sample path

1968, Leo Breiman, Probability, 1992, Society for Industrial and Applied Mathematics, Unabridged corrected republication, page 298,
These processes cannot have versions with continuous sample paths, otherwise the argument given in Chapter 12 forces them to be Brownian motion.

1981 [Wiley], Robert J. Adler, The Geometry of Random Fields, 2010, Society for Industrial and Applied Mathematics, Unabridged republication, page 39,
In order to study the 'finite' situation it is necessary to impose a number of conditions on the sample paths of the fields. Indeed, what we shall later generally assume is that sample paths belong, with probability one, to the special class of functions we now define.

A sample path or stochastic realization of the stochastic process #92;#123;#92;mathcal#123;I#125;(t)#92;#125;#92;infty#95;#123;t#61;0#125; for t#92;in#92;#123;0,#92;Deltat,2#92;Deltat,...#92;#125; is an assignment of a possible value to #92;mathcal#123;I#125;(t) based on the probability vector p(t).[…]Three sample paths of the stochastic model are compared to the deterministic solution in Fig.3.4. One of the sample paths is absorbed before 200 time steps (the population following this path becomes disease-free) but two sample paths are not absorbed during 2,000 time steps.

2008, Linda J. S. Allen, “Chapter 3: An Introduction to Stochastic Epidemic Models”, in Fred Brauer, Pauline van den Driessche, Jianhong Wu, editors, Mathematical Epidemiology, Springer, page 90:

(statistics) Any set of possible values to which the appropriate random variables determined by a stochastic process might map a given point in the sample space, taken over all values of the index space (often regarded as time).

stochastic realisation

set of values for a given point of sample space


conditional probability distribution

en-noun

conditional probability distribution (plural conditional probability distributions)

conditional probability distributions (plural)

1999, Anders L. Madsen, Bruce D'Ambrosio, Lazy Propagation and Independence of Causal Influence, Anthony Hunter, Simon Parsons, Symbolic and Quantitative Approaches to Reasoning and Uncertainty, Springer, page 293,
A Bayesian network consists of a directed, acyclic graph and a set of conditional probability distributions. For each variable in the graph there is a conditional probability distribution of the variable given its parents.

2010, Philipp Koehn, Statistical Machine Translation, Cambridge University Press, page 69,
Note that for two independent random variables X and Y, the conditional probability distribution p(x|y) is the same as simply p(y).

2015, Iman Behmanesh, Babak Moaveni, Geert Lombaert, Costas Papadimitriou, Chapter 6: Hierarchical Bayesian Model Updating for Probabilistic Damage Identification, H. Sezer Atamturktur, Babak Moaveni, Costas Papadimitriou, Tyler Schoenherr (editors), Model Validation and Uncertainty Quantification, Volume 3: Proceedings of the 33rd IMAC, Springer, page 58,
Alternatively, Laplace asymptotic approximation can be used to approximate the conditional probability distributions of θ₁ in Eq. (6.12) as a Gaussian distribution to simplify the sampling process.

probability distribution
probability distribution

(probability theory) The probability distribution of a random variable given the known outcome of another random variable.


normal random variable

normal random variable (plural normal random variables)

Since this density function cannot be integrated directly, in order to find probabilities involving normal random variables, we must resort to tables giving us various percentile points of the distribution.

1986, Mark C. K. Yang, David H. Robinson, Understanding and Learning Statistics by Computer, World Scientific, page 28:

Although normal random variables are continuous, they are often used to model discrete quantities.[…]The probability density function of a normal random variable depends on two parameters, denoted μ and σ, which are, respectively, the mean and standard deviation of the random variable.

2016, Geoffrey C. Berresford, Andrew M. Rockett, Applied Calculus, 7th edition, Cengage Learning, page 731:

A special type of continuous random variable, known as a normal random variable, is studied.[…]We present the additive property of normal random variables. The percentiles of normal random variables are studied.

2017, Sheldon M. Ross, Introductory Statistics, 4th edition, Elsevier (Academic Press), page 263:

(probability theory, statistics) A random variable whose probability distribution is a normal distribution.


arithmetization

arithmetization (countable and uncountable, plural arithmetizations)

(mathematics) The treatment of mathematics by methods involving only the fundamental concepts and operations of arithmetic.


cross-countries

diminishing cross-countries disparities in Internet use

(statistics, of comparative data) Across or between countries.


standard normal random variable

standard normal random variable (plural standard normal random variables)

standard normal random variables (plural)

Suppose X#92;simN(#92;mu,#92;sigma). Then
Z#61;#92;frac#123;x-#92;mu#125;#123;#92;sigma#125;
is a standard normal random variable. In other words, any normal random variable can be standardized by subtracting the mean from the variable and dividing the difference by the standard deviation.

2000, George E. Monahan, Management Decision Making: Spreadsheet modeling, analysis, and applications, volume 1, Cambridge University Press, page 405:

This technique is based on a polynomial transformation that proceeds by taking the sum of a linear combination of a standard normal random variable, its square, and its cube.

2010, Todd C. Headrick, Statistical Simulation: Power Method Polynomials and Other Transformations, Taylor & Francis (CRC Press), page 1:

2016, Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt, Probability and Statistics with R, 2nd Edition, Taylor & Francis (CRC Press), page 297,
A normal random variable with μ = 0 and σ = 1, often denoted Z, is called a standard normal random variable. […] Further, any normal random variable can be converted to a standard normal random variable using (4.25).

standard normal distribution
standard normal distribution

(probability theory, statistics) A random variable whose probability distribution is a standard normal distribution.


circle of antisimilitude

circle of antisimilitude (plural circles of antisimilitude)

(geometry) A reference circle about which two other circles are inverses of each other


arithmology

(mathematics) The science of numbers; basic arithmetic


bivariance

(mathematics, computing) The state of being bivariant.


Bourbakian

It is the notion of structure that is truly Bourbakian.

Claude Chevalley, quoted in: 2006, Amir D. Aczel, The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed, High Stakes Publishing, p. 94


Bourbakian

(mathematics) Any of the mathematicians involved in the formulation of Bourbakism.


percentilation

(statistics) The division of a distribution into percentiles


cocyclomatic

(mathematics) Having the same cyclomatic number


cyclomatic number

cyclomatic number (plural cyclomatic numbers)


circuit rank

(mathematics) The number of independent cycles in a graph


cobwebbing

(mathematics) A graphical technique for solving discrete-time systems, useful in finding the stationary points.


W-shaped

A-shaped - B-shaped - C-shaped - D-shaped - E-shaped - F-shaped - G-shaped - H-shaped - I-shaped - J-shaped - K-shaped - L-shaped - M-shaped - N-shaped - O-shaped - P-shaped - Q-shaped - R-shaped - S-shaped - T-shaped - U-shaped - V-shaped - W-shaped - X-shaped - Y-shaped - Z-shaped


cubohemioctahedron

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(geometry) A faceted form of the cuboctahedron.

立方半八面体

りっぽうはんはちめんたい


digonal

(mathematics) symmetrical about a 180° turn about an axis


P-shaped

Shaped like the letter "P" or its mirror image


H-shaped

Н-обра́зный

having a shape similar to a capital letter H


millimeter of mercury

millimeter of mercury (plural millimeters of mercury)

A unit of pressure equal to the amount of fluid pressure one millimeter deep in mercury at zero degrees Celsius on Earth.


hypergeometric random variable

hypergeometric random variable (plural hypergeometric random variables)

hypergeometric random variables (plural)

Computer generation of classical hypergeometric random variables has been discussed in detail by Kachitvichyanukul and Schmeiser (1985).

1992, Norman Lloyd Johnson, Samuel Kotz, Adrienne W. Kemp, Univariate Discrete Distributions, page 67:

Chapter 6 introduces testing of hypotheses immediately after the study of binomial and hypergeometric random variables.

2005, Martin Buntinas, Gerald Marlowe Funk, Statistics for the Sciences, page xv:

If #92;textstyleX is a hypergeometric random variable, then the variance is #92;textstyleVar(X)#61;#92;left(#92;frac#123;M#125;#123;N#125;#92;right)n#92;left(1-#92;frac#123;M#125;#123;N#125;#92;right)#92;left(1-#92;frac#123;n-1#125;#123;N-1#125;#92;right).

2007, Purna Chandra Biswal, Probability and Statistics, India: Prentice-Hall, published 2008, page 87:

Thus, whereas the expected value of the hypergeometric random variable with parameters n, N, p is the same as that of the binomial random variable with parameters n, p, its variance is smaller than that of the binomial by the factor (N − n)/(N − 1).

2017, Sheldon M. Ross, Introductory Statistics, Elsevier (Academic Press), page 250:

hypergeometric distribution
hypergeometric distribution

(probability theory, statistics) A random variable whose probability distribution is a hypergeometric distribution.


epsilonize

en-verb

epsilonize (third-person singular simple present epsilonizes, present participle epsilonizing, simple past and past participle epsilonized)

epsilonizes (present singular third-person)
epsilonizing (participle present)
epsilonized (participle past)
epsilonized (past)

(mathematics) To convert small errors to zero or to treat them as arbitrarily small


exinscribed

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(geometry) inscribed inside a polygon (and touching every side of it)


factorialize

factorialize (third-person singular simple present factorializes, present participle factorializing, simple past and past participle factorialized)

factorializes (present singular third-person)
factorializing (participle present)
factorialized (participle past)
factorialized (past)

(mathematics, transitive) To perform factor analysis on (a set of data).


golyhedron

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(geometry) A three-dimensional generalization of a golygon.


herpolhode

(mathematics) The surface created by the rotation of a rigid body


F-shaped

Shaped like the letter "F" or its mirror image


J-shaped

Shaped like the capital letter "J" or its mirror image


K-shaped

Shaped like the letter "K" or its mirror image


Q-shaped

What I want here to carry away from the painting of balloons is a certain ontological observation. What both indicates a balloon and portrays it as, say, red is a single object – a red, Q-shaped pigment patch.

2012, J. F. Rosenberg, Linguistic Representation, page 100:

Shaped like the capital letter Q or its mirror image


hypercomplexity

en-noun

hypercomplexity (countable and uncountable, plural hypercomplexities)

(mathematics) The state of being a hypercomplex number.


Holderian

(mathematics) Relating to either of the German mathematicians Otto Hölder or Ernst Hölder


Jordanize

en-verb

Jordanize (third-person singular simple present Jordanizes, present participle Jordanizing, simple past and past participle Jordanized)

Jordanizes (present singular third-person)
Jordanizing (participle present)
Jordanized (participle past)
Jordanized (past)

(mathematics) To convert to, or represent as a Jordan curve


Jordanization

en-noun

Jordanization (usually uncountable, plural Jordanizations)

In this section, which is not necessary for the following, we elaborate on the Jordanization of M, paying attention successively to the symplectic structure, the 'energy-scaling' block and the 'horizontal-vertical' decomposition...

2005 March, Alain Chenciner, Jacques Féjoz, Richard Montgomery, “Rotating Eights: I. the three Γᵢ families”, in Nonlinearity, volume 8, number 3:

Let us recall here some known results [3; 7] on the "Jordanization" of diffeomorphisms (directly issued from the corresponding ones for vector fields) and on the relationship between unipotent diffeomorphisms and nilpotent vector fields.

2005, Javier Ribòn, “Families of diffeomorphisms without periodic curves”, in The Michigan Mathematical Journal, volume 53, number 2, page 244:

This will allow us to understand this “Jordanization” phenomenon in very clear terms and prepare us to the analysis of the interacting theory (2)

2006 January, A.V. Smilga, “Ghost-free higher-derivative theory”, in Physics Letters B, volume 632, numbers 2-3:

(mathematics) Conversion to, or representation as a Jordan curve.


millile

(statistics) Any of the values in a series that divides the distribution of individuals in that series into a thousand groups of equal frequency.


FIR

finite impulse response
w:finite impulse response

(signal processing, of a filter) Initialism of finite impulse response; that has impulse response (or response to any finite length input) of finite duration.


D-shaped

Shaped like the letter "D" or its mirror image


G-shaped

Shaped like the capital letter "G" or its mirror image


E-shaped

Shaped like the capital letter "E" or its mirror image


HPF

(signal processing) Initialism of high-pass filter.


BPF

(signal processing) Initialism of band-pass filter.


N-shaped

Shaped like the capital letter "N" or its mirror image.


B-shaped

Shaped like the capital letter "B" or its mirror image


R-shaped

Shaped like the capital letter "R" or its mirror image


O-shaped

Shaped like the capital letter "O", circular.

O-förmig

shaped like the 15th letter of the alphabet


I-shaped

工字

having a shape similar to a capital letter I with serifs

Shaped like the capital letter "I" with serifs.

Shaped like the capital letter "I" without serifs.


multizeta

(mathematics) Involving multiple zeta values


zeta value

(mathematics) The value of a particular zeta function


pivot operation

pivot operation (plural pivot operations)

(mathematics) The interchange of suitable rows and/or columns of a matrix in order to place a particular element on the diagonal (prior to some other operation)


mirror symmetry

(algebraic geometry, theoretical physics) A relationship between Calabi-Yau manifolds where two such manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.


cup-shaped

Having the hollow form of a cup (but without the handle).


mushroom-shaped

Near-synonyms: mushroomoid, mushroomlike; mushroomy; fungiform

The deadman's pedals, which can be switched for use with either fitted or unfitted stock, are of a new type, mushroom-shaped and designed for operation with minimum fatigue.

1962 June, “The design of the S.R. electro-diesels”, in Modern Railways, page 395:

Having the characteristic shape of a mushroom (as a pillar with a broad, circular top).


Dedekind cut

Named after German mathematician Richard Dedekind (1831–1916), who introduced the concept (although a similar construction was used in Euclid's Elements to define proportional segments).

More than one formal solution was offered; the one which is easiest to work with (all of the solutions are provably equivalent) is the method of Dedekind cuts.

1990, Judith Roitman, Introduction to Modern Set Theory, Wiley, page 70:

To identify Dedekind cuts as the sought-for "real number system," we must show that they include all the rationals and irrationals—all the numbers that can be approximated with arbitrary accuracy by rationals.

1997, Reuben Hersh, What is Mathematics, Really?, Oxford University Press, page 274:

But there are, of course, many set theoretic ways of expressing Riemannian manifolds, just as there are many set theoretic ways of expressing real numbers (one can express them as pairs of lower and upper Dedekind cuts, or as lower Dedekind cuts, or as upper Dedekind cuts, or as equivalence classes of Cauchy sequences, and so on).

2011, Alexander R. Pruss, Actuality, Possibility, and Worlds, Continuum Books, page 55:

(mathematics) Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number.


topological property

topological property (plural topological properties)

(topology) A property of a topological space that is invariant under homeomorphisms.


Brouwer fixed-point theorem

the Brouwer fixed-point theorem (canonical)

Named after L. E. J. Brouwer (1881–1966), Dutch mathematician and philosopher.

(topology) A theorem which (in one of its simplest forms) states that a continuous function from an n-dimensional unit cube (in ℝⁿ) to itself has a fixed point.

Brouwer fixed-point theorem
L. E. J. Brouwer


rectanguloid

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(geometry) A rectangular object; A three-dimensional form related to the rectangle


reorthogonalize

en-verb

reorthogonalize (third-person singular simple present reorthogonalizes, present participle reorthogonalizing, simple past and past participle reorthogonalized)

reorthogonalizes (present singular third-person)
reorthogonalizing (participle present)
reorthogonalized (participle past)
reorthogonalized (past)

(mathematics) To produce, or to undergo reorthogonalization


spheroform

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(geometry) A three-dimensional, convex form that has a constant thickness


teragon

(mathematics) A polygon that has fractal sides


nonhypotenuse

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(geometry) Either of the two smaller sides of a right-angled triangle


pentatope

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hemiperfect

(mathematics) Having a half-integral abundancy index


abundancy index

en-noun

abundancy index (plural abundance indices or abundancy indexes)

abundance indices (plural)
abundancy indexes (plural)

The abundancy index of 20 is 2.1 because 1 + 2 + 4 + 5 + 10 + 20 = 42 and 42/20 = 2.1.

(mathematics) The divisor function of an integer divided by the integer.


Cullen number

(mathematics) A natural number of the form n·2ⁿ+1.


Carol number

First studied by Cletus Emmanuel, who named it after a friend, Carol G. Kirnon.

Mathematically, it is actually interesting in all sorts of ways, being both a Keith number and a Carol number.

2009, The Philosophers' Magazine - Issue 47, page 4:

[…] 7. There are 47 occurrences of 47 in the first thousand prime numbers. 8. The atomic number of Silver is 47. 9. It is a Carol number. […]

2014, Rajesh Kumar Thakur, The Power of Mathematical Numbers, page 131:

In 2007 the largest Carol number was found for n = 253987, consisting of 152916 digits, by using the programs MultiSieve and PrimeFormGW.

2019, Dr. Ir. Maarten Looijen, Over getallen gesproken - Talking about numbers, page 70:

For ngt;2, the binary representation of the n-th Carol number is n-2 consecutive unities, a single zero in the middle, and n#43;1 more consecutive unities, or to put it algebraically, #92;textstyle#123;CN(n)#61;#92;sum#95;#123;i#61;1,i#92;nen#43;2#125;#123;2n#125;2#123;i-1

2021, Elena Deza, Mersenne Numbers And Fermat Numbers, page 136:

(mathematics) An integer of the form 4ⁿ-2ⁿ⁺¹-1.


Cauchy-Riemann equation

Cauchy-Riemann equation (plural Cauchy-Riemann equations)

Augustin Cauchy
Bernhard Riemann
Cauchy-Riemann equation

Named after mathematicians Augustin Cauchy (1789-1857) and Bernhard Riemann (1826-1866).

(mathematics, complex analysis, always plural) Given a complex-valued function f and real-valued functions u and v such that f(z) = u(z) + iv(z), either of the equations (∂u)/(∂x)=(∂v)/(∂y) or (∂u)/(∂y)=-(∂v)/(∂x), which together form part of the criteria that f be complex-differentiable.

(complex analysis) The equivalent single equation (∂f)/(∂x)+i(∂f)/(∂y)=0.


pentatope number

pentatope number (plural pentatope numbers)

(mathematics) A number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left.


hemiperfect number

hemiperfect number (plural hemiperfect numbers)

(mathematics) A positive integer with a half-integral abundancy index


half-integral

(mathematics) Having a value that is half of an odd integer


divisor function

divisor function (plural divisor functions)

The value of the divisor function for 20 is 1 + 2 + 4 + 5 + 10 + 20 = 42, denoted as σ(20) = 42.

(mathematics) A function yielding the sum of the divisors of an integer. It is denoted by the Greek letter σ (sigma).


polydivisible

(mathematics) Describing an integer whose first digits are divisible by the numbers of digits concerned


Wronskian

From Wroński + -an. It was introduced in 1812 by the Polish mathematician Józef Hoene-Wroński.

differential equation
differential equation

(mathematics) A determinant that is related to the linear independence of a set of solutions of a linear differential equation.


epimorion

(obsolete, mathematics) The ratio equal to an integer divided by one more or less than that integer (e.g. 4/5 or 5/4)


Cauchy problem

After French mathematician Augustin Louis Cauchy.

In this chapter we formulate and in many cases prove results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations.

2006, Victor Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, page 41:

To emphasize the similarity between the characteristic method and the probabilistic approach we recall that the method of characteristics allows [one] to reduce the Cauchy problem for a first order PDE to a Cauchy problem for an ODE while the probabilistic approach allows [one] to reduce the Cauchy problem for a second order PDE to a Cauchy problem for an SDE (stochastic differential equation).

2006, S. Albeverio, Ya. Belopolskaya, “Probabilistic Interpretation of the VV-Method for PDE Systems”, in Olga S. Rozanova, editor, Analytical Approaches to Multidimensional Balance Laws, Nova Science Publishers, page 2:

In this monograph we discuss the C#92;infty well-posedness of the Cauchy problem for hyperbolic systems.

2014, Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v:

mathematical analysis
mathematical analysis

partial differential equation
partial differential equation

(mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary conditions that, for a smooth manifold S⊂ℝⁿ, u(x)=f_0(x) and (∂ᵏu(x))/(∂nᵏ)=f_k(x), ∀x∈S, k=1…m-1, given specified functions f_k defined on, and vector n normal to, the manifold.

problema di Cauchy

class of problem involving simultaneous PDEs


formal power series

formal power series (plural formal power series)

1995 Alberto Bertoni, Massimiliano Goldwurm, Giancarlo Mauri, Nicoletta Sabatini, Chapter 5: Counting Techniques for Inclusion, Equivalence, and Membership Problems, Volker Diekert, Grzegorz Rozenberg, The Book of Traces, World Scientific, page 138,
Moreover, it is useful to observe that in this case the definition of rational formal power series can be simplified: a f.p.s. r∈ℚ⟨!⟨z⟩!⟩ is rational if and only if r is the quotient of two polynomials.

1997, Greg Marks, Direct Product and Power Series Formations over 2-Primal Rings, Surender Kumar Jain, S. Tariq Rizvi, Advances in Ring Theory, Springer, page 239,
We also show that the ring of formal power series over a 2-primal ring (or even a ring satisfying (PS I)) need not be 2-primal.

This chapter gives a rigorous development of the algebraic properties of formal power series. Our goal is to extend the familiar operations on polynomial functions (like addition, multiplication, composition, and differentiation) to the setting of formal power series. In certain situations we will even be able to define infinite sums and products of formal power series.[…]In combinatorics, it usually suffices to consider formal power series whose coefficients are integers, rational numbers, or complex numbers.

2011, Nicholas Loehr, Bijective Combinatorics, Taylor & Francis (CRC Press), page 243:

(mathematics, algebra) Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x.


equioscillation

equioscillation (countable and uncountable, plural equioscillations)

(mathematics) An oscillation of equal to other oscillations amplitude or phase.


Bolzano-Weierstrass property

the Bolzano-Weierstrass property (canonical)

Named after Bernard Bolzano (1781–1848), a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian descent, and Karl Weierstrass (1815–1897), a German mathematician often cited as the "father of modern analysis".

(topology) The property held by some topological spaces that if a subset of such a space has an infinite quantity of points then the subset has at least one accumulation point.


Dobson unit

Named after Gordon Dobson, a researcher at the University of Oxford who in the 1920s built the first instrument to measure total ozone from the ground.

A unit of measurement of the amount of a trace gas (typically ozone) in a vertical column through the Earth's atmosphere.


Turán's theorem

Named after the Hungarian mathematician Pál Turán.

(graph theory) A theorem stating that, among the n-vertex simple graphs with no (r + 1)-cliques, T(n, r) has the maximum number of edges.


Hindman's theorem

(mathematics) A theorem stating that, if S, is an IP set and S=C_1∪C_2∪...∪C_n, then at least one C_i, contains an IP set.


IP set

Disputed. Perhaps an abbreviation of infinite-dimensional parallelepiped, or of idempotent.

(mathematics) A set of natural numbers that contains all finite sums of some infinite set.


microdegree

Map locations in the Android mapping classes are represented by GeoPoint objects, which contain latitude and longitude measured in microdegrees.

2010, Reto Meier, Professional Android 2 Application Development, page 264:

One millionth of a degree (unit of measurement).


Fermat's Last Theorem

Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665), who famously claimed to have a proof, although it was not successfully proven until 1994 by Andrew Wiles.

named-after

Another theorem, distinguished as Fermat's last Theorem, has obtained great celebrity on account of the numerous attempts that have been made to demonstrate it.

1872, William Thomas Brande, editor, A Dictionary of Science, Literature, & Art:

A lot has been written about Fermat's Last Theorem since its proof was announced in 1993.

2002, Peter Hilton, Derek Holton, Jean Pedersen, Mathematical Vistas: From a Room with Many Windows, Springer, page 23:

2002, Brendan Kelly, Algebra with the TI-83 Plus & TI-83 Plus SE, Brendan Kelly Publishing, page 36,
It appeared that Dr. Wiles had proved Fermat's Last Theorem, the most famous conjecture in Number Theory which had eluded the greatest mathematicians for over 350 years!

Then, almost casually, Dr. Wiles ended his lecture with these words: "And by the way, this means that Fermat's Last Theorem was true. Q.E.D."

2007, Eli Maor, The Pythagorean Theorem: A 4,000-year History, Princeton University Press, page 1:

Diophantine equation
Diophantine equation

(number theory) The theorem that the Diophantine equation aⁿ+bⁿ=cⁿ has no solutions for positive integers a,b,c,n, where n>2.

dernier théorème de Fermat

theorem that an + bn equal to cn has no positive integer solutions for a, b, c, n with n > 2


Fermat's little theorem

Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665), who stated a version of the theorem in a letter in 1640. Called little to distinguish it from Fermat's Last Theorem.

1999, John Stillwell, Translator's introduction, Peter Gustav Lejeune Dirichlet, Richard Dedekind (supplements), Lectures on Number Theory, [1863, P. G. Lejeune Dirichlet, R. Dedekind, Vorlesungen über Zahlentheorie], American Mathematical Society, page xi,
When combined with the historical remarks made by Gauss himself, they give a bird's eye view of number theory from approximately 1640 to 1840 - from Fermat's little theorem to L-functions - the period which produced the problems and ideas which are still at the center of the subject.

1999, Siguna Müller, On the Combined Fermat/Lucas Probable Prime Test, Michael Walker (editor), Cryptography and Coding: 7th IMA International Conference, Springer, LNCS 1746, page 222,
Most of the pseudoprimality tests originate in some sense on Fermat's Little Theorem aⁿ⁻¹ ≡ 1 mod n.

Incidentally, the special case of Fermat's little theorem for a = 2 was known to the Chinese as early as 500 B.C.
The first proof of Fermat's little theorem was given by Euler in 1736, almost a century after Fermat's announcement.

2007, Thomas Koshy, Elementary Number Theory with Applications, 2nd edition, Elsevier (Academic Press), page 327:

Fermat's little theorem says that if n is prime and gcd(a,n) = 1, then aⁿ⁻¹ ≡ 1 (mod n), so it follows that n must be composite, even though we have not produced a factor.

2008, Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, 2nd edition, Taylor & Francis (Chapman & Hall/CRC), page 189:

(number theory) The theorem that, for any prime number p and integer a, aᵖ-a is an integer multiple of p.


unmultiply

en-verb

unmultiply (third-person singular simple present unmultiplies, present participle unmultiplying, simple past and past participle unmultiplied)

unmultiplies (present singular third-person)
unmultiplying (participle present)
unmultiplied (participle past)
unmultiplied (past)

(mathematics) To reverse the process of multiplication; to find the factors of.


unode

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

(geometry) A double point of a surface whose tangent cone consists of one double plane.


binomial series

en-noun

binomial series (plural binomial series)

2010, James Stewart, Calculus: Concepts and Contexts, Cengage Learning, page 612,
Thus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1.

While applying his willful kind of calculus to Newton's binomial series, Euler not only constructed the exponential function, but found out that its inverse had already been in the minds of the 17th century and thus detected the very nature of logarithms.

2015, Hans-Heinrich Körle, Infinite Series in a History of Analysis: Stages up to the Verge of Summability, Walter de Gruyter, page 36:

(mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲α choose kxᵏ, where α choose k=(α(α-1)(α-2)…(α-k+1))/(k!).

Maclaurin series expansion of (1 + x)α

Maclaurin series, power series Taylor series

A particular useful series is termed the binomial series (or binomial theorem) and is frequently used to simplify engineering expressions. In this chapter we consider the arithmetic, geometric and binomial series.

2000, W. Bolton, Mathematics for Engineering, page 92:

The binomial series or binomial theorem is a formula for raising a binomial expression to any power without lengthy multiplication.

2017, John Bird, Higher Engineering Mathematics, page 60:

(mathematics, loosely) The binomial theorem.


Maclaurin series

Maclaurin series (plural Maclaurin series)

Named after Scottish mathematician Colin Maclaurin (1698-1746), who made extensive use of the series.

Analytically there are, of course, two Jacobi series branching off the Maclaurin series, but they are geometrically and physically identical, and involve only an interchange of a and b.

1953, Raymond Lyttleton, The Stability of Rotating Liquid Masses, Paperback edition, Cambridge University Press, published 2013, page 42:

1995, Ralph P. Boas, Gerald L. Alexanderson (editor), Dale H. Mugler (editor), Lion Hunting and Other Mathematical Pursuits, Mathematical Association of America, page 88,
If the Maclaurin series of f and g converge for |z| < r and g(z) ≠ 0 for 0 ≤ |z| < r, then if the Maclaurin series for f is divided by the Maclaurin series for g by long division (as if the series were polynomials), the resulting series represents f / g for |z| < r.

It was almost as a by-product of this work that, in the first Turin memoir, he proved the convergence of the Maclaurin series of a function up to the singularity nearest to the origin (Section 7.5); it was in this context that he created what he called 'calculus of limits', later known as the method of majorants.

1997, Frank Smithies, Cauchy and the Creation of Complex Function Theory, Cambridge University Press, page 203:

(calculus) Any Taylor series that is centred at 0 (i.e., for which the origin is the reference point used to derive the series from its associated function); for a given infinitely differentiable complex function f, the power series f(0)+(f'(0))/(1!)x+(f(0))/(2!)x²+(f'(0))/(3!)x³+⋯=∑ₙ₌₀ ᪲(f⁽ⁿ⁾(0))/(n!),xⁿ.


quadratic form

Quadratic form
Quadratic form (statistics)

Historically, quadratic form theory has been treated as a rich but misunderstood uncle.

1980, John Gregory, Quadratic Form Theory and Differential Equations, Academic Press, page ix:

2004, Nikita A. Karpenko, Izhboldin's Results on Stably Birational of Equivalence Quadrics, Oleg T. Izhboldin, Geometric Methods in the Algebraic Theory of Quadratic Forms, Springer, Lecture Notes in Mathematics 1835, page 156,
We claim that this quadratic form is isotropic, and this gives what we need according to [2, Proposition 4.4].

2009, R. Parimala, V. Suresh, J.-P. Tignol, On the Pfister Number of Quadratic Forms, Ricardo Baeza, Quadratic Forms—Algebra, Arithmetic, and Geometry, American Mathematical Society, page 327,
The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of −1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n − 2 terms and not less. The number of Pfister forms required to express a quadratic form of degree 6 with trivial discriminant is determined in various cases.

The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange's theorem that every positive integer is a sum of four squares, was continued in the 19th century by Dirichlet, Hermite, H. J. S. Smith, Minkowski and others.

2009, W. A. Coppel, Number Theory: An Introduction to Mathematics, 2nd edition, Springer, page 291:

homogeneous polynomial
homogeneous polynomial

(mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.

Quadratic forms of normal random vectors are of great importance in many branches of statistics, such as least-squares methods, the analysis of variance, regression analysis, and experimental design.

2009, Allan Gut, An Intermediate Course in Probability, 2nd edition, Springer, page 136:

(statistics, multivariate statistics) A scalar quantity of the form εᵀΛε, where ε is a vector of n random variables, and Λ is an n-dimensional symmetric matrix.


special linear group

special linear group (plural special linear groups)

The complex special linear group SL(n, C) is the subgroup of GL(n, C) consisting of matrices with determinant one.[…]The special linear groups are sometimes also called unimodular groups.

1972, Johan G. F. Belinfante, Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods, Society for Industrial and Applied Mathematics, page 6:

1998, F. Celler, C. R. Leedham-Green, A constructive recognition algorithm for the special linear group, Robert Curtis, Robert Wilson (editors), The Atlas of Finite Groups: Ten Years On, Cambridge University Press, 2003 Digitally Printed Edition, page 11,
In the first part of this note we present an algorithm to recognise constructively the special linear group.

Hence, all congruent bases induced from a game context belong to the special linear group SL(m).

2014, Holger Ingmar Meinhardt, The Pre-Kernel as a Tractable Solution for Cooperative Games: An Exercise in Algorithmic Game Theory, Springer, page 100:

(group theory) For given field F and order n, the group of n×n matrices with determinant 1, with the group operations of matrix multiplication and matrix inversion.


general linear group

general linear group (plural general linear groups)

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, Springer, 2000, Softcover Reprint, page 17,
Often Lie groups arise as subgroups of certain larger Lie groups; for example, the orthogonal groups are subgroups of the general linear groups of all invertible matrices.

We will again call this group the general linear group and denote it by GL₃.
In just the same way, the set of all nonsingular linear transformations of the plane L₂ is a group denoted by GL₂ and called the general linear group of order two.

2003, Maks Aizikovich Akivis, translated by Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 119:

We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups.

2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 1:

(group theory) For given field F and order n, the group of invertible n×n matrices, with the group operation of matrix multiplication.


logarithmic spiral

logarithmic spiral (plural logarithmic spirals)

(geometry) A self-similar spiral curve which often appears in nature.


golden spiral

(geometry) A logarithmic spiral whose growth factor is φ, the golden ratio.


HMAC

hash-based message authentication code
w:HMAC

(cryptography) Initialism of hash-based message authentication code (a specific type of the message authentication code (MAC) involving a cryptographic hash function and a secret cryptographic key)


matrix group

The properties of matrix groups are of great importance for representation theory.

1976, S. Califano, Vibrational States, Wiley, page 112:

2011, László Babai, Finite Groups and Complexity Theory: From Leningrad to St Petersburg via Las Vegas, Alexander Kulikov, Nikolay Vereshchagin (editors), Computer Science – Theory and Applications: 6th International Computer Symposium, Springer, Lecture Notes in Computer Science 6651, page 162,
This paper is a personal account of the author's journey through the evolution of some of these interconnections, culminating in recent definitive results on the matrix group membership problem.

2014, Peter Sarnak, Notes on thin matrix groups, Emmanuel Breuillard, Hee Oh (editors), Thin Groups and Superstrong Approximation, Cambridge University Press, page 343,
Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given.

(group theory) Any group of invertible matrices over a specified field, with the group operation of matrix multiplication.


linear group

In this section we shall describe the linear groups and in the next two sections we deal with the symplectic and orthogonal groups, collectively usually known under the name of classical groups since the time of Hermann Weyl (1939).

2003, Paul M. Cohn, Further Algebra and Applications, Springer, page 116:

2007, Peter Schmid, The Solution of the k(GV) Problem, Imperial College Press, page vii,
So finite groups often appear as subgroups of permutation groups or linear groups. The “geometry” of a group (as permutation group or linear group, or as a group of Lie type etc.) should be used in order to describe basic invariants.

The standard examples are linear groups such as the orthogonal and unitary groups, or the additive groups of p-adic integers, and this confirms that the concept of a compact group is natural.

2007, Karl Heinrich Hofmann, Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups, European Mathematical Society, page 1:

(group theory) Any group that is isomorphic to a matrix group.


position vector

position vector (plural position vectors)

Coordinate term: direction vector

(geometry) Of a point, a vector going from the origin to that point; equivalently, the transpose of the point.


unitary group

The symmetric group is closely related to the unitary group.[…]The latest developments in spin free quantum chemistry have been the use of the unitary group for the calculation of the matrix elements of the Hamiltonian in the configuration interaction treatment.

1995, Ruben Pauncz, The Symmetric Group in Quantum Chemistry, CRC Press, page 195:

Two of the most important types of groups that occur in physics are unitary groups and orthogonal groups.

2004, [1999], Shoon K. Kim, Group Theoretical Methods and Applications to Molecules and Crystals, Cambridge University Press, page 53:

(group theory) For given n, the group of n×n unitary matrices, with the group operation of matrix multiplication.


conference matrix

conference matrix (plural conference matrices)

Named by Vitold Belevitch, from their relevance to a problem in telephony.

(mathematics) A square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CᵀC is a multiple of the identity matrix I.


epogdoon

Borrowed from Ancient Greek ἐπόγδοον (epógdoon), from ἐπι- (epi-, “greater [by]”) + ὄγδοον (ógdoon, “an eighth part”).

Eighteen... bears the epogdoon relation to sixteen, which is broken up by the intervention of seventeen, an odd number.

1936, Frank Cole Babbitt translating Plutarch as "Isis and Osiris" in Moralia, Vol. V, p. 103, n.

We cannot say that 256 is the epogdoon of 243, for in the number 256 is not contained the whole of 243 and an eighth thereof, since an eighth of 243 is approximately 30½...

2016, John Magee translating Calcidius as On Plato's Timaeus, p. 201

(mathematics, Ancient Greece, rare) A number equal to another plus an eighth thereof; a number 1.125 times another.


beta distribution

beta distribution (plural beta distributions)

In particular, I illustrate how a hierarchical model in which the allele frequency distribution among populations is approximated by a Beta distribution and distribution of mean allele frequencies across loci is also approximated by a Beta distribution, accounts for the time-correlation of allele frequencies that existing methods of inference ignore.

2006, Kent E. Holsinger, “Chapter 2: Bayesian hierarchical models in geographical genetics”, in James S. Clark, Alan E. Gelfand, editors, Hierarchical Modelling for the Environmental Sciences: Statistical Methods and Applications, Oxford University Press, page 25:

2014, Stephen F. Bush, Smart Grid: Communication-Enabled Intelligence for the Electric Power Grid, Wiley, 2015, Reprint with corrections, page 366,
When more than two mutually exclusive events are involved in a frame of reference, the beta distributions form the marginals of a Dirichlet distribution.

2014, Nikolay Tcholtchev, Ina Schieferdecker, Framework for Ensuring Runtime Stability of Control Loops in Multi-agent Networked Environments, Marina Gavrilova, C. J. Kenneth Tan (editors), Transactions on Computational Science XXII, Springer, LNCS 8360, page 80,
We developed a Matlab script that implements the machine learning procedures as specified by (11), as well as by (13) with applying the beta-distribution (14) as a probability measure.

(probability, statistics) Any of a family of continuous probability distributions defined on the interval [0, 1] whose shape is parametrised by two positive parameters, denoted α and β, which appear as exponents in the associated random variable.


symplectic matrix

symplectic matrix (plural symplectic matrices)

Then we define the Krein signature of the eigenvalues of modulus one of a symplectic matrix and use this concept to build the rotation function on the symplectic group.

2001, Alberto Abbondandolo, Morse Theory for Hamiltonian Systems, CRC Press (Chapman & Hall/CRC), page x:

In the previous chapter algorithms for computing the eigenvalues of symplectic matrices have been considered that are based on an elimination process for computing the butterfly form of symplectic matrix. Unfortunately, this approach is not suitable when dealing with large and sparse symplectic matrices as an elimination process can not make full use of the sparsity.

2002, Heike Fassbender, Symplectic Methods for the Symplectic Eigenproblem, Kluwer Academic, page 173:

In order to fully exploit these structures we rely on the properties of Hamiltonian, skew-Hamiltonian, and symplectic matrices discussed in Section 1.5. In Section 4.1 we introduce the elementary symplectic matrices which are used by the structured algorithms.

2012, Dario A. Bini, Bruno Iannazzo, Beatrice Meini, Numerical Solution of Algebraic Riccati Equations, Society for Industrial and Applied Mathematics, page 121:

(linear algebra) For given field F (especially the real numbers), even order 2n and nonsingular skew-symmetric matrix Ω, any 2n×2n matrix M with elements in F such that MᵀΩM = Ω (where Mᵀ denotes the transpose of M).


Jacobi identity

Jacobi identity (countable and uncountable, plural Jacobi identities)

After German mathematician Carl Gustav Jakob Jacobi (1804-1851).

We give here two proofs of the Jacobi identity for the generalized Poisson bracket defined in Eq. (13.69a).

1995, Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, page 535:

2004, ames Lepowsky, Haisheng Li, Introduction to Vertex Operator Algebras and Their Representations, Springer (Birkhäuser), page 12,
As we have already mentioned, the Jacobi identity is actually the generating function of an infinite list of generally highly nontrivial identities, and one needs many of these individual componenent identities in working with the theory.

It is sufficient to test the Jacobi identity against three linear functions [354] (this reference also provides a code for evaluating Jacobi identities).

2005, Martin Kröger, Models for Polymeric and Anisotropic Liquids, Springer, page 113:

(mathematics) Given a binary operation × defined on a set S which also has additive operation + and additive identity 0, the property that a × (b×c) + b × (c×a) + c × (a×b) = 0 for all a, b, c in S.

Carl Gustav Jakob Jacobi
Jacobi identity


hugantic

en-adj

hugantic (comparative more hugantic, superlative most hugantic)

more hugantic (comparative)
most hugantic (superlative)


condensation point

en-noun

condensation point (plural condensation points)

(topology) (said to be "of" a given subset of a topological space) a point whose every neighborhood contains uncountably many points belonging to the given subset


Lindelöf covering theorem

Named after Ernst Leonard Lindelöf (1870–1946), a Finnish mathematician.

(topology) A theorem which states that for any open covering of some given set in Euclidean n-space there is an open subcovering (of that same set) which is countable.


grossone

(mathematics) A proposed form of integer notation that would be able to handle arithmetic of infinities


tangent space

This gives a degenerate norm on the tangent space of Hₙ at the origin, and one can use left translations to define similar norms at the tangent spaces of Hₙ at all other points.

2001, Stephen Semmes, Some Novel Types of Fractal Geometry, Oxford University Press, page 124:

In a Lie group G, because left translation by an element g #92;in G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance.
Moreover, one can give the tangent space TₑG a Lie bracket [ , ], so that in addition to being a vector space, it becomes a Lie algebra, called the Lie algebra of the Lie group.

2011, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer, page 178:

More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit!

2014, Peter K. Friz, Martin Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, page 7:

(topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x.


topological field

topological field (plural topological fields)

(mathematics) A field that is also a topological space in which addition and multiplication are both continuous.


topological vector space

topological vector space (plural topological vector spaces)

(functional analysis) A vector space over a topological field endowed with a topology (often the real or complex numbers with standard topology) such that vector addition and scalar multiplication are continuous functions.

linear topological space

vector space with topology such that addition and scalar multiplication are continuous


function value

(mathematics) An element of the range of a function. (Or, employing the "machine" metaphor, it is one of the "outputs" of a function.)


quadratic residue

quadratic residue (plural quadratic residues)

quadratic nonresidue

antonym(s) of “integer expressible as a square modulo n”

There are many small tables of quadratic residues giving for the first few primes p the positive quadratic residues of p arranged in order of their size.

1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 52:

1991, John Stillwell (translator), Peter Gustav Lejeune Dirichlet, Lectures on Number Theory, [1863, Vorlesungen über Zahlentheorie], American Mathematical Society, London Mathematical Society, page 82,
If we now make the assumption that q is a quadratic residue of all odd primes z not greater than 2m + 1, then it follows from earlier theorems (§37) that the prime q, since it is ≡ 1 (mod 8) and hence a quadratic residue of each power of 2, is also a quadratic residue of each number which has no odd prime factors except the prime numbers z

1999, Dinakar Ramakrishnan, Robert J. Valenza, Tsinghua University Press [清华大学出版社有限公司], Fourier Analysis on Number Fields, page 213,
Of special importance here is the quadratic reciprocity law, which for primes p and q gives a precise relationship between the status of p as a quadratic residue mod q and the status of q as a quadratic residue mod p.

(number theory, modular arithmetic) For given positive integer n, any integer that is congruent to some square m² modulo n.

kvadratisk rest

integer that is congruent to the square of some other number, to a given modulus


quadricone

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


modular arithmetic

modular arithmetic (countable and uncountable, plural modular arithmetics)

The reader now has examined, in some detail, several specific modular arithmetics, namely, #92;mathbb#123;Z#125;#95;2,#92;mathbb#123;Z#125;#95;5,#92;mathbb#123;Z#125;#95;6,#92;mathbb#123;Z#125;#95;7,#92;mathbb#123;Z#125;#95;8 and #92;mathbb#123;Z#125;#95;9.

1969, Joseph Landin, An Introduction to Algebraic Structures, Dover, page 153:

In this section we use two historical ciphers to introduce modular arithmetic with integers. Even though the historical ciphers are no longer relevant, modular arithmetic is extremely important in modern cryptography, especially for asymmetric algorithms.

2010, Christof Paar, Jan Pelzl, Understanding Cryptography: A Textbook for Students and Practitioners, Springer, page 13:

Hence for prime moduli, modular arithmetic is very similar to regular rational arithmetic with all four operations defined.

1997, Robert E. Jamison, “Rhythm and Pattern: Discrete Mathematics with an Artistic Connection for Elementary School Teachers”, in Joseph G. Rosenstein, Deborah S. Franzblau, Fred S. Roberts, editors, Discrete Mathematics in the Schools, American Mathematical Society, page 215:

(number theory) Any system of arithmetic for integers which, for some given positive integer n, is equivalent to the set of integers being mapped onto the finite set {0, ... n} according to congruence modulo n, and in which addition and multiplication are defined consistently with the results of ordinary arithmetic being so mapped.


quotient map

If the square #91;0,1#93;#92;times#91;0,1#93; (with the subspace topology) in #92;mathbb#123;R#125;² (with the standard topology) is mapped onto the set #91;0,1)#92;times#91;0,1) with the quotient map f(x,y)#61;(#92;lfloorx#92;rfloor,#92;lfloory#92;rfloor), then the target space (with its induced quotient topology) is topologically equivalent to a torus.

(topology) A surjective, continuous function from one topological space to another one, such that the latter one's topology has the property that if the inverse image (under the said function) of some subset of it is open in the function's domain, then the subset is open in the target space.


identification map

identification map (plural identification maps)


block graph

(graph theory) A type of undirected graph in which every biconnected component (block) is a clique.

(mathematics, education) A simple two-dimensional bar chart that shows amounts or quantities as blocks.


Floyd-Warshall algorithm

the Floyd-Warshall algorithm (canonical)

Named after Robert W. Floyd and Stephen Warshall.

(computer science, graph theory) An algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles).


dot and carry

en-verb

dot and carry (third-person singular simple present dots and carries, present participle dotting and carrying, simple past and past participle dotted and carried)

dots and carries (present singular third-person)
dotting and carrying (participle present)
dotted and carried (participle past)
dotted and carried (past)

(arithmetic) In addition, to set down the units and carry over the tens to the next column.


orthogonal group

en-noun

orthogonal group (plural orthogonal groups)

The symbol O^ε(n,q) for orthogonal groups has been well established in finite group theory as and, throughout the mathematics community, O(n, K) stands for an orthogonal group when K is the real or complex field.

1998, Robert L. Griess, Jr., Twelve Sporadic Groups, Springer, page 4:

Theorem 7.4. Let n ≥ 1. For odd primes p#92;not#92;equiv#92;pm 1#92;pmod#123;24#125; the odd-dimensional orthogonal groups O#95;#123;2n#43;1#125;(p) possess GA-realizations over #92;mathbb#123;Q#125;.

1999, Gunter Malle, B.H. Matzat, Inverse Galois Theory, Springer, page 146:

The normalizer of the full orthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.

2007, Marcelo Epstein, Marek Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Springer, page 106:

(group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication.


Gauss map

Named after German mathematician Carl Friedrich Gauss.

1969 [Van Nostrand], Robert Osserman, A Survey of Minimal Surfaces, 2014, Dover, Unabridged republication, page 73,
There exist complete generalized minimal surfaces, not lying in a plane, whose Gauss map lies in an arbitrarily small neighborhood on the sphere.

1985, R. G. Burns (translator), B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry― Methods and Applications: Part II: The Geometry and Topology of Manifolds, Springer, Graduate Texts in Mathematics, page 114,
14.2.2 Theorem The integral of the Gaussian curvature over a closed hypersurface in Euclidean n-space is equal to the degree of the Gauss map of the surface, multiplied by γₙ (the Euclidean volume of the unit (n-1)-sphere).

In §2 we use results of §1 for the study of Gauss maps of projective varieties. The classical Gauss map associates to each point of a nonsingular real affine hypersurface the unit vector of the external normal at this point.

2005, F. L. Zak, Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 5:

differential geometry
differential geometry

(geometry, differential geometry) A map from a given oriented surface in Euclidean space to the unit sphere which maps each point on the surface to a unit vector orthogonal to the surface at that point.


turning number

This program illustrates the Whitney-Graustein theorem, that two closed curves (immersions of the circle) in the plane can be deformed into one another without letting go of the immersion property if and only if their turning numbers are the same.

1989, Wayne Sewell, Weaving a Program: Literate Programming in WEB, page 516:

Moreover, l gives the turning number of the k-gon one sees in the horizontal symmetry plane when the axes of the ends are deleted.[…]Moreover, every even k ≥ 30 except for 32, 36, 40, 44, 48 52 an 56 gives an example, while for every even k > 102 there is more than one example since different turning numbers are possible.

1998, Karsten Groβe-Brauckmann, Robert B. Kusner, John M. Sullivan, “Constant Mean Curvature Surfaces with Cylindrical Ends”, in Hans-Christian Hege, Konrad Polthier, editors, Mathematical Visualization: Algorithms, Applications and Numerics, Springer, page 114:

2002, Journal of Physics: Mathematical and general, page 6187,
For any oriented curve β(τ),τ∈I,I=[0,Λ] of class C¹ (τ is the arclength parametrization) lying in an oriented Euclidean plane E, the turning number 𝒯_(n_β) may be defined as
𝒯_(n_β)=1/(2π)∫₀^Λκ_β(τ)dτ
where κ_β(τ) is the signed curvature of β.

(geometry, topology, mathematical analysis) A version of winding number in which the number of rotations is counted with respect to the tangent of the curve rather than a fixed point.


directed set

(algebra, set theory) A set equipped with a preorder such that for any two elements a and b there is an element c such that a ≤ c and b ≤ c.


Lipschitz condition

Lipschitz condition (plural Lipschitz conditions)

Named after Rudolf Lipschitz (1832–1903), a German mathematician. It is called a "condition" because it is a sufficient (but not necessary) condition for continuity of a function.

(mathematical analysis) A property which can be said to be held by some point in the domain of a real-valued function if there exists a neighborhood of that point and a certain constant such that for any other point in that neighborhood, the absolute value of the difference of their function values is less than the product of the constant and the absolute value of the difference between the two points.


Tauberian

From Tauber + -ian, specifically referring to Austrian and Slovak mathematician Alfred Tauber (1866-1942).

In the third chapter we obtain characterizations of Tauberian convergence classes in the limiting case and representations of Fourier and Fourier-Stieltjes coefficients.

1992, Dissertation Abstracts International, page 326:

Lorentz has developed a general method for obtaining Tauberian conditions for absolute summability.

2003, B. N. Prasad, “1966, Chandigarh - B. N. Prasad”, in The Shaping of Indian Science, Indian Science Congress Association Presidential Addresses, Volume 2: 1948-1981, Universities Press, page 916:

In this section we use ideas from the proof of Proposition 4.3 to obtain a convergence theorem involving more general Tauberian conditions.

2004, Jacob Korevaar, Tauberian Theory: A Century of Developments, Springer, page 142:

(mathematical analysis) Being or relating to Tauberian theorems, a class of theorems that are partial converses to Abelian theorems.

Abelian and tauberian theorems
Alfred Tauber


Hopf link

From Hopf + link. From being an interlinkage named after Heinz Hopf.

(topology) The simplest interlinkage of two loops; two interlinked rings.


Whitehead link

(topology) One of the most basic interlinkages of two loops; a link with one ring passing through the two lobes of a singly twisted loop.


Tauberian theorem

Tauberian theorem (plural Tauberian theorems)

After Austrian and Slovak mathematician Alfred Tauber (1866-1942).

Paragraph five deals with (A*)-algebras and contains a proof of a vector-valued variant of Wiener's Tauberian theorem.

1957, Einar Hille, Ralph Saul Phillips, Functional Analysis and Semi-groups, Part 1, American Mathematical Society, page 117:

1988, Staff writer, Foreword, [1933, Norbert Wiener, The Fourier Integral and Certain of Its Applications], Cambridge University Press, 1988 reissue, page xi,
Not only did the general Tauberian theorem give a unifying view on questions involving summations and limits, but it introduced a paradigm for what was called abstract harmonic analysis a few years later. […] Generalized harmonic analysis is the subject-matter of the last chapter, though it was conceived before the Tauberian theorems.

We should emphasize that our main concern is — besides the presentation of Tauberian theorems in the case of special summability methods — to put, by way of examples, different methods in the hands of the reader to prove Tauberian theorems in the case of special summability methods.

2000, Johann Boos, F. Peter Cass, Classical and Modern Methods in Summability, Oxford University Press, page 167:

(mathematical analysis) Any of a class of theorems which, for a given Abelian theorem, specifies conditions such that any series whose Abel sums converge (as stipulated by the Abelian theorem) is in fact convergent.

Abelian and tauberian theorems
Alfred Tauber
G. H. Hardy


Abel sum

After Norwegian mathematician Niels Henrik Abel (1802-1829).

lim _(x→1-0)∑ₙ₌₀ ᪲a_nxⁿ.
The existence of the Abel sum is ascertained when the series in question is known to be summable (C, r) for some value of r.

1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102, The Abel sum of ∑a_n is defined as the limit of the corresponding power series

Jacobi in his Vorlesungen über Dynamik [1884] had used Abel sums to separate variables in the Hamilton-Jacobi equation in connection with the geodesic flow on the surface of a 3-dimensional ellipsoid, etc.

2005, Bulletin of the American Mathematical Society, page 81:

Also, Abel's theorem guarantees that the Abel sum of a convergent series exists and is equal to the ordinary sum.

2012, Peter L. Duren, Invitation to Classical Analysis, American Mathematical Society, page 180:

Abel summation method
Abel summation method

(mathematical analysis) Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method).


totitive

(mathematics) Alternative form of totative


decategorification

For the n-levels systems approach we may associate categorification with increasing the level and decategorification with decreasing the level of complexity.

2013, Octavian Iordache, Polytope Projects, →ISBN, page 6:

(mathematics) The reverse process of categorification: The mapping of a category theoretic category to a set by using a systematic process in which isomorphic objects in a category are identified as equal.


RNN

random neural network
random neural network#English

recurrent neural network
recurrent neural network#English

recursive neural network
recursive neural network#English


recurrent neural network

recurrent neural network (plural recurrent neural networks)

(machine learning) An artificial neural network where the connections between units give them a kind of internal memory useful for processing arbitrary sequences of inputs as in speech recognition.


Jordan curve theorem

There is a vast generalization of the Jordan curve theorem to higher dimensions.

1995, William Fulton, Algebraic Topology: A First Course, Springer, page 343:

The Jordan curve theorem asserts that a simple closed curve in the complex plane divides the plane into exactly two connected components, one bounded (the "inside") and the other unbounded (the "outside" ).

2001, Theodore Gamelin, Complex Analysis, Springer, page 249:

The principal reason that orbit behavior in the plane is more limited than in higher dimensions is the Jordan curve theorem. Local sections will be the primary tool used to exploit the Jordan curve theorem.

2004, Nelson G. Markley, Principles of Differential Equations, Wiley, page 193:

(topology) The theorem that states that a simple closed curve (Jordan curve) divides the plane into precisely two distinct areas.


field of quotients

field of quotients (plural fields of quotients)

(algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b)+(a',b')=(ab'+a'b,bb'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b)≡(a',b') if and only if ab'=a'b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).


convex envelope

convex envelope (plural convex envelopes)

1965 [Holt Rinehart & Winston], Robert E. Edwards, Functional Analysis: Theory and Applications, Dover, 1995, Unabridged Corrected Edition, page 561,
In E the closed convex envelope of a compact (resp. weakly compact) set is τ(E,E')-complete.

1987, H. G. Eggleston, S. Madan (translators), Nicolas Bourbaki, Topological Vector Spaces: Chapters 1–5, [1981, N. Bourbaki, Espaces Vectoriels Topologiques], Springer, page IR-10,
Corollary 1. — The convex envelope of a subset A of E is identical with the set of linear combinations ∑ᵢλᵢx_i, where (x_i) is any finite family of points in A, the numbers λᵢ>0 for all i and ∑ᵢλᵢ=1.

The polytopes are, by definition, the convex envelopes of finite sets of points of an affine space.

2010, Marcel Berger, Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry, Springer, page 505:

(mathematical analysis, of a set) Convex hull.

One of the main reasons for the interest in convex envelopes is the fact that the set of global minimum points of f on S is contained in the set of global minimum points of conv_S(f) on S and the two minimum values coincide (see, e.g., [11, 17]). Hence, if the convex envelope were efficiently computable or available in closed form, one could replace the nonconvex problem of minimizing f on S with the convex problem of minimizing f on conv_S(f).

2004, Fabio Tardella, “On the existence of polyhedral convex envelopes”, in Christodoulos A. Floudas, Panos M. Pardalos, editors, Frontiers in Global Optimization, Springer (Kluwer Academic), page 564:

Tawarmalani and Sahinidis (2001) developed the convex envelope and concave envelope for x/y over a unit hypercube, compared it to the convex relaxation proposed by Zamora and Grosmmann (1998a), (1998b), (1999), proposed a semidefinite relaxation of x/y, and suggested convex envelopes for functions of the form f(x)y² and f(x)/y.

2004, C. A. Floudas, I. G. Akrotiriankis, S. Caratzoulas, C. A. Meyer, J. Kallrath, “Global Optimization in the 21st Century: Advances and Challenges”, in Ana Paula Barbosa-Póvoa, Henrique Matos, editors, European Symposium on Computer Aided Process Engineering-14: 37th European Symposium of the Working Party, Elsevier, page 25:

The concept of convex envelopes of nonconvex functions is a basic tool in theory and algorithms of global optimization, see e.g., Falk and Hoffman (1976), Horst and Tuy (1996), Horst et al. (2000).

2005, Nguyen Van Thoai, “4: General Quadratic Programing”, in Charles Audet, Pierre Hansen, Giles Savard, editors, Essays and Surveys in Global Optimization, Springer, page 120:

Comparing the use of convex envelopes for bilinear and trilinear forms in building convex approximations for MINLPs motivated the study in [6], and comparisons involving more general functional forms motivate the present article.

2012, Petro Bilotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew J. Miller, On the Composition of Convex Envelopes for Quadrilinear Terms:

(mathematics, optimisation theory, of a function on a set) For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the highest-valued convex function that underestimates or equals f over S.


concave envelope

concave envelope (plural concave envelopes)

No immediate use of Theorem 4 can be made computationally since, with the exception of a few special cases (e.g. rectangular S and separable f) it is very hard to construct concave envelopes and convex hulls.

1988, Ferenc Forgó, Nonconvex Programming, page 33:

Explicit expressions defining the facets of the convex and concave envelopes for trilinear monomials, with positive or negative bounded domains for each variable, are derived in this paper.

2004, C. A. Meyer, C. A. Floudas, “Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes”, in Christodoulos A. Floudas, Panos M. Pardalos, editors, Frontiers in Global Optimization, Springer,, page 327:

The cost of this order plan is linear in price, and the associated line must form a segment of the piecewise linear concave envelope.

2012, Joseph Geunes, Demand Flexibility in Supply Chain Planning, Springer (Kluwer Academic), page 20:

(mathematics, optimisation theory, of a function on a set) For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S.


spherical cone

(geometry) A cone with a spherical base.


octodecimal

Octodecimal 1249 is equivalent to decimal 6561 or nonary 10000.

The decimal scheme should have been made a duodecimal, or better a quinquedecimal or octodecimal one; the expansive system would need a more than India-rubber elasticity to include all the whimsical creations of an author's brain.

By studying other number system such as unodecimal (base-11), duodecimal (base-12), tridecimal (base-13), quadrodecimal (base-14), pentadecimal (base-15), heptadecimal (base-17), octodecimal (base-18), nonadecimal (base-19) and vigesimal (base-20), researcher will gain a better understanding of how number systems work in general.

2012 January, Debasis Das, U.A. Lanjewar, “Realistic Approach of Strange Number System from Unodecimal to Vigesimal”, in International Journal of Computer Science and Telecommunications, volume 3, number 1:

Traditional number system — binary, octal, decimal and hexadecimal, in regular use and Strange number system, with an extra edge of memorized information with greater density, zero redundancy problem, avoiding sign problem and reducing complexity of interconnections, - unodecimal, duodecimal, tridecimal, quadrodecimal, pentadecimal, heptadecimal, octodecimal, nona decimal, vigesimal and further are discussed through compliments.

2013 May, “Subtraction in Traditional and Strange Number System by r's and r-1's Compliments”, in International Journal of Computer Applications, volume 70, number 23:

(arithmetic) Using eighteen as the base.


xennial

I was born in 1980. According to some sources, this makes me a Gen Xer. According to others, I’m a Millennial. That makes me what then, a Xennial? I take online quizzes, like Pew Research Center’s “How Millennial Are You?”, and land dead between Gen X and Millennial due to my personal habits, body piercings, and so many more reasons.

2014 September 25, Sarah Stankorb, “Glad to be a Xennial”, in GOOD:

Xennial is a term that I first used in a 2014 story for GOOD magazine, following on the heels of writers like Doree Shafrir, who in 2011 called this cusp generation’s pop-cultural misfit status Generation Catalano, after crowdsourcing the My So-Called Life–inspired term from Danielle Nussbaum on Twitter.

2017 July 6, Sarah Stankorb, “I Made Up Xennial 3 Years Ago, So Why Is a Professor in Australia Getting All the Credit?”, in Vogue:

But there's this new group, you may know them — that microgeneration sandwiched between — who feel like they don't really belong to either group […] Love it or hate it, there's a newish name for these lost souls: the "xennials" (pronounced ZEE-knee-als).

2017 November 10, Dwight Adams, “You're a xennial — if you're caught between Generation X and millennials”, in Indianapolis Star:

I’m a reluctant millennial myself (or possibly — God forbid — an xennial), stuck in the pretty common category of not identifying with my so-called generation.

2023 April 20, Casey Schwartz, “Jean Twenge is ready to make you defend your generation again”, in The Washington Post:

Oregon Trail Generation
Oregon Trail Generation

(informal) A person born late in Generation X or early in Generation Y, that is, sometime in the late 1970s or early 1980s; a member of the Oregon Trail Generation.


cunca

From Galician cunca (“bowl, ¹⁄₆₀ Galician bushel”), from Old Galician-Portuguese cunca, from Latin concha (“mussel, vessel”), from Ancient Greek κόγχη (kónkhē, “shell, mussel”).

bor

bowl, <small>¹<big>⁄</big>₆₀</small> Galician bushel

der

(historical) A traditional Galician unit of dry measure, equivalent to about 1–1.5 L depending on the substance measured.


fundamental theorem

en-noun

fundamental theorem (plural fundamental theorems)

In this and the next chapter I return to the fundamental theorems of welfare economics. In this chapter I argue that the first fundamental theorem of welfare economics—asserting the efficiency of competitive economics—is fundamentally flawed.

1994, Joseph E. Stiglitz, Whither Socialism?, The MIT Press, page 27:

The first fundamental theorem of calculus states that the differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is obtained[…].

2010, Vasily E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, page 247:

There are two fundamental theorems of calculus. Both relate differentiation to integration. In most calculus courses, the Second Fundamental Theorem is usually proved first and then used to prove the First Fundamental Theorem. Unfortunately, this results in a First Fundamental Theorem that is weaker than it could be.

2012, Joseph L. Taylor, Foundations of Analysis, American Mathematical Society, page 114:

(chiefly mathematics) A theorem (or, in non-mathematical fields, a commonly accepted hypothesis) considered to be of central importance to a specified field.


polynomial form

polynomial form (plural polynomial forms)

(algebra) A linear combination of powers of an indeterminate (or products of powers of more than one indeterminate), with coefficients belonging to an integral domain or a field. (The indeterminate is thought of as an element extraneous to the set of coefficients, instead of as a variable element of it (as in the case of polynomial functions), just as, say, the square root of negative one is an element extraneous to the set of integers when it is adjoined to them to form the domain of Gaussian integers. The indeterminate forms a free commutative monoid, to which all powers of it belong, and the unity of it can also show up implicitly in the constant term of a polynomial form.)


axoid

(anatomy, mathematics) Relating to an axis


axoid

(mathematics) A helix surrounding an axis


transflection

transflection (countable and uncountable, plural transflections)

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Clearly, A", B", C" are not images of A, B, C, under either a translation, rotation, or reflection. Hopefully, it will be possible to find a unique transflection which does the job.

1968, Myron Frederick Rosskopf, Joan L. Levine, Bruce Ramon Vogeli, Geometry; a Perspective View, page 236:

During the sequence of considering translations, rotations, reflections, and transflections of a plane α, we show that ...

1972, Merlyn J. Behr, Dale G. Jungst, Fundamentals of elementary mathematics; geometry, volume 2, page 261:

Visualize these operations by considering the simple screw and transflection. Transflection can be imagined as seeing yourself in a mirror even as the mirror is being moved parallel to itself.

2015, C. S. Sunandana, Introduction to Solid State Ionics: Phenomenology and Applications, →ISBN:

(geometry) A transfiguration consisting of reflection about a line combined with translation along the same line.


boundary condition

boundary condition (plural boundary conditions)

(mathematics) Any of a set of constraints that limit the solutions of a differential equation


well-behavedness

(mathematics) The quality or condition of being well-behaved.


Bernstein basis polynomial

Bernstein basis polynomial (plural Bernstein basis polynomials)

Named after Sergei Natanovich Bernstein (1880–1968), a Russian and Soviet mathematician of Jewish origin.

(mathematical analysis, numerical analysis) Any polynomial of the form b_i,n(t)=n choose itⁱ(1-t)ⁿ⁻ⁱ!!, being the iᵗʰ term of the binomial expansion of (t+(1-t))ⁿ!.


Lagrange's interpolation formula

Lagrange's interpolation formula (uncountable)

Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.

(mathematics) A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n.

Joseph Louis Lagrange
Lagrange polynomial


HAIFA

(cryptography) Abbreviation of hash iterative framework: a cryptographic structure used in the design of hash functions.


trapezian

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(geometry) Having opposed trapeziform faces.


symmedian

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(geometry) Any of three lines associated with every triangle, constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side) and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half).


regular black hole

regular black hole (plural regular black holes)

From regular (“continuous”) + black hole (“region enclosed by an event horizon”). From having a spacetime formulation that is without discontinuities, that is, regular.

compound

(mathematics, physics) A mathematical formulation for a black hole which does not contain a singularity. The environment outside the event horizon is the same as other formulations for black holes.


Lp space

(mathematics) Function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.


hodl

en-verb

hodl (third-person singular simple present hodls, present participle hodling, simple past and past participle hodled)

hodls (present singular third-person)
hodling (participle present)
hodled (participle past)
hodled (past)

Originally a mistyping of hold, appearing in a forum post on December 18, 2013. The backronym "Hold On for Dear Life" is sometimes used.

Bitcoin price continues decline. If you're hodling, bite your government passbook.

2016 July 22, Venzen Khaosan, “Bitcoin Price Decline 100% Confirmed”, in Cryptocoins News:

I believe that the individuals who have the fortitude and insight to be hodling onto their coins deserve to be rewarded for their risk. It is also my hope that I am hodling more coins if/when the price skyrockets.

2017 April 22, “Bitcoin Is Not Immune to All Fiat Problems”, in The Merkle:

Cryptocurrency veterans love to “hodl,” aka store their virtual loot as a long-term investment. But digital currency wasn't only made for hodling.

2017 November 10, Leigh Cuen, “Cyber Monday Guide: How To Buy Christmas Presents With Bitcoin”, in International Business Times:

Buy and hold strategies for Bitcoin ("hodling" as it called) is likely to have far lower returns, even if the future is positive, and traders will lose the volatility that has made Bitcoin trading so profitable.

2017 November 26, Lynn Sebastian Purcell, “Why Futures Spell The End Of Bitcoin Mania”, in Toronto Star:

But even those with the stomach for hodling must avoid an even bigger issue — hacking.

2021 April 25, Erin Griffith, “We’re All Crypto People Now”, in The New York Times, →ISSN:

(neologism) To hoard a kind of cryptocurrency, especially bitcoin.


automated machine learning

en-noun

automated machine learning (uncountable)

artificial intelligence
artificial intelligence

(artificial intelligence) A field of study concerned with the design and development of algorithms and techniques of automating the end-to-end process of machine learning and artificial intelligence.

自動化された機械学習

じどうかされたきかいがくしゅう


beard-second

(humorous) A unit of length about the distance beard hair grows in one second: defined as five nanometers, or 10 nm according to some.


representation theory

representation theory (countable and uncountable, plural representation theories)

1989 [Clarendon Press], Robert J. Baston, Michael G. Eastwood, The Penrose Transform: Its Interaction with Representation Theory, 2016 Dover edition.

1998, A. Broer, A. Daigneault, Representation Theories and Algebraic Geometry, Springer:

2005, Andrea Brini, Francesco Regonati, Antonio Teolis, Combinatorics and Representation Theory of Lie Superalgebras over Letterplace Superalgebras, Hongbo Li, Peter J. Olver, Gerald Sommer (editors), Computer Algebra and Geometric Algebra with Applications: 6th International Workshop IWWM and International Workshop GIAE, Springer, LNCS 3519, page 239,
As an application, we describe in detail the way to specialize general results to the representation theory of the symmetric group and to classical invariant theory.

From his numerous contributions through research articles, books, and conference volumes we just mention his^([Vlastimil Dlab's]) work on the representation theory of hereditary algebras in the mid-seventies of the last century, joint with C.M. Ringel, laying the foundations of, in particular, tame hereditary representation theory over arbitrary fields, a theory still at the centre of interest.

2005, Ragnar-Olaf Buchweitz, Helmut Lenzing, Representations of Algebras and Related Topics, American Mathematical Society, page ix:

linear transformation
linear transformation

(mathematics, algebra) A branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.


approximand

Each shape affects how well F approximates a given approximand and how quickly an adaptive fuzzy system F approximates when learning tunes the parameters of Aⱼ.

1998, International Conference on Fuzzy Systems, Proceedings of the IEEE International Conference on Fuzzy Systems, Volumes 1-3, page 1237:

In effect, this result is ideal, as it gives the same rate m^(-(s-1/p)) we could hope to obtain by knowing that the underlying approximand was a ridge function in a specific direction […]

2000, Royal Society (Great Britain), Wavelets: The Key to Intermittent Information?, page 256:

The key task of approximating with granules is seeking for the tightest envelope for the approximand.

2012, Moamar Sayed-Mouchaweh, Edwin Lughofer, Learning in Non-Stationary Environments: Methods and Applications, page 275:

(mathematics) The solution to function, series, etc. that is to be approximated.


syzygant

(mathematics) A rational integral function of the invariants of a quantic that vanishes when expressed as a function of the coefficients.


Laplace's equation

Laplace's equation (plural Laplace's equations)

In practical problems, since the charges are confined to small regions while major part of the space is charge-free, it is obvious that Laplace's Equation has far greater utility than Poisson's equation.

1993, V. V. Sarwate, Electromagnetic Fields and Waves, New Age International Publishers, page 182:

Numerous problems in electrical engineering require a solution of Laplace's equation in two dimensions. This section outlines a classic problem that leads to Laplace's equation, then develops a finite element method for its solution.

1996, Peter P. Silvester, Ronald L. Ferrari, Finite elements for electrical engineers, 3rd edition, Cambridge University Press, page 29:

2002, Gerald D. Mahan, Applied Mathematics, Kluwer Academic / Plenum, page 141,
Laplace's equation appears in a variety of physics problems and several examples are provided below. The relevance of Laplace's equation to complex variables is provided by the following important theorem.

(potential theory) The partial differential equation (∂²φ)/(∂x_1²)+(∂²φ)/(∂x_2²)+⋯+(∂²φ)/(∂x_n²)=0, commonly written Δφ=0 or ∇²φ=0, where Δ(=∇²) is the Laplace operator and φ is a scalar function.


hemitropic

By surface analysis, hemi- + -trope + -ic

surf

a hemitropic function

(mathematics) Invariant relative to a proper orthogonal group.


Bloch's theorem

Named after the French mathematician André Bloch.

(mathematics) A theorem that gives a lower bound on the size of a disc in which an inverse to a holomorphic function exists.


Bloch function

(mathematics) Any of the elements of the Bloch space.


Bloch space

Named after the mathematician André Bloch.

(mathematics) The space, denoted by ℬ or ℬ, of holomorphic functions f defined on the open unit disc D in the complex plane, such that the function (1-|z|²)|f^′(z)| is bounded.


cuartillo

From Spanish cuartillo (“¹⁄₄₈ Castilian bushel, &c.”), from Old Spanish quartillo (“little fourth”) in reference to its forming a fourth celemin, from quarto + -illo (“-elle: forming diminutives”), from Latin quartus (“one-fourth”). Doublet of cuartilla and quartilho.

bor

<small>¹<big>⁄</big>₄₈</small> Castilian bushel, &c.

(historical) A traditional Castilian unit of dry measure, equivalent to about 1.2 L.


KAP

en-noun

KAP (countable and uncountable, plural KAPs)

(statistics) Initialism of knowledge, attitude(s), and practice(s): a type of representative survey, originated in the 1950s, that examines the level of knowledge, beliefs, and behaviours of a population in the context of a specific topic.


Taylor polynomial

Taylor polynomial (plural Taylor polynomials)

For example, if Mⁿ#61;#92;mathbb#123;R#125;ⁿ and Nᵏ#61;#92;mathbb#123;R#125;, a singularity means the vanishing of the gradient, or equivalently, the linear terms of the Taylor's polynomial of some degree greater than 1. If one has a singularity, one may look at the second-order terms in the Taylor's polynomial.

1980, Donald W. Kahn, Introduction to Global Analysis, Academic Press, page 314:

The polynomial approximation above for exp(x) is an example of a Taylor polynomial of degree 2 and in fact the polynomial approximations to sin(x) and cos(x) that we wrote down in Chapter 3 are also Taylor polynomials.

1989, J. Berry, John Stephen Berry, A. Norcliffe, S. Humble, Introductory Mathematics Through Science Applications, Cambridge University Press, page 211:

Then, we introduce and investigate the Taylor polynomials and we prove suitable versions of the Lagrange mean value theorem and of the Taylor formula.

2007, Andrea Bonfiglioli, Ermanno Lanconelli, Francesco Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer, page 733:

(mathematical analysis) A truncated Taylor series; the sum of the first n terms of a Taylor series.


Maclaurin polynomial

Maclaurin polynomial (plural Maclaurin polynomials)

Use the formula for the nth Maclaurin polynomial for a function f(x) given below to find the 5th Maclaurin polynomial for f(x) = ln(l + x). Show all of your work.

1996, Arthur Wayne Roberts, Calculus: The Dynamics of Change, Mathematical Association of America, page 109:

Use the Maclaurin polynomial for the function defined by f(x) = ln(l + x) to compute the value of In 1.2 accurate to four decimal places.

1996, Louis Leithold, The Calculus 7, HarperCollins, page 681:

In the Maclaurin polynomials for sin x, only the odd powers of x appear explicitly.

1998, Howard Anton, Calculus, Volume 2, Wiley, page 654:

Or, it could be reinforcement of material on Taylor and Maclaurin polynomials that the students have had.

2010, Stephen Hillbert, Diane D. Schwartz, Stan Seltzer, John Maceli, Eric Robinson, Calculus: An Active Approach with Projects, Mathematical Association of America, page 294:

(mathematics) A truncated Maclaurin series; the sum of the first n terms of a Maclaurin series.


Generation Alpha

Originated from a 2008 survey conducted by the Australian consulting agency McCrindle Research, according to founder Mark McCrindle, who is generally credited with the term. McCrindle describes how his team arrived at the name in a 2015 interview:
When I was researching my book The ABC of XYZ: Understanding the Global Generations (published in 2009) it became apparent that a new generation was about to commence and there was no name for them. So I conducted a survey (we're researchers after all) to find out what people think the generation after Z should be called and while many names emerged, and Generation A was the most mentioned, Generation Alpha got some mentions too and so I settled on that for the title of the chapter Beyond Z: Meet Generation Alpha. It just made sense as it is in keeping with scientific nomenclature of using the Greek alphabet in lieu of the Latin and it didn't make sense to go back to A, after all they are the first generation wholly born in the 21st Century and so they are the start of something new not a return to the old.
McCrindle Research also took inspiration from the naming of hurricanes, specifically the 2005 Atlantic hurricane season in which the names beginning with the letters of the Latin alphabet were exhausted, and the last six storms were named with the Greek letters alpha through zeta.

Generation Alpha members will be the most diverse of all the generations. In 2011, Generation Alpha reached a demographic milestone: there were more Generation Alpha babies born to minority families than white families in the United States […]

2016, Jeanne Meister, Kevin J. Mulcahy, The Future Workplace Experience, McGraw Hill Professional, →ISBN, page 160:

(demographics) The generation following Generation Z, born between the early to mid-2010s and the mid-2020s.


tetrahedralize

en-verb

tetrahedralize (third-person singular simple present tetrahedralizes, present participle tetrahedralizing, simple past and past participle tetrahedralized)

tetrahedralizes (present singular third-person)
tetrahedralizing (participle present)
tetrahedralized (participle past)
tetrahedralized (past)

(mathematics, usually of a surface or a mesh) To convert to tetrahedrons.


characteristic root

en-noun

characteristic root (plural characteristic roots)

Calque of French racine caractéristique, the term preferred by French mathematician Augustin-Louis Cauchy (1789–1857).

1962 [Prentice Hall], Leonard E. Fuller, Basic Matrix Theory, 2017, Dover, page 217,
There is an important relationship between the characteristic roots of a symmetric matrix and the r and p of the associated quadratic form. It is true that the number of positive characteristic roots is p, whereas the number of nonzero characteristic roots is r.

The determinant of a matrix is the product of its characteristic roots.

1985, Takeshi Amemiya, Advanced Econometrics, Harvard University Press, page 459:

2009, Babu Ram, Engineering Mathematics, Pearson Education India, page 13.56,
Theorem 13.54. The characteristic roots of a Hermitian matrix are real.


rhizoristic

From Ancient Greek ῥίζα (rhíza, “root”) + ὁρίζω (horízō, “to limit”).

(archaic, mathematics) Relating to the separation of the roots of an equation.


input neuron

(machine learning) A neuron in the input layer of a neural network, outputting only a constant.


hidden layer

(machine learning) The layer or layers of neurons between the input layer and the output layer in a neural network.


multilayer perceptron

multilayer perceptron (plural multilayer perceptrons)

(machine learning) A neural network having at least one hidden layer, and whose neurons use a nonlinear activation function (e.g. sigmoid).


feedforward neural network

feedforward neural network (plural feedforward neural networks)

(machine learning) A neural network in which the outputs are not used as input in a previous layer.


π Day

en-proper noun

<i class="Grek mention" lang="mul">π</i> Day

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As Quakers, we do not as a matter of religion celebrate any holidays (more accurately, all days are to be considered holidays because they have equal potential for revealing the Divine to us), but as a cultural matter we celebrate all kinds: Hindu, Christian, Jewish, ethnic, birthdays of famous composers, artists, writers, and scientists, π Day (March 14th – we bake several pies and divide them into fractions before eating), even some we just make up ourselves.

1999, David H. Albert, “Exits and Entrances”, in And the Skylark Sings with Me: Adventures in Homeschooling and Community-based Education, Gabriola Island, B.C.: New Society Publishers in cooperation with Holt Associates/Growing Without Schooling, →ISBN, page 128:

Likewise, March 14 in North America has been π Day, since in the USA the month is written before the day ('314'). In schools throughout North America, it has become a reason for mathematical projects, especially focussing on π.

2012 June 19, Jonathan M[ichael] Borwein, “The Life of Pi: From Archimedes to Eniac and Beyond”, in David H[arold] Bailey, Jonathan M. Borwein, Pi: The Next Generation: A Sourcebook on the Recent History of Pi and Its Computation, Switzerland: Springer Nature, published 2016, →DOI, →ISBN, page 446:


agrimetric

Under Ambassador's direction and in light new aid concepts, team will undertake such exploration and will review US assistance to Thailand, including specifically such items as […] (e) community development, including agrimetric centers in border areas, particularly Northeast.

1994, “Thailand”, in Foreign Relations of the United States, 1961–1963: Southeast Asia, volume XXIII, Washington: United States Department of State: United States Government Printing Office, →ISBN, page 878:

Data and technology have a central role to play in increasing the productivity and competitiveness of British farming. Last October, I launched the first of our agri-tech centres of excellence, the agrimetric centre at Rothamsted. They will develop new software models to improve our ability to understand and utilise the huge volume of data that exist.

2016 May 5, The Minister of State, Department for Environment, Food and Rural Affairs (George Eustice), “Commons Chamber: Oral Answers to Questions: Department for Environment, Food and Rural Affairs: British Farming”, in House of Commons Hansard (House of Commons), volume 609, column 293:

(statistics, agriculture) Relating to agrimetrics.


agrimetrics

We're South Africa's science councils Agricultural Research Council— Made up of 13 institutes covering topics from agrimetrics and plant biotechnology, to research on crops, soil and pests.

1994, New Scientist, volume 141, numbers 1908-1918, page 13:

For the third straight year Hybrid Turkeys leads in both independent strain studies and agrimetrics summaries.

1997, International Hatchery Practice, volumes 12-13:

Quantifiable comparisons are done sequentially, examining: • within peer group comparisons, such as agrimetrics; • within system farm-level mean measures; • within system measures of variance; and • within production flow measures using descriptive measures.

2003, David E. Reeves, “Benchmarking reproductive performance within production systems”, in 2003 Allen D. Leman Swine Conference:

• Mapping and enhancing different decision making tools for farmers • Influencing EC to develop agrimetrics data • Adopting and developing emerging technologies and infrastructures

2016, Report of the kick off meeting of the Knowledge Network on Sustainable Intensification (KNSI), 3 June 2016, Wageningen:

(statistics, agriculture) The measurement of fields or agricultural data.


rectahedron

(mathematics) A six-sided polyhedron whose faces are all rectangular (like a brick, shoe box or matchbox)


rhizic

(mathematics, archaic) Relating to the root of an equation.


unitation

(mathematics, archaic) The process of unitating.


online learning

stochastic gradient descent
stochastic gradient descent

(machine learning) The training of a neural network using stochastic gradient descent with a mini-batch size of one; only one input is used for training at a time.


chakravala method

From Sanskrit चक्रवाल (cakravāla, “cycle”).

(mathematics) The use of a cyclic algorithm to solve indeterminate quadratic equations.


curse of dimensionality

Coined by American mathematician Richard E. Bellman.

coinage

Coordinate term: blessing of dimensionality

The curse of dimensionality is that vectors will get exponentially farther and farther away from one another, in Euclidean distance, as the dimensionality increases. A lot of simple operations become impractical above 10 or 20 dimensions, like sorting a large list of vectors based on their distance[…]

2019, Hannes Hapke, Cole Howard, Hobson Lane, Natural Language Processing in Action, Simon and Schuster, →ISBN:

(mathematics, computing) A problem in the analysis of multidimensional systems caused by the sparse nature of the available data.