Book:David Nelson/The Penguin Dictionary of Mathematics/Fourth Edition/Errata
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Errata for 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.)
Group Action of Cyclic Group on Polygon
- action 3.
- For example, the $n$-element ${}^*$cyclic group whose elements are $e, a, a^2, \ldots, a^{n - 1}$ acts on the vertices of a ${}^*$regular polygon by the map for which $e x$ is $x$ for each vertex $x$, and $a^k x$ is the vertex obtained when $x$ is rotated through $2 \pi k / n$ radians about the center of the polygon.
Directed Angle
- angle
- Often an angle is regarded as the measure of the rotation involved in moving from one initial axis to another final axis (termed a direction angle).
Atom
- atom
- An element $A$ of a *lattice such that if $B < A$ then $B = A$ (the null element).
Frequency of Beats
- beats
- The frequency at which the amplitude fluctuates is the beat frequency, which is equal to the difference in frequency of the combining waves, $\size {f_1 - f_2}$.
Eugenio Beltrami
- Beltrami, Eugenio (1835-99)
Paradoxien des Unendlichen
- Bolzano, Bernhard Placidus (1781-1848)
- He also published an influential work, Paradoxes of the Infinite (1850), in which he anticipated some of the later results of *Cantor.
Boolean Algebra
- Boolean algebra
- An algebraic system consisting of a *set of elements $S$ together with two *binary operations, denoted by (the Boolean product) and $+$ (the Boolean sum) obeying certain *axioms ...
William Brouncker
- Brouncker, William, Viscount (1620-85)
Cauchy-Crofton Formula
- Cauchy-Crompton formula
Warwick Tucker
- chaos
- In $2002$ the Swedish mathematician Warwick Tucker showed that [the Lorenz attractor] is a strange attractor.
Chu Shih-Chieh
- Chu Shih-chich
Edward Cocker
- Cocker, Edward (1631-75)
Jan de Witt
- conic (conic section)
- The treatment of conic sections by coordinate geometry was begun in the $17$th century, notably by Jan de Witt $\text {1629}$ – $\text {1672}$ ...
Cybernetics
- cybernetics
- The subject was developed in 1946 by Norbert Wiener, who coined the name from the Greek word kuberne?te?s, meaning 'pilot' or 'steersman'.
Finite Dissection of Polyhedra
- dissection proof
- ... the German mathematician Max Dehn showed in $1901$ that there are two polyhedra of the same volume for which it is impossible to reconstruct one from the other by dissection.
Division Sign
- division sign
- The alternative sign $/$ ... was recommended by William de Morgan in $1845$ to simplify the printing of fractions.
Eccentric Angle
- eccentric angle
- The angle that a radius of the *auxiliary circle makes with the positive $x$-axis, used in forming the parametric equations of an *ellipse or *hyperbola.
Eccentricity of Ellipse in terms of Semi-Major and Semi-Minor Axes
- ellipse
Alternatively [the eccentricity] is given by
- $e^2 - 1 - \dfrac {b^2} {a^2}$
Elliptic Curve: Arbitrary Example
- elliptic curve
For example, the elliptic curve $y^2 = x^3 + 17$ has a rational point $\tuple {-2, 3}$, but it also has many others, e.g. $\tuple {2, 3}$, $\tuple {\frac 1 4, \frac {33} 8}$, and $\tuple {-1, 4}$.
Historical Note on Elliptic Functions
- elliptic functions
- Functions first derived from *elliptic integrals by Abel in $1826$.
Expansion: $\paren {x + 1}^2$
- expansion 1.
\(\ds \paren {x + 1}^2\) | \(=\) | \(\ds \paren {x + 1}] \paren {x + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \paren {x + 1} + 1 \paren {x + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^2 + 2 x + 1\) |
Daniel Gabriel Fahrenheit
- Fahrenheit degree
- [After G.D. Fahrenheit (1686-1736)]
Farey Sequence
- Farey sequence (of order $n$)
- Farey sequence (of order $n$) (C. Haros, 1802; J. Farey; 1816)
Ad Locos Planos et Solidos Isagoge
- Fermat, Pierre de (1601-65)
- Further work in his Isagoge ad locus planos et solidos (1679, On the Plane and Solid Locus) foreshadowed the later analytic geometry of Descartes ...
Floating-Point Representation: $105.7$
- floating-point representation
- For example, $105.7 = 0.1057 \times 10^{-3}$.
Period of Rotation of Plane of Oscillation of Foucault's Pendulum
- Foucault's pendulum
- To an observer on earth the plane of oscillation [of Foucault's pendulum] makes one rotation every $24$ hours (approximately).
Historical Note on Fuzzy Logic
- fuzzy logic
- A system of logic proposed in $1965$ by Lofti Zadeh, an Iranian electrical engineer, ...
Gaussian Integration Rule
- Gaussian integration rule
- A *numerical integration rule of the form
- $\ds \int \limits_a^b \map w x \map {\text f} x \rd x \approx \sum_{i \mathop - 1}^n w_i \, \map {\text f} {x_i}$
Gauss-Seidel Method
- Gauss-Seidel method
- Now, writing $\mathbf x_n$ for the column vector of values of $x_1, x_2, x_3$ after the $n$th iteration, and
- $\mathbf L = \begin {pmatrix} a_{1 1} & 0 & 0 \\ a_{2 1} & a_{2 3} & 0 \\ a_{3 1} & a_{3 2} & a_{3 3} \end {pmatrix}, \mathbf U = \begin {pmatrix} 0 & a_{1 2} & a_{1 3} \\ 0 & 0 & a_{2 3} \\ 0 & 0 & 0 \end {pmatrix}$
- the iterative relationship is
- $\mathbf x_{n + 1} = \mathbf L^{-1} \paren {\mathbf b - \mathbf U \mathbf x_n}$
Jørgen Pedersen Gram
- Gram-Schmidt method
- It is named after Jorgen Pederson Gram (1850-1916) and Erhard Schmidt (1876-1959).
Determinant of Order $3$ Hilbert Matrix
- Hilbert matrix
- For $n = 3$,
- $H_3 = \begin {pmatrix} 1 & \tfrac 1 2 & \tfrac 1 3 \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 \\ \end {pmatrix}$
- and $\map \det {H_3} = 1 / 12 \, 160$.
Hypergeometric Differential Equation
- hypergeometric differential equation
- A second-order differential equation of the form
- $x \paren {1 - x} \dfrac {\d^2 \phi} {\d x^2} + \sqbrk {c - \paren {a + b - 1} x} \dfrac {\d \phi} {\d x} - a b \phi = 0$
- where $a$, $b$ and $c$ are constants.
Improper Integral on $\hointr a b$
- infinite integral (improper integral)
- An integral ... whose integrand is a function $\map {\mathrm f} x$ that is finite for $a \le x < b$, but infinite for $x = b$, is
- $\ds \int \limits_a^b \map {\mathrm f} x \rd x$
- which is short for
- $\ds \lim_{\delta \mathop \to \infty} \int \limits_a^{b - \delta} \map {\mathrm f} x \rd x$
- where $\delta > 0$.
Inverse of Curve under Inversive Transformation
- inversion 1.
- A curve $\map f {x, y} = 0$ has an inverse $\map f {x', y'} = 0$, where
- $x' = \dfrac {r^2 x} {x^2 + y^2} \qquad y' = \dfrac {r^2 y} {x^2 + y^2}$
Maurice Henry Quenouille
- jack-knife (M.L. Quenouille, 1949)
Julia Set
- Julia set
- (b) Filled Julia set for $c = -0.75$
Kepler's Conjecture
- Kepler's conjecture (J. Kepler, 1611)
- The conjecture was proved in $2006$ by T.C. Hales.
Levi ben Gerson
- Levi ben Gerson (1288-1344)
- Jewish mathematician and astronomer who produced in his Sefer ha mispar (1321, Book of Number) one of the first texts to establish simple rules for calculating permutations and combinations ...
Marius Sophus Lie
- Lie, Marius Sophus (1842-99)
- ... noted for his work on transformation groups, which he described in his major treatise, Die Transformationsgruppen (1888-93).
Linear Function of Two Variables
- linear function
- A linear function of two variables has the form
- $\map {\mathrm f} {x, y} = a_0 + a_1 x + a_2 y + a_3 x y$
- where $a_0$, $a_1$, and $a_2$ are constants.
Nikolai Ivanovich Lobachevsky
- Lobachevsky, Nikolai Ivanovich (1793-1856)
Lowess
- lowess
- The name lowess is an acronym derived from 'locally weighted smoothing scatterplots'.
Pietro Mengoli
- Mengoli, Pietro (1626-82)
Emmy Noether
- Noether, Amalie (Emmy) (1883-1935)
William Oughtred
- Oughtred, William (1575-1660)
Reducible Polynomial
- reducible polynomial
- A *polynomial is reducible over a *field $F$ if it can be factored into two polynomials having coefficients in $F$.
Rhombohedron
- rhombohedron
- A hexagonal prism.
Willebrord van Royen Snell
- Snell, Willebrord van Royen (1580-1626)
Alfred Tarski
- Tarski, Alfred (1902-85)
Wronskian
- Wronskian
- It is named after the Polish mathematician Józef Maria Hoene-Wroński (1776-1853).
Primitive of $\sqrt {x^2 - a^2}$
- $\ds \int \sqrt {x^2 - a^2} \rd x = \frac 1 2 a^2 \cosh^{-1} \frac x a + \frac 1 2 {x \sqrt {x^2 - a^2} } + C$