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 $\map \det {H_3} = 1 / 12 \, 160$.
Nikolai Ivanovich Lobachevsky
- Lobachevsky, Nikolai Ivanovich (1793-1856)
Pietro Mengoli
- Mengoli, Pietro (1626-82)
Emmy Noether
- Noether, Amalie (Emmy) (1883-1935)
William Oughtred
- Oughtred, William (1575-1660)
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 $\sqrt {x^2 - a^2}$
- $\ds \int \sqrt {x^2 - a^2} \rd x = \frac 1 2 a^2 \cosh^{-1} \frac x a + \frac 1 2 {x \sqrt {x^2 - a^2} } + C$