algebraic number field

algebraic number field (plural algebraic number fields)

The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

Although we shall be concerned in the sequel only with quadratic fields, we shall nevertheless begin with a short discussion of the more general concept of an algebraic number field.

1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 61:

In this section we consider the maximal order of an algebraic number field and discuss the most important consequences which follow from the fact of it being a Dedekind ring.

1989, M. Pohst, H. Zassenhaus, Algorithmic Algebraic Number Theory, 1st paperback edition, Cambridge University Press, published 1997, page 381:

2006, Andres Iglesias, Nobuki Takayama (editors), Mathematical Software - ICMS 2006: 2nd International Congress, Proceedings, Masayuki Noro, An Efficient Implementation for Computing Gröbner Bases over Algebraic Number Fields, Springer, LNCS 4151, page 99,
If the operations over an algebraic number field are provided, we can apply [the] usual Buchberger algorithm over the field.

(mathematics, algebraic number theory) A field which includes the rational numbers and has finite dimension as a vector space over the rational numbers.


principal ideal ring

en-noun

principal ideal ring (plural principal ideal rings)

(algebra, ring theory) A commutative ring in which every ideal is a principal ideal.

(algebra, ring theory) A noncommutative ring in which every left ideal is a principal ideal and so is every right ideal.


commutative ring

commutative ring (plural commutative rings)

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.

1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129:

As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.

2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295:

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47:

(algebra, ring theory) A ring whose multiplicative operation is commutative.

anello commutativo

a ring whose multiplicative operation is commutative


nonstandard number

nonstandard number (plural nonstandard numbers)

mathematical analysis
mathematical analysis

(mathematical analysis, nonstandard analysis) A hyperreal number which is not a real number.


Dedekind domain

Dedekind domain (plural Dedekind domains)

Named after German mathematician Richard Dedekind (1831–1916).

It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible.

In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.

1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Elsevier (Academic Press), page 201:

As we can see every principal ideal domain is a Dedekind domain.

2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55:

Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals.

2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 266:

(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations).

anneau de Dedekind

integral domain whose prime ideals factorise uniquely


hypertensor

(mathematics) A vector or tensor whose components are themselves tensors.


supertensor

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eigengene

1999, Orly Alter et al., ‘‘Singular Value Decomposition for Gene Expression Data Processing and Modeling.’’ In: After the Genome V (October 6–10, 1999, Jackson Hole, WY), published by the U.S. Department of Energy (DOE) Office of Scientific and Technical Information (OSTI).
We describe the use of singular value decomposition in transforming gene expression data from genes/arrays space to ‘‘eigengenes’’/‘‘eigenarrays’’ space, where the eigengenes and eigenarrays are unique orthonormal superpositions of the genes and arrays, respectively.

(genetics, mathematics, physics) One of a set of right singular vectors of a genes x samples matrix that tabulates, e.g., the mRNA or gene expression of the genes across the samples.


Euler's criterion

Named after Leonhard Euler (1707–1783), Swiss mathematician and physicist, who devised the criterion in 1748.

(number theory) A theorem which states that, given a positive integer a which is coprime to an odd prime number p, a is a quadratic residue of p if and only if a^((p-1)/2) is congruent to 1 modulo p.


proper class

Thus, what is required is not that all proper classes be the size of the class of all sets, but that every proper class has at least as many members in #92;Omega#95;p as there are sets in #92;Omega#95;2, whatever is in #92;Omega#95;2.

1994, Shaughan Lavine, Understanding the Infinite, Harvard University Press, page 322:

2004, Hartry Field, The Consistency of the Naive Theory of Properties, Godehard Link, One Hundred Years of Russell's Paradox: Mathematics, Logic, Philosophy, Walter de Gruyter, page 308,
It is true that the absence of proper classes in ZF is sometimes awkward. It is also true that adding proper classes in the usual ways (either predicative classes as in Gödel—Bernays, or impredicative ones as in Morse-Kelley is conceptually unsettling: in each case (and especially in the more convenient Morse-Kelley case) they "look too much like just another level of sets", and the fact that there is no entity that captures the extension of predicates true of proper classes suggests the introduction of still further entities ("super-classes" that can have proper classes as members), and so on ad infinitum.

Any class is either a set or else a proper class. Every set is a class, but not all classes are sets. Specifically, the proper classes are not sets.

2009, Eric Steinhart, More Precisely: The Math You Need to Do Philosophy, Broadview Press, page 63:

(set theory) A class which is not a set.


cuartilla

From Spanish cuartilla, from cuarto and cuarta (“fourth, quarter”) + -illa (“-elle: forming diminutives”), from Latin quartus (“fourth”). Doublet of cuartillo and quartilho.

(historical) A traditional Spanish unit of dry measure, equivalent to about 13.9 L.


hyperconfluent

(mathematics, rare) A Confluent hypergeometric function


GDS

Graph Dynamical System
graph dynamical system#English

(mathematics) Initialism of Graph Dynamical System.


outpaint

en-verb

outpaint (third-person singular simple present outpaints, present participle outpainting, simple past and past participle outpainted)

outpaints (present singular third-person)
outpainting (participle present)
outpainted (participle past)
outpainted (past)

Our system treats this challenge using a generative adversarial network (GAN), a deep learning technique that outpaints the cropped image.

2022, Karim Yasser et al., “Egyart classify: an approach to classify outpainted Egyptian monuments images using GAN and ResNet”, in 2022 2nd International Mobile, Intelligent, and Ubiquitous Computing Conference, →DOI, page 166:

(transitive, intransitive) To extend an image by guessing how it might look beyond its borders; to perform the inverse of a crop (chiefly in the context of AI image models).


mixtilinear

en-adj

mixtilinear (comparative more mixtilinear, superlative most mixtilinear)

more mixtilinear (comparative)
most mixtilinear (superlative)

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Latin mixtus (“mixed”) + English linear.

A mixtilinear angle is an angle contained by a straight line and a curve.

(geometry) Containing, or consisting of, lines of different kinds (straight, curved, etc.).


homeoid

Steady-state solutions for the motion of a passively gravitating globular cluster (GC) inside an inhomogeneous, rotating, ellipsoidal elliptical galaxy (EG) are considered. It is assumed that an EG with a halo is comprised of a triaxial ellipsoid consisting of two layers. The first is formed by an inner, uniform ellipsoid representing the luminous part of the galaxy, while the second corresponds to the space between an inner and outer ellipsoid, which is uniformly filled with dark matter. The triaxial ellipsoids are taken to be homothetic and to have a common center; the space between them is called a homeoid. The outer boundary of the homeoid is the boundary of the galaxy halo. The densities of the luminous part of the EG and the homeoid are different. This picture of an EG is in agreement with our current understanding of galactic structure.

2012, S. A. Gasanov, “Steady-state solutions for the motion of a globular cluster in an inhomogeneous, rotating elliptical galaxy”, in Astronomy Reports, volume 56, number 6, SPRINGER, pages 469–482:

(mathematics) A shell bounded by two similar ellipsoids having a constant ratio of axes http://mathworld.wolfram.com/Homeoid.html.


relative pseudo-complement

en-noun

relative pseudo-complement (plural relative pseudo-complements)

(mathematics) The residual operation of a Heyting algebra when considered as a residuated lattice whose monoid operation is the meet operation. Equivalently, the relative pseudo-complement of a with respect to b is the supremum of the set of all z such that z∧a⩽b, where ∧ denotes the meet operation of the given Heyting algebra.


affine variety

affine variety (plural affine varieties)

(algebraic geometry) A set of points (in n-dimensional space) which satisfy a set of equations which have a polynomial of n variables on one side and a zero on the other side.


De Morgan algebra

De Morgan algebra (plural De Morgan algebras)

Named after British mathematician and logician Augustus De Morgan (1806–1871). The notion was introduced by Grigore Moisil.

Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice.

1980, H. P. Sankappanavar, “A Characterization of Principal Congruences of De Morgan Algebras and its Applications”, in A. I. Arruda, R. Chuaqui, N. C. A. Da Costa, editors, Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic, page 341:

By a topological de Morgan algebra we shall mean an abstract algebra (A,#92;land,#92;lor,#92;',l) where (A,#92;land,#92;lor,l) is a de Morgan algebra,

2000, Luo Congwen, Topological De Morgan Algebras and Kleene-Stone Algebras: The Journal of Fuzzy Mathematics, Volume 8, Pages 1-524, page 268:

If (L,#92;le,#123;⁻#125;) is a bounded distributive lattice with negation function (resp. a De Morgan algebra), then (L#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle,#92;le,#123;⁻#125;) constitutes also a bounded distributive lattice with negation function (resp. a De Morgan algebra); for every r#92;inL#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle its negation #92;overline#123;r#125;#92;inL#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle is defined by (#92;overline#123;r#125;,s)#61;#92;overline#123;(r,s)#125; for every s#92;inS.

2009, George Rahonis, “Chapter 12: Fuzzy Languages”, in Manfred Droste, Werner Kuich, Heiko Vogler, editors, Handbook of Weighted Automata, Springer, page 486:

distributive lattice
distributive lattice

(algebra, order theory) A bounded distributive lattice equipped with an involution (typically denoted ¬ or ~) which satisfies De Morgan's laws.

Augustus De Morgan
De Morgan algebra
Grigore Moisil


Kleene algebra

(algebra) A De Morgan algebra which also satisfies the inequation x∧∼x⩽y,∨∼y for all x and y, where "∼" here denotes the De Morgan involution.


nonstandard analysis

To be sure, the epsilon-delta methods arose in part because mathematicians became unsure of the foundational validity of infinitesimals. But since nonstandard analysis exactly provides the missing legitimacy, the original motivation for adopting epsilon-delta arguments appears to fall away.

2010 February 25, Joel David Hamkins, “Answer to "How helpful is non-standard analysis?"”, in mathoverflow.net, retrieved 2012-11-13:

(mathematical analysis) Analysis founded upon the hyperreal number system, i.e., which formally defines and uses infinitesimals, especially differentials.


principal ultrafilter

principal ultrafilter (plural principal ultrafilters)

(mathematics) An ultrafilter such that every set belonging to it is the superset of some fixed singleton set.


free ultrafilter

free ultrafilter (plural free ultrafilters)

(mathematical analysis, especially nonstandard analysis) An ultrafilter which contains all cofinite subsets of the set which is being "filtered" by it.


ordered integral domain

ordered integral domain (plural ordered integral domains)

(algebra) An integral domain which has a subset whose elements are said to be "positive", such that this subset is closed under addition, closed under multiplication, and all elements of the integral domain satisfy a law of trichotomy; namely, that either that element is in the said subset, or it is the zero (additive identity), or its product with −1 (the additive inverse of the multiplicative identity) belongs to the said subset.


Archimedean ordered field

Archimedean ordered field (plural Archimedean ordered fields)

The next axiom concerns [ x ], the integer part function. We have seen that this function plays a useful role in our theory, but it turns out that not all ordered fields have it defined. A field for which integer part is defined is called an Archimedean Ordered Field.

2006, Richard W. Kaye, “Archimedean ordered fields”, in web.mat.bham.ac.uk, retrieved 2012-11-13:

(mathematics) An ordered field which satisfies the axiom of Archimedes.

corpo ordenado arquimediano

an ordered field which satisfies the axiom of Archimedes


Russian copulation

(cryptography) A method of rearranging plaintext before encryption by breaking it into two parts and inverting their order, so as to conceal obvious headers, salutations, signatures, etc.


almost all

Alternative form: a.a.

(mathematics) All, except for a negligible set, i.e. finite, countable or of measure zero.


zero matrix

zero matrix (plural zero matrices or zero matrixes)

zero matrices (plural)
zero matrixes (plural)

(mathematics) A matrix whose entries are all zero.


triexponential

(mathematics) Having three summed exponential terms.


operadic

(mathematics) Of or pertaining to operads.


multidistance

(mathematics) An extension of the concept of distance between a pair of points in n-dimensional space


spanning tree

(graph theory) A tree structure which includes all vertices of a graph.

virityspuu

tree which includes all vertices of a graph


partial out

en-verb

partial out (third-person singular simple present partials out, present participle partialing out or partialling out, simple past and past participle partialed out or partialled out)

partials out (present singular third-person)
partialing out (participle present)
partialling out (participle present)
partialed out (participle past)
partialed out (past)
partialled out (participle past)
partialled out (past)

(mathematics, in a system of multiple variables) To remove one variable (by giving it a fixed value) in order to identify any correlation between others


ploughgate

Not having one ploughgate of land.

1818 July 25, Jedadiah Cleishbotham [pseudonym; Walter Scott], Tales of My Landlord, Second Series, […] (The Heart of Mid-Lothian), volume (please specify |volume=I to IV), Edinburgh: […] [James Ballantyne and Co.] for Archibald Constable and Company, →OCLC:

(Scotland) The Scottish carucate or hide: a unit of land area and tax assessment intended to support a household.


quotient group

This paper shows how to compute the quotient groups Jⁿ/Jⁿ⁺¹ (as well as the multiplicative structure of the graded ring consisting of these quotient groups).

1975, John R. Stallings, “Quotients of the Powers of the Augmentation Ideal in a Group Ring”, in Lee Paul Neuwirth, editor, Knots, Groups, and 3-manifolds: Papers Dedicated to the Memory of R. H. Fox, Princeton University Press, page 101:

The Weyl group is the quotient group N_H/T_H, and in the present case the Weyl group is simply the permutation group S₃.

1983, David H. Sattinger, Branching in the Presence of Symmetry, Society for Industrial and Applied Mathematics, page 33:

The class of reflexive groups doesn't behave nicely with regards to operations: a closed subgroup of a reflexive group need not be reflexive, and a quotient group of a reflexive group need not be reflexive.

2002, Alexander Arhangel'skii, “Topological Invariants in Algebraic Environment”, in Miroslav Hušek, Jan van Mill, editors, Recent Progress in General Topology II, North-Holland: Elsevier, page 39:

equivalence relation
equivalence relation

(group theory) A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure.


simple group

1989, Roger W. Carter, Simple Groups of Lie Type, John Wiley & Sons, Wiley Classics Edition, page 1,
Since the first edition of this book was published in 1972, great progress has been made in the theory of finite simple groups. Above all, the classification of the finite simple groups was finally completed in 1981. […] In addition, the five Mathieu groups have been supplemented by the discovery between 1965 and 1981 of 21 further 'sporadic' simple groups.

Let #92;mathcal#123;K#125; be the list of finite simple groups appearing in section 47.[…]
Classification Theorem. Every finite simple group is isomorphic to a member of #92;mathcal#123;K#125;.

2000, Michael Aschbacher, Finite Group Theory, 2nd edition, Cambridge University Press, page 260:

However, classification of the finite simple groups was much harder than could have been foreseen in the 19th century. It turned out to be easier (though still very hard) to classify continuous simple groups.[…]Each continuous simple group is the symmetry group of a space with hypercomplex coordinates, either from #92;mathbb#123;R#125;, #92;mathbb#123;C#125;, #92;mathbb#123;H#125;, or #92;mathbb#123;O#125;.

2010, John Stillwell, Mathematics and Its History, 3rd edition, Springer, page 495:

normal subgroups
normal subgroup#English

(group theory) A group which has no normal subgroups apart from the trivial group and itself.

gruppo semplice

group having no normal subgroups other than the trivial group and itself


trivial group

identity element
identity element#English

(group theory) The unique group (up to isomorphism) consisting of a single element (which is the identity element).


sub-Gaussian

moment-generating function
moment-generating function

(mathematics, of a distribution) whose moment-generating function is bounded by that of a Gaussian


moment-generating function

en-noun

moment-generating function (plural moment-generating functions)

probability distribution
probability distribution

(mathematics) A function derived from the probability distribution of a random variable.


eigenpair

(linear algebra) The mathematical pair of an eigenvector (ẕ), and its associated eigenvalue (λ). This is written as (λ,ẕ)

(mathematics, by extension) The mathematical pair of an eigenelement (ẕ), and its associated eigenvalue (λ). This is written as (λ,ẕ)


eigenprojection

(linear algebra) The linear operator projecting onto an eigenspace.


multiexponential

en-adj

multiexponential (comparative more multiexponential, superlative most multiexponential)

more multiexponential (comparative)
most multiexponential (superlative)

(mathematics) Derived from multiple exponents


nonpositivity

(mathematics) The condition of being nonpositive


nonsymmetrized

(mathematics) Not symmetrized (remaining nonsymmetric)


semijoin

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(algebra, databases) A type of join in relational algebra, denoted by ⋉ or ⋊.


majorize

en-verb

majorize (third-person singular simple present majorizes, present participle majorizing, simple past and past participle majorized)

majorizes (present singular third-person)
majorizing (participle present)
majorized (participle past)
majorized (past)

(mathematics, transitive) To subject to majorization.


familywise

(statistics) In terms of a family of related inferences.


autoconvolution

en-noun

autoconvolution (usually uncountable, plural autoconvolutions)

A convolution of a function with itself.


pseudometric

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(mathematics) Describing a generalization of a metric space in which the distance between two distinct points can be zero.


supercube

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(mathematics, rare) An analogue of the square or cube in arbitrary dimension: 0,1ⁿ along with the (n-1)-dimensional balls joining every 2ⁿ⁻¹ such points that share a coordinate, or any space equivalent.


arcwise-connected

For quotations using this term, see Citations:arcwise-connected.


explementary

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Two angles whose sum is a circle (360°) are explementary angles, and either is the explement of the other.

2002, Nathaniel Bowditch, American Practical Navigator, page 317:

(geometry) Of or relating to the explement


bilinearization

bilinearization (usually uncountable, plural bilinearizations)

(mathematics) A process that makes a function bilinear.


subsphere

The self-gravitational correction to a localized spherically symmetric static energy distribution is obtained from the negative Newtonian gravitational work done to successively assemble each spherical energy shell in the gravitational field of the already energy-corrected subsphere which that shell abuts.

2012, Steven Kenneth Kauffmann, “A Self-Gravitational Upper Bound on Localized Energy, Including that of Virtual Particles and Quantum Fields, which Yields a Passable "Dark Energy" Density Estimate”, in arXiv:

(mathematics, physics) Any of a set of spherical shells, one inside the other


well-posedness

(mathematics) The condition of being well-posed


well-posed

(mathematics) Having a unique solution whose value changes only slightly if initial conditions change slightly


intraobserver

(statistics) Describing variation in the scores, responses etc obtained by the same observer on different occasions


nonnormally

(statistics) Not showing a normal distribution


bidifferential

differential operator
differential operator

A differential operator having two arguments.


coassociator

An operator that measures nonassociativity of a coproduct.


adarme

From Spanish adarme (“¹⁄₁₆ Spanish ounce, peso”), from Andalusian Arabic الدرهم (ad-dárham), from Arabic دِرْهَم (dirham, “dirham”), from Middle Persian 𐭦𐭥𐭦𐭭 (drahm), from Ancient Greek δραχμή (drakhmḗ, “drachma”). Doublet of dram, drachma, diram, dirham, and dirhem.

bor

<small>¹<big>⁄</big>₁₆</small> Spanish ounce, peso

doublet

(historical) A traditional Spanish unit of mass, equivalent to about 1.8 g.


hysteron

(physics, mathematics) A two-valued operator that is used to model hysteresis.


preregular

(topology, of a space) In which any two topologically distinguishable points can be separated by neighbourhoods


gyro-

From Ancient Greek γῦρος (gûros, “circle”).


correntropy

(mathematics) A nonlinear measure of the similarity between two random variables


hyperforest

(mathematics) A related group of hypertrees


hypertree

(graph theory) A form of hypergraph based on trees.


rotopulsator

(mathematics) Any of a class of solutions to the n-body problem, in a curved space, whose configuration rotates during the motion.


n-body problem

(mathematics) The problem of predicting the motion of a group of celestial objects that interact with each other gravitationally.


hyperface

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(geometry) A hyperplane that forms a boundary of a polytope


varentropy

(mathematics) A form of entropy that is a measure of the uncertainty of a random variable.


Kurepa tree

Named after Yugoslav mathematician Đuro Kurepa.

named-after

(mathematics, set theory) A tree (T, <) of height ω₁, each of whose levels is at most countable, and has at least ℵ₂ branches.


keypair

public-key cryptography
public-key cryptography

(cryptography) A public key and its associated private key (in public-key cryptography).


quasipotential

(mathematics, physics) A complex function of momentum and energy applied to an individual attractor in a complex system of multiple attractors


plane tree

(mathematics) An ordered tree, a type of graph, in which the child nodes of each vertex have a definite number assigned.


maxterm

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All the maxterms in a product of maxterms should have the same variables, although each maxterm should differ from every other one by the pattern of complementation of those variables.

(algebra) A sum that is a logical OR of a set of variables where each individual variable only appears once in the sum, either in complemented or uncomplemented form, so that the value of the sum becomes 0.


tileable

(mathematics, of a shape) That is able to tile the plane


tileability

(mathematics) The ability of a shape to tile the plane


LCT

limit comparison test
limit comparison test#English

(mathematics) Initialism of limit comparison test.


submonoid

(mathematics) A subset of a monoid that is itself a monoid


sawtooth wave

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

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

鋸齒波 /锯齿波

function or waveform that ramps upwards and then sharply drops


triangle wave

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

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


rhombille

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Zeckendorf's theorem

Named after Belgian mathematician Edouard Zeckendorf.

(mathematics) A theorem stating that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.


pseudocodeword

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We examine the structure of pseudocodewords in Tanner graphs and derive lower bounds of pseudocodeword weights.

2004, IEEE International Symposium on Information Theory, page 67:

Other notions of pseudocodeword-like structures such as stopping sets are also known to coincide with these polytope vertices.

2008, Ming-Yang Kao, Encyclopedia of Algorithms, page 481:

(mathematics) The result obtained when a decoder fails to converge on a codeword, but instead converges on some other code structure.


pseudo-complement

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relative pseudo-complement
relative pseudo-complement

(mathematics) The relative pseudo-complement of a given element (of a Heyting algebra) with respect to the least element — the "zero" of that algebra.


culeus

From Latin culeus (“large leather sack, punishment of drowning within a sack, unit of bulk liquid measure”), from Ancient Greek κολεός (koleós, “sheath”), from Proto-Indo-European *ḱel- (“to cover”). Doublet of cullion and cojones.

bor

(historical) A Roman unit of liquid measure reckoned as the volume of 1600 Roman pounds of wine and equivalent to about 520 L although differing slightly over time.


heterocentric

en-adj

heterocentric (comparative more heterocentric, superlative most heterocentric)

more heterocentric (comparative)
most heterocentric (superlative)

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The purpose of this paper is to develop a complete and general theory of aperture referral in linear optics for dioptric systems that may be heterocentric and astigmatic. Apertures may be elliptical and decentred. Referral transforms the geometry of an aperture.

2011, William F. Harris, “Aperture referral in heterocentric astigmatic systems”, in Ophthalmic and Physiological Optics, volume 31, number 6, →DOI, →ISSN, page 603:

(geometry) Having different or multiple centres.


triforce

From Japanese トライフォース (toraifōsu), from English tri- + force. First used in the Legend of Zelda video game franchise for a magical artefact in that fictional universe.

A shape composed of three equilateral triangles arranged in such a manner that a fourth triangle of identical size, but pointing the opposite direction, could fit between them.

三角之力

Shape composed of three triangles in a specific alignment


column vector

(linear algebra) An n×1 matrix that represents a vector.

列向量

matrix with one column that represents a vector


row vector

(linear algebra) A 1×n matrix that represents a vector.

行向量

matrix with one row that represents a vector


sesquitertian

sesqui- + Latin tertianus (“belonging to the third”). Compare tertian.

a sesquitertian ratio

a sesquitertian relationship

(mathematics) Having the ratio of one and one third to one (as 4:3).


unitary matrix

en-noun

unitary matrix (plural unitary matrices or unitary matrixes)

unitary matrices (plural)
unitary matrixes (plural)

(linear algebra) A matrix which when multiplied by its conjugate transpose yields the identity matrix.


singular matrix

singular matrix (plural singular matrices or singular matrixes)

singular matrices (plural)
singular matrixes (plural)

(linear algebra) A square matrix which is not invertible.


conjugate transpose

conjugate transpose (plural conjugate transposes)

(mathematics) The transpose of a matrix, after replacing each element with its complex conjugate.

Conjugate transpose
Moore–Penrose pseudoinverse

adjoint matrix

transpose of matrix of complex conjugates

adjugate matrix

adjoint; historically also sometimes called


transpose conjugate

transpose conjugate (plural transpose conjugates)

Further constraints upon the four-port are discovered when the elements equating to zero in the product of the unitary S-matrix with its transpose conjugate are considered.

2006, David Andrews, Lumped Element Quadrature Hybrids, page 22:

In that case the inverse of the matrix is just the transpose conjugate and solving for the original signal x from the truncated outputs {y⁽⁰⁾(2n)},{y⁽¹⁾(2n)} is simple.

2012, Steven B. Damelin, Willard Miller, Jr., The Mathematics of Signal Processing, page 299:

Throughout this text, the transpose conjugate of a matrix or vector is given by A⁺ or b⁺, respectively.

2012, Dennis M. Sullivan, Quantum Mechanics for Electrical Engineers, page 85:

conjugate transpose
conjugate transpose#English

(mathematics) Synonym of conjugate transpose


inner product space

inner product space (plural inner product spaces)

(mathematics) A vector space which is additionally equipped with an inner product.


normed vector space

normed vector space (plural normed vector spaces)

(mathematics) A vector space which is additionally equipped with a norm.


Hermitian conjugate

Hermitian conjugate (plural Hermitian conjugates)

Named after Charles Hermite (1822–1901), French mathematician.


Euclidean domain

Euclidean domain (plural Euclidean domains)

So called because for such domains it is possible to apply the Euclidean algorithm in order to obtain greatest common divisors.

(algebra) an integral domain in which division with remainder is possible


linear form

In #92;mathbb#123;R#125;ⁿ, if vectors are represented as column vectors, then linear forms are represented as row vectors, and their action on vectors is given by the matrix product, with the row vector on the left and the column vector on the right.

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

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

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

In order to better understand the effect of reducing a K-vector subspace V#92;subseteqP#95;d modulo a generic linear form, we generalize Proposition 5.5.18 as follows.

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


bilinear form

(linear algebra) A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to either one of its arguments when the other one is held constant.


Pauli vector

The eigenvectors of the first Pauli matrix "encode" — as it were — the +X and −X directions. Likewise, the eigenvectors of the second Pauli matrix "encode" the +Y and −Y directions and those of the third Pauli matrix, the +Z and −Z directions. The dot product of the Pauli vector with any unit vector ⃑N yields a matrix which likewise has eigenvalues +1 and −1 and a pair of eigenvectors; the eigenvector with positive eigenvalue "encodes" the +N direction and the eigenvector with negative eigenvalue "encodes" the −N direction. The dot product of any two such "encoded" directions yields the probability amplitude that the spin of an electron prepared in the first direction will collapse to the second direction when measured along that second direction.

(mathematics) A vector whose components are Pauli matrices; e.g., ⃑σ=σ₁̂i+σ₂̂j+σ₃̂k.


orthocomplement

(mathematics, especially lattice theory) An element of an ortholattice which is the result of applying an orthocomplementation function to a given element (of that ortholattice) (of which it is said to be the orthocomplement).


orthogonal complement

orthogonal complement (plural orthogonal complements)

(linear algebra, functional analysis) The set of all vectors which are orthogonal to a given set of vectors.


chalder

[T]he valuation of lands, tenor of leaſes, the rents, the entails, rent charges, life rents, and payments for or out of land revenue, are all reckoned in Scotland by the chalder, boll, firlot, and lippy, and cannot be altered; [...]

1786, Daniel De Foe [i.e., Daniel Defoe], “Appendix. Part I. Containing an Account of Transactions in both Parts of the United Kingdoms, subsequent to the Union; […]”, in A History of the Union between England and Scotland, with a Collection of Original Papers Relating thereto. […], London: Printed for John Stockdale, […], →OCLC, page 600:

An old Scottish dry measure, equal to 16 bolls.


VAE

variational autoencoder
w:variational autoencoder

(machine learning) Initialism of variational autoencoder.


selective service
w:Selective Service System

A selective service class denoting that the registrant is already a reserve member of the armed forces or a student taking military training, as in ROTC or an accepted aviation cadet applicant.


dihedral group

From di- and -hedron; so called because the polygon, when it can be reflected along certain axes of symmetry, has two faces: its upper side and its underside.

(group theory) a group of symmetries of a regular polygon


alternating group

alternating group (plural alternating groups)

even permutations
even permutation#English

(group theory) a group of even permutations of a finite set


sporadic group

The earliest usage is thought to be that of English mathematician William Burnside in 1911, W. Burnside, Theory of Groups of Finite Order, 2nd Edition, in a comment about the Mathieu groups.

By the classification of finite simple groups in 1981 it is now known that every finite simple group is either cyclic of prime order, or an alternating group of degree 5 or bigger, or a simple group of Lie type as above, or one of 26 sporadic groups.

1998, Meinolf Geck, “Finite Groups of Lie Type”, in Roger William Carter, Meinolf Geck, editors, Representations of Reductive Groups, Cambridge University Press, page 63:

The first five sporadic groups were discovered by Mathieu in the late nineteenth century. The remaining twenty-one were discovered in the nineteen sixties and seventies.

1998, David J. Benson, “Cohomology of Sporadic Groups, Finite Loop Spaces, and the Dickson Invariants”, in Peter H. Kropholler, Graham A. Niblo, Ralph Stöhr, editors, Geometry and Cohomology in Group Theory, Cambridge University Press, page 10:

It is natural to expect that the sporadic groups should play a role in the structure of the Coker(J) space, though we are only beginning to understand some of the smaller sporadic groups in this framework.

2004, Alejandro Adem, R. James Milgram, Cohomology of Finite Groups, 2nd edition, Springer, page 245:

(group theory) Any one of the 26 exceptional finite simple groups, which do not belong to any of the general, infinite categories specified by the classification theorem for finite simple groups.

Mathieu groups
William Burnside
sporadic group


permutation group

permutation group (plural permutation groups)

In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.

1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80:

The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).

1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47:

Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group.[…]The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.

2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86:

(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.

transformation group

group whose elements are permutations of a set


braid group

(group theory) A group whose elements represent ways to weave some number of strings into braids.


catenarian

en-adj

catenarian (comparative more catenarian, superlative most catenarian)

more catenarian (comparative)
most catenarian (superlative)

The curve so formed is a catenarian curve, and, when inverted, delineates the curve of a type of arch better able to resist forces of destruction than the earlier semicircular arch. Investigators who followed Galilei and studied the catenarian arch mathematically were able to show that a simple catenarian outline was formed by the chan suspeneded as already described, the length of the chain depending upon the required span and rise of the arch.

1957, Bernard E. Jones, Freemason's Book of the Royal Arch, page 105:

a catenarian curve/arch

(geometry) Shaped like a catenary; following the curve made by a chain or rope when suspended from both ends.


cyclic group

1986, N. S. Gopalakrishnan, University Algebra, New Age International, 2nd Edition, page 22,
Proposition 1.5.6. Any subgroup of an infinite cyclic group is also an infinite cyclic group.

If u#58;G#95;1#92;rightarrow#92;mathbb#123;Z#125;#47;m#92;mathbb#123;Z#125; and v#58;G#95;2#92;rightarrow#92;mathbb#123;Z#125;#47;m#92;mathbb#123;Z#125; are isomorphisms of two cyclic groups with #92;mathbb#123;Z#125;#47;m#92;mathbb#123;Z#125;, then v#123;-1#125;#92;circu#58;G#95;1#92;rightarrowG#95;2 is an isomorphism.

2002, Serge Lang, Algebra, 3rd edition, Springer, page 24:

2003, Alexander Retakh (translator), Ėrnest Borisovich Vinberg, A Course in Algebra, [2001, Э. Б. Винберг, Курс алгебры, Factorial Press] American Mathematical Society, page 152,
Cyclic groups are the simplest groups imaginable.

(group theory) A group generated by a single element.


go into

en-verb

go into (third-person singular simple present goes into, present participle going into, simple past went into, past participle gone into)

goes into (present singular third-person)
going into (participle present)
went into (past)
gone into (participle past)

11 goes into 88 and 99 but not 100.

7 goes into 46 six times, with remainder 4.

(mathematics) To divide, to be a factor of.


preshared

(cryptography, of a key or secret) Shared between the communicating parties by some secure channel before it needs to be used.


sphere packing

en-noun

sphere packing (countable and uncountable, plural sphere packings)

(mathematics) The arranging of non-overlapping spheres in space

pallojen pakkaaminen

mathematics: arranging of non-overlapping spheres in space


self-affinity

(mathematics) The condition of a fractal whose pieces are scaled by different amounts in the x and y directions, so that self-similarity becomes apparent only after an anisotropic transformation.


desingularize

en-verb

desingularize (third-person singular simple present desingularizes, present participle desingularizing, simple past and past participle desingularized)

desingularizes (present singular third-person)
desingularizing (participle present)
desingularized (participle past)
desingularized (past)

(mathematics, transitive) To resolve (a singularity) by means of desingularization.


cotyla

From Latin cotyla, from Ancient Greek κοτύλη (kotúlē, “cup, half-pint”).

der

Alternative form of cotyle (“in all senses”)


quasicoherent

(mathematics, of a sheaf) Belonging to a generalization of coherent sheaves, including the locally free sheaves of infinite rank.


fanga

From Portuguese fanga, from Andalusian Arabic فَنِيقَة (faníqa, “sack”), from Arabic فَنِيقَة (fanīqa). Doublet of fanega.

(historical) A traditional Portuguese dry measure, equal to about 50–75 liters at different places and times.


crabling

The juvenile members of most of our seaside communities are much given to crab-fishing, and may be seen from early morn to evening late, dangling their legs over some convenient rock, sea-wall, or landing-steps, and with a piece of twine to which a dainty morsel of fish offal has been tied, doing their best to induce some greedy young crabling to grasp it with his nippers, [...]

1867, William Barry Lord, Crab, shrimp, and lobster lore:

This was, after all, the easiest kind of life a simple-minded crabling such as he could hope to achieve, and so Galoo's words did well to play upon his fears. the sea-swindler went on to say, giving the crabling a slight push to go.

2009, R. Y. Gheith, The Complete Isaac Trilogy:


lobsterling

In the first few days of the fourth stage the lobsterlings are good swimmers — this is their "redeeming vice" — but the swimming is strong and bears no comparison to the aimless drifting movement characteristic of larval stages.

1910, Albert Davis Mead, A method of lobster culture:

The approach of the molt seems to start the lobsterling on its course to the bottom; accordingly when this is delayed until after the fourth stage, it probably does not often occur until the approach of the succeeding molt.

1911, Francis Hobart Herrick, Natural history of the American lobster:


fawnling

Slay not that my musk-deer fawnling, Hunter! Prithee, have thou shame Of her night-black eye nor bind her With thy lasso long and strait.

2012, John Payne Hafiz, The Collected Poems of Hafiz:


monsterling

"They will know something is wrong, for the brute would not willingly have left her side. Odd that it did not fight you." The blunt-nose, slope-browed face of the monsterling was inscrutable, scarlet eyes inhuman and feral in the glow of the fire.

“My apologies, good monsterling. It was an accident.”

2006, Martin H. Greenberg, Brittiany A. Koren, Fantasy Gone Wrong:


bitcoin

From bit (“binary digit”) + coin, coined by Satoshi Nakamoto, believed to be a pseudonym for one or more people, in the domain name bitcoin.org registered on 18 August 2008; and in the title of the white paper “Bitcoin: A Peer-to-Peer Electronic Cash System”, a link to which was posted on the cryptography mailing list metzdowd.com on 31 October 2008: see the quotation.

We do not accept bitcoin at this store.

Bitcoin: A Peer-to-Peer Electronic Cash System [title] […] What is needed is an electronic payment system based on cryptographic proof instead of trust, allowing any two willing parties to transact directly with each other without the need for a trusted third party. […] In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions.

2008 October 31, Satoshi Nakamoto [pseudonym], “Bitcoin: A Peer-to-Peer Electronic Cash System”, in Bitcoin.org, archived from the original on 2025-01-04:

[A] man named Satoshi Nakamoto posted a research paper to an obscure cryptography listserv describing his design for a new digital currency that he called bitcoin.

2011 December, Benjamin Wallace, “The Rise and Fall of Bitcoin”, in Wired, San Francisco, Calif.: Condé Nast Publications, →ISSN, →OCLC, archived from the original on 2025-01-02:

[Bitcoins] do not depend on the state banks, but are created through mining. Mining is also called as hashing. It is a process of solving complex math problems using those computers which run bitcoin software.

2014, Dale Blake, “Getting and Selling Bitcoins”, in Bitcoin Guide for Beginners: Bitcoin Trading and Mining Made Easy, La Vergne, Tenn.: Mihails Konoplovs, →ISBN:

Other e-commerce sites that accept bitcoin are: Bitcoinshop.us which offers products like watches and air conditioners and ships to people within the United States; […] Travel booking websites also start to accept bitcoin sales on flights, hotel bookings to at least 200,000 hotels and railway Amtrak through their platforms.

2015, Scott Green, “Bitcoin Growth and Trends”, in Bitcoin: The Ultimate A–Z of Profitable Bitcoin Trading & Mining Guide Exposed! (The Blokehead Series), Milan: Scott Green, →ISBN:

Individual bitcoin users are not the only driving force in terms of securing the bitcoin network. Over the past few years, dedicated bitcoin nodes have been added to the network with the sole purpose of receiving and broadcasting new bitcoin transactions to other users and nodes on the network.

2016, Prypto [pseudonym], “Bitcoin Security”, in Bitcoin for Dummies (For Dummies), Hoboken, N.J.: John Wiley & Sons, →ISBN, part III (Using Bitcoin in Business), page 137:

Find a friend who has bitcoin and buy some from him or her directly. Many bitcoin users start this way. This method is the least complicated. One way to meet people with bitcoin is to attend a local bitcoin meetup listed at Meetup.com (https://meetup.com).

2017 June, Andreas M[arkos] Antonopoulos, “Introduction”, in Tim McGovern, editor, Mastering Bitcoin: Programming the Open Blockchain, 2nd edition, Sebastopol, Calif.: O’Reilly Media, →ISBN, page 11:

(uncountable, computing, finance) A decentralized cryptocurrency using blockchain technology.


bitcoin

This item costs one bitcoin.

The value of fiat money is determined by legal means (legal tender laws). In other words, fiat money is anything the government says in money. For instance, if the government says an IOU or a bitcoin is money, then it is money. The major currencies of the contemporary world are fiat money.

2008 November, Alex Marchand, “What is Money?”, in Dissolving Dollars: Exposing the Debt-based Insanity behind Modern Money, [S.l.]: 451° Publishing, published April 2015, →ISBN, page 12:

Users willing to devote CPU power to running a special piece of software would be called miners and would form a network to maintain the block chain collectively. […] The first miner to solve each puzzle would be awarded 50 new bitcoins, and the associated block of transactions would be added to the chain. […] [Satoshi] Nakamoto himself mined the first 50 bitcoins—which came to be called the genesis block—on January 3, 2009.

Each time you succeed in creating a hash, you are rewarded with 25 bitcoins, the block chain is updated and all the users on the network are notified. This serves as an incentive to keep mining the bitcoins and keep the bitcoin transactions working.

2013, Elda Watulo, John Davidson, “Making a Hash of It”, in How to Make Money Online With Digital Currency Bitcoins (Entrepreneur Book Series; 8), [S.l.]: JD-Biz Publishing, →ISBN:

They buy bitcoins at a lower price then sell them at a higher value.

2014, Kinsley Wallace, “How to Invest and Trade Bitcoins”, in Bitcoins: How to Invest, Buy and Sell: A Guide to Using the Bitcoin, [United States]: Speedy Publishing, →ISBN:

But consider that a significant amount of bitcoin is also being held by investors. Those investors do not plan to sell their bitcoin for some time because they are speculating that due to its utility as MoIP, demand will continue to rise and so too will its value. Currently, roughly 5.5 million bitcoin, or US$5.5 billion worth at the price of US$1,000 per coin, is held by the top 1,000 addresses recorded in Bitcoin's blockchain.

2017, Chris Burniske, Jack Tatar, Cryptoassets: The Innovative Investor’s Guide to Bitcoin and beyond, New York, N.Y.: McGraw-Hill Education, →ISBN, page 177:

(countable, computing, finance) A unit of the bitcoin (proper noun sense 1) cryptocurrency.

բիթքոին

decentralized cryptocurrency using blockchain technology; unit of the cryptocurrency


Turingery

From Turing + -ery, after its inventor Alan Turing.

(cryptography) A manual cryptanalysis technique used in breaking the Lorenz cipher during World War II, based on deducing the wheel settings of the machine that did the encryption.


adfected

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

From the Latin adfectus, variant spelling of affectus (“endowed with, possessed of; affected; impaired, weakened”), whence affect and, indirectly, affected.

(algebra, archaic, of an equation) Having different powers of the unknown quantity in its various terms.


heptacontagon

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

(geometry) A polygon with seventy edges.


tetracontagon

Blend of tetra- (“4”) + Ancient Greek τεσσαράκοντα (tessarákonta, “40”) + -gon (“-sided”), from γωνία (gōnía, “corner”).

τεσσαράκοντα


nonsingularity

en-noun

nonsingularity (countable and uncountable, plural nonsingularities)

(mathematics, physics) Absence of singularity.


plasterable

(mathematics) Describing a particular type of cone.


quasiidentity

(mathematics) A kind of Horn clause, a generalization of identities.


eigencurve

(mathematics, number theory) A rigid analytic curve that parametrizes certain p-adic families of modular forms.


eigenvariety

(mathematics, number theory) A higher-dimensional generalization of an eigencurve.


eigenvariable

(mathematics, logic) A kind of term in natural deduction.


micromort

From micro- (“one millionth”) + mort (“death”).

prefix

Coordinate term: microlife

This is described as 10 micromorts per operation, and we can compare this figure with other average risks, such as skydiving (also 10 micromorts), or riding a motorcycle in the UK (yoou^([sic]) have to go around 60 miles to rack up 10 micromorts).

2012 February 16, David Spiegelhalter, “Microlives: A lesson in risk taking”, in BBC Future:

Taking one pill from the jar amounts to being exposed to exactly one micromort. The amount you are willing to pay to avoid this risk is called your micromort value. Micromorts help us reason about decisions with small probabilities but very serious outcomes, such as deciding whether to undergo surgery that eliminates life-long pain but fails with a 1% probability, resulting in immediate death.

2020, Gregor Hohpe, The Software Architect Elevator […] , O'Reilly, →ISBN, page 49:

A unit of risk equivalent to a one-in-a-million probability of death.


antimaximum

(mathematics) A maximum that has changed sign, caused by certain values of parameters.


Bethe lattice

(graph theory) A kind of connected cycle-free graph.


binary system

二進制 /二进制

math, computing: numeric system with base 2

Quantum cryptography is one of the first applications of quantum physics, which is expected to replace today's binary system of computing, in which data is represented in series of ones and zeros.

2004 January 28, Jennifer L. Schenker, “A quantum leap in codes for secure transmissions”, in New York Times, retrieved 2013-04-10:

(mathematics, computing) A numeric system with a base of 2, using only the numerals 0 and 1.


genon

The genon in this case is defined as the branch-cut point where the branch-cut line ends.

2013, M Barkeshli, CM Jian, XL Qi, “Theory of defects in Abelian topological states”, in Physical Review B:

(mathematics) A branch cut between two complex planes.


K-theory

From circa 1960. The K stands for German Klasse (“class”).
The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

algebraic K-theory

study of rings generated by vector bundles

K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.

1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

K'''-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K'''-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

topological K-theory
topological K-theory

(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;

The study of rings R generated by the set of vector bundles over some topological space or scheme;
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.


diffuson

(mathematics, physics) A mathematical object relating to diffusive motion, used in the theory of electron transport in disordered systems.


cooperon

(mathematics, physics) A special Green's function used in modelling superconductivity.


antihyperbolic

inverse hyperbolic
inverse hyperbolic function


Minkowski space

en-noun

Minkowski space (plural Minkowski spaces)

Named after Hermann Minkowski (1864–1909), a German mathematician and professor.

special relativity
special relativity#English

(mathematics, physics) A four-dimensional real vector space equipped with an inner product of signature (−,+,+,+) or (+,−,−,−), in which special relativity is formulated

ミンコフスキー空間

four-dimensional vector space in which special relativity is formulated


uniform space

uniform continuity
uniform continuity#English

uniform convergence
uniform convergence#English

(topology) A space for which properties like uniform continuity and uniform convergence can be formulated.

一様空間

a space for which uniform continuity can be formulated


Cauchy space

(topology) A space for which notions of a Cauchy sequence and completeness exist.


pedal curve

(geometry) For a plane curve C and a given fixed point P, the locus of points X such that PX is perpendicular to a tangent to the curve passing through X.


positive-sum

Morality […] is a consequence of the interchangeability of perspectives and the opportunity the world provides for positive-sum games.

2011, Steven Pinker, The Better Angels of Our Nature, Penguin, published 2012, page 220:

(game theory) Pertaining to a game or situation where overall benefits to all participants can increase.


Ramsey number

Named after Frank Plumpton Ramsey (1903–1930), British mathematician, philosopher and economist.

(combinatorics) Any one of a certain set of numbers which are guaranteed to exist by Ramsey's theorem; a positive integer which is a certain function of some given multiset of positive integers, where that "certain function" is that which yields the minimal number guaranteed to exist by Ramsey's theorem.


small-worldness

A paper by Watts and his thesis advisor Steve Strogatz, largely free of mathematics, set ablaze huge interest in the phenomenon of small-worldness with its architectural fingerprints being found in diverse fields as ecosystems, natural language and the World Wide Web.

2009, Mihaela Ulieru, Peter Palensky, René Doursat, IT Revolution: First International ICST Conference, page 92:

Topologically, functional brain networks had reduced clustering and small-worldness, reduced probability of high-degree hubs, and increased robustness in the schizophrenic group.

2012, Issues in Neuroscience Research and Application: 2011 Edition, page 2353:

(mathematics) The property of being a small-world network.


small-world network

small-world network (plural small-world networks)

(mathematics) A kind of graph in which most nodes are not neighbors but most can be reached from every other by a small number of steps, i.e. the typical distance between two randomly chosen nodes grows proportionally to the logarithm of the number of nodes in the network.


daugh

(historical) An obsolete Scots unit equal to 4 ploughgates, notionally comprising 400 Scottish acres.


markland

An old Scots unit of measure equal to eight ouncelands.


ounceland

(historical, Scotland) An old Scots unit of measure equal to 20 pennylands.


pennyland

(Scotland, historical) An old Scots unit of measure of land.


Greek foot

Greek foot (countable and uncountable, plural Greek feet)

From Greek artists' preference for bare feet exhibiting a second toe longer than the hallux, as opposed to the Egyptian statues which used a longer big toe.

(countable, uncountable, medicine) Morton's toe


superexponentiation

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


power tower

(mathematics) A chain of exponentiations, or raising to powers, of the form a^(aa...); tetration.


ergodic theory

ergodic theory (countable and uncountable, plural ergodic theories)

(mathematics, uncountable) The study of the properties of dynamical systems over large timescales.

They pointed out that the microscopic ergodic theories of John von Neumann and Eberhard Hopf are not relevant to the problem of ergodic behavior in molecules.

1994, Ahmed Zewail, Femtochemistry: Ultrafast Dynamics of the Chemical Bond - Volume 1:

(mathematics) Any of a number of different formulations or schools of thought within ergodic theory as a whole.


orthocomplementation

(mathematics) An involution on a complemented lattice which is order-reversing and maps each element to a complement.


orthocomplemented

(mathematics, of a lattice) Having orthocomplementation.


antilattice

(mathematics) An orthomodular lattice in which a ∨ b = 1 for any a, b ≠ 0.


wedge sum

(topology) A one-point union of a family of topological spaces.

yhden pisteen yhdiste

a one-point union of a family of topological spaces


smash product

(mathematics) For two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y, the quotient of the product space X × Y under the identifications (x, y₀) ∼ (x₀, y) for all x ∈ X and y ∈ Y; usually denoted X ∧ Y.


polytetrahedron

A uniform convex polychoron made up of 600 tetrahedral cells; a hexacosichoron.


In every vertex of the 5-cell, 8-cell and 120-cell there are 4 concurring edges; in every vertex of the 24-cell, 8 edges.

1896, American Journal of Mathematics, page 181:

1988, Unnamed translator, L. A. Sidorov, Confuguration, article in Michiel Hazewinkel (editor) Encyclopaedia of Mathematics: An updated and annotated translation of the Soviet ‘Mathematical Encyclopaedia’, Volume 2: C, page 309,
For example, a 5-cell is bounded by five three-dimensional tetrahedra, an 8-cell by eight cubes, etc., the 5-cell and the 24-cell being duals of each other (the points correspond to spaces and lines to planes).

If we call a regular polytope with m facets an m-cell, then in E⁴ we have the 5-cell with tetrahedral facets (the simplex), the 8-cell with cubic facets (the hypercube), the 16-cell with tetrahedral facets (the regular cross-polytope), the 24-cell with octahedral facets, the 120-cell with dodecahedral facets, and the 600-cell with tetrahedral facets.

2007, Steven R. Lay, Convex Sets and Their Applications, page 230:

Later in the same document, President Hall listed all of the university's four-dimensional models: four Brill models—the 5-cell, 8-cell, 16-cell, and 24-cell—along with seven spun glass models by T. P. Hall that "illustrated rotations."

2008, Tony Robbin, Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought, page 16:

(geometry) A four-dimensional polytope, analogous to a tetrahedron, whose five bounding facets are tetrahedra.

kvinĉelo

four-dimensional polytope analogous to a tetrahedron

4-dimensional polytope analogous to a tetrahedron


hexadecachoron

hexadecachoron (plural hexadecachorons or hexadecachora)

hexadecachorons (plural)
hexadecachora (plural)

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From hexadeca- (“sixteen”) + -choron (“room”), from Ancient Greek ἕξ (héx, “six”), δέκα (déka, “ten”) and χῶρος (khôros, “room”).

confix

In a world with four spatial dimensions, for example, we can construct only six regular solids, viz pentatope, tesseract, hexadecachoron, icositetrachoron, hecatonicosachoron and hexacosichoron.

2005, V H Satheesh Kumar, P K Suresh, Are We Living in a Higher Dimensional Universe?, page 3:

We have chosen the surfaces of a pentatope (5-cell) as well as a hexadecachoron (16-cell), which are simple standard triangulations of a 3-sphere, shown in figure (7), as our underlying lattices.

2009, P. Khavari, C. C. Dyer, Aspects of Causality in Parallelisable Implicit Evolution Scheme, page 7:

In four dimensions, surprisingly, there are three space-filling convex regular polychora: the tesseract (the hypercube in ℝ⁴), the hexadecachoron, and the icositetrachoron.

2011, Jin Akiyama, Ikuro Sato, The element number of the convex regular polytopes, page 271:

(mathematics) A four-dimensional object analogous to an octahedron, constructed out of sixteen tetrahedra.


The remaining 8, as before, are the vertices of the complementary 16-cell (see Figure 9). In Figure 7, these 8 points have been blackened, but not joined, because the 16-cell has longer edges; diameters of octahedral facets of the 24-cell.

1995, Harold Scott Macdonald Coxeter, “Paper Three: Two Aspects Of The Regular 24-Cell In Four Dimensions”, in F. Arthur Sherk, editor, Kaleidoscopes: Selected Writings of H.S.M. Coxeter, page 32:

This polytope has 16 tetrahedral cells, and for this reason it is called the 16-cell.[…]The 16-cell has 24 edges that are the diagonals of the 24 square faces of the hypercube. (Since we are selecting alternate vertices of the hypercube, we only get one diagonal for each square face.)

2002, Gabor Toth, Glimpses of Algebra and Geometry, 2nd edition, page 385:

We will now consider the dual 16-cell. In some ways the 16-cell is a bit easier because it has fewer vertices and edges. But on the other hand the hypercube is perhaps easier to understand intuitively because we can readily (perhaps) see the various cubes of which it is composed.

2012, John Barnes, Gems of Geometry, 2nd edition, page 76:

(geometry) A four-dimensional polytope, analogous to an octahedron, whose sixteen bounding facets are tetrahedra.

4-dimensional polytope analogous to an octahedron


octachoron

en-noun

octachoron (plural octachorons or octachora)

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From octa- (“eight”) + -choron (“room”), from Ancient Greek ὀκτά- (oktá-, “eight”) and χῶρος (khôros, “room”).

Our 16-point 4D constellation is described by the set {(±1,±1,±1,±1)}, which is indeed the set of vertices of a tesseract (i.e., a regular octachoron) [4].

2012, Yequn Zhang, Murat Arabaci, Ivan B. Djordjevic, Rate-Adaptive Four-Dimensional Nonbinary LDPC-Coded Modulation for Long-Haul Optical Transport Networks, page 3:

The Schlegel diagram from this figure is a projection of a regular octachoron from the 4-D space to the 3-D space through a point beyond one of its facets.

2013, Milorad Cvijetic, Ivan B. Djordjevic, Advanced Optical Communication Systems and Networks, page 357:

...completing the notation for the vertices of an elementary lattice octachoron.

2013, Paul Jennings, Frank Nijhoff, On an elliptic extension of the Kadomtsev-Petviashvili equation, page 4:

(mathematics) A four-dimensional object analogous to a cube, constructed out of eight cubes.


According to this table the tetrahedron corresponds to the 5-cell, the cube to the 8-cell ("hypercube"), the octahedron to the 16-cell, the dodecahedron to the 120-cell and the icosahedron to the 600-cell. The 24-cell is an additional body in four-dimensional space which is dual to itself.

1988, Klaus Mainzer, Symmetries of Nature: A Handbook for Philosophy of Nature and Science, published 1996, page 159:

Similarly in dimension eight there is the quaternionic projective plane which we can think of as a 4-sphere with an 8-cell attached, and in dimension sixteen there is a Cayley projective plane which has similar properties.

1996, John Milnor, “Growing Up in the Old Fine Hall”, in Hugo Rossi, editor, Prospects in Mathematics: Invited Talks on the Occasion of the 250th Anniversary of Princeton University, page 6:

Of these nine, five are finite structures in Euclidean space and include the simplex, {3,3,3} (or 5-cell), the four-dimensional cube {4,3,3} (or 8-cell) and its dual {3,3,4} (or 16-cell), and the 120-cell {5,3,3} and its dual {3,3,5} (or 600-cell).

1997, J. François Gabriel, editor, Beyond the Cube: The Architecture of Space Frames and Polyhedra, page 412:

(geometry) A four-dimensional polytope, analogous to a cube, whose eight bounding facets are cubes; a tesseract.

4-dimensional polytope analogous to a cube


icositetrachoron

icositetrachoron (plural icositetrachorons or icositetrachora)

icositetrachorons (plural)
icositetrachora (plural)

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From icosi- (“twenty”) + tetra- (“four”) + -choron (“room”), from Ancient Greek εἴκοσι (eíkosi, “twenty”) τέτταρες (téttares, “four”) and χῶρος (khôros, “room”).

There are six convex regular 4-polytopes; five of these are the higher dimensional analogs of the Platonic solids, and one additional figure, the icositetrachoron, has no three-dimensional equivalent.

2011, Jin Akiyama, Ikuro Sato, The element number of the convex regular polytopes, page 270:

The icositetrachoron (hyperdiamond, octaplex or 24-cell) is the most unique polychoron, because it has no direct analog in any other dimension.

2012, Andrzej Katunin, Fractals based on regular convex polytopes, page 56:

Shape of Nature comes from a unique, optimally symmetric geometric object: the sixth platonic element, the icositetrachoron (or 24-cell regular polytope), crystallized as a hyperdiamond network, mathematically and physically made of loops and connections.

2012, Raymond Aschheim, Spin foam with topologically encoded tetrad on trivalent spin networks:

(mathematics) A four-dimensional object with no regular three-dimensional analogue, constructed out of twenty-four octahedra.


The octagonal projection of the regular 24-cell {3,4,3} reveals that the 24 vertices of this 4-dimensional polytope can be distributed as 16 + 8: the 16 vertices of the 4-cube γ₄ = {4,3,3} and the 8 vertices of its dual, the 16-cell β₄ = {3,3,4}. This view of the 24-cell is less well-known than the dodecagonal projection, in which the β₄ appears as two squares of different sizes joined by 8 equilateral triangles.

1995, Harold Scott Macdonald Coxeter, “Paper Three: Two Aspects of the Regular 24-Cell in Four Dimensions”, in F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Iviċ Weiss, editors, Kaleidoscopes: Selected Writings of H.S.M. Coxeter, page 25:

2002, T. Robbin, Formian for art and mathematics, G. A. R. Parke, P. Disney (editors), Space Structures 5, Proceedings of the 5th International Conference on Space Structures, Volume 1, page 445,
One more example of a four dimensional tessellation is given using the 24-cell, see Fig 2.

The regular polytope with Schläfli symbol {3,4,3}, the so-called 24-cell, can be obtained from the 16-cell as follows. The vertices of the 24-cell are the midpoints of the 24 edges of the 16-cell.

(geometry) A four-dimensional polytope whose twenty-four bounding facets are octahedra and which has no three-dimensional analogue.

hyper-diamond

4-dimensional polytope with 24 octahedral facets


hecatonicosachoron

hecatonicosachoron (plural hecatonicosachorons or hecatonicosachora)

hecatonicosachorons (plural)
hecatonicosachora (plural)

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From hecaton- (“one hundred”) + icosa- (“twenty”) + -choron (“room”), from Ancient Greek ἑκατόν (hekatón, “one hundred”) εἴκοσι (eíkosi, “twenty”) and χῶρος (khôros, “room”).

uder

The pentachoron, the hecatonicosachoron, and the hexacosichoron all have dichoral angles which are not divisors of 2π, and consequently, they are not space-filling.

The hecatonicosachoron (dodecaplex or 120-cell) and the hexacosichoron (tet- raplex or 600-cell) are also unique polychora, which have no analogs in higher dimensions.

2012, Andrzej Katunin, Fractals based on regular convex polytopes, page 57:

The pentachoron, hypercube, orthoplex, hecatonicosachoron, and hexacosichoron are analogues of the ordinary Platonic solids.

2012, Richard Elwes, How to Solve the Da Vinci Code:

(geometry) A four-dimensional object analogous to a dodecahedron, constructed out of one hundred and twenty dodecahedra, arranged 4 to a vertex.

正百二十胞体

せいひゃくにじゅうほうたい


The triacontagonal projection of the 120-cell was first drawn by W. A. Wythoff, but not in reproducible form.

1964, Canadian Mathematical Bulletin, page 394:

In dimension four there are ten star polytopes which together with the 120-cell and 600-cell form the family of pentagonal polytopes (those regular 4-polytopes possessing axes of fivefold symmetry); all twelve have the same symmetry group.

1987, Jeffrey Hurst Butler, Hyperplane sections of regular star polytopes, page 28:

In 1985, M. Davis [3] gave the first explicit geometric construction of a closed hyperbolic 4-manifold by gluing together the opposite sides of a regular hyperbolic 120-cell P with dihedral angle 72°.

2006, John G. Radcliffe, “The Geometry of Hyperbolic Manifolds of Dimension a least 4”, in András Prékopa, Emil Molnár, editors, Non-Euclidean Geometries: János Bolyai Memorial Volume, page 270:

(geometry) A four-dimensional polytope, analogous to a dodecahedron, whose 120 bounding facets are dodecahedra.

正百二十胞体

せいひゃくにじゅうほうたい

Davis 120-cell

4-dimensional manifold in hyperbolic geometry constructed from a hyperbolic 120-cell


hexacosichoron

en-noun

hexacosichoron (plural hexacosichorons or hexacosichora)

hexacosichorons (plural)
hexacosichora (plural)

From hexacosi- (“six hundred”) + -choron (“room”), from Ancient Greek ἑξακόσιοι (hexakósioi, “six hundred”) and χῶρος (khôros, “room”).

uder

Fig. 5 shows a polar representation of 5880 rotations generated by subsampling the hexacosichoron.

2007, J.R. Plaisier, L. Jiang, J.P. Abrahams, Cyclops: New modular software suite for cryo-EM:

The present paper deals with a collection of 60 rays in a 4-dimensional real vector space represented by the vertices of a hexacosichoron (i.e. a 4-dimensional analogue of the icosahedron).
Comment: the reviewer uses the word, the reviewed paper uses 600-cell only.

2011. Artur E. Ruuge, review MR2594002 (2011d:81017) of Waegell, Mordecai and Aravind, P. K., "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem.", Journal of Physics A 43 (2010), no. 10, 105304

The hexacosichoron with an edge-length of 2φ⁻¹ could be constructed from the next permutations of its 120 vertices.

(mathematics) A four-dimensional object analogous to an icosahedron, constructed out of six hundred tetrahedra, arranged 20 to a vertex.


1995, J. Bokowski, On the Construction of Equifacetted 3-Spheres, Neil L. White, Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curaçao Conference, page 303,
P. McMullen has pointed out to the author that he has constructed an interesting equifacetted 3-sphere in his Ph.D.Thesis by starting with the 600-cell.

The dual to the 120-cell is the 600-cell. The 600-cell is a 4-dimensional convex regular polytope with 600 facets. Its Schläfli symbol of the^([sic]) 600-cell is {3, 3, 5}. It is a 4-dimensional analogue of the icosahedron, constructed by the golden ratio on the edges of an octohedron.

2005, Lawrence B. Crowell, Quantum Fluctuations of Spacetime, volume 25, World Scientific Series in Contemporary Chemical Physics, page 329:

The other two figures are the 120-cell and 600-cell which form a dual pair as is clear from their Schläfli symbols being the reverse of each other.

2012, John Barnes, Gems of Geometry, 2nd edition, page 79:

(geometry) A four-dimensional polytope, analogous to an icosahedron, whose 600 bounding facets are tetrahedra.


apeirohedron

en-noun

apeirohedron (plural apeirohedrons or apeirohedra)

apeirohedrons (plural)
apeirohedra (plural)

/əˌpiːɹɵˈhiːdɹən/

/əˌpeɪ̯ɹɵˈhiːdɹən/

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There are exactly 12 regular apeirohedra that in some sense are reducible and have components that are regular figures of dimensions 1 and 2. These apeirohedra are blends of a planar regular apeirohedron, and a line segment { } or linear apeirogon {∞}. This explains why there are 12 = 6·2 blended (or non-pure) apeirohedra. For example, the blend of the standard square tessellation {4,4} and the infinite apeirogon {∞}, denoted {4,4}#{∞}, is an apeirohedron whose faces are helical apeirogons (over squares), rising above the squares of {4,4}, such that 4 meet at each vertex; the orthogonal projections of {4,4}#{∞} onto their component subspaces recover the original components, the square tessellation and the linear apeirogon.

2014, Daniel Pellicer with Egon Schulte, “Polygonal Complexes and Graphs for Crystallographic Groups”, in Robert Connelly, Asia Ivić Weiss, Walter Whiteley, editors, Rigidity and Symmetry, New York, N.Y.: Springer, →ISBN, page 331:

(mathematics, geometry) A polyhedron with an infinite number of faces.


logistic growth

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


lost generation

Attributed to Gertrude Stein and popularized by Ernest Hemingway in the 1920s.

The generation that came of age during World War I.


semilattice

partially ordered set
partially ordered set

(mathematics) A partially ordered set that either has a join (a least upper bound) for any nonempty finite subset (a join-semilattice or upper semilattice) or has a meet (or greatest lower bound) for any nonempty finite subset (a meet-semilattice or lower semilattice). Equivalently, an underlying set which has a binary operation which is associative, commutative, and idempotent.


FSVO

(mathematics, programming, Internet slang) Initialism of for some values of.


disjoint union

(mathematics) A union of sets forced to be disjoint by attaching information referring to the original sets to their elements (i.e., by using indexing).

(mathematics) A union of sets which are already pairwise disjoint.

(topology) The disjoint union of the underlying sets of a given family of topological spaces, equipped with a topology.

coproduct

union of topological spaces equipped with a topology


ratio scale

(measurement, statistics) A scale of measurement of data having a fixed zero value, which permits the quantitative comparison of differences of values.


zolotnik

zolotnik (plural zolotniks or zolotniki)

Borrowed from Russian золотни́к (zolotník).

An old Russian unit of weight, equal to 1/96 of a pound or approximately 4.2658 grams.


desiatina

An old Russian unit of area, approximately 10,925 square metres (the Treasury/official desiatina) or 14,567 square metres (proprietor's desiatina).


chetverik

Borrowed from Russian четвери́к (četverík), from че́тверть (čétvertʹ, “quarter”) + -и́к (-ík).

bor+

An old Russian dry measure, approximately 26.24 litres; one fourth of an osmina.


vershok

An old Russian unit of length, approximately 4.4 centimetres.


shkalik

An old Russian liquid measure, equal to 61.5 millilitres.


shtof

An old Russian liquid measure, equal to 1.23 litres.


bochka

(historical) An old Russian liquid measure, approximately 492 litres.


overdisperse

en-verb

overdisperse (third-person singular simple present overdisperses, present participle overdispersing, simple past and past participle overdispersed)

overdisperses (present singular third-person)
overdispersing (participle present)
overdispersed (participle past)
overdispersed (past)


Grassmannian

From Grassmann + -ian, after Hermann Grassmann (1809–1877), German mathematician and polymath.

(mathematics) A space which parameterizes all linear subspaces of a vector space of a given dimension.


algebraic notation

en-noun

algebraic notation (usually uncountable, plural algebraic notations)

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notação algébrica

a method for recording chess moves using letters and numbers

(algebra) A method for recording domain and range of a function.


superprime

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(mathematics) Any of the subsequence of prime numbers that occupy prime-numbered positions within the sequence of prime numbers.


continuum theory

(topology) The branch of topology devoted to the study of continua, a type of metric space.


microbundle

(mathematics) A kind of generalization of a vector bundle.


direct sum

topological spaces
topological space#English

(mathematics) coproduct in some categories, like abelian groups, topological spaces or modules

(linear algebra) A linear sum in which the intersection of the summands has dimension zero. Equivalently, a linear sum of two subspaces, any vector of which can be expressed uniquely as a sum of two vectors: one vector belonging to the first subspace, and the other vector belonging to the second subspace.

block diagonal matrix
block diagonal matrix

(linear algebra) A block diagonal matrix interpreted as having its non-zero blocks (which are square matrices) as summands.


Frey curve

(mathematics) The elliptic curve associated with a solution of Fermat's equation.


seminormality

(mathematics) The quality of being seminormal.


seminormal

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(algebra, of a ring) Being a commutative reduced ring in which, whenever x, y satisfy x³=y², there is s with s²=x and s³=y.

(group theory) Of a subgroup A of a group G, having a subgroup B such that AB = G, and for any proper subgroup C of B, AC is a proper subgroup of G.


arithmetic-geometric mean

arithmetic-geometric mean (plural arithmetic-geometric means)

(mathematics) A mean of two numbers which is the common limit of a pair of sequences, whose terms are defined by taking the arithmetic and geometric means of the previous pair of terms.


dual number

If f(x) is a polynomial or a power series then its derivative can be obtained “automatically” using dual numbers as follows: f'(x)#61;#123;f(x#43;#92;epsilon)-f(x)#92;over#92;epsilon#125; where f(x#43;#92;epsilon) should be expanded using the algebra of dual numbers. The role of the ε is therein similar to that of an infinitesimal.

In the special case of the algebra #92;mathbf#123;D#125; of dual numbers we get the vertical tangent bundle V.

1993, Ivan Kolář, Peter W. Michor, Jan Slovák, Natural Operations in Differential Geometry, Springer, page 336:

These numbers are now called dual numbers (a term due to Study) and can be defined as expressions of the form a#43;b#92;epsilon, #92;epsilon²#61;0. When this is done as Kotel'nikov and Study showed, the dual distance between two points of this sphere representing two lines, forming an angle #92;varphi and having shortest distance d, is equal to the dual number #92;varphi#43;#92;epsilond, while the motions of Euclidean space are represented by rotations of the dual sphere, and consequently depend on three dual parameters.

1996, Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich, translated by Roger Cooke, Mathematics of the 19th Century: Geometry, Analytic Function Theory, Birkhäuser, page 87:

Thus we investigate natural T-functions defined on T#42;T#123;#92;mathbf#123;D#125;#92;otimesA#125;M to determine all natural operators T#92;toTT#42;Tᴬ for #92;mathbf#123;D#125; denoting the algebra of dual numbers.

2002, Jiří Tomáš, “Some Classification Problems on Natural Bundles Related to Weil Bundles”, in Olga Gil-Medrano, Vicente Miquel, editors, Differential Geometry, Valencia 2001: Proceedings of the International Conference, World Scientific, page 297:

(algebra) An element of an algebra (the algebra of dual numbers) which includes the real numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.


alqueire

From Portuguese alqueire, from Arabic الكَيْل (al-kayl, “a measure”), from ك ي ل (k y l).

(historical) A traditional Portuguese unit of dry measure, equal to 13.1–19.3 liters in different 19th-century contexts.


convolver

(signal processing) An algorithm that performs convolution on a signal.


split-complex number

split-complex number (plural split-complex numbers)

algebra over the field
algebra over a field#English

(mathematics) An element of the two-dimensional algebra over the field of real numbers which contains an element ȷ ≠ 1 satisfying ȷ² = 1.


canada

From Portuguese canada, from Latin cannatus.

(historical, measure) A traditional Portuguese unit of liquid volume equal to 1.7–2.1 liters depending on the area of Portugal, used particularly for wine.


pseudopolynomial

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(mathematics) A function where f(n+k) ≡ f(n) (mod k) for all integers n ≥ 0, k ≥ 1.


erfc

(mathematics) complementary error function, i.e. 1 - erf(x)


maltitude

(mathematics) Any of the four line segments perpendicular to the sides of a cyclic quadrilateral and passing through the opposite side's midpoint.


ill-conditioned

en-adj

ill-conditioned (comparative more ill-conditioned, superlative most ill-conditioned)

more ill-conditioned (comparative)
most ill-conditioned (superlative)

(numerical analysis) Having a high condition number.


hyperconnection

en-noun

hyperconnection (countable and uncountable, plural hyperconnections)

(mathematics) A set of sets that contains the empty set, all one-element sets for any element that is included in any of the sets, and the union of any overlapping group of sets that are elements.


teraparsec

(astronomy) An astronomical unit of distance equal to 10¹² parsecs.


antimedial

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(geometry) Being a triangle formed by three lines parallel to the sides of an original triangle ABC: the parallel to AB through C, the parallel to AC through B, and the parallel to BC through A.


hidden Markov model

hidden Markov model (plural hidden Markov models)

A statistical Markov model in which the system being modelled is assumed to be a Markov process with unobserved (hidden) states.


UST

UST (countable and uncountable, plural USTs)

(combinatorics) Initialism of uniform spanning tree.


coordinate axis

coordinate axis (plural coordinate axes)

(mathematics, geometry) One of the lines in the coordinate system that represents one of the dimensions.


co-ordinate axis

co-ordinate axis (plural co-ordinate axes)

(mathematics) Alternative form of coordinate axis


p-value

The larger the p-value the smaller the chance of a difference. A p-value of 1.00 means 0 % chance of a difference, while a p-value of 0.95 means a chance of difference close to 0. A p-value of > 0.95 literally means that we have > 95 per cent chance of finding a result less close to expectation, which means a chance of < (1-0.95) , i.e., < 0.05 of finding a result this close or closer. Using the traditional 5 per cent decision level, this would mean, that we have a strong argument that such […]

2008, Ton J. Cleophas, A.H. Zwinderman, Toine F. Cleophas, Eugene P. Cleophas, Statistics Applied to Clinical Trials, Springer Science & Business Media, →ISBN, page 145:

(statistics) In statistical significance testing, the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.


damping ratio

(physics, mathematics) A dimensionless measure of the damping in second-order linear dynamic system, appearing as the term ζ in the generalised equation of motion for such a system, ̈q+2ζωₙ̇q+ωₙ²q=F.


semipermutable

Every permutable subgroup of a finite group is semipermutable. The converse, however, is not necessarily true.

(group theory, of a subgroup) That commutes with every subgroup whose order is relatively prime to its own.


septile

(statistics) Any of the quantiles that divide an ordered sample population into seven equally numerous subsets.


Eisenstein integer

Eisenstein integer (plural Eisenstein integers)

Named after Gotthold Eisenstein (1823–1852), German mathematician.

To divide an Eisenstein integer a#43;b#92;omega by another Eisenstein integer c#43;d#92;omega, notice that (c#43;d#92;omega)(c#43;d)(c#43;d#92;omega²)#61;c³#43;d³; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by (c#43;d)(c#43;d#92;omega²), then simplify.

(algebra) A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω].


coordinatewise

(mathematics) By or with respect to coordinates or an individual coordinate.


semiflow

(mathematics) A form of matrix annuler in a Petri net.


superprojective

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(mathematics, of a Banach space X) Having every closed, infinite-codimensional subspace of X contained in an infinite-codimensional subspace which is complemented in X


unisolvent

(mathematics) solvable in one unique manner


power series

Thus no single choice of z₀ will give a power series expansion of f(z) valid for all #92;textstylez#92;in#92;mathbb#123;C#125;#92;setminus#92;left#92;#123;-1,1#92;right#92;#125; even though f is analytic on this set.

1983, Ian Stewart, David Tall, Complex Analysis, Cambridge University Press, page 259:

In this we may use as our starting-point a general discussion of the theory of power series with complex variables and complex coefficients. The construction of such a theory of power series offers no difficulty once we define the concept of limit in the domain of complex numbers; in fact, it follows the theory of real power series almost exactly.

1988, Richard Courant, translated by E. J. McShane, Differential and Integral Calculus, 2nd edition, volume 1, Wiley, page 413:

1899, Oskar Bolza, The Theory of Functions, 2013, Reprint, Books on Demand, page 69,
From this theorem (for which in many cases Cauchy's theorem on double sums may be substituted) follow easily the rules for the multiplication and division of power series, Taylor's theorem for power series along with the rules for differentiation of power series and series of power series.

mathematical analysis
mathematical analysis

(mathematics, mathematical analysis) Any infinite series of the general form ∑ᵢ₌₀ ᪲a_i(x-c)ⁱ.


cocone

(category theory) An object V together with an arrow going from each object of a diagram to V such that for any arrow A in the diagram, the pair of arrows to V which form a triangle with A also commute with it.


differentiably

(mathematics) In a differentiable manner


supersequence

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We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction.

2016, Frederic Sala, Ryan Gabrys, Clayton Schoeny, Lara Dolecek, “Exact Reconstruction from Insertions in Synchronization Codes”, in arXiv:


Bernoulli number

Bernoulli number (plural Bernoulli numbers)

Named after Swiss mathematician Jacob Bernoulli (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Seki Kōwa.
The numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts.

1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418,
The assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely
t/(eᵗ-1)=∑ₙ₌₀ ᪲B_n(tⁿ)/(n!)

In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.

2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman & Hall/CRC), page 287:

This done, he extends occurrence of Bernoulli numbers in the expansion of #92;textstyle#92;frac 1 2#92;cot(#92;frac 1 2x) to the more general form #92;textstyle#92;frac#92;pin#92;cot(#92;frac#123;m#92;pi#125;n) and uses that to relate Bernoulli numbers to the values of #92;zeta(2n). To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series.

2007, C. Edward Sandifer, How Euler Did It, Mathematical Association of America, page 175:

(mathematical analysis, number theory) Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions.

Bernoulli number
Jacob Bernoulli
Seki Takakazu


arithmetic density

arithmetic density (plural arithmetic densities)

(geography, statistics) The population density measured as the number of people per unit area of land.


orthoptic

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(geometry) The isoptic whose given angle is a right angle.


isoptic

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(geometry) The set of points for which two tangents of a given curve meet at a given angle.


indicator function

indicator function (plural indicator functions)

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

indikaattorifunktio

function equal to 1 on elements of a given set


well-orderable

(set theory, order theory) Capable of being well-ordered.


special unitary group

special unitary group (plural special unitary groups)

1992 [Prentice-Hall], George H. Duffey, Applied Group Theory: For Physicists and Chemists, 2015, Dover, Unabridged Republication, page 284,
The special unitary group in two dimensions is represented by the 2 X 2 unitary matrices whose determinants equal 1.

The group is called the special unitary group in two dimensions, or SU(2), because it acts on matrices of degree 2.

2000, Herbert S. Green, Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process, Springer, page 26:

2004, Roger Cooke (translator), Vladimir I. Arnold, Lectures on Partial Differential Equations, [1997, Lekstii ob uravneniyakh s chastnymi proizvodnymi], Springer, page 81, When n = 3, the group of rotations SO(3) is isomorphic to the real three-dimensional projective space ℝP³. It has a two-sheeted covering by the three-dimensional sphere (the group of unit quaternions), which in turn is isomorphic to the special unitary group SU(2), also known as the spin group of order 3, as in the following diagram

(linear algebra, group theory) For given n, the group of n×n unitary matrices with complex elements and determinant equal to one.

erityinen unitaarinen ryhmä

group of n×n matrices with complex elements and determinant 1


Pythagorean trigonometric identity

Pythagorean trigonometric identity (plural Pythagorean trigonometric identities)

Pythagorean trigonometric identities (plural)

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

jedynka trygonometryczna

statement that sum of squares of sine and cosine equals one


noncommutative geometry

noncommutative geometry (countable and uncountable, plural noncommutative geometries)

(mathematics, physics) A version of C*-algebra which does away with commutativity, and the geometry which would be generated by/associated with such a modified algebra.


biquaternion

(algebra) Any of the numbers w+xi+yj+zk! where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group.


crude birth rate

crude birth rate (plural crude birth rates)

(demography) Number of childbirths in a society per 1,000 people per year


crude death rate

crude death rate (plural crude death rates)

(demography) Number of deaths in a society per 1,000 people per year


natural increase rate

natural increase rate (plural natural increase rates)

(demography) The difference between the crude birth rate and the crude death rate of a population


demographic transition

demographic transition (countable and uncountable, plural demographic transitions)

(demography) The process that represents the transition from high birth and death rates to low birth and death rates as a country develops from a pre-industrial to an industrialized economic system.


dyadic fraction

dyadic fraction (plural dyadic fractions)

(mathematics) A vulgar fraction whose denominator is a power of two, or a rational number representable as such a fraction.


digit

From Middle English digit, from Latin digitus (“a fingerbreadth; a number”). Doublet of digitus and toe .

bit

position in a sequence of numerals, symbol in a positional number system

The base-10 number 123.4 has four digits: the hundreds digit is 1, the tens digit is 2, the units digit is 3, and the tenths digit is 4.

figure

informal, usually in discussion of money

(mathematics) A position in a sequence of numerals representing a place value in a positional number system.

Hexadecimal numeration (Base sixteen) includes the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 but also A (=10 decimal), B, C, D, E, and F. Sixteen itself is written as the two-digit number 10.

(mathematics) A distinct symbol representing a natural number in a positional number system.

(historical units of measure) A unit of length notionally based upon the width of an adult human finger, standardized differently in various places and times, (especially) the English digit of ¹⁄₁₆ foot, now equivalent to about 1.9 cm.

(geometry, rare, obsolete) Synonym of degree: ¹⁄₃₆₀ of a circle.


digit

en-verb

digit (third-person singular simple present digits, present participle digiting, simple past and past participle digited)

digits (present singular third-person)
digiting (participle present)
digited (participle past)
digited (past)

(archaic, transitive) To point at or point out with the finger.


S plane

(mathematics, engineering) The complex plane on which Laplace transforms are graphed; a mathematical domain where processes are viewed in the frequency domain rather than the time domain; used as a graphical analysis tool in engineering and physics.


antisymmetrize

antisymmetrize (third-person singular simple present antisymmetrizes, present participle antisymmetrizing, simple past and past participle antisymmetrized)

antisymmetrizes (present singular third-person)
antisymmetrizing (participle present)
antisymmetrized (participle past)
antisymmetrized (past)


hypercell

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(geometry) A four-dimensional face of a polytope.


hyperlattice

(mathematics) A lattice generalized to any number of dimensions.


nephroid

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From nephro- + -oid, i.e. "kidney-shaped".

(geometry) A plane curve resembling two intersecting ellipses


alysoid

Ancient Greek ἅλυσις (hálusis, “chain”).


anguinea

(mathematics) A cubic curve of the form xy² + ey = ax³ + bx² + cx + d


cubo-cubic

Cremona transformation
Cremona transformation

(geometry, of a transformation) Being a Cremona transformation such that the homaloids of the transformation and its inverse all have degree 3.


tacnode

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A tacnode is defined to be a singularity equivalent to the origin in the curve y²-x²#61;0, that is, the union of two smooth arcs meeting with contact of order two. Similarly, an oscnode is a singularity equivalent to y²-x⁶#61;0, i.e., consisting of two smooth branched with contact of order three, and so on.

2012, Joe Harris, Algebraic Geometry: A First Course, page 262:

(geometry) A point of a curve where two or more osculating circles to the curve at that point are tangent, so that two branches of the curve have ordinary tangency at the double point.


antipoint

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(geometry) One of a pair of points constructed from two foci of a curve.


Aronhold set

Named after Siegfried Heinrich Aronhold.

(geometry) One of the 288 sets of seven of the 28 bitangents of a quartic curve corresponding to the seven odd theta characteristics of a normal set.


azygetic

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azygetic triad; azygetic tetrad; azygetic set


bicuspidal

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bielliptic

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(geometry, of a curve) Being a branched double cover of an elliptic curve.