minimal ideal

Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.

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

Theorem 12. A semisimple ring R has only a finite number of minimal ideals and is their direct sum. Moreover, each minimal ideal is a simple ring.

1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33:

Note that if a semigroup has a zero, then its minimal ideal is {0}. In this context, the notion of minimal ideal is not so useful, and so we introduce the notion of a 0-minimal ideal.

2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596:

(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.


parsimonization

(mathematics, computing) Conversion to a parsimonious form


kidney-shaped

en-adj

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

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

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


radical ideal

1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,
Any intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime.

It follows at once that R#47;I is a reduced ring iff I is a radical ideal. Thus, the ideals I(X) are all radical ideals.

1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33:

1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,
Thus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals […] .

(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I).

commutative algebra; ring theory; commutative algebra; ring theory


reconvolution

(mathematics) A second or subsequent convolution, typically performed after deconvolution


undercoverage

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

(statistics) Insufficient coverage (inadequate representation in a sample)


ratiuncule

/ɹeɪʃˈʌŋkjuːl/

Learned borrowing from Latin ratiuncula.

(mathematics, obsolete) A ratio very close to unity.


eigendimension

(mathematics) The dimension of an eigenspace


reduced ring

We do this for reduced rings in Corollary 10, and for rings with nonzero nilpotents in Corollary 15.

1997, Thomas G. Lucas, “Characterizing When R(X) is Completely Integrally Closed”, in Daniel Anderson, editor, Factorization in Integral Domains, Marcel Dekker, page 401:

Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m#92;inM and a#92;inR, ma#61;0 implies mR#92;capMa#61;0. Various results of reduced rings are extended to reduced modules.

2004, Tsiu-Kwen Lee, Yiqiang Zhou, “Reduced Modules”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, Marcel Dekker, page 365:

In general, the first case of importance is the normalization of a reduced ring R in its quotient ring K(R).

2005, David Eisenbud, The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry, Springer, page 210:

(algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0.

reduzierter Ring

ring that has no nonzero nilpotent elements


reduced algebra

en-noun

reduced algebra (plural reduced algebras)

(algebra, ring theory) An algebra over a reduced ring.


summation method

summation method (plural summation methods)

mathematical analysis
mathematical analysis

(mathematical analysis) Any of various methods for constructing generalized sums of series


eigenmap

Interestingly, if we compute the Laplace-Beltrami eigenmap of the sphere, its Reeb graph also has a chain structure as in Fig.2(c).

2008, Dimitris Metaxas, Leon Axel, Gabor Fichtinger, Medical Image Computing and Computer-Assisted Intervention, page 150:

In this paper, a method of Semantic-based Laplacian Eigenmap (SBLE) is proposed.

2016, Zhiqiu Huang, Xingming Sun, Junzhou Luo, Cloud Computing and Security, page 551:

(mathematics) The graph in which data points function as nodes and connectivity between nodes is governed by the proximity or correlation between data points.


lattice theory

lattice theory (countable and uncountable, plural lattice theories)

Apart from an introduction of effective techniques, a most significant contribution of our work has been the elevation of lattice theory in computational intelligence – soft computing.

2006, Vassilis G. Kaburlasos, Towards a Unified Modeling and Knowledge-Representation based on Lattice Theory, Springer, page 4:

2008, Rudolf Wille, Formal Concept Analysis as Applied Lattice Theory, Sadok Ben Yahia, Engelbert Mephu Nguifo, Radim Belohlacek (editors), Concept Lattices and Their Applications: Fourth International Conference, Selected Papers, Springer, LNAI 4923, page 42,
Formal Concept Analysis is a mathematical theory of concept hierarchies which is based on Lattice Theory.

(mathematics, uncountable) The branch of mathematics concerned with lattices (partially ordered sets).


metaparameter

(mathematics) A parameter that controls the value of one or more others.


summability method

summability method (plural summability methods)

(mathematical analysis) A summation method.


costructure

costructure (countable and uncountable, plural costructures)

(mathematics) The structure produced by the intersection of multiple related curves or shapes.


transitive set

(set theory) Any set X such that for any x ∈ X, if y ∈ x then y ∈ X; equivalently, such that if x ∈ X and x is not an urelement then x ⊆ X.


transitive closure

transitive closure (plural transitive closures)

Transitive closure
Transitive set#Transitive closure

(set theory, order theory, of a binary relation R on a set X) The smallest binary relation on X that includes R and is transitive.

(set theory, of a set X) The smallest transitive set that contains X.


Alexandrov topology

Alexandrov topology (plural Alexandrov topologies)

Along with the closely related concept of Alexandrov-discrete space, after Russian mathematician Pavel Sergeyevich Alexandrov.

4.24. THEOREM. The following three restrictions on a space-time M are equivalent:
(a) M is strongly causal;
(b) the Alexandrov topology agrees with the manifold topology;
(c) the Alexandrov topology is Hausdorff.

1972, Roger Penrose, Techniques of Differential Topology in Relativity, SIAM, page 34:

In Minkowski space-time there is, indeed, a topology definable solely in terms of the causal connectivity among events, the Alexandrov topology, which is provably equivalent to the ordinary manifold topology.

1990, Palle Yourgrau, Demonstratives, Oxford University Press, page 252:

2016, Piero Pagliani, Covering Rough Sets and Formal Topology — A Uniform Approach Through Intensional and Extensional Operators, James F. Peters, Andrzej Skowron (editors), Transactions on Rough Sets XX, Springer, LNCS 10020, page 134,
(6) S_0,2(P(R_C)) is the family of open sets of the Alexandrov topology induced by R_C and S(P(R_C)) is the family of closed sets of the Alexandrov topology induced by R^⌣_C;

(topology) A topology in which the intersection of any family of open sets is an open set.

Alexandrov topology
Pavel Sergeyevich Alexandrov


studentize

en-verb

studentize (third-person singular simple present studentizes, present participle studentizing, simple past and past participle studentized)

studentizes (present singular third-person)
studentizing (participle present)
studentized (participle past)
studentized (past)

(statistics, transitive) To perform studentization on.


pointless topology

en-noun

pointless topology (usually uncountable, plural pointless topologies)

The works of Ehresmann [1957/58], Benabou [1957/58], Papert and Papert [1957/58] give rise to pointless topology, i.e. an abstract treatment of a class of lattices (the frames) extending the class whose members are the lattices of open sets of the topological spaces.

1995, Francis Buekenhout, editor, Handbook of Incidence Geometry: Buildings and Foundations, North-Holland: Elsevier, page 1018:

1997, Stephen C. Hirtle, Andrew U. Frank (editors), Spatial Information Theory A Theoretical Basis for GIS: International Conference COSIT '97, Proceedings, Springer, LNCS 1329, page 170,
Pointless topology is more general than point-set topology in that there are complete Heyting algebras which do not come from the open set lattices of topological spaces.

2001, Peter Johnstone, Elements of the History of Locale Theory, C.E. Aull, R. Lowen (editors), Handbook of the History of General Topology, Volume 3, Kluwer Academic, page 841,
Banaschewski was to become one of the most prolific authors on pointless topology, and in particular on its constructive and choice-free aspects, in the next three decades.

(mathematics) An approach to topology that avoids mentioning points.


gigadeath

They'll have no reason to hit the datasphere now. There will be no gigadeaths.

2008, Ken MacLeod, Fractions: The First Half of The Fall Revolution, page 285:


BFS

Coordinate term: DFS

breadth-first search
breadth-first search#English

(graph theory) Initialism of breadth-first search.


DFS

Coordinate term: BFS

depth-first search
depth-first search#English

(graph theory) Initialism of depth-first search.


iterative deepening search

iterative deepening search (countable and uncountable, plural iterative deepening searches)

breadth-first search
breadth-first search

(computer science, graph theory) A type of depth-first search in which each row of the tree is searched incrementally, simulating a breadth-first search with less memory usage.


totally ordered

1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,
(A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above).

The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.

1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:

G is a transitive group of order automorphisms of a totally ordered set X,
θ is a convex G-congruence on the totally ordered set X,
ξ is the order type of the totally ordered set X,
◌̅η is the order type of some class Y=xη of the congruence η,

1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation

ζ is the order type of the totally ordered quotient set X/θ of the totally ordered set X by the congruence θ.

partially ordered set
partially ordered set

(set theory, order theory) That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset.


cofilament

(category theory) A diagram in a category that is indexed by a directed set.


Brun's constant

(number theory) The finite value, denoted by B₂, to which the sum of the reciprocals of the twin primes converge, according to Brun's theorem.


Brun's theorem

(number theory) A theorem stating that the sum of the reciprocals of the twin primes converges to a finite value (Brun's constant).


transformation matrix

transformation matrix (plural transformation matrices or transformation matrixes)

transformation matrices (plural)
transformation matrixes (plural)

Given a linear transformation T(x) in functional form, its transformation matrix can be constructed by applying T to each vector of the standard basis, then inserting the results into the columns of the new matrix.

A transformation matrix of dimension n×m operates on a column vector of dimension m×1 to produce a row vector of dimension 1×n.

1963 [McGraw-Hill], Lawrence P. Huelsman, Circuits, Matrices and Linear Vector Spaces, 2011, Dover, page 191,
We would like to make as many as possible of the elements of the transformation matrix equal zero.

1968 [McGraw-Hill], Granino A. Korn, Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, 2000, Dover, page 414,
Refer to Sec. 14.8-6 for a procedure yielding transformation matrices T with the desired properties.

The 4x4 transformation matrices are conveniently used to convert various entities expressed in different coordinate systems into another.

2005, Gerard Kim, Designing Virtual Reality Systems: The Structured Approach, Volume 1, Springer, page 47:

linear transformation
linear transformation

(linear algebra) A matrix (of dimension n×m) that represents some linear transformation from ℝᵐ→ℝⁿ.

kuvausmatriisi

matrix that represents a linear transformation


offdiagonal

(mathematics) That is not an element on the diagonal of a square matrix


projection matrix

projection matrix (plural projection matrices)

2006, H. Watari, Y. Shimoyama, Chapter 5: Density Matrix Formalism of Angular Momentum in Multi-Quantum Magnetic Resonance, Christopher J. Bender, Lawrence Berliner (editors, Computational and Instrumental Methods in EPR, Springer, page 191,
The resultant eigenvalues of the angular momentum are independent of direction, but the projection matrices are dependent.

We use the GLU utility function gluPerspective() to set the projection matrix and define the whole image to be the view port (essentially what is to be shown).

2012, Jan Erik Solem, Programming Computer Vision with Python, O'Reilly, page 91:

transformation matrix
transformation matrix

(linear algebra) A transformation matrix whose associated transformation is a projection.

Caswell (1989) pointed out that most linear stage-structured models have both primitive and irreducible projection matrices.[…]The projection matrix M shown in the table is then calculated as PS + R.

1999, Terrance J. Quinn II, Richard B. Deriso, Quantitative Fish Dynamics, Oxford University Press, page 372:

We then turn to applications of population projection matrices both for natural resource management and for conservation.

2013, John H. Vandermeer, Deborah E. Goldberg, Population Ecology: First Principles, 2nd edition, Princeton University Press, page 62:

(statistics) A matrix that maps a vector of response (dependent variable) values to a vector of fitted or predicted values.


least squares

(mathematics) A method of fitting a straight line through a series of points such that the sum of the squares of the residuals is minimized


bisquare

(mathematics) An extension of the least squares method that removes or downweights extreme outliers


axiom of power set

The axiom of choice differs from other axioms of ZF by stating existence of a set (i.e., a choice function) without defining it (unlike, for instance, the axiom of pairing or the axiom of power set).

1978, Thomas Jech, Set Theory, Academic Press, page 38:

Verifying that the axiom of power set is in #92;#123;#92;phi#92;in#92;mathcal#123;L#125;#42;#58;S#92;vdash#92;mathcal#123;I#125;(#92;phi)#92;#125; relies on some rudimentary comprehension axioms.

2003, Thomas Forster, Reasoning About Theoretical Entities, World Scientific, page 51:

But the ZF axioms of which the hierarchy is an intuitive model involve impredicative quantifications. Most striking is the axiom of power set in tandem with the axiom of separation.

2011, Adam Rieger, “9: Paradox, ZF, and the Axiom of Foundation”, in David DeVidi, Michael Hallett, Peter Clark, editors, Logic, Mathematics, Philosophy: Vintage Enthusiasms: Essays in Honour of John L. Bell, Springer, page 183:

2012, A. H. Lightstone, H. B. Enderton (editor), Mathematical Logic: An Introduction to Model Theory, Plenum Press, Softcover, page 292,
The Axiom of Power Set asserts that the collection of all subsets of a set is a set. […] Adding the Axiom of Power Set compels the collection empty to be a set.

(set theory) The axiom that the power set of any set exists and is a valid set, which appears in the standard axiomatisation of set theory, ZFC.


nullsurface

(mathematics) A surface whose normal at all points is a null vector


Motzkin number

(mathematics) For a given number n, the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). They have diverse applications in geometry, combinatorics, and number theory.


Narayana number

Narayana number (plural Narayana numbers)

Named after Canadian mathematician T. V. Narayana (1930–1987).

(combinatorics) Any of the triangular array of natural numbers defined by N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, occurring in various counting problems.


Delannoy number

Delannoy number (plural Delannoy numbers)

Named after French army officer and amateur mathematician Henri Delannoy.

(mathematics) The number of possible paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east.


Schröder number

Schröder number (plural Schröder numbers)

Named after the German mathematician Ernst Schröder.

(mathematics) The number of possible lattice paths from the southwest corner (0, 0) of an n × n grid to the northeast corner (n, n), using only single steps north, northeast or east, and not rising above the southwest–northeast diagonal.


Gabor transform

Gabor transform (plural Gabor transforms)

(signal processing) A special case of the short-time Fourier transform, used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time.


Gabor filter

(signal processing) A linear filter used for image texture analysis.


Gabor wavelet

(signal processing) A form of wavelet designed so as to minimize the product of its standard deviations in the time and frequency domain.


Gabor atom

(signal processing) A kind of function used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function.


Kolakoski sequence

Kolakoski sequence (plural Kolakoski sequences)

Named after the recreational mathematician William Kolakoski (1944–97), who discussed it in 1965, though subsequent research has revealed that it first appeared in a paper by Rufus Oldenburger in 1939.

(mathematics) An infinite sequence of symbols {1,2} that is its own run-length encoding and the prototype for an infinite family of related sequences.


anticycle

(graph theory) A cycle in the complement of a given graph.


order type

1965 [John Wiley], Raymond L. Wilder, Introduction to the Foundations of Mathematics: 2nd Edition, 2012, Dover, page 116,
Another way of putting this is to state that the order type is that aspect of the arrangement of the elements of a simply ordered set, which remains unchanged when any two elements are exchanged. […] As in the case of cardinal numbers, order types may be denoted by suitable symbols called ordinal numerals.

In [13] Chajoth studied how the order type of a chain can alter if we change the position of elements in a linearly ordered set, resp. if we introduce a new element in a linearly ordered set.

2005, Egbert Harzheim, Ordered Sets, Springer, page 332:

2011, Douglas Cenzer, Valentina Harizanov, Jeffrey B. Remmel, Effective Categoricity of Injection Structures, Benedikt Löwe, Dag Normann, Ivan Soskov, Alexandra Soskova (editors, Models of Computation in Context: 7th Conference on Computability in Europe, CiE 2011, Proceedings, Springer, LNCS 6735, page 51,
We let ω denote the order type of N under the usual ordering and Z denote the order type of Z under the usual ordering.

equivalence relation
equivalence relation

(set theory) In the context of sets equipped with an order (especially, the context of totally ordered sets), the characteristic of being a member of some equivalence class of such sets under the equivalence relation "existence of an order-preserving bijection".

järjestystyyppi

characteristic of being a member of an equivalence class of ordered sets


eigenvariate

These activities are usually taken to be the first local eigenvariate around the voxel of interest (a regional mean, weighted in accordance with the local covariance structure).

2004, Kenneth Hugdahl, Richard J. Davidson, The Asymmetrical Brain, page 183:

(mathematics) A principal component obtained from a multivariate dataset through principal components analysis.


axiom of countable choice

The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable. One example is the axiom of countable choice, which states that if A#95;0,A#95;1,#92;dotsA#95;n#92;dots form a denumerable set of nonempty sets, their product is nonempty.[…]The axiom of countable choice is constantly used in analysis; it is often hidden so as not to sow confusion in the minds of the students (who are inclined to accept anything desired) or of the professors (who do not like to shake the foundations of the discipline).

2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, page 169:

we have no way to infer
∃R∀n[P0n→∀x(R_nx→Fx)∧n=Nx:R_nx)]
without an axiom of countable choice.

2012, Richard G. Heck, Jr., Reading Frege's Grundgesetze, Oxford University Press, page 271, But, once again, while we can easily prove ∀n[P0n→∃G(∀x(Gx→Fx)∧n=Nx:Gx)]

2013, Valentin Blot, Colin Riba, On Bar Recursion and Choice in a Classical Setting, Chung-chien Shan (editor), Programming Languages and Systems: 11th International Symposium, APLAS 2013, Proceedings, Springer, LNCS 8301, page 349,
We show how Modified Bar-Recursion, a variant of Spector's Bar-Recursion due to Berger and Oliva can be used to realize the Axiom of Countable Choice in Parigot's Lambda-Mu-calculus, a direct-style language for the representation and evaluation of classical proofs.

(set theory) A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function; equivalently, the statement that the direct product of a countable collection of nonempty sets is nonempty.


de Moivre's formula

Named after Abraham de Moivre (1667–1754), French mathematician.

(mathematics) A formula that connects trigonometry and complex numbers, stating that, for any complex number (and, in particular, for any real number) x and integer n, big ( cos (x)+i sin (x) big )ⁿ= cos (nx)+i sin (nx), where i is the imaginary unit.


Størmer's theorem

(number theory) A theorem that gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations.


arithmetic geometry

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

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

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

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

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

Arakelov geometry is one of the branches in arithmetic geometry.

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

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


diophantine geometry

In this article, we shall recall how transcendence and diophantine geometry have grown into beautiful and far-reaching theories which now enhance many different aspects of mathematics, and we shall describe some of the principal problems that have emerged as a consequence of wide-ranging research in the field.

2000, A. Baker, G. Wüstholz, “Number Theory, Transcendence and Diophantine Geometry in the Next Millennium”, in V. Arnold, M. Atiyah, P. Lax, B. Mazur, editors, Mathematics: Frontiers and Perspectives, American Mathematical Society, page 1:

This reflects the general principle in Diophantine geometry that in attempting to study any significant problem, it is essential to have a thorough understanding of the geometry before one can hope to make progress on the number theory.

2009, Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd edition, Springer, page 41:

The topics of the talks extended from algebraic number theory to diophantine geometry, curves and abelian varieties over finite fields and applications to codes, boolean functions or cryptography.

2012, Yves Aubry, Christophe Ritzenthaler, Alexey Zykin, editors, Arithmetic, Geometry, Cryptography and Coding Theory: 13th Conference, American Mathematical Society, page vii:

Diophantine equation
Diophantine equation

(mathematics) A developing branch of mathematics in which techniques of algebraic geometry are applied to number theory (specifically the theory of Diophantine equations), in particular being concerned with algebraic varieties over fields that are finitely generated over their prime fields and over local fields.


kaneh

From Hebrew קָנֶה (qané, “rod”, literally “cane”). Doublet of cane and canna.

bor

(historical) A Hebrew measure of length, equal to six cubits.


Zermelo set theory

After German mathematician Ernst Zermelo, who introduced the theory in a paper in 1908.

1998, Geoffrey Hellman, 8: Beyond Definitionism—But Not Too Far Beyond, Matthias Schirn (editor), 2003, Paperback, The Philosophy of Mathematics Today, Oxford University Press (Clarendon Press), page 215,
The system can be claimed to transcend certain critical limitations of predicative analysis and certain awkwardness of known nominalistic approaches to mathematics at once, and would seem to provide a well-motivated alternative to Zermelo set theory as a framework for scientifically applicable mathematics.

According to the great logician Alonzo Church, in spite of superficial differences, these two "widely accepted symbolic systems," Zermelo set theory and simple type theory, are "in their currently accepted forms essentially similar" [1939, 69-70].

2001, José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Springer (Birkhäuser), page 338:

Z is a formal specification language founded on Zermelo set theory.

2006, Gerard O'Regan, Mathematical Approaches to Software Quality, Springer, page 20:

(set theory) An early and influential axiomatisation of set theory.


skew-Hermitian

(linear algebra) Of a matrix T, such that T^†=-T.


p-adic

From p + -adic. The letter p follows common number theory usage in representing an arbitrary prime number; the suffix -adic signals that properties of the number p (generally, those due to its being prime, and specifically those of a specified p) are fundamental to the theory and determine the properties of the resulting construction.

The theory of p'''-adic analytic functions in domains other than simple disks is not very well known yet, although such kind of functions happens to intervene in questions linked to p'''-adic functional analysis, number theory, and others.

1995, Alain Escassut, Analytic Elements in P-adic Analysis, World Scientific, page v:

1997, Fernando Q. Gouvêa, p-adic Numbers: An Introduction, Springer, 2nd Edition, page 60,
Definition 3.3.3 The ring of p'''-adic integers is the valuation ring
Z ₚ=x∈ Q ₚ:|x|ₚ<1.

2. Preliminaries
3. A p'''-adic Cauchy problem
4. The p'''-adic Hilbert transform
5. The boundaries in the p'''-adic space L^q
[…]

2007, Nguyen Minh Cheung, Nguyen Van Co, Le Quang Thuan, §12: Harmonic Analysis Over P-adic Field I: Some Equations and Singular Field Operators, N. M. Chuong, Yu V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (editors), Harmonic, Wavelet and P-adic Analysis, World Scientific, page 272, The paper is organized as follows

(number theory) Of, pertaining to, (ultimately) derived from or defined in the context of p-adic numbers.


combinatorialize

en-verb

combinatorialize (third-person singular simple present combinatorializes, present participle combinatorializing, simple past and past participle combinatorialized)

combinatorializes (present singular third-person)
combinatorializing (participle present)
combinatorialized (participle past)
combinatorialized (past)

(mathematics) To generate combinatorial sequences


Lychrel number

Coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend's first name.

In base ten (decimal), no Lychrel numbers have been yet proved to exist, but many are suspected.

(mathematics) A natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers.


rug-pull

A back-formation from pull the rug out from under.

back-formation

Coordinate term: pump and dump

The anonymous founders of a crypto collective called AnubisDAO raised nearly $60 million in a few hours last year; less than a day later, the funds disappeared in the second-largest rug pull of 2021, according to the blockchain-tracking firm Chainalysis.

2022 March 2, David Yaffe-Bellany, “Millions for Crypto Start-Ups, No Real Names Necessary”, in The New York Times, →ISSN:

(cryptocurrencies) A fraud scheme where anonymous founders trick people into investing money for a cryptocurrency project which is then abandoned.


semiparabolic

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

(geometry) Having the form of a semiparabola.


Solèr's theorem

The theorem was originally proved by Maria Pia Solèr.

(mathematics) A result concerning certain infinite-dimensional vector spaces, stating that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions.


bounded lattice

en-noun

bounded lattice (plural bounded lattices)

2004, Anna Maria Radzikowska, Etienne Kerre, On L-Fuzzy Rough Sets, Leszek Rutkowski, Jörg Siekmann, Ryszard Tadeusiewicz, Lotfi A. Zadeh (editors), Artificial Intelligence and Soft Computing — ICAISC 2004: 7th International Conference, Proceedings, Springer, LNAI 3070, page 526,
A residuated lattice is an extension of a bounded lattice by a monoid operation and its residuum, which are abstract counterparts of a triangular norm and a fuzzy residual implication, respectively.

2006, Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne Kerre, Triangle Lattices: Towards an Axiomatization of Interval-Valued Residuated Lattices, Salavatore Greco, Yukata Hata, Shoji Hirano, Masahiro Inuiguchi, Sadaaki Miyamoto, Hung Son Nguyen, Roman Słowiński (editors), Rough Sets and Current Trends in Computing: 5th International Conference, Proceedings, Springer, LNAI 4259, page 117,
Indeed, in the scope of these logics, formulas can be assigned not only 0 and 1 as truth values, but also elements of [0,1], or, more generally, of a bounded lattice ℒ.

In this paper, we study idempotent nullnorms on bounded lattices. We prove that there is no idempotent nullnorm on a distributive bounded lattice L different from the proposal in [13].

2018, Gül Deniz Çayli, Funda Karaçal, “Some Remarks on Idempotent Nullnorms on Bounded Lattices”, in Vicenç Torra, Radko Mesiar, Bernard De Baets, editors, Aggregation Functions in Theory and in Practice, Springer,, page 32:

(algebra, order theory) Any lattice (type of partially ordered set) that has both a greatest and a least element.


cocircularity

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(geometry) The condition of being cocircular


cocircular

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


tridigital

(mathematics) With three numerical digits.


unprimeable

In the year 200 Diophantus, noted Greek mathematician was born. 200 million is approximately 2/3 of the total population of this country we call home. 200 is also the smallest base 10 unprimeable number.

2007 May 26, “Episode #200”, in Game Show Newsnet:

6,292,015 is a composite, Duffinian, odd, semiprime, square-free and unprimeable number.

2015 June 29, “Brain-O Quiz Answers”, in Mathematical Mystery Tour:

200 is the smallest unprimable number

2015 November 1, Adam Spencer, Adam Spencer's World of Numbers, Xoum Publishing, page 200:

There are infinitely many unprimeable numbers, although the first unprimeable number ending in an odd number is not until 212159.

2019, Julia Collins, Numbers in Minutes:

(arithmetic) That cannot be turned into a prime number by changing just one of its digits to any other digit.


logarithmic derivative

logarithmic derivative (plural logarithmic derivatives)

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

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

In economics, it is popular to consider the logarithmic derivative (#92;ln#92;varphi(t))'#61;#92;varphi'(t)#47;#92;varphi(t), as a convenient growth measure.

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

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

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

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


aranzada

Borrowed from Spanish aranzada, from aranzata (“ration”), from Old Spanish arenzo and arienço (“Spanish coin”), from Latin argenteus (“silvery, silver piece”).

der

(historical) A traditional Spanish unit of land area, equivalent to about 4472 m², chiefly used for vinyards and olive groves.


rug pull

en-verb

rug pull (third-person singular simple present rug pulls, present participle rug pulling, simple past and past participle rug pulled)

rug pulls (present singular third-person)
rug pulling (participle present)
rug pulled (participle past)
rug pulled (past)

To prevent rugging, projects have implemented measures to commit liquidity through locking contracts and token burn related to liquidity pools. DeFi100 coin creators have rug pulled close to USD 32 million. SushiSwap has also suffered[…]

2021 August 31, Rajesh Dhuddu, Srinivas Mahankali, Blockchain A to Z Explained: Become a Blockchain Pro with 400+ Terms (English Edition), BPB Publications, →ISBN:

For that reason, you need to do your research before deciding on a cryptocurrency to yield farm. ... former Google engineers with visible faces and active Twitter accounts, it is unlikely that you will be “rug pulled” by that project.

2022 May 10, Robert Ross, A Beginner's Guide to High-Risk, High-Reward Investing: From Cryptocurrencies and Short Selling to SPACs and NFTs, an Essential Guide to the Next Big Investment, Simon and Schuster, →ISBN, page 244:

“Until the crypto market had a drop and some of the bigger exchanges cratered,” he explained. “People got nervous and wanted their money out of crypto. It should have been no big deal to ... “And then you got rug pulled?” I guessed.

2022 December 14, Roxie Rivera, When The Westons Went To Moscow, Night Works Books:

(cryptocurrencies) to carry out a rug-pull


Maths B

(Australia, Queensland) abbreviation of Mathematics B.


topologization

Many of the topological concepts that will be defined later can be incorporated into methods of topologization in the same general way.

1963, Donald Bushaw, Elements of General Topology, Wiley, page 29:

1968, A. F. Mutylin, An Example of Nontrivial Topologization of the Field of Rational Numbers: Complete Locally Bounded Fields, V. P. Kompaniec (editor), Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Society Translations, Series 2, Volume 73, page 164,
The p-adic topologization also does not coincide with ours and is not stronger than it since any Uₙ contains Pₖ which is an integer and which is relatively prime with p.

The problem on the non-discrete Hausdorff topologization of rings was investigated in a series of works.

2005, Vladimir Arnautov, Kirill Filippov, “On Group Topologies on an Abelian Group Preceding One Another”, in Svetlana Cojocaru, Gerhard Pfister, Victor Ufnarovski, editors, Computational Commutative and Non-commutative Algebraic Geometry, IOS Press, page 251:

(topology) The act or process of topologizing; the result of a process of topologizing, especially the topology so produced.


prime number theorem

en-noun

prime number theorem (plural prime number theorems)

But we cannot infer from them the equivalence in any sense of these two propositions, since we have used in our proof of the prime number theorem a subsidiary theorem on the order of magnitude of #92;zeta'(s)#47;#92;zeta(s).

1932, A. E. Ingham, Albert Edward Ingham, The Distribution of Prime Numbers, Cambridge University Press, page 39:

1974 [Academic Press], Harold M. Edwards, Riemann's Zeta Function, 2001, Dover, page 182,
The problem of locating the roots p of ζ, and consequently the problem of estimating the error in the prime number theorem, is closely related to the problem of estimating the growth of ζ in the critical strip 0<Res<1 as Ims→∞.

In 1859, Riemann attempted to give a complete proof of the prime number theorem using the zeta function for complex variables s. Although he was not successful in proving the prime number theorem he established many properties of the zeta function and showed that the prime number theorem depended on the zeros of the zeta function.

2016, Benjamin Fine, Gerhard Rosenberger, Number Theory: An Introduction via the Density of Primes, 2nd edition, Springer (Birkhäuser), page 145:

(number theory) The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.

In [6], [11], abstract prime number theorems are proved under a variety of conditions.

1996, Illinois Journal of Mathematics, volume 40, University of Illinois Press, page 245:

(number theory) Any theorem that concerns the distribution of prime numbers.


field extension

field extension (plural field extensions)

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

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

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

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

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

kuntalaajennus

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


monovariant

(mathematics) A variable whose value changes in one direction only (gets either larger or smaller)


prime field

1968, J. F. Schuh, Mathematical Tools for Modern Physics, Philips Technical Library, page 91,
Theorem 2. Each field contains a unique prime field.

2004, H. Gopalkrishna Gadiyar, K. M. Sangeeta Maini, R. Padma, Cryptography, Connections, Cocycles and Crystals: A p-adic Exploration of the Discrete Logarithm Problem, Anne Canteaut, Kapaleeswaran Viswanathan (editors), Progress in Cryptology - INDOCRYPT 2004: 5th International Conference, Proceedings, Springer, LNCS 3348, page 305,
Applying Hensel's lemma to the discrete logarithm problem over prime fields reveals the rich geometric and algebraic structure underlying the problem.

Since every field contains a prime field, by studying prime fields, and the way properties are passed over to extensions, Steinitz would attain a full picture of the structure of all possible fields.

2004, Leo Corry, Modern Algebra and the Rise of Mathematical Structures, 2nd edition, Springer (Birkhäuser), page 194:

(algebra, field theory) A field that contains no proper subfields.


disjoint set

(mathematics) A family of sets sharing no elements in common; sets whose intersection is the empty set.


algebraic extension

algebraic extension (plural algebraic extensions)

What we now have to prove is that: if an overring R#42; of R is such that R#42; is an integral domain, R#42; is integral over R and the quotient field #92;mathfrak#123;L#125;#42; of R#42; is a finite algebraic extension of #92;mathfrak#123;L#125;, then R#42; is a local ring and a finite R-module.

1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200:

It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.

2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110:

We describe the correspondence between subfields of the field P#95;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #92;pi#95;1(X#92;setminusO).

2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65:

(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K).


Plücker coordinate

Plücker coordinate (plural Plücker coordinates)

Introduced by Julius Plücker in the 19th century.

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


discrete valuation

discrete valuation (plural discrete valuations)

Therefore the theory branches in two essential parts : first the study of discrete valuations of the Weyl field D#95;1(K) extending a discrete valuation of K, secondly the discrete valuations of D#95;1(K) having the purely transcendental field K(t) as a residue field.

2000, Freddy Van Oystaeyen, Algebraic Geometry for Associative Algebras, CRC Press, page 173:

One of the few possibilities to extend some arithmetic aspects of number theory to more abstract fields is via valuation theory and, as a first step, we may restrict attention to discrete valuations.

2004, Farahat S. Aly, Freddy Van Oystaeyen, “Hopf Order Module Algebra Orders”, in Stefaan Caenepeel, Fred Van Oystaeyen, editors, Hopf Algebras in Noncommutative Geometry and Physics, CRC Press, page 21:

Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field.

2011, Ricardo Portilla, “Parametrizing nilpotent orbits in p-adic symmetric spaces”, in Robert S. Doran, Paul J. Sally, Jr., Loren Spice, editors, Harmonic Analysis on Reductive, p-adic Groups, American Mathematical Society, page 201:

(algebra, field theory) A valuation (on some field) that takes integer values (including infinity).


quasicyclic

(mathematics) Describing a certain type of cyclic group or cyclic code


combinatorial commutative algebra

combinatorial commutative algebra (uncountable)

This construction has become a fundamental tool in combinatorial commutative algebra due to the work of Stanley, Hochster and Reisner ([Ho77], [S75], [S96], [R76]).

2011, Viviana Ene, Jũrgen Herzog, Gröbner Bases in Commutative Algebra, American Mathematical Society, page 88:

2013, Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón Introduction, Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón (editors), Monomial Ideals, Computations and Applications, Springer, page vii,
Monomial ideals and algebras are among the simplest structures in commutative algebra and the main objects in combinatorial commutative algebra.

Using techniques from combinatorial commutative algebra [MS05, KM], they showed that certain lattice games admit a particular type of algebraic decomposition known as affine stratifications, from which a finitely representable winning strategy can be recovered.

2013, Aaron N. Siegel, Combinatorial Game Theory, American Mathematical Society, page 269:

(algebra) A relatively new discipline in mathematics that combines techniques and concepts from combinatorics and commutative algebra, and in which the geometry of convex polytopes also plays a significant role.


stereographic projection

stereographic projection (plural stereographic projections)

1974 [Prentice-Hall], Richard A. Silverman, Complex Analysis with Applications, Dover, 1984, Unaltered republication, page 19,
Thus we see that stereographic projection establishes a one-to-one correspondence between the extended complex plane and the Riemann sphere.

1991, E. J. F. Primrose (translator), Simeon Ivanov (editor of translation), A. T. Fomenko, A. A. Tuzhilin, Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space, American Mathematical Society, page 67,
Proposition 2. The stereographic projection π_N:S²→Oxy of the sphere S² onto the xy-plane from the North Pole N preserves the angles between tangent vectors. Thus, the coordinates specified on S²⧵N by the stereographic projection π_N are isothermal.

Before we can recognize such patterns confidently, we need to know whether the stereographic projection faithfully represents the true clustering of directions in space.

2004, Richard J. Lisle, Peter R. Leyshon, Stereographic Projection Techniques for Geologists and Civil Engineers, 2nd edition, Cambridge University Press, page 40:

(projective geometry, complex analysis, cartography) A function that maps a sphere onto a plane; especially, the map generated by projecting each point of the sphere from the sphere's (designated) north pole to a point on the plane tangent to the south pole.


comb filter

So called because its frequency response consists of a series of regularly spaced notches, giving the appearance of a comb.

(signal processing) A filter that adds a delayed version of a signal to itself, causing constructive and destructive interference.


Malfatti circle

Malfatti circle (plural Malfatti circles)

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

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


rootfinding

The algorithms in 26, 28 yield (up to polylog factors) these nearly optimal precision and Boolean cost bounds for the worst case polynomial rootfinding.

2002, Felipe Cucker, Joseph Maurice Rojas, Foundations of Computational Mathematics: Proceedings of the SMALEFEST 2000:

(mathematics) The process of finding the root of an equation.


vertical angle

vertex angle

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

vrcholový úhel

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


zero divisor

regular element

antonym(s) of “any element whose product with some nonzero element is zero”

trivial zero divisor

An idempotent element e#92;ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)#61;0#61;(1-e)e.

Linnell [25, 1977] proved that if G is a torsion-free abelian by locally finite by super-solvable group and K is any field, then K[G] has no nontrivial zero divisors.

1984, J. B. Srivastava, “23: Projective Modules, Zero Divisors, and Noetherian Group Algebras”, in Dinesh N. Manocha, editor, Algebra and its Applications, CRC Press, page 170:

In the ring of integers, there are no zero divisors except 0. In a ring obtained from a Boolean algebra, on the other hand, every element except the identity is a zero-divisor.
The concept of a zero-divisor is intimately related to cancellation law as we see n the following proposition.
1.7 Proposition: Let R be a ring and x#92;inR. Then for all y,x#92;inR, either of the equations xy#61;xz or yx#61;zx implies y#61;z if and only if x is not a zero divisor. In other words, cancellation by an element is possible iff it is not a zero-divisor.

1989, K. D. Joshi, Foundations of Discrete Mathematics, New Age International, page 390:

In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors.

2010, Mitsuo Kanemitsu, “The Number of Distinct 4-Cycles and 2-Matchings of Some Zero Divisor Graphs”, in Masami Ito, Yuji Kobayashi, Kunitaka Shoji, editors, Automata, Formal Languages and Algebraic Systems: Proceedings of AFLAS 2008, World Scientific, page 63:

(algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

If R is an integral domain, that is, has no zero divisors, then R#91;x#93; also has no zero divisors.

2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234:

An element a of a ring is called a zero-divisor if a#92;ne 0 and ab#61;0 or ba#61;0 for some b#92;ne 0; if a is neither 0 nor a zero-divisor, it is said to be regular (see Section 7.1). A non-trivial ring without zero-divisors is called an integral domain; this term is not taken to imply commutativity.

2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi:

If a and b are non-zero elements of R such that ab#61;0, then aandb are both called zero divisors. If R is non-trivial and has no zero divisors, then it is called an integral domain. Note that if a is a unit in R, it cannot be a zero divisor (if ab#61;0, then multiplying both sides of this equation by a#123;-1#125; yields b#61;0.

2009, Victor Shoup, A Computational Introduction to Number Theory and Algebra, 2nd edition, Cambridge University Press, page 171:

(algebra, ring theory) A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.


finite geometry

finite geometry (countable and uncountable, plural finite geometries)

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

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

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

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

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

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

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

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

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

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

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

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

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

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


tetrahelix

tetrahelix (plural tetrahelixes or tetrahelices)

tetrahelixes (plural)
tetrahelices (plural)

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

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


biexponentiality

(mathematics) The quality of being biexponential.


Scaramucci

Anthony Scaramucci
White House Director of Communications

Borrowed from Italian Scaramucci, the measurement comes from Anthony Scaramucci's short time as White House Director of Communications.
Doublet of Scaramouche.

Ardern's peroration began, “It's been three weeks now since I was asked to take this job ...” From the crowd, a laugh of astonishment. In the space of only two Scaramuccis, to steal the unit of measurement of the day, she had transformed the fortunes of the Labour Party.

2017, Toby Manhire, “The rapture”, in Toby Manhire, editor, The Spinoff Book, New Zealand: Penguin Random House, published 2019, →ISBN:

My White House tenure lasted for approximately 220 “Scaramuccis.”

2018, Dan Pfeiffer, Yes We (Still) Can, London: Biteback Publishing, →ISBN:

If Trump himself had been more experienced at governing, or if he'd kept some senior officials longer than one or two Scaramuccis, or if he'd surrounded himself with true pros, not sycophants and sons-in-law, things might have been different.

2020, Paul Begala, You're Fired, New York: Simon and Schuster, →ISBN, page 14:

(humorous, US politics) A unit of time equal to the duration of Anthony Scaramucci's tenure as White House Director of Communications, variously defined as 10 or 11 days.


ordered ring

1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
If < is an ordering on A compatible with its ring structure, we shall say that (A,+,·,<) is an ordered ring. An element x of an ordered ring A is positive if x>0, and x is strictly positive if x>0.
The set of all positive elements of an ordered ring A is denoted by A_+, and the set of all strictly positive elements of A is denoted by A^*₊.
If (A,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.

(OR) The relations x>0 and y>0 imply xy>0.
The ring A, together with such an ordering, is called an ordered ring.
Examples. — 1) The rings Q and Z , with the usual orderings, are ordered rings.
2) A product of ordered rings, equipped with the product ordering, is an ordered ring. In particular, the ring Aᴱ of mappings from a set E to an ordered ring A is an ordered ring.
3) A subring of an ordered ring, with the induced ordering, is an ordered ring.

1990, P. M. Cohn, J. Howie (translators), Nicolas Bourbaki, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19, DEFINITION 1. — Given a commutative ring A, we say that an ordering on A is compatible with the ring structure on A if it is compatible with the additive group structure of A, and if it satisfies the following axiom

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4gt;2 because 4-2gt;0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b#92;inR, then we know by trichotomy that exactly one of the following must be true: a-bgt;0, a-b#61;0, or -(a-b)gt;0.

2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253:

(1) The set R⁺ is closed under addition and multiplication.
(2) If x∈R then exactly one of the following is true: (trichotomy law)
(a) x=0,
(b) x∈R⁺,
(c) -x∈R⁺.
If further R is an integral domain we call R an ordered integral domain.
[…]
Lemma 3.5.9. If R is an ordered ring and a∈R is a positive element, then the set na:n∈ N⊂R⁺.
[…]
Theorem 3.5.2. An ordered ring must be infinite.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Johns Hopkins University Press, page 77, Definition 3.5.4. A ring R is an ordered ring if there exists a distinguished set R⁺, R⁺⊂R, called the set of positive elements, with the properties that

(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.


quasiseparable

(mathematics, of a graph) That becomes disconnected when a suitable subgraph is removed


profinite group

profinite group (plural profinite groups)

This proposition shows that to have a representation with big image of a profinite group, for example, a Galois group, we need to take a profinite topological ring A as a coefficient ring, like the p-adic integer ring #92;Z#95;p.

2000, Haruzo Hida, Modular Forms and Galois Cohomology, Cambridge University Press, page 28:

We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a "quasi-free profinite group" (Section 10.6).

2011, Moshe Jarden, Algebraic Patching, Springer, page 207:

2014, David Barnes, Constanze Roitzheim, Rational equivariant rigidity, Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Lück, Jérôme Scherer, An Alpine Expedition through Algebraic Topology, American Mathematical Society, page 14,
Recall that a profinite group is an inverse limit of an inverse system of finite groups, with the p-adic numbers being the canonical example. Any finite group is, of course, profinite, but when we talk of a profinite group we assume that the group is infinite.

(topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space.

groupe profini

topological group isomorphic to the inverse limit of an inverse system of discrete finite groups


invariant theory

invariant theory (countable and uncountable, plural invariant theories)

1993, Bernd Sturmfels, Introduction, David Hilbert, Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, Cambridge University Press, page xi,
Today, invariant theory is often understood as a branch of representation theory, algebraic geometry, commutative algebra, and algebraic combinatorics. Each of these four disciplines has roots in nineteenth-century invariant theory. […] In modern terms, the basic problem of invariant theory can be categorized as follows. Let V be a K-vector space on which a group G acts linearly. In the ring of polynomial functions K[V] consider the subring K[V]ᴳ consisting of all polynomial functions on V which are invariant under the action of the group G. The basic problem is to describe the invariant ring K[V]ᴳ. In particular, we would like to know whether K[V]ᴳ is finitely generated as a K-algebra and, if so, to give an algorithm for computing generators.

Invariant theory is the great romantic story of mathematics.[…]In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are all offsprings of invariant theory.

2001, Gian-Carlo Rota, “What is invariant theory, really?”, in H. Crapo, D. Senato, editors, Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota, Springer,, page 41:

For a linear algebraic group G and a regular representation (#92;rho,V) of G, the basic problem of invariant theory is to describe the G-invariant elements (#123;#92;bigotimes#125;ᵏⱽ)ᴳ of the k-fold tensor product for all k.

2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 225:

(algebra, representation theory) The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.


pseudosequence

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

(mathematics) An apparent sequence used as a key in data compression


sequantitative

An initial sequantitative determination of EPM—1 in 20 sera of healthy donors and 28 patients with gastrointestinal tumors revealed that all sera of healthy donors contain high levels, in contrast to the majority of tumor sera (20/28) which had low or no detectable levels.

1987, Gastroenterology - Volume 92, page 1371:

The favorable reliability and objectiveness of this evaluation system appear to be due to partly, at least, introducing a sequantitative mycological assessment criterion.

2000, Kenji Nagino, Hiroshi Shimohira, Masatoshi Ogawa, Katsuhisa Uchida, Hideyo Yamaguchi, “Comparison of the therapeutic efficacy of oral doses of fluconazole and griseofulvin in a guinea pig model of dermatophytosis”, in The Journal of Antibiotics, volume 53, number 2:

Figure 1 Confirmation of microarray results by sequantitative RT-PCR analysis.

2005 January, Sophie Vasseur, Cédric Malicet, Ezequiel L Calvo, Jean Charles Dagorn, Juan L Iovanna, “Gene expression profiling of tumours derived from rasⱽ¹²/E1A-transformed mouse embryonic fibroblasts to identify genes required for tumour development”, in Molecular Cancer, volume 4, number 4:

(statistics) Pertaining to the quantitative analysis of autocorrelation in sequence data.


cofunctionality

(mathematics, computing) The condition of being a cofunction


kinetostatic

(mathematics, physics) Relating to a pair of opposing kinematic vectors


directed acyclic graph

directed acyclic graph (plural directed acyclic graphs)

1995, Volker Turan, Weimin Chen, GLB-closures in Directed Acyclic Graphs and Their Applications, Ernst W. Mayr, Gunther Schmidt, Gottfried Tinhofer (editors), Graph-Theoretic Concepts in Computer Science: 20th International Workshop, Proceedings, Springer, LNCS 903, page 121,
A subset S of the vertices of a directed acyclic graph is called glb-closed, if it contains the greatest lower bounds of all pairs of vertices of S. […] Directed acyclic graphs are widely used in different areas of computer science.

In studying these properties, a distinction is made between directed acyclic graphs with directed edges and simple acyclic graphs with undirected edges.[…]A directed acyclic graph G#61;(V#59;R) is asymmetric and does not contain cycles.

2001, Peter J. Pahl, Rudolf Damrath, Mathematical Foundations of Computational Engineering: A Handbook, Springer, page 574:

2011, Michael Donders, Sara Miner More, Pavel Naumov, Information Flow on Directed Acyclic Graphs, Lev D. Beklemishev, Ruy de Queiroz (editors), Logic, Language, Information, and Computation: 18th International Workshop, Proceedings, Springer, LNAI 6642, page 95,
A logical system that describes the properties of this relation for an arbitrary fixed directed acyclic graph is introduced and proven to be complete and decidable.

(graph theory, computer science) A finite directed graph that contains no directed cycles.


acyclic digraph

acyclic digraph (plural acyclic digraphs)

1993, Suh-Ryung-Kim, The Competition Number and Its Variants, J. Gimbel, J.W. Kennedy, L.V. Quintas (editors), Quo Vadis, Graph Theory?: A Source Book for Challenges and Directions, Elsevier (North-Holland), page 313,
In the study of the competition graph of an acyclic digraph, there are two fundamental questions which were proposed by Roberts in 1978: One is to characterize the acyclic digraphs that have interval competition graphs and the other is to characterize the graphs which are competition graphs of acyclic digraphs.

2009, Tamara Mchedlidze, Antonios Symvonis, Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs, Hung Q. Ngo (editor), Computing and Combinatorics: 15th Annual International Conference, Proceedings, Springer, LNCS 5609, page 76,
Determining whether a graph has a hamiltonian path or circuit is NP-complete [3]. The problem remains NP-complete for cubic planar graphs [3], for maximal planar graphs [9], and for planar digraphs [3]. It can be trivially solved in polynomial time for planar acyclic digraphs.

2018, Gregory Gutin, 3. Acyclic Digraphs, Jørgen Bang-Jensen, Gregory Gutin, Classes of Directed Graphs, Springer, page 125,
Acyclic digraphs form a well-studied family of digraphs of great interest in graph theory, algorithms and applications.

directed acyclic graph
directed acyclic graph

(graph theory, computer science) A directed acyclic graph, a finite directed graph that contains no directed cycles.


Robertson-Seymour theorem

the Robertson-Seymour theorem (canonical)

Named after mathematicians Neil Robertson and Paul D. Seymour, who proved it in a series of papers from 1983 to 2004.

(graph theory) A theorem stating that the undirected graphs, partially ordered by the graph-minor relationship, form a well-quasi-ordering.


elementary algebra

After 1820 we find elementary algebra commencing to be a required subject for admission. Columbia required it in 1821, Yale in 1847, and Princeton in 1848.

1920, Inez Dale Powelson, The History of the Teaching of Elementary Algebra, University of California, Berkeley, page 28:

1939, Regina Grosswege, Learning Activities in Elementary Algebra, College Entrance Book Company, unnumbered page,
A perusal of the following outstanding features of this workbook will show how this book has been designed to enable the student to master elementary algebra in a thoroughly efficient and interesting manner.

2006, Richard N. Aufmann, Vernon C. Barker, Joanne S. Lockwood, Essentials of Elementary Algebra, Houghton Mifflin:

(mathematics, education) An informally defined area of algebra considered suitable to be taught to secondary school students.

beginning algebra

algebra deemed suitable for secondary students


provector

(mathematics) A contravariant operator formed by substituting signs of partial differentiation for the facients of a quantic.


Hurwitz algebra

Hurwitz algebra (plural Hurwitz algebras)

Named after German mathematician Adolf Hurwitz (1859–1919); see also Hurwitz problem, Hurwitz's theorem.

As we recall, a Hurwitz algebra is a composition algebra with unit element.

1995, Susumo Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, page 40:

Observe that the unital composition algebra associated with (C,#92;star) by the construction given in the proof of Proposition (33.27) is the Hurwitz algebra (C,#92;diamond) if we set a#61;1.

1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 465:

2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 15,
Unital composition algebras (or Hurwitz algebras) form a well-known class of algebras. Any Hurwitz algebra has a finite dimension equal to either 1, 2, 4, or 8. The two-dimensional Hurwitz algebras are the quadratic étale algebras over the ground field F, the four dimensional ones are the generalized quaternion algebras, and the eight dimensional Hurwitz algebras are called Cayley algebras, and are analog to the classical algebra of octonions (for a good survey of the latter, see [Bae02]).

(algebra) Any one of the unital composition algebras identified by Hurwitz's theorem (on composition algebras) as solutions to the Hurwitz problem.


trigonometrical station

trigonometrical station (plural trigonometrical stations)

triangulation pillar
triangulation pillar#English


imaginary geometry

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

calque

мни́мая геометрия

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

Claim 7. The Euclidean law of cosines holds for any infinitesimal triangle #92;triangleABC in imaginary geometry.

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

non-Euclidean geometries
non-Euclidean geometry

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

мни́мая геометрия

geometry with no parallel postulate — see also absolute geometry


superabundant number

en-noun

superabundant number (plural superabundant numbers)

1984, Richard K. Guy (editor), Reviews in Number Theory 1973-83, Volume 4, Part 1, As printed in Mathematical Reviews, American Mathematical Society, page 173,
The authors prove a theorem: If Q(X) is the number of superabundant numbers ≦X, then for c<5/48,Q(X)≧( log X)¹⁺ᶜ for sufficiently large X.

1995, József Sándor, Dragoslav S. Mitrinović, Borislav Crstici, Handbook of Number Theory I, Springer, page 111,
1) A number n is called superabundant if (σ(n))/n>(σ(m))/m for all m with 1Corollary. Q(x)>c log x log log x/( log log log x)².

It follows that there is a superabundant number in each real interval #91;x,2x#93; and that the number of superabundant numbers is thus infinite.

2017, Kevin Broughan, Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents, Cambridge University Press, page 151:

(number theory) A positive integer whose abundancy index is greater than that of any lesser positive integer.


pseudoinverted

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

(mathematics) Converted into a pseudoinverse (by pseudoinversion)


transdifferential

(mathematics, biology) Relating to transdifferentiation


multenion

(mathematics) A tuple consisting of multiple numeric parts, such as a quaternion.


toesa

From Spanish and Portuguese toesa, from French toise. Doublet of toise.

(historical) Synonym of estado, Spanish toise or fathom, a traditional unit of length, equivalent to about 1.67 m.


autologistic

(mathematics) Pertaining to a logistic model that is based on autocorrelation.


integral element

integral element (plural integral elements)

1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, American Mathematical Society Translations, Series 2, Volume 3, page 21,
A subring of 𝔅 containing the ring o of integral elements of the field k_0(π), distinct from 𝔅, and not contained in any other subring of 𝔅 distinct from 𝔅, is called a maximal ring of the algebra 𝔅. In a division algebra, the only maximal ring is the ring of integral elements.

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,
If 𝔖 is the ring of integral elements in a commutative ring 𝔗 (over a subring R) and if the element t of 𝔗 is integral over 𝔖, then t is also integral over R (that is, contained in 𝔖).

In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^([Richard Dedekind's]) theory.

2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:

(algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R.

elemento intero

element of a given ring that is a root of some monic polynomial with coefficients in a given subring


Caccioppoli set

Caccioppoli set (plural Caccioppoli sets)

The concept was introduced by the Italian mathematician Renato Caccioppoli in a paper in 1927.

(mathematics) A set whose boundary is measurable and has (at least locally) finite measure.


probable error

probable error (countable and uncountable, plural probable errors)

(statistics) The half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.


Kleisli category

Kleisli category (plural Kleisli categories)

Named after the Swiss mathematician Heinrich Kleisli (1930–2011).

(category theory) A category naturally associated to any monad T, and equivalent to the category of free T-algebras.


premine

en-verb

premine (third-person singular simple present premines, present participle premining, simple past and past participle premined)

premines (present singular third-person)
premining (participle present)
premined (participle past)
premined (past)

"Premining" is where the currency creator mines a certain amount of coins for themselves before they release the currency for other miners.

2014 June 13, Black Market Cryptocurrencies:

Ethereum and other projects that premine their coins are criticized because control of the cryptocurrency’s supply is potentially more centralized among “insiders” who participated in the presale and could manipulate the price or influence governance decisions. […] The sale documents said that once the Ethereum blockchain launched and the premined ether was issued, miners would generate new ether initially at an annual rate of 26% of the amount of ether issued in the crowdsale – the issuance rate isn’t fixed and is capped at 18 million ETH minted per year.

2020 November 7, Camila Russo, “Sale of the Century: The Inside Story of Ethereum's 2014 Premine”, in CoinDesk, archived from the original on 2024-06-17:

(transitive) To mine (units of a cryptocurrency) in advance of that currency being made available to the public.


logicomathematical

(mathematics) Relating to logic and mathematics (or to mathematical logic)


algebraic poset

en-noun

algebraic poset (plural algebraic posets)

1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
A poset (P,<) is said to be algebraic if and only if
i) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists,
ii) for every x∈P, the set
K_X:=y∈P|ycompact,y<x
is non-empty and up-directed, and
x= operatorname supK_x.
A poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<y for some compact element c of P.
Concerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets K_x fail to be up-directed ([50], 4.2 or [49], 4.5). Even when "enough" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets K_x.
The concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) "algebraic lattice" (cf. [9], [20], I-4).

ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions.
A structure (D,#92;le#95;D,#92;cup#95;D) where (D,#92;le#95;D) is an algebraic poset and #92;cup#95;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#92;cup#95;Dd#61;d for all d#92;inD) will be called a nondeterministic algebraic poset.

1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:

It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #123;A#42;#92;cupA#42;#125;#92;sqrt#123;#125; in the prefix order; this poset enjoys the added condition that every element is compact.

1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:

The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.

1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:

(algebra, order theory) A partially ordered set (poset) in which every element is the supremum of the compact elements below it.


Dirichlet series

Dirichlet series (countable and uncountable, plural Dirichlet series)

Named after German mathematician Peter Gustav Lejeune Dirichlet.

named-after

Traditionally, starting from Euler, multiplicativity of arithmetic sequences is customarily expressed in the form of an Euler product factorization of the generating Dirichlet series. It turns out that in the situation of modular forms, suitable Dirichlet series constructed by Fourier coefficients of eigenfunctions of Hecke operators can be expressed through Dirichlet series formed by the corresponding eigenvalues.

2009, Anatoli Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Springer (Birkhäuser), page 137:

We have now given heuristically a large family of multiple Dirichlet series, one for each simply laced Dynkin diagram.

2012, Daniel Bump, “Chapter 1: Introduction: Multiple Dirichlet Series”, in Daniel Bump, Solomon Friedberg, Dorian Goldfeld, editors, Multiple Dirichlet Series, L-functions and Automorphic Forms, Springer, page 6:

2014, Marius Overholt, A Course in Analytic Number Theory, American Mathematical Society, page 157,
The sum
A(s)=∑ₙ₌₁ ᪲a_nn⁻ˢ
of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane σ>σ_c(A), and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of σ>σ_c(A) by Proposition 3.3.

(number theory) Any infinite series of the form ∑ₙ₌₁ ᪲(a_n)/(nˢ), where s and each a_n are complex numbers.


quasi-satellite

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


semimeridian

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

surface of revolution
surface of revolution

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


dollar auction

Among the most terrifying of all dollar auctions followed World War II, when the United States and the Soviet Union entered into an arms race to produce more sophisticated and destructive weapons.

2012, Robert Dodge, Schelling's Game Theory: How to Make Decisions, page 117:

(game theory) A non-zero-sum sequential game designed to illustrate a paradox brought about by traditional rational-choice theory in which players are compelled to make an ultimately irrational decision based entirely on a sequence of apparently rational choices made throughout the game. It involves an auction for a dollar bill where the second-highest bidder loses the amount that they bid, making them the biggest loser in the auction.


beanplot

(mathematics) An alternative form of boxplot in which individual observations are shown as small lines in a one-dimensional scatter plot.


Chinese postman problem

en-noun

Chinese postman problem (plural Chinese postman problems)

Originally studied by the Chinese mathematician Kwan Mei-Ko in 1960, whose Chinese paper was translated into English in 1962. The name "Chinese postman problem" was coined in his honour, and is varyingly ascribed to Alan J. Goldman or Jack Edmonds, both of the US National Bureau of Standards.

(graph theory) The problem of finding the shortest closed path or circuit that visits every edge of a (connected) undirected graph.


polynomial ring

polynomial ring (plural polynomial rings)

It then follows that if A is a graded ring over a local ring, A is a homomorphic image of a polynomial ring over a regular local ring. For the sake of brevity, we refer to a graded polynomial ring over a regular local ring simply as a graded polynomial ring.

1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270:

In Section 3.2 we shall study the special properties of a polynomial ring over a field; for the moment we note a property of polynomial rings which applies quite generally, the Hilbert basis theorem (after David Hilbert, 1862-1943):
Theorem 3.3
If R is any right Noetherian ring, the polynomial ring R#91;x#93; is again right Noetherian.

2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106:

Because they possess a rich theory and provide an excellent source of examples and counterexamples, integer-valued polynomial rings have attained some prominence in the theory of non-Noetherian commutative rings.

2009, Jesse Elliott, “Some new approaches to integer-valued polynomial rings”, in Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson, editors, Commutative Algebra and Its Applications: Proceedings of the 5th International Fez Conference, Walter de Gruyter, page 223:

(algebra) A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K.


phase angle

(mathematics) Angular component of a polar coordinate representation.

(mathematics) Angular component of a vector.


multideterminantal

en-adj

multideterminantal (comparative more multideterminantal, superlative most multideterminantal)

more multideterminantal (comparative)
most multideterminantal (superlative)

(mathematics, chemistry) That has multiple determinants.


complex projective line

en-noun

complex projective line (plural complex projective lines)

By way of an example, we consider in detail the complex projective line #92;C#92;mathbb#123;P#125;¹.[…]Thus via the function w#95;0 the complex projective line #92;C#92;mathbb#123;P#125;¹ becomes identified with the "extended complex plane" (i.e. the ordinary complex plane with an additional "point at infinity").
2..2.1 Theorem The complex projective line #92;C#92;mathbb#123;P#125;¹ is diffeomorphic to the 2-dimensional sphere S².

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

Finally, C is interested in the geometry of the complex projective line P¹(#92;C).

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

Hence, C(#92;Gamma) is an uncountable union of complex projective lines.

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

As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere #92;dot#123;S#125;² ̇provides a conformal model for the complex projective line #92;C#92;mathbb#123;P#125;¹.

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

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


complex line

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

A complex line in #92;Cⁿ is a set of the form l#61;#92;#123;z#58;z#61;a#43;#92;lambdab,#92;lambda#92;in#92;C#92;#125;, where a and b are fixed points in #92;Cⁿ, with b#92;ne 0. Let us say that l is the complex line through a in the “direction” b.

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

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

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

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

complex plane

affine subspace that is 1-dimensional over the complex numbers


projective line

projective line (plural projective lines)

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

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

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

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

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

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


back substitution

back substitution (countable and uncountable, plural back substitutions)

(linear algebra) A method of solving linear systems that have been transformed into row echelon form.

(combinatorics) A method of solving recurrence relations.


Gaussian elimination

Gaussian elimination (countable and uncountable, plural Gaussian eliminations)

1976, Thomas E. Rosmond, Frank D. Faulkner, Direct Solution of Elliptic Equations by Block Cyclic Reduction and Factorization, Monthly Weather Review, Volume 104, Issues 1-6, American Meteorological Society, page 644,
We then repeat the Gaussian elimination procedure for the ( boldsymbol E-α₂⁽²⁾) factor in (35), obtaining another intermediate solution vector. It is immediately apparent in this example that four Gaussian eliminations are necessary to obtain ψ_(i,4).

Theoretical partitioning and resource allocation for the Gaussian elimination method used to solve the matrix are discussed.

1989, Richard A. Blech, “Parallel Gaussian Elimination of a Block Tridiagonal Matrix Using Multiple Microcomputers”, in Technical Report, NASA, page 1:

The use of Gaussian eliminations limits the range of functionality of the algorithm, which operates only with square matrices A.

1996, E. D'Hollander, G. R. Joubert, F. J. Peters, D. Trystram, Parallel Computing: State-of-the-art and Perspectives, Elsevier, page 642:

(linear algebra) A method of reducing an augmented matrix to row echelon form.


Gauss-Jordan elimination

Gauss-Jordan elimination (plural Gauss-Jordan eliminations)

(linear algebra) A method of reducing an augmented matrix to reduced row echelon form.


row echelon form

row echelon form (countable and uncountable, plural row echelon forms)

(linear algebra) The stepped appearance of a matrix that has undergone Gaussian elimination with the result that the leading coefficient or pivot (that is, the first nonzero number from the left) of a nonzero row is to the right of the pivot of the row above it.


reduced row echelon form

reduced row echelon form (plural reduced row echelon forms)

(linear algebra) The form of a row-reduced matrix which is in row echelon form.


augmented matrix

augmented matrix (plural augmented matrices)

(linear algebra) Matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.


coefficient matrix

coefficient matrix (plural coefficient matrices)

(linear algebra) A matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.


regulize

en-verb

regulize (third-person singular simple present regulizes, present participle regulizing, simple past and past participle regulized)

regulizes (present singular third-person)
regulizing (participle present)
regulized (participle past)
regulized (past)

From Latin regulare (“to direct, rule, regulate”), from regula (“rule”), from regere (“to keep straight, direct, govern, rule”) + -ize.

der

Let p be the threshold wherefrom output signal begins to appear and P the saturation threshold, beyond which the output signal values remain constant. We regulize this interval by considering it is equal to the unity.

1988, Academia de Studii Economice (Romania), Economic Computation and Economic Cybernetics Studies and Research:

The development of the boundary integral equations exploits (as usual) Somigliana's identity but a special manipulation is carried out to "regulize" certain integrals associated with the crack line.

1996, Applied Mechanics Reviews - Volume 49, Issues 7-9, page 40:

We regulize by dropout (p = 0.25) just before the fully connected layer.

2018, Danail Stoyanov, Understanding and Interpreting Machine Learning in Medical Image Computing Applications, →ISBN:

(statistics) To manipulate data so that it is on a comparable scale; normalize.


binwidth

(statistics) The numerical width of a bin


durgy

en-adj

durgy (comparative more durgy, superlative most durgy)

more durgy (comparative)
most durgy (superlative)

From Middle English *dwerghy, from Middle English dwergh (“dwarf”), equivalent to dwarf + -y.


moio

From Portuguese moio, from Latin modius (“Roman peck of about 9 L”), from modus (“measure”) + -ius (“-y: forming adjectives”). Doublet of moyo.

der

(historical) A traditional Portuguese dry measure, equal to about 0.78–1.15 m³ at different places and times.


dynamic programming

Coined by American mathematician Richard E. Bellman in the 1940s.

coinage

The advantage of dynamic programming as a procedure for solving optimization problems is the simplification obtained by decomposition.

1975, Sven Danø, Nonlinear and Dynamic Programming: An Introduction, Springer Science & Business Media, →ISBN, page 73:

Dynamic programming, developed by Richard Bellmann, is a powerful method for solving optimization problems. It has the attractive feature of breaking up a complex optimization problem into a number of simpler problems.

2000, Rein Luus, “Preface”, in Iterative Dynamic Programming, CRC Press, →ISBN:

optimal substructure
optimal substructure

(mathematics, computer science) An optimization method by which a problem having optimal substructure is recursively broken into simpler subproblems which are solved to produce the globally optimal solution.


sulung

Learned borrowing from Old English sulung, from sulh (“plough, ploughland”).

lbor

The counting of sulungs (as of hides) is a horrible task on which no two scholars agree, and it is not surprising that before the age of the computer Jolliffe made slips and that his desire to find eighty-sulung units sometimes overrode the evidence or the geographical probabilities.

2000, Nicholas Brooks, Anglo-Saxon Myths: State and Church, 400–1066, →ISBN, page 57:

(historical) A unit of land in medieval Kent, comparable to the hide and the carucate.


composition algebra

en-noun

composition algebra (plural composition algebras)

1993, F. L. Zak (translator and original author), Simeon Ivanov (editor), Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 11,
More precisely, Xⁿ⊂ℙᴺ is a Severi variety if and only if ℙᴺ=ℙ(𝔍), where 𝔍 is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra 𝔄, and X corresponds to the cone of Hermitian matrices of rank <1 (in that case SX corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, X is a Severi variety if and only if X is the “Veronese surface” over one of the composition algebras over the field K (Theorem 4.9).

We call a composition algebra with an associative norm a symmetric composition algebra and denote the full subcategory of #92;mathsf#123;Comp#125;#95;m consisting of symmetric composition algebras by #92;mathsf#123;Scomp#125;#95;m.

1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 464:

2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 150,
At least in the split cases, this is a construction that depends on two unital composition algebras, since the Jordan algebra involved consists of the 3 x 3-hermitian matrices over a unital composition algebra.

(algebra) A non-associative (not necessarily associative) algebra, A, over some field, together with a nondegenerate quadratic form, N, such that N(xy) = N(x)N(y) for all x, y ∈ A.