primitive element

primitive element (plural primitive elements)

primitiivinen alkio

element of a coalgebra satisfying a particular condition

An algebraic extension L#47;K is called simple if L#61;K(#92;alpha) for some #92;alpha#92;inL. The element #92;alpha is called a primitive element for L#47;K. Every finite separable algebraic field extension is simple.
Suppose that L#61;K(#92;alpha#95;1,#92;dots#92;alpha#95;r) is a finite separable extension and K#95;0#92;subseteqK is an infinite subset of K. Then there exists a primitive element #92;alpha of the form #92;textstyle#92;alpha#61;#92;sum#95;#123;i#61;1#125;ʳc#95;i#92;alpha#95;i with c#95;i#92;inK#95;0.

2009, Henning Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer, page 330:

(algebra, field theory) An element that generates a simple extension.

primitiivinen alkio

element that generates a field extension

Furthermore, if the irreducible polynomial has a primitive element α (where α=1) that is a root, then the polynomial is termed a primitive polynomial and corresponds to the polynomial for a maximal length feedback shift register.

1996, J. J. Spilker, Jr. Chapter 3: GPS Signal Structure and Theoretical Performance, Bradford W. Parkinson, James J. Spilker (editors), Global Positioning System: Theory and Applications, Volume 1, AIAA, page 114, Likewise, α=α, etc., namely, the elements are all expressed as powers of α and because α=1, α is termed a primitive element of GF(2ⁿ). […]

2003, Soonhak Kwon, Chang Hoon Kim, Chun Pyu Hong, Efficient Exponentiation for a Class of Finite Fields GF(2ⁿ) Determined by Gauss Periods, Colin D. Walter, Çetin K. Koç, Christof Paar (editors), Cryptographic Hardware and Embedded Systems, CHES 2003: 5th International Workshop, Proceedings, Springer, LNCS 2779, page 228,
Also in the case of a Gauss period of type (n,1), i.e. a type I optimal normal element, we find a primitive element in GF(2ⁿ) which is a sparse polynomial of a type I optimal normal element and we propose a fast exponentiation algorithm which is applicable for both software and hardware purposes.

Here, necessarily, c must be a primitive element of 𝔽_𝕢, since this is the norm of a root of the polynomial.

2008, Stephen D. Cohen, Mateja Preśern, The Hansen-Mullen Primitivity Comjecture: Completion of Proof, James McKee, James Fraser McKee, Chris Smyth (editors, Number Theory and Polynomials, Cambridge University Press, page 89, For q a power of a prime p, let 𝔽_𝕢 be the finite field of order q. Its multiplicative group 𝔽^*_q is cyclic of order q-1 and a generator of 𝔽^*_q is called a primitive element of F_q. More generally, a primitive element γ of F_qⁿ, the unique extension of degree n of 𝔽_𝕢, is the root of a (necessarily monic and automatically irreducible) primitive polynomial f(x)∈ 𝔽_𝕢[x] of degree n. […]

(algebra, field theory, of a finite field) An element that generates the multiplicative group of a given Galois field (finite field).

primitiivinen alkio

element of a finite field that generates its multiplicative group

primitive root of unity

element that generates the multiplicative group of a finite field

1972, W. Wesley Peterson, E. J. Weldon, Jr., Error-correcting Codes, The MIT Press, 2nd Edition, page 457,
Let A be a prime number for which 2 is a primitive element. Then 2ᴬ⁻¹-1 is divisible by A.

multiplicative field
multiplicative field

(number theory) Given a modulus n, a number g such that every number coprime to n is congruent (modulo n) to some power of g; equivalently, a generator of the multiplicative field of integers modulo n.

But suppose L'#92;inC#95;#92;nu(S#95;0) so that #92;operatorname#123;det#125;(L')#61;#92;eta'#92;pi#92;blacktriangleright 0 for some totally positive unit #92;eta' and so that L' is everywhere locally a primitive''' element of the #92;mathfrako-lattice R#95;#92;nu.

1985, Revista Matemática Iberoamericana, Volume 1, Real Sociedad Matemática Española, page 111:

(algebra, lattice theory, of a lattice) An element that is not a positive integer multiple of another element of the lattice.

primitiivinen alkio

element of a lattice that is not a positive multiple of another element

2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29,
A primitive element of a Hopf algebra is an element h∈H such that
Δh=1⊗h+h⊗1.
It is easily seen that the bracket [x,y]:=xy-yx of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra. For H=U(g) any element of g is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of U(g) coincides with the Lie algebra g.

multiplicative identity
multiplicative identity

(algebra, of a coalgebra over an element g) An element x ∈ C such that μ(x) = x ⊗ g + g ⊗ x, where μ is the comultiplication and g is an element that maps to the multiplicative identity 1 of the base field under the counit (in particular, if C is a bialgebra, g = 1).

In this paper we apply regression models and other pattern recognition techniques to the task of classifying primitive elements of a free group.

2004, Dmitry Y. Bormotov, “Experimenting with Primitive Elements in F₂”, in Alexandre Borovik, Alexei G. Myasnikov, editors, Computational and Experimental Group Theory: AMS-ASL Joint Special Session, American Mathematical Society, page 215:

(group theory, of a free group) An element of a free generating set of a given free group.


branching process

en-noun

branching process (plural branching processes)

(mathematics, probability theory) A stochastic process which typically models reproduction of objects, with the assumption that the reproductions of individual particles are mutually independent.


primitive polynomial

primitive polynomial (plural primitive polynomials)

We claim that every primitive polynomial can be written as a product of irreducible elements in #92;mathbfD#91;x#93;.[…]By induction on the degree of the primitive polynomials, we conclude that both g(x),h(x) can be written as product of irreducible elements in #92;mathbfD#91;x#93;.

1992, T. T. Moh, Algebra, World Scientific, page 124:

If f(x)#92;in#92;Z#91;x#93;, the ring of polynomials with coefficients in #92;Z, then the content of f, denoted by c(f), is the greatest common divisor of the coefficients of f. The polynomial f(x) is called a primitive polynomial if c(f)#61;1. Since c(fg)#61;c(f)c(g), by Gauss's lemma [Hungerford, 74], the set S of primitive polynomials in #92;Z#91;x#93; is a multiplicatively closed set. Define #92;Lambda#61;#92;Z#91;x#93;#95;S, the localization of #92;Z#91;x#93; at S, a subring of the field of quotients #92;Q(x) of #92;Z#91;x#93;. Elements of #92;Lambda are of the form f(x)#47;g(x) with f(x),g(x)#92;in#92;Z#91;x#93; and g(x) a primitive polynomial.

2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114:

According to the Gauss lemma, the product of primitive polynomials is primitive. Therefore if f(x),g(x) are primitive and f(x)#61;g(x)h(x) with h(x) in F#91;x#93;, then necessarily h(x) is in A#91;x#93; and primitive. […]The irreducible elements of A#91;x#93; are irreducible elements of A and primitive polynomials which are irreducible in F#91;x#93;.

2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1:

greatest common divisor
greatest common divisor

(algebra, ring theory) A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.

monic polynomial

polynomial over an integral domain such that no noninvertible element divides all of its coefficients

Primitive polynomials make the initialization of LFSRs a simpler task since any nonzero state guarantees that all non-zero states will be visited in the maximum length sequence.

2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69:

Definition 7.2 Let f(x) be a monic polynomial of degree n over #92;mathbb#123;F#125;#95;q. If f(x) has a primitive element of #92;mathbb#123;F#125;#95;#123;qⁿ#125; as one of its roots, f(x) is called a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q.
Theorem 7.7 For any positive integer n there always exist primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q. All the n roots of a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q are primitive elements of #92;mathbb#123;F#125;#95;#123;qⁿ#125;. All primitive polynomials of degree n over #92;mathbbF#95;q are irreducible over #92;mathbb#123;F#125;#95;q. The number of primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q is equal to #92;phi(qⁿ-1)#47;n.

2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145:

This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when n = 5, 6, 7 or 8. The HMPC (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed.

2008, Stephen D. Cohen, Mateja Preŝern, “The Hansen-Mullen Primitivity Conjecture: Completion of Proof”, in James McKee, Chris Smyth, editors, Number Theory and Polynomials, Cambridge University Press, page 89:

(algebra, field theory) A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.


trilobular

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

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


Hopfield network

Hopfield network (plural Hopfield networks)

Popularised by John Hopfield in 1982 (but described earlier by Little in 1974).

recurrent neural network
recurrent neural network

A kind of recurrent neural network, providing a model for understanding human memory.


polynomial basis

polynomial basis (plural polynomial bases)

2007, Nicos Karcanias, Efstathios Milonidis, 2: Structural Methods for Linear Systems: An Introduction, Matthew C. Turner, Declan G. Bates (editors), Mathematical Methods for Robust and Nonlinear Control: EPSRC Summer School, Springer, Lecture Notes in Control and Information Sciences 367, page 89,
If T(s)=M(s)D(s)⁻¹ is a RCMFD of T(s), then M(s) is a polynomial basis for 𝔛_t. If Q(s) is a greatest right divisor of M(s) then T(s)=◌̅M(s)Q(s)D(s)⁻¹, where ◌̅M(s) is a least degree polynomial basis of 𝔛_t [15].

When dealing with various polynomial bases up to a certain degree and with corresponding moments, any moment (with respect to any basis) can be expressed as a function of moments of the same or fewer orders with respect to an arbitrary basis.[…]As we already saw in Chapter 1, OG^([orthogonal]) moments are, unlike geometric and all other moments, coordinates of f in the polynomial basis in the common sense used in linear algebra.

2009, Jan Flusser, Barbara Zitova, Tomas Suk, Moments and Moment Invariants in Pattern Recognition, Wiley, page 166:

For this purpose, a polynomial basis is used here, which consists of all monomials in the scheduling signals up to a fixed total order. Such a polynomial basis can be interpreted as the polynomial terms, which would appear in a multivariate Taylor approximation of the unknown scheduling function.

2012, A. Kominek et al., “Ident. of Low-Compl. LPV-IO Models for Engine Control”, in Javad Mohammadpour, Carsten W. Scherer, editors, Control of Linear Parameter Varying Systems with Applications, Springer, page 452:

(algebra, ring theory) A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients).

When using polynomial bases, the first stage in computing the product of two elements of #92;mathbbF#95;#123;2ⁿ#125; is the multiplication of two polynomials of degree at most n-1 in #92;mathbbF#95;2#91;x#93;.

1999, I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, page 20:

2010, Vladimir Tujillo-Olaya, Jaime Velasco-Medina, Hardware Architectures for Elliptic Curve Cryptoprocessors Using Polynomial and Gaussian Normal Basis Over GF(2²³³), Marina L. Gravrilova, C. J. Kenneth Tan, Edward David Moreno (editors), Transactions on Computational Science XI: Special Issue on Security in Computing, Part 2, Springer, LNCS 6480, page 79,
In this case, the GF(2ᵐ) multiplication is implemented in hardware using three algorithms for polynomial basis (PB) and three for gaussian normal basis (GNB).

In contrast to the case of normal bases considered in Theorem 5.1.9, the dual basis of a polynomial basis is usually not a polynomial basis.
[…]
5.1.13 Theorem [1265, 1298] Let #92;theta be a root of a monic irreducible polynomial f of degree m over K#61;#92;mathbbF#95;q, and let B#61;#92;#123;1,#92;theta,#92;theta²,#92;dots#92;theta#123;m-1#125;#92;#125; be the corresponding polynomial basis of F#61;#92;mathbbF#95;#123;qᵐ#125; over K. Then the dual basis B#42; of B is a polynomial basis if and only if f is a binomial and m#92;equiv 1#92;#33;#92;#33;#92;#33;#92;#33;#92;pmodp, where q is a power of the prime p.
5.1.14 Corollary There exists a dual pair of polynomial bases of #92;mathbbF#95;#123;qᵐ#125; over #92;mathbbF#95;q if and only if the following three conditions are satisfied:[…].

2013, Gary L. Mullen, Daniel Panario, Handbook of Finite Fields, Taylor & Francis (Chapman & Hall / CRC Press), page 103:

primitive polynomial
primitive polynomial

(algebra, field theory, cryptography, of a finite field) Specifically, a basis, of the form {1, α, ..., αⁿ⁻¹}, of a finite extension F_(qⁿ) of a Galois field F_q, where α is a primitive element of F_(qⁿ) (i.e., a root of a degree-n primitive polynomial over F_q).


quadrisector

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

(geometry) A line or curve that quadrisects something.


perfect field

1984, Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press, Addison-Wesley, page 10,
If K is a perfect field of prime characteristic p, and if n is a nonnegative integer, then the mapping α→α from K to K is an automorphism.

So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.

2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116:

a) K is a perfect field;
b) any irreducible polynomial of K[X] is separable;
c) any element of an algebraic closure of K is separable over K;
d) any algebraic extension of K is separable;
e) for any finite extension K→L, the number of K-homomrphisms from K to an algebraically closed extension of K is equal to [L:K].
Corollary 3.1.8. Any algebraic extension of a perfect field is again a perfect field.

2005, Antoine Chambert-Loir, A Field Guide to Algebra, Springer, page 57, Definition 3.1.7. One says a field K is perfect if any irreducible polynomial in K[X] has as many distinct roots in an algebraic closure as its degree. By the very definition of a perfect field, Theorem 3.1.6 implies that the following properties are equivalent

(algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots.

corps parfait

field such that every irreducible polynomial over it has distinct roots


tridecimal

(arithmetic, music) Of a number, expressed in base thirteen.


Frobenius

Frobenius endomorphism
Frobenius endomorphism

(mathematics) Used attributively to attribute a concept to the work of Ferdinand Georg Frobenius or more directly to associate it with the Frobenius endomorphism (commutative algebra, field theory) or Frobenius morphism (category theory).


Frobenius endomorphism

Frobenius endomorphism (plural Frobenius endomorphisms)

Named after German mathematician Ferdinand Georg Frobenius.

Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.

2003, Claudia Miller, “The Frobenius endomorphism and homological dimensions”, in Luchezar L. Avramov, Marc Chardin, Marcel Morales, Claudia Polini, editors, Commutative Algebra: Interactions with Algebraic Geometry: International Conference, American Mathematical Society, page 208:

2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11,
Let k=◌̅ 𝔽_𝕡, and let q be a power of p such that the group G is defined over 𝔽_𝕢. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra 𝒢 and the adjoint action of G on 𝒢 are also defined over 𝔽_𝕢 and we still denote by F:𝒢→𝒢 the Frobenius endomorphism on 𝒢.
[…] Assume that H,X and the action of H over X are all defined over 𝔽_𝕢. Let F:X→X and F:H→H be the corresponding Frobenius endomorphisms.

2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356,
The first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (MEVA 1900) using the curve
E:y²+y=x³.
In this case, the characteristic polynomial of the Frobenius endomorphism denoted by ϕ₂ (cf. Example 4.87 and Section 13.1.8), which sends P_∞ to itself and (x_1,y_1) to (x,y), is
χ_E(T)=T²+2.
Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P∈E( 𝔽_2ᵈ), we have ϕ=-[2]P.

(algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.

commutative algebra; field theory; commutative algebra; field theory

Ferdinand Georg Frobenius
Frobenius endomorphism

endomorphisme de Frobenius

particular endomorphism on a commutative ring with prime characteristic


PAPR

(signal processing) Initialism of peak-to-average power ratio.


ones and zeroes

/ˈwʌnz n̩ ˈzɪəɹəʊz/

From the fact that the numerals 1 (one) and 0 (zero) are used to indicate binary numbers.

The incoming signals were broken down into digital code—ones and zeroes—by a relatively simple computer and downlinked to Fort Huachuca, where another computer of vastly greater power examined the bits of raw information and tried to make sense of them.

1989 August 17, Tom Clancy, Clear and Present Danger, New York, N.Y.: G[eorge] P[almer] Putnam’s Sons, →ISBN; Berkley premium tie-in edition, New York, N.Y.: Berkley Books, November 2018, →ISBN, page 360:

The so-called higher level programming languages, like the various dialects of BASIC and PASCAL familiar to students in the 1980s, greatly simplified the business of programming, since it is no longer necessary—at least for certain purposes—to do what is now called "machine language programming," or what in earlier days was programming, i.e., employing the ones and zeroes now frequently viewed as rather tedious and theoretical [...]

1990, G. E. Scott, “Malleable Persons”, in Moral Personhood: An Essay in the Philosophy of Moral Psychology (SUNY Series in Ethical Theory), Albany, N.Y.: State University of New York Press, →ISBN, page 40:

Hiro Protagonist [in Snow Crash (1992) by Neal Stephenson] must win, one might say, on both Surf and Turf against the wonderfully evil Evangelist Minister R. Bob Riff and his allies, both in the world of violence, swordplay, and aircraft carriers and in the world of ones and zeros.

1996, R[obert] James Woolsey [Jr.], “Resilience and Vulnerability in the Digital Age”, in Stuart J. D. Schwartzstein, editor, The Information Revolution and National Security: Dimensions and Directions (Significant Issues Series; vol. XVIII, no. 3), Washington, D.C.: Center for Strategic and International Studies, →ISBN, →ISSN, page 79:

Now one must also: [...] be able to discern or possess a device that is able to discern the ones and zeroes on the storage medium, [...]

1998, Doug Carlston, “Storing Knowledge”, in Margaret MacLean, Ben H. Davis, editors, Time & Bits: Managing Digital Continuity, Los Angeles, Calif.: Getty Conservation Institute, →ISBN, page 24:

But suppose there is a special and universal and innate language for thinking. Suppose, in particular, that just as computers have a language of ones and zeroes that underlies all of their programs and applications, there is an equally fundamental language shared by all humans: a language of thought (lot) [...].

2001, Andrew Brook, Robert J. Stainton, “Knowledge of Language”, in Knowledge and Mind: A Philosophical Introduction (A Bradford Book), 1st paperback edition, Cambridge, Mass.: MIT Press, →ISBN, pages 54–55:

You once said that the "ones and zeros" in the world's computers were demons that could trick the human race into losing all its history.

2007, Henning Mankell, chapter 8, in Laurie Thompson, transl., Kennedy’s Brain: Translated from the Swedish, New York, N.Y.: The New Press, →ISBN, part 1 (Christ’s Cul-de-sac), page 83:

If you are a developer or an architect, you are one for a good reason: Your magic and craft is turning ones and zeros into a solution that makes life and work better.

2008, Steven C. Seow, “Preface”, in Karen Gettman, editor, Designing and Engineering Time: The Psychology of Time Perception in Software, Upper Saddle River, N.J., Boston, Mass.: Addison-Wesley, →ISBN, page xvii:

Neo has been hacking since he was in elementary school. He is more at ease with ones and zeros than his own language.

2012, Wayne M. Smith, “The Wraith of Frank—Cyber War Style”, in The Sand Castle, [Bloomington, Ind.]: Xlibris, →ISBN, pages 99–100:

If the movie [Ralph Breaks the Internet] never falls into a feedback loop, credit should probably go to [John C.] Reilly and especially [Sarah] Silverman, who bring human vulnerability to little bits of ultra-branded ones and zeroes.

2018 November 14, Jesse Hassenger, “Disney Goes Viral with an Ambitious, Overstuffed Wreck-It Ralph Sequel”, in The A.V. Club, archived from the original on 2019-11-21:

(computing, informal) Binary code; on and off bits.


normal basis

A normal basis is generated by the repeated action of the Frobenius endomorphism on a suitable element β; it is the orbit of β for that endomorphism.

It is a characterising property of normal bases that #92;beta#123;qᵐ#125;#61;#92;beta.

1989, Willi Geiselmann, Dieter Gollmann, Symmetry and Duality in Normal Basis Multiplication, T. Mora (editor), Applied Algebra, Algebraic Algorithms, and Error-correcting Codes: 6th International Conference, Proceedings, Springer, LNCS 357, page 230,
We also combine dual basis and normal basis techniques. The duality of normal bases is shown to be equivalent to the symmetry of the logic array of the serial input / parallel output architectures proposed in this paper.

12.3 In the finite field E#61;#92;mathbbF#95;#123;3³#125;, find: (a) a primitive root of E whose conjugates do not form a normal basis of E#47;#92;mathbbF#95;3; (b) a normal basis that does not consist of primitive roots of E.
For an arbitrary field E with prime field #92;mathbbF#95;p, the extension E#47;#92;mathbbF#95;p does always have at least one normal basis consisting of primitive roots.

2006, Falko Lorenz, translated by Silvio Levy, Algebra: Volume I: Fields and Galois Theory, Springer, page 260:

2015, Sergey Abrahamyan, Melsik Kyureghyan, New recursive construction of normal polynomials over finite fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott (editors, Topics in Finite Fields, American Mathematical Society, page 1,
The set of conjugates of normal element is called normal basis. A monic irreducible polynomial F∈ 𝔽_𝕢[x] is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over 𝔽_𝕢. The minimal polynomial of an element in a normal basis α,α^q,…,α is m(x)=∏ᵢ₌₀ⁿ⁻¹(x-α)∈ 𝔽_𝕢[x] which is irreducible over 𝔽_𝕢. The elements of a normal basis are exactly the roots of some N-polynomial. Hence an N-polynomial is just another way of describing a normal basis.

(algebra, field theory) For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}.

base normale

particular type of basis of a finite field


emergent counting

the mathematical concept of matching the sequence of counting words with objects where each object is allocated a number in turn, with the final number word corresponding to the total.


wheat and chessboard problem

(mathematics) A mathematical problem where one must determine how many grains of wheat are on a chessboard, assuming one grain of wheat is placed on the first square, and each subsequent square having twice as many grains of wheat as the previous square.


polyexponential

(mathematics) Involving many exponentials


polyexponential

(mathematics) A function of multiple exponential terms.


pseudovalue

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

Each observer's data generates a pseudovalue of perceptual accuracy and the set of pseudovalues are treated as if they are independent and identically distributed.

2013, William Thomson, Peter N. T. Wells, The Perception of Visual Information, page 264:

In traditional databases, missing information is mostly handled by means of a pseudovalue, called a null value […]

2015, Janusz Kacprzyk, Witold Pedrycz, Springer Handbook of Computational Intelligence, page 300:

(mathematics, computing) Something that resembles a value but is not a true value or does not really belong to the set of data being studied.


Galois extension

Galois extension (plural Galois extensions)

Named for its connection with Galois theory and after French mathematician Évariste Galois.

The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.

The fundamental theorem of Galois theory states that there is a one-to-one correspondence between the subfields of a Galois extension and the subgroups of its Galois group.

Corollary If L#58;K is a Galois extension, there exists an irreducible polynomial f in K#91;x#93; such that L#58;K is a splitting field extension for f over K.

1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108:

First, arithmetic obstructions against constructing central extensions of a fixed finite base Galois extension are analyzed with the local-global principle to give some quantitative estimates of them.

1989, Katsuya Miyake, “On central extensions”, in Jean-Marie De Koninck, Claude Levesque, editors, Number Theory, Walter de Gruyter, page 642:

With the help of the results in Section 7.5 it is not hard to describe all Galois extensions.
Proposition 7.6.1. Let E#47;F be a finite field extension. Then (i) E#47;F is a Galois extension if and only if it is normal and separable; (ii) E#47;F is contained in a Galois extension if and only if it is separable.

2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211:

(algebra, Galois theory) An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F.

extension de Galois

algebraic extension that is normal and separable


separable polynomial

separable polynomial (plural separable polynomials)

Over a perfect field, the separable polynomials are precisely the square-free polynomials.

The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.

We know that F(#92;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).

1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:

Proposition 1.4.2 A finite field extension M#47;K is Galois if and only if M is the splitting field over K of a separable polynomial.

2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:

If #92;pi#95;P is a separable polynomial in K#91;t#93;, then the derivative #92;partial#95;t#92;pi#95;P is prime to #92;pi#95;P in K#91;t#93;, and therefore a unit in K#91;t#93;#95;P.[…]In the case when #92;pi#95;P is an inseparable polynomial we may write #92;pi#95;P#61;f(t#123;pʳ#125;) for a suitable rgt;0 and separable polynomial f.

2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:

(algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial).


presmooth

en-verb

presmooth (third-person singular simple present presmooths, present participle presmoothing, simple past and past participle presmoothed)

presmooths (present singular third-person)
presmoothing (participle present)
presmoothed (participle past)
presmoothed (past)

to presmooth a distribution

(transitive, statistics) To smooth in advance.


Borel transform

en-noun

Borel transform (plural Borel transforms)

(mathematics) A transform used in Borel summation.

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


Nachbin's theorem

(mathematics) A theorem used in complex analysis to establish a bound on the growth rates for an analytic function.


Borel summation

Borel summation (usually uncountable, plural Borel summations)

(mathematics) A summation method for divergent series, particularly useful for summing divergent asymptotic series, and in some sense giving the best possible sum for such series.


notch filter

(signal processing) A bandstop filter with a narrow stopband.


metagraph

(mathematics) A graphical representation of a set of objects and the morphisms relating them


splitting field

splitting field (plural splitting fields)

Theorem 3.2. If K is a field and f#92;inK#91;x#93; has degree n#92;ge 1, then there exists a splitting field F of f with #91;F#58;K#93;#92;len#33;

1974, Thomas W. Hungerford, Algebra, Springer, page 258:

In the case of the polynomial t²#43;1#92;in#92;R#91;t#93;, the situation is quite clear; its splitting field is #92;C since t²#43;1#61;(t#43;i)(t-i) where i#61;#92;sqrt#123;-1#125;.

2003, Derek J. S. Robinson, An Introduction to Abstract Algebra, Walter de Gruyter, page 130:

Example 4.3. By definition, #92;Q(i) is the splitting field of x²#43;1 over #92;Q, and #92;C is the splitting field for the same polynomial, over #92;R.
Example 4.4. The splitting field F of x⁸-1 over #92;Q is generated by #92;zeta#58;#61;2#123;2#92;pii#47;8#125;; indeed, the roots of x⁸-1 are all the 8-th roots of 1, and all of them are powers of #92;zeta:[…]In fact, #92;zeta is a root of the polynomial x⁴#43;1, which is irreducible over #92;Q; therefore F#61;#92;Q(#92;zeta) is 'already' the splitting field of x⁴#43;1.

2009, Paolo Aluffi, Algebra: Chapter 0, American Mathematical Society, page 430:

(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.

deskonposizio gorputz

(Galois theory, of a polynomial) smallest field containing all roots

The terminology "splitting field of a K-algebra" is motivated by the same terminology regarding a polynomial. A splitting field of a K-algebra A is a field extension K#92;mapstoL such that A#92;otimes#95;KL is split; in the special case A#61;K#91;x#93;#47;f(x) this is the same as a splitting field of the polynomial f(x).

2001, T. Y. Lam, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117,
Ex. 7.6. For a finite-dimensional k-algebra R, let T(R)= operatorname radR+[R,R], where [R,R] denotes the subgroup of R generated by ab-ba for all a,b∈R. Assume that k has characteristic p>0. Show that
T(R)⊆a∈R:aforsomem>1,
with equality if k is a splitting field for R.

Group algebras are defined over the prime field #92;Q or #92;mathbbF#95;p (depending on the characteristic), and by what we have just proved #92;QG and #92;mathbbF#95;pG have splitting fields that are finite degree extensions of the prime field.[…]
Some other basic facts about splitting fields are left to the exercises at the end of this chapter. Thus, if A is a finite-dimensional algebra over a field F that is a splitting field for A and E#92;supsetF is a field extension, it is the case that every simple E#92;otimes#95;FA-module can be written in F (Exercises 4 and 8).

2016, Peter Webb, A Course in Finite Group Representation Theory, Cambridge University Press, page 162:

(algebra, ring theory, of a K-algebra) Given a finite-dimensional K-algebra (algebra over a field), an extension field whose every simple (indecomposable) module is absolutely simple (remains simple after the scalar field has been extended to said extension field).

Every finite dimensional central simple algebra has a splitting field: moreover, if said CSA is a division algebra, then a maximal subfield of it is a splitting field.

1955, Shimshon A. Amitsur, Generic Splitting Fields of Central Simple Algebras, Annals of Mathematics, Volume 62, Number 1, Reprinted in 2001, Avinoam Mann, Amitai Regev, Louis Rowen, David J. Saltman, Lance W. Small (editors), Selected Papers of S. A. Amitsur with Commentary, Part 2, American Mathematical Society, page 199,
The main tool in studying the structure of division algebras, or more generally, of central simple algebras (c.s.as) over a field ℭ are the extensions of ℭ that split the algebras. A field 𝔉⊇ℭ is said to split a c.s.a. 𝔄 if 𝔄⊗𝔉 is a total matrix ring over 𝔉. The present study is devoted to the study of the set of all splitting fields of a given c.s.a. 𝔄.

central simple algebra
central simple algebra

(algebra, ring theory, of a central simple algebra) Given a central simple algebra A over a field K, another field, E, such that the tensor product A⊗E is isomorphic to a matrix ring over E.

1999, P. Shumyatsky, V. Zobina (translators), David Louvish (editor of translation), Ya. G. Berkovich, E. M. Zhmud’, Characters of Finite Groups, Volume 2, American Mathematical Society, page 165,
DEFINITION 2. A field K is called a splitting field of a character χ of a group G if χ∈ operatorname Char_K(G), i.e., χ is afforded by a K-representation of G.
Let T be a representation of G affording the character χ. It follows from Definition 2 that K is a splitting field of χ if and only if T is equivalent to Δ, where Δ is a K-representation of G. In other words, K is a splitting field of a character χ if and only if a representation T affording χ is realized over K. Every character of G has a splitting field (for example, C is a splitting field of any character of G). If K is a splitting field of both characters χ₁,χ₂, then K is a splitting field of χ₁+χ₂, Therefore, in studying splitting fields, we may consider irreducible characters only.
DEFINITION 3. A field K is called a splitting field of a group G if it is a splitting field for every χ∈ operatorname Irr(G).

(algebra, character theory) (of a character χ of a representation of a group G) A field K over which a K-representation of G exists which includes the character χ; (of a group G) a field over which a K-representation of G exists which includes every irreducible character in G.


central simple algebra

central simple algebra (plural central simple algebras)

The complex numbers #92;C form a central simple algebra over themselves, but not over the real numbers #92;R (the centre of #92;C is all of #92;C, not just #92;R). The quaternions #92;mathbbH form a 4-dimensional central simple algebra over #92;R.

The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).

This crossed product E#92;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.

1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 151:

Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra D.

2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:

Let A#95;1,A#95;2 be central simple algebras over a field K. Then A#95;1#92;otimes#95;KA#95;2 can be shown to be a central simple algebra over K. Further, if A is a central simple algebra over a field K, then A#92;otimes#95;KA#92;operatorname#123;op#125;#92;cong#92;operatorname#123;Aut#125;#95;#123;K#92;operatorname#123;-Vect

2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:

associative algebra
associative algebra#English

(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.

centrale enkelvoudige algebra

type of associative algebra over a field


pseudoisotropic

(mathematics) That is isotropic in the three physical dimensions but not in some higher dimension


valuation ring

(algebra) An integral domain D such that for every element x of its field of fractions F, at least one of x or x⁻¹ belongs to D.


Monsky's theorem

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


equidissectable

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


equidissection

equidissection (countable and uncountable, plural equidissections)

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


equiquantal

(statistics) Relating to equal quantities


panmagic

(mathematics) Being a pandiagonal magic square.


pandiagonal

(mathematics) Of a magic square: having the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.


Eisenstein prime

Eisenstein prime (plural Eisenstein primes)

(mathematics) Any Eisenstein integer that is prime.


harmonic progression

harmonic progression (plural harmonic progressions)

arithmetic progression
arithmetic progression

(mathematics) A sequence of numbers whose reciprocals form an arithmetic progression.


coherentization

coherentization (usually uncountable, plural coherentizations)

(mathematics) conversion to a coherent form


automorphism group

automorphism group (plural automorphism groups)

The automorphism group of a set is the set's symmetric group.

The automorphism group of a finite-dimensional vector space (respectively, projective space) is the space's general linear group (respectively, projective linear group).

Moreover, we think that this approach to the automorphism groups of Banach space domains may also serve as motivating and illustrative material in introducing students to the theory of Lie groups and complex manifolds.

1984, José M. Isidro, László L. Stachó, Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, Elsevier (North-Holland), page vi:

Recently Droste, Holland and Macpherson in [DHM1], [DHM2] and [DHM3] investigated various questions concerning the automorphism groups of homogeneous #92;aleph#95;0-categorical trees.

1993, Matatyahu Rubin, The Reconstruction of Trees from Their Automorphism Groups, American Mathematical Society, page 5:

Unlike the classical integration theory, the non-commutative theory comes with the intrinsic dynamics of the modular automorphism group. This chapter is devoted to the elementary theory of the modular automorphism groups.

2002, Masamichi Takesaki, Theory of Operator Algebras II, Springer, page 91:

function composition
function composition

(group theory) The group of automorphisms of a given set or other mathematical object, with the group operation being function composition.


hyperdistribution

hyperdistribution (countable and uncountable, plural hyperdistributions)

The simplest approach to estimating hyperdistribution is to plot the frequency distribution of parameter estimates available for all stocks of interest.

1998, National Research Council, Division on Earth and Life Studies, Ocean Studies Board, Improving Fish Stock Assessments, →ISBN, page 34:

Prior distributions for the hyperdistribution parameters are considered to be relatively uninformative distributions.

2008, David L. Thomson, Evan G. Cooch, Michael J. Conroy, Modeling Demographic Processes in Marked Populations, →ISBN, page 1126:

In superstatistics, the hyperdistribution f(η) is determined by the physical problem that one wants to model.

2011, Jan Naudts, Generalised Thermostatistics, →ISBN, page 74:

(statistics, countable) A parameterized set of probability distributions.


pentadiagonal

(mathematics, of a matrix) In which the only nonzero entries are on the main diagonal, and the first two diagonals above and below it


blocktime

(Philippines) The units of air time sold by a broadcaster sold for use by another entity, often an advertiser or politician.


stochastic calculus

(mathematics) A branch of mathematics that operates on stochastic processes.


Parrondo's paradox

(game theory) The paradox whereby a combination of losing strategies (such as alternating between two games that each, played individually, would result in an overall loss) may give a winning strategy.


ship-pound

(historical) A unit of weight once used in the Baltic ports.


TOI

TOI (countable and uncountable, plural TOIs)

(ice hockey statistics) Icetime; Abbreviation of time on ice.


monogeneity

(mathematics) The condition of being monogenous


spiric

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


log-log

(mathematics) Describing a relationship (or a graph) between logarithmic quantities (or scales)


deregression

deregression (countable and uncountable, plural deregressions)

(mathematics) The removal of association between variables


root of unity

In the case of the field of complex numbers, it follows from de Moivre's formula that the n nth roots of unity are #92;textstyle#92;cos#92;left(k#92;frac#123;2#92;pi#125;#123;n#125;#92;right)#43;i#92;sin#92;left(k#92;frac#123;2#92;pi#125;#123;n#125;#92;right), where k#61;1,#92;dots,n.

We now show that the primitive n-th roots of unity generate the other n-th roots of unity.

2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89:

A nice application of Hensel's Lemma is to determine which roots of unity can be found in #92;Q#95;p.

2003, Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, page 72:

We have seen that, for a fixed value of n, the multiplicative group (U#95;n,#92;dot) is generated by any primitive nth root of unity. In particular, if #92;omega is a primitive 6th root of unity, then #92;omega⁶#61;1, six is the smallest positive integer for which this is true, and U#95;6#61;#92;#123;#92;omega⁰,#92;omega,#92;omega²,#92;omega³,#92;omega⁴,#92;omega⁵#92;#125;. It is easy to see that #92;omega², which is a 6th root of unity, is also a cube root of unity. The same is true of #92;omega⁴. The element #92;omega³ is a square root of unity, whereas #92;omega⁵ is primitive.

2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150:

(number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1.

racine de l’unité

field element, some positive power of which equals 1


Bondareva-Shapley theorem

Formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s.

characteristic function
characteristic function

(game theory) A theorem that describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. The game's core is non-empty if and only if the game is balanced.


transcendence degree

transcendence degree (plural transcendence degrees)

2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263,
Let F_1/k and F_2/k denote algebraic function fields of transcendence degree one.

Lemma 7.19 Suppose that L is a field extension^([sic – meaning extension field]) of transcendence degree r over a field K and that L is not separably generated over K. If x#95;1,#92;dots,x#95;n are elements of L such that L#61;K(x#95;1,#92;dots,x#95;n), then for a suitable relabeling of the x#95;i's, the subfield K(x#95;1,#92;dots,x#95;#123;r#43;1#125;) of L is of transcendence degree r and is not separably generated over K.

2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422:

Proposition 2.1.1 Every finite matrix group #92;Gamma#92;subset#92;operatorname#123;GL#125;(#92;Cⁿ) has n algebraically independent invariants, i.e., the ring #92;C#91;x#93;#92;Gamma has transcendence degree n over #92;C.

2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24:

algebraically independent
algebraically independent

(algebra, field theory, of a field extension) Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.

transcendental degree

cardinality of largest algebraically independent subset of a given extension field


algebraically independent

en-adj

algebraically independent (not comparable)

algebraically dependent

antonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”

The singleton set #92;#123;#92;alpha#92;#125; is algebraically independent over K if and only if the element #92;alpha is transcendental over K.

A subset S#92;subsetL is algebraically independent over K if every element of S is transcendental over K and over each of the extension fields over K generated by the remaining elements of S.

1999, David Mumford, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Lecture Notes in Mathematics 1358, 2nd Edition, Expanded, page 40,
If the statement is false, there are n elements x_1,…,x_n in R such that their images ◌̅x_i in R/P are algebraically independent. Let 0 ne p∈P. Then p,x_1,…,x_n cannot be algebraically independent over k, so there is a polynomial P(Y,X_,…,X_n) over k such that P(p,x_,…,x_n)=0.

Setting y#95;i#61;y#95;#123;i,1#125; (where 1 denotes the identity of the semigroup T) we obtain a #92;sigma-algebraically independent over R set #92;#123;y#95;i#92;verti#92;inI#92;#125; such that S#61;R#92;#123;(y#95;i)#95;#123;i#92;inI#125;#92;#125;.

2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251:

If α ne 0,1 is algebraic and β is an algebraic irrational of degree d>2, then
αᵝ,…,α
are algebraically independent.

2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138, Let us begin with the following conjecture of Schneider

(algebra, field theory) (Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K.


scree plot

(mathematics) A graph of the eigenvalues of factors


DSA

Digital Signature Algorithm
w:Digital Signature Algorithm

(cryptography) Initialism of Digital Signature Algorithm.


stabilizer subgroup

en-noun

stabilizer subgroup (plural stabilizer subgroups)

(algebra) Said of a group acting on a set, and with respect to an element of that set: the subgroup of the given group whose action on the given element of the set that the group acts on leaves that element fixed.


orbit-stabilizer theorem

(algebra) A theorem which states that for each element of a given set that a given group acts on, there is a natural bijection between the orbit of that element and the cosets of the stabilizer subgroup with respect to that element.


twin prime pair

twin prime pair (plural twin prime pairs)


oversizedness

The state or quality of being oversized.


constant morphism

constant morphism (plural constant morphisms)

Any morphism to a zero object must be constant.

(category theory) A morphism which yields the same morphism when post-composed with either one of a parallel pair of morphisms.


coconstant morphism

coconstant morphism (plural coconstant morphisms)

(category theory) A morphism which yields the same morphism when pre-composed with either one of a parallel pair of morphisms.


zero morphism

The composition of a zero morphism with any morphism (that it is composable with) yields another zero morphism.

(category theory) A morphism which is both a constant morphism and a coconstant morphism.


axiom of regularity

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that every non-empty set contains a member that is disjoint from that set.


axiom of pairing

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that if two sets exist, there exists a set with those two sets as its sole elements.


axiom of union

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that a union exists for any set, containing exactly all the elements contained within the sets within that set.


axiom of infinity

(set theory) One of the axioms in axiomatic set theory that guarantees the existence of an infinite set.


multiperfect

(mathematics) Being a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (or k-fold perfect) iff the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn.


triperfect

(mathematics) multiperfect such that σ(n) = 3n


backcalculation

(mathematics) A form of calculation in which input parameters are adjusted until the calculated result coincides with the actual result found


metanode

(mathematics) A node, in a graph, that represents a subgraph that is at a lower level of abstraction


Feigenbaum constant

Feigenbaum constant (plural Feigenbaum constants)

Named after the mathematician Mitchell Feigenbaum.

(mathematics) Either of two mathematical constants that express ratios in a bifurcation diagram for a nonlinear map.


phinary

Blend of phi + binary, in reference to the Greek letter phi (φ).

(mathematics, colloquial) A numeral system that uses the golden ratio as its base.


Ostrowski numeration

(mathematics) Either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers, and a non-integer representation of real numbers.


backcalculate

en-verb

backcalculate (third-person singular simple present backcalculates, present participle backcalculating, simple past and past participle backcalculated)

backcalculates (present singular third-person)
backcalculating (participle present)
backcalculated (participle past)
backcalculated (past)

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(mathematics) To perform backcalculation


superpermutation

(mathematics) For some value n, a string that contains each permutation of n symbols as a substring.


reciprocantive

(mathematics) Relating to a reciprocant.


reciprocant

suf

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

(mathematics) A contravariant expressing a certain condition of tangency; a differential invariant.


linear span

(linear algebra) A linear subspace generated by all linear combinations of a given subset of vectors of a given vector space.


linear subspace

en-noun

linear subspace (plural linear subspaces)

scalar multiplication
scalar multiplication

(linear algebra) A subset of vectors of a vector space which is closed under the addition and scalar multiplication of that vector space.


linear sum

(linear algebra) A linear subspace which is the linear span of the union of two subspaces (of some vector space).


deorphaned

(mathematics, computing) From which orphans have been removed


rank-nullity theorem

If — for a homogeneous system of linear equations — there are V unknowns and R (linearly independent) equations then, according to the rank-nullity theorem, the solution space is N equals V − R dimensional.

linear transformation
linear transformation

(linear algebra) A theorem about linear transformations (or the matrices that represent them) stating that the rank plus the nullity equals the dimension of the entire vector space (which is the linear transformation’s domain).


row-reduced

(linear algebra) Said of a matrix which has undergone row reduction; i.e., of a matrix which has a 1 as the leading entry of each one of its non-zero rows and which has zeroes as all the other entries of any column containing such a leading entry (of any non-zero row).


prime triplet

(mathematics) An ordered set of three prime numbers in which the smallest and largest of the three differ by six.


GAN

GAN (countable and uncountable, plural GANs)

The GAN we look at in the next exercise is a vanilla GAN with convolution.

2020, Micheal Lanham, Practical AI on the Google Cloud Platform, O'Reilly Media, →ISBN, page 248:

generative adversarial network
generative adversarial network#English

(machine learning) Acronym of generative adversarial network.


pseudodivision

pseudodivision (countable and uncountable, plural pseudodivisions)

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(mathematics) Any of several methods of division that employ addition and shifting of digits.


pseudoremainder

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(mathematics) A value in pseudodivision more or less corresponding to the remainder in normal division.


pseudoquotient

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(mathematics) A value in pseudodivision more or less corresponding to the quotient in normal division.


hypernormal

(mathematics) A matrix M is hypernormal if, given any n by n diagonal matrix G with entries of 1 or -1, MᵀGM = G.


associative algebra

associative algebra (plural associative algebras)

algebra over a field
algebra over a field

(algebra) An algebra over a ring (or more narrowly, an algebra over a field) whose bilinear product is associative.


complex conjugation

complex conjugation (countable and uncountable, plural complex conjugations)

(mathematics) The unary operation which maps complex numbers to their complex conjugates.


algebra over a field

algebra over a field (plural algebras over a field)

(algebra) An algebra over a ring such that the ring is in fact a field. As a module over a field, such an algebra is in particular a vector space.


algebra over a ring

algebra over a ring (plural algebras over rings)

(algebra) A module (over a commutative ring R) equipped with a bilinear binary operation taking values in R.


non-associative algebra

non-associative algebra (plural non-associative algebras)

(algebra) An algebra over a ring (or more narrowly, an algebra over a field) whose bilinear product is not necessarily associative.


geoadditive

(statistics, of a model) Obtained by the addition of multiple geographically separate models


multicategorical

(mathematics) Relating to a multicategory


tetraexponential

(mathematics) Having four summed exponential terms.


hyperfairness

(game theory) The property of being hyperfair.


cubic kilometre

cubic kilometre (plural cubic kilometres)

a unit of volume, km³, equal to 1,000,000,000 cubic metres


running meter

length of one meter of a product that is sold by meter, such as fabric or board


sigmoid function

sigmoid function (plural sigmoid functions)

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

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

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

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

mathematical analysis
mathematical analysis

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


Haugh unit

The effectiveness of spray oiling one, six, 24, 48, 72, and 96 hours after lay was compared with no oiling with respect to albumen pH and Haugh units after seven, 14, and 21 days storage of the eggs at a temperature of about 10°C (50°F).

2017, William J Stadelman, Egg Science and Technology:

A unit of measure of egg protein quality, based on the egg's weight and the height of the albumen immediately around the yolk.


shape operator

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

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

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

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

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

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

differential geometry
differential geometry

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


cake number

By analogy with slices being made by a knife through a cube-shaped cake.

(mathematics) For some number n, the maximum number of regions into which a cube can be partitioned by exactly n planes.


free module

(algebra) A module that has a basis. Equivalently, a module consisting of n-tuples of ring elements with no extra identities. (Then the number n is said to be its rank.) Equivalently, a direct sum whose summands are all the same ring. (The quantity of summands may be more than two.)


cone-shaped

en-adj

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

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

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


Ockham algebra

Named after the English Franciscan friar and scholastic philosopher William of Ockham (1287–1347) by Alasdair Urquhart in 1979. The notion was introduced by Joel Berman in 1977.

(algebra) A bounded distributive lattice with a dual endomorphism (where “dual” means that it satisfies De Morgan’s laws).

Alasdair Urquhart
Ockham algebra
William of Ockham


Smarandache function

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

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


conjugate axis

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


criticoid

These criticoids are obtained in consequence of the analogy between the linear transformation of a quotic in one variable and the factorial transformation of a quotoid in one dependent and one independent variable.

1871, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science:

differential equation
differential equation

(mathematics, obsolete) An invariant function of a differential equation


pseudoresidual

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(statistics) The difference between the observed value and the median


division algebra

en-noun

division algebra (plural division algebras)

(algebra) A linear algebra such that its non-zero vectors form a multiplicative group.

(algebra) An algebra over a field such that every non-zero element of it has a multiplicative inverse. (It is not required to have a unity element.)


Sperner's lemma

The initial result of this kind was proved by Emanuel Sperner.

Brouwer fixed-point theorem
Brouwer fixed-point theorem

(mathematics) A combinatorial analog of the Brouwer fixed-point theorem, stating that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors.


Sperner's theorem

Named after Emanuel Sperner, who published it in 1928.

(set theory) A theorem that describes the largest possible families of finite sets none of which contain any other sets in the family.


eigenplane

(mathematics) A two-dimensional invariant subspace in a given vector space


column echelon form

column echelon form (plural column echelon forms)

(linear algebra) The transpose of the row echelon form of the transpose of a given matrix.


left nullspace

(linear algebra) The vector space of all row vectors whose dot products with the columns of a given matrix are zero; the nullspace of the transpose of a given matrix.


geometric analysis

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

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

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

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

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

partial differential equation
partial differential equation

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


sexagesm

From Latin sexāgēsima (“sixtieth”), either directly or via French.

(mathematics, obsolete) Synonym of sexagesimal: a sixtieth part of something.

(mathematics, obsolete) Synonym of minute: a sixtieth of a degree of arc.


Dirichlet energy

Dirichlet energy (plural Dirichlet energies)

Named after German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), who made significant contributions to the theory of Fourier series.

Variational principles play an important role in both geometry and physics, and one of the key problems with applications in both fields is the variational problem associated to the Dirichlet energy of maps between Riemannian manifolds.

2005, Roger Moser, Partial Regularity for Harmonic Maps and Related Problems, World Scientific, page 1:

The Dirichlet energy of a function f#92;inW#123;1,2#125; can be recovered, moreover, as the energy of the composition #92;xi#95;#123;BW#125;#92;circf, where #92;xi#95;#123;BW#125; is the biLipschitz embedding in Corollary 2.2 (compare with Theorem 2.4).

2011, Camillo De Lellis, Emanuele Nunzio Spadaro, Q-valued Functions Revisited, American Mathematical Society, page 28:

Since the Dirichlet energy is weakly lower semicontinuous and strongly continuous, the linear lower-order terms are weakly continuous on H#95;D¹(#92;Omega), and since the finite element spaces are dense in H#95;D¹(#92;Omega), we verify that I#95;h#92;to#92;GammaI as h#92;to 0.

2015, Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, page 89:

(mathematical analysis, functional analysis, Fourier analysis) A quadratic functional which, given a real function defined on an open subset of ℝⁿ, yields a real number that is a measure of how variable said function is.

functional analysis; Fourier analysis; functional analysis; Fourier analysis

Dirichlet energy
Peter Gustav Lejeune Dirichlet

Dirichlet-energie

functional that maps a function to a real number representing its variability


decangle

/ˈdɛkˌæŋɡəl/

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

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

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

1658, John Newton, Trigonometria Britanica, page 9:

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

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


quinquangle

/ˈkwɪŋˌkwæŋɡəl/

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

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

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

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

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


biangle

/ˈbaɪˌæŋɡəl/

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

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

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

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

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

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

non-Euclidean geometry
non-Euclidean geometry#English

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


diangle

/ˈdaɪˌæŋɡəl/

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

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

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

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

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

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


witch of Agnesi

witch of Agnesi (plural witches of Agnesi)

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

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


Sørensen-Dice coefficient

Sørensen–Dice coefficient (plural Sørensen-Dice coefficients)

Sørensen–Dice coefficient (canonical)
Sørensen-Dice coefficients (plural)

Developed independently by the botanists Thorvald Sørensen and Lee Raymond Dice in the 1940s.

(statistics) A statistic used to gauge the similarity of two samples. It is equal to twice the number of elements common to both sets, divided by the sum of the number of elements in each set.


Jaccard index

Jaccard index (plural Jaccard indexes or Jaccard indices)

Jaccard indexes (plural)
Jaccard indices (plural)

Developed by Swiss botanist Paul Jaccard (1868–1944), who gave it the French name coefficient de communauté.

(statistics) A statistic for gauging the similarity and diversity of sample sets, based on the size of the intersection between the sets divided by the size of the union of the sets.


Azumaya algebra

Azumaya algebra (plural Azumaya algebras)

Introduced in a 1951 paper by Goro Azumaya, for the case where R is a commutative local ring.

(mathematics) A generalization of central simple algebras to R-algebras where R need not be a field.


Brauer group

Defined by the algebraist Richard Brauer.

(mathematics) For a field K, an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras.


Kronecker symbol

Kronecker symbol (plural Kronecker symbols)

(number theory) (a/n) or (a|n), a generalization of the Jacobi symbol to all integers n.


precomposition

precomposition (countable and uncountable, plural precompositions)

(mathematics) A form of pullback where a function of a variable y can be rewritten as a function of x if y itself is a function of x.


affinize

en-verb

affinize (third-person singular simple present affinizes, present participle affinizing, simple past and past participle affinized)

affinizes (present singular third-person)
affinizing (participle present)
affinized (participle past)
affinized (past)


AdS

(physics, mathematics, cosmology) Initialism of anti de Sitter.


birthday paradox

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(probability theory) The counterintuitive observation that a random group of people needs to have only 23 members before there is a 50% chance of two of them having the same birthday.


hypersection

(mathematics) section through a hypersurface


regular graph

(graph theory) A graph whose vertices all have the same degree.


plane graph

(graph theory) An embedding of a planar graph in a plane.


cubic graph

(graph theory) A 3-regular graph; i.e., a graph whose every vertex has degree three.


dual graph

The edges of a dual graph are perpendicular, in some sense, to the edges of its original graph. The dual graph also has as many vertices and faces as its original graph has faces and vertices, respectively. Therefore the dual graph has the same Euler characteristic as its original graph.

(graph theory) A graph derived from some plane graph in such a way that the derived graph has a vertex corresponding to each face of the given graph, an edge corresponding to each edge of the given graph that is shared by a pair of distinct faces, and a self-loop for each edge of the given graph that is a border of the same face on both of its sides.


planar graph

(graph theory) A graph which can be embedded in a plane in such a way that its edges only intersect at vertices, i.e., they do not cross each other.


medial graph

The number of vertices of a medial graph is equal to the number of edges of its original graph, and the number of edges of a medial graph is equal to the sum of the degrees of the vertices of the original graph.

(graph theory) A graph derived from a given plane graph such that this derived graph has a vertex corresponding to each edge of the given graph, and such that for every “angle” (consecutive trio of edge, vertex, edge) along the border of a face of the given graph there is a corresponding edge which connects the vertices corresponding to the two edges that are part of that “angle”.


eigenbrain

(mathematics, computing) One of a set of eigenvectors used in visual recognition systems to represent a person's brain.


elementary matrix

en-noun

elementary matrix (plural elementary matrices)

When an elementary matrix multiplies a square matrix from the left, it applies to that matrix an elementary row operation homologous to the one that created its own self. (Likewise, when an elementary matrix multiplies a square matrix from the right, it applies to that matrix an elementary column operation homologous to the one that created its own self.)

If a series of elementary matrices applied to a given square matrix row-reduce it to the identity matrix, then the same series of elementary matrices applied to the identity matrix yield the inverse of the given matrix.

(linear algebra) A square matrix which is obtained from the identity matrix by a single elementary row (or column) operation.


characteristic vector

characteristic vector (plural characteristic vectors)


angle of torsion

angle of torsion (plural angles of torsion)

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


submarking

(graph theory) A subset of a marking of a Petri net.


WHP

(mathematics, computer science) Initialism of with high probability.


numericist

(mathematics) A mathematician whose speciality is numerical methods


sweep line

(mathematics) A line that sweeps over a plane or surface and interacts with points therein


Ulam spiral

Devised by mathematician Stanislaw Ulam in 1963.

(mathematics) A graphical depiction of the set of prime numbers, constructed by writing the positive integers in a square spiral and specially marking the primes.


Robinson-Schensted correspondence

Robinson-Schensted correspondence (plural Robinson-Schensted correspondences)

Named after two mathematicians who worked in this area, Robinson (in 1938) and Schensted (1961).

(mathematics) A bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. It has applications in combinatorics and other areas.


Young tableau

Young tableau (plural Young tableaus or Young tableaux)

Young tableaus (plural)
Young tableaux (plural)

(mathematics) A diagram obtained by filling in the boxes of the Young diagram with symbols from some alphabet, usually required to be a totally ordered set.


Young diagram

Introduced by Alfred Young, a Cambridge mathematician.

(mathematics) A finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a partition λ of a non-negative integer n, the total number of boxes of the diagram.


Schensted algorithm

Robinson-Schensted correspondence
Robinson-Schensted correspondence

(mathematics) The simplest description of the Robinson-Schensted correspondence; a procedure that constructs one tableau by successively inserting the values of the permutation according to a specific rule, while the other tableau records the evolution of the shape during construction.


Schensted insertion

Schensted insertion (countable and uncountable, plural Schensted insertions)

(mathematics) The insertion step of the Schensted algorithm.


exterior derivative

exterior derivative (plural exterior derivatives)

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The exterior derivative of a “scalar”, i.e., a function f=f(x¹,x²,...,xⁿ) where the xⁱ’s are coordinates of ℝⁿ, is df=∂f/∂x¹dx¹+∂f/∂x²dx²+...+∂f/∂xⁿdxⁿ.

The exterior derivative of a k-blade f,dx∧dx∧...∧dx is df∧dx∧dx∧...∧dx.

The exterior derivative d may be though of as a differential operator del wedge: ∇∧, where ∇=∂/∂x¹dx¹+∂/∂x_2dx²+...+∂/∂xⁿdxⁿ. Then the square of the exterior derivative is d²=∇∧∇∧=(∇∧∇)∧=0∧=0 because the wedge product is alternating. (If u is a blade and f a scalar (function), then fu≡f∧u, so d(fu)=∇∧(fu)=∇∧(f∧u)=(∇∧f)∧u=df∧u.) Another way to show that d²=0 is that partial derivatives commute and wedge products of 1-forms anti-commute (so when d² is applied to a blade then the distributed parts end up canceling to zero.)

differential operator
differential operator

(calculus) A differential operator which acts on a differential k-form to yield a differential (k+1)-form, unless the k-form is a pseudoscalar, in which case it yields 0.


photon sphere

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

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

The redshift of light leaking outward from the photon sphere is #92;sqrt#123;3#125;-1#61;0.732. All light rays approaching a black hole closer than #92;sqrt#123;3#125; times the radius of the photon sphere spiral inwards and are captured (see Figure 13.5).

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

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

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

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

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

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

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

sphère de photons

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


photon circle

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

(of a black hole) Synonym of photon sphere


last photon orbit

last photon orbit (plural last photon orbits)

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


semiordinate

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


subposterior

(mathematics) An independent part of a posterior


constantive

Thermodynamic properties are additive quantities. Spectroscopic properties are constantive quantities. Wavefunctions, statistical mechanical partition functions, and probabilities are multiplicative quantities.

1993, Jerry Ray Dias, Molecular orbital calculations using chemical graph theory, page 105:

(mathematics) Composed of elements that are each expressible by a unary polynomial.


eigenprojector

(mathematics) An operator that forms an eigenprojection


positive-definite

(linear algebra) Said of a real, square matrix: that the product of it with a column vector on its right side and the transpose of that column vector on its left side is greater or equal to zero, only equaling zero if the vector itself is zero.

(linear algebra) Said of a quadratic form: that the application of it to a tuple is greater or equal to zero, only equaling zero if the tuple itself is zero.


exterior product

exterior product (plural exterior products)

The exterior product of two vectors is a bivector.

The exterior product between two vectors is anti-commutative; therefore the exterior product between a vector and itself is zero.

The exterior product between a multivector of grade k and a multivector of grade n is a multivector of grade k+n, unless k+n is larger than the dimension of the vector space to which the vectors belong (out of which the multivectors are constructed), in which case their product is zero.

The exterior product between a multivector of grade k and a multivector of grade n is commutative if k times n is even and anti-commutative if k times n is odd. (This is related to what is meant when it is said that the exterior product is alternating. It means that a permutation of the factors of a wedge product of vectors changes the sign of the product if and only if the permutation is odd.)

(algebra) A kind of product between vectors and/or multivectors which is associative, linear, and alternating.


evaluatable

(mathematics) That can (or should) be evaluated


seminvariant

(mathematics) Alternative form of semi-invariant


biprime

(mathematics) Having exactly two prime factors.


disattenuation

disattenuation (usually uncountable, plural disattenuations)

(statistics) A correction to account for attenuation


Klein quadric

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


Klein quartic

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

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


Voronoi pole

(mathematics) The vertices in a Voronoi diagram that are farthest from the point defining a cell.


Voronoi formula

(mathematics) An equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side.


lie over

en-verb

lie over (third-person singular simple present lies over, present participle lying over, simple past lay over, past participle lain over)

lies over (present singular third-person)
lying over (participle present)
lay over (past)
lain over (participle past)

Let R be a subring of a ring R′ which is integral over R. If P is a prime ideal of R, then there is a prime ideal P′ of R′ that lies over P. Moreover, if P′ and P″ are prime ideals of R′ that lie over P and if P′ ⊆ P″, then P′ = P″.

1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Academic Press, page 84:

Let R ⊆ S be integral domains such that R is integrally closed and S is integral over R. Suppose that P′ ⊆ P are prime ideals of R. Let Q be a prime ideal of S which lies over P. Then Q contains a prime ideal Q′ of S that lies over P′.

2012, A. Barlotti, E.W. Ellers, P. Plaumann, K. Strambach, Generators and Relations in Groups and Geometries, Springer Science & Business Media, page 369:

(transitive, algebra, ring theory, of a prime ideal q of a ring R' with subring R) To intersect with R at some other prime ideal p; q lies over p if and only if q∩R=p.


eeny-weeny

en-adj

eeny-weeny (comparative more eeny-weeny, superlative most eeny-weeny)

more eeny-weeny (comparative)
most eeny-weeny (superlative)

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common logarithm

en-noun

common logarithm (plural common logarithms)


lake of Wada

Calque of Japanese 和田の湖. Introduced by Kunizo Yoneyama (in 1917), who credited the discovery to Takeo Wada.

(mathematics) Any of three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary.

Kunizo Yoneyama
Lakes of Wada
Takeo Wada


Hurwitz surface

Hurwitz surface (plural Hurwitz surfaces)

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


Torr

Clipping of Torricelli. Named after Italian physicist wplink

(metrology) The unit symbol for a torr (“unit of pressure”)


megacorpse

One million corpses, as a unit of measure in reference to nuclear warfare.


Chern class

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


Chow group

Named after Wei-Liang Chow by Claude Chevalley (1958).

(mathematics) The Chow groups of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes.


Gauss-Bonnet theorem

Named after Carl Friedrich Gauss, who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet, who published a special case in 1848.

Euler characteristic
Euler characteristic

(mathematics) An important statement about surfaces in differential geometry, connecting their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).


Plücker matrix

Plücker matrix (plural Plücker matrices)

Named after the German mathematician Julius Plücker.

(mathematics) A special skew-symmetric 4×4 matrix that characterizes a straight line in projective space, and is defined by six Plücker coordinates with four degrees of freedom.


Plücker embedding

Plücker embedding (plural Plücker embeddings)

(mathematics) A method of realizing the Grassmannian Gr_k(V) of all k-dimensional subspaces of an n-dimensional vector space V as a subvariety of a projective space.


Plücker relation

Plücker relation (plural Plücker relations)

(mathematics) Any of a group of simple quadratic relations that are satisfied by the Plücker embedding.


Gaussian curvature

Gaussian curvature (countable and uncountable, plural Gaussian curvatures)

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


triangle group

(mathematics) A group that can be realized geometrically by sequences of reflections across the sides of a triangle.


nonblocker

(mathematics) A subset of vertices in an undirected graph, all of which are adjacent to vertices outside of the subset


ultrametricized

(mathematics) Converted to ultrametric form


Kirby move

After Robion Kirby (born 1938), American topologist.

(mathematics) Any of the moves used in the Kirby calculus for modifying framed links in the 3-sphere.


semiconcentric

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

(geometry) Having the form of concentric semicircles


rahmonic

From harmonic with the first syllable reversed, like cepstrum from spectrum.

(signal processing) A harmonic of a cepstrum.


gamnitude

From magnitude with the first syllable reversed, like cepstrum from spectrum.

(signal processing) A magnitude of a cepstrum.


transfinite induction

transfinite induction (plural transfinite inductions)

The validity of the principle of transfinite induction for well-ordered sets enables us to carry out proofs by transfinite induction and definitions by transfinite induction. A proof by transfinite induction is a direct application of the principle when it is required to show that each element of a well-ordered set A has a property P.[…]To understand the method of definition by transfinite induction some preparation is necessary.

1967, Kam-Tim Leung, Doris Lai-chue Chen, Elementary Set Theory, Parts I and II, Hong Kong University Press, page 108:

1970 [Addison-Wesley], Howard DeLong, A Profile of Mathematical Logic, Dover, 2004, page 218,
Just what kinds of transfinite inductions are to be considered finitary is debatable. Transfinite induction up to an arbitrary ordinal is certainly not finitary. However, it can be shown that certain transfinite inductions are reducible to ordinary mathematical inductions. For example, induction up to ω^ω is reducible to ordinary induction. Gentzen in his proof used transfinite induction up to ε₀.

The published version of 1936 had a different proof based on the famous principle of transfinite induction up to the ordinal #92;varepsilon#95;0 by which consistency^([of Peano arithmetic]) followed. A third proof of 1938 used transfinite induction but the logical system was a sequent calculus.

2009, Jan von Plato, “Gentzen's Logic”, in Dov M. Gabbay, John Woods, editors, Handbook of the History of Logic, Volume 5: Logic from Russell to Church, Elsevier (North-Holland), page 667:

mathematical induction
mathematical induction

(mathematics, set theory) An extension of mathematical induction to well-ordered sets of transfinite cardinality, such as sets of ordinal numbers or cardinal numbers.


matrix algebra

matrix algebra (usually uncountable, plural matrix algebras)

(algebra) Generically, the arithmetic of matrices, involving matrix addition, scalar multiplication of matrices, matrix multiplication, and matrix inversion.

(algebra) A matrix ring over a commutative ring.


projective space

projective space (plural projective spaces)

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


epsilon number

From the Greek letter ε (“epsilon”), used to denote the numbers.

More generally, the epsilon numbers are the ordinals #92;varepsilon for which #92;varepsilon#61;#92;omega#92;varepsilon. The smallest epsilon number is #92;varepsilon#95;0. It is a countable ordinal, being the countable union of countable sets. By the Veblen fixed-point theorem, the class of epsilon numbers is unbounded.

1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240:

We show that the associated ordinals are the #92;varepsilon#95;0th epsilon number and the first #92;varepsilon#95;0-critical number, respectively.

1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17:

2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, [Revision of 1971 Addison-Wesley edition], page 203,
Thus there is at least one epsilon number, namely ε₀; we can easily show, in fact, that ε₀ is the least epsilon number.

(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.


multitapered

(signal processing) Employing the multitaper technique.


affinely independent

linearly independent
linearly independent

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


projective frame

projective frame (plural projective frames)

affinely independent
affinely independent

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


real projective plane

real projective plane (plural real projective planes)

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


homogeneous coordinate

homogeneous coordinate (plural homogeneous coordinates)

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


Proth prime

(mathematics) A Proth number that is prime.


Schubert calculus

Named after German mathematician Hermann Schubert (1848–1911), who introduced the theory in the nineteenth century.

Hall appears to have been first with this observation, too, for in a lecture he gave at a 1959 Canadian Mathematical Congress conference in Banff on the algebra of symmetric polynomials he noted that the Schubert calculus has combinatorics similar to that of the symmetric polynomials [9].

1986, Christopher I. Byrnes, Anders Lindquist, Frequency Domain and State Space methods for Linear Systems, North-Holland, page 77:

2014, Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki, k-Schur Functions and Affine Schubert Calculus, Springer, Fields Institute for Research in the Mathematical Sciences, page 2,
The rich combinatorial backbone of the theory of Schur functions, including the Robinson–Schensted algorithm, jeu-de-taquin, the plactic monoid (see for example [139]), crystal bases [127], and puzzles [74], now underlies Schubert calculus and in particular produces a direct formula for the Littlewood-Richardson coefficients.

2016, Letterio Gatto, Parham Salehyan, Hasse-Schmidt Derivations on Grassmann Algebras, IMPA, Springer, page 117,
This point of view was extensively developed by Laksov–Thorup [96–98] and Laksov [93, 94] in the case of equivariant Schubert calculus.

(mathematics) A branch of algebraic geometry concerned with solving certain types of counting problem in projective geometry; a symbolic calculus used to represent and solve such problems;

A branch of algebraic geometry concerned with solving certain types of counting problem in projective geometry; a symbolic calculus used to represent and solve such problems;
(by generalisation) the enumerative geometry of linear subspaces; the study of analogous questions in generalised cohomology theories.


circle group

(algebra) The group of all complex numbers of unit modulus, with the group operation being complex multiplication.


double number

split-complex number
split-complex number


gyroline

(mathematics) A line in gyrovector space.


pretopology

pretopology (countable and uncountable, plural pretopologies)

(mathematics) The field of study of pretopological spaces (in plural for different prespaces).

προτοπολογία

field of study of generalized topological space(s)

(mathematics) A filter subset or a preclosure operator which acts as a distinct prespace (pre-topological space).


cross term

(linear algebra) In a quadratic form, a term which is not a squared variable; i.e., a term with more than one variable.


polynomial algebra

polynomial algebra (plural polynomial algebras)


lamplighter group

lamplighter group (plural lamplighter groups)

The group can be viewed as acting on a doubly infinite sequence of street lamps …,l_-2,l_-1,l_0,l_1,l_2,l_3,… each of which may be on or off, with a lamplighter standing at some lamp l_k.

(group theory) The restricted wreath product mathbf Z₂≀ mathbf Z.


rototranslation

(mathematics) Any rigid transformation (Euclidean isometry) that involves rotation and translation


concavification

concavification (countable and uncountable, plural concavifications)

(mathematics) The process of converting to concave form.


concavify

en-verb

concavify (third-person singular simple present concavifies, present participle concavifying, simple past and past participle concavified)

concavifies (present singular third-person)
concavifying (participle present)
concavified (participle past)
concavified (past)


trans-lunar injection

en-noun

trans-lunar injection (plural trans-lunar injections)

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


supergaussian

(mathematics) Describing a graph, distribution or equation between Gaussian and rectangular