p-hacking

Blend of p-value + hacking, from "p" for probability. Coined by researchers Uri Simonsohn, Leif D. Nelson, and Joseph P. Simmons in 2014.

coinage

Cherry-picking promising findings, also known by such terms as data dredging, significance chasing, significance questing, selective inference, and “p-hacking,” leads to a spurious excess of statistically significant results in the published literature and should be vigorously avoided.

[2016, Ronald L. Wasserstein, Nicole A. Lazar, “The ASA Statement on p-Values: Context, Process, and Purpose”, in The American Statistician, volume 70, number 2, →DOI:

statistically significant
statistically significant#English

(sciences, statistics) The practice of reanalysis of data until a desired result is obtained which is statistically significant


deorbit burn

(astronautics) A firing of the engines of a spacecraft to enable it to re-enter the Earth's atmosphere.


rhombic dodecahedron

en-noun

rhombic dodecahedron (plural rhombic dodecahedra or rhombic dodecahedrons)

rhombic dodecahedra (plural)
rhombic dodecahedrons (plural)

Doing this we would find that the network which is dual to the space filling array of rhombic dodecahedra is the space filling array of tetrahedra and octahedra in which 12 edges meet at the vertices which fall at the centers of the original rhombic dodecahedra.

1990, [1978], Peter Pearce, Structure in Nature Is a Strategy for Design, page 5:

Crystallographers are familiar with the rhombic dodecahedron as a domain of reference with which to account for the growth and structure of natural crystals.

2007, Scott Eastham, American Dreamer: Bucky Fuller and the Sacred Geometry of Nature, page 160:

Rhombic Dodecahedrons appear as well in the unit cells of diamonds, four vertices absent with the chemical bonds on the remaining edges. The Rhombic Dodecahedron can be used to tessellate 3D space in a manner similar to the Cube.

2013, L. Dennis, Brender McNair, N. J. Woolf, L. H. Kauffman, “6: The Mereon 120/80 — Form Informing Function”, in Lynnclaire Dennis, Jytte Brender McNair, Louis H. Kauffman, editors, The Mereon Matrix: Unity, Perspective and Paradox, page 135:

(geometry) A convex polyhedron that has 12 congruent rhombic faces, 24 edges and 14 vertices of two types (eight 3-edge and six 4-edge) and is a Catalan solid.


Schläfli symbol

Schläfli symbol (plural Schläfli symbols)

From Schläfli (“a surname”) + symbol, after 19th-century Swiss mathematician Ludwig Schläfli (1814–1895).

1995, Unnamed translator, E. B. Vinberg, Polynomial Group, article in M. Hazewinkel (editor), Encyclopaedia of Mathematics, Volume 4: Monge–Ampère Equation — Rings and Algebras, 2013, International Edition, page 481,
The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron — {3, 3}, the cube — {4, 3}, the octahedron — {3, 4} — the dodecahedron — {5, 3}, and the icosahedron — {3, 5}.

For the regular tiling of squares the Schläfli symbol is {4,4}, indicating that there are four squares surrounding each vertex. And for the regular tiling of hexagons the Schläfli symbol is {6,3}, indicating that there are three hexagons surrounding each vertex.

2007, Michael E. Mortenson, chapter L, in Geometric Transformations for 3D Modeling, page 248:

The 24-cell has has 24 octahedral cells, 24 vertices, 96 triangular faces and 96 edges. Its Schläfli symbol is {3, 4, 3} from which we see that the vertex figure is a cube.

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

(geometry) A notation that recursively encodes certain properties of a specified regular polytope or tessellation.

Schläflin symboli

notation encoding certain properties of a regular polytope or tessellation


tauberian

mathematical analysis
mathematical analysis

(mathematical analysis) Alternative letter-case form of Tauberian


Perhaps the most obvious next step is to investigate 4-polytopes with respect to symmetry equivalence.

1984, London Mathematical Society lecture note series, numbers 88-90, page 101:

The situation for 4-polytopes is fundamentally different from that for 3-dimensional polytopes. One reason is that there is no similar reduction of 4-polytope theory to a combinatorial (graph) problem.

2004, Martin Henk, Jürgen Richter-Gebert, Günter M. Ziegler, “16: Basic properties of convex polytopes”, in Joseph O'Rourke, Jacob E. Goodman, editors, Handbook of Discrete and Computational Geometry, 2nd edition, page 369:

For d = 4, Hamiltonian cycles in the regular 4-polytopes are known to exist. However, it seems that so far no decision about the existence or non-existence of 1-Hamiltonian surfaces in the 2-skeleton of any of the three sporadic regular 4-polytopes can be made, compare [120].

2011, Felix Effenberger, Hamiltonian Submanifolds of Regular Polytopes, page 40:


tetrakis hexahedron

tetrakis hexahedron (plural tetrakis hexahedra or tetrakis hexahedrons)

tetrakis hexahedra (plural)
tetrakis hexahedrons (plural)

Of the fourteen crystals examined, five were from the Karnul district (four tetrakis-hexahedra and one octahedron with tetrakis-hexahedron), one from Sambalpur (tetrakis-hexahedron with octahedron, four from Panna (much distorted tetrakis-hexahedra), the remaining four said to have come from Simla.

1904, Max Bauer, translated by L. J. Spencer, Precious Stones, volume 1, published 1968, page 154:

In the Isometric Tribe can be found the families of the Tetrakis Hexahedrons and Octahedrons, with their various descendants.

1909, Marshman Edward Wadsworth, Crystallography: An Elementary Manual for the Laboratory, page 189:

The goniometrical angle over the longer edges, that is, the angle between the normals to each pair of faces intersecting in a longer edge, in the cases of the two tetrakis hexahedra most frequently met with, has the following values:

1922, Alfred Edwin Howard Tutton, Crystallography and Practical Crystal Measurement, volume 1, page 153:

Not one exhibits plane faces or straight edges, being mostly distorted triakis-octahedra or tetrakis-hexahedra with finely striated curved faces.

1946, Mineralogical Society of America, Mineralogical Abstracts, volume 9, page 109:

Because of this another polyhedron – the tetrakis hexahedron – that has 14 vertices and 24 faces might be also a good model. The tetrakis hexahedron is slightly less regular than an icosahedron.

2010, Sumudu P. Leelananda, Yaping Feng, Pawel Gniewek, Andrzej Kloczkowski, Robert L. Jernigan, “6: Statistical Contact Potentials in Protein Coarse-Grained Modeling: From Pair to Multi-body Potentials”, in Andrzej Kolinski, editor, Multiscale Approaches to Protein Modeling, page 150:

(geometry) A Catalan solid that has 24 congruent triangular faces and 36 edges and can be imagined as a cube with a square pyramid on each face.

tetracisesaedro

polyhedron with 24 triangular faces and 36 edges


Schlegel diagram

Schlegel diagram (plural Schlegel diagrams)

From Schlegel (“a surname”) + diagram, after German mathematician Victor Schlegel, who introduced the diagram in 1886.

The location of the point x₀ can always be chosen so that the edges in the Schlegel diagram can be drawn as non-intersecting straight lines.

1999, R. B. King, “1: Topology in Chemistry”, in D Bonchev, D.H Rouvray, editors, Chemical Topology: Introduction and Fundamentals, page 21:

Schlegel diagrams are a useful way to explore how these polyhedra can be linked (Hoppe and Köhler 1988).

2002, Ian David Brown, chapter L, in The Chemical Bond in Inorganic Chemistry: The Bond Valence, page 150:

These structures may all be represented on a planar surface by their Schlegel diagrams. A polyhedron Schlegel diagram is its networks of edges and vertices drawn in a special way: if you hold the polyhedron so close to your face that one of faces frames the entire polyhedron and you see all the other edges meeting inside that frame, then you have a Schlegel diagram.

2013, Arthur L. Loeb, “5: Polyhedra: Surfaces or Solids?”, in Marjorie Senechal, editor, Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, page 69:

(geometry) A projection of a polytope from n-dimensional space to n-1 dimensions through a point beyond one of its faces; especially such a projection (itself represented in 2 dimensions) of a 3- or 4-dimensional polytope.


quantaloid

(mathematics) A category enriched over the category Sup of suplattices.


quantale

(mathematics) A kind of partially ordered algebraic structure that generalizes locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis.


fewnomial

(mathematics) A polynomial with few monomials.


ringoid

(mathematics) A ring-like algebraic structure.


anticommute

en-verb

anticommute (third-person singular simple present anticommutes, present participle anticommuting, simple past and past participle anticommuted)

anticommutes (present singular third-person)
anticommuting (participle present)
anticommuted (participle past)
anticommuted (past)

(mathematics, intransitive) To operate anticommutatively.


Josephus problem

Named after Flavius Josephus, a Jewish historian of the 1st century, who describes such a situation in his account of the siege of Yodfat.

(mathematics) A theoretical problem in which every nth person, counting around a circle of people, is chosen for execution, and this process repeats after the removal of each chosen person, with the last person left being freed. The problem is to select one's position so as to be the last to be chosen.


Diffie-Hellman

Named after Whitfield Diffie and Martin Hellman.

(cryptography) A particular mechanism for key exchange in cryptographic communications.


Legendre's differential equation

Named after French mathematician Adrien-Marie Legendre (1752–1833).

(mathematics) The equation that defines the Legendre polynomials: d/dx[(1-x²)d/dxP_n(x)]+n(n+1)P_n(x)=0.


copresheaf

(category theory) The dual of a presheaf.


Kerckhoffs's principle

(cryptography) The idea that a cryptosystem should be designed to be secure even if everything about the system, except the secret key, is public knowledge.

Auguste Kerckhoffs
Kerckhoffs's principle


angular defect

Prop. XVI. If the angular defects of two triangles are equal, the areas of the triangles are equal.

1864, Transactions of the Royal Society of Edinburgh, volume 23, page 444:

We have seen that under any reasonable definition of area, the angular defect of a triangle is proportional to its area.

1995, Arlan Ramsay, Robert D. Richtmyer, Introduction to Hyperbolic Geometry, page 120:

In hyperbolic geometry, similar triangles do not exist and the area of a triangle is directly proportional to its angular defect[…].

1998, Kappa Mu Epsilon, The Pentagon: A Mathematics Magazine for Students, page 7:

(geometry, non-Euclidean geometry) The amount by which the sum of the interior angles of a triangle is less than 180° (π radians); the amount by which the sum of the internal angles of a polygon is less than what would be expected on the Euclidean plane.

angular deficiency

amount by which the total of the interior angles of a polygon is less than on the Euclidean plane

The angular defect clearly has something to do with curvature, because the larger it is, the more pointed the surface is at the vertex.

2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, page 604:

2014, C. R. Calladine, The Static-geometric Analogy in the Equations of Thin Shell Structures, W. Olszak, Thin Shell Theory: New Trends and Applications, page 294,
Figure 4(c) shows a flattened view of a small part of the undeformed polygonalized S-surface, consisting of the triangles surrounding a particular vertex. When the S-surface is strained there will be a consequent change of angular defect, which we wish to calculate.

2015, Jan Guichelaar (translator and editor), Alex Van Den Brandhof, Arnout Jaspers (editors), Half a Century of Pythagoras Magazine, page 164,
Theorem. For a spherical polyhedron the total angular defect equals 720°.

(geometry) The amount by which the total of the angles around a vertex of a polyhedron is less than 360° (2π radians).


coiflet

From Coifman + -let; they were designed by Ingrid Daubechies at the request of Ronald Coifman.

(mathematics) A discrete wavelet that has a scaling function with a vanishing moment.


symlet

(mathematics) A kind of modified Daubechies wavelet with increased symmetry.


symmetry group

[The] situation is more delicate in the case of higher dimensional symmetry groups; it is not in general possible to reduce the order of an equation invariant under an r-parameter symmetry group by r using only quadratures.

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, published 2000, page 76:

It is often assumed that the design unit inside each fundamental region (particularly for monotranslational symmetry groups p111 and p1a1 and ditranslational symmetry groups p1 and pg) is asymmetric.

2000, C. E. Horne, Geometric Symmetry in Patterns and Tilings, page 79:

The structure of the symmetry group of a molecule affects several spectroscopic aspects and vice versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.
In this chapter, we study symmetry groups of fulleroids.

2010, Stanislav Jendrol, František Kardoš, “28: Symmetry of Fulleroids”, in Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 28–1:

(geometry, algebra, group theory) A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.

full symmetry group

emphasises the inclusion of orientation-reversing transformations


first cosmic velocity

From first + cosmic + velocity, from being the first velocity greater than that able to be held upon the Earth, thus entering the realm of the Cosmos.

The velocity in a circular orbit about the Earth.


second cosmic velocity

From second + cosmic + velocity, from being the next threshold velocity greater than the first cosmic velocity.

Escape velocity of Earth (interplanetary speed).


second cosmic velocity

en-noun

second cosmic velocity (plural second cosmic velocities)

(by extension) Escape velocity of the home body.


third cosmic velocity

From third + cosmic + velocity, from being the next threshold velocity greater than the second cosmic velocity.

The escape velocity of the Solar System (interstellar speed).


third cosmic velocity

third cosmic velocity (plural third cosmic velocities)

(by extension) The escape velocity of a star system.


quintinomial

(mathematics) Any expression that has five terms.


cross-polytope

The case p = 1, which is actually important to us, is illustrated in Figure 1.14; we are looking at the intersection of a cross-polytope with the probability simplex.

2006, Ingemar Bengtsson, Karol Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, page 25:

2008, Robert Erdahl, Andrei Ordine, Konstantin Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices, Matthias Beck, et al. (editors), Contemporary Mathematics 452: Integer Points in Polyhedra — Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS-IMS-SIAM Joint Summer Conference, June 11-15, 2006, page 104,
The convex hulls of such crosses often appear as cells in Delaunay tilings – cross polytopes are examples, as are the more spectacular symmetric perfect Delaunay polytopes.

The cross polytope βₙ is the regular polytope in n dimensions corresponding to the convex hull of the points formed by permuting the coordinates (±1,0,0,...,0), and has therefore 2n vertices. It is named so because its vertices are located equidistant from the origin, along the Cartesian axes in n-space. The cross polytope in n dimensions is bounded by 2ⁿ (n - 1)-simplices, has 2n vertices and 2ⁿ(n - 1) edges.

2009, Herbert W. Hamber, Quantum Gravitation: The Feynman Path Integral Approach, Springer, page 263:

If one works with the l₁ norm, a ball (the set of all points whose distance lies within a certain radius around a point of interest) has the shape of a cross-polytope. A one-dimensional cross-polytope is a line segment, a two-dimensional cross-polytope is a square, for three dimensions, an octahedron, and so forth.

2010, Rainer Typke, Agatha Walczak-Typke, “Indexing Techniques for Non-metric Music Dissimilarity Measures”, in Zbigniew W. Raś, Alicja A. Wieczorkowska, editors, Advances in Music Information Retrieval, Springer, page 10:

While centrally symmetric d-polytopes with 2d vertices are affinely equivalent to the d-dimensional cross-polytope, a complete classification of the centrally symmetric d-polytopes with 2d + 2 vertices is out of reach.

2003, Branko Grünbaum, Convex Polytopes, 2nd edition, Springer, page 121b:

(geometry) A polytope that is the convex hull of points, two on each Cartesian axis of a Euclidean space, that are equidistant from the origin; an orthoplex.


Coxeter diagram

Coxeter diagram (plural Coxeter diagrams)

It suffices to make use of the tables above and observe that the Coxeter diagram of a direct sum of C-matrices is the disconnected union of the Coxeter diagrams of the summands.
For the description of C⁻-matrices satisfying condition (2) we shall use Coxeter diagrams with the additional stipulation that in case aᵢⱼ < −1 the vertices vᵢ and vⱼ are joined by a dotted line with index −aᵢⱼ. By the Coxeter diagram of a Coxeter polyhedron and a lattice generated by reflections we mean the Coxeter diagram of the corresponding C⁻-matrix.

1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups, page 358:

An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type M. We introduce admissible partitions of Coxeter diagrams in order to generalize this situation.

1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. Buekenhout, J. Doyen, F. de Clerck, J. A. Thas, J. W. P. Hirschfeld, editors, Finite Geometries and Combinatorics, 2nd International Conference, page 277:

The Coxeter diagrams of type E₇ and E₈ have no nontrivial automorphisms, so σ₀ is trivial in those cases.[…]The Coxeter diagram of type F₄ has a unique nontrivial automorphism, but σ₀ is trivial in this case. One can see this by using the Dynkin diagram instead of the Coxeter diagram and noting that it does not have any nontrivial automorphisms.

2008, Peter Abramenko, Kenneth S. Brown, Buildings: Theory and Applications, page 259:

Coxeter-Dynkin diagram
Coxeter-Dynkin diagram

(geometry, algebra) A Coxeter-Dynkin diagram.


evolutionarily stable strategy

evolutionarily stable strategy (plural evolutionarily stable strategies)

evolutionarily stable strategies (plural)

Coined by British biologist John Maynard Smith in 1972 in his paper Game Theory and The Evolution of Fighting.

coin

To be an evolutionarily stable strategy, remember, a strategy must not be invadable, when it is common, by a rare mutant strategy.

1976, Richard Dawkins, The Selfish Gene:

An evolutionarily stable strategy is a strategy that does well against copies of itself.

(evolutionary theory, game theory) A strategy that, when adopted by a population, is effective and unlikely to be replaced by another strategy.


mathematical sociology

The use of mathematics to construct formal theories about society.


hyperellipsoidal

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

A regularized Mahalanobis distance can be used for extracting hyperellipsoidal clusters [30 ].

2015 December 12, “A Comparison Study on Similarity and Dissimilarity Measures in Clustering Continuous Data”, in PLOS ONE, →DOI:

(geometry) Relating to, or having the form of a hyperellipsoid


hyperellipsoid

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

(geometry) A figure, related to the hypersphere, having ellipsoidal rather than spherical symmetry


snub disphenoid

en-noun

snub disphenoid (plural snub disphenoids)

So named in 1966 by mathematician Norman Johnson in his classification of the Johnson solids. See also Snub disphenoid on Wikipedia.Wikipedia

For the snub disphenoid, our existence proof used the intermediate value theorem in the real numbers to argue that as AB decreases and CD increases there is a point where they become equal.

2005, Robin Hartshorne, Geometry: Euclid and Beyond, page 457:

In this structure-type, the eight-coordinate polyhedra (snub disphenoids) share edges to form a framework composed of cross-linked chains that extend along the [100] and [010] directions.

2006, Andrew J. Locock, “Chapter 6: Crystal chemistry of actinide phosphates and arsenates”, in Sergey Krivovichev, Peter Burns, Ivan Tananaev, editors, Structural Chemistry of Inorganic Actinide Compounds, page 225:

Thus, at a basic level this activity is about exploring such polyhedra, starting with the regular cases of the tetrahedron, octahedron, and icosahedron and moving into other solids like the triangular dipyramid and snub disphenoid.

2013, Thomas Hull, Project Origami: Activities for Exploring Mathematics, 2nd edition, page 153:

equilateral triangle
equilateral triangle

A non-regular convex polyhedron that has 12 equilateral triangles as faces and 18 edges and is a Johnson solid.

Norman Johnson (mathematician)
snub disphenoid

Siamese dodecahedron

polyhedron with 12 equilaterally triangular faces

deltahedron

polyhedron with faces that are all equilateral triangles


REF

row echelon form
row echelon form#English

(mathematics, linear algebra) Initialism of row echelon form.


multitwist

A Leininger-Reid subgroup cannot be convex cocompact, because it contains a multitwist.

2007, Athanase Papadopoulos, Handbook of Teichmüller Theory - Volume 1, →ISBN, page 407:

(topology) An automorphism that induces a set of Dehn twists around a surface.


simsun permutation

simsun permutation (plural simsun permutations)

From the initial syllables of the surnames of Rodica Simion and Sheila Sundaram, who introduced them.

(mathematics) A permutation in which, for all k, the subsequence of the smallest k elements has no three consecutive elements in decreasing order.


equitangential

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

(geometry) Of a curve: having tangential segments that are all of the same length.


triacontagon

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

From Ancient Greek τρῐᾱ́κοντᾰ (trĭā́kontă) + -gon.

suffix

After projection onto the (x₁, x₂) plane, the A’s, B’s, C’s, and D’s form four concentric triacontagons, inscribed in circles of radii a, b, c, and d.

1947, H. S. M. Coxeter, Regular Polytopes, published 2012, unnumbered page:

1964 July, B. L. Chilton, On the Projection of the Regular Polytope {5, 3, 3} into a Regular Triacontagon, Canadian Mathematical Bulletin, Volume 7, Number 3, page 386,
In the projection the 600 points can be classified in 12 sets: each set consists of either of 30 or of 60 vertices which lie on a circle concentric with the outer triacontagon.

1997, Johannes Kepler, E. J. Aiton, Alistair Matheson Duncan, Judith Veronica Field (translators), The Harmony of the World, [1619, Harmonices Mundi], American Philosophical Society, Volume 309, page 59,
What should the decision now be about the sides of the Triacontagon?


deltoidal icositetrahedron

en-noun

deltoidal icositetrahedron (plural deltoidal icositetrahedra or deltoidal icositetrahedrons)

deltoidal icositetrahedra (plural)
deltoidal icositetrahedrons (plural)

These correspond to the foregoing U- and F-vertices of the deltoidal icositetrahedron whose three-edged vertices are "completely truncated"; that is to say, the truncating planes pass through the three end points of the edges radiating from the truncated vertex.

1958, The Mathematical Gazette, volume 42, page 20:

The mineral displays a well-developed habit of deltoidal icositetrahedra with dull faces.

1994, Proceedings of the Ninth Quadrennial IAGOD Symposium, International Association on the Genesis of Ore Deposits, page 593:

At the Olympus Mons Casino on Mar's they have craps games where the die are rhombic dodecahedron (12 sided), or tetrakis hexahedron (24 sides) or deltoidal icositetrahedron (24 sides). That's for your benefit, Gabby said to me.

2012, Ivan Pasztor, Of All The Things I've Lost, page 69:

(geometry) A convex polyhedron that has 24 congruent kite-shaped faces and 48 edges and is a Catalan solid, being the dual polyhedron of the rhombicuboctahedron.


Menger sponge

(mathematics) A fractal curve, a three-dimensional generalization of the Cantor set and Sierpinski carpet, formed by repeated subdivision of a cube, and having infinite surface area and zero volume.


Sierpinski carpet

Sierpinski carpet (plural Sierpinski carpets)

First described by Wacław Sierpiński in 1916.

(mathematics) A plane fractal, a generalization of the Cantor set to two dimensions, formed by repeated subdivision of a square.


Cantor dust

(mathematics) A multidimensional version of the Cantor set, formed by taking a finite Cartesian product of the Cantor set with itself.


Sierpinski sponge

Sierpinski sponge (plural Sierpinski sponges)

Named after Wacław Sierpiński, Polish mathematician.

(mathematics) The three-dimensional extension of the Sierpinski carpet.


Sierpinski number

Sierpinski number (plural Sierpinski numbers)

Named after Polish mathematician Wacław Sierpiński.

named-after

(number theory) An odd natural number k such that k×2ⁿ+1 is composite for all natural numbers n.


Sierpinski curve

en-noun

Sierpinski curve (plural Sierpinski curves)

(mathematics) Any of a recursively defined sequence of continuous closed plane fractal curves which, in the limit n→∞, completely fill the unit square.


truncated tetrahedron

truncated tetrahedron (plural truncated tetrahedra or truncated tetrahedrons)

truncated tetrahedra (plural)
truncated tetrahedrons (plural)

The space of the El Al pyramid is created essentially by the packing of truncated tetrahedrons.

1968, Cindy Marriott, A Critical Analysis of Israeli Architecture, 1948-1967: A Study in Cultural Conflict, page 63:

The tendons now form the edges of a slightly distorted truncated tetrahedron.

1976, Anthony Pugh, An Introduction to Tensegrity, page 35:

This ideal net gives rise to the formation of B₁₂ truncated tetrahedra, where the A cations are lodged.

2000, Angel Vegas, Cations in Inorganic Solids: Crystallography Reviews, volume 7, number 3, page 248:

2009, Xiaotian Lin, Miao Jin, Feng Luo, Xiangfen David Gu, Discrete Curvature Flows for Surfaces and 3-Manifolds, Frank Nielsen (editor), Emerging Trends in Visual Computing: LIX Fall Colloquium, ETVC 2008, page 69,
Once the edge lengths of the tetrahedral mesh have been obtained, we can realize it in the hyperbolic space ℍ³ First, we introduce how to construct a single truncated tetrahedron; then we explain how to glue multiple truncated tetrahedra by hyperbolic rigid motion.

2010, K.M. Kosuda, J. M. Bingham, K. L. Wustholz, R. P. Van Dyne, 9: Nanostructures and Surface-Enhanced Raman Spectroscopy, Gary P. Wiederrecht (efitor), Handbook of Nanoscale Optics and Electronics, page 318,
Without any further modification, the shapes of the nanoparticles fabricated by NSL are truncated tetrahedrons. However, by thermal annealing the truncated tetrahedrons are transformed into hemispheres.

(geometry) A convex polyhedron, with four triangular and four hexagonal faces, that is an Archimedean solid and can be constructed by truncating a regular tetrahedron at each vertex by one-third the length of each adjoining edge.


truncated cube

The truncation procedure that produced the truncated cube I can be applied to arbitrary polyhedra so as to obtain new polyhedra.

2003, Saul Stahl, Geometry from Euclid to Knots, published 2010, page 278:

The truncated cubes are obtained by symmetrically edging the eight vertices of the cube by l × t, where l is the length of the cubes side and 0 < t ≤ 1/2. When t = 1/2, a cuboctahedron is obtained.

2012, Cecilia Noguez, Ana L. González, “11: Localized Surface Plasmons of Multifaceted Metal Nanoparticles”, in Tapan K. Sau, Andrey L. Rogach, editors, Complex-shaped Metal Nanoparticles: Bottom-Up Syntheses and Applications, page 384:

In this subsection the optical behavior of different truncated cubes will be analyzed and compared to those of a cube and of a sphere of the same dimensions.

2013, S. D'Agostino, “Chapter 3: Size and Shape Dependence of Localized Surface Plasmon Resonances”, in Fabio Della Sala, Stefania D'Agostino, editors, Handbook of Molecular Plasmonics, page 152:

Truncated cubes can be packed sharing their octagonal faces, the remaining voids are filled by octahedra (Figure 10(b)).

2014, Walter Steurer, “1: Crystal Structures of Metallic Elements and Compounds”, in David E. Laughlin, Kazuhiro Hono, editors, Physical Metallurgy, page 20:

(geometry) A polyhedron with eight triangular and six octagonal faces that is an Archimedean solid and can be constructed by truncating a cube at each vertex along each adjoining edge.


snub dodecahedron

snub dodecahedron (plural snub dodecahedra or snub dodecahedrons)

snub dodecahedra (plural)
snub dodecahedrons (plural)

From dodecahedron simum, Kepler's name for this solid in is 1619 work Harmonices Mundi.

This gives two points, A’, A₁’, which are projections of vertices of the two enantiomorphous snub dodecahedra that can be inscribed in the regular dodecahedron.

1935, L. Lines, Solid Geometry, page 182:

The form can be derived by triangulating the pentagonal faces of the snub dodecahedron (Fig. 19).

1992, Gábor Gévay, “Icosahedral Morphology”, in István Hargittai, editor, Fivefold Symmetry, page 198:

In the case of the latter, two chiral members, the snub cube and the snub dodecahedron, are realized.

2001, Ehud Keinan, Israel Schechter, Chemistry for the 21st Century, page 138:

The snub dodecahedron has 60 vertices, 150 edges and 92 faces of which 12 are pentagonal and 80 triangular. This polyhedron has I symmetry and can therefore be constructed in left- and right-handed forms.

2006, P. W. Fowler, D. E. Manolopoulos, An Atlas of Fullerenes, page 50:

The result is a structure of two nested snub cubes rotated by an angle of 45° against each other (n = 24) and two nested snub dodecahedra, where the rotation angle amounts to 36° (n = 60).

2010, Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 51–8:

(geometry) A polyhedron that has 12 pentagonal and 80 triangular faces and is an Archimedean solid.

snub icosidodecahedron

polyhedron with 12 pentagonal and 80 triangular faces


great rhombicuboctahedron

great rhombicuboctahedron (plural great rhombicuboctahedra or great rhombicuboctahedrons)

great rhombicuboctahedra (plural)
great rhombicuboctahedrons (plural)

From great + rhombicuboctahedron, referring to the fact that, similarly to the rhombicuboctahedron, the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron, the dual of the cuboctahedron.

Figure 244 — the structure for this complicated design is the great rhombicuboctahedron, which consists of octagonal, hexagonal, and square faces.

1977, Wucius Wong, Principles of Three Dimensional Design, page 70:

Zeolite A consists of truncated octahedra flanking the edges of a larger cube; they are connected along the edges by smaller cubes, and the truncated octahedra and cubes flank a great rhombicuboctahedron at the center (28).

2005, M. Cahay, Nanoscale Devices, Materials, and Biological Systems: Fundamentals and Applications : Proceedings of the International Symposium, page 513:

The great rhombicuboctahedron can be made by truncating the cuboctahedron's vertices.

2012, Edward S. Popko, Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, page 169:

(geometry) An Archimedean solid that is isogonal and has twenty-six regular faces (twelve square, eight hexagonal and six octagonal).

cubottaedro troncato

polyhedron with 12 square, 8 hexagonal and 6 octagonal faces


truncated cuboctahedron

truncated cuboctahedron (plural truncated cuboctahedra or truncated cuboctahedrons)

truncated cuboctahedra (plural)
truncated cuboctahedrons (plural)

From truncated + cuboctahedron, referring to a partial construction (the resultant figure needs further distortion to make the faces regular).

The H'(S-S) type of linked truncated octahedra and truncated cuboctahedra is analogous to the fluorite structure with the truncated octahedra in the positions of the fluorine atoms and the truncated cuboctahedra in the positions of the calcium atoms.

1962, The Mineralogical Magazine and Journal of the Mineralogical Society, volume 33, page 1013:

A three-dimensional network of linked cavities, each with a truncated cuboctahedron shape (Figure 6.5) through the structure, is thus formed; the truncated cuboctahedra are also space-filling (each shares its octagonal face with six others), forming channels that run parallel to the three axial cubic directions through these large cavities.

2012, Lesley E. Smart, Elaine A. Moore, Solid State Chemistry: An Introduction, 4th edition, page 274:

Its pore structure consists of truncated octahedra linked to other cavities through 6-membered rings and of truncated cuboctahedra linked together through 8-membered rings.

2013, István Palinkó, Zoltán Kónya, Ákos Kukovecz, Imre Kiricsi, “22: Zeolites”, in Robert Vajtai, editor, Springer Handbook of Nanomaterials, page 822:

great rhombicuboctahedron
great rhombicuboctahedron

(geometry) An Archimedean solid that is isogonal and has twenty-six regular faces (twelve square, eight hexagonal and six octagonal); a great rhombicuboctahedron.


Gibbs sampling

Named after the physicist Josiah Willard Gibbs, in reference to an analogy between the sampling algorithm and statistical physics.

probability distribution
probability distribution

(statistics) A Markov chain Monte Carlo algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult.


semiartinian

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

(algebra) Of a module _RM: such that, for every M→N epimorphism, where N≠0, the socle of N is essential in N.


subanalytic

We present a complex analytic proof of the Pila-Wilkie theorem for subanalytic sets.

2016, Gal Binyamini, Dmitry Novikov, “The Pila-Wilkie theorem for subanalytic families: a complex analytic approach”, in arXiv:

(mathematics) Of a set of points: defined in a way broader than for semianalytic sets.


superdiagonal

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

We classify all free spectrahedra that are circular, that is, closed under multiplication by exp(i t): up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal.

2016, Eric Evert, J. William Helton, Igor Klep, Scott McCullough, “Circular Free Spectrahedra”, in arXiv:

(mathematics) The diagonal of a matrix that lies directly above and to the right of the main diagonal.


semihereditary

The classical ring of quotients of a commutative Bezout ring is a (von Neumann) regular local ring if and only if R is a commutative semihereditary local ring.

2016, Bohdan Zabavsky, “Type conditions of stable range for identification of qualitative generalized classes of rings”, in arXiv:

(mathematics) Of a ring: hereditary for finitely generated submodules.


oscnode

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

Three nodes may coincide as consecutive points of a curve of finite curvature, giving rise, not to a triple point, but to the singularity called an oscnode; this is, in fact, an osculation or three-pointic contact of two branches of the curve.

1879, George Salmon, A Treatise on the Higher Plane Curves, page 214:

Rhamphoid cuspidal lines possess two kinds of singular points which affect the class of the surface, and occur (i) when the rhamphoid cusp changes into an oscnode, (ii) when there are cubic nodes on the surface.

1908, A. B. Basset, “Singular liens and curves on surfaces”, in The Quarterly Journal of Pure and Applied Mathematics, volume 39, page 338:

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

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

(geometry) A node of a plane curve where two branches meet with contact order 3.


survivorship bias

survivorship bias (countable and uncountable, plural survivorship biases)

A form of selection bias in which only the people or things that "survived" some process are considered, while those that did not are inadvertently overlooked because of their lack of visibility.


uniruled

(mathematics) Of a variety: covered by a family of rational curves.


truncated dodecahedron

truncated dodecahedron (plural truncated dodecahedra or truncated dodecahedrons)

truncated dodecahedra (plural)
truncated dodecahedrons (plural)

From truncated + dodecahedron, referring to a construction method for the figure.

1813, Spinell, entry in John Mason Good, Olinthus Gregory, Newton Bosworth (editors), Pantologia: A New Cyclopaedia, Volume XI: SPA — TZE, unnumbered page,
This, which may be regarded as a brown garnet or schoerl in truncated dodecahedrons of Rome de Lisle, was originally found in the island of Ceylon, among tourmalines and other crystalline substances, with which it was confounded.

The morphologies commonly observed in bcc materials include truncated dodecahedrons with {110} faces and {100} truncations and cubes with {100} faces.

1983, Electron Microscopy Society of America, Proceedings: 41st Annual Meeting, page 232:

Void shapes were found to be generally truncated dodecahedra with {011} faces and {001} truncations.

1990, David S. Gelles, “Neutron Irradiation Damage in Ferritic Fe-Cr Alloys”, in Nicholas H. Packan, R. E. Stoller, A. S. Kumar, editors, Effects of Radiation on Materials: 14th International Symposium, Volume 1, page 90:

1996, T. A. Fournelle, K. W. Weston, Generalizations of Steinberg Groups, World Scientific Publishing, Series in Algebra, Volume 4, page 176,
The edges of the linkages lie on the edges of a truncated dodecahedron. Note that the group of orientation preserving symmetries of a truncated dodecahedron is precisely A₅.

(geometry) An Archimedean solid with thirty-two regular faces (twelve decagons and twenty triangles) and ninety edges.

dodecaedro troncato

polyhedron with 12 decagonal faces, 20 triangular faces and 90 edges


truncated icosidodecahedron

truncated icosidodecahedron (plural truncated icosidodecahedra or truncated icosidodecahedrons)

truncated icosidodecahedra (plural)
truncated icosidodecahedrons (plural)

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

From truncated + icosidodecahedron, referring to a partial construction method for the figure. (The resultant figure requires further distortion to make the faces regular.)

The Solid Shapes Lab, another Science Materials Center contribution to future mathematicians, together to form not only regular polyhedrons but rhombicosidodecahedrons, truncated icosidodecahedrons, and such.

1961, The New Yorker, Volume 37, Part 4, page 172:

The truncated icosahedron and truncated icosidodecahedron have fivefold symmetry and cannot four four-connected frameworks with lattice symmetry.

1962, The Mineralogical Magazine and Journal of the Mineralogical Society, volume 33, page 1008:

It is also a Catalan solid, the disdyakis triacontahedron, dual of the truncated icosidodecahedron.

2007, Numerical Analysis and Applied Mathematics: International Conference of Numerical Analysis and Applied Mathematics, page 95:

An Archimedean solid with 62 regular faces (12 decagons, 20 hexagons and 30 squares) and 180 edges.

icosidodecaedro troncato

polyhedron with 12 decagonal, 30 hexagonal and 30 square faces


midsphere

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

The edges are omitted as irrelevant to the present argument, but the midspheres are still included because Kepler rather casually substitutes the midsphere for the insphere of the octahedron, since it happens to give a value closer to the observed distance.

1999, Richard Padovan, Proportion: Science, Philosophy, Architecture, page 260:

A face of a Catalan solid can be simply constructed by the Dorman-Luke construction [3, 4, 6, 16], which is based on a dualization process known as polar reciprocation [6], with respect to the midsphere.

2003, Rona Gurkewitz, Bennett Arnstein, Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality, page 4:

The Archimedean solids can all be inscribed in a sphere and in one of the Platonic solids. Their duals are the Catalan solids, with faces that are congruent but not regular (face-transitive); indtead of circumspheres like the Archimedean solids, they have inspheres. The midspheres, touching the edges, are common to both of them.

2014, Walter Steurer, “Crystal Structures of Metallic Elements and Compounds”, in David E. Laughlin, Kazuhiro Hono, editors, Physical Metallurgy, 5th edition, volume 1, page 20:

(geometry, solid geometry) The sphere which, if it exists, is tangent to every edge of a given polyhedron and with respect to which the polyhedron's vertices are the inversion poles of the planes of the faces of its dual polyhedron (and vice versa).

intersphere

sphere tangent to every edge of a polyhedron

1994, Tim Gallagher, Bruce Piper, Chapter 7: Convexity Preserving Surface Interpolation, Nickolas S. Sapidis, Designing Fair Curves and Surfaces, page 179,
For any two circles (spheres) there always exists at least one midcircle (midsphere) which inverts the two given circles (spheres) into each other. in the older literature this is also known as the circle (sphere) of antisimilitude. […] The center of a midcircle (midsphere) is the center of similitude of two given circles (spheres). […] Clearly the midcircle (midsphere) of two equal circles (spheres) is the midline (midplane) between the two, which is partial justification for calling inversion in a circle or sphere reflection in a circle or sphere.

(geometry) The sphere, between two given spheres, by which each of the latter may be inverted into the other.

sphere of antisimilitude

intermediate sphere by which each of two given spheres can be inverted into the other


coorbit theory

For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces. Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces.

2016, Hans Georg Feichtinger, Felix Voigtlaender, “From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces”, in arXiv:

(mathematics) A theory for atomic decomposition of a range of Banach spaces of distributions, including among others the wavelet transform and the short-time Fourier transform.


noiselet

Further, we introduce a modification to the complex fast noiselet transform, which enables computationally-efficient real-time generation of the binary noiselet-based patterns using efficient integer calculations on bundled patterns.

2016, Anna Pastuszczak, Bartłomiej Szczygieł, Michał Mikołajczyk, Rafał Kotyński, “Efficient adaptation of complex-valued noiselet sensing matrices for compressed single-pixel imaging”, in arXiv:

(signal processing) Any of a family of functions related to wavelets, analogous to the way that the Fourier basis is related to a time domain signal.


drunkard's walk

(mathematics) A type of random walk in which termination conditions lead to a biased ending state.


walktrap

We show that fastgreedy and walktrap community detection methods on weighted label co-occurence^([sic]) graphs are 85-92% more likely to yield better F1 scores than random partitioning.

2016, Piotr Szymański, Tomasz Kajdanowicz, Kristian Kersting, “How is a data-driven approach better than random choice in label space division for multi-label classification?”, in arXiv:

(graph theory) An algorithm for identifying communities in large networks using random walks.


Triebel-Lizorkin space

Triebel-Lizorkin space (plural Triebel-Lizorkin spaces)

Named after Hans Triebel and Petr Ivanovich Lizorkin.

(functional analysis) A generalization of many standard function spaces such as Lᵖ spaces and Sobolev spaces.


matricization

matricization (countable and uncountable, plural matricizations)

(mathematics) The operation that turns a tensor into a matrix.


matricize

en-verb

matricize (third-person singular simple present matricizes, present participle matricizing, simple past and past participle matricized)

matricizes (present singular third-person)
matricizing (participle present)
matricized (participle past)
matricized (past)

(mathematics, transitive) To convert (a tensor) into a matrix by means of matricization.


quartary

From Latin quārtārius (“quarter-sextarius, quarter-pint”), from quārtus (“fourth”) + -ārius (“-ary: forming adj.”). Equivalent to quart + -ary. Doublet of quartarius.

der

(historical, rare, obsolete) Synonym of quartarius, quarter-sextarius, a Roman unit of liquid measure equivalent to about 0.14 L

quartarius

(historical, rare, obsolete) A Roman unit of mass, one-fourth of a Roman pound.


quartary

From Latin quārtus (“fourth”) + -ary (adjective-forming suffix) within English in place of the correct form quaternary.

(proscribed, obsolete) Synonym of quaternary: Of the fourth rank or order, fourth in a series.


prewhiten

en-verb

prewhiten (third-person singular simple present prewhitens, present participle prewhitening, simple past and past participle prewhitened)

prewhitens (present singular third-person)
prewhitening (participle present)
prewhitened (participle past)
prewhitened (past)

(signal processing, transitive) To modify (a data set) to make the samples statistically independent, analogous to white noise.


orthohedron

en-noun

orthohedron (plural orthohedrons or orthohedra)

orthohedrons (plural)
orthohedra (plural)

(mathematics) A figure, somewhat like a cube but with two square faces and four rectangular faces


petaparsec

(astronomy) An astronomical unit of distance equal to 10¹⁵ parsecs.


wine gallon

A pre-imperial unit of capacity, roughly equivalent to 3.785 litres; used to measure wine, honey, oils, and other commercial liquids.


DCT

DCT (countable and uncountable, plural DCTs)

discrete cosine transform
discrete cosine transform#English

(mathematics) Initialism of discrete cosine transform.


kmph

kilometres per hour
kilometres per hour#English


cutwidth

(graph theory) The minimum number of edges that cross any cut between lower-numbered and higher-numbered vertices in an optimal linear arrangement of the vertices of a graph.


rowmotion

(mathematics) A certain operation on the order ideals (or antichains) of a finite poset.


Whittle likelihood

Whittle likelihood (plural Whittle likelihoods)

Named after the mathematician and statistician Peter Whittle, who introduced it in his PhD thesis in 1951.

(signal processing) An approximation to the likelihood function of a stationary Gaussian time series.


polygamma function

polygamma function (plural polygamma functions)

(mathematics) A meromorphic function on ℂ, defined as the (m+1)-th derivative (for a given order m) of the logarithm of the gamma function.


digamma function

digamma function (plural digamma functions)

(mathematics) The first of the polygamma functions, being the logarithmic derivative of the gamma function.


trigamma function

trigamma function (plural trigamma functions)

(mathematics) The second of the polygamma functions, after the digamma function.


circle of Apollonius

circle of Apollonius (plural circles of Apollonius)

Apollonius of Perga
Circles of Apollonius

Named for the ancient Greek geometer and astronomer Apollonius of Perga (ca 262—ca 190 BCE).

In the present paper an attempt is made to find for the tetrahedron the analogues of the circles of Apollonius of the triangle.

1934, Tôhoku Mathematical Journal, volumes 39-40, page 264:

It follows that O lies on the circle of Apollonius for A and C and ratio OT/OC, and on the circle of Apollonius for B and C and ratio OT/OC. Point O is thus found at the intersections, if any exist, of these two circles of Apollonius. The details are left to the reader.

1995, Howard Whitley Eves, College Geometry, page 165:

More generally, the locus of points z such that #92;left#92;vertx-z#92;right#92;vert#61;#92;alpha#92;left#92;verty-z#92;right#92;vert is a circle of Apollonius which has x and y as inverse points and which cuts each circle through x and y orthogonally.[…]They show loci analogous to circles of Apollonius and the locus of points equidistant from two given points when the norm is an #92;mathcal#123;l#125;#95;p-norm.

1996, A. C. Thompson, Minkowski Geometry, page 126:

(geometry) The figure (a generalised circle) definable as the locus of points P such that, for given points A, B and C, =/(\frac).

(geometry) Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the problem of Apollonius (i.e., intersect each circle tangentially).


bicomposite

(mathematics) composite in two different ways


WLS

(statistics) Abbreviation of weighted least squares.


semidirect product

semidirect product (plural semidirect products)

Refers to the fact that the criteria are less strict than for the direct product; compare semidirect.

1998, Hernán Cendra, Darryl D. Holm, Jerrold E. Marsden, Tudor S. Ratiu, Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products, A. G. Khovanskiĭ, A. Varchenko, V. Vassiliev (editors), Geometry of Differential Equations, American Mathematical Society, Translations, Series 2, Volume 186, Advances in the Mathematical Sciences 39, page 8,
The preceding result is a special case of a general theorem on reduction by stages for semidirect products acting on a symplectic manifold […] .

Also, by Lemma 3.1, every group G with a normal subgroup N and complement H is isomorphic to a semidirect product of N by H, and so once we prove Theorem 3.2, it will be fair to say that we have constructed all possible split extensions (up to isomorphism).

2008, I. Martin Isaacs, Finite Group Theory, American Mathematical Society, page 69:

Theorem 4.8.5 Let G be a group, H,Klt;G,H◅ G. Then G is the internal semidirect product of H and K if and only if H#92;capK#61;1 and G#61;HK.
Many groups can be understood as semidirect products.

2012, Fernando Q. Gouvêa, A Guide to Groups, Rings, and Fields, page 61:

(group theory) A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup.


undersimplification

undersimplification (countable and uncountable, plural undersimplifications)

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

Both oversimplification (e.g., explanatory variables are incomplete) and undersimplification (e.g., explanatory variables are redundant) of reality can lead to trustless explanation and unfaithful prediction.

2016 January 23, “Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression”, in PLOS ONE, →DOI:


parametrix

fundamental solution
fundamental solution

partial differential equation
partial differential equation

(mathematics) An approximation to a fundamental solution of a partial differential equation, and is essentially an approximate inverse to a differential operator.


Picard-Lindelöf theorem

Named after Émile Picard and Ernst Lindelöf.

(mathematics) An important theorem on the existence and uniqueness of solutions to first-order equations with given initial conditions.


biconnectivity

(graph theory) The property of being biconnected.


homeomorphous

en-adj

homeomorphous (comparative more homeomorphous, superlative most homeomorphous)

more homeomorphous (comparative)
most homeomorphous (superlative)


Spearman's rank correlation coefficient

en-noun

Spearman's rank correlation coefficient (plural Spearman's rank correlation coefficients)

A statistical measure that expresses two variables by correlation on a scale from 1 to -1.

Charles Spearman
Spearman's rank correlation coefficient


generalized circle

generalized circle (plural generalized circles)

We shall speak of lines and circles as generalized circles. A line is a generalized circle that passes through the point at infinity. Under the inversion #92;zeta#61;1#47;z the generalized circle (53.3) goes into
(53-4) β|ζ|²-pζ-◌̅p◌̅ζ+α=0,

1965, H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, page 247

which is a generalized circle in the ζ-plane.

For any Apollonian family of circles defined by the point circles A and B, the Coaxal Circles Theorem states that every generalized circle through A and B meets each of the Apollonian circles at right angles.

1999, David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, page 252:

This is why it makes sense to call every circle and every straight line a generalized circle. Thus the geodesics of ds consist of all arcs of generalized circles which lie wholly within Ω and are orthogonal to w (see Figure 2).

2001, Elie Zahar, Poincaré's Philosophy: From Conventionalism to Phenomenology, page 84:

(geometry, inversive geometry) A circle or a line, the two being regarded as types of a single form.


contour integral

contour integral (plural contour integrals)

(mathematics) The integral of a function around a contour on the complex plane.


Möbius transformation

Möbius transformation (plural Möbius transformations)

Named for German mathematician and theoretical astronomer August Ferdinand Möbius (1790–1868).

2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97,
Jørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete.

2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323,
Modular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant.

Classical Kleinian groups are discrete subgroups of Möbius transformations which act on the Riemann sphere with a nonempty region of discontinuity.

2013, Angel Cano, Juan Pablo Navarrete, Seade Kuri José Antonio, Complex Kleinian Groups, page 1:

complex projective line
complex projective line

(geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line.

August Ferdinand Möbius
Möbius transformation


point at infinity

point at infinity (plural points at infinity)

1960, Roger A. Johnson, Advanced Euclidean Geometry, Republished 2007, page 45,
In the geometry of inversion, therefore, it is usual to sacrifice the line of points at infinity which is so useful in other fields, and adopt the convention of a single point at infinity, the inverse of the center of the circle of inversion.

As the degree form Y² or X² of a parabola has only a single root, it has only one point at infinity.

1990, Shreeram Shankar Abhyankar, Algebraic Geometry for Scientists and Engineers, page 10:

1997, Susan Addington, Stuart Levy, Lost in the Fun House: An Application of Dynamic Projective Geometry, James King, Doris Schattschneider, Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, page 159,
To unify the treatment of concurrent and parallel lines, define a point at infinity for each family of parallel lines, and declare each point at infinity to lie on each of the parallel lines that define it. […] Now parallel lines meet at infinity, and vanishing points are the images of points at infinity.

Projective space is, by definition, the collection of the points in affine space together with the points at infinity.

2003, Ron Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, page 18:

(geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion.

Hyperbolic geometry provides each line with two points at infinity. Whether there is a piece of the line beyond the points at infinity, completing the piece lying in the finite to a closed curve, we cannot say, since we cannot move as far as the points at infinity, let alone beyond them.

1996, John Stillwell, Sources of Hyperbolic Geometry, page 82:

2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,
As we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.
This representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré, […] .

To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l’ and we call ω their common point at infinity.

2012, Norbert A'Campo, Athanase Papadopoulos, “Notes on non-Euclidean Geometry”, in Athanase Papadopoulos, editor, Strasbourg Master Class on Geometry, page 126:

(geometry, hyperbolic geometry) An ideal point.


circle inversion

circle inversion (plural circle inversions)

The first of these is obtained by the method of circle inversion first discovered by Peaucellier.

1875, A. B. Kempe, On Some New Limkages: Messenger of Mathematics, volume 4, page 121:

Nikolai Joukowski (1847-1921) was a Russian mathematician who did research in aerodynamics (James and James, 1992). He used circle inversion in the complex number plane to study airfoil shapes.

1997, Mathematical Association of America, MAA Notes, numbers 41-42, page 169:

As is common with nonlinear projections, it is possible to come up with many variants of circle inversions.[…]An obvious (but perhaps not very useful) extension of circle inversion is sphere inversion, where the spaces inside and outside a sphere are swapped.

2011, David Salomon, The Computer Graphics Manual, page 88:

(geometry, inversive geometry, for a given circle with centre O and radius r) A transformation that maps any point P ≠ O to the point P’ on the ray from O through P for which OP × OP’ = r², and which maps O to the point at infinity.


inverse semigroup

inverse semigroup (plural inverse semigroups)

Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.

2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35:

Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.

2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v:

Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.

2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249:

(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.

inverse Halbgruppe

semigroup in which every element has an inverse


cryptovariable

(cryptography) The key used for encryption and decryption.


block cipher

(cryptography) Any deterministic cipher that operates on fixed-size groups of bits (binary digits), called blocks, with an unvarying transformation specified by a symmetric key.


cryptofunction

(cryptography) A mathematical function used to carry out cryptography.


TFR

Coordinate term: fertility rate

(demography) Initialism of total fertility rate.

Coordinate term: NOTAM

(US, aviation) Initialism of temporary flight restriction.

time-frequency representation
time-frequency representation#English

(signal processing) Initialism of time-frequency representation.


perceptronium

Coined by the physicist Max Tegmark, from perceptron + -ium.

Max Tegmark, a physicist at the Massachusetts Institute of Physics (he also spoke at the New York event), has proposed that there is a state of matter — like solid, liquid and gas — that he calls perceptronium: atoms arranged so they can process information and give rise to subjectivity.

2016 July 4, George Johnson, “Consciousness: The Mind Messing With the Mind”, in New York Times:

A hypothetical state of matter capable of giving rise to self-awareness and subjectivity.


differential operator

differential operator (plural differential operators)

Define P(D) to be the differential operator given by P(D)#61;D²#43;5D#43;6, where D is the differentiation operator. Then P(D)(#92;sin(x))#61;-#92;sin(x)#43;5#92;cos(x)#43;6.

[…]these conditions appeared in the very first papers on ordinary differential operators.

2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328:

A differential operator P#92;left(D#92;right) is hypoelliptic if for any domain ⊂ #92;mathbb#123;R#125;ⁿ any solution u of the equation P#92;left(D#92;right)u#61;0 from the class #92;mathfrak#123;D#125;'#92;left(#92;Omega#92;right) is a function from C#92;infty(ω) for any open set ω ⊂⊂ #92;Omega.
A complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].

2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27:

Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.

1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, page 251:

(mathematics, mathematical analysis) An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives).


digit sum

The digit sum of 517 is 5 + 1 + 7 = 13.

Given an integer X written in base 10 (as is usual), compute the sum of the digits of X: call the result the digit sum of X. If the digit sum is greater than 9, we form the digit sum again. Continue in this way to obtain the iterated digit sum which is at most 9.

2004, J. F. Humphreys, M. Y. Prest, Numbers, Groups and Codes, Cambridge University Press, →ISBN, page 49:

(mathematics, of a positive integer) The sum of all digits.

The digit sum of 517 is 4, since 5 + 1 + 7 = 13 and 1 + 3 = 4.

The definition is that to find the digit sum of any number, add all the digits in the number. If number is a one digit no. then stop else add all the digits in this new number. Continue this process till we get one digit number.

2012, Kiran Parulekar, Amazing Properties of Squares & Their Calculations, Kiran Anil Parulekar, page 17:

Every number has a digit sum. The digit sum of 7 is seven. The digit sum of 62 is eight (6 + 2). The digit sum of 728 is also eight (7 + 2 + 8=17, 1 + 7 = 8). The digit sum of a number is found by adding all the digits in the number until a single digit is left.

2002, Paul Swan, Investigating Patterns in Number Relationships, R.I.C. Publications, →ISBN, page 1:

(mathematics, of a positive integer) the number obtained when repeatedly applying above described operation until arriving at a single-digit number.


Morley triangle

Morley triangle (plural Morley triangles)

Named for Anglo-American mathematician Frank Morley.

Show that the sides of the Morley triangle of T (cf. 10.3.10) lie on three of those lines (notice that each vertex of the Morley triangle is the center of a cardioid lying inside T and tangent to all its sides, one of them twice.)

2009, Marcel Berger, Geometry I, page 276:

The length of a side of the Morley triangle of a given triangle T is less than one third the length of the smallest side of T.
Remark. The Morley triangle of T is the equilateral triangle formed by the intersection in pairs of the angle trisectors of T.

2013, Dragoslav S. Mitrinovic, J. Pecaric, V. Volenec, Recent Advances in Geometric Inequalities, page 276:

There are a total of 18 Morley triangles that can be constructed. One amusing article I spotted on the web was from a math enthusiast who wrote a computer program trying to illustrate the central Morley triangle we started with, but due to a bug actually trisected the exterior angles in come cases... and was surprised to produce equilateral Morley triangles anyway!

2016, Erik Seligman, Math Mutation Classics, page 27:

Morley's trisector theorem
Morley's trisector theorem

(geometry) Any of a number of triangles described in Morley's trisector theorem, especially the equilateral triangle formed from the points of intersection of adjacent angle trisectors of a given triangle.

triangle de Morley

first Morley triangle, Morley's triangle

equilateral triangle formed by intersection points of adjacent trisectors


hyperperfect number

hyperperfect number (plural hyperperfect numbers)

1966, American Mathematical Society Translations, page 258,
[…] the asymptotic density of all hyperperfect numbers, that is, numbers m for which m | σ(m), is equal to zero.

The rank of a hyperperfect number N is the ratio of divisor sum to N (which equals 2 for perfect numbers).

1974, William Judson LeVeque, editor, Reviews in number theory, as printed in Mathematical reviews, 1940 through 1972, volumes 1-44 inclusive, volume 1, page 107:

In 1974, Daniel Minoli and Robert Bear described a number of properties of hyperperfect numbers.

1999, James J. Tattersall, Elementary Number Theory in Nine Chapters, page 144:

(mathematics, number theory) Any natural number n for which, for some positive integer k, n = 1 + k(σ(n) - n - 1), where σ(n) is the sum of the positive divisors of n.


hyperperfect

A square-free number is hyperperfect if and only if it is unitary hyperperfect.

1981, Mathematics of computation, volume 36, page 299:

(mathematics) Being a hyperperfect number.


MLP

multilayer perceptron
multilayer perceptron#English

(machine learning) Initialism of multilayer perceptron.


Rule 110

The binary number sequence 01101110, summarising the rule, is equivalent to the decimal number 110. This is the rule's Wolfram code.

(cellular automata) A simple cellular automaton, unusual in being Turing-complete, where each cell's value changes depending on its current value and those of two neighbouring cells.


Euler line

Named for Swiss mathematician Leonhard Euler.

Since the line joining the circumcenter and orthocenter of a triangle is its Euler line, we see that this parabola is the envelope of the Euler lines of the triangles Aᵢ.

1983, The American Mathematical Monthly, volume 40, page 199:

The Euler line in the preceding figure is OH. N, the center of the nine-point circle, not only lies on the Euler line, but is also its midpoint.

2000, Alfred S. Posamentier, Making Geometry Come Alive: Student Activities and Teacher Notes, page 147:

2011, Derek Allan Holton, A Second Step to Mathematical Olympiad Problems, World Scientific Publishing, Mathematical Olympiad Series, Volume 7, page 57,
Show that the perpendicular bisector of LM in Figure 2.5 meets the Euler line halfway between the orthocentre and the circumcentre of ΔABC.

(geometry) A line that, for a given triangle, passes through several important points, including the circumcentre, orthocentre and centroid; an analogous line for certain other 2- and 3-dimensional geometric figures.

Theorem 2.1 A connected graph with even local degrees has an Euler line.

1961, Yale University, Graphs and Their Uses, page 25:

In defining an Euler graph, some authors drop the requirement that the Euler line be closed.

2009, J. P. Chauhan, Krishna's Applied Discrete Mathematics, page 279:

Euler lines mainly deal with the nature of connectivity in graphs. The concept of an Euler line is used to solve several puzzles and games.[…]A closed walk running through every edge of the graph G exactly once is called an Euler line.

2013, Bhavanari Satyanarayana, Kuncham Syam Prasad, Near Rings, Fuzzy Ideals, and Graph Theory, page 372:

(graph theory) An Eulerian path, a looped path through a graph that passes along every edge exactly once.


Exeter point

Named for the Phillips Exeter Academy (in the town of Exeter, New Hampshire), where it was discovered during a computers-in-mathematics workshop in 1986.

It ^([the Euler line]) passes through the centroid, the Exeter point and the center of the nine-point circle of the triangle.

2015, Gerard Venema, Just the Facts 101: Foundations of Geometry, Cram101 Textbook Reviews, unnumbered page:

(geometry, given a triangle ΔABC whose medians intersect its circumcircle at {A, A’}, {B, B’} and {C, C’} and ΔDEF constructed from the tangent lines at A, B and C such that D, E and F are respectively opposite A, B and C) The point of concurrence of the rays DA’, EB’ and FC’.

B and C such that D; E and F are respectively opposite A; B and C; B and C such that D; E and F are respectively opposite A; B and C

Clark Kimberling
Encyclopedia of Triangle Centers
Exeter point
Exeter, New Hampshire
Phillips Exeter Academy


inversive geometry

inversive geometry (countable and uncountable, plural inversive geometries)

1983, John Willard Milnor, On the Geometry of the Kepler Problem, The American Mathematical Monthly, Volume 90, reprinted in 1994, John Milnor, Collected Papers, I: Geometry, page 261,
In particular, the non-Euclidean geometry of Lobachevsky and Bolyai, and the inversive geometry of Steiner play an important role.

The study of properties of such families of circles gave rise to a new geometry, called inversive geometry, which was able to provide particularly striking proofs of previously known results in Euclidean geometry as well as new results.

1999, David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, page 2:

Even after that, despite the need identified by Klein, Kasner, and J. W. Young, foundations of inversive geometry did not receive much research attention.
David Hilbert's school at Göttingen spawned an axiomatization of inversive geometry by B. L. van der Waerden and Lucas J. Smid in 1935.

2007, Elena Anne Marchisotto, James T. Smith, The Legacy of Mario Pieri in Geometry and Arithmetic, page 143:

(geometry) The branch of geometry concerned with inversion transformations, specifically circle inversions in the Euclidean plane, but also as generalised in non-Euclidean and higher-dimensional spaces.


right trapezoid

right trapezoid (plural right trapezoids)

Right trapezoids are used in the trapezoidal rule for estimating areas under a curve.

(geometry) A trapezoid with two adjacent right angles


midcircle

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

Every triangle has a midcircle, and its center is the fourth kind of middle point that a triangle can have. In 1767 Leonhard Euler proved that for all triangles, the orthocenter, the circumcenter, the centroid and the center of the midcircle are always on the same line.

2010, Alex Bellos, Here's Looking at Euclid, page 61:

(geometry) The circle (if one exists) that passes through the midpoint of each side of a given polygon, especially a triangle.


difference quotient

difference quotient (plural difference quotients)

(mathematical analysis) The slope of a secant as obtained by dividing the vertical difference between two intersections of the secant and a curve with their horizontal distance. When the horizontal distance approaches zero the slope of the secant approaches the derivative of the curve.


estadal

From Spanish estadal, from estado + -al, from Latin stātus (“state, status, standing”). Doublet of state, status, and estado.

der

(historical) A traditional Spanish unit of length, equivalent to about 3.3 m.

(historical) A traditional Spanish unit of area, equivalent to about 11.2 m².


semimean

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

The planes in which lie the semimajor and semimean axes of both the pebbles and quartz-feldspar aggregates coincide with those of the crystalline schistosity (Sj, which follows the bedding (S0).

1970, Akademii͡a︡ nauk SSSR, Doklady of the Academy of Sciences of the U.S.S.R:

(geometry) Pertaining to half the length of the middle axis on an ellipse (halfway between the major and minor axes).


semimean

Moreover, semimeans unify many of the more common measures.

1983, The Engineering Economist - Volume 29, page 12:

The third captures the mean squared wealth and the lower semimean squared wealth.

2010, Gerd Infanger, Stochastic Programming:

(statistics) The result obtained by taking either the largest or smallest half of the values and calculating the mean.


product rule

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

(mathematics) A formula used to find the derivatives of products of two or more functions: (f·g)'=f'·g+f·g',!


Helly family

Named after the mathematician Eduard Helly (1884-1943).

(combinatorics) A family of order k of sets, such that any minimal subfamily with an empty intersection has k or fewer sets in it.


geometric multiplicity

en-noun

geometric multiplicity (plural geometric multiplicities)

(linear algebra) The dimension of the eigenspace of a given eigenvalue (of a given square matrix).


de Sitter space

de Sitter space (plural de Sitter spaces)

Named after Willem de Sitter (1872–1934), professor of astronomy.

A Lorentzian manifold in Minkowski space which is the analogue of the sphere in ordinary Euclidean space.


anti-de Sitter space

anti-de Sitter space (plural anti-de Sitter spaces)

A Lorentzian manifold in Minkowski space which is the analogue of a hyperbola in ordinary Euclidean space.


syndeticity

From syndetic + -ity. Equivalent to syndeton + -icity.

The state or quality of being syndetic, particularly

(grammar) The use of syndeton, the state or quality of using a conjunction or equivalent conjunctive structure.

Values within affordances were ranked and assigned metric values. Standardized canonical discriminant function coefficients of .835, .515, and .298 for complexity, concreteness, and syndeticity.

1994, Mary Esther Brown, Children's Naming of Subject Categories..., Drexel University, p. 79

(library science) The use of crossreferences or the ability to be crossreferenced.

For topological recurrence we need a different notion of largeness for subsets of S, namely that of syndeticity. A subset of E of N is said to be syndetic if it has bounded gaps. Equivalently, E C N is syndetic if N is the union of finitely many shifts.

1988, Robert Ellis & al., ed., Topological Dynamics and Applications... American Mathematical Society, p. 211

(mathematics) The condition of having bounded gaps between members.


polyzeta

The paper ends by the combinatorial setting of systems of local systems of coordinates on the group of group-like series. * The present work is part of a series of papers devoted to the study of the renormalization of divergent polyzetas (at positive and at non-positive indices) via the factorization of the non commutative generating series of polylogarithms and of harmonic sums and via the effective construction of pairs of dual bases in duality in #92;phi-deformed shuffle algebras.

2015, Gérard H.E. Duchamp, Vincel Hoang Ngoc Minh, Christophe Tollu, Van Chiên Bui, Quoc Hoan Ngô, “(Pure) transcendence bases in ϕ-deformed shuffle bialgebras”, in Seminaire Lotharingien de Combinatoire, volume Universit\'e Louis:

(mathematics) A complex function related to zeta functions


supercuspidal

A theorem of Bernstein states that for any compact open subgroup K of G(k), there are, up to unramified twists, only finitely many K-spherical supercuspidal representations of G(k).

2015, Manish Mishra, “A Galois side analogue of a theorem of Bernstein”, in arXiv:

(mathematics) That has a zero Jacquet functor for every proper parabolic subgroup


ham sandwich theorem

LL-Q1860 (eng)-Flame, not lame-ham sandwich theorem.wav

By analogy with cutting a sandwich having three layers (two of bread and one of ham), for the case where n=3.

(mathematics) A theorem stating that, given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n−1)-dimensional hyperplane.


sedenionic

(mathematics) Of or pertaining to sedenions.


Morita-equivalent

(mathematics) Exhibiting Morita equivalence.


Morita equivalence

Named after Japanese mathematician Kiiti Morita, who defined equivalence and a similar notion of duality in 1958.

(mathematics) A certain relationship defined between algebraic rings that preserves many ring-theoretic properties.


plane geometry

The study of figures having parts that lie in a singular plane.


MoE

GLaM [51] is an autoregressive mixture-of-experts (MoE) model with up to 1200B parameters.

2023, Gerhard Paaß, Sven Giesselbach, Foundation Models for Natural Language Processing: Pre-trained Language Models Integrating Media, Springer Nature, →ISBN, page 130:

(machine learning) Initialism of mixture of experts.

(statistics) Initialism of margin of error.


Hamiltonicity

(graph theory) The property of being Hamiltonian.


voudon

Blend of voudou + quaternion. Named after the 256 Ifá divination signs in West African voodoo.

Following the convention adopted by Tony Smith, who has called the 2⁸-ions Voudons after the 256 deities of the Ifa pantheon of Voodoo or Voudon

2002, Robert P. C. de Marrais, “Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2ⁿ-ions Beyond the Sedenions”, in arXiv, →DOI, page 7:

According to the logic of doubling the dimensions on which Cayley–Dickson algebra is built, after quaternions the so-called 8D octonions (#92;mathbbO—Octonions) can be obtained, followed by 16D sedenions, 32D pathions, 64D chingons, 128D routons, and 256D voudons.

2022, Zlatka Valkova-Jarvis, Vladimir Poulkov, Viktor Stoynov, Dimitriya Mihaylova, Georgi Iliev, “A Method for the Design of Bicomplex Orthogonal DSP Algorithms for Applications in Intelligent Radio Access Networks”, in Symmetry, volume 14, number 3, MDPI AG, →DOI, →ISSN:

There is no theoretical limit to multispace dimensional numbering as there are definitions for octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and voudon (256D) numbered space.

2023, Melanie Swan, Renato P. dos Santos, “Quantum Intelligence: Responsible Human-AI Entities”, in AAAI 2023 Spring Symposia, Socially Responsible AI for Wellbeing, March 27–29, 2023, USA, CEUR Workshop Proceedings, →DOI:

(mathematics) A 256-dimensional hypercomplex number.


semigroup homomorphism

semigroup homomorphism (plural semigroup homomorphisms)

(mathematics) a function that preserves semigroup structure


Bayesianism

(statistics) An approach to probability which quantifies expectations based on given data, following Bayes' theorem.


hypocube

A square, or any other geometric figure analogous to a cube but having fewer than three dimensions.


Rosser's trick

Introduced by J. Barkley Rosser in 1936.

Gödel's incompleteness theorem
Gödel's incompleteness theorem

(mathematics) A method for proving Gödel's incompleteness theorems without the assumption that the theory being considered is ω-consistent. While Gödel's original proof uses a sentence that states (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".


monotoneity

(mathematics) The property of being monotone.


Pochhammer symbol

Pochhammer symbol (plural Pochhammer symbols)

(mathematics) The notation (x)ₙ, where n is a non-negative integer, which depending on the context represents either the rising factorial or the falling factorial.


decycle

en-verb

decycle (third-person singular simple present decycles, present participle decycling, simple past and past participle decycled)

decycles (present singular third-person)
decycling (participle present)
decycled (participle past)
decycled (past)

In the decycling phase it removes or redirects some edges until the resulting graph is acyclic.

2009, Ioannis G. Tollis, Maurizio Patrignani, Graph Drawing: 16th International Symposium:

(graph theory, transitive) To remove the cycles from.


Herschel graph

From Herschel (“a surname”) + graph, after British astronomer Alexander Stewart Herschel (1836—1907), who identified the associated polyhedron (an enneahedron) as one for which there is no solution to the icosian game.

Select a suitable independent set / and use part (b) to show that the graph in Fig. 11.81 (known as the Herschel graph) has no Hamilton cycle.

1994, Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, page 587:

A bipartite graph like the Herschel graph of Figure 9.2 is also non-hamiltonian, but the algorithm is not likely to delete enough vertices to notice that it has a large separating set.

2004, William Kocay, Donald L. Kreher, Graphs, Algorithms, and Optimization, page 202:

2006, Michael S. Keane, Dee Denteneer, Frank Hollander, Evgeny Verbitskiy, Dynamics and Stochastics, Institute of Mathematical Statistics, Lecture Notes—Monograph Series, Volume 48, page 174,
It is difficult to control what loops may arise: for example the Herschel graph [3] shows that a convex polyhedron need not be Hamiltonian as a graph.

(mathematics, graph theory) A bipartite undirected graph with 11 vertices and 18 edges that is the smallest non-Hamiltonian polyhedral graph.


Cramer's rule

Named after the Swiss mathematician Gabriel Cramer, who originally published the formula in 1750.

linear equations
linear equation#English

(linear algebra) An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.


L'Hôpital's rule

Named after French mathematician Guillaume de l'Hôpital (1661–1704).

(mathematics) The rule that the limit of the ratio of two functions equals the limit of the ratio of their derivatives, usable when the former limit is indeterminate and the latter limit exists.

regla de L'Hôpital

a rule which uses derivatives to help evaluate limits involving indeterminate forms


ramifiable

One cannot prove in Z F C that "δ_D and cf(D) are ramifiable cardinals implies that D is weakly ramifiable".

1998, Travaux de Mathématiques - Issue 2:

We say that D is Θ-ramifiable if for each tree T on D, there is a branch h of T (we say that T has a branch).

2002, Reports on Mathematical Logic - Issues 36-40, page 25:

(mathematics) Capable of being subdivided into branches.


Rendulic

From a 1937 paper by L. Rendulic, who introduced a method for modelling triaxial stress.

Stress probes at different initial axi-symmetric (triaxial) stress states have been carried out to exhibit bifurcation domain in the Rendulic plane.

2007, Evelyn Regar, A.M.G.J. van Leeuwen, Patrick W. Serruys, Optical Coherence Tomography in Cardiovascular Research, →ISBN, page 124:

A Rendulic plot is a plot representing the stress path for triaxial tests originally suggested by Rendulic (1937) and later developed by Henkel (1960).

2007, Braja M. Das, Advanced Soil Mechanics, →ISBN, page 413:

The Rendulic contours in principal stress space showed that the stress paths taken were members of a family for both drained and undrained tests.

2013, Nayland G. Stanley-Wood, Enlargement and Compaction of Particulate Solids, →ISBN:

(mathematics, sciences) Pertaining to the two-dimensional modelling of three-dimensional stress paths that are symmetric about an axis.


congy

From French conge, from Latin congius, q.v. Only attested in English in the obsolete plural form congys.

(historical, obsolete) Synonym of congius, Roman units of liquid measure and mass.


Banach-Tarski paradox

Named after Polish mathematicians Stefan Banach and Alfred Tarski.

(mathematics) A theorem in set-theoretic geometry, which states that given a solid ball in three‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

Alfred Tarski
Banach-Tarski paradox
Stefan Banach


autocorrelate

autocorrelate (third-person singular simple present autocorrelates, present participle autocorrelating, simple past and past participle autocorrelated)

autocorrelates (present singular third-person)
autocorrelating (participle present)
autocorrelated (participle past)
autocorrelated (past)

(transitive, signal processing) To cross-correlate (a signal) with itself.


prosthaphaeretic

(mathematics, historical) Relating to prosthaphaeresis.


prosthaphaeresis

From Ancient Greek προσθαφαίρεσις (prosthaphaíresis), from πρόσθεν (prósthen, “before”) plus ἀφαίρεσις (aphaíresis, “taking away”).

uder

προσθαφαίρεσις

(mathematics, historical) Variant of prosthaphæresis


stoquastic

Stoquastic Hamiltonians, those for which all off-diagonal matrix elements in the standard basis are real and non-positive, are common in the physical world.

2015, Ian Kivlichan, “On the complexity of stoquastic Hamiltonians,” publication not named (termpaper?), 11 December 2015

We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic.

2016, Edward Farhi, Aram W Harrow, “Quantum Supremacy through the Quantum Approximate Optimization Algorithm”, in arXiv:

As stoquastic problems we use frustrated Ising Hamiltonians, having random local X and Z ﬁelds, and random σ_zσ_z couplings.

2016, R. Barends et al., “Digitized adiabatic quantum computing with a superconducting circuit”, in Nature, 534, 222-226, 9 June 2016:

quantum mechanics
quantum mechanics#English

stochastic matrices
stochastic matrices#English

(mathematics) Of a Hermitian matrix, having real non-positive off-diagonal elements. These matrices have elements of quantum mechanics and classical theory of stochastic matrices. For non-technical purposes ‘stoquastic’ is equivalent to “avoiding the sign problem”.


-megaly

From Ancient Greek μεγάλος (megálos), further μέγας (mégas).


Goddard-Thorn theorem

Named after Peter Goddard and Charles Thorn.

(mathematics, physics) A theorem about certain vector spaces, relating to string theory.


Kac-Moody algebra

en-noun

Kac-Moody algebra (plural Kac-Moody algebras)

Named after Victor Kac and Robert Moody, who independently discovered them.

(mathematics) A Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. They have applications in theoretical physics.


Killing form

(mathematics) A symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.


Albert algebra

Named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers.

(mathematics) A 27-dimensional exceptional Jordan algebra.


Jordan algebra

(mathematics) A non-associative algebra over a field whose product satisfies the axioms (i) xy=yx (the commutative law) and (ii) (xy)(xx)=x(y(xx)) (the Jordan identity).


Kantor-Koecher-Tits construction

Introduced by Jacques Tits (1962), Kantor (1964), and Koecher (1967).

(algebra) A method of constructing a Lie algebra from a Jordan algebra.


Cartan integer

(mathematics) Any of the scalar products a_ij=2(r_i,r_j)/(r_i,r_i) that make up the Cartan matrix of a simple Lie algebra.


rejection region

rejection region (plural rejection regions)

Now it turns out that practically speaking a given rejection function induces only two rejection regions.

2006, William A. Dembski, No Free Lunch, →ISBN:

Thus, any event in the rejection region leads to rejection of the null hypothesis.

2010, Chris Spatz, Basic Statistics: Tales of Distributions, →ISBN, page 180:

How does this affect our rejection region? In this case, we calculate the area of the rejection region on only one side of the sampling distribution.

2013, David Weisburd, Chester Britt, Statistics in Criminal Justice, →ISBN, page 185:

(statistics) A range of test values for which the null hypothesis is rejected.


ortholattice

(mathematics) An orthocomplemented lattice.


chainette

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


hexlet

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

(geometry) A chain of six spheres, each of which is tangent to both of its neighbours and also to three mutually tangent given spheres.


biquasiprimitive

The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient.

2015, Jehan A. Al-bar, Ahmad N. Al-kenani, Najat Mohammad Muthana, Cheryl E. Praeger, Pablo Spiga, “Finite edge-transitive oriented graphs of valency four: a global approach”, in arXiv:

(mathematics) quasiprimitive in two different ways


coreflective

We give sufficient conditions to obtain left determined model structures on a full reflective subcategory, on a full coreflective subcategory and on a comma category.

2015, Philippe Gaucher, “About left determined model categories”, in arXiv:


finitistic

Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1 or infinite..

2015, Francois Couchot, “Gaussian trivial ring extensions and fqp-rings”, in Communications in Algebra, volume Taylor \& Francis:


supersolution

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

We study the asymptotic, as {\epsilon} goes to 0, of the corresponding value functions, and show convergence, in the sense of weak semilimits, to sub and supersolution of a suitable limit equation containing the effective Hamiltonian.

2015, Thuong Nguyen, Antonio Siconolfi, “Singularly Perturbed Control Systems with Noncompact Fast Variable”, in arXiv:

(mathematics) The superset of a solution


Baire space

A topological space such that every intersection of a countable collection of open dense sets in the space is also dense.


key stretching

(cryptography) Techniques used to make a key more secure against brute force attacks by increasing the resources needed to test each possible key.


bicomplex number

bicomplex number (plural bicomplex numbers)

(algebra) A pair (w,z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^*=(w,-z), and the product of two bicomplex numbers as (u,v)(w,z)=(uw-vz,uz+vw).


highly composite number

highly composite number (plural highly composite numbers)

Plato
Srinivasa Ramanujan
highly composite number

Coined by Indian mathematician Srinivasa Ramanujan in 1915, although it has been suggested that Plato may have known of the concept, since he specified 5040 (a highly composite number) as the ideal number of citizens in a city.

In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q#95;#123;2k#125;(N),#92;1#92;lek#92;le 4, where Q#95;#123;2k#125;(N) denotes the number of representations of N as the sum of 2k squares, and to #92;sigma#95;#123;-s#125;(N), where #92;sigma#95;#123;-s#125;(N) denotes the sum of the (-s)th powers of the divisors of N.

2012, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, page 359:

Hardy has stated that a highly composite number is as unlike a prime as a number can be.

1998, K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Books, page 48:

Ramanujan devoted a section of his paper to the study of Q(x), the number of highly composite numbers #92;lex Since d(2n)gt;d(n), we see that between x and 2x, there is always a highly composite number.

2013, M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, page 144:

A highly composite number, then, was in Hardy's phrase "as unlike a prime as a number can be." Ramanujan had explored their properties for some time; in the earliest pages of his second notebook he'd listed about a hundred highly composite numbers−the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120−searching for patterns. He found them.

2013, Robert Kanigel, The Man Who Knew Infinity, Simon & Schuster, page 232:

(number theory) A positive integer that has more divisors than any smaller positive integer.

hochzusammengesetzte Zahl

positive integer with more divisors than any smaller positive integer


window function

window function (plural window functions)

(signal processing, statistics) A function that limits a signal to a restricted portion of time or frequencies.


tailgroup

(mathematics) The group of instances at the tail (low end) of a distribution curve


XXXL

XXXL (countable and uncountable, plural XXXLs)

extra extra extra large
extra extra extra large#English

(chiefly clothing) Initialism of extra extra extra large (the manufactured size or an item of that size).


XXXL

(chiefly clothing) Initialism of extra extra extra large (of the manufactured size).


four color theorem

(graph theory) A theorem stating that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions so that no two adjacent regions have the same color (adjacent being defined as two regions sharing a boundary, not counting corners, in which three or more regions share a boundary).


Goldbach's conjecture

“Goldbach’s conjecture, I finally remember it,” he says, as I’m standing to go. “And I want to make sure I tell you. It’s important: For every even number n>2, there exist two prime numbers p_1 and p_2 such that we can write: p_1 + p_2 = n…”

2022, Iain Reid, We Spread, Scribner, pages 255-256:

(number theory) A conjecture stating that every even integer greater than 2 can be expressed as the sum of two primes.


defuzzify

en-verb

defuzzify (third-person singular simple present defuzzifies, present participle defuzzifying, simple past and past participle defuzzified)

defuzzifies (present singular third-person)
defuzzifying (participle present)
defuzzified (participle past)
defuzzified (past)

In practice we usually defuzzify the output waveform B to a single numerical value y j in Y by computing the fuzzy centroid of B with respect to the output universe of discourse Y.

2007, W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral, W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral, Elementary Fuzzy Matrix Theory and Fuzzy Models for Social Scientists:

(mathematics) To subject to defuzzification.


Edwards curve

(mathematics) Any of a family of elliptic curves with applications in cryptography.


superjump

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

(mathematics) The jump operator for type two functionals.


place value

(arithmetic) The value of a digit of numbers. For example, ones, tens, hundreds in the decimal counting; ones, sixes, thirty-sixes in the senary counting.


number ray

(mathematics) A number line that has one endpoint (typically at zero)


semiconjugacy

(mathematics) A mapping that is semiconjugate.


Futurama theorem

From "The Prisoner of Benda" (2010), season 6, episode 10 of Futurama. The episode and the theorem were written by the mathematician Ken Keeler.

(group theory) A theorem which proves that, given a mind swap machine that can't be used on the same pair of people more than once, it is possible to reverse any number of mind swaps by introducing two more people that did not have their minds swapped with anyone before.

Futurama
Ken Keeler
The Prisoner of Benda


Hohmann transfer orbit

en-noun

Hohmann transfer orbit (plural Hohmann transfer orbits)

Named after Walter Hohmann, German scientist who described it in 1925.

(aerospace) An elliptical orbit used to transfer between two circular orbits of different radii in the same plane.


AWS

“The concern that these weapons might misidentify targets is the least of our worries,” he said. “More significant is the threat of an A.W.S. arms race and proliferation crisis.”

2021 June 3, Maria Cramer, quoting James Dawes, “A.I. Drone May Have Acted on Its Own in Attacking Fighters, U.N. Says”, in The New York Times, →ISSN:

artificial intelligence
artificial intelligence

(military, robotics, artificial intelligence) Initialism of Autonomous Weapons System.


cafiso

From Italian cafiso, from Sicilian cafisu, from Arabic قَفِيز (qafīz), from Aramaic קְפִיזָא and Classical Syriac ܩܦܝܙܐ (qəp̄īzā), of uncertain derivation from Egyptian, Akkadian, and similar units.

der

Aramaic קְפִיזָא

Alternative form of qafiz (particularly in Italian contexts)


prime gap

(mathematics) The difference between two successive prime numbers.


bit-compressed

(mathematics, computing) Represented by the minimum number of bits that can represent the maximum value (of an integer variable)


bitvector

(mathematics, computing) A vector (or similar array) of bits that represent Boolean variables.


hyperclique

(mathematics) An itemset containing items that are strongly correlated with each other


pseudocount

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

(mathematics, statistics) A number added to a number of observations so as to change a generated probability to one obtained by prior knowledge.


ultrametrics

(mathematics) The study and use of ultrametric systems


cafiz

(historical) Alternative form of qafiz, particularly in Tunisian contexts


quasireversible

(chemistry, mathematics) Apparently reversible


Lindemann-Weierstrass theorem

Named after Ferdinand von Lindemann and Karl Weierstrass, who worked on related proofs.

(number theory) A result that is useful in establishing the transcendence of numbers, stating that, if α₁, ..., αₙ are algebraic numbers which are linearly independent over the rational numbers ℚ, then e^(α₁), ..., e^(αₙ) are algebraically independent over ℚ.


Fary-Milnor theorem

Named after István Fáry and John Milnor, who proved it independently in 1949 and 1950.

(mathematics) In knot theory, a theorem stating that three-dimensional smooth curves with small total curvature must be unknotted.


outedge

(graph theory) An outgoing edge in a digraph, i.e. one that leaves a particular node.