey Theory, 2nd edition, American Mathematical Society, page vii:

(mathematics) A branch of mathematics which deals with patterns that inevitably arise in sufficiently large sets (i.e., subsets of some structure).


functional analysis

(functional analysis) The branch of mathematics dealing with infinite-dimensional vector spaces, whose elements are actually functions, as well as generalizations such as Banach spaces and Hilbert spaces.

functional analysis
simple:functional analysis


bifurcation theory

(mathematics) A branch of mathematics concerned with dynamical systems which suffer sudden qualitative changes from an infinitesimal change in a parameter.


catastrophe theory

(mathematics) The branch of mathematics dealing with dynamical systems which can undergo abrupt irreversible qualitative changes due to a tiny change in parameters.


man-hour

Coordinate term: machine-hour

Ten workers for two days—that's 160 man-hours.

[...] every man has tools and amenities always within one minute's walk; thus non-productive man-hours are kept to a minimum.

1960 December, “New G.E. Line diesel loco maintenance depot at Stratford”, in Trains Illustrated, page 766:

Since 1950 our agricultural output per man-hour has actually doubled!

1962 January 11, John F. Kennedy, Second State of the Union Address:


incenter

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(US, geometry) The point formed at the intersection of the three angle bisectors of a triangle; also the centre of the incircle.


skewed

(statistics, of information) Biased, distorted


superabundant

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superabundant number
superabundant number

(number theory) Being or relating to a superabundant number.


birthday attack

birthday attack (plural birthday attacks)

(cryptography) A covert method of decryption exploiting the so-called birthday paradox.


birthday problem

(probability theory) The mathematics problem that concerns the probability there exist two people in a random group who share a birthday.


rainbow table

Coined by cryptanalyst Philippe Oechslin in 2003.

coined

(mathematics, cryptography) A table of surjective functions used to decrypt a text that was coded using a hash table.


unsalted

(cryptography) Without a cryptographic salt.


long ton

Distinguishing this measure from the "short ton" of 2000 pounds.

(chiefly historical) A traditional unit of mass equal to 20 hundredweights of 112 pounds avoirdupois each, equivalent to approximately 1016 kilograms.

長噸 /长吨

ton of 20 hundredweights of 112 pounds each


commutativity

en-noun

commutativity (usually uncountable, plural commutativities)

(mathematics, physics) The state of being commutative.

kommutativitet

mathematics, physics: The state of being commutative.


anticommutativity

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(algebra, physics) The state of being anticommutative.


anticommutative

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(algebra) Of an operator * for which a * b = - (b * a)


cartload

From Middle English cartlode, cartelode, equivalent to cart + load.

(historical, specifically) A load: various English units of weight or volume based upon standardized cartloads of certain commodities.


homogeneous function

homogeneous function (plural homogeneous functions)

homogeneous polynomial
homogeneous polynomial

homogeneous polynomials
homogeneous polynomials

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

(mathematics) a function f(x) which has the property that for any c, f(cx)=cf(x).


homogeneous polynomial

homogeneous polynomial (plural homogeneous polynomials)

(mathematics) a polynomial such that the sum of the exponents of the variables is the same for every term.

齐次多项式

mathematics: polynomial such that the sum of the exponents of the variables is the same for every term


homogenous polynomial

homogenous polynomial (plural homogenous polynomials)

homogeneous polynomial
homogeneous polynomial#English

(mathematics) Alternative spelling of homogeneous polynomial


homogenous function

homogenous function (plural homogenous functions)

homogeneous function
homogeneous function#English

(mathematics) Alternative spelling of homogeneous function


measurement ton

measurement ton (plural measurement tons)

A unit of volume for cargo freight usually reckoned at 40 U.S. or Imperial cubic feet.


spheroidal

en-adj

spheroidal (comparative more spheroidal, superlative most spheroidal)

more spheroidal (comparative)
most spheroidal (superlative)

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The buccal mass in milacids is spheroidal, and contains an oxygnathic jaw.

2004, G. M. Barker, Natural Enemies of Terrestrial Molluscs, page 299:


nullary

(mathematics, of a function) Taking no entries; having trivial domain; having the arity of zero.

vakio-

mathematics, of a function: taking no entries


linear combination

en-noun

linear combination (plural linear combinations)

(linear algebra) a sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element)


summand

In the expression 1+2=3, the summands are 1 and 2.


polynomial time

(mathematics) The complexity class where the runtime can be bounded (from above) by a polynomial in the input size.


Euler's formula

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

named-after

eⁱᶿ= cos θ+i sin θ

trigonometric function
trigonometric function

(complex analysis) A formula which links complex exponentiation with trigonometric functions:

κₙ(θ)=κ₁ cos ²θ+κ₂ sin ²θ

differential geometry
differential geometry

(differential geometry) A formula which calculates the normal curvature of an arbitrary direction in the tangent plane in terms of the principal curvatures κ₁ and κ₂ and the angle θ which that direction makes with the first principal direction:


linearly independent

(linear algebra) (Of a set of vectors or ring elements) whose nontrivial linear combinations are nonzero


linear independence

The linear independence of a set of vectors can be determined by calculating the Gram determinant of those vectors; if their Gram determinant is zero, then they are linearly dependent, and if their Gram determinant is non-zero, then they are linearly independent. Incidentally, the same Gram determinant can be used to calculate the hyper-volume of a hyper-parallelepiped (whose edges which "radiate" from an "origin" vertex are described by the vectors).

linearly independent
linearly independent

(algebra) the state of being linearly independent


linearly dependent

(algebra, of a set of vectors or ring elements) Having a nontrivial linear combination which is zero.


linear dependence

en-noun

linear dependence (plural linear dependences)

(algebra, uncountable) The state of being linearly dependent

(mathematics, countable) A linear relation between some appropriate things


symmetric polynomial

symmetric polynomial (plural symmetric polynomials)

x+y+z

xyz

2x²+2y²+2z², etc.

An example of a non-symmetric polynomial is x+2y: if we interchange x and y, this polynomial changes to 2x+y.

(algebra) A polynomial which remains unchanged if its variables are permuted


elementary symmetric polynomial

elementary symmetric polynomial (plural elementary symmetric polynomials)

elementary symmetric polynomials (plural)

(algebra) a coefficient of some power of t in (t+x_1)(t+x_2)⋯(t+x_n), for some n∈ N


subring

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multiplicative identity
multiplicative identity

(algebra) a ring which is contained in a larger ring, such that the multiplication and addition on the former are a restriction of those on the latter. When rings are required to have a multiplicative identity (as is standard in most branches of algebra), the subring must also have the same multiplicative identity as the larger ring.


subfield

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Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form
(1)#92;qquadf(c)#61;a#95;0#43;a#95;1c#43;a#95;2c²#43;...#43;a#95;ncⁿ#92;qquad#92;qquad#92;mbox#123;(each#125;a#95;i#92;mbox#123;in#125;F#92;mbox#123;).#125;
[...]
 If f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.

1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394:

We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u]. [...] This shows that F[u] is a subfield of K.
 Since, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.

1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397:

(algebra) A subring of a field, containing the multiplicative identity and closed under inversion.


homomorphism

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Coordinate term: homology

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

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

A homomorphism of presheaves h#58;A#92;rightarrowB is a collection of homomorphisms h#95;U#58;A(U)#92;rightarrowB(U) commuting with restrictions.

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

Definition 1.15. Let G and H be matrix Lie groups. A map #92;Phi from G to H is called a Lie group homomorphism if (1) #92;Phi is a group homomorphism and (2) #92;Phi is continuous.

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

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

automorphism

structure-preserving map between algebraic structures


morphism

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

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

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

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

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

Briefly, the yellow morphic males can change their status from paired to satellite and from satellite to the paired one. However, they cannot cross into the status of the red morph.[…] The colour morphism of males is proved to be irreversible after its expression at an early stage of ontogeny.

2010, T.J. Pandian, Sexuality in Fishes, page 51:

(biology) Being or having distinct variants of a plant or animal species in the same locale; polymorphism.


absolutely summable

mathematical analysis
mathematical analysis

(mathematical analysis) Of an infinite series: such that the sum of the absolute values of its summands converges.


analytic continuation

analytic continuation (countable and uncountable, plural analytic continuations)

1968, [McGraw-Hill], Granino A. Korn, Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, 2000, Dover, Unabridged republication, page 206,
The standard method of analytic continuation starts with a function f(z) defined by its power-series expansion (7.5-4) inside some circle |z-a|=r.

Since there are potentially many different ways to carry out this analytic continuation process, there are questions of ambiguity and redundancy.

2008, Steven G. Krantz, A Guide to Complex Variables, Mathematical Association of America, page 125:

Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example, in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

2014, M. Mursaleen, Applied Summability Methods, Springer, page 31:

(mathematical analysis) The practice of extending analytic functions.

1975, S. Smith (translator), L. A. Muraveǐ Asymptotics for the Wave Equation, S. Smith (translator), Valentin P. Michaǐlov (editor), Boundary Value Problems for Differential Equations, Mathematical Association of America, page 107,
Using (2.6), (2.15)-(2.18) and (2.54) in the same way as in the cases considered earlier, we get that the series (0.14) determines the analytic continuation of the Green function of the second boundary value problem into the domain K_B∩|k|<β with the property (0.23).

The set of all the analytic continuations of the given element f(z) is called the complete analytic function.

1995, Valentin I. Ivanov, Michael K. Trubetskov, Handbook of Conformal Mapping with Computer-Aided Visualization, CRC Press, page 26:

(mathematical analysis) An extension of an analytic function which is itself analytic.


Banach space

Named after Polish mathematician Stefan Banach (1892–1945).

named-after

1962 [Prentice-Hall], Kenneth Hoffman, Banach Spaces of Analytic Functions, 2007, Dover, page 138,
Before taking up the extreme points for H¹ and H ᪲, let us make a few elementary observations about the unit ball Σ in the Banach space X.

Already in these cases there is convergence in Banach spaces that are not only infinite-dimensional but nonseparable.

1992, R. M. Dudley, M. G. Hahn, James Kuelbs, editors, Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference, Springer, page ix:

2013, R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, page 35,
[A] Banach space is a complete normed linear space X. Its dual space X' is the linear space of all continuous linear functionals f:X→ℝ, and it has norm ‖f‖_(X')≡sup|f(x)|:‖x‖<1; X' is also a Banach space.

(functional analysis) A normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.


continuous function

en-noun

continuous function (plural continuous functions)

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

անընդհատ ֆունկցիա

function whose value changes only slightly when its input changes slightly

(mathematical analysis, topology) a function from one topological space to another, such that the inverse image of any open set is open


convergent sequence

convergent sequence (plural convergent sequences)

(mathematical analysis) a sequence which converges


convergent series

convergent series (plural convergent series)

(mathematical analysis) An infinite series whose partial sums converge.


confluent

en-adj

confluent (comparative more confluent, superlative most confluent)

more confluent (comparative)
most confluent (superlative)

(geometry, of a triangle) Exactly the same size as another triangle.

(mathematics) Given a binary operation →_β on a set A, and its reflexive, transitive closure ↠_β , then, for all a1, a2, and a3 in A, if a1 →_β a2 and a1 →_β a3, then there must exist an a4 in A such that a2 ↠_β a4 and a3 ↠_β a4.


divergent series

en-noun

divergent series (plural divergent series)

(mathematical analysis) An infinite series whose partial sums are divergent.


infinite series

infinite series (plural infinite series)

In 1734, Leonhard Euler solved the Basel problem by summing the infinite series #92;textstyle#92;frac#123;1#125;#123;1²#125;#43;#92;frac#123;1#125;#123;2²#125;#43;#92;frac#123;1#125;#123;3²#125;#43;#92;cdots#61;#92;sum#95;#123;n#61;1#125;#123;#92;infty#125;#92;frac#123;1#125;#123;n²#125;#61;#92;frac#123;#92;pi²#125;#123;6#125;.

(mathematical analysis) Any expression that represents the addition a countably infinite number of ordered summands, often explicitly indexed by the positive or nonnegative integers.


harmonic analysis

trigonometric series
trigonometric series

almost periodic function
almost periodic function

(mathematical analysis) A study of the representation of functions or signals as the superposition of basic waves, involving the notions of harmonic functions, trigonometric series, Fourier series, Fourier transforms, almost periodic functions, and others.


noetherian

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(algebra) Satisfying some (usually ascending) chain condition:

(of a ring) In which any ascending chain of ideals eventually becomes constant.

(of a module) In which any ascending chain of submodules eventually becomes constant.

(of a group) In which any ascending chain of subgroups eventually becomes constant.

(of a topological space) In which any ascending chain of open subsets eventually becomes constant.

(mathematics, of a binary relation on a set) Such that the converse relation is well-founded (on that set).

(algebraic geometry, of a scheme) admitting a finite covering by open affine subsets operatorname SpecA_i, where each A_i is a noetherian ring;


Artinian

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From Artin + -ian, after Emil Artin, an Armenian-Austrian mathematician.

(algebra, of a ring) in which any descending chain of ideals eventually starts repeating.

(algebra, of a module) in which any descending chain of submodules eventually starts repeating.


submodule

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(algebra) A module contained in a larger module, both over the same ring, such that the ring multiplication in the former is a restriction of that in the latter.


Egyptian fraction

Egyptian fraction (plural Egyptian fractions)

The fraction #92;frac#123;7#125;#123;10#125; can be written as the Egyptian fraction #92;frac#123;1#125;#123;2#125;#43;#92;frac#123;1#125;#123;5#125;.

The calculation of Mᴺ where both M and N are real numbers can be done by expressing N as the sum of an integer h and an Egyptian fraction 1#47;g#95;1#43;1#47;g#95;2#43;1#47;g#95;3#43;.... Then Mᴺ#61;Mʰ#92;sqrt#91;g#95;1#93;#123;M#125;#92;sqrt#91;g#95;2#93;#123;M#125;#92;sqrt#91;g#95;3#93;#123;M#125;... where each of the roots can be calculated by means of the Newton–Raphson method.

1995, C. Pomerance, A. Sárkőzy, Combinatorial number theory, R. L. Graham, M. Grötschel, L. Lovász (edtors), Handbook of Combinatorics, Elsevier (North-Holland), page 1014,
However, a representation of r as a sum of distinct Egyptian fractions is certainly not unique and this fact leads to many questions.

In [27], Erdős also considers various questions relating to Egyptian fraction decompositions of #92;textstyle 1#61;#92;sumⁿ#95;#123;k#61;1#125;#92;frac#123;1#125;#123;x#95;k#125;.

2013, R. L. Graham, “Paul Erdős and Egyptian Fractions”, in László Lovász, Imre Ruzsa, Vera T. Sós, editors, Erdős Centennial, Springer,, page 293:

The modern description of Egyptian fraction reckoning being "restricted" to unit fractions is obviously anachronistic (indeed, the Egyptian concept of fractions did not include a numerator, but from a historian's point of view this cannot be criticized on the basis that our modern fractions consist of denominator and numerator). Furthermore, this criticism does not do justice to the development of Egyptian fractions.

2016, Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, page 5:

(number theory) A representation of a rational number as a sum of distinct unit fractions.

egyptiläinen murtoluku

representation of a rational number as sum of distinct unit fractions


constant of integration

constant of integration (plural constants of integration)

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

constant d’integració

constant to be added at the indefinite integral


cellular automaton

cellular automaton (plural cellular automata or cellular automatons)

cellular automata (plural)
cellular automatons (plural)

Or maybe just taking some precautions when setting up the division scheme for a particular cellular automaton might be good enough in other cases, depending on how dynamic and non-homogeneous the behavior over the grid will be.

1989 March 21, Dave Hiebeler, “Fast LIFE (was: Re: summing 9-cell neighborhoods)”, in comp.theory.cell-automata (Usenet):

For fixed n this is, of course, a cellular automaton. You can allow any positive integer states for each of the n cells. The transition rule is such that if a cell has state x >=2 then the state never exceeds x and if it has state x < 2 then it either stays the same or increase by 1 or 2.

2003 October 8, Edwin Clark, “a puzzle related to artinian group”, in comp.theory.cell-automata (Usenet):

The paper concludes:
The Busy Boxes variant of the SALT family of reversible cellular automata has shown itself to possess an intriguing number of features that are worthy of further study and analysis. Physicist Gerard T'Hooft has written about the possibility that reversible CA's such as this one may help elucidate some aspects of physics in the quantum realm [5]. We believe that the study of emergent, qualitative, and statistically predictable behavior of these kinds of discrete, deterministic systems can indeed shed light on questions in physics, and may also help to clarify the deeper principles of entropy and the origin of complex, self-propagating systems such as those found in our biosphere

2012 July 3, Mike James, “A New Computational Universe - Fredkin's SALT CA”, in www.i-programmer.info, retrieved 2012-10-25:

Hello, I have recently constructed a cellular automaton in Golly, with a starting population of roughly 899 million, that produces Shierpinski's triangles.

2013 June 19, Aaron Ball, “Sierpinski's triangles generated from cellular automaton in Golly”, in comp.theory.cell-automata (Usenet):

(computing theory) An automaton consisting of cells arranged in a regular grid, in one or more dimensions. Each application of an associated rule creates a new generation, where the cells have a new configuration of values.


coplanar

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(geometry, of at least two things, usually lines or plane figures) Within the same plane.


characteristic polynomial

characteristic polynomial (plural characteristic polynomials)

The characteristic polynomial of (cc 1 4\3-5) is |cc 1-x 4\3-5-x|=x²+4x-17.

The characteristic polynomial of a 2×2 matrix M is λ²- mbox tr(M)λ+ mbox det(M), where mbox tr(M) denotes the trace of M and mbox det(M) denotes the determinant of M.

The characteristic polynomial of a 3×3 matrix M is -λ³+ mbox tr(M)λ²- mbox tr( mbox adj(M))λ+ mbox det(M), where mbox adj(M) denotes the adjugate of M.

(linear algebra) The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant.

ordinary differential equation
ordinary differential equation

differential operator
differential operator

(mathematics) A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t).


indeterminant

(mathematics) A variable with no value assigned to it; for example the variables in a polynomial.


linear transformation

linear transformation (plural linear transformations)

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

lineêre transformasie

a map between vector spaces preserving vector addition and scalar multiplication


self-adjoint

(functional analysis) Being a Hilbert space operator which is Hermitian and also whose eigenvectors span the entire Hilbert space.


skew-symmetric

(ccc 0 2 3\-2 0-4\-3 4 0)

(linear algebra) Of a matrix, satisfying A^( textsf )T=-A, i.e. having entries on one side of the diagonal that are the additive inverses of their correspondents on the other side of the diagonal and having only zeroes on the main diagonal.

schiefsymmetrisch

whose entries on one side of the diagonal are the additive inverses of their correspondents on the other side of the diagonal and whose elements on the main diagonal are zero


skew-symmetry

(linear algebra) The state of being skew-symmetric


unit vector

(linear algebra, mathematical analysis) A vector with the length of 1, defined as mathbf ̂u=(mathbf u)/(‖ mathbf u‖).


convex hull

1994, David Eppstein, Chapter 10: Average Case Analysis of Dynamic Geometric Optimization, Association of Computing Machinery, Society for Industrial and Applied Mathematics, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, page 83,
This problem can be solved in time O(nᶜ) by combining a weight-balanced tree of the convex hull vertices with a farthest neighbor data structure of Agarwal and Matoušek [2].

Thus, the polyhedron and the convex hull of the set of its vertices coincide, which was to be proved.
Since the convex hull of a set M is determined uniquely from M, Theorem 1 implies that a bounded solid polyhedron and its surface, i.e., a closed convex polyhedron, are determined uniquely from its vertices.

2005, A.D. Alexandrov, Convex Polyhedra, page 23:

The convex hull of a compact subset of an infinite dimensional topological vector space need not be a compact set.

2007, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, page 185:

(mathematics) The smallest convex set of points in which a given set of points is contained.

convex envelope

smallest convex set of points containing a given set


unlimited register machine

unlimited register machine (plural unlimited register machines)

(computing theory) A particular type of theoretical computer, with infinitely many memory cells, called registers, and formal rules to determine the machine's behavior based on their contents.


pushdown automaton

pushdown automaton (plural pushdown automata or pushdown automatons)

pushdown automata (plural)
pushdown automatons (plural)

(computing theory) An automaton with finitely many states that can also use one unbounded stack of memory; the automaton may only push, pop, or read the top of the stack. Abbreviation: PDA.


context-free

(linguistics) (Of a formal language) which can be decided by a pushdown automaton or a context-free grammar (sense 1)

yhteydetön

linguistics, formal language: which can be decided by a pushdown automaton or a context-free grammar


formal language

formal language (plural formal languages)

Holonym: formal system

Meronyms: alphabet, symbol, word

(computing theory) A set of finite strings (called words) made of symbols (from a finite set of symbols, called an alphabet).


Diophantine equation

Diophantine equation (plural Diophantine equations)

1993, Ross Talent, Alf van der Poorten (translation editors), Vladimir G. Sprindžuk, Classical Diophantine Equations, Springer, Lecture Notes in Mathematics 1559, page viii,
Nevertheless, the fundamental idea of Skolem's method, the reduction of algebraic diophantine equations to exponential equations, has shown exceptional vitality in recent episodes of the theory of diophantine equations.

Based on lectures he^([Davenport]) gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities.

2005, Harold Davenport, edited by T. D. Browning, Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd edition, Cambridge University Press, page i:

2020, Hasan Sankari, Mohammad Abobala, Neutrosophic Linear Diophantine Equations With Two Variables, in Neutrosophic Sets and Systems, Volume 38, University of New Mexico, page 402,
The following theorem determines the criteria for the solvability of [a] neutrosophic linear Diophantine equation.

(number theory) A polynomial equation whose variables are only permitted to assume integer values.

Diophantine equation
simple:Diophantine equation

equació diofàntica

polynomial equation of integer variables


overdetermined

en-adj

overdetermined (comparative more overdetermined, superlative most overdetermined)

more overdetermined (comparative)
most overdetermined (superlative)

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(linear algebra, of a system of linear equations) Having more equations than variables.


phase space

en-noun

phase space (countable and uncountable, plural phase spaces)

Henri Poincaré
Josiah Willard Gibbs
Ludwig Boltzmann
phase space

The concept was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré and Josiah Willard Gibbs.

For mechanical systems, the phase space usually consists of all possible values of position and momentum variables.

The phase space of a single material point is a manifold M×V*, and the Newton state-space is identifical^([sic]) with M×V.

1991, Jan J. Slawianowski, Geometry of Phase Spaces, Wiley, page 485:

2006, Matthew Charles Cargo, Phase Space Approaches to Higher Order WKB Approximations, University of California, Berkeley, page 8,
Their energies are those of the level sets p²/2+V(x)=E that enclose an area of 2π hbar (n+1/2) in the classical phase space. Despite embracing a configuration space point of view, these methods have produced a phase space result.

The distribution of particles in phase space is given either by the injector characteristics and injection process or in the case of electron beams by the equilibrium of quantum excitation due to synchrotron radiation and damping.

2007, Helmut Wiedemann, Particle Accelerator Physics, Springer, page 377:

(mathematics, physics, dynamical system theory) Given a (dynamical) system, any topological space such that every point in the space's underlying set uniquely represents a state of the system and every possible state is represented by some point;

Given a (dynamical) system, any topological space such that every point in the space's underlying set uniquely represents a state of the system and every possible state is represented by some point;
(less formally) the underlying set of such a topological space.


analytic mechanics

(mathematics, physics) The application of calculus to classical mechanics.

解析力学

the application of calculus to classical mechanics


wave mechanics

(physics, mathematics) The mathematical basis for quantum mechanics in which light and subatomic particles have the characteristics of both waves and particles.


intuitionist

A person who studies intuitionistic mathematics


intuitionistic

Intuitionistic type theory is based on a certain analogy or isomorphism between propositions and types: a proposition is identified with the type of its proofs. This identification is usually called the Curry–Howard isomorphism, which was originally formulated for intuitionistic logic and simply typed lambda calculus. Type Theory extends this identification to predicate logic by introducing dependent types, that is types which contain values. Type Theory internalizes the interpretation of intuitionistic logic proposed by Brouwer, Heyting and Kolmogorov, the so called BHK interpretation. The types of Type Theory play a similar role to sets in set theory but functions definable in Type Theory are always computable.ᵂᴾ

The system, which has come to be known as IZF, or Intuitionistic Zermelo–Fraenkel (ZF refers to ZFC without the axiom of choice), has the usual axioms of extensionality, pairing, union, infinity, separation and power set. The axiom of regularity is stated in the form of an axiom schema of set induction. Also, while Myhill used the axiom schema of replacement in his system, IZF usually stands for the version with collection.ᵂᴾ

proof by contradiction
proof by contradiction

(mathematics, logic) Dealing strictly in constructive proofs, abstaining from proof by contradiction


intuitionism

intuitionism (countable and uncountable, plural intuitionisms)

(mathematics) An approach to mathematics/logic which avoids proof by contradiction, and which requires that, in order to prove that something exists, one must construct it.

ինտուիցիոնիզմ

approach to mathematics which avoids proof by contradiction


local ring

The unique maximal ideal of a (commutative) local ring contains all of the zero divisors of such ring, and all elements of the ring outside of it are units. Then in a local ring, the sum of any two zero divisors is also a zero divisor. Contrapositively, if two ring elements add up to a unit then one of them must be a unit as well. A simple example of a local ring is #92;mathbb#123;Z#125;#95;8.

(algebra) A commutative ring with a unique maximal ideal, or a noncommutative ring with a unique maximal left ideal or (equivalently) a unique maximal right ideal.


nilradical

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(algebra) The set of nilpotent elements of an algebraic structure such as an ideal


combinatorialist

A mathematician who specializes in combinatorics.


supplementary angle

supplementary angle (plural supplementary angles)

Angle A is 40°, and its supplementary angle, B, is 140°.

(geometry) An angle that, when added to another referenced angle, produces an angle of 180 degrees.


sextile

From Latin sextīlis (“one sixth”), from sextus (“sixth”) + īlis (“-ile”).

(statistics) A quantile of six equal proportions; any of the subsets thus obtained.


quantile

Back-formation from quantity, modeled on percentile.

(statistics) One of the class of values of a variate which divides the members of a batch or sample into equal-sized subgroups of adjacent values or a probability distribution into distributions of equal probability.


graph theorist

(graph theory) A mathematician or computer scientist who specializes in graph theory


number theorist

number theorist (plural number theorists)

A mathematician who specializes in number theory


group theorist

An algebraist who specializes in group theory


ring theorist

An algebraist who specializes in ring theory


representation theorist

representation theorist (plural representation theorists)

representation theory
representation theory

An algebraist who specializes in representation theory


cokernel

(category theory) For a category with zero morphisms: the coequalizer between a given morphism and the zero morphism which is parallel to that given morphism.


oblique angle

(geometry) Any angle that is not a right angle or multiple of right angles.

vino kulma

any angle that is not a right angle or multiple of right angles


right-angled

(not comparable, geometry) Having a right angle.


chordal

They play very well together, with Vogl providing the melodic and chordal framework and Shanbaum riffing over it.

1985 December 21, Nan Donald, “Flat-picking up a Storm”, in Gay Community News, volume 13, number 23, page 11:

(mathematics, music) Of, pertaining to, or consisting of chords.

(graph theory) For a graph, in which all cycles of four or more vertices have a chord.


ZFC

Zermelo-Fraenkel set theory with Choice
w:Zermelo–Fraenkel set theory

(set theory) Initialism of Zermelo-Fraenkel set theory with Choice; the standard axiomatization of set theory, including the axiom of choice.


axiom of choice

axiom of choice (countable and uncountable, plural axioms of choice)

A calque of German Axiom der Auswahl (now more commonly Auswahlaxiom), which first appeared in print with a description of the axiom in 1908, Ernst Zermelo, Untersuchungen über die Grundlagen der Mengenlehre I ["Investigations in the foundations of set theory I"], Mathematische Annalen, 65 (although the paper was dated 1907).

The axiom of choice is logically equivalent to the assertion that every vector space has a basis.

If V = L then the axioms of choice and the continuum hypothesis are both true, and the assertion that a measurable cardinal exists is false.

1993, Thomas Tymoczko, editor, Penelope Maddy: Does V Equal L?: New Directions in the Philosophy of Mathematics: An Anthology, page 357:

Throughout this work we adopt the Zermelo–Fraenkel (ZF) axioms of set theory with the Axiom of Choice, commonly abbreviated as ZFC. It follows from the work of Gödel and Cohen that if the ZF axioms are consistent, the Axiom of Choice can be neither proved nor disproved from the ZF axioms.

1993, Gary L. Wise, Eric B. Hall, Counterexamples in Probability and Real Analysis, page vii:

To clarify these ideas for the reader, let us show, without the axiom of choice, that a product of finitely many nonempty sets is nonempty: This is done by induction on the number n of sets.[…]The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable.

2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, page 169:

Perhaps what this debate about whether to accept the axiom of choice indicates is that the disjunction between regularity and randomness is as fundamental to our conception of the world as that between discreteness and continuity.

2004, Michael Potter, Set Theory and its Philosophy: A Critical Introduction, page 259:

There is a uniform status quo almost with just a few exceptions of a handful of mathematicians around the world. Most people accept the “infinite choice, infinite decimals” approach to real numbers, and the justification that has been created to substantiate this, and to overcome some of the difficulties that I have shown you is through an elaborate axiomatic framework. Mathematicians are not stupid! They realize that this theory is dubious, and so what they have done is they have created a rather elaborate axiomatic framework. And that axiomatic framework is something that we can study in set theory and logic; and its purported aim is to create a framework for mathematics; but its initial aims were very much directed towards solving the problems, overcoming the difficulties with real numbers as infinite decimals. Prominent amongst these axioms is this Axiom of Choice! The Axiom of Choice manifests itself in mathematics in many ways; but its primary role — OK — its primary role is exactly here at the level of infinite decimals and real numbers. It essentially asserts, as a matter of faith or belief, that it is possible to choose an infinite number of digits arbitrarily and independently, and that a legitimate mathematical object results. This is the key philosophical point. Are we, or are we not, in a position to be able to specify a first digit, a second digit, a third digit, a fourth digit, and so on to infinity? And are we allowed to call that specification a new mathematical object, a real number? The Axiom of Choice says that “Yes we are!”, so it is a statement of belief and its primary objective is to allow the building of this real number system.

2012, 47:31 – 49:50 from the start, in Difficulties with real numbers as infinite decimals (I) Real numbers + limits Math Foundations 91, episode 91, njwildberger (N. J. Wildberger), via YouTube:

(set theory) One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.

ընտրության աքսիոմ

axiom that any product of non-empty sets is non-empty


direct product

If A and B are sets, their direct product is the set of ordered pairs (a,b) with a in A and b in B.

(set theory) The set of all possible tuples whose elements are elements of given, separately specified, sets.

Theorem 3.2.3. A periodic Abelian group is the direct product of its Sylow subgroups, S(p).

1976, Marshall Hall, Jr., The Theory of Groups, 2nd edition, page 40:

(group theory) Such a set of tuples formed from two or more groups, forming another group whose group operation is the component-wise application of the original group operations and of which the original groups are normal subgroups.

(topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.

suora tulo

product of sets — see also Cartesian product

Let us start with the definition of the direct product of two probability measures. Let P#95;1 and P#95;2 be probability measures on #92;Omega#95;1 and #92;Omega#95;2, respectively, and denote #92;Omega#95;1#92;times#92;Omega#95;2 by #92;Omega. A probability measure P on #92;Omega with #92;mathfrak#123;D#125;#92;left(P#92;right)#61;#92;mathfrak#123;D#125;#92;left(P#95;1#92;right)#92;times#92;mathfrak#123;D#125;#92;left(P#95;2#92;right) is called the direct product of P#95;1 and P#95;2 (written P#95;1#92;timesP#95;2) if
P(B_1×B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2.
The probability space (Ω,P) is called the direct product of (Ω₁,P_1) and (Ω₂,P_2), written
(Ω,P)=(Ω₁,P_1)×(Ω₂,P_2).
For example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.

1978, K. Itô, An Introduction to Probability Theory, page 53

(mathematics) Any of a number of mathematical objects analogously derived from a given ordered set of objects.

(category theory) A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects.


universal property

universal property (plural universal properties)

(mathematics) The property of there being a unique morphism from an arbitrary object to the universal morphism of a comma category which is from a functor to a fixed object; or, dually, the property of there being a unique morphism from the universal morphism to an arbitrary object of a comma category which is from a fixed object to a functor. (Caveat: the uniqueness is up to isomorphism.)


x-direction

(algebraic geometry) The direction aligned with the x-axis of a coordinate system.


y-direction

(algebraic geometry) The direction aligned with the y-axis of a coordinate system.


z-direction

(algebraic geometry) The direction aligned with the z-axis of a coordinate system.


heart-shaped

en-adj

heart-shaped (comparative more heart-shaped, superlative most heart-shaped)

more heart-shaped (comparative)
most heart-shaped (superlative)

[…] Passing, I leave thee lilac with heart-shaped leaves, / I leave thee there in the door-yard, blooming, returning with spring.

1865, Walt Whitman, “When Lilacs Last in the Dooryard Bloom’d”, in Sequel to Drum-Taps: When Lilacs Last in the Dooryard Bloom’d and other poems:

Having the traditional shape of a heart (♥), that is pointed at one end and indented at the opposite side.


splicer

uniform spanning trees
uniform spanning tree

(graph theory) A union of uniform spanning trees.


round off

en-verb

round off (third-person singular simple present rounds off, present participle rounding off, simple past and past participle rounded off)

rounds off (present singular third-person)
rounding off (participle present)
rounded off (participle past)
rounded off (past)

The instructions say that we are not to round off until we reach the final figure. [in a calculation with interim steps]

Round off 15.4 to 15, round off 15.51 to 15.5 or to 16, round off 0.499 to 0, and round off 970,000 to 1 million.

"This product contains no PCBs" is a typical commercial distortion if it actually contains 0.498 of the measurement unit, rounded off to "0"

en:make rounder by removing significant digits

(mathematics, transitive, intransitive) To change (a number) to an approximation having fewer significant digits.

arredondiar

to change a number into an approximation having fewer significant digits


semifinite

(mathematical analysis) (Of a measure space) in which every nonzero measurable set has a subset with finite nonzero measure


covariant

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(category theory, of a functor) Which preserves the order of morphism composition: formally, which associates each morphism f:X→Y to a morphism F(f):F(X)→F(Y).


covariant

(algebra) A bihomogeneous polynomial in x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations.

(algebra) The variety defined by a covariant.


contravariant

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(category theory, of a functor) Which reverses the order of morphism composition: formally, which associates each morphism f:X→Y to a morphism F(f):F(Y)→F(X).

(linear algebra, of a vector) Scaling inversely with a change of basis.


contravariant

(algebra) A bihomogeneous polynomial in dual variables of x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations.


spinor

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(algebra) An element of the fundamental representation of a Clifford algebra that transforms to its negative when the space is rotated through a complete turn from 0° to 360°.


time-dependent

(mathematics, physics) Determined by the value of a variable representing time.

time-dependent density functional theory


time-independent

(mathematics, physics) Not time-dependent; not determined by the value of a variable representing time.


time domain

(mathematics, physics) The set of values, representing time, over which a function dependent on time is defined.


petameter

(US) (metrology) An SI unit of length equal to 10¹⁵ meters; alternative spelling of petametre.


abstract nonsense

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Coined by Norman Steenrod, popularized by Serge Lang.

category theoretical
category theoretical

(mathematics, humorous) Details which are straightforward but so tedious that an author or lecturer would rather skip them (especially those of a category theoretical nature).

³Lang’s Algebra [Lan02, p. 759] supports the general consensus that this was not intended as an epithet:
 In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod, see [CE56]), it was realized that there was a systematic way of developing certain relations of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretic, and were affectionately called abstract nonsense by Steenrod.

2016, Emily Riehl, Category Theory in Context (Aurora: Dover modern math originals), New York: Dover, →ISBN, →OCLC, page x:

(mathematics, humorous) Details which involve diagram chasing.

(mathematics, humorous) Category theory in general.


available seat-mile

en-noun

available seat-mile (plural available seat-miles)

(aviation) A measure of an airline flight's passenger carrying capacity, equal to the number of seats onboard an aircraft multiplied by the distance flown in miles. Seats that are not available for sale to revenue-paying passengers (e.g., seats reserved for crew rest, etc.) are excluded from this calculation.


hulking

A hulking shape burst through the doorway and hurtled down the corridor, leaving a maelstrom of air currents in his wake.

2001, Eoin Colfer, Artemis Fowl, page 212:

Large and bulky, heavily built; massive.


decimal place

The answer is 1.245, to three decimal places

(arithmetic) the position of a digit to the right of the decimal point in a decimal number.

小數位 /小数位

position of digit to the right of the decimal point


place of decimals

place of decimals (plural places of decimals)

It will always be better to calculate one place of decimals more than are required by the question.

1841, Charles Hutton, Thomas Stephens Davies, A Course of Mathematics in Two Volumes, Composed for the Use of the Royal Military Academy, page 56:

Find the positive root of the equation eˣ – 3x = 0, correct to 3 places of decimals.

1980, Michael H. Stone, The Borderline Syndromes: Constitution, Personality, and Adaptation, page 101:

In consequence of the need of Tables of Jₙ (x) with fairly large values of n and x for Astronomical purposes, Hansen constructed a Table of J₀ (x) and Jᵢ (x) to six places of decimals with a range from x = 0 to x = 10.0 with interval 0.1.

1995, G. N. Watson, A Treatise on the Theory of Bessel Functions, page 655:

To round a decimal correct to one place of decimals (1 d.p.) you look at the second place of decimals.

2001, Keith Pledger, Edexcel GCSE Mathematics, page 103:

Use Newton Raphson method to find the values of (i) VT2 and (ii) —, correct to four places of decimals.

2005, Veerarajan & Ramachandran, Numerical Methods: With Programs In C, pages 3–54:

In the application of Simpson's rule, when the number of places of decimals to which the answer is required, is not mentioned, it is usual to calculate the answer correct to three places of decimals.

2006, Bansi Lal, Topics in Integral Calculus, page 272:

(arithmetic, dated, chiefly UK, usually in the plural) A decimal place.


coset

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From co- + set; apparently first used 1910 by American mathematician George Abram Miller.

1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597,
Theorem 10.5. The collection consisting of an invariant subgroup H and all its distinct cosets is itself a group, called the factor group of G, usually denoted by G/H. (Remember that the left and right cosets of an invariant subgroup are identical.) Multiplication of two cosets aH and bH is defined as the set of all distinct products z = xy, with x ∈ aH and y ∈ bH; the identity element of the factor group is the subgroup H itself.

1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614,
In general, the coset in row x consists of all the elements xh as h runs through the various elements of H.

Example 3. Let G#61;#92;mathbb#123;Z#125; (the operation is #43;), H#61;2#92;mathbb#123;Z#125;. Then the coset 1#43;2#92;mathbb#123;Z#125; is the set of integers of the form 1#43;2k where k runs through all elements of #92;mathbb#123;Z#125;.

2009, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer, page 231:

(algebra, group theory) The set that results from applying a group's binary operation with a given fixed element of the group on each element of a given subgroup.

kõrvalklass

result of applying group operation with fixed element from the parent group on each element of a subgroup


left coset

(algebra) A left coset of a subgroup is a copy of that subgroup, multiplied on the left by some element from the parent group


right coset

(algebra) A copy of a subgroup, multiplied on the right by some element from the parent group.


epimorphism

(category theory) A morphism p such that for any other pair of morphisms f and g, if f∘p=g∘p, then f = g.

Epimorphismus

mathematics: map f such that whenever g composed with f equals h composed with f, then g=h


codomain

domain

antonym(s) of “target set of a function”

Definition 2.5. A function is onto if each element of the codomain has at least one element of the domain assigned to it. In other words, a function is onto if the range equals the codomain.

1994, Richard A. Holmgren, A First Course in Discrete Dynamical Systems, Springer, page 11:

2006, Robert L. Causey, Logic, Sets, and Recursion, 2nd Edition, Jones & Bartlett Learning, page 192,
Once we have described f as a function from A to B, by convention we will call B the codomain, even though other sets, of which B is a subset, could have been used. […] If y is an element of the codomain, then y∈ mathit Img(f,A) iff there is some x in the domain such that f maps x to y.

For example, the codomain of g(X)#61;X³ consists of all real numbers. A function links each element in its domain to some element in its codomain. Each domain element is linked to exactly one codomain element.

2017, Alan Garfinkel, Jane Shevtsov, Yina Guo, Modeling Life: The Mathematics of Biological Systems, Springer, page 12:

(mathematics, mathematical analysis) The target set into which a function is formally defined to map elements of its domain; the set denoted Y in the notation f : X → Y.

(more generally, of a binary relation R between A and B) The set B.


Cauchy sequence

Cauchy sequence (plural Cauchy sequences)

Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.

However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.

1955, [Van Nostrand], John L. Kelley, General Topology, Springer, published 1975, page 174:

In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. In the general case, however, this is not so. If a metric space does have the property that every Cauchy sequence converges, the space is called a complete metric space.

2000, George Bachman, Lawrence Narici, Functional Analysis, page 52:

Cantor first redefined Cauchy sequences using rational numbers only.[…]Cantor's idea was to define the real number line as the collection of all (rational) Cauchy sequences.

2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153:

(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<ϵ.


predictive

Borrowed from Latin praedictivus, from praedico. Equivalent to predict + -ive.

However, the posterior predictives combine two sources of information: what we might term the structural effect of WIC participation as well as an unobserved correlation between the errors of the participation and outcome equations.

2008, Siddhartha Chib, William Griffiths, Bayesian Econometrics, page 160:

Alternatively, we can use prior predictives to help define prior distributions.

2018, Simon Farrell, Stephan Lewandowsky, Computational Modeling of Cognition and Behavior, page 308:

(statistics) Simulated data generated from a statistical model, based on the estimates for the real data.


unit fraction

(arithmetic) A fraction whose numerator is 1 and whose denominator is a positive integer; for example, 1/2 or 1/7.

yksikkömurtoluku

fraction whose numerator is 1 and denominator a positive integer


icosihenagon

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

From Ancient Greek εἰκοσιέν (eikosién, “twenty-one”) + -gon.

uder

(geometry) A polygon with twenty-one edges and twenty-one angles.


timelike

en-adj

timelike (comparative more timelike, superlative most timelike)

more timelike (comparative)
most timelike (superlative)

(mathematics, physics) Of a four-vector in representing a point in spacetime, having a positive or negative four-vector norm for the metric signature (+, -, -, -) and (-, +, +, +) respectively.


token economy

The supply of any token is also important in the operation of a token economy, as well as for the more fundamental issue of token valuation.

2019, Chris Brummer, Cryptoassets: Legal, Regulatory, and Monetary Perspectives, Oxford University Press, →ISBN, page 165:

(cryptocurrencies, economics) An economy based on cryptocurrency tokens.


homeomorphism

(topology) a continuous bijection from one topological space to another, with continuous inverse.


one-size-fits-all

(of a garment) Designed to be worn with comfort by people with a wide range of sizes.

均碼 /均码

designed to be worn with comfort by people with a wide range of sizes


primality

en-noun

primality (usually uncountable, plural primalities)

(mathematics) The condition of being a prime number.


Grubbs' test

Named after Frank E. Grubbs, who published the test in 1950.

(statistics) A statistical test for deciding whether a data point is an outlier.


coiner

From Middle English coynour, from Middle French coigneur; equivalent to coin + -er (agent noun suffix). In the cryptocurrency sense, coin + -er (“supporter”).

(cryptocurrencies) A person who invests in or advocates for cryptocurrencies.


half dozen

The half-dozen pieces […] were painted white and carved with festoons of flowers, birds and cupids. To display them the walls had been tinted a vivid blue which had now faded, but the carpet, which had evidently been stored and recently relaid, retained its original turquoise.

1963, Margery Allingham, chapter 1, in The China Governess: A Mystery, London: Chatto & Windus, →OCLC:


sigmate

en-adj

sigmate (comparative more sigmate, superlative most sigmate)

more sigmate (comparative)
most sigmate (superlative)

From sigma + -ate (adjective-forming suffix).


petawatt

(metrology) An SI unit of power equal to 10¹⁵ watts. Symbol: PW


polynomiograph

(mathematics) A two-dimensional coloured image, prepared using polynomiography, that is the representation of a polynomial


polynomiography

(mathematics) An iterative process used to create two-dimensional coloured images (polynomiographs) that are the representations of polynomials


Kelvin function

en-noun

Kelvin function (plural Kelvin functions)

Kelvin differential equation
Kelvin differential equation

Bessel function of the first kind
Bessel function of the first kind

(mathematics) Any of class of special functions, usually denoted as two pairs of functions berₙ(x), beiₙ(x), kerₙ(x) and keiₙ(x) with variable x and given order number n. The former two functions berₙ(x) and beiₙ(x) respectively correspond to the real part and the imaginary part of the Kelvin differential equation's solution that can be expressed with the Bessel function of the first kind Jₙ(x), and the latter kerₙ(x) and keiₙ(x) correspond to those that can be expressed with the modified Bessel function of the second kind Kₙ(x).


Bessel function

Bessel function (plural Bessel functions)

For Friedrich Bessel, German mathematician.

differential equation
differential equation

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


cumulant

(mathematics) Any of a set of parameters of a one-dimensional probability distribution of a certain form


maximal

In the sequence (1, 2, 10, 5, 12, 6), the fifth term, f(5) = 12, is a maximal term, as each of the first 4 terms are smaller than 12.

(mathematics, of a sequence) Larger than any previous term in the sequence.

With respect to the ordering induced by set-theoretic inclusion, the set #92;#123;#92;#123;2#92;#125;,#92;#123;1,2#92;#125;,#92;#123;2,3#92;#125;#92;#125; has two maximal elements: #92;#123;1,2#92;#125; and #92;#123;2,3#92;#125;

(mathematics, of an element of a poset) Such that no other element is greater (with respect to the given partial order).


maximal

(mathematics) The element of a set with the greatest magnitude.

(mathematics) Said of an ideal of a ring or a filter of a lattice: that it is as large as it can be without being trivial (improper).


berkovets

From Russian бе́рковец (bérkovec); derived from бе́рковеск (bérkovesk), Old East Slavic бьрковьскъ (bĭrkovĭskŭ, “of Birka”), ultimately Old Norse Birka, a port city in mediaeval Sweden.

der

(historical) An old Russian unit of weight, approximately 164 kilograms or 10 poods.


floored

en-adj

floored (comparative more floored, superlative most floored)

more floored (comparative)
most floored (superlative)


multiplicative

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

From 6th-century Latin multiplicativus, from Classical Latin multiplicāre (“to multiply”).

Matrix determinants are multiplicative; if A and B are square matrices of the same size, then #92;mbox#123;det#125;(AB)#61;#92;mbox#123;det#125;(A)#92;,#92;mbox#123;det#125;(B).

(mathematics, of a function, etc.) Distributive over multiplication.

(algebra) Having multiplication as an operator.


hairy ball theorem

(mathematics) A theorem stating that there is no nonvanishing continuous tangent vector field on the sphere; colloquially, that "you can't comb a hairy ball flat without creating a cowlick".


vector field

(mathematics) a function mapping a subset of ℝⁿ into a subset of ℝⁿ

differential geometry
differential geometry

(differential geometry) a function which associates, to each point on a surface, a vector in the tangent plane of that point


nonvanishing

For each nonvanishing function h, which is a solution of (2.1), we consider #92;Omega as above and we define
#92;Omega#95;i#61;d#92;Omega(e#95;i),#92;#59;#92;#59;#92;#59;W#61;#92;frac#123;#92;Omega#125;#123;h#125;

2001 January 1, A. A. Coley, Bäcklund and Darboux Transformations: The Geometry of Solitons : AARMS-CRM Workshop, June 4-9, 1999, Halifax, N.S., Canada (CRM proceedings & lecture notes), American Mathematical Soc., →ISBN, page 153:

This means that the vector space of solutions of (2.25) near #92;lambda#61;0 is generated by
#92;sigma#95;1(#92;lambda) holomorphic and nonvanishing at 0,
#92;left(#92;lambda#92;sigma#95;2(#92;lambda)#43;(log#92;lambda)#92;sigma#95;1(#92;lambda)#92;right) where a #92;sigma#95;2 is holomorphic and nonvanishing at 0.

2013 November 11, C. Herbert Clemens, A Scrapbook of Complex Curve Theory (University Series in Mathematics), Springer Science & Business Media, →ISBN, →OCLC, page 61:

(mathematics) Of an expression, especially a function, being nonzero at a value, everywhere on a specified set, or on the entire domain.


nonzero

nonzero Hopf number

(mathematics, of a quantity) Not equal to zero.

ei-nolla (more common to say joka ei ole nolla "that is not zero")

nichtnull (only predicatively or compounding)


nonpositive

(mathematics, of a quantity) Not positive; either zero or negative.


retraction

en-noun

retraction (countable and uncountable, plural retractions)

/ɹɪˈtɹækʃən/

(mathematics) A continuous function from a topological space onto a subspace which is the identity on that subspace.


convolve

en-verb

convolve (third-person singular simple present convolves, present participle convolving, simple past and past participle convolved)

convolves (present singular third-person)
convolving (participle present)
convolved (participle past)
convolved (past)

Borrowed from Latin convolvere (“roll together”), from con- (“together”) + volvō (“roll”); see voluble, volute, and compare involve, evolve, revolve.

(mathematics) To form the convolution of something with something else


Euclidian space

en-noun

Euclidian space (countable and uncountable, plural Euclidian spaces)

(mathematics, geometry) A space where the principles of Euclidean geometry apply.


celestial equator

celestial equator (plural celestial equators)

(astronomy) A great circle on the celestial sphere, coincident with the plane of Earth's Equator.


sequent

1550s; borrowed from Middle French sequent, from Old French sequent, itself borrowed from Latin sequentem, present participle of sequī (“to follow”).


tetradecimal

(mathematics, computing) Based on the number 14; using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C and D.


pentadecimal

Kleene [13] uses a similar approach, based on Turing machine encodings over an alphabet of fifteen symbols, which are then translated into pentadecimal numbers.

1980 April, Norman H Cohen, “Gödel numbers: a new approach to structured programming”, in ACM SIGPLAN Notices:

(computing, mathematics) Based on the number 15; using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D and E.


simplexity

(mathematics) The number of simplexes into which a structure can be divided


contractible

en-adj

contractible (comparative more contractible, superlative most contractible)

more contractible (comparative)
most contractible (superlative)

(mathematics) (of a topological set) Able to be reduced to one of its points by a continuous deformation


convergent

convergent (comparative more convergent, superlative most convergent)

more convergent (comparative)
most convergent (superlative)

(mathematical analysis, topology) Of a (typically infinite) sequence or series in a topological space, converging to a (finite, proper) limit.


convergent

(mathematics) The rational number obtained when a continued fraction is terminated after a finite number of terms.


continued fraction

en-noun

continued fraction (plural continued fractions)

Several results focus on the famous Rogers–Ramanujan continued fraction [47], [48, pp. 214-215], the only continued fraction appearing in Ramanujan's published papers.

1992, G. E. Andrews, B. C. Berndt, L. Jacobsen, R. L. Lamphere, “The Continued Fractions Found in the Unorganized Portions of Ramanujan's Notebooks”, in Memoirs of the American Mathematical Society, volume 99, number 477, page 1:

2000, Andrew Zardecki, 18: Continued Fractions in Time Series Forec[a]sting, Da Ruan (editor), Fuzzy Systems and Soft Computing in Nuclear Engineering, Physica-Verlag, Studies in Fuzziness and Soft Computing, page 397,
We achieve this by using well-known examples from the number theory pertaining to the continued fractions. Any sequence of natural numbers drawn from the probability distribution of the quotients of the continued fraction corresponding to an irrational number represents a typical sequence, in the sense that almost all sequences of quotients have this distribution.

We have seen that all rational numbers, and no other number, can be expressed as finite simple continued fractions.
The main reason of interest of continued fractions, however, is in their application to the representation of irrational numbers. To that end we shall need infinite simple continued fractions.

2009, M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo, translated by Daniele Gewurz, Elementary Number Theory, Cryptography and Codes, page 48:

(mathematics, number theory) A compound numerical expression consisting of an integer plus a fraction whose numerator is a positive integer and whose denominator is a continued fraction (an integer plus a fraction), and so on, with finite or infinite recursion.


quinary

From the Latin quīnārius (“containing five each”), from quīnī (“five each”, “five at a time”) + -ārius (whence the English suffix -ary); compare the French quinaire, the Italian quinario, and the Portuguese quinário. Doublet of quinarius.

der

(arithmetic) To the base of five; represented by the numerals 0, 1, 2, 3 and 4.


curlicue fractal

en-noun

curlicue fractal (plural curlicue fractals)

(mathematics) Any of a family of fractal shapes generated iteratively from a given irrational number


keychain

(cryptography) A digital repository for the storage of cryptographic keys.


reachable

en-adj

reachable (comparative more reachable, superlative most reachable)

more reachable (comparative)
most reachable (superlative)

/ˈɹiːtʃəbəl/

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

(mathematics) (of a node) That may be reached from another node in a graph by passing along one or more lines


centroid

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

From centre + -oid. From 1844, used as a replacement for the older terms "centre of gravity" and "centre of mass" in situations described in purely geometrical terms, and subsequently used for further generalisations.

形心

mean position of points in a geometrical figure — see also center of gravity, barycenter

barycentre

centre of gravity of a uniform, rigid object

The center of gravity of any body or geometrical magnitude is by definition the same as the centroid of a certain system of parallel forces. It will be convenient, therefore, to use the word centroid in most cases instead of center of gravity.[…]The centroid of any area may be found by the following method: Divide the area into parts such that the area and centroid of each part are known. Take the centroids of the partial areas as the points of application of forces proportional respectively to those areas. The centroid of this system of forces is the centroid of the total area, and may be found by the method of Art. 172.

1892, Leander Miller Hoskins, The Elements of Graphic Statics, MacMillan and Co., pages 151–152:

For example, if a postal code area (PCA) has 1000 distinct residences, we might suppose all 1000 residences are at the centroid of the PCA. Centroids are commonly used, for example, with geographic information systems and CD-ROM phone books (Francis, Lowe, Rushton and Rayco 1999).

2004, Richard L. Francis, Timothy J. Lowe, Arie Tamir, “7: Demand Point Aggregation for Local Models”, in Zvi Drezner, Horst W. Hamacher, editors, Facility Location: Applications and Theory, Springer-Verlag, page 207:

operatorname Centroid(𝒳)=(∫xg(x)dx)/(∫g(x)dx)      (6)
where the integrals are taken over the whole space ℝⁿ, and g is the characteristic function of the subset, which is 1 inside 𝒳 and 0 outside it [27].

2020, Cheng Zhang, Qiuchi Li, Lingyu Hua, Dawei Song, Assessing the Memory Ability of Recurrent Neural Networks, Giuseppe De Giacomo, et al. (editors), ECAI 2020: 24th European Conference on Artificial Intelligence, IOS Press, page 1660, In ℝⁿ, a centroid is the mean position of all the points in all of the coordinate directions. The centroid of a subset 𝒳 of ℝⁿ is computed as follows

(geometry, physics, engineering, of an object or a geometrical figure) The point at which gravitational force (or other universally and uniformly acting force) may be supposed to act on a given rigid, uniformly dense body; the centre of gravity or centre of mass.

(geometry, specifically, of a triangle) The point of intersection of the three medians of a given triangle; the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of the three vertices.

The centroid of an arbitrary function f is given by #92;frac#123;#92;intxf(x)dx#125;#123;#92;intf(x)dx#125;, where the integrals are calculated over the domain of f.

(mathematical analysis, of a function) An analogue of the centre of gravity of a nonuniform body in which local density is replaced by a specified function (which can take negative values) and the place of the body's shape is taken by the function's domain.

barycentre

centre of gravity of a set of points of equal mass

The k-means procedure classifies a given data set by using a user defined number of clusters, k, a priori. The centroids can be placed randomly, or algorithmically, but it should be noted that the initial placement will affect the result. The next step is to analyze each point within the data set and group it with the nearest centroid according to some distance metric. When all points have been assigned to a group, a new centroid is calculated for each group as a barycenter of the cluster, resulting from the previous step. Once the k new centroids are calculated, the algorithm reiterates through the data set, and each sample is again assigned to a cluster based on its distance to the new centroids. This process is continued until the position^([sic]) of the centroids no longer change.

2011, Ross Maciejewski, Data Representations, Transformations, and Statistics for Visual Reasoning, Morgan & Claypool Publishers, page 34:

The k-means clustering algorithm (MacQueen, 1967) is a widely used clustering method that applies relocation of points to find a locally optimal partitioning of a dataset. In k-means, the total distance between each data point and a representative point (centroid) of the cluster to which it is assigned is minimized. Each iteration of k-means has two steps. In the cluster assignment step, k-means assigns each point to a cluster such that, of all the current cluster centroids, the point is closest to the centroid of that cluster. In the cluster re-estimation step, k-means re-estimates the new cluster centroids based on the reassignments of points to clusters in the previous step. The cluster re-assignment and centroid re-estimation steps proceed in iterations until a specified convergence criterion is reached, such as when the total distance between clusters and centroids does not change substantially from one iteration to another.

2012, Biswanath Panda, Joshua S. Herbach, Sugato Basu, Roberto J. Bayardo, “2: MapReduce and its Application to Massively Parallel Learning of Decision Tree Ensembles”, in Ron Bekkerman, Mikhail Bilenko, John Langford, editors, Scaling Up Machine Learning, Cambridge University Press, page 26:

(statistics, cluster analysis, of a cluster of points) the arithmetic mean (alternatively, median) position of a cluster of points in a coordinate system based on some application-dependent measure of distance.

1974 [Prentice-Hall], Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, 2017, Dover, page 248,
Just as in the case of centers of a tree (Section 3-4), it can be shown that every tree has either one centroid or two centroids. It can also be shown that if a tree has two centroids, the centroids are adjacent.

2009, Hao Yuan, Patrick Eugster, An Efficient Algorithm for Solving the Dyck-CFL Reachability Problem on Trees, Giuseppe Castagna (editor), Programming Languages and Systems: 18th European Symposium, Proceedings, Springer, LNCS 5502, page 186,
A node x in a tree T is called a centroid of T if the removal of x will make the size of each remaining connected component no greater than |T|/2. A tree may have at most two centroids, and if there are two then one must be a neighbor of the other [6, 5]. Throughout this paper, we specify the centroid to be the one whose numbering is lexicographically smaller (i.e, we number the nodes from 1 to n). There exists a linear time algorithm to compute the centroid of a tree due to the work of Goldman [21]. We use operatorname CT(T) to denote the centroid of T computed by the linear time algorithm.

(graph theory, of a tree) Given a tree of n nodes, either (1) a unique node whose removal would split the tree into subtrees of fewer than n/2 nodes, or (2) either of a pair of adjacent nodes such that removal of the edge connecting them would split the tree into two subtrees of exactly n/2 nodes.


centre of mass

(mathematics, physics) A point, near, or within a body at which the object's mass can be assumed to be concentrated; it coincides with the centroid for a body of uniform density, and with the centre of gravity in a uniform gravitational field.

質心 /质心

point where the mass can be considered concentrated


barycentre

(mathematics, physics, astronomy) The point at the centre of a system; an average point, weighted according to mass or other attribute. The term is usually used in astronomy for the centre of mass about which a system rotates, for example, the moon and the earth rotate about a common point within the earth but not near the centre. Jupiter and the Sun rotate about a common point just outside the surface of the Sun.


row-equivalence

equivalence relation
equivalence relation

(linear algebra) A relation between two matrices of the same size, such that every row of one matrix is a linear combination of the rows of the other matrix, and vice versa. It is an equivalence relation.


binary function

en-noun

binary function (plural binary functions)

(mathematics) a function of the type f:V×V→V, or more generally, a function of the type f:U_1×U_2×...×U_n→V in which exactly two of the sets Uᵢ are equal to V.


first fundamental form

(differential geometry) the Riemannian metric for 2-dimensional manifolds, i.e. given a surface with regular parametrization x(u,v), the first fundamental form is a set of three functions, {E, F, G}, dependent on u and v, which give information about local intrinsic curvature of the surface. These functions are given by


Christoffel symbol

Christoffel symbol (plural Christoffel symbols)

Named for Elwin Bruno Christoffel (1829–1900).

⃑x_uu\⃑x_uv\⃑x_vv=ΓᵤᵤᵘΓᵤᵤᵛl\ΓᵤᵥᵘΓᵤᵥᵛm\ΓᵥᵥᵘΓᵥᵥᵛn⃑x_u\⃑x_v\⃑n

(differential geometry) For a surface with parametrization ⃑x(u,v), and letting i,j,k∈u,v, the Christoffel symbol Γᵢⱼᵏ is the component of the second derivative ⃑x_ij in the direction of the first derivative ⃑x_k, and it encodes information about the surface's curvature. Thus,


holonomy

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

(differential geometry) Given a smooth closed curve C on a surface M, and picking any point P on that curve, the holonomy of C in M is the angle by which some vector turns as it is parallel transported along the curve C from point P all the way around and back to point P.


tangent plane

(differential geometry) The plane passing through a given point which contains the tangent lines of all the curves on a surface passing through that same point.


surface normal

(differential geometry) A unit vector at any given point P of a surface M which is perpendicular to the tangent plane T_P(M) of M at P. For a surface with parametrization ⃑x(u,v), it is given by

A unit vector at any given point P of a surface M which is perpendicular to the tangent plane T_P(M) of M at P. For a surface with parametrization ⃑x(u,v), it is given by
⃑n=±⃑x_u×⃑x_v/ lVert ⃑x_u×⃑x_v rVert
where the sign can be chosen at will.


Green's theorem

LL-Q1860 (eng)-Flame, not lame-Green's theorem.wav

George Green (mathematician)
Green's theorem

Named after the mathematician George Green.

fundamental theorem of calculus
fundamental theorem of calculus

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

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

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

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


surface of revolution

surface of revolution (plural surfaces of revolution)

A sphere is a surface of revolution of a circle around an axis which runs through the center of the circle.

A torus is a surface of revolution of a circle around an axis which is outside the circle.

(mathematics) A surface formed when a given curve is revolved around a given axis. If the resulting surface is a closed one, it also defines a solid of revolution.


tetrakaidekahedron

tetrakaidekahedron (plural tetrakaidekahedrons or tetrakaidekahedra)

tetrakaidekahedrons (plural)
tetrakaidekahedra (plural)

Two large flat surfaces and twelve edge facets make each squame a fourteen-sided solid or tetrakaidecahedron.

2011, Terence Allen, Graham Cowling, The Cell: A Very Short Introduction, Oxford, page 81:

(mathematics) A solid figure having fourteen faces.


hyperdimensional

A game of tic-tac-toe in five dimensions is hyperdimensional.

Of or pertaining to a system having more dimensions than naturally observed in our universe.


megaohm

(metrology) An SI unit of resistance equal to 10⁶ ohms. Symbol: MΩ


axis of symmetry

axis of symmetry (plural axes of symmetry)

(geometry) Any of one or more lines about which a geometric figure is symmetric

о́с на симе́трия

a line about which a geometric figure is symmetric


dime bag

A small bag containing ten dollars worth (usually one gram) of marijuana. This may refer to the unit of measurement or the bag itself.


Floydian

en-adj

Floydian (comparative more Floydian, superlative most Floydian)

more Floydian (comparative)
most Floydian (superlative)

(mathematics) Of or pertaining to the work of Robert Floyd, American mathematician and computer scientist.


square pyramid

Great Pyramid of Giza
Great Pyramid of Giza

(geometry) A three-dimensional geometric figure with a square base and four triangular sides that connect at one point. An example is the Great Pyramid of Giza.


integrally

en-adv

integrally (comparative more integrally, superlative most integrally)

more integrally (comparative)
most integrally (superlative)


Q.E.D.

From Late Latin QED, from Latin quod erat demonstrandum.

[A]fter the same manner S and U are proved to be equal, therefore the square of CB is equal to the square of the 2 other sides Q E D.

1684 August 30, Mr. Ash, “A New and Easy Way of Demonstrating Some Propositions in Euclid”, in Philosophical Transactions: Giving Some Accompt of the Present Undertakings, Studies and Labours of the Ingenious in Many Considerable Parts of the World, volume XIV, number 162, London: Printed by T. R. for John Martyn, printer to the Royal Society; […] , →OCLC, page 674:

Now let any right line meet four harmonicals in A, C, B, D, and if those harmonicals be parallel, the thing is evident; but, if they intersect in V, draw ECF parallel to VD, the line most remote from C; then, by the first part, EC#61;CF, and AD#58;AC#58;#58;VD#58;EC or CF. Therefore AD#58;AC#58;#58;BD#58;CB. q. e. d.

1803, “The Mathematical Repository, Nº 63”, in The Gentleman’s Diary, or The Mathematical Repository; an Almanack for the Year of Our Lord 1803: […], London: Printed for the Company of Stationers by Nichols and Son, […]; [a]nd sold by George Greenhill, […], →OCLC, page 33:

By Cases 1 and 2, we see that any finite collection of 4k-1 primes cannot contain all such primes. Thus there are infinitely many primes of this type. ¶ Q.E.D.

1999, William Dunham, “Euler and Analytic Number Theory”, in Euler: The Master of Us All (The Dolciani Mathematical Expositions; 22), [Washington, D.C.]: Mathematical Association of America, →ISBN, page 64:

quod erat demonstrandum
quod erat demonstrandum#English

(mathematics, dated) Initialism of quod erat demonstrandum (“what was to be proved; what was to be demonstrated”): placed at the end of a mathematical proof to show that the theorem under discussion is proved.


stere

From French stère, from Ancient Greek στερεός (stereós, “solid”), because the unit was used for solid commodities such as firewood.

der

It will be observed, that in this system it is only necessary to remember the metre, are, litre, and stere, all the others having certain relations to these, being equal to them taken 10, 100, 1000, &c. times, […]

1814, Peter Barlow, A New Mathematical and Philosophical Dictionary:

(obsolete) A measure of volume used e.g. for cut wood, equal to one cubic metre.

motti

measure of volume for cut wood equalling one cu-m


triangularly

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Lie group

Named for Norwegian mathematician Sophus Lie.

1994, Silvio Levy (translator), Albert S. Schwarz, Topology for Physicists, [1989, A. S. Shvarts, Kvantovaya teoriya polya i topologiya], Springer, 1996, 2nd Printing, page 233,
Every connected Lie group is homotopically equivalent to its maximal compact subgroup. This reduces the study of the homotopy and homology of Lie groups to the compact case.

A Lie group is a manifold on which the group manipulations, product and inverse, arc defined. Lie groups play an extremely important role in the theory of fibre bundles and also find vast applications in physics.

2009, Mikio Nakahara, Geometry, Topology and Physics, 2nd edition, Taylor & Francis, page 207:

As is well known, in finite dimensions each Lie group is, at least locally near the identity, completely described by its Lie algebra.

2009, Boris Khesin, Robert Wendt, The Geometry of Infinite-Dimensional Groups, Springer, page 1:

(topology) Any group that is a smooth manifold and whose group operations are differentiable.

Liegruppe

analytic group that is also a smooth manifold


factoring

(mathematics) The process of factorization.


factored

A radical in final factored form is written last, proceeded by the other factors listed with numbers before letters (variables); and variables are usually listed in alphabetical order.

1998, Walter W. Leffin, George L. Henderson, Mary Van Beck Voelker, Introduction to Technical Mathematics:

An alternative representation (to sum-of-product form) of a logic function which is closer to the physical implementation of the logic is the factored form.

2005, Gary D. Hachtel, Fabio Somenzi, Logic Synthesis and Verification Algorithms, page 417:

It follows from this definition that in factored expressions, complements are permitted only for the literals.

2018, Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski, Computer Arithmetics for Nanoelectronics, page 333:

(mathematics, logic) Expressed in terms of factors (values that are multiplied together to form a whole)

To study the predictive efficiency of the M-Scales, the design was based on correlational analysis while the factored dimensions of the M-Scales were studied through factor analysis.

1963, Mark K. Kipfmueller, The Predictibility (sic) and Factored Dimensions of the M-Scales for Eleventh Grade Parochial School Students, page 57:

Few of the studies cited in this chapter report the intercorrelations of the factored inventory.

1985, Richard Borden, David B. Gray, Russell H. Weigel, Ecological Beliefs and Behaviors: Assessment and Change, page 44:

These authors have investigated current practices in the use of factor analysis and have found them lacking in six areas: ratio of variable to factors, average reliability of the factored variables, sample size, type of analysis, factor retention, and factor rotation.

2017, Dragoş Iliescu, Adapting Tests in Linguistic and Cultural Situations, page 263:

(statistics) Having been subjected to, or resulting from a factor analysis.


subvariety

(mathematics) A subset of a variety which is itself a variety.


fifty-fifty

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(sports) A challenge between two opposing players, where both have an equal chance of taking possession.


central angle

(geometry) An angle positioned with its vertex at the center of a circle.

keskuskulma

an angle positioned with its vertex at the center of a circle


betweenness

en-noun

betweenness (countable and uncountable, plural betweennesses)

This is done by computing all the betweennesses of all edges and then looking for the ones with the highest values.

2010, David Easley, Networks, Crowds, and Markets: Reasoning About a Highly Connected World:

(graph theory, countable) A measure of the number of geodesic paths through a node or edge.


parallelepipedal

(geometry) Having the form of a parallelepiped


distributivity

distributivity (countable and uncountable, plural distributivities)

(mathematics) the fact of being distributive

分配法則 (bunpai hōsoku, literally “distributive law”)


Lipschitz

(mathematics) (Of a real-valued real function f) Such that there exists a constant K such that whenever x_1 and x_2 are in the domain of f, |f(x_1)-f(x_2)|≤K|x_1-x_2|.


hyperspatial

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Of or pertaining to multi-dimensional geometry.


sesquilinear

From sesqui- (“one and a half”) + linear; “conjugate linear” being considered metaphorically “half linear”, hence one and a half times linear, as opposed to bilinear (“linear in both variables”).

(mathematics) (of a map of a complex vector space) Linear in one coordinate and conjugate linear in the other

półtoraliniowy

(mathematics) linear in one variable and conjugate liner in the other


reducible

en-adj

reducible (comparative more reducible, superlative most reducible)

more reducible (comparative)
most reducible (superlative)

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(mathematics, of a polynomial) Able to be factored into polynomials of lower degree, as x²-1.

reduktebla

able to be factored into polynomials of lower degree

(mathematics, of an integer) Able to be factored into smaller integers; composite.

可約の

able to be factored into smaller integers; composite

(topology, of a manifold) Containing a sphere of codimension 1 that is not the boundary of a ball.


endpoint

(mathematics) Either of the two nodes of a graph of degree 1.