group functor

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

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

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

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

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

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

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


rectangular hyperbola

en-noun

rectangular hyperbola (plural rectangular hyperbolas)

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


Fibonacci spiral

Fibonacci spiral (plural Fibonacci spirals)

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


linear algebraic group

linear algebraic group (plural linear algebraic groups)

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

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

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

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

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

general linear group
general linear group

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


algebraic group

algebraic group (plural algebraic groups)

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

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

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

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

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

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


regular map

Morphism of algebraic varieties
Regular map (graph theory)

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

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

Since one expects that the canonical projection f#x3A;X#x5C;rightarrowX#x2F;G is a regular map of algebraic varieties and so has closed fibres, all orbits must be closed subsets in the Zariski topology of X.

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

(algebraic geometry) A morphism between algebraic varieties.


nonacoptic

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

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

Branko Grünbaum
small stellated dodecahedron


acoptic

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

affix

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

(geometry, uncommon) Not self-intersecting.


Atiyah-Singer index theorem

Proved by Michael Atiyah and Isadore Singer in 1963.

differential geometry
differential geometry

elliptic differential operator
elliptic operator

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


Dirichlet eta function

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

mathematical analysis
mathematical analysis

Riemann zeta function
Riemann zeta function

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


kidney-shaped

en-adj

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

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

Having a shape like a kidney, that is, an approximately circular or elongated ovoid shape with an inward curve or indentation on one side.


arithmetic geometry

1994, Nancy Childress, John W. Jones (editors), Arithmetic Geometry: Conference on Arithmetic Geometry with an Emphasis on Iwasawa Theory, American Mathematical Society, back cover,
This book resulted from a research conference in arithmetic geometry held at Arizona State University in March 1993.

Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties over arbitrary rings, in particular over nonalgebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory.

2010, Jean-Louis Colliot-Thélène, Peter Swinnerton-Dyer, Paul Vojta, Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School, Springer, page v:

Their^([semistable models']) occurrence in arithmetic geometry is ubiquitous for the study of ℓ-adic or p-adic cohomology, and of Galois representations.

2013, Matthieu Romagny, “Models of Curves”, in Pierre Dèbes, Michel Emsalem, Matthieu Romagny, A. Muhammed Uludağ, editors, Arithmetic and Geometry Around Galois Theory, Springer, page 149:

Arakelov geometry is one of the branches in arithmetic geometry.

2014, Atsushi Moriwaki, Arakelov Geometry, American Mathematical Society, page vii:

(mathematics) A developing branch of mathematics in which techniques of algebraic geometry are applied to number theory, in particular being concerned with schemes of finite type over the spectrum of the ring of integers, Spec(ℤ).


semiparabolic

(geometry) Having the form of a semiparabola.


cocircularity

(geometry) The condition of being cocircular


cocircular

(geometry, of a set of points) Lying on the same circle


logarithmic derivative

en-noun

logarithmic derivative (plural logarithmic derivatives)

From this Coulomb function, we can then calculate the logarithmic derivative of U²(r) at the eigenvalue for any value of r outside the core. If there exists a radius R lying in the Coulomb region and sufficiently small that the logarithmic derivative at R is a smooth function of the energy, we can obtain the logarithmic derivative for arbitrary energies within a reasonable range by interpolating it between the eigenvalues.

1955, Frank S. Ham, “The Quantum Defect Method”, in Frederick Seitz, David Turnbull, editors, Solid State Physics, Volume 1, Academic Press, page 138:

In economics, it is popular to consider the logarithmic derivative (#x5C;ln#x5C;varphi(t))'#x3D;#x5C;varphi'(t)#x2F;#x5C;varphi(t), as a convenient growth measure.

1994, Bjarne S. Jensen, The Dynamic Systems of Basic Economic Growth Models, Kluwer Academic, page 329:

In this section we consider the R-matrix method in which a solution is first found in an inner region 0 ≤ r ≤ a by expanding in a basis set yielding the logarithmic derivative of the wave function on the boundary r = a.

1995, Philip G. Burke, Charles J. Joachain, Theory of Electron-Atom Collisions: Part 1: Potential Scattering, Plenum Press, page 86:

(calculus, mathematical analysis) Given a real or complex function f, the ratio of the value of the derivative to the value of the function, (f')/f, regarded as a function.


field extension

field extension (plural field extensions)

A Galois field extension may be defined in terms of its Galois group (Section 2) or in terms of the internal structure of the extension (Section 3).

1974, Thomas W. Hungerford, Algebra, Springer, page 230:

1998, David Goss, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
Note that the extension of L obtained by adjoining all division points of ψ includes at most a finite constant field extension.

2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.

(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.

kuntalaajennus

pair of fields such that one is a subfield of the other


Plücker coordinate

Plücker coordinate (plural Plücker coordinates)

Introduced by Julius Plücker in the 19th century.

(geometry) Any of six homogeneous coordinates assigned to a line in projective 3-space, used in geometric algebra and computer graphics.


Malfatti circle

Malfatti circle (plural Malfatti circles)

Named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.

(geometry) Any of the three circles inside a given triangle such that each circle is tangent to the others and to two sides of the triangle.


vertical angle

vertex angle

(geometry) Any of the opposite angles formed by intersecting lines.

vrcholový úhel

(geometry) The angle at the apex of an isosceles triangle or cone.


tetrahelix

tetrahelix (plural tetrahelixes or tetrahelices)

tetrahelixes (plural)
tetrahelices (plural)

(geometry) A helix formed from tetrahedra linked face to face.


finite geometry

finite geometry (countable and uncountable, plural finite geometries)

But the work of von Staudt, Fano, and Veblen led even to the construction of finite geometries having only finitely many points and lines. Originally having only incidence relations, these finite geometries have been extensively developed through the introduction of suitable axioms of order and congruence by Järnefelt and Kustaanheimo, who have proposed their use in physics to solve certain paradoxes due apparently to the use of continuous variables.

1980, Judson Chambers Webb, Mechanism, Mentalism and Metamathematics: An Essay on Finitism, Springer, page 39:

A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed.

1999, S. Ball, “Polynomials in Finite Geometries”, in J. D. Lamb, D. A. Preece, editors, Surveys in Combinatorics, 1999, Cambridge University Press, page 17:

Finite geometries, such as Euclidean and projective geometries, are powerful mathematical tools for constructing error-control codes.

2009, William Ryan, Shu Lin, Channel Codes: Classical and Modern, Cambridge University Press, page 430:

(geometry) Any geometric system that has only a finite number of points.

The coordinates of geometry were as incompatible with representation theory as were the splitting fields of representation theory with finite geometry.

1983, Robert A. Liebler, “Combinatorial Representation Theory and Translation Planes”, in Norman L. Johnson, Michael J. Kallaher, Calvin T. Long, editors, Finite Geometries: Proceedings of a Conference, CRC Press, page 307:

We give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets.

1996, Dieter Jungnickel, “Maximal Sets of Mutually Orthogonal Latin Squares”, in S. Cohen, H. Niederreiter, editors, Finite Fields and Applications: Proceedings of the 3rd International Conference, Cambridge University Press, page 129:

Finite geometry is a broad area of research, but much of it is not Euclidean in flavour and no theory seems to match that of oriented matroids in establishing combinatorial Euclidean geometry.

2007, Michael B. Smyth, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian E. Pratt-Hartmann, Johan F. A. K. van Benthem, editors, Handbook of Spatial Logics, Springer, page 787:

(geometry, uncountable) The branch of geometry that concerns geometric systems with only a finite number of points.


imaginary geometry

Calque of Russian мни́мая геометрия (mnímaja geometrija), coined by mathematician Nicolai Lobachevsky, probably influenced by мни́мое чи́сло (mnímoje číslo, “imaginary number”), although his precise reasoning is unknown.

calque

мни́мая геометрия

1879, Frederick Pollock, Introduction, Part 1: Biographical, Leslie Stephen, Frederick Pollock (editors), William Kingdon Clifford, Lectures and Essays, Volume 1, Cambridge University Press, page 14,
He liked talking about imaginary geometry, as a matter of pure amusement, to anyone interested in it. But at the same time he attached a serious import to it.

Claim 7. The Euclidean law of cosines holds for any infinitesimal triangle #x5C;triangleABC in imaginary geometry.

2011, Seth Braver, Lobachevski Illuminated, Mathematical Association of America, page 188:

non-Euclidean geometries
non-Euclidean geometry

(geometry) Absolute geometry, an axiomatised geometry in which the parallel postulate is absent and not replaced by an alternative, and of which Euclidean geometry and some non-Euclidean geometries are subtypes.

мни́мая геометрия

geometry with no parallel postulate — see also absolute geometry


quasi-satellite

(astronomy) An object in a specific type of coorbital configuration (1:1 orbital resonance) with a planet, causing it to stay close to the planet over many orbital periods, despite not being a true satellite.


semimeridian

surface of revolution
surface of revolution

(geometry) The intersection of a surface of revolution and a half-plane through its centre


complex projective line

en-noun

complex projective line (plural complex projective lines)

By way of an example, we consider in detail the complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹.[…]Thus via the function w#x5F;0 the complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹ becomes identified with the "extended complex plane" (i.e. the ordinary complex plane with an additional "point at infinity").
2..2.1 Theorem The complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹ is diffeomorphic to the 2-dimensional sphere S².

1985, B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, translated by R. G. Burns, Modern Geometry — Methods and Applications: Part II, Springer, page 18:

Finally, C is interested in the geometry of the complex projective line P¹(#x5C;C).

2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14:

Hence, C(#x5C;Gamma) is an uncountable union of complex projective lines.

2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups, Springer (Birkhäuser), page 142:

As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere #x5C;dot#x7B;S#x7D;² ̇provides a conformal model for the complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹.

2018, Norman W. Johnson, Geometries and Transformations, Cambridge University Press, page 194:

equivalence relation
equivalence relation

(analytic geometry, projective geometry) A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation "(α, β) ≡ (λα, λβ) for all nonzero complex λ".


complex line

1990, R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Volume 1 Function Theory, Wadsworth & Brooks/Cole, page 102,
However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in C ⁿ through a point A∈ C ⁿ in the direction of a vector B∈ C ⁿ is the one-dimensional complex submanifold of C ⁿ described parametrically as A+tB:t∈ C.

A complex line in #x5C;Cⁿ is a set of the form l#x3D;#x5C;#x7B;z#x3A;z#x3D;a#x2B;#x5C;lambdab,#x5C;lambda#x5C;in#x5C;C#x5C;#x7D;, where a and b are fixed points in #x5C;Cⁿ, with b#x5C;ne 0. Let us say that l is the complex line through a in the “direction” b.

2014, Paul M. Gauthier, Lectures on Several Complex Variables, Springer (Birkhäuser), page 39:

The question of finding different familes^([sic]) of complex lines sufficient for holomorphic extension was put in [12]. Clearly, the family of complex lines passing through one point is not enough. As shown in [16], the family of complex lines passing through a finite number of points also, generally speaking, is not sufficient.

2018, Bairambay Otemuratov, “A Mulitidimensional Analogue of Hartogs's Theorem on n-Circular Domains for Integrable Functions”, in Zair Ibragimov, Norman Levenberg, Utkir Rozikov, Azimbay Sadullaev, editors, Algebra, Complex Analysis, and Pluripotential Theory: 2 USUZCAMP, 2017, Springer,, page 110:

(complex analysis, analytic geometry) A 1-dimensional affine subspace of a vector space over the complex numbers.

complex plane

affine subspace that is 1-dimensional over the complex numbers


projective line

projective line (plural projective lines)

A projective line in P²( R ), consists of the projective points defined by coplanar directions in R ³.
In other words, just as the points in P²( R ) correspond to one-dimensional subspaces in R ³, the projective lines correspond to two-dimensional subspaces in R ³.

2007, Unnamed translator, Ana Irene Ramírez Galarza, José Seade, Introduction to Classical Geometries, [2002, Introducciòn a la Geometria Avanzada], Springer (Birkhäuser), page 97, In P2( R ), the projective lines are defined by two projective points, that is, by two linearly independent directions of R ³; if we take one vector for each direction, the two vectors generate a plane through the origin in R ³, that is, a subspace of dimension 2, and a projective line can be defined as follows

2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique] Springer, page 37,
Consider the projective line mathbf P¹, with homogeneous coordinates x and t and open sets U_0(x ne 0) and U_1(t ne 0).

The projective line is related to the Kronecker algebra,[…]In general one has to deal with the so-called weighted case which leads to the study of the canonical algebras and to the weighted projective lines (as Ringel pointed out in his survey [93]).

2009, Dirk Kussin, Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras, American Mathematical Society, page 13:

(projective geometry) A line that includes a point at infinity; a line in a projective space; a projective space of dimension 1.


trilobular

(geometry) Having a shape that is broadly triangular but is rounded, with three lobular segments conjoined.


quadrisector

(geometry) A line or curve that quadrisects something.


Borel transform

Borel transform (plural Borel transforms)

(mathematics) A transform used in Borel summation.

(mathematics) A generalization of this in Nachbin's theorem.


equidissectable

(geometry) Of a polygon: able to be divided into triangles of equal area.


equidissection

equidissection (countable and uncountable, plural equidissections)

(geometry) The division of a polygon into triangles of equal area.


Monsky's theorem

(geometry) A theorem stating that it is not possible to dissect a square into an odd number of triangles of equal area.


spiric

(geometry) A curve, the plane section of a torus.


sigmoid function

sigmoid function (plural sigmoid functions)

1995, Jun Han, Claudio Moraga, The Influence of the Sigmoid Function Parameters on the Speed of Backpropagation Learning, José Mira, Francisco Sandoval (editors), From Natural to Artificial Neural Computation: International Workshop on Artificial Neural Networks, Proceedings, Springer, LNCS 930, page 195,
[The] [s]igmoid function is the most commonly known function used in feed forward neural networks because of its nonlinearity and the computational simplicity of its derivative.

The first significant new insight from gamma spatial analysis emerged on re-examination of the sigmoid function representing bilateral saturation.

2012, Walter Freeman, Neurodynamics: An Exploration in Mesoscopic Brain Dynamics, Springer, page 241:

2016, Roumen Anguelov, Svetoslav Markov, Hausdorff Continuous Interval Functions and Applications, Marco Nehmeier, Jürgen Wolff von Gudenberg, Warwick Tucker (editors), Scientific Computing, Computer Arithmetic, and Validated Numerics: 16th International Symposium, SCAN 2014, Springer, LNCS 9553, page 10,
Sigmoid functions find multiple applications to neural networks and cell growth population models [14,20]. A sigmoid function on R with a range [a,b] is defined as a monotone function s(t): R →[a,b] such that lim _(t→-∞)s(t)=a, and lim _(t→∞)s(t)=b.

(mathematical analysis, statistics) Any of various real functions whose graph resembles an elongated letter "S"; specifically, the logistic function y=(eˣ)/(eˣ+1)=1/(1+e⁻ˣ).


shape operator

Gaussian curvature is a prime example; although defined in terms of shape operators, it belongs to this intrinsic geometry, since it passes the test of isometric invariance.

1966, Barrett O'Neill, Elementary Differential Geometry, Academic Press, page 304:

Timelike surfaces have symmetric shape operators which can be put into one of three canonical forms on a fixed tangent space with respect to an orthonormal basis:[…].

2006, Balkan Journal of Geometry and Its Applications, volumes 11-12, page 41:

We have already seen that a plane has zero shape operator. Intuitively, since the shape operator detects the change in the unit normal U, a zero shape operator for a surface M should imply that M is a plane. This is verified by the following result.

2007, John Oprea, Differential Geometry and Its Applications, Mathematical Association of America, page 85:

(geometry, differential geometry) The differential of the Gauss map of an oriented surface at a given point on the surface.


cone-shaped

en-adj

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

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

Having the shape of or resembling the shape of a cone.


Smarandache function

Named after Florentin Smarandache, who rediscovered the function in 1980.

(number theory) A function, denoted by S(n) for some positive integer n, that yields the smallest number s such that n divides the factorial s!. For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, so S(8) = 4.


conjugate axis

(geometry) Either of two axes in a conic section, such that each is parallel to the tangent at the extremity of the other.


geometric analysis

Papers are submitted for publication by sending three hard (printed) copies to The Journal of Geometric Analysis, Department of Mathematics, Washington University,[…].

1999, The Journal of Geometric Analysis, volume 9, CRC Press, page 273:

2013, Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, Fernando Schwartz, Gilbert Weindtein (editors) Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, American Mathematical Society.

In 2012, thematic programs on Geometric Analysis and Spectral Theory and on Moduli spaces, Extremality and Global Invariants were held at the CRM. There were several workshops on cutting-edge topics in geometric analysis and spectral theory, notably on the Geometry of Eigenvalues and Eigenfunctions, on Manifolds of Metrics and Probabilistic Methods in Geometry and Analysis and on Spectral invariants on non-compact and singular spaces.

2014, Pierre Albin, Dmitry Jakobson, Frédéric Rochon, “Preface”, in Geometric and Spectral Analysis, American Mathematical Society, page vii:

partial differential equation
partial differential equation

(mathematical analysis, geometry, topology) A branch of mathematics in which techniques from differential geometry are used in the analysis and solution of partial differential equations, and vice versa.


decangle

/ˈdɛkˌæŋɡəl/

And from hence also shall be the geodesy of the Icosaedrum. For the finding out of the heighth of the pyramis, there is the semidiagony of the side of the decangle and the halfe ray of the circle: But the side of the decangle is a right line subtending the halfe periphery of the side of the quinquangle, or else the greater segment of the ray proportionally cut.

1636, Peter Ramus, translated by Peter Bedwell, The Way To Geometry: [Being Necessary and Usefull for Astronomers, Enginees, Geographers,. Architects, Land-meaters, Carpenters, Sea-men & Etc.], pages 277–278:

The ſide of a Hexagon or Radius being given, the ſide of a Decangle or ſubtenſe of 30 deg. is alſo given ; for by the 19 of the ſecond, the Semi-radius being deducted from the ſquare root of the ſquares of Radius and the half Radius added together, the remainer is the ſide of a Decangle.

1658, John Newton, Trigonometria Britanica, page 9:

Subtract the ſide of a decangle from the diameter, the remainer multiplied by the radius ,ſhall be the ſquare of one ſide of a Pentagon, whoſe ſquare root ſhall be the ſide itſelf, or ſubtense of 72 degrees.

1671, John Newton, Institutio mathematica: or, A mathematical institution, page 54:


quinquangle

/ˈkwɪŋˌkwæŋɡəl/

It is about 2,500 yards in circuit, is built of red stone, and, according to Von Orlich, is now " a bastioned quinquangle ; the ancient walls with semicircular bastions face the two streams ; the land side is quite regular, and consists of two bastions, and a half-bastion with three ravelins," and stands higher than any ground in face of it.

1858, Edward Thornton, A Gazetteer of the Territories Under the Government of the East India Company and of the Native States on the Continent of India, W. H. Allen & Co., page 22:

Kenett states that the military works still known by the name of Tadmarten Camp and Hook-Norton Barrow were cast up at this time ; the former, large and round, is judged to be a fortification of the Danes, and the latter, being smaller and rather a quinquangle than a square, of the Saxons.

1881, John Kirby Hedges, The history of Wallingford, volume 1, page 170:

(archaic) Synonym of pentagon, a regular 5-sided figure.


diangle

/ˈdaɪˌæŋɡəl/

The diangle ABCD indicates the entire duration of life, A and C being its commencement and its termination.

1853, “The Science of Medicine”, in The Journal of Psychological Medicine and Mental Pathology, page 241:

Now the wedge 1,2 is mapped into the exterior of the circular diangle C,S₁,1,O,2,S₂,C; both regions are indicated by a shading of the boundary. We now seek the images of the reflected wedges.

1949, Arnold Sommerfeld, translated by Ernst G. Strauss, Partial Differential Equations in Physics, volume 1, Academic Press, page 319:

Let B be a region of F bounded by a closed polygon comprised of segments of geodesics of surface F, and let hₙ be a sequence of nondegenerate diangles of region B whose sides are geodesics of region B and which converge to diangle hₙ.

1969, Siberian Mathematical Journal, Kluwer/Plenum Publishers, page 107:


biangle

/ˈbaɪˌæŋɡəl/

If n planes be drawn through any axis of the right-vector, each of which makes angles π/n with the planes on either side of it, the whole of space is divided into n congruent figures which may be called biangles, the space between any two adjacent planes being easily seen to be continuous with the vertically opposite space between them.

1895, W. Burnside, “Kinematics of Non-Euclidean Space”, in Proceedings of the Edinburgh Mathematical Society, page 46:

Then the right biangle CABD and the oblique biangle CPND are equivalent since the triangles API and BNI are congruent.

1912, Proceedings of the Edinburgh Mathematical Society, volumes 30-32, Scottish Academic Press, page 35:

It is therefore natural to consider whether precisely this concept of a straight line, with which Kant¹¹ too was familiar, is the reason why Kant in the one passage says that the concept of a straight biangle is free of contradiction.

2012 December 6, Gerold Prauss, “Kant and the Straight Biangle”, in Enno Rudolph, Ion-Olimpiu Stamatescu, editors, Philosophy, Mathematics and Modern Physics: A Dialogue, page 228:

non-Euclidean geometry
non-Euclidean geometry#English

A digon or bigon; a two-sided shape (especially in non-Euclidean geometry)


witch of Agnesi

en-noun

witch of Agnesi (plural witches of Agnesi)

Named after Italian mathematician Maria Gaetana Agnesi, who wrote about such curves, and from a mistranslation of Italian versiera, the term she used for them which means "rope", but which is spelled the same as versiera (“witch”).

(mathematics) A cubic plane curve defined from two diametrically opposite points of a circle.


angle of torsion

angle of torsion (plural angles of torsion)

(geometry) The indefinitely small angle between two consecutive osculating planes of a curve of double curvature.


photon sphere

The radius of the photon sphere is also the lower bound for any stable orbit.

For a Schwarzschild (non-spinning) black hole, the radius of the photon sphere is 1.5 times the Schwarzschild radius.

The redshift of light leaking outward from the photon sphere is #x5C;sqrt#x7B;3#x7D;-1#x3D;0.732. All light rays approaching a black hole closer than #x5C;sqrt#x7B;3#x7D; times the radius of the photon sphere spiral inwards and are captured (see Figure 13.5).

2000, Edward Harrison, Cosmology, 2nd edition, Cambridge University Press, page 250:

For a Kerr black hole though, one that has spin, the situation is different for the orbits near the black hole. In particular, there are two photon spheres, in contrast with the one photon sphere around a stationary Schwarzschild black hole. The outermost sphere is for photons that are orbiting oppositely to the direction of rotation of the black hole (the ones we say are on retrograde orbits). Inside this is the photon sphere for photons travelling in the same sense around the black hole as it is rotating (on prograde orbits). For a very slowly rotating black hole that isn't so very different from a Schwarzschild black hole, these two photon spheres are very nearly co-spatial. For black holes of increasing spin, these surfaces are increasingly further apart.

2015, Katherine Blundell, Black Holes: A Very Short Introduction, Oxford University Press, page 44:

2017, Lori Gardi, The Mandelbrot Set as a Quasi-Black Hole, David de Hilster (editor), John Chappell Natural Philosophy Society: 2017 Proceedings, John Chappell Natural Philosophy Society, page 65,
The lesser known region just outside the black hole is referred to as the photon sphere (outer grey region in Figure 9). The photon sphere is a region of space just outside the black hole where photons are forced to travel in complex orbits due to the extreme curvature of space within this region [2]. […] Finally, the boundary that exactly separates the black hole from the photon sphere is referred to as the event horizon (white circle surrounding the black region in Figure 9).

The region inside the photon sphere has a diameter of 1.5× Schwarzschild diameter D(s), but due to the curvature, this region gets magnified to 2.6× D(s).

2022, Wouter Schmitz, Understanding Relativity, Springer, page 277:

(of a black hole) A spherical limit around a black hole at such distance that any photon meeting it tangentially is constrained to travel in a circular orbit; (more loosely) the region between this limit and the event horizon.

sphère de photons

spherical boundary surrounding a black hole in which photons travel in circular orbits


photon circle

From the path of photon becoming circular, instead of the usual hyperbolic orbits.

(of a black hole) Synonym of photon sphere


last photon orbit

last photon orbit (plural last photon orbits)

From being the closest tangential approach to a black hole, where a photon's path does not eventually fall through the event horizon.


semiordinate

(geometry) Half a chord bisected by the transverse diameter of a conic.


Klein quadric

(mathematics) The lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a hyperbolic quadric, Q, known as the Klein quadric.


Klein quartic

Named after German mathematician Felix Klein, who described it in 1878.

A compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed.


common logarithm

common logarithm (plural common logarithms)


Hurwitz surface

Hurwitz surface (plural Hurwitz surfaces)

(geometry) A compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms.


Chern class

(mathematics, topology, geometry) Any of a group of characteristic classes associated with complex vector bundles, with applications in physics, string theory, etc.


Gaussian curvature

Gaussian curvature (countable and uncountable, plural Gaussian curvatures)

(differential geometry) The product of the principal curvatures of a surface at a given point.


semiconcentric

(geometry) Having the form of concentric semicircles


projective space

projective space (plural projective spaces)

(geometry) A kind of geometrical space in which only relation(s) of incidence between points and lines matter; in such a space distances or angles do not matter and parallelism (as in an affine space) does not matter either.


affinely independent

linearly independent
linearly independent

(geometry) Said of a set of points in an affine space: the property that the vectors issuing from an arbitrarily chosen point to the rest of the points are linearly independent.


projective frame

projective frame (plural projective frames)

affinely independent
affinely independent

(geometry) The projection onto an n − 1 dimensional hyperplane (which is extended so as to be projective) of a set of n + 1 affinely independent n-dimensional points (called an affine frame).


real projective plane

real projective plane (plural real projective planes)

(geometry) A three-dimensional vector space over the field of real numbers in which one-dimensional linear subspaces are considered as “points” and two-dimensional linear subspaces are considered as “lines”. Equivalently, a sphere in which pairs of antipodal points are considered as “points” and great circles are considered as “lines”.


homogeneous coordinate

homogeneous coordinate (plural homogeneous coordinates)

(geometry) Any one of a set of n + 1 coordinates which non-uniquely identify a point in an n-dimensional projective space; when such coordinates are all multiplied by the same scalar, then the set of them still identifies the same point.


trans-lunar injection

trans-lunar injection (plural trans-lunar injections)

A maneuver that moves a spacecraft from orbit around the Earth onto a trajectory towards the Moon.


rhombidodecahedral

(geometry) Relating to a rhombidodecahedron


contact manifold

contact manifold (plural contact manifolds)

(mathematics, contact geometry) A smooth manifold of odd dimensionality in contact geometry.


basis spline

Basis splines can be used for curve-fitting and numerical differentiation of experimental data, and for shaping curves in computer-aided design.

(mathematics) A spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of basis splines of that degree.


Napierian logarithm

Napierian logarithm (plural Napierian logarithms)

(mathematics) The logarithm to the base e, named after John Napier.


Miquel's theorem

(geometry) A result in geometry concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides.


Kepler-Poinsot polyhedron

Kepler-Poinsot polyhedron (plural Kepler-Poinsot polyhedra)

Named after Johannes Kepler and Louis Poinsot; Poinsot rediscovered Kepler's figures in 1809.


unduloid

(geometry) A surface of revolution of an elliptic catenary


orthotomic circle

orthotomic circle (plural orthotomic circles)

(geometry) A circle that cuts three other given circles at right angles.


biarc

A geometric figure consisting of two circular arcs tangent at their common endpoint.


great-circle arc

great-circle arc (plural great-circle arcs)

This causes a greater deviation of the ground trace from a great-circle arc due to the increased effect of Earth's rotation.

2006, Irfan Ali, Pierino G. Bonanni, Naofal Al-Dhahir, John E. Hershey, Doppler Applications in LEO Satellite Communication Systems, Springer Science & Business Media, →ISBN, page 66:

The two axes connected by the solid but incomplete great-circle arc were measured along a single sahlfold axial surface on opposite sides of a tubular fold ; ...

2012, E. Hansen, Strain Facies, Springer Science & Business Media, →ISBN, page 85:

... it may be swung in a meridian-plane, and a great-circle arc having a radial pivot in said meridian-plane, ...

1898, Official Gazette of the United States Patent Office:

Any great-circle arc of length #x5C;theta between zero and #x5C;pi is uniquely identified by its two end points, say a and b.

2002, Leo Dorst, Chris Doran, Joan Lasenby, Applications of Geometric Algebra in Computer Science and Engineering, Springer Science & Business Media, →ISBN, page 81:


Walsh function

Named after the American mathematician Joseph L. Walsh.

(mathematics) In harmonic analysis, any of a complete orthogonal set of functions that can be used to represent any discrete function.


duoprism

(geometry) A polytope resulting from the Cartesian product of two polytopes.


pseudorhombic

(geometry, crystallography) describing a structure that is almost rhombohedral


Möbius function

Named after August Ferdinand Möbius (1790–1868), a German mathematician and theoretical astronomer who introduced it in 1832.

(combinatorics) A function from the natural numbers to the set {−1, 0, 1} which maps perfect squares to 0, prime numbers to −1, and is multiplicative.


Radon point

(geometry) A point in the intersection of convex hulls specified by Radon's theorem.


Radon's theorem

(geometry) A theorem on convex sets, stating that any set of d + 2 points in Rᵈ can be partitioned into two sets whose convex hulls intersect.


Helly's theorem

Named after Eduard Helly, who made the discovery in 1913.

(geometry) A basic result in discrete geometry on the intersection of convex sets, which gave rise to the notion of a Helly family.


morphogeometry

The geometry of form (in any of various contexts)


dragon curve

Any of a family of self-similar fractal curves that can be approximated by recursive methods.


twindragon

A form of dragon curve, constructed by placing two Heighway dragon curves back to back.


vanishing ideal

vanishing ideal (plural vanishing ideals)

(algebraic geometry) An ideal consisting of all polynomials (belonging to a certain polynomial ring) which zero out within a given algebraic variety.


Lissajous orbit

Lissajous orbit (plural Lissajous orbits)

A quasiperiodic orbital trajectory that an object can follow around a Lagrangian point of a three-body system without requiring any propulsion.


Lyapunov orbit

In the restricted three-body problem, a curved path around a Lagrangian point that lies entirely in the plane of the two primary bodies.


orbital mechanics

Astrodynamics; the study of the movement of vehicles in space.


Kessler syndrome

Proposed by NASA scientist Donald J. Kessler in 1978.

A theoretical scenario in which the density of objects in low Earth orbit due to space pollution is high enough that collisions between objects could cause a cascade in which each collision generates debris that increases the likelihood of further collisions.


Baire function

(mathematics) A function obtained from a continuous function by transfinite iteration of the operation of forming pointwise limits of sequences of functions.


Delaunay triangulation

Delaunay triangulation (countable and uncountable, plural Delaunay triangulations)

Named after Boris Delaunay for his work on this topic from 1934.

(geometry, computer graphics) For a given set P of discrete points in a plane: a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P).


rotational symmetry

rotational symmetry (plural rotational symmetries)

(geometry) The property a shape has when it looks the same after some rotation by a partial turn.


ISCO

w:en:International Standard Classification of Occupations

Abbreviation of International Standard Classification of Occupations.


tridecahedron

tridecahedron (plural tridecahedra or tridecahedrons)

tridecahedra (plural)
tridecahedrons (plural)

(geometry) A polyhedron with thirteen faces.


enneacontagon

(geometry) A polygon with 90 sides and 90 angles.


octacontagon

(geometry) A polygon with 80 sides and 80 angles.


subtendent

Lastly, if you put BC the lesser of the sides, including the right angle for Radius, the greater side, AC, including the right angle is the tangent of ABC, the greater acute angle opposite; and the subtendent A B, is the secant of the same acute angle.

1642, Bartholomäus Pitiscus, Trigonometry: Or, The Doctrine of Triangles., page 91:

The very Subtendent line is found of an odde part; but the very subtendent of an even part is not found at one operation, but onely the Square of the Subtendent; and the greater the number of parts be, so much harder it willbe to finde out the subtendent.

1657, Johannes Jonstonus, An History of the Constancy of Nature:

The first figure, Vale, hath but one mood, and therefore of as great extent as it selfe, which is Upalem; whose nature is to let us know, whan a plane right angled triangle is given us to resolve, whose subtendent and one of the obliques is proposed, and one of the ambients required, that we must have recourse unto its resolver, which being Rad_U_Sapy→Yr sheweth, that if we joyne the artificiall sine of the angle opposite to the die demanded with the Logarithm of the subtendent, the summe searched in the canon of absolute numbers will afford us the Logarithm of the side required.

1834, Thomas Urquhart, Lord Thomas Maitland Dundrennan, The Works of Sir Thomas Urquhart of Cromarty, Knight, page 81:

For example, even if there were no possible quantity, mathematicians could say that if it was possible to make a right angled triangle, it was something assured that the hypotenuse or the subtendent of the right angle would make a square equal to the squares of the other two sides

1994, Alistair Cameron Crombie, Styles of Scientific Thinking in the European Tradition, page 840:

(geometry, trigonometry) The side that subtends (is opposite) a specified angle;


subtendent

en-adj

subtendent (comparative more subtendent, superlative most subtendent)

more subtendent (comparative)
most subtendent (superlative)

Captain Herschel expresses doubts of this, but it is evident that as the cusps closed both must have been within the space subtendent by half the length of the slit.

1875, Lieutenant-Colonel J.F. Tennant, “Report on Observations of the Total Elipse of the Sun”, in Memoirs of the Royal Astronomical Society, volume 42, page 9:

Make subtendent lines to them, and cut away the arches; then erect plain Looking-glasses, that may be of the same latitude, and of the same parallel lines, and the same longitude; glew them fast together, and fit them lo, that they may not be pulled asunder, as they are joyned long-ways, and erected upon a plain superficies.

1957, Giambattista della Porta, Natural Magick, page 359:

An outermost, “ballooning” membrane was seen to form dilated vesicles above the subtendent “armour” layer.

1988, British Phycological Journal, page 53:

(geometry) That subtends; that lies opposite to a specified angle or point.


Sylvester-Gallai theorem

Named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.

(geometry) A theorem stating that, given a finite number of points in the Euclidean plane, either all the points lie on a single line, or there is at least one line which contains exactly two of the points.

James Joseph Sylvester
Sylvester-Gallai theorem
Tibor Gallai


de Bruijn-Erdős theorem

Named after the mathematicians Nicolaas Govert de Bruijn and Paul Erdős.

(geometry) A theorem that states a lower bound on the number of lines determined by n points in a projective plane.


Pasch's theorem

Stated in 1882 by the German mathematician Moritz Pasch.

(geometry) A result in plane geometry which cannot be derived from Euclid's postulates. It states that, given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).


Pasch's axiom

Its essential role was discovered in 1882 by the German mathematician Moritz Pasch.

(geometry) A statement in plane geometry, used implicitly by Euclid, which cannot be derived from Euclid's postulates. It states that, if a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.


vertex angle

(geometry) An angle associated with a vertex of an n-dimensional polytope.


cutplane

(geometry) A plane that intersects a solid shape


Cauchy horizon

(physics) A light-like boundary of the domain of validity of a Cauchy problem. One side of the horizon contains closed space-like geodesics and the other side contains closed time-like geodesics.


ball and stick

The handwriting sense derives from the resemblance of the circular and linear elements of the letters to a ball and a stick.

(education) A style of handwriting in which the letters are not joined and feature straight lines and circular arcs; block letters.

ball-and-stick model
ball-and-stick model

(physical chemistry) The representation of chemical substances in a three-dimensional way using ball-and-stick models.


bifolium

Cartesian coordinate
Cartesian coordinate

(geometry) A quartic plane curve whose equation in Cartesian coordinates is (x²+y²)²=ax²ʸ.


dodecagram

A star polygon that has twelve vertices.


heptadecahedron

en-noun

heptadecahedron (plural heptadecahedra or heptadecahedrons)

heptadecahedra (plural)
heptadecahedrons (plural)

(geometry) A polyhedron with seventeen faces.


hexadecahedron

hexadecahedron (plural hexadecahedra or hexadecahedrons)

hexadecahedra (plural)
hexadecahedrons (plural)

(geometry) A polyhedron with sixteen faces.


pentadecahedron

pentadecahedron (plural pentadecahedra or pentadecahedrons)

pentadecahedra (plural)
pentadecahedrons (plural)

(geometry) A polyhedron with fifteen faces.


enneadecahedron

enneadecahedron (plural enneadecahedra or enneadecahedrons)

enneadecahedra (plural)
enneadecahedrons (plural)

Guy [24] and Knowlton have independently discovered a uniform unistable enneadecahedron.

Klamkin noted that [1] contains an improved solution by R.K. Guy to the polyhedron problem, consisting of a doubly truncated 17-sided convex prism, making it an enneadecahedron (19 faces). […] Ogilvy referred to his book [4], where he describes Guy's enneadecahedron.

1980, Crux Mathematicorum, pages 28, 29:

Our solid is a 17-sided prism, half of whose section is illustrated in Fig. 5, truncated obliquely as in the side view of Fig. 6, so it is an enneadecahedron.

1990, Murray S. Klamkin, Problems in Applied Mathematics: Selections from SIAM Review, Society for Industrial and Applied Mathematics, →ISBN, page 478:

(geometry) A polyhedron with nineteen faces.


base angle

(geometry) Either of the two angles that intersect with the base of a triangle.


rhombidodecahedron

(geometry) A nonconvex uniform polyhedron with square faces and sharing symmetry of a dodecahedron.


Gougu

Transliteration of Chinese 勾股 (gōugǔ), literally the longer and shorter legs of a right triangle.


Gougu theorem

Borrowed from Chinese 勾股定理 (gōugǔ dìnglǐ), literally the longer and shorter legs of a right triangle theorem.


TREE

Kruskal's tree theorem
Kruskal's tree theorem

(mathematics) An extremely fast-growing function based on Kruskal's tree theorem.


apeirotope

(geometry) A generalized polytope having infinitely many facets.


triacontadigon

(geometry) A polygon with thirty-two sides and thirty-two angles.


dihectagon

Dihectagon — 200 Sides

2014, Charles W. Nigh, Geometric Shapes & Forms:

A Dihectagon is a 200 sided shape. Made using the polygon tool.

2016 February 29, Greg N., “Dihectagon”, in 3D Warehouse:

Figure 4.57 A dihectagon looks very similar to a circle. Figure 4.57 shows a 200-sided shape (a dihectagon). It looks very much like a circle now, but it isn't

2017, Ed Southall, Yes, But Why? Teaching for Understanding in Mathematics, page 90:

I understand what a hectogon (a one hundred-sided figure) is for instance, but cannot imagine one, at least not in such a way that my image would be distinct from my image of a dihectagon (see Descartes 2008 [1641]: 51-52).

2019, Kevin Lynch, “Knowing What an Experience Is Like and the Reductive Theory of Knowledge‐wh”, in Analytic Philosophy, volume 61, number 3, →DOI:

(geometry, rare) A polygon with two hundred sides and two hundred angles.

200-угольник

двухсотугольник


octadecahedron

octadecahedron (plural octadecahedra or octadecahedrons)

octadecahedra (plural)
octadecahedrons (plural)

The cell structure was considered to consist of spherical holes but for the purposes of calculation, the cells were assumed to be octadecahedra. ln this model, a compressive force along the principal axis of the octadecahedron and parallel to.

1974, International Polymer Science and Technology, page 202:

In fact the structure of 2 can be related to the octadecahedron and the pentacapped trigonal prism in an interesting fashion

(geometry) A polyhedron with eighteen faces.


tetracontadigon

Construct an edge-to-edge tessellation with a triangle, a heptagon, and a tetracontadigon. Is it possible to continue such a tessellation to have the full petal around the tetracontadigon?

2010, Sergei Abramovich, Topics in Mathematics for Elementary Teachers, page 90:

[…] (edge-to-edge tessellation with triangle, heptagon, and tetracontadigon, see Fig. 25). Fig. 25 Edge-to-edge tessellation with triangle, heptagon, and tetracontagon

2021, Sergei Abramovich, Michael L. Connell, Developing Deep Knowledge in Middle School Mathematics:

Using GeoGebra, construct an edge-to-edge tessellation with triangle, heptagon, and tetracontadigon. Is it possible to continue such a tessellation to have the full petal around the tetracontadigon?

2024, Sergei Abramovich, Developing Deep Knowledge in Middle School Mathematics, page 82:

(geometry) A polygon with forty two sides and forty two angles.

42-угольник

сорокадвухуго́льник


enneacontahexagon

Eventually, he worked his way up to a ninety-six-sided figure, an enneacontahexagon.

(geometry) A polygon with ninety six sides and ninety six angles.


icosikaidigon

The exterior wall is a 22-sided polygon (an icosikaidigon) and contains three large double-hung windows per bay.

2005, Baltimore Civil Engineering History, page 209:

A 22—sided shape once came up in relation to Activity 10.17, and various internet sources suggested we could call it an icosikaidigon!

2013, Essential Primary Mathematics, page 269:

(geometry) A polygon with twenty two sides and twenty two angles.


micropolygon

(computer graphics) A very small polygon.


tetracontakaihexagon

For example, a 46-sided polygon could be called a tetracontakaihexagon, or a combination of “tetracontagon” “kai” (often used in combining names) “hexagon.”

2012, Patricia Barnes-Svarney, Thomas E Svarney, The Handy Math Answer Book, page 184:

(geometry) A polygon with forty six sides and forty six angles.


triacontakaitetragon

For instance a 34-sided polygon is called a triacontakaitetragon, and a 52-sided polygon is called a pentacontakaidigon.

2013, Mark A. Freitag, Mathematics for Elementary School Teachers, page 586:

(geometry) A polygon with thirty four sides and thirty four angles.


icosagonal

(geometry) Having the form of a icosagon


hendecagram

hendecagram (11-pointed) dodecagram (12-pointed)


d'Alembert operator

d'Alembert operator (plural d'Alembert operators)

Named after Jean le Rond d'Alembert (1717–1783), a French mathematician, mechanician, physicist, philosopher, and music theorist.

differential operator
differential operator

(physics) A differential operator which may be expressed as ∂_μ∂^μ=∑_(μ=0)³∂/∂x^μ∂/∂x_μ; it is the four-dimensional (Minkowski space) equivalent of the three-dimensional Laplace operator.

Jean le Rond d'Alembert
d'Alembert operator


linear triplet

(geometry) The three supplementary adjacent angles formed by three intersecting lines


equidiagonal

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


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#x3D;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#x3D;0 can be considered a Ford circle with infinite radius, corresponding to the case p#x3D;1,#x5C;q#x3D;0.

Thus the Ford circle [1], drawn tangent to the real axis at #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, and having radius #x5C;textstylea#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D;, must contain in its interior some points belonging to #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, such as #x5C;textstyle#x5C;theta#x2B;ia#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D; whose imaginary part lies between #x5C;textstyleq#x5F;n#x7B;-2#x7D; and #x5C;textstyleq#x5F;#x7B;n#x2B;1#x7D;#x7B;-2#x7D;.

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

A Farey fraction #x5C;textstyle#x5C;frackc defines a Ford circle #x5C;textstyle#x5C;mathcal#x7B;C#x7D;(k,c) in #x5C;textstyle#x5C;mathbbC. Its center is given by #x5C;textstylekc#x2B;i#x5C;frac 1#x7B;2c²#x7D; and its radius is #x5C;textstyle#x5C;frac1#x7B;2c²#x7D;. Two Ford circles #x5C;textstyle#x5C;mathcalC(a,b) and #x5C;textstyle#x5C;mathcalC(c,d) are tangent whenever #x5C;textstylead-bc#x3D;#x5C;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #x5C;textstyleF#x5F;N.

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

Ford circles are a collection of horocycles in #x5C;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#x5F;#x7B;a#x2F;b#x7D; and C#x5F;#x7B;c#x2F;d#x7D; are tangent if and only if #x5C;vertad-bc#x5C;vert#x3D;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


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.


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.


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, #x5C;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.


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.


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.


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 #x5C;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


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#x7B;2k#x7D;.

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).


arcdegree

/ˈɑːkdɪˌɡɹiː/

/ˈɑɹkdɪˌɡɹiː/

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


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 #x5C;mathcal#x7B;O#x7D;. 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


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 #x5C;boldsymbol#x7B;a#x7D;.

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


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.

(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).


equiangulation

(geometry) Conversion to equiangular form


subsimplicial

(geometry) Relating to, or composed of subsimplexes


subsimplex

(geometry) A polyhedron at the intersection of simplexes.


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.


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.


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#x5F;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.


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 #x5C;mathbbZ#x5F;6, while Lee and Park ([LP09]) constructed one with algebraic fundamental group #x5C;mathbbZ#x5F;2, and one with algebraic fundamental group #x5C;mathbbZ#x5F;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 #x5C;pi#x5F;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).


trisection point

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.


pentalateral

(geometry) A decagon whose opposite sides are parallel.


cocked hat curve

cocked hat curve (plural cocked hat curves)

From the resemblance to the hat also called a bicorn.


plane of symmetry

plane of symmetry (plural planes of symmetry)

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


circumsphere

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


cochleoid

(mathematics) A strophoid shaped somewhat like a snail


circumconic

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


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.


Weyl tensor

pseudo-Riemannian manifold
pseudo-Riemannian manifold

(differential geometry) A measure of the curvature of spacetime (or, more generally, a pseudo-Riemannian manifold), differing from the Riemann tensor in that it does not convey information on how the volume of the body changes, but only how its shape is distorted by the tidal force.


cubicuboctahedron

(geometry) Either of two star polyhedra whose square faces are parallel to a square and whose triangular faces are parallel to those of an octahedron


star polyhedron

(geometry) A polyhedron that is partially nonconvex.


great cubicuboctahedron

en-noun

great cubicuboctahedron (plural great cubicuboctahedra)

(geometry) A nonconvex uniform polyhedron that has faces of eight triangles, six squares and six octagrams


small cubicuboctahedron

small cubicuboctahedron (plural small cubicuboctahedra)

(geometry) A uniform star polyhedron that has faces of eight triangles, six squares and six octagrams


space-filling

(geometry) That fills a closed (typically 2-dimensional or 3-dimensional) space.


MEGA

(orbital dynamics, space flight, astronautics) Abbreviation of Moon-Earth gravity assist.

orbital dynamics; space flight; orbital dynamics; space flight

(orbital dynamics, space flight, astronautics) Abbreviation of Mars-Earth gravity assist.


Euler angle

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

Euler angles can also represent the orientation of a mobile frame of reference in physics or the orientation of a basis in 3-dimensional linear algebra.

The use of Euler angles or Quaternions to represent entity (i.e., vehicle) orientation in the draft military standard for Distributed Interactive Simulation has drawn fierce debate.

1991, “AIAA Flight Simulation Technologies Conference: A Collection of Technical Papers”, in AIAA, page 271:

We have not drawn a picture, since Euler angle pictures of three-dimensional rotations are notoriously difficult to comprehend; what is very useful is for the reader to make a set of wire axes with labels to help follow the sequence of rotations.

2006, Don Koks, Explorations in Mathematical Physics: The Concepts Behind an Elegant Language, Springer, page 171:

Why do we need to know what Euler angles are? Because we use them. Many commercial 3D computer animation packages that are available today use Euler angles to describe and change orientations of objects.

2008, Midori Kitagawa, Brian Windsor, MoCap for Artists: Workflow and Techniques for Motion Capture, Elsevier (Focal Press), page 188:

(geometry) Any one of three angles which together describe the orientation of a rigid body with respect to a fixed three-dimensional coordinate system, according to a formalism introduced by Leonhard Euler.

eulerscher Winkel

any one of three angles that describe orientation in a 3-d coordinate system


lithocapture

lithocapture (countable and uncountable, plural lithocaptures)

/ˈlɪ.θoʊˌkæp.t͡ʃɚ/

/ˈlɪ.θəˌkæp.t͡ʃɚ/

/ˈlɪ.θoʊ.kæpˌt͡ʃɚ/

/ˈlɪ.θə.kæpˌt͡ʃɚ/

/ˈlɪ.θoʊˌkæp.t͡ʃə/

From litho- (“pertaining to stone”) + capture, coined as a pun on aerocapture and lithobraking.

prefix

If the initial orbit is hyperbolic, lithobraking would properly be termed lithocapture.

2020, Jonathan C. McDowell, “Astronautical Glossary”, in Jonathan's Space Report, retrieved 2021-02-21:

(astronautics, humorous) A form of lithobraking in which a spacecraft or other object on a hyperbolic trajectory inbound to a planet, moon, asteroid, or other celestial body is decelerated by impact with the surface of the body.


lithocapture

en-verb

lithocapture (third-person singular simple present lithocaptures, present participle lithocapturing, simple past and past participle lithocaptured)

lithocaptures (present singular third-person)
lithocapturing (participle present)
lithocaptured (participle past)
lithocaptured (past)

/ˈlɪ.θoʊˌkæp.t͡ʃɚ/

/ˈlɪ.θəˌkæp.t͡ʃɚ/

/ˈlɪ.θoʊ.kæpˌt͡ʃɚ/

/ˈlɪ.θə.kæpˌt͡ʃɚ/

/ˈlɪ.θoʊˌkæp.t͡ʃə/

(astronautics, humorous) To perform lithocapture.