
On locally convex curves
V. S. Klimov^{} ^{} P.G. Demidov Yaroslavl State University,
14 Sovetskaya str., Yaroslavl 150003, Russia
Abstract:
We introduce the definition of locally convex curves and establish some properties of such curves. In the section 1, we consider the curve $K$ allowing the parametric representation $x = u(t), y = v(t), (a \leqslant t \leqslant b)$, where $u(t)$, $v(t)$ are continuously differentiable on $[a,b]$ functions such that $u'(t) + v'(t) > 0 \forall t \in [a,b]$. A continuous on $[a,b]$ function $\theta(t)$ is called the angle function of the curve $K$ if the following conditions hold: $u'(t) = \sqrt{(u'(t))^2 + (v'(t))^2} \cos \theta(t), \quad v'(t) = \sqrt{(u'(t))^2 + (v'(t))^2} \sin \theta(t)$. The curve $K$ is called locally convex if its angle function $\theta(t)$ is strictly monotonous on $[a,b]$. For a closed curve $K$ the number $deg K= \cfrac{\theta(b) \theta(a)}{2 \pi}$ is whole. This number is equal to the number of rotations that the speed vector $(u'(t),v'(t))$ performs around the origin. The main result of the first section is the statement: if the curve $K$ is locally convex, then for any straight line $G$ the number $N(K;G)$ of intersections of $K$ and $G$ is finite and the estimate $N(K;G) \leqslant 2 deg K$ holds. We discuss versions of this estimate for closed and nonclosed curves. In the sections 2 and 3, we consider curves arising in the investigation of a linear homogeneous differential equation of the form $L(x) \equiv x^{(n)} + p_1(t) x^{(n1)} + \cdots p_n(t) x = 0 $ with locally summable coefficients $p_i(t) (i = 1, \cdots,n)$. We demonstrate how conditions of disconjugacy of the differential operator $L$ that were established in works of G. A. Bessmertnyh and A. Yu. Levin, can be applied.
Keywords:
regular curve, corner function, degree, straight line, differential equation, polyline.
DOI:
https://doi.org/10.18255/1818101520175567577
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UDC:
513.7 Received: 27.02.2017
Citation:
V. S. Klimov, “On locally convex curves”, Model. Anal. Inform. Sist., 24:5 (2017), 567–577
Citation in format AMSBIB
\Bibitem{Kli17}
\by V.~S.~Klimov
\paper On locally convex curves
\jour Model. Anal. Inform. Sist.
\yr 2017
\vol 24
\issue 5
\pages 567577
\mathnet{http://mi.mathnet.ru/mais584}
\crossref{https://doi.org/10.18255/1818101520175567577}
\elib{http://elibrary.ru/item.asp?id=30353168}
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