Book:David Nelson/The Penguin Dictionary of Mathematics/Fourth Edition/Errata

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 $\delta > 0$.

Book:David Nelson/The Penguin Dictionary of Mathematics/Fourth Edition/Errata

Errata for 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.)

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