An element $e$ in a set $M$ is called a left (right) unit (left (right) identity) with respect to a [[binary operation]] $*$ defined
...unit is used for only one of these operations, usually multiplication. The unit with respect to addition is called the [[zero]] element.
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...m''' asks how large the values of such a polynomial must be on the [[unit circle]] in the [[complex plane]]. The answer to this would yield information ab
...= \pm 1$. Let $\Vert p \Vert$ denote the supremum of $p(z)$ on the unit circle. ''Littlewood's problem'' asks for constants $c_1$ and $c_2$ such that th
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...le has 180 degrees. Degrees are also used to measure circular arcs (a full circle has 360 degrees).
...ee of irrationality, and many more. It is also, of course, the name of the unit for temperature measurements in various scales.
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...$ radians; it is approximately $57^\circ17'44''$. A radian is taken as the unit of measurement of angles in the so-called circular, or radian, measurement
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...e real axis (the equation is stable), then the multipliers lie on the unit circle. Consider a canonical equation $\dot x = i \lambda J H(t) x$ with a real pa
...ability it is necessary and sufficient that all multipliers be on the unit circle and that there be no coincident multipliers of different kinds.
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then the circle $\lvert z\rvert=R$ is a [[natural boundary]]: all points of the cicle are s
...t sequence $p_n$, radius of convergence equal to 1, but for which the unit circle is not a natural boundary.
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Szegö quadrature formulas are the analogues on the unit circle $\bf T$ in the complex plane of the Gauss quadrature formulas on an interva
...i_n$ as nodes (as in Gaussian formulas), because these are all in the open unit disc $\mathbf D$ (cf. also [[Szegö polynomial|Szegö polynomial]]). Theref
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...ert kernel and the [[Cauchy kernel|Cauchy kernel]] in the case of the unit circle:
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...[[Multipliers|Multipliers]]) of the variational equation lies on the unit circle. There are also results about the local structural stability of certain hyp
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...nces|[4]]]). There are various generalizations of a self-perimeter for the unit sphere $S$ in a normed space of dimension greater than two (see [[#Referenc
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...al power dilation with spectrum in $\partial S$. The minimal radius of the circle which is a spectral set for every contraction in a Banach space is equal to
...> J. von Neumann, "Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes" ''Math. Nachr.'' , '''4''' (1951) pp. 258–281</TD></TR><T
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...a closed curve $L'$ with a cusp at the point $w=1$, touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. ban
The function $w=\lambda(\rho t+\alpha)$ maps the exterior of the unit circle in the $t$-plane to the exterior of $L'$. To obtain a Zhukovskii profile of
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be a bounded regular analytic function in the unit disc $ \Omega = \{ {z \in \mathbf C } : {| z | < 1 } \} $,
...s serve as Lobachevskii straight lines, these being orthogonal to the unit circle (Poincaré's model), and
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...omeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.
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...n a so-called great circle is obtained as the intersection. A unique great circle can be drawn through any two points $A$ and $B$ on the sphere (Fig. a), exc
...a straight line is the shortest curve between its ends, an arc of a great circle on a sphere is only the shortest curve when it is shorter than the compleme
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...fmath.org/legacyimages/a/a013/a013980/a0139808.png" /> lie inside the unit circle, then equation (*) has the solution
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...ally onto a standard pair $(D^n,D^m)$ or $(D^n,D_+^m)$, where $D^k$ is the unit ball of the space $\mathbf R^k$ with centre at the origin and $D_+^k$ is th
...imbedding of a circle and an arc into a plane is locally flat; however, a circle or an arc can be imbedded in $\mathbf R^k$ with $k\geq3$ in a manner that i
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...nd bijectively. The circle property: Under a fractional-linear mapping any circle in $ \overline{\mathbf C}\; $(
i.e. a circle in $ \mathbf C $
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...axis of revolution and the circle described by the centre of its rotating circle.
...f $u$ and $v$, $r$ is the radius vector of the surface $F$, and $n$ is the unit normal to $F$.
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in the open unit disc whose $ H ^ \infty $-
...hur algorithm can also be used to obtain a Routh or Jury test for the open unit disc, that is, the Schur algorithm can be used to determine whether or not
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is a bounded regular [[Analytic function|analytic function]] in the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
of the circle $ \Gamma = \{ {z } : {| z | = 1 } \} $
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of measure zero on the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $,
that is regular, analytic and bounded in the unit disc $ D = \{ {z } : {| z | < 1 } \} $
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...p"> P.E. Blanksby, H.L. Montgomery, "Algebraic integers near the unit circle" ''Acta Arith.'' , '''18''' (1971) pp. 355–369</TD></TR>
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...o, then $f(z)=0$ in $D$. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set $E$ of the given t
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...mages/c/c026/c026230/c0262306.png" /> rotates only in one direction as the circle is traversed. The following inequality expresses a necessary and sufficient
....org/legacyimages/c/c026/c026230/c02623054.png" />, and the radius of this circle cannot be increased without imposing additional restrictions on the class o
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...nction $u(z)$, $z=r\mathrm{e}^{\mathrm{i}\phi}$, can be represented in the unit disc $U=\{ z\in\C : \abs{z} < 1 \}$ by a Poisson–Stieltjes integral
where $\mu$ is a Borel measure concentrated on the unit circle $T=\{ z\in\C : \abs{z} = 1 \}$, $\int\rd\mu(\xi)=1$. Then almost-everywhere
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in the unit disc $ D $
on the boundary circle $ C $(
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...n|holomorphic functions]] of one complex variable. In the case of the unit circle one has the following relationship between the Cauchy kernel and the [[Hilb
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...nt lifting theorem and a certain contractive analytic function in the open unit disc. This characterization of all solutions has several different network
on the unit circle whose norm $ \| g \| _ \infty = { \mathop{\rm ess} \sup } \{ {| {g ( e
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inside the unit circle on the complex plane. Similarly, the dynamical
polynomial $w(z)$ has $m$ roots inside and $n-m$ roots outside the unit
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lies on the unit circle and there is the spectral decomposition $ U = \int _ {0} ^ {2 \pi } e ^ {
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on the open unit disc $ | z | < 1 $
...I. Schur, "On power series which are bounded in the interior of the unit circle. II" I. Gohberg (ed.) , ''Methods in Operator Theory and Signal Processing
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under which the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $
is mapped onto itself, the maximum diameter of the image of the circle $ \Gamma _ {R} = \{ {z } : {| z | = R } \} $
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the circle $ | z | = 1 $
A hyperbolic circle in $ D $
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where $z\rightarrow 1$ along any path not tangent to the unit circle.
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...ta$ are real, namely $\theta$ and $1/\theta$, and the rest lie on the unit circle. The field $\mathbf{Q}(\theta)$ is thus a quadratic extension (cf. [[Extens
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...$ can be characterized as the space of analytic functions $f ( z )$ on the unit disc having [[Taylor series|Taylor series]] representation
...ions (and, more generally, of contractive analytic matrix-functions on the unit disc) in terms of Schur parameters (see [[#References|[a9]]] and [[#Referen
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be a meromorphic function in the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
on the circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $
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Along with these, there are examples of bounded analytic functions in the unit disc $ D $
of measure zero on the unit circle $ \Gamma $.
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The general name for polynomials orthogonal on the circle, over a contour or over an area. Unlike the case of orthogonality in a real
==Orthogonal polynomials on the circle.==
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...e unit disc $D = \{ z \in \mathbf{C} : |z| < 1 \}$; then all points of the circle $\Gamma = \{ z \in \mathbf{C} : |z| = 1 \}$ except, possibly, for a [[First
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...aofmath.org/legacyimages/m/m064/m064430/m06443021.png" /> is an arc of the
circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...ar group (1) is then replaced by the modular group of automorphisms of the
unit disc. For example, it is convenient to apply the fractional-linear transfor
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...explains the connection of roots of unity with the problem of squaring the circle (construction of polygons, cf. [[Geometric constructions|Geometric construc
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of functions of bounded type in the unit disc $ \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $:
almost-everywhere on the unit circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $;
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...(cf. [[Spectrum of an operator|Spectrum of an operator]]) lies on the unit circle, and $U$ has a representation
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analytic on the open unit disc $ | z | < 1 $(
...s the [[Banach algebra|Banach algebra]] of complex-valued functions on the unit disc having a [[Fourier series|Fourier series]]
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...opediaofmath.org/legacyimages/b/b110/b110130/b11013061.png" /> on the
unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...w.encyclopediaofmath.org/legacyimages/b/b110/b110130/b11013075.png" /> and
unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
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...ametrized by a unit vector $\alpha \in S ^ { 1 }$, $S ^ { 1 }$ is the unit circle in $\mathbf{R} ^ { 2 }$ and $p \in \mathbf R _ { + } : = [ 0 , \infty )$. B
If $n_0$ is a unit vector normal to $S$ at the point $x _ { 0 }$, then for an arbitrary $\gamm
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Let $a$ be a complex-valued function defined on the complex unit circle $\bf T$, with [[Fourier coefficients|Fourier coefficients]]
...he exponentials of continuous complex-valued functions defined on the unit circle.
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is the unit $ n \times n $
...style="background-color:white;" colspan="1">Situated inside or on the unit circle; in the latter case simple elementary divisors of the monodromy matrix corr
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