is the imaginary unit; $ 1 $
is a unit vector in $ {\mathcal H} $(
8 KB (1,210 words) - 08:23, 6 June 2020
...e of the pencil. The power of the centre of the pencil with respect to any circle of the pencil is the same and is called the power of the pencil. If the $
axis is the radical axis of the pencil, then the equation of an arbitrary circle of the pencil has the form
20 KB (3,056 words) - 08:25, 6 June 2020
...g when the curve $C$ can be rationally parametrized. For example, the unit circle $X ^ { 2 } + Y ^ { 2 } = 1$ has the rational parametrization
...ine of slope $t$ through the point $( - 1,0 )$ and notes that it meets the circle in the variable point
16 KB (2,510 words) - 17:43, 1 July 2020
...f non-intersecting circles (and here a straight line is considered to be a circle of infinite radius) onto a domain $ G _ {2} $
enters in its description. In particular, the general form of mappings of the unit disc $ D = \{ {z } : {| z | < 1 } \} $
28 KB (4,208 words) - 17:46, 4 June 2020
is the circle of radius $ \rho $
is equal to the arithmetic average of its values on any sufficiently small circle $ L \subset D $
26 KB (3,804 words) - 08:11, 13 February 2022
...lde{A}$ is the algebra with involution obtained from $A$ by addition of a unit element, there exists a unique norm on $\tilde{A} $ which converts $\tilde
...tructed. Any $\Cstar$-algebra $A$ has an approximate unit, located in the unit ball of $A$ and formed by positive elements of $A$. If $I$, $J$ are closed
15 KB (2,316 words) - 21:05, 20 April 2012
1) $n = 1$. Let $D$ be the domain bounded by a part of the unit circle $\gamma = \{ z _ { 1 } : | z _ { 1 } | = 1 \}$ and a smooth open arc $\Gamm
6 KB (978 words) - 17:44, 1 July 2020
...ed space; if a point is removed, then the remainder is connected, but if a circle that does not reduce to a point is removed, then the remainder is disconnec
...' if for any two points $x,y \in X$ there is a continuous map $p$ from the unit interval $[0,1]$ with the usual topology to $X$ such that $p(0) = x$ and $p
8 KB (1,209 words) - 06:00, 22 April 2023
is a regular [[Analytic function|analytic function]] in the unit disc $ D= \{ {z = re ^ {i \theta } \in \mathbf C } : {| z | < 1 } \} $
is an arbitrary set of positive measure on the circle $ \Gamma = \{ {z = e ^ {i \theta } } : {| z | = 1 } \} $,
6 KB (965 words) - 08:11, 6 June 2020
...n circular optical elements can be quantified by mean-square deviation per unit area. Given a function $f$ on $D$ and $n \in \mathbf{N} _ { 0 } = \{ 0,1,2
...mathbf{N} _ { 0 }$ and $m = n - 2 j$ with $j = 0 , \dots , n$, the Zernike circle polynomial is
8 KB (1,192 words) - 08:49, 10 November 2023
...[#References|[a14]]]. Let $f$ be a bounded continuous function on the open unit disc $U$ in $\mathbf{R} ^ { 2 }$. Suppose that for each point in $U$ there
...ns integrable with respect to the (normalized) Lebesgue measure $m$ on the unit ball $B$ in $\mathbf{C} ^ { n }$. If $f$ is harmonic with respect to the in
17 KB (2,607 words) - 16:38, 2 February 2024
...$\mathbf{C}$) is rational, analytic within a circle containing the closed unit disc, $\operatorname { det } k ( z ) \neq 0$, $| z | < 1$, $k ( 0 ) = I$ (a
9 KB (1,297 words) - 09:30, 3 February 2024
These ideas have been applied as well to circle diffeomorphisms [[#References|[a10]]],
..., M.L. Stein, P.R. Stein, "On finite limit sets for transformations on the unit interval" ''J. Comb. Theory'' , '''15A''' (1973) pp. 25–44 {{MR|0316636}}
12 KB (1,685 words) - 08:27, 6 June 2020
be the end point of the arc on the unit circle $ x ^ {2} + y ^ {2} = 1 $(
12 KB (1,667 words) - 08:26, 6 June 2020
...while the statement "the ratio of the length of the circumference of the circle to its diameter is greater than three and ten seventy-firsts and less than
...$\bar{\eta}=8$, $\bar{\epsilon}=5$ and $\mu^0\bar{\alpha}$ means that the unit $\bar{\alpha}$ is not to be multiplied by a power of the unknown).
18 KB (2,697 words) - 13:11, 13 December 2013
Owing to the widespread use of electronic computers the circle of customers keeps increasing; hence the tendency to establish automatizati
...se considerably simplified the programming and substantially increased the circle of customers.
13 KB (1,892 words) - 20:11, 31 December 2018
5) Fix a pairing $P$. Consider the knot $K$ as an oriented circle and connect the points $( z _ { j } , t _ { j } )$ and $( z _ { j } ^ { \pr
...urther chords attached to their points, while on the solid portions of the circle all the endpoints are explicitly shown); and
10 KB (1,481 words) - 19:05, 23 January 2024
then this reduces to the unit circle $ | \lambda | = 1 $.
7 KB (1,085 words) - 08:13, 6 June 2020
...such as the real line $\mathbf{R}$, the unit interval $\mathbf{I}$ and the circle $S ^ { 1 }$ are absolute (neighbourhood) extensors. An absolute (neighbourh
8 KB (1,172 words) - 19:57, 15 March 2023
...-known tool for computing numerically the [[conformal mapping]] $g$ of the unit disc $D$ onto a star-like region $\Delta$ given by the polar coordinates $\
...wing cross-section into the exterior of a Jordan curve that is close to a circle; see [[#References|[a6]]], Chapt. 10. Moreover, for this application, the e
7 KB (1,076 words) - 07:40, 7 February 2024