On Locally Convex Curves

Modelirovanie i Analiz Informacionnyh Sistem

Vladimir Klimov¹ 

[1] P.G. Demidov Yaroslavl State University,;

关键词: regular curve;    corner function;    degree;    straight line;    differential equation;    polyline;   

DOI  :  10.18255/1818-1015-2017-5-567-577

来源: DOAJ

【 摘 要 】

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  it  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 it 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 non-closed curves. In the sections 2 and 3, we consider curves arising in the investigation of a linear homogeneous differential equation of the form \(L(x) \equiv x^{(n)} + p_1(t) x^{(n-1)} + \cdots p_n(t) x = 0 \) 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.

【 授权许可】

Unknown