principal square root

principal square root (plural principal square roots)

(mathematics) The positive square root of a number; the positive number which, when squared, yields another number


Poincaré map

(mathematics) The intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace (the Poincaré section), transversal to the flow of the system.


Cartan subalgebra

en-noun

Cartan subalgebra (plural Cartan subalgebras)

Introduced by Élie Cartan in his doctoral thesis.

(mathematics) A certain kind of nilpotent subalgebra of a Lie algebra.


Duffing equation

Duffing equation (plural Duffing equations)

differential equation
differential equation

(mathematics) A nonlinear second-order differential equation used to model certain damped and driven oscillators.


effectivize

en-verb

effectivize (third-person singular simple present effectivizes, present participle effectivizing, simple past and past participle effectivized)

effectivizes (present singular third-person)
effectivizing (participle present)
effectivized (participle past)
effectivized (past)

(chiefly mathematics) To make effective.


equidiagonal

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

(geometry) Of a convex quadrilateral: having the two diagonals of equal length.


Goldbach number

en-noun

Goldbach number (plural Goldbach numbers)

Goldbach's conjecture
Goldbach's conjecture

(mathematics) A positive even integer that can be expressed as the sum of two odd primes; involved in Goldbach's conjecture.


Goldbach partition

Goldbach partition (plural Goldbach partitions)

(mathematics) The expression of a given even number as the sum of two primes; involved in Goldbach's conjecture.


Lefschetz fixed-point theorem

the Lefschetz fixed-point theorem (canonical)

Named after Solomon Lefschetz, who first stated it in 1926.

(mathematics) A formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X.


Ford circle

Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.

Every Ford circle is tangent to the horizontal axis y#61;0, and any two Ford circles are either disjoint or meet at a tangent.

There is a unique Ford circle associated with every rational number. Additionally, the axis y#61;0 can be considered a Ford circle with infinite radius, corresponding to the case p#61;1,#92;q#61;0.

Thus the Ford circle [1], drawn tangent to the real axis at #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, and having radius #92;textstylea#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125;, must contain in its interior some points belonging to #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, such as #92;textstyle#92;theta#43;ia#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125; whose imaginary part lies between #92;textstyleq#95;n#123;-2#125; and #92;textstyleq#95;#123;n#43;1#125;#123;-2#125;.

1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:

A Farey fraction #92;textstyle#92;frackc defines a Ford circle #92;textstyle#92;mathcal#123;C#125;(k,c) in #92;textstyle#92;mathbbC. Its center is given by #92;textstylekc#43;i#92;frac 1#123;2c²#125; and its radius is #92;textstyle#92;frac1#123;2c²#125;. Two Ford circles #92;textstyle#92;mathcalC(a,b) and #92;textstyle#92;mathcalC(c,d) are tangent whenever #92;textstylead-bc#61;#92;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #92;textstyleF#95;N.

2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:

Ford circles are a collection of horocycles in #92;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#95;#123;a#47;b#125; and C#95;#123;c#47;d#125; are tangent if and only if #92;vertad-bc#92;vert#61;1.

2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:

(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers).

Ford-Kreis

any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle


snowball sampling

(sociology, statistics) A nonprobability sampling technique where existing study subjects recruit future subjects from among their acquaintances.


Legendre transformation

Legendre transformation (plural Legendre transformations)

Adrien-Marie Legendre
Legendre transformation

Named after Adrien-Marie Legendre (1752–1852), a French mathematician.

(mathematics) Given a function f(x,y,z,...) which is concave up with respect to x (i.e., its second derivative with respect to x is greater than zero), an involutive procedure for replacing x with another variable, say p=∂f/∂x thus yielding another function, say F=F(p,y,z,...). This new function contains all of the information of the original f encoded, as it were, within it so that ∂F/∂p=x and applying a similar transformation to F yields the original f. The formula is: F(p,y,z,...)=p·x(p)-f(x(p),y,z,...) where x must be expressed as a function of p. (Note: The concave upwardness means that ∂f/∂x is monotonically increasing, which means that p as a function of x is invertible, so x should be expressible as a function of p.)


stopping time

(mathematics) A random variable representing the time required for a stochastic process to satisfy a certain condition, which must be determined using only information available up to such time and must occur almost surely in finite time.


Euclid's lemma

Named after ancient Greek mathematician Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Elements.

I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that #92;sqrt 2 is irrational. I said there that if 2 is a factor of a² then a itself must be even. This follows from Euclid's Lemma upon taking p#61;2, the only even prime, and taking b#61;a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing #92;sqrt 2 to be irrational to prove that #92;sqrtp is irrational for any prime p.

1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78:

If a and b are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: 12#92;mid 9#92;cdot 8, but 12#92;nmid 9 and 12#92;nmid 8.

2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179:

In our discussion of Euclid's lemma (Corollary 2.18), we noted that the uniqueness of factorization of integers is a fact that we often take for granted given the way it is introduced in school.

2008, Martin Erickson, Anthony Vazzana, Introduction to Number Theory, Taylor & Francis (Chapman & Hall / CRC Press), page 42:

principal ideal domain
principal ideal domain

(number theory) The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;

The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;
slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
(algebra, by generalisation) the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.


chilogramme

The distillation is continued by putting fire under the retort; and as soon as two chilogrammes have passed over, another ten chilogrammes of fresh alcohol at 40 degrees are introduced into the mixture, drop by drop, regulating as much as possible the quantity introduced, by that which passes into the receiver. This process is to be continued until there have been drawn off 15 chilogrammes of liquid, which will give, on being rectified in the water bath, 8 chilogrammes of pure ether, and some alcohol of an ethereal scent very proper to be made use of over again.

1809, The Repertory of Arts, Manufactures, and Agriculture, volume XV, London, page 118:

It appears, however, from an official Report, obligingly communicated to us by Dr. Kelly, that the actual standard chilogramme has been found to contain only 15433 English grains.

1821, Alexander Tilloch, The Philosophical Magazine and Journal: Comprehending the Various Branches of Science, the Liberal and Fine Arts, Geology, Agriculture, Manufactures, and Commerce, volume LVII, London: Richard and Arthur Taylor, page 421:

A. 20. Signor Lana. / The Sardinian standards have been compared with the metre and the chilogramme.

1840, Extracts of Papers Printed and Manuscript, Laid Before the Commission Appointed to Consider the Steps to Be Taken for Restoration of the Standards of Weight and Measure, and the Subjects Connected Therewith, London: W. Clowes & Sons, page 3:

The Gramme is equal to 15.434 grains Troy, / […] / A chilogramme is 1000 grammes, = 15434.0000 / A myriagramme is 10000 grammes, = 154340.0000 / All the preceding French weights and measures are deduced from some decimal proportion of the metre. Thus, the chilogramme corresponds wit the contents of a cubic vessel of pure water at the lowest temperature, the side of which vessel is the tenth part of the metre, (the decimetre,) and the gramme answers to the like contents of a cubic vessel, the side of which is the hundredth part of the metre, (the centimetre;) for the contents of all cubic vessels are to each other in the triplicate ratio of their sides.

1840, The Merchants’ Magazine, and Commercial Review, volume I, New York, page 258:

If a chilogramme (kilogramme) of water fall through 772 feet, its temperature will rise one degree Fahrenheit. To discover, then, from what greater height a chilogramme of water must fall to raise its temperature to one degree centigrade, the following proportion will suffice:—100 : 180 : : 772 : 1389.6. But 1389 feet are equal to 423 metres; thus the mechanical equivalent of heat, according to the French system, is 423 metres, by falling through which a chilogramme of water rises through one degree centigrade. […] In the ascending scale a thousand grammes is called a kilogramme or chilogramme. The chilogramme is something less than 2+¹⁄₄ lb. imperial, or consists of 15432.3 English grains. The chilogramme is the commercial unit of weight in France, as the gramme is the basis of scientific weights.

1867, Physiology at the Farm in Aid of Rearing and Feeding the Live Stock, Edinburgh and London: William Blackwood and Sons, pages 599, 614:

Six weights in she-goat forms with a ring on their back; they were used to weigh the meat of this animal. The largest is worth our chilogramme. (Pompeii).

1876, V. J., transl., A New General Hand-book to the National Museum in Naples According to Its Latest Arrangement with an Appendix Relative to Pompeii and a Vade Mecum to the Certosa of S. Martino, Naples: Cav. Gennaro De Angelis and son, translation of original by Achilles Migliozzi, page 281:

We did this by multiplying together the three measurements (L., B., and T.) in millimetres of the print and calculating the value of this parallelopiped, in relation to a chilogramme of the body weight.

This is the aes rude called also aes infectum (unwrought bronze); many of these pieces are preserved in our museums varying in weight from a few grammes to a chilogramme and more. These are the pieces called raudera, rauduscula, rudera.

1900, The Numismatic Circular, volume 8 or 9:


absolute geometry

absolute geometry (plural absolute geometries)

From 1832; introduced by Hungarian mathematician János Bolyai (1802—1860).

絶対幾何学

1993 [Princeton University Press], Donald M. Davis, The Nature and Power of Mathematics, 2004, Dover, page 85,
Recall that absolute geometry is the set of statements that can be deduced from Euclid's first four postulates. Then existence of parallel lines is certainly a theorem of absolute geometry, while the question being addressed by most of the mathematicians discussed in this section is whether uniqueness of parallels is also a theorem of absolute geometry.

(geometry) The single (up to logical equivalence) geometry whose axiomatic system is equivalent to that of Euclidean geometry without the parallel postulate or any alternative.

1970, J. F. Rigby, Axioms for Absolute Geometry, III, Canadian Journal of Mathematics, Vol. XXII, No. 1, Canadian Mathematical Congress, University of Toronto Press, page 185,
A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper.

(geometry) Any geometry whose axiomatic system extends that of absolute geometry (in the singular sense) and neither assumes nor contradicts the parallel postulate.


Harshad number

Coined by Indian recreational mathematician D. R. Kaprekar, from Sanskrit हर्षद (harṣada, “giving joy”).

coin

The sum of digits of 24 is 2 + 4 = 6, and 24 is divisible by 6, so 24 is a Harshad number.

(mathematics) A positive integer which is divisible by the sum of all of its digits.


pseudocomplex

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

(mathematics) Relating to the square root of ±1


tangentoid

(mathematics) A graph of the tan (tangent) function


birational geometry

Birational geometry amounts to the study of mappings given by rational functions rather than polynomials.

1999, Yongbin Ruan, Surgery, Quantum Cohomology and Birational Geometry, Ya. Eliashberg, D. Fuchs, T. Ratiu, Alan Weinstein (editors), Northern California Symplectic Geometry Seminar, American Mathematical Society, Translations, Series 2, Volume 196, page 183,
Recently, some amazing relations between quantum cohomology and birational geometry have been discovered.

2008, Paltin Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, Ciro Ciliberto, et al., Projective Varieties with Unexpected Properties, Walter de Gruyter, page 317,
This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ℙⁿ (quasi-lines).

2009, Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Springer (Birkhäuser), Progress in Mathematics 276, page 233,
In this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform.

(algebraic geometry) A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets.


fundamental theorem of arithmetic

en-noun

fundamental theorem of arithmetic (uncountable)

(number theory) the theorem that states that every integer greater than one is uniquely expressible as a product of prime numbers, which is called its prime factorization


complex geometry

Included in complex geometry are the application of transcendental methods to algebraic geometry, as well as the more geometric aspects of complex analysis.

I tried to teach the subject in a way that would enable the students to follow recent developments in complex geometry and in particular some of the exciting aspects of the interplay between complex geometry and string theory.

2004, Daniel Huybrechts, Complex Geometry: An Introduction, Springer, page vi:

2004, Yum-Tong-Siu, Hyperbolicity in Complex Geometry, Olav Arnfinn Lauda, Ragni Piene (editors), The Legacy of Niels Henrik Abel, Springer, page 543,
The hyperbolicity problem in complex geometry studies the conditions for a given complex manifold X to be hyperbolic.

Professor Phillip Griffiths is a giant in complex geometry.

2014, Robert S. Doran, Greg Friedman, Scott Nollet, Hodge Theory, Complex Geometry, and Representation Theory, American Mathematical Society, page ix:

(mathematics) The study of complex manifolds, complex algebraic varieties, and functions of several complex variables.


combinatorial geometry

combinatorial geometry (countable and uncountable, plural combinatorial geometries)

The generation of combinatorial types of point configurations and hyperplane arrangements point configurations and hyperplane arrangements has long been an outstanding problem of combinatorial geometry.

2003, Lukas Finschi, Komei Fukuda, “Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements”, in Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, editors, Discrete and Computational Geometry, Springer,, page 425:

The following problem of Erdős [Er46] is possibly the best known (and simplest to explain) problem in combinatorial geometry. How often can the same distance occur among n points in the plane?

2006, Peter Brass, William O. J. Moser, János Pach, Research Problems in Discrete Geometry, Springer, page 183:

2012, Mohammed Mostefa Mesmmoudi, et al., Discrete Curvature Estimation Methods for Triangulated Surfaces, Ullrich Köthe, Annick Montanvert, Pierre Soille (editors), Applications of Discrete Geometry and Mathematical Morphology, Springer, LNCS 7346, page 28,
In combinatorial geometry, the most common discrete representation for a surface is a triangle mesh.

(geometry, uncountable) The field of mathematics which examines extremal problems of a combinatorial nature expressed geometrically.

For the generation of these combinatorial types no direct method is known, and it appears to be necessary to use combinatorial abstractions — allowable sequences of permutations, #92;lambda-functions, chirotopes, combinatorial geometries, or oriented matroids; in our work we will use oriented matroids [BLVS⁺99].

2003, Lukas Finschi, Komei Fukuda, “Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements”, in Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, editors, Discrete and Computational Geometry, Springer, page 425:

Our purpose in this chapter is to derive fundamental properties of combinatorial geometries, and to show how these properties strengthen our intuitive understandings of figures.[…]In this section, we give an axiom system for combinatorial geometries and then prove that each combinatorial figure is a combinatorial geometry.

2012, Don Row, Talmage James Reid, Geometry, Perspective Drawing, and Mechanisms, World Scientific, page 15:

(geometry, theory of matroids, countable) A simple matroid.


computational geometry

1991, Herbert Edelsbrunner, Lines in Space—A Collection of Results, Jacob E. Goodman, Richard D. Pollack, William L. Steiger (editors), Discrete and Computational Geometry: Papers from the DIMACS Special Year, American Mathematical Society, Association for Computing Machinery, page 77,
In computational geometry it is a truism that problems in three-dimensional space, ℝ³, are significantly more difficult and complex than the corresponding problems in the two-dimensional plane, ℝ².

2003, Ren-Hong Wang, Multivariate Spline and Geometry, Ren-Hong Wang (editor), Computational Geometry, American Mathematical Society, International Press, page 195,
In this survey, we will be concerned with the relationship between multivariate splines and geometries including computational geometry, differential geometry, and algebraic geometry.

Computational geometry processing is about practical algorithms that operate on geometric data sets, and these data sets tend to be rather large if they are to be useful.

2012, J. Andreas Bærentzen, Jens Gravesen, François Anton, Henrik Aanæs, Guide to Computational Geometry Processing, Springer, page 1:

(computer science, geometry) A field of study, within computer science, devoted to algorithms that solve problems formulated in geometric terms.


cu in

Alternative form: cu. in.


conformal geometry

conformal geometry (countable and uncountable, plural conformal geometries)

2000, Boris N. Apanasov, Conformal Geometry of Discrete Groups and Manifolds, Walther de Gruyter, page vii,
Generally, conformal geometries naturally appear at infinity for negatively curved Riemannian geometries. Moreover, due to M. Gromov's [5, 6] geometric approach to infinite groups, conformal geometry invents new fruitful methods in combinatorial group theory.

The lectures concerned the study of some non-linear partial differential equations related to curvature invariants in conformal geometry.

2004, Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry, European Mathematical Society, page 2:

This chapter gives a brief introduction of the basic theoretic foundations and profound and beautiful structures of conformal geometry.

2018, Miao Jin, Xianfeng Gu, Ying He, Yalin Wang, Conformal Geometry: Computational Algorithms and Engineering Applications, Springer, page 1:

conformal transformation
conformal transformation

(geometry) The study of conformal (angle-preserving) transformations on a space;

The study of conformal (angle-preserving) transformations on a space;
(formally) an ordered pair comprising a space and the group of conformal transformations that act on the space.


square number

36 is a square number being the product of 6 and 6 itself.

(mathematics) Any integer which is the product of some integer with the integer itself.

kvadrattal

an integer which is the product of some integer with itself


contact geometry

contact geometry (countable and uncountable, plural contact geometries)

Kontaktgeometrie

subset of the tangent bundle of a manifold

The defining conditions for a contact geometry are opposite to two equivalent conditions for complete integrability of a hyperplane distribution: i.e. that it be tangent to a codimension 1 foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

The contact geometry is in many ways an odd-dimensional counterpart of the symplectic geometry, a structure on certain even-dimensional manifolds. The concepts of contact geometry and symplectic geometry are both motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the constant-energy hypersurface, which, being of codimension 1, has odd dimension.

2004, Ko Honda, 3-Dimensional Methods in Contact Geometry, Simon Donaldson, Yakov Eliashberg, Misha Gromov (editors), Different Faces of Geometry, Springer (Kluwer Academic), page 47,
A contact manifold (M,ζ) is a (2n+1)-dimensional manifold M equipped with a smooth maximally nonintegrable hyperplane field ζ⊂TM, i.e., locally, ζ= ker α, where α is a 1-form which satisfies α∧(dα)ⁿ ne 0. Since dα is a nondegenerate 2-form when restricted to ζ, contact geometry is customarily viewed as the odd-dimensional sibling of symplectic geometry. Although contact geometry in dimensions > 5 is still in an incipient state, contact structures in dimension 3 are much better understood, largely due to the fact that symplectic geometry in two dimensions is just the study of area.

differential geometry
differential geometry

(differential geometry, countable) Given a smooth manifold of odd dimensionality, a distribution (subset) of the tangent bundle that satisfies the condition of complete nonintegrability, or equivalently may be locally defined as the kernel of a maximally nondegenerate differential 1-form;

Given a smooth manifold of odd dimensionality, a distribution (subset) of the tangent bundle that satisfies the condition of complete nonintegrability, or equivalently may be locally defined as the kernel of a maximally nondegenerate differential 1-form;
(uncountable) the study of such structures.


pentadal

(statistics) Of or related to a 5-year period.


Gen X

If the MTV innovation is one hallmark of Gen X, another important attribute is this generation's sense of humor, which is self-conscious, self-deprecating, and filled with irony, but also tends to be somewhat subversive and antiestablishment.

1999, Rob Owen, Gen X TV: The Brady Bunch to Melrose Place, Syracuse University Press, →ISBN, page 54:


Gen Z

But on streaming platforms like TikTok — where many Gen Z stars can simply upload content of themselves in sweatpants and a sport bra dancing to the latest trending audio from the confines of their own bathrooms and generate hundreds of thousands, if not millions, of likes — creators say they’ve been experiencing tremendous growth.

2020 March 24, EJ Dickson, “Amid Coronavirus Fears, TikTok Collab Houses Still Going Strong”, in Rolling Stone:

Though Gen Z isn’t the first group to employ loud and unapologetic tactics — some older activists and writers, including Katha Pollitt, have been pushing these ideas for years — they may be doing it in larger numbers.

2020 December 10, Jessica Grose, “The New Abortion Rights Advocates Are on TikTok”, in The New York Times, →ISSN:


nonzeroary

(computing, mathematics, rare, nonstandard) Not zeroary.


differential form

differential form (plural differential forms)

From early 20th century.
The concept was clarified chiefly by French mathematician Élie Cartan (1869–1951). In his fundamental paper, 1899, Sur certaines expressions différentielles et le problème de Pfaff, Annales Scientifiques de l'École Normale Supérieure (3), tome 16, Cartan used the (French) term expression différentielle, and in his 1922, Leçons sur les invariants intégraux, Hermann, he used the terms exterior differential form and exterior derivative.

The notion of differential form combines the concepts of multilinear form (itself an extension of linear form) and smooth function.

The choice of a Riemannian manifold - roughly speaking, a differentiable manifold whose every point has a tangent space with a defined metric - means it is possible to define a differential form over it.

Differential forms provide a unified approach to defining integrands over curves, surfaces and higher-dimensional manifolds, as well as providing an approach to multivariable calculus that is independent of coordinates.

Now that the length of tangent vector is defined, the magnitude of a differential form is also determined. By making use of this fact we can give a more precise statement to the theorem of de Rham. That is, from the point of view of magnitude, we may prove that within the set of all closed forms representing a de Rham cohomology class, there is one and only one differential form that has the best shape. Such a form is called a harmonic form, and it can be characterized by using a differential operator called the Laplacian.

2001, Shigeyuki Morita, translated by Teruko Nagase and Katsumi Nomizu, Geometry of Differential Forms, American Mathematical Society, page 145:

2004, Ismo V. Lindell, Differential Forms in Electromagnetics, IEEE Press, page xiii,
The present text attempts to serve as an introduction to the differential form formalism applicable to electromagnetic field theory. A glance at Figure 1.2 on page 18, presenting the Maxwell equations and the medium equation in terms of differential forms, gives the impression that there cannot exist a simpler way to express these equations, and so differential forms should serve as a natural language for electromagnetism.

In this first chapter, we discuss various versions of the Hardy-Littlewood inequality for differential forms, including the local cases, the global cases, one weight cases, and two-weight cases. We know that differential forms are generalizations of the functions, which have been widely used in many fields, including potential theory, partial differential equations, quasiconformal mappings, nonlinear analysis, electromagnetism, and control theory; see [1-19], for example. During recent years new interest has developed in the study of the Lᵖ theory of differential forms on manifolds [20, 21].[…]The development of the Lᵖ theory of differential forms has made it possible to transport all notations of differential calculus in #92;mathbbRⁿ to the field of differential forms.

2009, Ravi P. Agarwal, Shusen Ding, Craig Nolder, Inequalities for Differential Forms, Springer, page 1:

(differential geometry, tensor calculus, sometimes as "differential p-form") A completely antisymmetric tensor (of order p) that is defined on a Riemannian manifold; an expression, derived by applying a formalism to said tensor, that represents an integrand over the manifold.

differentiaalimuoto

antisymmetric tensor defined on a Riemannian manifold; the same tensor expressed as an integrand


Hjelmslev's theorem

(geometry) a theorem stating that if a set of points P_i lying on a line is isometrically mapped to points P_i' on a different line, then the midpoints of the segments P_iP_i' also lie on a line.

Johannes Hjelmslev
en:Hjelmslev's theorem


field of scalars

field of scalars (plural fields of scalars)


transcendental number theory

transcendental number theory (uncountable)

1975, Alan Baker, Transcendental Number Theory, Cambridge University Press (Cambridge Mathematical Library), page 48,
Nevertheless, transcendental number theory has led to new results in several associated subjects.

1998, Transcendental Number Theory and Related Topics: Proceedings of Japan-Korea Joint Seminar:

The transcendental number theory was originated by Liouville in 1844 when he showed a method to construct transcendental numbers.

2020, Saradha Natarajan, Ravindranathan Thangadurai, Pillars of Transcendental Number Theory, Springer, page 21:

transcendental number
transcendental number

(number theory) The branch of number theory that deals with transcendental numbers.


nude number

(mathematics) A positive integer that is divisible by every one of its digits.


superior highly composite number

superior highly composite number (plural superior highly composite numbers)

superior highly composite numbers (plural)

(mathematics) A positive integer n which there is a positive integer e such that d(n)/nᵉ ≥ d(k)/kᵉ for all k > 1, where d(n) is defined to be the divisor function of n


multilinear form

multilinear form (plural multilinear forms)

forme multilinéaire

multiply linear mapping to a field of scalars

1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163,
Finally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (and Hankel operators too) really should be viewed as forms, not operators.

1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454,
A multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate m-multilinear form can more concisely be written as M!f.

Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).

2003, Maks A. Akivis, Vladislav V. Goldberg, translated by Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 55:

(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;

Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;
(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.


Tukey's range test

(statistics) A single-step multiple-comparison procedure and statistical test that can be used to find means that are significantly different from each other.


Tukey lambda distribution

Tukey lambda distribution (plural Tukey lambda distributions)

probability distribution
probability distribution

(statistics) A continuous, symmetric probability distribution defined in terms of its quantile function, typically used to identify an appropriate distribution.


Teichmüller-Tukey lemma

Named after John Tukey and Oswald Teichmüller.

(mathematics) A lemma stating that every non-empty collection of finite character has a maximal element with respect to inclusion.

John Tukey
Oswald Teichmüller
Teichmüller-Tukey lemma


differentiable manifold

differentiable manifold (plural differentiable manifolds)

variété différentiable

manifold locally similar enough to a Euclidean space for calculus to be done

The charts (homeomorphisms) that make up any differentiable structure of a differentiable manifold are required to be such that the transition from one to another is differentiable.

Whitney showed that a differentiable manifold Mᵏ can be regularly mapped on the vector space R#123;2k#125;.

1950, L. S. Pontryagin, Characteristic Cycles on Differentiable Manifolds, American Mathematical Society, page 33:

The phase space of a dynamical system will be taken to be a differentiable manifold.

1994, N. Aoki, K. Hiraide, Topological Theory of Dynamical Systems: Recent Advances, Elsevier (North-Holland), page 1:

In this chapter we introduce differentiable manifolds and smooth maps. A differentiable manifold' is a topological space on which there are defined coordinates allowing basic notions of differentiability.

2009, Jeffrey Marc Lee, Manifolds and Differential Geometry, American Mathematical Society, page 1:

(differential geometry) A manifold that is locally similar enough to a Euclidean space (ℝⁿ) to allow one to do calculus;

A manifold that is locally similar enough to a Euclidean space (ℝⁿ) to allow one to do calculus;
(more formally) a manifold that can be equipped with a differentiable structure (an atlas of ℝⁿ-compatible charts).


Maharam's theorem

(mathematics) A result about the decomposability of measure spaces, playing an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval [0,1] on the reals), and "purely atomic parts", using the counting measure on some discrete space.


Maharam algebra

Maharam algebra (plural Maharam algebras)

(mathematics) A complete Boolean algebra having a continuous submeasure (also known as a Maharam submeasure).


near-ring

(mathematics) An algebraic structure similar to a ring but satisfying fewer axioms. They arise naturally from functions on groups.


arcdegree

/ˈɑːkdɪˌɡɹiː/

/ˈɑɹkdɪˌɡɹiː/

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

(geometry) A unit of measurement of angle equal to ¹⁄₃₆₀ of a circle's circumference.


inclusion map

(mathematics) A function between two sets A and B, with the former being a subset of the latter, that maps elements from A to the corresponding elements of B (ι:A→B,ι(x)=x).

inkluusiokuvaus

function that maps elements from a subset to the corresponding elements of its superset


semicone

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


semisum

(mathematics) half the result of an addition


absolute differential calculus

absolute differential calculus (uncountable)

1926 [Blackie & Son], Marjorie Long (translator), Tullio Levi-Civita, The Absolute Differential Calculus (Calculus of Tensors), 1977, Dover https://books.google.com.au/books?id=8PzDAgAAQBAJ&printsec=frontcover&dq=%22absolute+differential+calculus%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwi60s2Pq5DsAhX94zgGHf40DQ0Q6AEwAHoECAAQAg#v=onepage&q=%22absolute%20differential%20calculus%22&f=false.

1931 [Blackie Company], A. J. McConnell, Applications of Tensor Analysis, 1957, Dover, page v,
The absolute differential calculus came into prominence as the instrument best fitted for dealing with the general theory of relativity and it has also been found indispensable for the differential geometry of hyperspaces.

Instead, it is of interest to try to look at the emergence of the absolute differential calculus (ADC in the rest of this chapter) from the viewpoint of the mathematical community of the time.

1999, Umberto Bottazzini, “8: Ricci and Levi-Civita: from differential invariants to general relativity”, in Jeremy Gray, editor, The Symbolic Universe: Geometry and Physics 1890-1930, Oxford University Press, page 241:

mathematical analysis
mathematical analysis

(mathematical analysis, dated or historical) Ricci calculus; the rules of index notation and manipulation for tensors and tensor fields, as developed by Gregorio Ricci-Curbastro.


image matting

(machine learning) Extraction of an object in the foreground of an image from its background.


Laplace expansion

Laplace expansion (plural Laplace expansions)

(mathematics) An expression for the determinant |B| of an n × n matrix B that is a weighted sum of the determinants of n submatrices (or minors) of B, each of size (n − 1) × (n − 1).


analytic number theory

1987, F. X. Gallagher, Applications of Guinand's Formula, A.C. Adolphson, A. Ghosh, J.B. Conrey, R.I. Yager (editors), Analytic Number Theory and Diophantine Problems: Proceedings of a Conference, Birkhäuser, page 135,
According to Weil [22], analytic number theory, which deals with inequalities and asymptotic formulas, is not number theory but analysis.

2002, Chaohua Jia, Kohji Matsumoto (editors), Analytic Number Theory, Kluwer Academic, page ix,
Talks covered various research fields including analytic number theory, algebraic number theory, modular forms and transcendental number theory.

Computational number theory, however, studies problems from elementary, algebraic geometric and analytic number theory which require the help of fast computers (particularly vector and parallel systems) and fast algorithms (particularly deterministic polynomial-time algorithms).

2002, Song Y. Yan, Number Theory for Computing, 2nd edition, Springer, page 174:

(mathematics, number theory) A branch of number theory that uses methods from mathematical analysis to solve problems about the integers.


abstract analytic number theory

abstract analytic number theory (uncountable)

The prime number theorem serves as a prototypical example of an idea generalised via abstract analytic number theory and adapted to other fields; the emphasis is on results pertaining to asymptotic analysis.

1975, John Knopfmacher (editor), Abstract Analytic Number Theory, North-Holland https://books.google.com.au/books?id=HIusKnF2kmEC&printsec=frontcover&dq=%22abstract+analytic+number+theory%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjdm__u-5rsAhURIDQIHS14AjsQ6AEwAHoECAYQAg#v=onepage&q=%22abstract%20analytic%20number%20theory%22&f=false.

Only recently the authors completed the monograph [16] which contains a thorough presentation of the algebraic, combinatorial and analytic aspects of the theory of non-unique factorizations, together with self-contained introductions to additive group theory, to the theory of v-ideals and to abstract analytic number theory.

2006, Alfred Geroldinger, Franz Halter-Koch, “Non-unique factorizations: a survey”, in James W. Brewer, Sarah Glaz, William Heinzer, Bruce Olberding, editors, Multiplicative Ideal Theory in Commutative Algebra, Springer, page 207:

In [11], J. Knopfmacher gives an interesting exposition of abstract analytic number theory wherein many of the results of classical number theory are generalized in a suitable context of semigroups satisfying certain axioms.

2006, R. Sivaramakrishnan, Certain Number-Theoretic Episodes In Algebra, Taylor & Francis (Chapman & Hall/CRC), page 339:

analytic number theory
analytic number theory

(number theory) A branch of number theory in which suitable ideas and techniques from analytic number theory are generalised and applied to a variety of mathematical fields.


rotoidal

(mathematics) Having the form of a rotoid


subdecimal

When we examine the units to note how many are founded upon natural magnitudes, we recognize that the second is not a subdecimal part of the solar day, but only the 36400 of that natural unit.

1888, Proceedings of the Physical Society, volumes 9-10, page 56:

The detection of decimal points involves three purposes: First, recognition is concerned with the integer part alone, and subdecimal digits are not recognized in most cases.

1994, Horst Bunke, Patrick S. P. Wang, Henry S. Baird, Document Image Analysis, page 168:

The fact that nearly all Central American languages have a vigesimal numeral system (partly beside a subdecimal system of counting) seems to suggest that the vigesimal system only spread in recent times, […]

1997, Stefan Liedtke, The Languages of the 'First Nations' (page 119)

(mathematics) That results from a division by ten or by a multiple of ten)


locally compact group

locally compact group (plural locally compact groups)

Indeed, if there is one property of locally compact groups more responsible than any other for the rich positive content of their representation theory, it is their possession of left-invariant and right-invariant (Haar) measures. The connection between the representation theory of a locally compact group and that of general Banach algebras proceeds directly from Lebesgue integration with respect to Haar measure on the group.

1988, J. M. G. Fell, R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Volume 1, Academic Press, page 63:

A locally compact group G is called Lie projective if it is the projective limit #92;textstyle#92;underset#123;#92;leftarrow#125;#123;#92;lim#125;#95;#123;#92;alpha#92;in#92;mathbb#123;A

2012, H. Heyer, Probability Measures on Locally Compact Groups, Springer, page 12:

Let G be a locally compact group and H a closed subgroup of G, and suppose that #92;pi and #92;tau are irreducible representations of G and H, respectively.

2013, Eberhard Kaniuth, Keith F. Taylor, Induced Representations of Locally Compact Groups, Cambridge University Press, page 269:

(topology) A topological group whose underlying topology is both locally compact and Hausdorff.

lokalkompakte Gruppe

topological group whose topology is locally compact and Hausdorff


homotopy theory

homotopy theory (countable and uncountable, plural homotopy theories)

Homotopy theory originated as a topic in algebraic topology, but is now studied as an independent discipline in its own right, and as a tool in fields including category theory and algebraic geometry.

1988, Alex Heller, Homotopy Theories, American Mathematical Society, https://books.google.com.au/books?id=hkXUCQAAQBAJ&printsec=frontcover&dq=%22homotopy+theories%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiAzeb7qKfsAhUayTgGHeEyBZQQ6AEwAHoECAAQAg#v=onepage&q=%22homotopy%20theories%22&f=false.

This is our main result, deduced from the axioms of a cofibration category, which leads to many new theorems on the homotopy classification problems in topology and in various algebraic homotopy theories.

1989, Hans Joachim Baues, Algebraic Homotopy, Cambridge University Press, page 371:

They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories.

2010, Julia E. Bergner, “A Survey of (∞,1)-Categories”, in John C. Baez, J. Peter May, editors, Towards Higher Categories, Springer,, page 69:

(mathematics) The systematic study of the situation of maps' having a homotopy between them; the study of equivalence classes (called homotopy classes) of maps.


arithmetic combinatorics

For instance, progressions and lattices play a similar role in arithmetic combinatorics that balls and subspaces play in the theory of normed vector spaces.

2006, Terence Tao, Van H. Vu, Additive Combinatorics, Cambridge University Press, page 112:

Some of the techniques involved^([with character sum problems]) belong to arithmetic combinatorics.

2016, Mei-Chu Chang, “Character sums and arithmetic combinatorics”, in Andrew Beveridge, Jerrold R. Griggs, Leslie Hogben, Gregg Musiker, Prasad Tetali, editors, Recent Trends in Combinatorics, Springer,, page 405:

Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis.[…]One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems.

2019, Cain Mckay, Probability and Statistics, ED-Tech Press, page 23:

(mathematics) A field of mathematics in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.


additive number theory

Abstractly, additive number theory includes the study of abelian groups and commutative semigroups with an addition operation.

Two principal objects of study in additive number theory are the sumset A#43;B#61;#92;#123;a#43;b#58;a#92;inA,b#92;inB#92;#125; of two subsets A and B of elements from an abelian group G, and the h-fold sumset of A, hA#61;#92;underset#123;h#125;#123;#92;underbrace#123;A#43;#92;cdots#43;A,.}}

Additive number theory has close ties to combinatorial number theory and the geometry of numbers.

1966 [Macmillan], W. E. Deskins, Abstract Algebra, 1995, Dover, page 152,
One of the famous theorems of additive number theory states that each positive integer is expressible in at least one way as the sum of the squares of not more than four positive integers.

The Circle Method is a beautiful idea for studying many problems in additive number theory. It originated in investigations by Hardy and Ramanujan [HR] on the partition function P(n), the number of ways we can write n as a sum of positive integers. Since then it has been used to study problems in additive number theory ranging from writing numbers as sums of primes or k#92;mathsf#123;th#125; powers (for fixed k) to trying to count how many twin primes there are less than x.

2006, Steven J. Miller, Ramin Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, page 303:

Additive number theory is a very broad area within which lies the theory of partitions, and the subject of representation of integers as sums of squares, cubes, etc. There is a vast literature in additive number theory on problems of the following type: Given a subset A of the integers, when does it form a basis for the integers? That is, when can every positive integer be expressed as the sum of at most a (fixed) bounded number of elements of A?

2008, Krishnaswami Alladi, editor, Surveys in Number Theory, Springer, page ix:

(number theory) The subfield of number theory concerning the behaviour of sumsets (integer sets added to each other elementwise).


geometry of numbers

The field was initiated by German mathematician Hermann Minkowski (1910, Geometrie der Zahlen).

A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in #92;textstyle#92;mathbb#123;R#125;ⁿ; the study of these lattices provides fundamental information on algebraic numbers.

1969, C. G. Lekkerkerker, Geometry of Numbers, Wolters-Noordoff, North-Holland, page 1,
The geometry of numbers to which this book is devoted deals with arbitrary bodies and arbitrary lattices in the n-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice.

2000, C. D. Olds, Anneli Lax, Giuliana Davidoff, The Geometry of Numbers, Mathematical Association of America:

2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 181,
The easy proof is obtained applying the pigeon-hole principle to
α₁x_1+…+αₙx_n(mod 1)|x_i=0,…N,
or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2ⁿ⁺¹ given by
|X_0+α₁X_1+…αₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N.

(number theory) The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers.


additive combinatorics

Coined circa early 2000s by Australian-American mathematician Terence Tao for a rapidly developing field growing out of combinatorial number theory, named differently to reflect a changed emphasis in the problems being studied.

One major area of study in additive combinatorics is that of inverse problems: for instance, given the sumset A#43;B is small in size, what can we say about the structures of A and B? In the case of integer sumsets, Freiman's theorem provides a partial answer.

2007, Andrew Granville, Additive Combinatorics, American Mathematical Society, https://books.google.com.au/books?id=SOVPnwEACAAJ&dq=%22Additive+combinatorics%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj5zJbJ0bLsAhUQX30KHTZYAZwQ6AEwRnoECBwQAg.

We now discuss what appears at first glance to be an unrelated topic, namely that of additive combinatorics (and its noncommutative counterpart, multiplicative combinatorics). One of the main objects of study in either additive or multiplicative combinatorics are approximate groups — sets A (typically finite) contained in an additive or multiplicative ambient group G that are "almost groups" in the sense that they are "almost" closed under either addition or multiplication.

2014, Terence Tao, Hilbert's Fifth Problem and Related Topics, American Mathematical Society, page 14:

This book deals with additive combinatorics, a vibrant area of current mathematical research. Additive combinatorics—an offspring of combinatorial number theory and additive number theory—can be described as the study of combinatorial properties of sumsets (collections of sums with terms from given subsets) in additive structures.

2016, Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, Taylor & Francis (CRC Press), page xi:

(mathematics) A subbranch of combinatorics that concerns additive problems expressed using sumsets.

Bulletin of the American Mathematical Society

discipline that concerns combinatorics problems expressed using sumsets


semiorder

(mathematics) A type of order that is not wholly comparable


affine differential geometry

affine differential geometry (uncountable)

The term reflects the categorisation developed by German mathematician Felix Klein for his Erlangen programme (1872, Vergleichende Betrachtungen über neuere geometrische Forschungen), in which he found a useful distinction between projective, affine and Euclidean geometry (in order of increasing restrictiveness). (Riemannian geometry was not initially included.)

The basic difference between Riemannian and affine differential geometry is that in the affine case we introduce volume forms over a manifold instead of metrics.

1999, Alexander I. Bobenko, Wolfgang K. Schief, 5: Discrete Indefinite Affine Spheres, Alexander I. Bobenko, Ruedi Seiler (editors), Discrete Integrable Geometry and Physics, Oxford University Press (Clarendon Press), page 113,
Tzitzeica's classical papers are believed to have initiated a new area in mathematics, namely affine differential geometry. […] The present paper extends the above-mentioned approach to affine differential geometry.

The Tzitzeica surfaces are the analogues of spheres in affine differential geometry and, indeed, are known as affine spheres or affinsphären [39]. According to Nomizu and Sasaki [227], the origins of affine differential geometry reside in this work of Tzitzeica at the turn of the nineteenth century.

2002, C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, Cambridge University Press, page 88:

In Chapter 4, we study questions related to real affine differential geometry. The structure group in Riemannian geometry is the orthogonal group #92;mathcal#123;O#125;. The structure group in affine differential geometry is the affine group.

2012, Miguel Brozos-Vázquez, Peter B. Gilkey, Stana Nikcevic, Geometric Realizations of Curvature, World Scientific (Imperial College Press), page 89:

affine transformation
affine transformation

(differential geometry) A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations.

Erlangen programme
Felix Klein
affine differential geometry


pseudoangle

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

(mathematics) The equivalent of an angle (or solid angle) in a non-Euclidean space


Nevanlinna theory

Named after Finnish mathematician Rolf Nevanlinna (1895–1980), who published the theory in 1925.

A key tool in Nevanlinna theory is the Nevanlinna characteristic, T(r,f), which measures the rate of growth of a meromorphic function.

Precisely, our aim has been to show how the Nevanlinna theory may be applied to get insight into the properties of solutions of complex differential equations.

1992, Ilpo Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, page 1:

Motivated by an analogy between Nevanlinna theory and Diophantine approximation theory, discovered independently by C. F. Osgood [Osg 1985] and P. Vojta [Vojt 1987], S. Lang recognized that the careful study of the error term in Nevanlinna'a Second Main Theorem would be of interest in itself.

2001, William Cherry, Zhuan Ye, Nevanlinna's Theory of Value Distribution, Springer, page vi:

2010, Paul Vojta, Diophantine Approximation and Nevanlinna theory, Jean-Louis Colliot-Thélène, Peter Swinnerton-Dyer, Paul Vojta (editors), Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School, Springer, Lecture Notes in Mathematics 2009, page 111,
Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth's theorem on diophantine approximation. […] The circle of ideas has developed further in the last 20 years: Lang's conjecture on sharpening the error term in Roth's was carried over to a conjecture in Nevanlinna theory which was proved in many cases.

(complex analysis) A part of the theory of meromorphic functions that describes the asymptotic distribution of solutions to the equation ƒ(z) = a, as a varies.

Acta Mathematica
Nevanlinna theory
Rolf Nevanlinna


Ahlfors theory

Named after Finnish mathematician Lars Ahlfors (1907—1996), who published the theory in 1935.

In this chapter we use Ahlfors' theory of covering surfaces to obtain results on the functional n(r,a).

1968, Joseph Belsley Miles, The Asymptotic Behavior of the Counting Function for the A-values of a Meromorphic Function, University of Wisconsin-Madison, page 29:

Terms of the form o(A(r)) in Ahlfors theory are given in the form cD(r) where c is a constant.

1986, Pacific Journal of Mathematics, volumes 122-123, page 441:

The Ahlfors theory itself describes covering of curves or domains, but not covering of distinct, complex values #92;boldsymbol#123;a#125;.

2004, G. Barsegian, “A new program of investigations in analysis: Gamma-lines approaches”, in G. Barsegian, I. Laine, C. C. Yang, editors, Value Distribution Theory and Related Topics, Kluwer Academic, page 43:

bordered Riemann surface
bordered Riemann surface

(complex analysis, differential geometry) A geometric counterpart to Nevanlinna theory that extends the applicability of the concept of covering surface (of a topological space) by defining a covering number (a generalised "degree of covering") applicable to any bordered Riemann surface equipped with a conformal Riemannian metric.

Acta Mathematica
Ahlfors theory
Lars Ahlfors


supervertex

(mathematics) A vertex formed by merging two existing verticies of a graph by removing their connecting edge


algebraic analysis

The area of study in algebraic geometry began as a research programme, instigated in 1959 by Japanese mathematician Mikio Sato.

1986, Goro Kato (translator), Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of Algebraic Analysis, Princeton Universty Press, https://books.google.com.au/books?id=qYUrDgAAQBAJ&printsec=frontcover&dq=%22algebraic+analysis%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiK5MSlvsLsAhWXbn0KHbPuBPQQ6AEwAnoECAYQAg#v=onepage&q=%22algebraic%20analysis%22&f=false.

In their classical work [123], Kashiwara, Kawai, and Kimura pointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we may add that what we mean by algebraic microlocal analysis is the successful attempt to adapt the methods of abstract algebraic geometry to the non-commutative setting in which the base ring is now the ring of variable coefficients^([sic]) partial differential operators.

1999, Goro Kato, Daniele C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, page v:

2008, Yoshitsugu Takei, The work of T. Kawai on exact WKB analysis, T. Aoki, H. Majima, Y. Takei, N. Tose (editors), Algebraic Analysis of Differential Equations, Springer, page 19,
Thus Kawai has been engaged mainly in the research of exact WKB analysis or algebraic analysis of singular perturbation theory since 1989.

partial differential equation
partial differential equation

(algebraic geometry, algebraic topology) The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions and certain generalisations of functions (such as hyperfunctions and microfunctions).


autodependency

(mathematics) The dependency of the value of a variable on its value at a previous time


algebraic combinatorics

1982, Trevor Evans, Finite Representations of Two-variable Identities, E. Mendelsohn, Algebraic and Geometric Combinatorics, North-Holland, page 135,
It is part of the folklore of algebraic combinatorics that “most' two-variable groupoid identities have non-trivial models in finite fields, the groupoid operation being represented by a linear function ax+by.

This volume arose through the initiative of Kluwer Academic Publishers in an attempt to introduce some areas of research in algebraic combinatorics which originally appeared in Russian to a wider mathematical community.

1993, I. A. Faradžev, A. A. Ivanov, M. Klin, A. J. Woldar, editors, Investigations in Algebraic Theory of Combinatorial Objects, Kluwer Academic, page vii:

This book is intended primarily as a one-semester undergraduate text for a course in algebraic combinatorics.[…]Algebraic combinatorics is a huge subject, so some selection process was necessary to obtain the present text.

2013, Richard P. Stanley, Algebraic Combinatorics, 2nd edition, Springer, page xi:

(algebra, combinatorics) A branch of mathematics in which techniques from abstract algebra are applied to problems in combinatorics, and vice versa.

combinatoire algébrique

branch of mathematics at the intersection of combinatorics and algebra


bifiltration

(mathematics) A two-dimensional filtration


endotactic

en-adj

endotactic (comparative more endotactic, superlative most endotactic)

more endotactic (comparative)
most endotactic (superlative)

(mathematics, of a network) That is in contact with an internal network


equiangulation

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

(geometry) Conversion to equiangular form


subsimplex

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

(geometry) A polyhedron at the intersection of simplexes.


subsimplicial

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

(geometry) Relating to, or composed of subsimplexes


subtensor

(mathematics) A constituent part of a tensor


numerical integration

en-noun

numerical integration (countable and uncountable, plural numerical integrations)

(mathematics, computing) The study of numerical analysis algorithms to obtain approximate values of definite integrals.

numerical integration
simple:numerical integration


eigengenome

(genetics) (mathematics) One of a set of left singular vectors of a sequences x genomes matrix that tabulates, e.g., the DNA copy numbers of the genomes across the sequences.


eigendecompose

en-verb

eigendecompose (third-person singular simple present eigendecomposes, present participle eigendecomposing, simple past and past participle eigendecomposed)

eigendecomposes (present singular third-person)
eigendecomposing (participle present)
eigendecomposed (participle past)
eigendecomposed (past)

(mathematics) To form the eigendecomposition of a matrix


numerical linear algebra

(computing) The study of numerical analysis algorithms to solve mathematical problems in the field of linear algebra.

numerical linear algebra
simple:numerical linear algebra


algebraic graph theory

en-noun

algebraic graph theory (countable and uncountable, plural algebraic graph theories)

The three main branches of algebraic graph theory involve the use of linear algebra, the use of group theory and the study of graph invariants.

The last section introduces elementary algebraic graph theory which is then applied to electrical networks.

1979, Béla Bollobás, Graph Theory: An Introductory Course, Springer, page 26:

Several famous results in algebraic graph theory, including one of the oldest, are formulae for the numbers of spanning trees of certain graphs.

1993, Norman Biggs, Algebraic Graph Theory, 2nd edition, Cambridge University Press, page 38:

Graphs with diameter d and girth 2d#43;1 are known as Moore graphs. They were introduced by Hoffman and Singleton in a paper that can be viewed as one of the prime sources of algebraic graph theory.

2013, Chris Godsil, Gordon F. Royle, Algebraic Graph Theory, Springer, page 77:

(uncountable, mathematics, graph theory) The subbranch of graph theory in which algebraic methods are applied to problems about graphs.


computational fluid dynamics

computational fluid dynamics (uncountable)

(computing, mechanics) The study of computational algorithms to solve mathematical problems in the field of fluid dynamics.


supervariety

(mathematics, linguistics) A variety having subordinate varieties; the parent of a subvariety.


Gen Alpha

(demographics, informal) Generation Alpha


MDCT

MDCT (countable and uncountable, plural MDCTs)

modified discrete cosine transform
w:modified discrete cosine transform

(mathematical analysis) Initialism of modified discrete cosine transform.


Gbit

(computing, informal) Abbreviation of gibibit.


Tbit

(computing, informal) Abbreviation of tebibit.


MiBps

You can add 100 MiBps increments of peak data throughput up to your system's hardware capacity by using FC 5268. [...] Host throughput on TS7700 clusters with 25 increments installed are not constrained at 2500 MiBps.

2017 April 11, IBM TS7700 Release 3.3, IBM Redbooks, →ISBN, pages 219–220:

The gp2 volumes have a throughput limit of 128 MiBps - 250 MiBps, depending on volume size.

2020 July 8, Implementing IBM Spectrum Virtualize for Public Cloud on AWS Version 8.3.1, IBM Redbooks, →ISBN:


Clifford algebra

Clifford algebra (plural Clifford algebras)

Named after William Kingdon Clifford (1845–1879), an English mathematician and philosopher.

(algebra, mathematical physics) A unital associative algebra which generalizes the algebra of quaternions but which is not necessarily a division algebra; it is generated by a set of γᵢ (with i ranging from, say, 1 to n) such that the square of each γᵢ is fixed to be either +1 or −1, depending on each i, and such that any product γᵢγⱼ anticommutes when its factors are distinct (i.e., when i ne j).

Clifford algebra
William Kingdon Clifford


spigot algorithm

spigot algorithm (plural spigot algorithms)

(mathematics) An algorithm for computing the value of a transcendental number, generating its digits sequentially with increasing precision.


Bailey-Borwein-Plouffe formula

the Bailey-Borwein-Plouffe formula (canonical)

Named after the authors of the article in which it was published: David H. Bailey, Peter Borwein, and Simon Plouffe.

(mathematics) A formula for calculating pi, giving rise to a spigot algorithm for computing the nth base-16 digit without computing the preceding digits.


Fourier analysis

After French mathematician and physicist Jean-Baptiste Joseph Fourier (1768–1830), a pioneer of the field.

trigonometric function
trigonometric function

(mathematics) A field of study concerned with the representation or approximation of general functions, such as arbitrary waveforms, by sums of trigonometric functions.


Meyer set

Named after Yves Meyer, who introduced the concept.

Minkowski difference
Minkowski difference

(mathematics) A set relatively dense X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. They are now best known as a mathematical model for quasicrystals.


Heesch number

(geometry) The maximum number of layers of copies of the same shape that can surround a given shape.


Heesch's problem

(geometry) The mathematical problem of determining the set of numbers that can be Heesch numbers.


QTM

QTM (countable and uncountable, plural QTMs)

quaternary triangular mesh
quaternary triangular mesh#English

(geometry) Abbreviation of quaternary triangular mesh.


pseudocircle

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

(mathematics) The finite topological space X consisting of four distinct points { a, b, c, d } with the non-Hausdorff topology a,b,c,d,a,b,c,a,b,d,a,b,a,b,∅.


medcouple

From med (“median”) + couple, as the median of a kernel function over pairs ("couples") of data points. The original paper also introduced a similar concept medtriple that has not seen wide use.

(statistics) A scaled median difference between the two halves of a distribution.


homological algebra

The notion of chain complex is central to homological algebra.

The development of homological algebra was closely intertwined with the emergence of category theory.

Homological algebra for our purposes consists of studying certain abelian groups Hⁿ(G,A) and H#95;n(G,A), known as the nth cohomology and homology groups of the group G with coefficients in the G-module A.

2002, C. R. Leedham-Green, S. McKay, The Structure of Groups of Prime Power Order, Oxford University Press, page 171:

A similar shift of interpretation of CT took place at about the same time in another context, namely homological algebra.

2007, Ralph Krömer, Tool and Object: A History and Philosophy of Category Theory, Springer (Birkhäuser), page 93:

In Section 4.1, we shall introduce some minimal amount of homological algebra needed to launch singular homology.

2014, Anant R. Shastri, Basic Algebraic Topology, Taylor & Francis (CRC Press / Chapman & Hall), page 169:

(algebra) The branch of mathematics dealing with homology and in particular with homological functors and the algebraic structures they entail.


Morley's trisector theorem

Named for Anglo-American mathematician Frank Morley, who discovered it in 1899.

Among the theorems proved by Wu's method are Simon's theorem, the nine point circle theorem, Pascal's theorem, Feuerbach's theorem, Morley's trisector theorem and Thébault's conjecture which was open over forty years and was proved only in 1983.

1989, Paul GochetEric Grégoireet al., From Modal Logic to Deductive Databases, Wiley, page 8:

1996, Dongming Wang, GEOTHER: A Geometry Theorem Prover, M. A. McRobbie, J. K. Slaney (editors), Automated Deduction - Cade-13: 13th International Conference, Proceedings, Springer, LNAI 1104, page 166,
The provers are practically efficient: a number of difficult theorems including Morley's trisector theorem, Thebault's recently confirmed conjecture and MacLane's non-existence of real 83 configuration have been proved by them in the matter of seconds.

equilateral triangle
equilateral triangle

(geometry) A theorem stating that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.


illegal prime

(mathematics) A prime number that encodes information that is illegal to possess or distribute.


metafinite

(mathematics) Describing a structure composed of a finite part, an infinite part, and a form of mapping between the two


algebraic K-theory

Algebraic cycles are typically studied by imposing one of several equivalence relations. The equivalence relation most relevant for algebraic K-theory is rational equivalence.

1999, E. M. Friedlander, “Lecture VII. Beilinson's vision”, in H. Bass, A. O. Kuku, C. Pedrini, editors, Algebraic K-theory And Its Applications, World Scientific, page 61:

Algebraic K-theory has two components: the classical theory which centers around the Grothendieck group K#95;0 of a category and uses explicit algebraic presentations and higher algebraic K-theory which requires topological or homological machinery to define.

2013, Charles A. Weibel, The K-book: An Introduction to Algebraic K-theory, American Mathematical Society, page ix:

2014, Daniel Scott Farley, Ivonne Johanna Ortiz, Algebraic K-theory of Crystallographic Groups, Springer, Lecture Notes in Mathematics 2113, page 1,
Algebraic K-theory is a branch of algebra dealing with linear algebra over a general ring R instead of a field. […] Algebraic K-theory plays an important part in many areas of mathematics, especially number theory, algebraic topology and algebraic geometry.

(algebraic geometry) K-theory studied from the point of view of algebra.

K-théorie algébrique

K theory studied from the point of view of algebra


Witt group

Named after German mathematician Ernst Witt (1911–1991), who introduced the concept in 1937.

groupe de Witt

group of equivalence classes found within a field, variety or category

A general method of studying the Witt group of a smooth variety is through the graded group associated to the filtration induced by the filtration of the Witt group of the function field by powers of the fundamental ideal of even rank forms.

1995, Raman Parimala, “Study of quadratic forms - Some connections with geometry”, in S. D. Chatterji, editor, Proceedings of the International Congress of Mathematicians, Zürich m94, Birkhäuser, page 328:

2011, Marco Schlichting, Higher Algebraic K-theory, Guillermo Cortiñas (editor), Topics in Algebraic and Topological K-Theory, Springer, Lecture Notes in Mathematics 2008, page 167,
The second reason for this emphasis is that an analog of the Thomason-Waldhausen Localization Theorem also holds for many other (co-) homology theories besides K-theory, among which Hochschild homology, (negative, periodic, ordinary) cyclic homology [49], topological Hochschild (and cyclic) homology [2], triangular Witt groups [6] and higher Grothendieck–Witt groups [77].

The Witt group of a category with duality is given as the quotient of the isometry classes of symmetric spaces modulo metabolic spaces.[…]For coherent and derived Witt groups, the derived tensor product of complexes gives rise to duality-preserving functions and consequently to pairings in triangular Witt groups, cf. [GN03].

2020, Matthias Wendt, “Oriented Schubert calculus in Chow-Witt rings of Grassmannians”, in Federico Binda, Marc Levine, Manh Toan Nguyen, Oliver Röndigs, editors, Motivic Homotopy Theory and Refined Enumerative Geometry, American Mathematical Society, pages 239–240:

equivalence relation
equivalence relation

metabolic quadratic space
metabolic quadratic space

(algebra) Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;

Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;
(algebraic geometry, by extension) given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum, modulo the subgroup generated by metabolic spaces
(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.


Reuleaux polygon

Reuleaux polygon (plural Reuleaux polygons)

Named after Franz Reuleaux, 19th-century German engineer.

(geometry) A curve of constant width made up of circular arcs of constant radius.


L-fuzzy

(set theory) Relating to a generalized variant of the notion of fuzzy sets, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind, usually at least a poset or lattice.


Goguen category

Goguen category (plural Goguen categories)

(set theory) A fuzzy kind of category in which membership is determined not in binary (true-or-false) fashion but by a more general interval.


antimorphism

The transform u^λ of an element u of A by an automorphism or antimorphism of A has the same spectrum Sᵤ and the same minimal polynomial as u, and is semisimple if u is so.

1960, Indian Mathematical Society, Journal: New series, volume 24, page 602:

The meet of a set of antimorphisms is again an antimorphism.

2007, Michael Winter, Goguen Categories: A Categorical Approach to L-fuzzy Relations, page 26:

(mathematics) An isomorphism of a mathematical object or system of objects such that the image of a product of two values is the same as the product of the images of the two values in reverse order.


mile a minute

He talks a mile a minute.


pyramid inch

(historical, pyramidology) A unit of measure claimed by pyramidologists to have been used in ancient times. Supposedly one twenty-fifth of a "sacred cubit", 1.00106 imperial inches, or 2.5426924 centimetres.


Zermelo's theorem

Ernst Zermelo
Zermelo's theorem
Zermelo's theorem (game theory)

(game theory) A theorem about finite two-person games of perfect information in which the players move alternately and chance does not affect the decision-making process. It states that if the game cannot end in a draw, then one of the two players must have a winning strategy, i.e. force a win.


ternary logarithm

ternary logarithm (plural ternary logarithms)

(mathematics) The logarithm to the base three


surface integral

surface integral (plural surface integrals)

(mathematics) An integral over a 2-dimensional domain


Fermat pseudoprime

Fermat pseudoprime (plural Fermat pseudoprimes)

341 is a Fermat pseudoprime to the base two.

(mathematics) With respect to an integer base b, with b > 1, a composite integer n such that bⁿ⁻¹ is congruent to one modulo n.


sumtree

(mathematics) A tree in which the value of a node is equal to the sum of its subnodes


algebraic statistics

հանրահաշվական վիճակագրություն

discipline that uses algebra in the study of statistics

That workshop was built on the algebraic statistics paradigm, which is that statistical models for discrete data can be regarded as solutions to systems of polynomial equations.

2005, Lior Pachter, Bernd Sturmfels, editors, Algebraic Statistics for Computational Biology, Volume 13, Cambridge University Press, page ix:

Algebraic statistics is concerned with the development of techniques in algebraic geometry, commutative algebra, and combinatorics, to address problems in statistics and its applications.

2009, Mathias Drton, Bernd Sturmfels, Seth Sullivant, Lectures on Algebraic Statistics, Springer (Birkhäuser), page vii:

We present a brief review of classical experimental design in the spirit of algebraic statistics.

2010, G. Pistone, E. Riccomagno, M. P. Rogantin, “5: Methods in Algebraic Statistics for the Design of Experiments”, in Luc Pronzato, Anatoly Zhigljavsky, editors, Optimal Design and Related Areas in Optimization and Statistics, Springer,, page 97:

mathematical statistics
mathematical statistics

(mathematics, statistics) A discipline within mathematical statistics in which algebra is used to describe and analyse problems;

A discipline within mathematical statistics in which algebra is used to describe and analyse problems;
specifically, the use of algebraic geometry and commutative algebra in statistics.


algebraic fundamental group

algebraic fundamental group (plural algebraic fundamental groups)

1997, Frans Oort, The Algebraic Fundamental Group, Leila Schneps, Pierre Lochak (editors), Geometric Galois Actions 1: Around Grothendieck's Esquisse D'un Programme, Cambridge University Press, London Mathematical Society, page 78,
Serre constructed an example of an algebraic variety X over a number field K plus two embeddings of K into ℂ such that the geometric fundamental groups of X_1( ℂ) and X_2( ℂ) are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).

Neves and Papadakis ([NP09]) constructed a numerical Campedelli surface with algebraic fundamental group #92;mathbbZ#95;6, while Lee and Park ([LP09]) constructed one with algebraic fundamental group #92;mathbbZ#95;2, and one with algebraic fundamental group #92;mathbbZ#95;3 was added in the second version of the same paper.

2011, Ingrid Bauer, Fabrizio Catanese, Roberto Pignatelli, “Surfaces of general type with geometric genus zero: a survey”, in Wolfgang Ebeling, Klaus Hulek, Knut Smoczyk, editors, Complex and Differential Geometry, Springer, page 11:

In order to work in arbitrary characteristic, we should use the algebraic fundamental group, which we denote by #92;pi#95;1. (A very good introduction to algebraic fundamental groups is [24].[…])

2013, János Kollár, “Rationally Connected Varieties and Connected Groups”, in Károly Böröczky, Jr., János Kollár, Szamuely Tamas, editors, Higher Dimensional Varieties and Rational Points, Springer, page 70:

(algebraic geometry) A group which is an analogue for schemes of the fundamental group for topological spaces.


anabelian geometry

anabelian geometry (countable and uncountable, plural anabelian geometries)

We shall say more about the noncommutative nature of anabelian geometries after we state the main theorem (Theorem 1.17 below) on these geometries.

1999, Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, American Mathematical Society, page 63:

The basic idea of Grothendieck's anabelian geometry is that under certain "anabelian hypotheses" the étale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme is functorially encoded in its étale fundamental group.
Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves.

2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 519:

Therefore, the time seems right to encourage a much broader understanding of the arithmetic issues surrounding anabelian geometry and its ramifications.
While the overall importance of the theorems of anabelian geometry appears to be widely acknowledged, there is as yet not much specific knowledge within the arithmetic geometry community of its coherent body of concepts and philosophy, and of the new technology that yields actual results.

2012, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii:

algebraic fundamental group
algebraic fundamental group

(mathematics, algebraic geometry, arithmetic geometry) A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being abelian (commutative).


submass

(mathematics) The mass of a substring of a weighted alphabet


weighted alphabet

weighted alphabet (plural weighted alphabets)

(mathematics) An alphabet, every symbol of which is assigned a weight (positive real number)


heterokurtosis

(statistics) The property of a series of random variables of not every variable having the same finite kurtosis


homokurtosis

2010, Jeffrey M. Wooldridge, Econometric Analysis of Cross-Section and Panel Data (2nd ed.), MIT Press
In terms of the original error u_i, this assumption implies that E(u_i⁴∣ mathbf xᵢ)= mathit constant≡κ² under H_0. This is called the homokurtosis (constant conditional fourth moment) assumption. Homokurtosis always holds when u is independent of mathbf x, but there are conditional distributions for which E(u∣ mathbf x)=0 and Var(u∣ mathbf x)=σ² but E(u⁴∣ mathbf x)=0 depends on mathbf x.

(statistics) The property of a series of random variables of every variable having the same finite kurtosis


heterokurtosis-robust

en-adj

heterokurtosis-robust (comparative more heterokurtosis-robust, superlative most heterokurtosis-robust)

more heterokurtosis-robust (comparative)
most heterokurtosis-robust (superlative)

This test […] is sometimes called a heterokurtosis-robust test for heteroskedasticity.

2010, Jeffrey M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, 2nd edition, The MIT Press, page 141:

(statistics) Of an estimator or series of estimators: robust in the presence of heterokurtosis


heteroscedasticity-robust

heteroscedasticity-robust (comparative more heteroscedasticity-robust, superlative most heteroscedasticity-robust)

more heteroscedasticity-robust (comparative)
most heteroscedasticity-robust (superlative)

(statistics) Of an estimator or series of estimators: robust in the presence of heteroscedasticity.


trisection point

en-noun

trisection point (plural trisection points)

The centroid of a triangle is, by definition, the point of intersection of the three medians: moreover, it is a trisection point of each of the medians.

1959 [Doubleday], William R. Gondin, Bernard Sohmer, Intermediate Algebra & Analytic Geometry Made Simple, 1967, W. H. Allen, page 147,
Find the trisection points of the line P_4P_5 in the same diagram.

1965, Paul A. White, Vector Analytic Geometry, Dickenson Publishing Company, page 73,
In triangle ABC, let E be the midpoint of BC and D the trisection point of AC nearest C.

1998, Paul J. Nahin, An Imaginary Tale: The Story of √, Princeton University Press, 2010, paperback, page 88,
And if λ=2, then P is a trisection point of P_1P_2 and
z=(z_1+2z_2)/3.
Notice that this trisection point is closer to P_2 than it is to P_1. The other trisection point, the one that is closer to P_1 than it is to P_2, occurs for λ= tfrac 1 2, which gives
z=(2z_1+z_2)/3.

(geometry) Either of two points which trisect a given line segment.


scagnostic

Coined by John W. Tukey as a blend of scatterplot + cognostic (another Tukey coinage).

(mathematics) A heuristic used to automatically analyse variables in a large dataset in order to determine which scatterplots are to be visualised in a scatterplot matrix.


Kummer's theorem

Named after Ernst Kummer, who proved it in an 1852 paper.

binomial coefficient
binomial coefficient

(mathematics) A formula for the exponent of the highest power of a prime number that divides a given binomial coefficient.


snake-in-the-box problem

the snake-in-the-box problem (canonical)

(graph theory) The problem of finding the longest possible induced path in a hypercube.


greatest generation

Popularized by the 1998 book The Greatest Generation.

Other members of the greatest generation found their life's calling in uniform, continuing to serve in the military after the war and helping it to adapt to the changing times.

1998, Tom Brokaw, The Greatest Generation, Random House, →ISBN, page 386:

My dad did the things that a boy remembers. […] He also taught me that we are here for something bigger than ourselves; he lived out the responsibilities and the sacrifices of the greatest generation to whom we owe so much.

2004 July 29, John Kerry, 2004 Democratic National Convention Acceptance Address:

What we have instead is the Greatest Generation, an idea that delineates one of the most important developments in the national mythos over the last several decades.

2013, Kenneth Rose, Myth and the Greatest Generation, Routledge, →ISBN, page 2:

The generation of people who came of age during the Great Depression and went on to fight in World War II, especially those born from the early 1900s to late 1920s.


G.I. Generation

greatest generation
greatest generation#English


allotrious

From Ancient Greek ἀλλότριος (allótrios, “foreign”) + -ous. Sense 2 arose by Folk etymology.

af

(mathematics) Pertaining to the distinct classes or roles into which an element can be allotted.


biternary

(mathematics) Having three variables and their duals


canonizant

(mathematics) A covariant that is the catalecticant of the penultimate emanant


cubicovariant

(mathematics) a covariant of a cubic form


cubinvariant

(mathematics) an invariant of a cubic form


differentiant

(obsolete, mathematics) A binary invariant


harmonizant

(mathematics) A form of bilinear invariant


involutant

(mathematics) Given two matrices M, N of order w, the involutant is the resultant of the w² scalar equations obtained by equating to zero a linear function with scalar coefficients of the w² matrices which result from multiplying 1, m m², ... mʷ⁻¹ into 1, n, n², ... nʷ⁻¹.


kenotheme

(mathematics) An intersection of n hypersurfaces in an n-dimensional projective space


meicatalecticizant

(obsolete, mathematics) Synonym of catalecticant


nullform

(mathematics) A vector that has no invariants.


pentalateral

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

(geometry) A decagon whose opposite sides are parallel.


semicovariant

(mathematics) A covariant form of a cumulant


tantipartite

(obsolete, mathematics) Synonym of multilinear


transvectant

suf

Latin <i class="Latn mention" lang="la">trānsvectus</i>

(mathematics) An invariant formed from n invariants in n variables using Cayley's Ω process.


undecic

(mathematics) Of the eleventh order or degree


coimage

(mathematics) The dual of the image of a morphism


subcanonical

(mathematics, of the topology of a category) in which every representable contravariant functor is a sheaf


wellpowered

(mathematics, of a category) in which each object has only a set of pairwise nonisomorphic subobjects


Gen-Zer

/d͡ʒɛn ˈzɛdə/

/d͡ʒɛn ˈziːɚ/

From Gen Z, shortened from Generation Z + -er.

Coordinate term: Gen-Xer

The track makes the 18-year-old Eilish the youngest artist ever to record a Bond theme, the latest in a string of achievements that has made the precocious Gen-Z-er a regular fixture on the Guinness World Records blog.

2020 February 19, Lindsay Zoladz, “A Bond Theme Suits Billie Eilish Quite Well”, in The New York Times:

Another super PAC that supports moderate Republicans, Defending Main Street, has spent over $1.2 million and is running an ad that describes Ms. Leavitt as a “woke Gen-Z’er” and plays a Snapchat video she once posted where she used crude language to refer to her viewers.

2022 September 12, Annie Karni, “Gun Shoot Will Conclude Former Trump Aide’s House Primary Race”, in The New York Times:


branchwidth

(mathematics) The minimum width of a branch-decomposition.


branch-decomposition

(mathematics) An unrooted binary tree that represents the hierarchical clustering of the edges of an undirected graph.


cocoloring

(graph theory) The assignment of a color to each vertex of a graph such that each color class forms an independent set in the graph or its complement.


diconnected

(mathematics, of a graph) Strongly connected: having every vertex reachable from every other vertex.


superconcentrator

(mathematics) A graph such that there exists one or more paths between pairs of vertices in each of two subsets of vertices


plagiogonal

(mathematics) Relating to the orthogonal group of a quadratic form


septimic

(mathematics) Synonym of septic (“of degree 7”)


supertransitive

(set theory) Describing a transitive set that contains all subsets of all its elements


cocked hat curve

en-noun

cocked hat curve (plural cocked hat curves)

From the resemblance to the hat also called a bicorn.


data dredging

(sciences, statistics) Synonym of p-hacking


affected equation

affected equation (plural affected equations)

(obsolete, mathematics) A polynomial equation in two variables


subround

(cryptography) A subsidiary stage of encryption, part of a round.


Lipschitz continuity

Lipschitz continuity (countable and uncountable, plural Lipschitz continuities)

(mathematical analysis) A strong form of uniform continuity for functions.


megahash

(computing, cryptocurrencies) A million hashes.


hashrate

Specialized hardware like a GPU is designed to do large numbers of small tasks very quickly, and so excel at hashing: typically a single GPU can produce a hashrate at least ten times faster than that of a single CPU.

2017, Adam Rothstein, The End of Money, John Murray Press, →ISBN:

(computing, cryptocurrencies) The number of proof of work hashes that a mining computer can make in a given period of time, usually a second.


terahash

(computing, cryptocurrencies) A trillion (10¹²) hashes.


outer measure

(mathematical analysis, measure theory) A measure-like function derived from some premeasure as follows: when it is applied to a set E, it yields the infimum of the premeasures of all possible premeasurable covers of E. More abstractly, a measure-like function whose domain is the power set of some underlying set, which has the properties of (1) yielding zero when applied to the empty set, (2) being monotonic (if A is subset of B then this function applied to A is less-than-or-equal to this function applied to B), (3) being subadditive (if A is a countable sum of Bⱼ's then this function applied to A is less-than-or-equal to the sum of the applications of this function to the Bⱼ's).


sliceness

(mathematics) The property of being a slice knot.


nonslice

(mathematics) Not being or relating to a slice knot.


slice knot

(mathematics) A mathematical knot in three-dimensional space that bounds a disc in four-dimensional space.


Conway knot

Kinoshita-Terasaka knot
Kinoshita-Terasaka knot

(mathematics) A particular knot with 11 crossings, related by mutation to the Kinoshita-Terasaka knot.


Jones polynomial

Jones polynomial (plural Jones polynomials)

(mathematics) A particular knot polynomial that is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^(1/2) with integer coefficients.


Chern-Simons theory

Named after mathematicians Shiing-Shen Chern and James Harris Simons.

quantum field theory
quantum field theory

(mathematics) A three-dimensional topological quantum field theory of Schwarz type, with applications in condensed-matter physics and quantum computing.


Laurent polynomial

Laurent polynomial (plural Laurent polynomials)

(mathematics) A linear combination of positive and negative powers of a variable with coefficients in a given field. They differ from ordinary polynomials in that they may have terms of negative degree.

Laurent polynomial
Pierre Alphonse Laurent


Alexander polynomial

Alexander polynomial (plural Alexander polynomials)

Discovered in 1923 by James Waddell Alexander.

(mathematics) A knot invariant which assigns a polynomial with integer coefficients to each knot type.


Alexander-Conway polynomial

Alexander-Conway polynomial (plural Alexander-Conway polynomials)

Named after James Waddell Alexander and John Horton Conway.

Alexander polynomial
Alexander polynomial

(mathematics) A version of the Alexander polynomial that can be computed using a skein relation.


Conway's constant

(mathematics) The constant 1.303577269034..., an algebraic number of degree 71, related to the limit of the growth of the look-and-say sequence.


Khovanov homology

Named after Mikhail Khovanov, who developed it in the late 1990s.

(mathematics) An oriented link invariant that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial.


double bubble conjecture

the double bubble conjecture (canonical)

(mathematics) A conjecture in the theory of minimal surfaces, stating that the shape that encloses and separates two given volumes and has the minimum possible surface area is a "standard double bubble": three spherical surfaces meeting at angles of 2π/3 on a common circle.


phase plane

(mathematics) A two-dimensional vector field of the phase between two variables


supercovariant

(mathematics, physics) Covariant in terms of supersymmetry.


superprojector

(mathematics, physics) A superspace projection operator


pseudorational

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

It is found that some pseudorational families also have the Cartesian property.

1991, Russian Mathematical Surveys, page 38:

(mathematics) Able to be expressed as the convolution ratio of distributions with compact support.


alpha-beta filter

alpha-beta filter (plural alpha-beta filters)

(signal processing) An algorithm for data estimation and smoothing, closely related to Kalman filters, based on an approximation model with two internal states, the first state obtained by integrating the value of the second state over time.


decentralized finance

An experimental form of finance that does not rely on central financial intermediaries, but instead uses smart contracts on blockchains.


DeFi

DeFi services use smart contracts, cryptocurrency, and blockchain to replace some of the services that banking has traditionally provided.

2020, Lorne Lantz, Daniel Cawrey, Mastering Blockchain, O'Reilly Media, →ISBN, page 158:

DeFi eliminates human intermediaries like brokers, bank clerks and traders, and instead uses algorithms to execute financial transactions, such as lending and borrowing.

2021 September 5, Eric Lipton, Ephrat Livni, “Crypto’s Rapid Move Into Banking Elicits Alarm in Washington”, in The New York Times, →ISSN:

decentralized finance
decentralized finance#English

(finance, cryptocurrencies) Acronym of decentralized finance.


Chinese remainder theorem

the Chinese remainder theorem (canonical)

The earliest known formulation of the theorem was in the Chinese Sun Zi's Mathematical Manual circa 400 AD.

(number theory) A theorem stating that, if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.

Chinese remainder theorem
Sunzi Suanjing


Euclidean division

Euclidean division (countable and uncountable, plural Euclidean divisions)

(arithmetic) The process of dividing one integer by another to yield a quotient and a remainder smaller than the divisor.


Turán graph

Named after the Hungarian mathematician Pál Turán.

(graph theory) A complete multipartite graph T(n,r) formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge if and only if they belong to different subsets.


basinhopping

(mathematics) A particular Monte Carlo method of optimization


dinatural

(mathematics) That can be transformed in two related ways.


dinaturality

(mathematics) The condition of being dinatural.


plane of symmetry

plane of symmetry (plural planes of symmetry)

(geometry) A planene about which a geometric figure is symmetric (by reflection)


Heun's method

ordinary differential equation
ordinary differential equation

(mathematics) A numerical procedure for solving ordinary differential equations with a given initial value, having two variants:

The improved or modified Euler method (that is, the explicit trapezoidal rule).


anarboricity

From an- + arboricity. Coined by Ronald C. Read; the etymologically inaccurate Greek prefix an- was chosen to achieve a pun on the name of Ann Arbor, where the University of Michigan is based.

(graph theory) The maximum number of edge-disjoint cyclic subgraphs of a graph G whose union is G.


rotoinversion

rotoinversion (countable and uncountable, plural rotoinversions)

Coordinate term: rotoreflection

(mathematics, crystallography) The combination of a rotation and an inversion in a point on the axis.


antiorthic

(mathematics) perpendicular to an orthic axis


antilaplacian

(mathematics) The number of which a given number is the Laplacian


circumconic

(mathematics) A conic section that passes through all the vertices of a triangle


circumconics

(mathematics) The study or construction of such shapes


circumscriptable

(mathematics, of a polygon) Having an inscribed circle that is tangent to each side


circumsphere

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

(geometry) A sphere that is circumscribed about a polyhedron and touches each of its verticies


cochleoid

(mathematics) A strophoid shaped somewhat like a snail


Euler's rotation theorem

Named after Swiss mathematician Leonhard Euler (1707–1783), who proved the theorem in 1775.

named-after

Otherwise, this approach will be valid subject to the previous rotation of the axes according to Euler's Rotation Theorem.

2003, On the Move to Meaningful Internet Systems 2003: CoopIS, DOA, and ODBASE, Proceedings, page 136:

2005, JSME International Journal: Mechanical systems, machine elements and manufacturing, Volume 48, Issue 4, Japan Society of Mechanical Engineers, page 501,
According to Euler's rotation theorem, any finite rotation transformations may be described using only three angles defined as Euler angles.

From Euler's rotation theorem we know that an arbitrary rotation in three dimensions can be parameterized by three independent variables called the Euler angles, each rotating around an orthogonal axis.

2006, Patricia J. Lee, Quantum Information Processing with Two Trapped Cadmium Ions, University of Michigan, page 40:

(geometry) A theorem which states that, in 3-dimensional space, any displacement of a rigid body such that some point on it remains fixed is equivalent to a single rotation about some axis that runs through said point.


coheight

(mathematics) The amplitude of a cosine function


coversed cosine

en-noun

coversed cosine (plural coversed cosines)

(mathematics) The trigonometric function 1 − cos(x)


cupolarotunda

(mathematics) A cupola adjoined to a rotunda


digitaddition

(mathematics) The addition of all the digits of a number; the sum so obtained. The repetition of this process eventually leads to a single digit: the digital root of the original number.


digital root

(mathematics) The single digit obtained from a number by repetitive digitaddition


pentacontaheptachoron

(mathematics) A four-dimensional polytope that has 57 cells


(mathematics) Synonym of pentacontaheptachoron


tricuspoid curve

tricuspoid curve (plural tricuspoid curves)


deltoid curve

(mathematics) A hypocycloid that has three cusps