stablecoin

Several other companies have recently introduced so-called stablecoins, linked to the value of the dollar. JPMorgan Chase even said it was experimenting with the concept last month.

2019 February 28, Nathaniel Popper, Mike Isaac, “Facebook and Telegram Are Hoping to Succeed Where Bitcoin Failed”, in The New York Times, →ISSN:

The first attempt to scale a stablecoin came from Maker. At the time of writing, there are about half a billion dollars worth of cryptocurrency tied up as collateral for this stablecoin—the Dai—whose value is pegged to the US dollar.

2020, David Birch, The Currency Cold War, London Publishing Partnership, →ISBN, page 98:

But the meltdown gathered momentum this week when TerraUSD, a stablecoin, imploded. Stablecoins, which are meant to be a more reliable means of exchange, are typically pegged to a stable asset such as the U.S. dollar and are intended not to fluctuate in value.

2022 May 12, David Yaffe-Bellany, Erin Griffith, Ephrat Livni, “Cryptocurrencies Melt Down in a ‘Perfect Storm’ of Fear and Panic”, in The New York Times, →ISSN:

A cryptocurrency with price stability by design, often pegged to a stable asset such as the US dollar.


Platt scaling

probability distribution
probability distribution

(machine learning) A technique for transforming the outputs of a classification model into a probability distribution over classes, based on a logistic regression model.


support vector machine

en-noun

support vector machine (plural support vector machines)

(machine learning) A supervised learning model with associated algorithms that analyse data used for classification and regression analysis.


artifacted

(computer graphics, signal processing) Having artifacts.


metric tensor

2015, 1:10:00–1:11:15 from the start, in 1 Exterior Product, Fermions (Mathematics for Theoretical Physicists - Hirosi Ooguri), spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):

differential geometry
differential geometry

(differential geometry) a symmetric bilinear form which is non-degenerate (i.e., having all non-zero eigenvalues); a differential of distance on a manifold


Laplace-de Rham operator

Laplace-de Rham operator (plural Laplace-de Rham operators)

(differential geometry) A generalization of the Laplace operator; the anticommutator of the exterior derivative and the codifferential.

Laplace operators in differential geometry


SSDL

2007, Fornasier: Soya: a programming model and runtime environment for component composition using SSDL, title:

1986, Grotjohn: Sample-set differential logic (SSDL) for complex high-speed VLSI, title:

Grotjohn and Hoefflinger [2] further improved the clocked CVSL and named it sample-set differential logic (SSDL)...

1988, Lu: Implementation of iterative networks with CMOS differential logic, page 1:

sample-set differential logic
sample-set differential logic#English

(mathematics) Initialism of sample-set differential logic.

2002, Pogrebnoy: High-speed low-power sense amplifying half-latch and apparatus thereof for small-swing differential logic (SSDL):

2002, Fiedler: High-speed differential sampling flip-flop, page 1:

...small swing differential logic (SSDL) whose outputs are not full-swing can be used...

1995, Matsui: A low-voltage 32/spl times/32-bit multiplier in dynamic differential logic, page 1:

small-swing differential logic
small-swing differential logic#English

(mathematics) Initialism of small-swing differential logic.


line of apsides

line of apsides (plural lines of apsides)

(geometry, astronomy) Synonym of apse line


fillability

(mathematics) The condition of being fillable


premarginalization

(mathematics) marginalization prior to some other procedure


submartingale

If a gambler repeatedly plays a game with positive expectation, his payoff over time is a submartingale.

(mathematics) A stochastic process for which the conditional expectation of future values given the sequence of all prior values is superior or equal to its current value.


cohomologicity

cohomologicity (countable and uncountable, plural cohomologicities)

(mathematics) The condition of being cohomological


imaginarity

(mathematics) The condition of being imaginary


majorisation

(mathematics) Alternative form of majorization


metaconjecture

(mathematics) A conjecture that holds for most, but not all, conditions


pseudospinor

(mathematics, physics) A pseudospin analog of a spinor.


quasiisometry

equivalence relation
equivalence relation

(mathematics) An equivalence relation that is an isometry when viewed at a large scale


quasinorm

(mathematics) A function that has all the properties of a norm except the triangle inequality


ultramodularity

(mathematics) The condition of being ultramodular


Fleiss' kappa

(statistics) A statistical measure for assessing the reliability of agreement between a fixed number of raters.


pseudomoment

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(mathematics) A numerical entity that combines the properties of moments and metrics


rhombiform

en-adj

rhombiform (comparative more rhombiform, superlative most rhombiform)

more rhombiform (comparative)
most rhombiform (superlative)

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ATH

(stock market and cryptocurrencies trading) Initialism of all-time high, referring to the value of an asset.


isotomic

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From iso- (“same”) + -tome (“cutting, section”) + -ic (“adjectival suffix”), ultimately from Ancient Greek τέμνω (témnō, “I cut”).

affix

(geometry) Having equal but reflected cevians.


cevian

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(geometry) A line from the vertex of a triangle to a point on the opposite side


suversed

The key insight was to replace the law of cosines by a law of versed sines or a law of suversed sines, where the suversed sine of an angle is the versed sine of the supplement of that angle.

2015, Maria Zack, Elaine Landry, Research in History and Philosophy of Mathematics, page 45:

(trigonometry) versed and belonging to the supplement


Ferrers diagram

en-noun

Ferrers diagram (plural Ferrers diagrams)

(mathematics) A diagram that shows the possible partitions of a positive integer as rows of dots.


nothing-up-my-sleeve number

nothing-up-my-sleeve number (plural nothing-up-my-sleeve numbers)

In reference to magic tricks in which the performer may have something concealed in a sleeve (or may demonstrate to the audience that they do not).

The key schedule, however, expanding the key using an essentially one-way method, is more complex, with the binary expansions of both e and the golden ratio of "nothing up my sleeve numbers".

2014 May, Mr. Ajay Bhaisare, “Data Protection Outsourcing of Cloud Data to Maintain Trust between Cloud Service and Data Owner Using RC5 Algorithm”, in IJCSMC, volume 3, number 5, page 608:

Anyway, they make the same point: “The BADA55-VPR curves illustrate the fact that 'verifiably pseudorandom' curves with 'systematic' seeds generated from 'nothing-up-my-sleeve numbers' also do not stop the attacker from generating a curve with a one-in-a-million weakness.”

(cryptography) A number which, by its construction, is above suspicion of having hidden properties; used in creating cryptographic functions such as hashes and ciphers.


transnormalization

This procedure is termed transnormalization. Using variables that are transnormalized is a necessary, but not sufficient, condition for the joint distribution to be multi-normal. However, tests to date have shown that joint normality is a good assumption whenever the variables have been selected by nature, as opposed to having been selected artificially by man. Predictors selected by man, such as time of day, are included in the TRP equation as "marginal" variables.

1977, Proceedings of the 7th Technical Exchange Conference, El Paso, Texas, 30 November-3 December 1976:

... such transnormalization.

1977, Raymond L. Joiner, Separation of Mixed Data Sets Into Homogeneous Sets:

The three main procedures that comprise this multivariate transnormalized method are transnormalization, correlation, and regression conditional probability. The transnormalization process (a nonlinear process) transforms the observed or raw data into the standard normal variable with the same cumulative climatological probability as the raw predictor. Transnormalization ensures normal distribution of the predictor.

1978, Meteorological and Geoastrophysical Abstracts:

(mathematics and sciences) The transformation of raw data such that the resulting cumulative distribution is normal.


enneadecagon

(geometry) A polygon with nineteen sides and nineteen angles.


hexadecagon

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(geometry) A polygon with sixteen sides and sixteen angles.


adjacency list

(mathematics) A collection of unordered lists (each representing the neighbours of a vertex) that represents a finite graph.


anisotomic

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RREF

(mathematics) Initialism of reduced row echelon form.


leaf-shaped

en-adj

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

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


icositetragon

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(geometry) A polygon with 24 sides and 24 angles.


hexacontagon

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From Ancient Greek ἑξήκοντα (hexḗkonta, “60”) + -gon, from Ancient Greek γωνία (gōnía, “corner”).


Sobel test

Based on the work of Michael E. Sobel, a statistics professor at Columbia University, New York.

(statistics) A kind of specialized t test that tests the significance of a mediation effect.


adjacent angle

(geometry) Either of a pair of angles that share a common side


angle of depression

en-noun

angle of depression (plural angles of depression)

The angle formed by the line of sight and the horizontal plane for an object below the horizontal.


opposite angle

(geometry) Either of the angles directly opposite to each other when two lines intersect in a plane.


tetradecahedron

tetradecahedron (plural tetradecahedra or tetradecahedrons)

tetradecahedra (plural)
tetradecahedrons (plural)

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(geometry) A polyhedron with fourteen faces.


Ramsey's theorem

Ramsey's theorem (countable and uncountable, plural Ramsey's theorems)

Named after British mathematician and philosopher Frank P. Ramsey.

(mathematics) A (version of a) theorem concerning the existence of cliques in a labelled complete graph.

The theorem that any graph labelling (with colours) of a sufficiently large complete graph contains monochromatic cliques.

The theorem that any graph labelling (with colours) of an infinite complete graph contains at least one infinite monochromatic clique.


applicand

(mathematics) A function, in lambda calculus, that is the subject of application


Menger's theorem

It was proved for edge-connectivity and vertex-connectivity by Karl Menger in 1927.

(graph theory) A characterization of the connectivity in finite undirected graphs in terms of the minimum number of disjoint paths that can be found between any pair of vertices.


GOE

About a year and a half ago, Schroeppel and Hickerson found a bunch of patterns with very small predecessor counts. These are used as modules in existence proofs for GOE's within various rectangles, some quite cozy.

1991 March 28, Bill Gosper, “on Life (and Death)”, in comp.theory.cell-automata (Usenet):

This describes briefly the discovery of the first GOL GOE by Roger Banks, saying: "[he] used sophisticated mathematical techniques to prove that a certain 9-by-33 rectangular pattern is itself a Garden-of-Eden pattern."

1999 April 1, Tim Tyler, “proof for garden of eden in GOL ?”, in comp.theory.cell-automata (Usenet):

As far as I know, even a definite result for the 6x6 case would be progress. The most recent investigation of GoEs that I know about was some interesting work that Donald Knuth recently did in passing for his upcoming Volume 4 of _The Art of Computer Programming_, using binary decision diagrams.

2008 October 22, Dave Greene, “State wrt "Garden of Eden"in Conway's "Game of Life"”, in comp.theory.cell-automata (Usenet):

(cellular automata) Initialism of Garden of Eden.


ARMA

(statistics) Initialism of autoregressive moving average


ARIMA

(statistics) Initialism of autoregressive integrated moving average.


ARCH

(statistics) Initialism of autoregressive conditional heteroskedasticity


GARCH

(statistics) Initialism of generalised autoregressive conditional heteroskedasticity


boundary term

If M is truncated to have toral boundary, there will be a boundary term after the integration by parts step.

1996, Kojima Sadayoshi, Saito Kyoji, Matsumoto Y, Topology And Teichmuller Spaces, →ISBN:

Thus the boundary term ambiguity is under physical control: Each distinct choice of the quasi-local expression given by the Hamiltonian boundary term is associated with a physically distinct boundary condition.

2017, Ni Wei-tou, One Hundred Years Of General Relativity, →ISBN:

(mathematics) A term that must be added to an equation to represent a boundary condition.


defective matrix

defective matrix (plural defective matrices)

1994, Sigal Ar, Jin-Yi Cai, Chapter 5: Reliable Benchmarks Using Numerical Stability, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, Society for Industrial and Applied Mathematics, page 41,
An n x n matrix A is a defective matrix if it has a defective eigenvalue.
Obviously, a Jordan block of dimension greater than 1, and a matrix whose Jordan canonical form has a Jordan block of dimension greater than 1, are defective matrices.

Therefore, the sum of the geometric multiplicities of a defective matrix will be less than n.

2007, Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, page 261:

Thus, defective matrices exhibit high sensitivity to imperfections.
It can be checked that the unperturbed matrices L* are in fact nearly-defective, because they have nearly-parallel eigenvectors. In particular, the matrices are themselves perturbations of order ε of exactly defective matrices L_(id).

2013, Angelo Luongo, “Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures”, in Alexander F. Vakakis, editor, Normal Modes and Localization in Nonlinear Systems, Springer, page 153:

linearly independent
linearly independent

(linear algebra) A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable.


free Boolean algebra

free Boolean algebra (plural free Boolean algebras)

(algebra) A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas). Starting with a set of n variables which are independent of each other and are called generators, the power set of this set has 2ⁿmembers which may be called atoms and are valuations of the n variables: a valuation can be considered to be a set of variables which are "true" under that valuation, or a conjunction of generators (such that variables not included in that set are included in negated form in the equivalent conjunction). Then the power set of the set of atoms yields a set of 2^(2ⁿ) members which are the elements of the said field of sets. These elements correspond to Boolean formulas: a formula can be considered to be a set of valuations which make the formula true, or a linear combination (i.e., a disjunction) of atoms.


elliptic operator

elliptic operator (plural elliptic operators)

From its association with a corresponding elliptic partial differential equation — a second-order linear partial differential equation with coefficients specified analogously to those of the equation of a conic section that generates an ellipse.

2001, L. Gross, Heat Kernel Analysis on Lie Groups, Laurent Decreusefond, Bernt Øksendal, Ali S. Üstünel (editors), Stochastic Analysis and Related Topics VII: Proceedings of the Seventh Silivri Workshop, Springer-Verlag (Birkhäuser), page 1,
The study of second order elliptic operators over some of these infinite dimensional manifolds has been the subject of much work in the past forty years, some of it for the purposes of quantum field theory and some of it for the purposes of stochastic analysis or infinite dimensional differential topology.

Just as in the usual elliptic theory, in the theory of nonlocal elliptic operators we need not restrict ourselves to individual operators but can also consider families of nonlocal elliptic operators.

2008, Vladimir E. Nazaikinskii, Anton Yu. Savin, Boris Yu. Sternin, Elliptic Theory and Noncommutative Geometry: Nonlocal Elliptic Operators, Springer, page 37:

The Dirichlet problem for an elliptic operator was solved in Chap. 10 by morphing a solution of the Dirichlet problem for the Laplacian on a ball into a solution of the Dirichlet problem for an elliptic operator on a ball.

2014, Lester L. Helms, Potential Theory, 2nd edition, Springer, page 412:

mathematical analysis
mathematical analysis

differential operator
differential operator

(mathematical analysis) Any of a particular class of second-order linear differential operators.


symplectic group

symplectic group (plural symplectic groups)

So named by German mathematician Hermann Weyl, replacing previous confusing names. More at symplectic.

This representation ω arises by virtue of the existence of an action by automorphisms of the symplectic group on a certain two-step nilpotent group, commonly called the Heisenberg group.
Weil was of course working in the adelic formalism of modern number theory, so he considered symplectic groups and Heisenberg groups with coefficients in a general local field.

1985, Roger Howe, “Dual Pairs in Physics: Harmonic Oscillators, Photons, Electrons, and Singletons”, in Mosh Flato, Paul Sally, Gregg Zuckerman, editors, Applications of Group Theory in Physics and Mathematical Physics, American Mathematical Society, page 179:

2001, G. Wassermann (translator), V. I. Arnol'd, A. B. Givental', Symplectic Geometry, V. I. Arnol'd, S. P. Novikov (editors), Dynamical Systems IV: Symplectic Geometry and its Applications, Springer, 2nd Edition, page 18,
The exponential of an operator gives the exponential mapping H↦exp(H)=∑Hᵏ/k! of the space of Hamiltonian operators to the symplectic group. The symplectic group acts by conjugation on itself and on its Lie algebra.

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group.

2012, Rolf Berndt, Ralf Schmidt, Elements of the Representation Theory of the Jacobi Group, Springer, page v:

(linear algebra, group theory) For given field F and positive integer n, the group of 2n×2n symplectic matrices with elements in F.


IMD

index of multiple deprivation
w:Indices of Multiple Deprivation

(British, demography) Initialism of index of multiple deprivation.


law of tangents

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


Wellsean

From the surname of chemist A. F. Wells, plus the suffix -ean, which forms adjectives from proper names.

The unit cell contains a single 8-connected cube-centered vertex, six 4-connected distorted square planar vertices and eight 4-connected distorted tetrahedral vertices. It is a Wellsean structure with a Wells point symbol given by (468)(48)(48) and a Schläfli symbol of (5, 4.2667).

2002, Michael J. Bucknum, “Jubilite: A 4-,8-Connected Cubic Structural Pattern in Space Group Pm3”, in Chemistry Preprint Archive, page 138:

(chemistry, mathematics) Having a structure within which both polygonality and connectivity are fractional.


ill-posed

an ill-posed problem


Tikhonov regularization

Tikhonov regularization (countable and uncountable, plural Tikhonov regularizations)

(mathematics) A particular method of regularization of ill-posed problems, or estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated.


product ring

For instance, let X be an infinite set, R the product ring K^(XY)#92;#33;, and A the set of homomorphisms R#92;rightarrowK given by evaluation at all the elements of X. Then the hypothesis of the Lemma holds, but #92;text#123;Int#125;#95;A(R) is the product ring k^(XY)#92;#33;, which is not in general free as a k-module.

1996, George M. Bergman, Adam O. Hausknecht, Cogroups and Co-rings in Categories of Associative Rings, American Mathematical Society, page 246:

Thus the set X#92;timesY becomes a ring with these operations and it is called the product ring. The identity element of the product ring with respect to the addition is obviously (0,0) where 0 and 0 are identity elements of addition in the rings X and Y#92;#33;, respectively.

2003, Erdoğan S. Şuhubi, Functional Analysis, Springer, page 63:

2007, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique], Springer, page 101,
Then A_λ is isomorphic to the product ring k×k via the homomorphism sending X to (α,-α).

(mathematics, ring theory) A ring that is the direct product of rings.


Grothendieckian

Of or relating to Alexander Grothendieck (1928–2014), German-born French mathematician who was instrumental in the creation of modern algebraic geometry.

grothendieckien

of or relating to Alexander Grothendieck


Teichmüller space

Teichmüller space (plural Teichmüller spaces)

(geometry) A space T(S) assigned to a (real) topological (or differential) surface S: it parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.


polysaturated

(mathematics, of an internal binary mapping on a countably infinite set) Having the property that for some cardinality, all subsets of that cardinality or smaller map to another element of the subset.


Grothendieck-Riemann-Roch theorem

Hirzebruch-Riemann-Roch theorem
Hirzebruch-Riemann-Roch theorem

(algebraic geometry) An important generalization of the Hirzebruch-Riemann-Roch theorem about complex manifolds.


quasi-invariant

For quotations using this term, see Citations:quasiinvariant.

(mathematics) Preserving a specified property under transformation.


Gelfond's constant

After Soviet mathematician Алекса́ндр О́сипович Ге́льфонд (Alexander Osipovich Gelfond).

2009, Samuel W. Gilbert, The Riemann Hypothesis and the Roots of the Riemann Zeta Function, BookSurge Publishing, page 93,
Gelfond's constant satisfies the identity
e^π=(-1)⁻ⁱ
Therefore, the roots of the Riemann zeta function are defined by geometrical constraints of the discrete partial sums of the Dirichlet series terms by continuous and geometrically equivalent envelopes defined by powers of Gelfond's constant.

and we have the promised appearance of Gelfond's constant.

2007, Julian Havil, Nonplussed!: Mathematical Proof of Implausible Ideas, Princeton University Press, page 143, This means that ∑ₙₑᵥₑₙV_n(1)=∑ₘ₌₁ ᪲V_2m(1)=∑ₘ₌₁ ᪲(πᵐ)/(m!)=e^π-1

2016, Ravi P. Agarwal, Hans Agarwal, Syamal K. Sen, Birth, growth and computation of pi to ten trillion digits, David H. Bailey, Jonathan M. Borwein (editors, Pi: The Next Generation, Springer, page 403,
Alexander Osipovich Gelfond (1906-1968) was a Soviet mathematician. He proved that e^π (Gelfond's constant) is transcendental, but nothing yet is known about the nature of the numbers π + e, πe, or πᵉ.

(mathematics) The value e^π (approximately 23.140692632779), known to be transcendental.


linear order

A set of numbers on a number line form a linear order.


faithful functor

faithful functor (plural faithful functors)

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is injective into the corresponding target hom-set.


full functor

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is surjective into the corresponding target hom-set.


antedependence

(mathematics) The dependence of a measure on some previous observations


prospectivity

prospectivity (countable and uncountable, plural prospectivities)

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Schur bases his definitions of equality, of addition and multiplication of projective segments, upon the correspondence known as ' prospectivity,' and, at first, avails himself only of the axioms of connection and of order [Shur's postulates 1. to 8.];

1919, Ludwik Silberstein, Projective Vector Algebra:

If a point O on I is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line I; in fact they are all the same prospectivity, however m, n, S, and S' may have been varied subject to these conditions.

2014, Alfred North Whitehead, Science and Philosophy, →ISBN:

Alternatively, Friedrich Schur (1881, p. 253) used Thomae's (1873, p. 11) definition of projectivity in terms of prospectivity to prove the fundamental theorem in a manner which is independent of the Archimedean axiom.

2016, Francesca Biagioli, Space, Number, and Geometry from Helmholtz to Cassirer, →ISBN, page 122:

(geometry) A projective relationship between a line and itself by projecting onto an intersecting line from a point on another line that intersects at the same point, and then back on to the first line from a different point on the first intersecting line.


Laplacian matrix

Laplacian matrix (plural Laplacian matrices)

Named after Pierre-Simon, marquis de Laplace (1749 – 1827), a French scholar whose work was important to the development of mathematics, statistics, physics and astronomy.

(graph theory) A square n×n matrix which describes an undirected graph of n vertices by letting rows and columns correspond to vertices, letting its diagonal elements contain the degrees of corresponding vertices and letting its non-diagonal elements contain either −1 or 0 depending on whether there is or there is not (respectively) an edge connecting the pair of corresponding vertices.


special function

special function (plural special functions)

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


Cayley's formula

(graph theory) A formula for the number of spanning trees of a complete graph of n vertices: nⁿ⁻².


Hann function

Named after the Austrian meteorologist Julius von Hann.

(signal processing) A discrete window function, w(n)=1/2;(1- cos ((2πn)/(N-1))), typically used to select a subset of a series of samples in order to perform a Fourier transform or other calculation.


trigintaduonion

The 32-dimensional compounding fields and their quantum interplays in the trigintaduonion space can be presented by analogy with octonion and sedenion electromagnetic, gravitational, strong and weak interactions.

2007, Zihua Weng, Compounding Fields and Their Quantum Equations in the Trigintaduonion Space:

In this paper we introduce efficient algorithm for the multiplication of trigintaduonions. The direct multiplication of two trigintaduonions requires 1024 real multiplications and 992 real additions.

2014, Aleksandr Cariow, Galina Cariowa, “An algorithm for multiplication of trigintaduonions”, in Journal of Theoretical and Applied Computer Science, volume 8, number 1:

The trigintaduonions form a 32-dimensional Cayley-Dickson algebra. In this paper, we introduce the k-Fibonacci and k-Lucas trigintaduonions. Moreover, we give some properties of these trigintaduonions and derive relationships between them.

2018, Kübra Gül, “On-Fibonacci and-Lucas Trigintaduonions”, in International Journal of Contemporary Mathematical Sciences, volume 13, number 1:

(mathematics) A 32-dimensional hypercomplex number that is a nonassociative extension of a sedenion.


sober space

For a reader more interested in function spaces than in functors, this concludes the description of content except to add that the classification of coadjoint G’s by sets B bearing a topological topology relativizes to T₀ spaces. For other readers: and trivially to sober spaces.

1975, Mathematica Scandinavica, Societates Mathematicae, page 318:

1983, Houston Journal of Mathematics, Volume 9, University of Houston, page 192,
In the Hofmann and Lawson paper, it is proved that the topological space Spec(L) is a locally quasicompact sober space […] .

Thus sober spaces are necessarily T₀.

2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, volume 2, Oxford University Press, page 492:

How rich are sober and L-sober spaces, are there important examples? It is well-known [25] that Hausdorff spaces, and hence compact T₀ (and so finite T₀ spaces), are sober spaces in the traditional setting. Furthermore, the soberification of a space—the spectrum of the topology of a space—is always sober; and if the original space is not Hausdorff, then its soberification is a sober space which is not Hausdorff. So there are many non-Hausdorff sober spaces as well.

2003, A. Pultr, S. E. Rodabaugh, “Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces”, in Stephen Ernest Rodabaugh, Erich Peter Klement, editors, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Springer, page 155:

When X is a noetherian sober space the construction of loc. cit. can be extended to an arbitrary lower-semicontinuous function for the constructible topology.

2004, Ofer Gabber, “Notes on some t-structures”, in Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser, editors, Geometric Aspects of Dwork Theory, volume 1, Walter de Gruyter, page 711:

(topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space.


Sierpinski space

Sierpinski space (plural Sierpinski spaces)

Named in honour of Wacław Sierpiński (1882–1969), a Polish mathematician.

Alexandrov-discrete space
Alexandrov-discrete space

(topology) A two-point Alexandrov-discrete space.


monoidal category

monoidal category (plural monoidal categories)

(category theory) A category 𝒞 with a bifunctor ⊗:𝒞×𝒞→𝒞 which may be called tensor product, an associativity isomorphism α_(A,B,C):(A⊗B)⊗C≃A⊗(B⊗C), an object I which may be called tensor unit, a left unit natural isomorphism λ_A:I⊗A≃A, a right unit natural isomorphism ρ_A:A⊗I≃A, and some "coherence conditions" (pentagon and triangle commutative diagrams for those isomorphisms).


algebraic variety

algebraic variety (plural algebraic varieties)

If #92;mathcal#123;W#125; is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x#92;in#92;mathcal#123;W#125; axe taken to be the germs of functions on #92;mathcal#123;W#125; induced by functions in the stalk #92;mathfrak#123;R#125;#95;x. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,#92;mathfrak#123;R#125;), by definition, is a local-ringed space.

2005, Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvili, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, page 87:

The quest for understanding the structure of algebraic varieties is a very natural one.[…]A landmark from that time was Hironaka's 1964 theorem on the existence of a smooth model for any complex algebraic variety.

2011, Carel Faber, Gerard van der Geer, Eduard Looijenga, Classification of Algebraic Varieties, European Mathematical Society, page vii:

Remark 2.8 Abstract algebraic varieties are obtained by gluing affine algebraic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3. This allows to develop the category of algebraic varieties with the usual constructions therein. All subsequent definitions could be formulated for abstract algebraic varieties, but will only be developed in the affine or quasi-affine case to keep things simple.

2014, Herwig Hauser, “Blowups and Resolution”, in David Ellwood, Herwig Hauser, Shigefumi Mori, Josef Schicho, editors, The Resolution of Singular Algebraic Varieties, American Mathematical Society, page 7:

(algebraic geometry) The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition.

varisto

set of solutions to a system of polynomial equations; any of certain generalisations of such a set

affine variety

set of solutions to a system of polynomial equations, any of certain generalisations of such a set


free abelian group

free abelian group (plural free abelian groups)

A free abelian group of rank n is isomorphic to ℤ⊕ℤ⊕...⊕ℤ=⨁ⁿℤ, where the ring of integers ℤ occurs n times as the summand. The rank of a free abelian group is the cardinality of its basis. The basis of a free abelian group is a subset of it such that any element of it can be expressed as a finite linear combination of elements of such basis, with the coefficients being integers. (For an element a of a free abelian group, 1a = a, 2a = a + a, 3a = a + a + a, etc., and 0a = 0, (−1)a = −a, (−2)a = −a + −a, (−3)a = −a + −a + −a, etc.)

(algebra) A free module over the ring of integers.


ring-shaped

en-adj

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

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

Having the shape of a ring; circular; annular.


cyclotomic polynomial

en-noun

cyclotomic polynomial (plural cyclotomic polynomials)

For a prime number p, the pᵗʰ cyclotomic polynomial is xᵖ-1/x-1=xᵖ⁻¹+xᵖ⁻²+...+x²+x+1.

Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x+1.

Euler's totient function
Euler's totient function

(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial.


almond-shaped

en-adj

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

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


top-shaped

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

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


algebraic subvariety

en-noun

algebraic subvariety (plural algebraic subvarieties)

(algebraic geometry) A subset of an algebraic variety that is itself an algebraic variety.


Galois connection

Galois connection (plural Galois connections)

From Galois (attributive form of Galois theory) + connection; ultimately after French mathematician Évariste Galois. Coined by Norwegian mathematician Øystein Ore in 1944, Galois connexions, Transactions of the American Mathematical Society, 55, pages 493-513.

1986, Horst Herrlich, Miroslav Hušek, Galois Connections, Austin Melton, Mathematical Foundation of Programming Semantics: International Conference, Proceedings, Springer, Lecture Notes in Computer Science: 239, page 122,
Define maps G:A→B and F:B→A by G(a)=y!∈!Y|∀x!∈!axρy and F(b)=x!∈!X|∀y!∈!bxρy. Then (F,G) is called a Galois connection of the first kind.

Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of "polarities," and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities.

2001, J. Michael Dunn, Gary M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, page 398:

Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair #92;left#92;langle#92;uparrow,#92;downarrow#92;right#92;rangle of mappings #92;uparrow assigning subcollections of Y to subcollections of X, and #92;downarrow assigning subcollections of X to subcollections of Y.

2006, Radim Bělohlávek, Taťána Funioková, Vilém Vychodil, “Galois connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes”, in Bernd Reusch, editor, Computational Intelligence, Theory and Applications: International Conference, Proceedings, Springer,, page 205:

partially ordered set
partially ordered set

(category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets.


circular angle

(geometry) An angle that defines a position on a circle.


generic property

generic property (plural generic properties)

(mathematics, measure theory) A property that is true almost everywhere in a given set (i.e., the set of points at which the property is not true is either of measure zero or a subset of a set of measure zero).

1989, Thomas S. Parker, Leon O. Chua, Appendix C: Differential Topology, Practical Numerical Algorithms for Chaotic Systems, Springer, Softcover reprint, page 314,
A property P that refers to members of a set Y is a generic property if the subset of Y whose members exhibit property P contains a dense open subset of Y.
Remark: If P₁ and P₂ are generic properties of a set Y, then so is the property "P₁ and P₂."

2001, Constantin I. Chueshov (translator), Igor Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, [1999, Russian edition] Acta Scientific Publishing House, page 117,
It should be noted that if a property of a dynamical system holds for the parameters from an open and dense set in the corresponding space, then it is frequently said that this property is a generic property.
However, it should be kept in mind that the generic property is not the one that holds almost always. […] Therefore, it should be remembered that generic properties are quite frequently encountered and stay stable during small perturbations of the properties of a system.

We now formulate the first generic property concerning the singularities of a Hamiltonian function.

2010, H. W. Broer, F. Takens, “Chapter 1: Preliminaries of Dynamical Systems Theory”, in B. Hasselblatt, H. W. Broer, F. Takens, editors, Handbook of Dynamical Systems, volume 3, Elsevier (North-Holland), page 24:

(topology, algebraic geometry) A property that is true in some dense open subset of a given set.


artificial idiocy

From the analogy with artificial intelligence.

artificial stupidity
artificial stupidity#English


ideal number

Apparently a calque of German ideale Zahl, a concept in number theory developed by German mathematician Ernst Kummer (1810—1893) and later incorporated by Richard Dedekind into the ring theory concept of ideal.

1861, H. J. Stephen Smith, Report on the Theory of Numbers—Part II, Report of the Thirtieth Meeting of the British Association for the Advancement of Science, June-July 1860, British Association for the Advancement of Science, page 133,
This symbolic representation of ideal numbers is very convenient, and tends to abbreviate many demonstrations.
Every ideal number is a divisor of an actual number, and, indeed, of an infinite number of actual numbers. Also, if the ideal number ϕ(α) be a divisor of the actual number F(α), the quotient ϕ₁(α)=F(α)÷ϕ(α) is always ideal; for if ϕ₁(α) were an actual number, ϕ(α), which is the quotient of F(α) divided by ϕ₁(α), ought also to be an actual number.

1992 [World Scientific], C. Y. Hsiung, Elementary Theory of Numbers, 1995, Allied Publishers, page 214,
One of the most important developments of algebraic number theory is the creation of ideal numbers. All ideal numbers can be classified into ideal number classes. Both the study of ideal numbers and the computation of ideal number classes are important problems of algebraic number theory. The introduction of ideal numbers was meant originally for the number theory, but now ideal numbers have become an important tool in algebra.

1995, L. V. Kuz'min, Ideal number, entry in M. Hazewinkel (editor) Encyclopaedia of Mathematics: Volume 3, Springer, page 125,
The semi-group D is a free commutative semi-group with identity; its free generators are called prime ideal numbers. In modern terminology, ideal numbers are known as integral divisors of A.

(number theory) An algebraic integer that represents an ideal in the ring of integers of a number field.

Ernst Kummer
Richard Dedekind
ideal number

ideale Zahl

algebraic number that represents an ideal in the ring of integers


wedge-shaped

en-adj

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

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


quasitrivial

(mathematics, computing theory) Trivial in a restricted sense or to some limited extent.


framing effect

The tendency of people to modify their choices depending on how the choices are presented.


alcohol by volume

A standard measure of the amount of alcohol contained in a given volume of an alcoholic beverage, usually defined as the number of millilitres of pure ethanol present in 100 millilitres of solution at 20 °C (68 °F).


scheme theory

This book develops Grothendieck's scheme theory as a method for studying algebraic geometry.[…]In the preface of EGA [ Éléments de géométrie algébrique ], Grothendieck even claimed that a knowledge of classical algebraic geometry may hinder the reader from studying scheme theory.

1997, Kenji Ueno, translated by Goro Kato, Algebraic Geometry 1: From Algebraic Varieties to Schemes, American Mathematical Society, page x:

There is another approach dealing with more ring theoretical objects, i.e. starting from a nice class of algebras, called schematic algebras, it is possible to define a space, a Zariski topology, points, lines etc..., cf. [VOW1], [AZ], [ATV]. then it becomes possible to develop a scheme theory, completely extending the theory of schemes in classical algebraic geometry.

1999, Freddy Van Oystaeyen, “A Deformation of Projective Schemes”, in Freddy Van Oystaeyen, editor, Commutative Algebra and Algebraic Geometry, Marcel Dekker, page 295:

The main point of scheme theory is to consider C as a covariant functor R#92;mapstoC(R) from the category #92;textit#123;B-ALG#125; (also written as #92;textit#123;ALG#125;#95;#123;#47;#92;mathfrak#123;B

2004, Haruzo Hida, p-Adic Automorphic Forms on Shimura Varieties, Springer, page 68:

locally ringed space
locally ringed space

(algebraic geometry) The branch of mathematics that concerns schemes (locally ringed spaces admitting coverings by open sets, each isomorphic to the spectrum of some commutative ring).


kemple

His Honour, ye see, being under hiding in thae sair times--the mair's the pity--he lies a' day, and whiles a' night, in the cove in the dern hag; but though it's a bieldy eneugh bit, and the auld gudeman o' Corse-Cleugh has panged it wi' a kemple o' strae amaist, yet when the country's quiet, and the night very cauld, his Honour whiles creeps doun here to get a warm at the ingle and a sleep amang the blankets, and gangs awa in the morning.

1814, Sir Walter Scott, Waverley, Or 'Tis Sixty Years Hence, Complete:

Hence the money acquired from the sale of the three kemples, at the average of the above prices, will purchase five cubic yards of this very efficient manure.

1829, George Robertson, Rural Recollections; or, the progress of improvement in agriculture and rural affairs, page 209:

From a crop of wheat, of 40 bushels to the acre, or of 2600 lb., at 65 lb. per bushel, the straw will weigh 9 kemples of 440 lb. each, or 3960 lb., affording just 1/2 more weight of straw than of grain.

1844, Henry Stephens, The Book of the Farm, Detailing the Labours of the Farmer, Farm-Steward, Ploughman, Shepherd, Hedger, Cattle-man, Field-worker, and Dairy-Maid:

(Scotland) The size of a standard load of straw, approximately 450 pounds.


arithmetic average

en-noun

arithmetic average (plural arithmetic averages)


prime ideal

By analogy with the notion of prime number in number theory.

Given a prime number p, there is only a finite number of prime ideals #92;mathfrak#123;p#125; in #92;mathfrak#123;o#125; such that #92;mathfrak#123;p#125;#92;capJ#61;p (they are the prime ideals of #92;mathfrak#123;o#125;p).

1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39:

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189,
In the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o on the basis of Axiom II; thus, in that section there are no lower prime ideals but o. Since every ideal a ne o is divisible by a prime ideal distinct from o (proof: from among all the divisors of a distinct from o choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o.

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, 2nd edition, Cambridge University Press, page 47:

(algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.


Pythagorean triangle

Pythagorean triangle (plural Pythagorean triangles)

(mathematics) A triangle whose sides form a Pythagorean triple, such as 3, 4, 5.


projective variety

projective variety (plural projective varieties)

Varieties are sometimes called closed sets and some authors call an open subset of a projective variety a quasiprojective variety. The latter term is in an attempt to unify the concept of affine and projective variety.

2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, Springer, page 724:

A closed subset V#92;subset#92;mathbb#123;C#125;Pⁿ is called a (complex) projective variety if, locally, V is defined by a set of complex polynomial equations. Outside of its singular locus, that is, away from the subset where the defining equations do not have maximal rank, the projective variety is a complex submanifold of #92;mathbb#123;C#125;Pⁿ.

2006, Werner Ballmann, Lectures on Kähler Manifolds, European Mathematical Society, page 16:

The central topic of this note is a famous open problem to characterize which projective varieties are Kobayashi hyperbolic.

2015, Katsutoshi Yamanoi, “Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory”, in Takushiro Ochiai, Toshiki Mabuchi, Yoshiaki Maeda, Junjiro Noguchi, Alan Weinstein, editors, Geometry and Analysis on Manifolds: In Memory of Professor Shoshichi Kobayashi, Springer (Birkhäuser), page 209:

homogeneous polynomial
homogeneous polynomial

(algebraic geometry) A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.


pack-year

(medicine, smoking) A unit for estimating the number of cigarettes a person has smoked previously; it is calculated by multiplying the average number of packs (each containing 20 cigarettes) the person smokes per day by the number of years the person has smoked.

paquet-année

unit for measuring the number of cigarettes a person has smoked previously


subformation

(mathematics) A formation that is a subset of a larger formation.


unit hyperbola

(geometry) The hyperbola that satisfies the equation: x²-y²=1


Euler's polyhedron formula

(mathematics) A mathematical formula relating the vertices, edges and faces of a polyhedron: V-E+F=2


Wilson score interval

Wilson score interval (plural Wilson score intervals)

Developed by Edwin Bidwell Wilson (1927).

(statistics) An improvement over the normal binomial approximation interval in that the actual coverage probability is closer to the nominal value.


golden triangle

golden triangle (plural golden triangles)

(geometry) An isosceles triangle whose duplicated side is in the golden ratio to the distinct side.


residual sum of squares

residual sum of squares (plural residual sums of squares)

(statistics) The sum of the squared residuals between some model and a data set; a measure of how well the model fits with the data.


relative frequency

relative frequency (plural relative frequencies)

(statistics) The number of data points satisfying something divided by the total number of data points.


absolute frequency

absolute frequency (plural absolute frequencies)

(statistics) the number of data points satisfying something


inclusion function

inclusion function (plural inclusion functions)

(mathematics) A function whose domain is a subset of its codomain, and for which all of the elements in its domain are fixed points.


degenerate conic

degenerate conic (plural degenerate conics)

(geometry) A conic section that is reducible as the union of two lines.


polysingularity

(mathematics) The condition of a function or equation that has multiple singularities


motor variable

split-complex number
split-complex number

(mathematics) A variable that can be a split-complex number.


wolog

without loss of generality
without loss of generality#English

(mathematics) Abbreviation of without loss of generality.


light hour

(astronomy, physics) A unit of length equal to the distance light travels in one hour


Kleene closure

Named in honor of Stephen Cole Kleene (1909–1994), an American mathematician. The “closure” part comes from the fact that a Kleene closure is closed with respect to concatenation; cf. free monoid.

(mathematics, computer science) The set of all strings of finite length made up of elements of a given set. (Then the Kleene closure is said to be of that given set. For a given set S, its Kleene closure may be denoted as S^*. The Kleene closure includes a string of zero length. Strings are equivalent to ordered tuples but written without the parentheses and commas.)


free monoid

(algebra) A monoid whose underlying set is the Kleene closure of some set of generators, and whose operator is concatenation.


group object

If H is another group object in C, a morphism f#58;G#92;rightarrowH is a morphism of group objects if it commutes with the three structure morphisms; as is standard for sets and true generally (again by Lemma 7.7), it is enough to check #92;mu. Thus we form the category #92;textit#123;Gp#125;(C) of all group objects in C; one important example is #92;textit#123;Gp#125;(#92;textit#123;Ho#125;).

1995, J. Michael Boardman, “Chapter 14: Stable Operations in Generalized Cohomology”, in I.M. James, editor, Handbook of Algebraic Topology, Elsevier (North-Holland), page 617:

2005, Angelo Vistoli, Part 1: Grothendieck typologies, fibered categories, and descent theory, Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental Algebraic Geometry: Grothendieck's FGA Explained, American Mathematical Society, page 20,
The identity is obviously a homomorphism from a group object to itself. Furthermore, the composite of homomorphisms of group objects is still a homomorphism; thus, group objects in a fixed category form a category, which we denote by Grp(C).

(category theory) Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element.


quotient ring

The third section covers a construct similar to the ring S°R — the maximal quotient ring, which exists for any ring. (When R is nonsingular, the maximal quotient ring is exactly S°R.) Finally, Section D provides an answer to the question of which right and left nonsingular rings have coinciding maximal right and left quotient rings.

1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39:

On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.

2006, Peter A. Linnell, “Noncommutative localization in group rings”, in Andrew Ranicki, editor, Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42:

2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain.

2012, Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, translated by Svetla Nikova, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10:

(algebra, ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R.

anneau quotient

ring whose elements are the cosets of an ideal


fractional ideal

fractional ideal (plural fractional ideals)

Products of fractional ideals are again fractional ideals, since if A#92;alpha#92;subseteqR and B#92;beta#92;subseteqR, then (AB)(#92;alpha#92;beta)#92;subseteqR.

1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476:

Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group.

2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10:

2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106,
A fractional ideal of F is a finitely generated 𝒪_F-submodule of F. Fractional ideals form a group together with the ideal multiplication. The neutral element is 𝒪_F and the inverse of a fractional ideal a⊂F is
a⁻¹=x∈F;xa⊂𝒪_F.
[…] Two fractional ideals a,b are called equivalent, if there is a r∈F such that a=rb.

(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.

R-submodule of Frac(R) such that for some nonzero r∈R


field of fractions

field of fractions (plural fields of fractions)

1971 [Wadsworth Publishing], Allan Clark, Elements of Abstract Algebra, 1984, Dover, page 175,
The general construction of the field of fractions ℚ_R out of R is an exact parallel of the construction of the field of rational numbers ℚ out of the ring of integers ℤ.

1989, Nicolas Bourbaki, Commutative Algebra: Chapters 1-7, [1985, Éléments de Mathématique Algèbre Commutative, 1-4 et 5-7, Masson], Springer, page 535,
In this no., A and B denote two integrally closed Noetherian domains such that A ⊂ B and B is a finitely generated A-module and K and L the fields of fractions of A and B respectively.

We denote by #92;mathcal#123;A#125;(#92;mathbb#123;K#125;) the #92;mathbb#123;K#125;-algebra of entire functions in #92;mathbb#123;K#125; i.e. the set of power series with coefficients in #92;mathbb#123;K#125; converging in all #92;mathbb#123;K#125; and we denote by #92;mathcal#123;M#125;(#92;mathbb#123;K#125;) the field of meromorphic functions in #92;mathbb#123;K#125;, i.e. the field of fractions of #92;mathcal#123;A#125;(#92;mathbb#123;K#125;).

2013, Jean-Paul Bézivin, Kamal Boussaf, Alain Escassut, “Some old and new results on the zeros of the derivative of a p-adic meromorphic function”, in Khodr Shamseddine, editor, Advances in Ultrametric Analysis: 12th International Conference on p-adic Functional Analysis, American Mathematical Society, page 23:

(algebra, ring theory) The smallest field in which a given ring can be embedded.


factor ideal

1965 [Prentice-Hall], Seth Warner, Modern Algebra, 1990, Dover, page 651,
Consequently, we may speak of the sequence of invariant factor ideals of a finitely generated unitary module over a principal ideal domain A.

1985 [W. H. Freeman and Company], Nathan Jacobson, Basic Algebra I, 2nd Edition, Dover, 2009, page 193,
We shall now call the sequence of order ideals, ann z₁, ann z₂, ..., whose uniqueness has just been proved, the invariant factor ideals of the module M.

Recall that a factor ideal I of a half-shell A comes from a factor congruence θ of A where I = 0/θ. Factor ideals and congruences will be needed whenever we want to show that we have embedded a half-shell into a Baer–Stone half-shell.[…]
2.1 Definition. Remember that a half-shell A is called a Baer–Stone half-shell if the right and left annihilators of every single element of A are factor ideals. It is called finitely Baer-Stone if the annihilators on both sides of any finite subset of A are factor ideals.

2012, Arthur Knoebel, Sheaves of Algebras over Boolean Spaces, Springer (Birkhäeser), page 246:

(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R).


Pythagorean quadruple

Pythagorean quadruple (plural Pythagorean quadruples)

(mathematics) A set of three positive integers a, b, c, and d, for which a²+b²+c²=d²


geopbyte

The term was in use on English-language websites by 2009 (see quotation below). Before that, it had begun to appear on Spanish-language sites by 2007. The earliest known use is in a February 2006 edit to Spanish Wikipedia's "Byte" article by an anonymous contributor, whose immediately preceding edits to another article make it clear that the term was inspired by Korean 겁 (劫, geop), presumably in the sense "kalpa" (an extremely long span of time in Hindu and Buddhist chronology).

I don't remember where I found this:[…]1000 Brontobytes = 1 Geopbyte[…]Geopbyte: Not sure why this term was created. It's doubtful that anyone alive today will ever see a Geopbyte hard drive.

2009 June 27, NightFire [comment author], “WinXP Overhaul Guide: How to Make It Look Like Vista, Run Like Lightning, and Stay Productive, Fast & Smooth”, in Instructables, comment:

Geopbyte: a geopbyte is 1,024 brontobytes, which is too much data to even comprehend.

2014, John G. Iannarelli, Michael O'Shaughnessy, Information Governance and Security Protecting and Managing Your Company's Proprietary Information, page 35:

Any substantial company that doesn't take advantage of its own geopbytes of available data is in danger of eventual extinction.

2015, Jenny Dearborn, Data Driven, page 41:

Chapter 16, “Healthcare Analytics”: Covering data analytics/big data/population management; effectively managing the geopbytes of data out there lurking in our electronic records […]

2015, Patricia Sengstack, Charles Boicey, Mastering Informatics: A Healthcare Handbook for Success, page 18:

You cannot wrap your head around how big a geopbyte is.

(computing, nonstandard) Roughly 10³⁰ bytes; 1024 brontobytes.


factor out

en-verb

factor out (third-person singular simple present factors out, present participle factoring out, simple past and past participle factored out)

factors out (present singular third-person)
factoring out (participle present)
factored out (participle past)
factored out (past)

(transitive, algebra) To isolate a common factor from an expression.

ottaa yhteinen tekijä

(algebra) to isolate a common factor from an expression


group scheme

It is instructive to give a few explicit examples of such^([affine]) group schemes:

1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6:

We will tend to use boldface characters, such as #92;mathbf#123;G#125;, for algebraic groups over k, as also for group schemes over #92;mathfrak#123;X#125; that are obtained from groups over k. A quintessential example is #92;mathbf#123;G#125;#95;#92;mathfrak#123;X#125;#61;#92;mathbf#123;G#125;#92;times#95;#123;k#125;#92;mathfrak#123;X#125;. For arbitrary group schemes, or more generally group functors, over #92;mathfrak#123;X#125; we shall tend to use german characters, such as #92;mathfrak#123;G#125;.

2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11:

Every finite flat group scheme of order p over S is of the form Gᴸ#95;#123;a,b#125; for some triple (L,a,b), and G#123;L#125;#95;#123;a,b#125; is isomorphic to G#123;L#95;1#125;#95;#123;a#95;1,b#95;1#125; if and only if there exists an isomorphism from L to L#95;1 carrying a to a#95;1 and b to b#95;1.

2014, J. S. Milne, “The Work of John Tate”, in Helge Holden, Ragni Piene, editors, The Abel Prize 2008-2012, Springer,, page 303:

(category theory, scheme theory) A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group.


group functor

In order to be able to efficiently state the main results of this chapter (in §4) we formulate here a group-functor version of the R-nilpotent tower lemma (Ch.III, 6.4).

1972, A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations, Springer, page 106:

A direct product of k–group functors is again a k–group functor; so is a fibre product if the morphisms used in its construction are homomorphisms of k–group functors.

2003, Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd edition, American Mathematical Society, page 20:

By a group functor we mean a functor from small k-algebras to groups. A homomorphism of group functors is a natural transformation. A subgroup functor of a group functor G is a subfunctor H such that H(R) is a subgroup of G(R) for all R; it is normal if H(R) is normal in G(R) for all R.

2017, J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, page 118:

(category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.


aparametric

(mathematics) Not derived from parameters


simple ring

Theorem 2. A semi-simple ring #92;mathfrak#123;A#125; satisfying the minimum condition can be decomposed in only one way as a direct sum of ideals which are simple rings.

1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page 43:

By theorem 7.7.1 any field is a simple ring.

1969, Frederick Michael Hall, An Introduction to Abstract Algebra, volume 2, Cambridge University Press, page 195:

A field is clearly a simple ring. Indeed, a commutative simple ring with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring is Fₙ, the n × n matrix ring over a field F, n > 1.

1994, P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, page 204:

Since M#95;n(D) has no nonzero proper two-sided ideals, it follows from the above discussion that M#95;n(D) is a left simple ring. We shall see that every simple ring is isomorphic to M#95;n(D) for some n, and for some division ring D.

2017, Ramji Lal, Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group Extensions and Schur Multiplier, Springer, page 335:

(algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself).

anneau simple

ring that contains no ideals other than the zero ideal and the ring itself


GRR

gross reproduction rate
gross reproduction rate#English

(demography) Initialism of gross reproduction rate.


simple algebra

1945 [McGraw-Hill], E. T. Bell, 2nd Edition, 1992, Dover, The Development of Mathematics, page 252,
A semi-simple algebra over F is either simple or the direct sum of simple algebras over F.

1989 [W. H Freeman & Company], Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215,
If A is a finite dimensional simple algebra over F, then A≅M_n(Δ) where Δ is a finite dimensional division algebra over Δ.

One proves that if #92;mathcal#123;L#125; is a simple algebra over #92;mathbb#123;R#125;, then L#95;C is either a simple algebra over #92;mathbb#123;C#125;, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over #92;mathbb#123;R#125; reduces to a similar problem over #92;mathbb#123;C#125;.

2005, Igor R. Shafarevich, translated by Miles Reid, Basic Notions of Algebra, Springer, page 198:

(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0).

algebra semplice

algebra that contains no nontrivial proper ideals and whose multiplication operation is not zero


setwise

(mathematics) In terms of a set or sets.


Kelly criterion

Described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956.

(probability theory) A formula used to determine the optimal size of a series of bets in order to maximize the logarithm of wealth.


prime ring

A ring is called a prime ring if the product of nonzero ideals in it remains nonzero. It is obvious that a prime ring is necessarily semi-prime.

1969, Taita Journal of Mathematics, volumes 1-2, page 56:

The ring R is said to be prime if for all nonzero ideals A, B of R we have AB≠0. An ideal P of R is called a prime ideal if R/P is a prime ring. Prime rings and prime ideals are important building blocks in noncommutative ring theory.

1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 223:

The so-called extended centroid of a prime ring, i.e., a field defined as the center of the Martindale ring of quotients, will enable us to extend a part of the theory of central simple algebras to general prime rings.

2014, Matej Brešar, Introduction to Noncommutative Algebra, Springer, page 163:

(algebra, ring theory) Any nonzero ring R such that for any two (two-sided) ideals P and Q in R, the product PQ = 0 (the zero ideal) if and only if P = 0 or Q = 0.

Moreover, the image of φ_A is the smallest subring of A, in the sense that it is contained in any subring of A, and it is called the prime ring of A.

2012, Sebastian Xambo-Descamps, Block Error-Correcting Codes: A Computational Primer, Springer Science & Business Media, page 110:

(algebra, ring theory) Synonym of prime subring

The image of ℤ or ℤ/mℤ, respectively, in R as described in the above proposition obviously consists of all sums n · 1 in R, where n ∈ ℤ. It is also called the prime ring of R. R itself is called a prime ring if it equals its own prime ring. If p is a prime number, then the field ℤ/pℤ is a also called the prime field of characteristic p.

2012, Thomas Becker, Volker Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer Science & Business Media, page 50:

Theorem 3.6.3. If R is a prime ring of characteristic zero then R is isomorphic to ℤ. If R is a prime ring of characteristic n then R is isomorphic to ℤₙ.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra: From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory, JHU Press, page 81:

(algebra, ring theory, uncommon) A ring which is equal to its own prime subring.


inner automorphism

en-noun

inner automorphism (plural inner automorphisms)

(algebra) An automorphism which is a conjugation (by a fixed element).


outer automorphism

outer automorphism (plural outer automorphisms)

(algebra) An automorphism which is not an inner automorphism.


maximal ideal

Denote the minimal prime ideal in (b) by J#95;#92;mathsf#123;max#125;(#92;lambda), the unique maximal ideal of infinitesimal character #92;lambda.

1994, William M. McGovern, Completely Prime Maximal Ideals and Quantization, American Mathematical Society, page 15:

2004, Ayman Badawi, Abstract Algebra Manual: Problems and Solutions, Nova Science, 2nd Edition, page 87,
Let S be the set of all prime ideals of B, and H be the set of all maximal ideals of B.

In particular, a prime ideal #92;mathfrak#123;p#125;#92;subsetA is a closed point of #92;mathrm#123;Spec#125;#92;A if and only if #92;mathfrak#123;p#125; is a maximal ideal.

2013, Igor R. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, 3rd edition, Springer, page 11:

(algebra, ring theory) An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).


superalignment

superalignment (countable and uncountable, plural superalignments)

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

artificial intelligence
artificial intelligence

(artificial intelligence) Aligning of a future superhuman artificial intelligence.


downweight

en-verb

downweight (third-person singular simple present downweights, present participle downweighting, simple past and past participle downweighted)

downweights (present singular third-person)
downweighting (participle present)
downweighted (participle past)
downweighted (past)

(statistics) To give something a lower weight


biomodel

(mathematics, biology) A mathematical model of a biological system (typically an ecosystem)


eigenvectorial

(mathematics) Relating to an eigenvector


annihilator method

en-noun

annihilator method (usually uncountable, plural annihilator methods)

From its use of an annihilator (in this case a differential operator) to render the equation more tractable.

Although the annihilator method is very efficient when applicable, it is limited to equations whose right members R have a constant-coefficient annihilator.

1969, Tom Apostol, Calculus, Volume 2: Multi-variable calculus and linear algebra with applications to differential equations and probability, page 166:

The annihilator method is just one convenient method to select the form of solution to try when applying the method of undetermined coefficients.

1997, Robert E. O'Malley, Jr, Thinking about Ordinary Differential Equations, Cambridge University Press, page 58:

Sometimes, instead of using the variation of parameters technique, it is easier to use the so-called annihilator method (also called the method of undetermined coefficients). We now discuss this annihilator method for solving equations of the form
(3.48) y#123;#92;Delta#92;Delta#125;#43;#92;alphay#92;Delta#43;#92;betay#61;f(t),
where #92;alpha and #92;beta are constants and f is a function that can be annihilated as defined below.

2001, Martin Bohner, Allan Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Springer, page 116:

Many ways exist to solve the particular solution. These include the method of undetermined coefficients, variation of parameters, annihilator methods, Laplace transform methods, and numerical methods.

2012, S. Graham Kelly, Mechanical Vibrations: Theory and Applications, Cengage Learning, page 206:

ordinary differential equation
ordinary differential equation

(mathematical analysis) A procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations.


sequency

The execution of a sequence may involve some or all of the three stages; if only amplitude gain is to be corrected and not sequency and target specification, then correction can be rapid enough to change an arpeggio-like run in “mid flight.”

2014, R. A.M. Gregson, Time Series in Psychology, →ISBN:

From this, we see that frequency can be regarded as a special measure of sequency applicable to sinusoidal waveforms only.

2014 -, Anish Deb, Suchismita Ghosh, Power Electronic Systems, →ISBN, page 16:

Spatial domain processing methods include all three types, but frequency domain operations, by nature of the frequency (and sequency) transforms, are global operations.

2016, Scott E Umbaugh, Digital Image Processing and Analysis, →ISBN:

Walsh functions with different sequencies are orthogonal, and cal and sal functions of the same sequency are orthogonal but differ only by a time offset.

2017, A. Richard Thompson, James M. Moran, George W. Swenson Jr., Interferometry and Synthesis in Radio Astronomy, →ISBN, page 293:

(mathematics) Half the number of zero crossings in the time base of a Walsh function.


primary ideal

The prime and primary ideals play roles which are (very roughly) similar to those played by prime numbers and by prime.power numbers in elementary arithmetic.

1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10:

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, page 189,
Thus all higher primary ideals are symbolic powers of higher prime ideals.
Prüfer has called the ideals a with the property a* = a v-ideals. The integral v-ideals are just those in whose primary ideal decomposition only higher primary ideals occur.

A primary ideal has a prime ideal as radical, so its corresponding algebraic set is irreducible. Primary ideals can, however, have multiplicity, so they give a finer description of the solution set.

1997, Ralf Fröberg, An Introduction to Gröbner Bases, John Wiley & Sons, page 71:

(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.


ground set

(mathematics) The underlying set of mathematical structures such as posets and matroids.


pseudocoronal

(astronomy, mathematics, organic chemistry) Relating to a pseudocorona


de Bruijn index

de Bruijn index (plural de Bruijn indexes)

Invented by the Dutch mathematician Nicolaas Govert de Bruijn.

(mathematics) A notation for representing terms in the lambda calculus with the purpose of eliminating the names of the variables from the notation.


Church-Rosser theorem

Introduced by Alonzo Church and J. Barkley-Rosser in a 1936 paper.

(mathematics, computing theory) A theorem stating that, when applying reduction rules to terms in the lambda calculus, the ordering in which the reductions are chosen makes no difference to the eventual result.


subiteration

(mathematics, computing) A subsidiary iteration


subsethood

(mathematics) The condition of being a subset


vector bundle

Let E and F be two vector bundles on a compact algebraic set X.

1995, Mauro Beltrametti, Andrew John Sommese, The Adjunction Theory of Complex Projective Varieties, Walter de Gruyter, page 8:

2012, Igor Burban, Bernd Kreussler, Vector Bundles on Degenerations of Elliptic Curves and Yang-Baxter Equations, American Mathematical Society, page 5,
This solutions actually depend on the choice of a trivialisation ξ of the universal family 𝒫(n,d) of stable vector bundles of rank n and degree d.

Definition 6.1.5 A non-zero vector bundle #92;mathcal#123;F#125; on X is said to be decomposable if it can be written as #92;mathcal#123;F#125;#95;1#92;oplus#92;mathcal#123;F#125;#95;2 with #92;mathcal#123;F#125;#95;1 and #92;mathcal#123;F#125;#95;2 non-zero vector bundles. Otherwise it is said to be indecomposable.

2017, David J. Benson, Representations of Elementary Abelian p-Groups and Vector Bundles, Cambridge University Press, page 130:

(topology, category theory) A fiber bundle for which the fiber is a vector space.


Moore graph

Named in 1960 by Hoffman and Singleton, after Edward F. Moore, who posed the question of describing and classifying these graphs.

(graph theory) A regular graph of degree d and diameter k whose number of vertices equals the upper bound 1+d∑ᵢ₌₀ᵏ⁻¹(d-1)ⁱ.


tangent bundle

(topology) A fiber bundle for which the base space is a differentiable manifold and each fiber over a point of that manifold is the tangent space of that point.


hedonium

Suppose that we agreed to allow almost the entire accessible universe to be converted into hedonium—everything except a small preserve, say the Milky Way, which would be set aside to accommodate our own needs.

2014, Nick Bostrom, Superintelligence, page 219:

It’s because we intuitively or pre-theoretically think that we shouldn’t give all our cookies to the utility monster or kill ourselves to tile the solar system with hedonium that we reject the straightforward extension of utilitarian happiness-maximizing theory to such cases and reach for a different solution.

2015 November 17, Eric Schwitzgebel, Mara Garza, “A Defense of the Rights of Artificial Intelligences”, in Midwest Studies in Philosophy:

Suppose the AI goes on to convert all matter in the accessible universe to hedonium, i.e., to rearrange matter so as to produce the greatest amount of pleasurable experience per unit mass.

2016, Olle Häggström, Here Be Dragons, page 122:

The Machine Intelligence Research Institute's Buck Shlegeris recently recounted a conversation where “someone said that after the Singularity, if there was a magic button that would turn all of humanity into homogenous goop optimized for happiness (aka hedonium), they'd press it.[…]”

2020, Brian Christian, “Uncertainty”, in The Alignment Problem, New York: W.W. Norton & Company, →ISBN:

(philosophy of artificial intelligence, rare) Matter arranged in a way that produces pleasure or happiness as efficiently as possible, as might be encouraged by philosophical hedonism.


differential structure

differential structure (plural differential structures)

First let (M,#92;tau) be a topological space. The sheaf #92;mathfrak#123;G#125; of real continuous functions on (M,#92;tau) is said to be a differential structure on M if for any open set U#92;in#92;tau, any functions f#95;i#92;in#92;mathfrak#123;G#125;(U), and any w#92;inC#92;infty(#92;mathbb#123;R#125;ⁿ), the superposition w#92;circ(f#95;1,#92;dotsf#95;n)#92;in#92;mathfrak#123;G#125;(U).

2002, R.W. Carroll, Calculus Revisited, Springer, pages 12–37:

It is important to emphasise that, among the various choices for #92;lambdaⁱ#95;#123;jk#125; and A#123;i#92;a#125;#95;#123;#92;j#92;b#125;, some are intrinsic to the differential structure of the manifold #92;mathcal#123;M#125;. In other words, among all the operators of D-differentiation, some arise from the differential structure of #92;mathcal#123;M#125;.[…]On the other hand, there exist operators of D-differentiation that do not follow from the differential structure of #92;mathcal#123;M#125;.

2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29:

This chapter is concerned with differentiable measures on general measurable spaces and on measurable spaces equipped with certain differential structures enabling us to consider differentiations along vector fields.

2010, Vladimir Igorevich Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, page 369:

Given a Hausdorff topological space M with differential structures #92;mathcal#123;A#125;#95;1 and #92;mathcal#123;A#125;#95;2 (these being maximal smooth atlases), we say that #92;mathcal#123;A#125;#95;1 and #92;mathcal#123;A#125;#95;1 are equivalent if there is a diffeomorphism #92;phi#58;(M,#92;mathcal#123;A#125;#95;1)#92;rightarrow(M,#92;mathcal#123;A#125;#95;2) from M with the first differential structure to M with the second differential structure. Note that #92;phi need not be the identity function.

2015, Stephen Bruce Sontz, Principal Bundles: The Classical Case, Springer, page 12:

(topology) A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it.


top-hat transform

top-hat transform (plural top-hat transforms)

So called because a graph of the function has a shape resembling a top hat.

(mathematics, signal processing) A particular operation that extracts small elements and details from images.


bottom-hat transform

bottom-hat transform (plural bottom-hat transforms)

black top-hat transform
black top-hat transform#English

(mathematics, signal processing) Synonym of black top-hat transform


black top-hat transform

black top-hat transform (plural black top-hat transforms)

(mathematics, signal processing) A form of the top-hat transform that finds the elements in the input image that are darker than their surroundings.


white top-hat transform

white top-hat transform (plural white top-hat transforms)

(mathematics, signal processing) A form of the top-hat transform that finds the elements in the input image that are brighter than their surroundings.


antiextensive

(mathematics) Of an operation in morphology: such that A∘B⊆A.


hit-or-miss transform

hit-or-miss transform (plural hit-or-miss transforms)

(mathematics) In morphology, an operation that detects a given configuration (or pattern) in a binary image, using the erosion operator and a pair of disjoint structuring elements.


residualizing

(mathematics) That reduces (something) to a residue


orthogonal matrix

orthogonal matrix (plural orthogonal matrices)

(linear algebra) A square matrix whose columns, considered as vectors, are orthonormal to each other. This implies that the transpose of such a matrix is also its inverse.


conjugacy class

conjugacy class (plural conjugacy classes)

If a group acts on itself through conjugation, then its orbits are its conjugacy classes and its stabilizer subgroups are its centralizers.

equivalence relation
equivalence relation

(algebra) A subset of a group which is an equivalence class in the quotient set of the group divided by conjugation as equivalence relation.


diskcyclicity

(mathematics, rare) The property of being diskcyclic.


Ludolphian number

From Ludolph van Ceulen, a German mathematician who improved on previous approximations of pi.

Pi... came to be called the Ludolphian number, after Ludolph van Ceulen, a German mathematician who approximated it to thirty-five decimal places... a calculation that took Ludolph most of his life to accomplish, and gave him such satisfaction that he had the digits engraved on his tombstone... the stone vanished, possibly to be turned into a sidewalk slab. Somewhere in Leiden, people may be walking over Ludolph's digits.

1992 March 2, Richard Preston, “The Mountains of Pi”, in The New Yorker:

(mathematics, chiefly historical) Synonym of pi, particularly the 35-decimal approximation discovered by Ludolph van Ceulen


Planck pressure

Planck pressure (plural Planck pressures)

(physics) The unit of pressure in the system of Planck units, equal to one Planck force in one Planck area (or one Planck energy in one Planck volume): p_p=(c⁷)/(hbar G²)≈4.633×10¹¹³ mbox Pa.


hyperbolic angle

hyperbolic angle (plural hyperbolic angles)

(geometry) An angle that describes a position along a hyperbola.


rectangular hyperbola

rectangular hyperbola (plural rectangular hyperbolas)

(mathematics) A hyperbola where the two asymptotes intersect at a right angle.


teraliter

(US) A unit of volume equivalent to 10¹² liters. Symbol: Tl


hyperbolic sector

hyperbolic sector (plural hyperbolic sectors)

rectangular hyperbola
rectangular hyperbola

(geometry) An area on a plane bounded by two rays from the origin and a rectangular hyperbola.


apomorphism

(mathematics) The dual of a paramorphism


disjoint union topology

disjoint union topology (plural disjoint union topologies)

The reader will recall that the morphism and object sets of A'#61;#92;coprod#95;#123;p,q#92;inP#125;A#95;c(q,p) were not necessarily given the disjoint union topologies.

1978, Maria Louise Shea Terrell, Topological 2-categories and Principal Topological Categories, University of Virginia, page 19:

Given any indexed collection of topological spaces #92;left#92;#123;X#95;#92;alpha#92;right#92;#125;#95;#123;#92;alpha#92;inA#125;, we define the disjoint union topology on #92;textstyle#92;coprod#95;#123;#92;alpha#92;inA#125;X#95;#92;alpha by declaring a subset of #92;textstyle#92;coprod#95;#123;#92;alpha#92;inA#125;X#95;#92;alpha to be open if and only if its intersection with each X#95;#92;alpha is open in X#95;#92;alpha.

2003, John M. Lee, Introduction to Smooth Manifolds, Springer, page 548:

And the set Gⁿ(#92;mathbb#123;F#125;) of all sub vector spaces of #92;mathbb#123;F#125;ⁿ, i.e. the disjoint union of the Gⁿ#95;k(#92;mathbb#123;F#125;), gets the disjoint union topology.

2005, Friedrich Ischebeck, Ravi A. Rao, Ideals and Reality: Projective Modules and Number of Generators of Ideals, Springer, page 118:

(topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union.


Fibonacci spiral

Fibonacci spiral (plural Fibonacci spirals)

(geometry) An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in a Fibonacci tiling; a spiral formed by connecting quarter-circles whose radii are Fibonacci number multiples of the initial radius.


algebraic group

algebraic group (plural algebraic groups)

The purpose of this article is to discuss various results concerning the subgroups of simple algebraic groups G and of the corresponding finite groups of Lie type G^F (where F is a Frobenius morphism).

1998, Martin W. Liebeck, “Introduction to the subgroup structure of algebraic groups”, in Roger William Carter, Meinolf Geck, H. K. Moffatt, editors, Representations of Reductive Groups, Cambridge University Press, page 129:

We begin by showing how to make the quotient of an affine algebraic group by a normal algebraic subgroup into an affine algebraic group, and we determine the structure of commutative affine algebraic groups.

1998, Roe Goodman, Nolan R. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press, page 464:

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page 75,
We point out that some properties of an algebraic group are more easily read in its Lie algebra. An important property of an algebraic group is its solvability.

(algebraic geometry) A group that is an algebraic variety.


linear algebraic group

linear algebraic group (plural linear algebraic groups)

Geometric invariant theory arises in an attempt to construct a quotient of an algebraic variety X by an algebraic action of a linear algebraic group G.

2003, Igor Dolgachev, Lectures on Invariant Theory, Cambridge University Press, page xiii:

The differential Galois group of a homogeneous linear differential equation has a structure of linear algebraic group; hence it is endowed, in particular, with the Zariski topology.[…]Kolchin used the differential algebra developed by Ritt and also built the foundations of the theory of linear algebraic groups.

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page xi:

2015, Willem A. de Graaf, Orbit Closures of Linear Algebraic Groups, Jaime Gutierrez, Josef Schicho, Martin Weimann (editors), Computer Algebra and Polynomials: Applications of Algebra and Number Theory, Springer, LNCS: 8942, page 76,
Actions of linear algebraic groups appear in many contexts. […] Throughout we assume that the base field is algebraically closed and of characteristic 0, as many constructions that we use (e.g., the correspondence between a linear algebraic group and its Lie algebra) only work well in characteristic 0.

general linear group
general linear group

(algebraic geometry, category theory) An algebraic group that is isomorphic to a subgroup of some general linear group.


regular map

Morphism of algebraic varieties
Regular map (graph theory)

We now find out how a regular map acts on the ring of regular functions on a closed set.[…]Moreover, regular maps can be characterised as the maps that take regular functions into regular functions.

1994, Igor R. Shafarevich, translated by Miles Reid, Basic Algebraic Geometry 1, 2nd edition, Springer, page 29:

Since one expects that the canonical projection f#58;X#92;rightarrowX#47;G is a regular map of algebraic varieties and so has closed fibres, all orbits must be closed subsets in the Zariski topology of X.

2017, José F. Fernando, José M. Gamboa, Carlos Ueno, Polynomial, regular and Nash images of Euclidean spaces, Fabrizio Broglia, Françoise Delon, Max Dickmann, Danielle Gondard-Cozette, Victoria Ann Powers (editors), Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 160,
The 1-dimensional semialgebraic set 𝒮:=x>0,xy=1 is the image of the regular map
f: R ²→ R ²,(x,y)↦((xy-1)²+x²,1/((xy-1)²+x²)).

(algebraic geometry) A morphism between algebraic varieties.

1990 [McGraw-Hill], Jay Kappraff, Connections: The Geometric Bridge Between Art and Science, 2001, World Scientific, page 141,
Just as there are only five regular maps on the sphere (or plane), there are only three classes of regular maps that can be created on a torus.

If n is finite, the regular map is a tessellation of the plane by congruent n-sided polygons, m of which meet at each vertex.

2010, Jozef Širáñ, Yan Wang, “Maps with highest level of symmetry that are even more symmetric than other such maps: Regular maps with largest exponent groups”, in Richard A. Brualdi, Samat Hedayat, Hadi Kharaghani, Gholamreza B. Khosrovshahi, Shahriar Shahriari, editors, Combinatorics and Graphs: The Twentieth Anniversary Conference of IPM Combinatorics, American Mathematical Society, page 98:

2013, Roman Nedela, Martin Škoviera, 7.6: Maps, Jonathan L. Gross, Jay Yellen, Ping Zhang (editors), Handbook of Graph Theory, 2nd Edition, CRC Press, page 845,
If M is a regular map of type p,q, then mathit Aut(M)≈Δ(p,q,2)/N for some normal subgroup N⊴Δ(p,q,2). Similar statements hold for the class of orientably regular maps and subgroups of Δ⁺ and Δ⁺(p,q,2).

(graph theory) A symmetric tessellation of a closed surface; a decomposition of a two-dimensional manifold into topological disks such that every flag (incident vertex-edge-face triple) can be transformed into any other flag by a symmetry (i.e., an automorphism) of the decomposition.


Hurewicz theorem

Hurewicz homomorphism
Hurewicz homomorphism

(mathematics) A basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism.


generalized continued fraction

generalized continued fraction (plural generalized continued fractions)

generalized continued fractions (plural)

(mathematics) A continued fraction were the numerators and denominators can assume arbitrary values.


general continued fraction

general continued fraction (plural general continued fractions)

generalized continued fraction
generalized continued fraction

(mathematics) A generalized continued fraction.


simple continued fraction

simple continued fraction (plural simple continued fractions)

(mathematics) A continued fraction were all of the numerators are 1.


regular continued fraction

regular continued fraction (plural regular continued fractions)

simple continued fraction
simple continued fraction

(mathematics) A simple continued fraction.


periodic continued fraction

periodic continued fraction (plural periodic continued fractions)

(mathematics) A continued fraction with a sequence of numerators and denominators that repeat indefinitely


mex

(mathematics) The mex of a subset of a well-ordered set is the smallest value from the whole set that does not belong to the subset.


isovariant

(mathematics) That commutes with the actions of a group on the domain and codomain, and preserves isotropy subgroups of elements that fix points in the domain and codomain.


nonacoptic

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

From non- + acoptic, introduced by the mathematician Branko Grünbaum to clarify terminology in the geometry of polygons and polyhedra. Initially he used the term coptic, but abandoned it in favor of nonacoptic in later writings.

1999, Grünbaum, Branko, "Acoptic polyhedra", Advances in discrete and computational geometry (South Hadley, MA, 1996) (PDF), Contemp. Math., 223, Amer. Math. Soc., Providence, RI, pp. 163–199, doi:10.1090/conm/223/03137, MR 1661382.
See Norgate [N2], where nonacoptic pentahedra are investigated as well.

Branko Grünbaum
small stellated dodecahedron


acoptic

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Introduced by the mathematician Branko Grünbaum to clarify terminology in the geometry of polygons and polyhedra. From Ancient Greek ἀ- (a-, “not”) + κόπτω (kóptō, “to cut”).

affix

1999, Branko Grünbaum, "Acoptic polyhedra", Advances in discrete and computational geometry (South Hadley, MA, 1996) (PDF), Contemp. Math., 223, Amer. Math. Soc., Providence, RI, pp. 163–199, doi:10.1090/conm/223/03137, MR 1661382.

(geometry, uncommon) Not self-intersecting.


spie chart

A pictorial graph where a radar chart is superimposed over a pie chart to compare two sets of data.


finite continued fraction

en-noun

finite continued fraction (plural finite continued fractions)

(mathematics) A continued fraction which terminates after a certain number of steps.


bincount

(statistics) A count of the number of things in each element of an array


binsize

(statistics) The number of things in each element of an array


infinite continued fraction

infinite continued fraction (plural infinite continued fractions)

(mathematics) A continued fraction that goes on forever


underbound

en-verb

underbound (third-person singular simple present underbounds, present participle underbounding, simple past and past participle underbounded)

underbounds (present singular third-person)
underbounding (participle present)
underbounded (participle past)
underbounded (past)

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It will be seen that our computation time underbounds his; and that, in fact, we can give a group for which the difference in computation time is arbitrarily large.

1968, Philip Martin Spira, On the Computational Complexity of Finite Functions, page 28:

(mathematics) To provide a lower bound to.


crossvalidated

(mathematics) assessed via cross-validation


difference set

(combinatorics) A subset of a group such that any element of the group can be expressed as the difference (or the quotient) of two elements in such subset.


Atiyah-Singer index theorem

Proved by Michael Atiyah and Isadore Singer in 1963.

differential geometry
differential geometry

elliptic differential operator
elliptic operator

(differential geometry) A theorem stating that, for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).


Euler's continued fraction formula

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

named-after

(mathematics) A formula connecting a finite sum of products to a finite continued fraction:


Grandi's series

Named after Luigi Guido Grandi (1671-1742), Italian mathematician.

(mathematics) The divergent infinite series ∑ₙ₌₀ ᪲(-1)ⁿ=1-1+1-1+⋯.


Dirichlet eta function

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

mathematical analysis
mathematical analysis

Riemann zeta function
Riemann zeta function

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


Riemann hypothesis

Named after German mathematician Bernhard Riemann (1826–1866), who first formulated and discussed the hypothesis.

The Riemann hypothesis has deep implications about the distribution of prime numbers.

1995, John Corning Carey, On Beurling's Approach to the Reimann Hypothesis, University of California, Berkeley, page 43,
But in the absence of such assumptions, the task of finding functions h∈ℱ₂ for which ‖1+h‖₂ is small is equivalent to proving the Riemann hypothesis, as we will now demonstrate.

2003, Marcus du Sautoy, The Music of the Primes, 2004, HarperCollins Publishers (Harper Perennial), page 10,
A solution of the Riemann Hypothesis will have huge implications for many other mathematical problems.

2010, Samuel J. Patterson, The Riemann Hypothesis – a short history, Gerrit Dijk, Masato Wakayama (editors), Casimir Force, Casimir Operators and the Riemann Hypothesis, Walter de Gruyter, page 30,
The one problem proposed in Riemann's paper which remained unproved, the only one Riemann put forward explicitly as a conjecture, was the Riemann Hypothesis.

2021, Naji Arwashan, The Riemann Hypothesis and the Distribution of Prime Numbers, Nova Science Publishers, page x,
The Riemann Hypothesis is considered by many accounts the single most important and difficult question in math today.

(mathematics, mathematical analysis, number theory) The conjecture that the zeros of the Riemann zeta function exist only at the negative even integers and certain complex numbers whose real part is ½.

hypothèse de Riemann

conjecture about the zeros of the Riemann zeta function


pairwise disjoint

Proposition 4.5. Every permutation is a product of pairwise disjoint cycles, and this decomposition is unique up to the order of the terms.

2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 61:

2009, John M. Franks, A (Terse) Introduction to Lebesgue Integration, American Mathematical Society, page 27,
For example, if we had a collection of pairwise disjoint intervals of length 1/2,1/4,1/8,…1/2ⁿ,…,etc., then we would certainly like to be able to say that the measure of their union we is the sum ∑1/2ⁿ=1 which would not follow from finite additivity.

To show that all #92;Gamma#95;i-translates of F#95;i, are pairwise disjoint, it suffices to show that all #92;Gamma#95;#123;i,0#125;-translates of F#95;i are pairwise disjoint, since then the argument as above will show inductively that the #92;Gamma#95;#123;i,m#125;-translates of F#95;i are pairwise disjoint for all mgt;0.

2015, Su Gao, Stephen C Jackson, Brandon Seward, Group Colorings and Bernoulli Subflows, American Mathematical Society, page 158:

(mathematics, set theory, of a collection of two or more sets) Let A_λ_(λ∈Λ) be any collection of sets indexed by a set Λ. We call the indexed collection pairwise disjoint if for any two distinct indices, λ,μ∈Λ, the sets A_λ and A_μ are disjoint.

mútuament disjunts

such that any two distinct sets are disjoint