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''simple arc, Jordan arc'' An intrinsic characterization is: A simple arc is a [[Line (curve)|line (curve)]] that has ramification index 1 at two poi

along a non-closed simple arc $ \gamma = \{ {z ( t) } : {0 \leq t \leq 1 } \} $'' The removal of the points of the arc $ \gamma $

...$ is non-trivial for the arc $L_1$ (Fig. a); this group is trivial for the arc $L_2$ (Fig. b), but $E^3\setminus L_2$ itself is not homeomorphic to the co

...]] in which any two points can be joined by a continuous image of a simple arc; that is, a space $ X $ ...ace is a Hausdorff space in which any two points can be joined by a simple arc, or (what amounts to the same thing) a Hausdorff space into which any mappi

...Cantor curve have the same finite branch index, then the Cantor curve is a simple closed line. The universal Cantor curve (the Menger curve) can be construct

...any loop in $V\setminus qM$ is homotopic to zero in $U\setminus qM$ (local simple connectedness). If $m=n-2$, then such a criterion holds for $n\neq4$, but i

...the least upper bound of the lengths of the broken lines inscribed in this arc. Any continuous curve has a length, finite or infinite. If its length is fi Then the length of an arc of the curve counted from the point corresponding to the parameter value $

be the end point of the arc on the unit circle $ x ^ {2} + y ^ {2} = 1 $( The arc from $ B $

...2$ be a quadratic differential on $R$; let $C$ be the set of all zeros and simple poles of $Q(z)\,dz^2$ and let $H$ be the set of all poles of $Q(z)\,dz^2$ o ...C\cup H)$ there passes a trajectory of $Q(z)\,dz^2$ that is either an open arc or a Jordan curve on $R$.

are contained in a simple arc lying in $ K $.

...l curvature of the surface on the set of points of the curve. For a simple arc $ L $,

...Jordan [[#References|[1]]]. Together with the similar assertion: A simple arc does not decompose the plane, this is the oldest theorem in set-theoretic t ...ounded component $A$ and every point $x_0\in\Gamma$, there exists a simple arc with ends $x_0$ and $x$ and all points of which, except $x_0$, are containe

...nd have the property that every proper subcontinuum is an [[Arc (topology)|arc]]. See [[#References|[a7]]]. ...sed curve is the only homogeneous bounded plane continuum that contains an arc" ''Canad. Math. J.'' , '''12''' (1960) pp. 209–230</TD></TR><TR><TD va

...cible continuum. Every locally connected irreducible continuum is a simple arc, that is, is homeomorphic to an interval.

...the same direction) and such edges (arcs) are then said to be multiple. An arc (or edge) can begin and end at the same vertex, in which case it is known a ...ne says that an edge $\{u,v\}$ connects two vertices $u$ and $v$, while an arc $(u,v)$ begins at the vertex $u$ and ends at the vertex $v$. Each graph can

is defined as any simple Jordan arc $ c = \overline{ {ab }}\; $, and such that the arc $ c $

...the line" (N.N. Luzin). There are no Peano curves at which every point is simple or two-fold, but there is a Peano curve with as multiple points (only count ...)$, $z=t$, where the first two functions give a Peano curve. Although this arc is impermeable to rain, it is by no means a continuous surface.

The following are some very simple solutions: ...nside $\gamma$ and $D$ is the domain bounded by a part of $\gamma$ and the arc $\Gamma$, with $0 \notin \overline { D }$, then for $z \in D$ and $f \in H

...there is a disc $D$ embedded in $M$ such that $D\cap F=\partial D$ and the simple closed curve $\partial D$ does not bound a disc in $F$. Otherwise, such a s ...\cap\partial M=\beta$, and $\alpha$ does not cobound a disc in $F$ with an arc in $\partial F$. If a properly embedded surface $F$ in a three-dimensional

plane, having as its only singularity a simple pole at the point $ s = 1 $ by a circular arc of radius $ R = 2 N + 1 $

...f a quadratic differential are called critical points of it. The zeros and simple poles are called finite critical points and their totality is denoted by $ ...nt for each choice of the sign of the integrand; furthermore, each maximal arc of a trajectory (or orthogonal trajectory) of $ U $

...images/r/r082/r082150/r08215077.png" /> is a parameter proportional to the arc length, takes the form ...mplete (geodesically complete) if it is complete as a metric space (if any arc of a geodesic can be extended indefinitely on both sides). A Riemannian spa

...mid 0 \leq s \leq S \} $, $ x = (x_{1},\ldots,x_{n}) $, where $ s $ is the arc-length; let $ F = F(x(s)) $ be a function defined on $ \gamma $. The curvil ...'line integral of the first kind''', or a '''line integral with respect to arc-length'''. It is the limit of suitable integral sums, which can be describe

...of them is also finite, then such a curve can be split into finitely many simple arcs having no common points except their end points. ...an curve|Jordan curve]] (a space homeomorphic to the circle, also called a simple closed curve).

onto the $ w $-plane with slit along an arc of the circle $ | w- \beta _ {0} | = | \zeta _ {0} | $ The estimates (3)–(5) are simple consequences of results on the ranges of the functionals

...l stems from the fact that they appeared first in the rectification of the arc of an ellipse and other second-order curves in work by Jacob and Johann Ber ...d kind and third kinds have a pole-type singularity with residue zero or a simple pole, respectively. Regarded as functions of the upper limit of integration

is the arc element. ...ecessary for the solution of each problem; hence there arises a method, as simple as it is general, for solving problems concerning the motions of bodies" (

Some very simple solutions are given below. ...it circle $\gamma = \{ z _ { 1 } : | z _ { 1 } | = 1 \}$ and a smooth open arc $\Gamma$ connecting two points of $\gamma$ and lying inside $\gamma$. Let $

...ch functions. At that time one basically studied three, geometrically most simple, cases in the theory of cluster sets: a) $ z _ {0} $ of the second category and metrically dense on some arc $ \gamma \subset \Gamma $,

...) \subseteq \mathbf C$, bounded by an arc of the unit circle and a smooth simple curve $\Gamma \subseteq B ( 0,1 )$ and assume that $f \in C ( \Gamma )$. Th

...{C} \backslash K$. In $\mathbf{C} ^ { n }$, $n &gt; 1$, there is no such a simple topological description. ...slash K$ (cf. also [[Analytic set|Analytic set]]). If $K$ is a rectifiable arc, $K$ is polynomially convex and $P ( K ) = C ( K )$.

If all the roots are simple, then the functions are real and equation (8) has a complex root, for example a simple root $ q _ {k} = \rho ( \cos \phi + i \sin \phi ) $,

...ic features of more general random walks. In particular, even in this very simple scheme there appear properties of "randomness" which are intuitively para \mathop{\rm arc} \sin \sqrt \alpha .

...y classes, since it is a Hilbert space with a reproducing kernel and has a simple description in terms of the Taylor coefficients: ...igher-dimensional case becomes manifest, first of all, in the absence of a simple characterization of the zero sets and of a factorization of functions in th

being the arc $ t ^ \prime tt ^ {\prime\prime} $ ...plex-valued continuous function that does not vanish on an oriented closed simple smooth contour $ \Gamma $,

bounded by a simple Jordan curve $ \sigma $ be a simply-connected mixed domain bounded by a simple Jordan curve $ \sigma $

...okhotskii formulas (2) and (3) remain valid for the interior points of the arc $ \Gamma $. The idea of representing a function on a simple closed curve as the jump of an analytic function across that curve has been

is a simple parametric arc (the homeomorphic image of a closed interval); 3) $ \overline{S}\; \setmi

be the length of the arc $ PQ $ The torsion and the curvature, as functions of the arc length, determine the curve up to its position in space.

denotes the arc length reckoned from some fixed point, let $ \zeta _ {0} = \zeta (s _ {0} regarded as a function of the arc length $ s $,

Alongside this most simple class of $ p $- Geometrically this means that the linear measure of the arc on the Riemann surface to which $ w = f ( z) $

...Of these, the so-called natural parametrization, in which the length of an arc of the curve, counted from some given point, serves as the parameter, is es ...in space, by specifying its curvature and its torsion as functions of the arc length $ s $

...admit one and only one invariant (up to a constant factor). In particular, simple transitive groups are of this type. ...one to determine the mean distance between two points inside the oval in a simple way. The kinematic measure of a set of figures is the measure of the set of

is a system of two non-intersecting smooth simple closed curves $ L $ is the element of arc length of a curve $ L ^ \prime $

is the elementary arc length along $ C $, allows the Maxwell equations to be written in the simple form (8). However, the potential $ A _ {k} $

...cess serves as one of the models of [[Brownian motion|Brownian motion]]. A simple transformation will convert a Wiener process into the "standard" Wiener pro } \mathop{\rm arc} \sin \sqrt {

r( t) = t ^ {-} 1 { \mathop{\rm arc} \mathop{\rm tan} } t,\ \ 1) Simple iteration: $ \phi _ {0} ^ {k+1}= \Phi _ {0} ^ {k} $,

of XX century when appeared computationally simple but {{Cite|Bah67}}, {{Cite|Vaart|&sect; 16.6}}, {{Cite|Arc}}.

...ongs to one of the following types: a) a smooth, simple, non-closed Jordan arc; b) a cycle, i.e. a curve diffeomorphic to a circle; or c) a point (an equi

...ties and differentiable structures), in group theory (algebraic groups and simple finite groups connected with Lie groups), in the theory of differential equ ...es. They found methods for transforming one curve into another of the same arc length, even though the respective arcs could not be brought into correspon

...f small elastic-plastic deformations [[#References|[3]]], which relates to simple deformation processes (zero curvature), (4) takes the form ...d and corresponding variational formulations have been given. A theorem on simple loading has been proved, defining the class of functions $ K _ {i} $,

Dalmatia, who developed in 1760 a simple geometrical method of fitting a Jesuit Christopher Maire were commissioned to measure an arc of

along some arc one obtains a digon with the symbol $ s = aa ^ {- 1} $ is called simply connected if any simple closed curve in $ R $

...hen hitting a recognizer one computes the predicate term and goes along an arc corresponding to the predicate value. When hitting a transformer with opera ...has been constructed, in particular, for programming languages containing simple variables, the assignment operator, compound and conditional operators, cyc

...n $f= \phi + i \psi$ along an (oriented [[Rectifiable curve|rectifiable]]) arc $\gamma$ parametrized by $z: [\alpha, \beta] \to \mathbb C$ may be defined ...connected with analytic functions to the corresponding problems for a very simple function — the [[Cauchy kernel]] $\frac{1}{\zeta-z}$ (where $\zeta\in \pa