# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, November 25, 2014

### Lecture 7 (Nov. 24)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 11:50
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## Geometric form of non-oscillation theorems

Solutions of linear systems $\dot x(t)=A(t)x(t), \ x\in\mathbb R^n,\ t\in[0,\ell]$ can be highly oscillating relatively to hyperplanes $(p,x)=0, \ p\in\mathbb R^{n*}\smallsetminus 0$. However, there exists a class of system for which one can produce such bounds.

Let $\Gamma:t\mapsto x(t)$ be a smooth parametrized curve. Its osculating frame is the tuple of vector functions $v_1(t)=\dot x(t)$ (velocity), $v_2(t)=\dot v_1(t)$ (acceleration), till $v_n(t)=\dot v_{n-1}(t)$. Generically these vectors are linear independent for all $t$ except isolated points. The differential equations defining the curve up to a rigid motion have a “companion form”,

$\dot v_k=v_{k+1},\quad k=1,\dots,n-1,\qquad \dot v_n=\sum_{i=1}^n\alpha_i(t)v_i,\quad \alpha_i\in\mathbb R.$

Note that this is a vector ODE with scalar coefficients, i.e., a tuple of identical scalar ODEs. Besides, it may exhibit singularities: if the osculating frame degenerates (which may well happen at isolated points of the curve), the coefficients of this equation exhibit a pole at the corresponding moments of time $t\in[0,\ell]$.

However, the osculating frame is not a natural object: it depends on the parametrization. The invariant notion is the osculating flag, the flag of subspaces spanned (in $T_x\mathbb R^n\simeq\mathbb R^n$) by the vectors $\mathbb R_1 v_1\subset \mathbb Rv_1+\mathbb Rv_2\subset\cdots$. The flag can be naturally parametrized by the orthogonalization procedure applied to the osculating frame: by construction, this means that we consider the $n$-tuple of orthonormal vectors $e_1(t),\dots,e_n(t)$ with the property that

$\mathrm{Span\ }(v_1,\dots,v_k)= \mathrm{Span\ }(e_1,\dots,e_k),\qquad \forall k=1,\dots, n-1.$

This new frame satisfies the Frenet equations: their structure follows from the invariance of the flag and the orthogonality of the frame.

$\dot e_k(t)=\varkappa_{k-1}(t)e_{k-1}(t)+\varkappa_{k}(t)e_{k+1}(t),\qquad \varkappa_0\equiv\varkappa_{n}\equiv0.$

The functions $\varkappa_1(t),\dots,\varkappa_{n-1}(t)$ are called Frenet curvatures: they are nonnegative except for the last one (hypertorsion) which has sign and may change it at isolated hyperinflection points.

Definitions. (Absolute) integral curvatures of a smooth (say, real analytic) curve $\Gamma:[0,\ell]\to\mathbb R^n$, parametrized by the arclength $t$, are the quantities $K_j=\int_0^\ell|\varkappa_j(t)|\,\mathrm dt$, $j=1,\dots,n-1$, and $K_n=\pi\#\{t:\ \varkappa_{n-1}(t)=0\}$ (the last quality, equal to the number of hyperinflection points up to the constant $\pi$, is called integral hyperinflection).

Let $\Gamma:[0,\ell]\to\mathbb R^n\smallsetminus\{0\}$ be a smooth curve avoiding the origin in the space. Its absolute rotation around the origin $\Omega(G,0)$ is defined as the length of its spherical projection on the unit sphere, $x\mapsto \frac x{\|x\|}$.  The absolute rotation $\Omega(\Gamma, a)$ around any other point $a\notin\Gamma$ is defined by translating this point to the origin.

If $L\subset\mathbb R^n$ is a $k$-dimensional affine subspace disjoint from $\Gamma$ and $P_L:\mathbb R^n\to L^\perp$ the orthogonal projection on the orthogonal complement $L^\perp$, the absolute rotation $\Omega(\Gamma, L)$ of $\latex \Gamma$ around $L$ is the absolute rotation of the curve $P_L\circ\Gamma$ around the point $P_L(L)\in L^\perp\simeq \mathbb R^{n-k}$.

The absolute rotation of $\Gamma$ around an affine hyperplane $L$ is defined as $\pi\cdot \#(\Gamma\cap L)$.

Formally the 0-sphere $\mathbb S^0=\{\pm 1\}\subset\mathbb R^1$ is not connected, but it is convenient to make it into the metric space with two “antipodal” points at the distance $\pi$, similarly to higher dimensional unit spheres with antipodal points always distanced at $\pi$.

Denote by $\Omega_k(\Gamma)$ the supremum $\sup_{\dim L=k}\Omega(\Gamma,L)$, where the supremum is taken over all affine subspaces $L$ of dimension $k$ in $\mathbb R^n$.

Main Theorem.

$\Omega_k(\Gamma)\leqslant n + 4\bigl(K_1(\Gamma)+\cdots+K_{k+1}(\Gamma)\bigr) \qquad \forall k=0,\dots,n-1$.

The proof of this theorem is based on a combination of arguments from integral geometry and the Frobenius formula for a differential operator vanishing on given, say, real analytic functions $f_1(t),\dots,f_n(t)$. Denote by $W_k(t)$ the Wronski determinant of the first $k$ functions $f_1,\dots,f_k$, adding for convenience $W_0\equiv 1,\ W_1\equiv f_1$. These Wronskians are real analytic, and assuming that $W_n$ does not vanish identically, we can construct the linear $n$th order differential operator

$\displaystyle \frac{W_n}{W_{n-1}}\,\partial\,\frac{W_{n-1}}{W_{n}}\cdot\frac{W_{n-1}}{W_{n-2}}\,\partial\,\frac{W_{n-2}}{W_{n-1}}\,\cdots\, \frac{W_2}{W_1}\,\partial\,\frac{W_1}{W_2}\cdot\frac{W_1}{W_0}\,\partial\,\frac{W_0}{W_1}.$

One can instantly see that this operator is monic (composition of monic operators of order 1) and by induction prove that it vanishes on all functions $f_1,\dots, f_n$.

The straightforward application of the Rolle theorem guarantees that if all the Wronskians are nonvanishing on $[0,\ell]$, then the operator is disconjugate and no linear combination of functions $\sum c_i f_i(t)$ can have more than $n-1$ isolated root.

In the case where the Wronskians $W_k(t)$ are allowed to have isolated roots, numbering $\nu_k$ if counted with multiplicity, then the maximal number of zeros that a linear combination as above may exhibit, is bounded by $(n-1)+4\sum_{k=1}^n \nu_k$.

References.

1. A. Khovanskii, S. Yakovenko, Generalized Rolle theorem in $\mathbb R^n$ and $\mathbb C$. Contains detailed description of the so called Voorhoeve index, the total variation of argument of an analytic function on the boundary of its domain and why this serves as a substitute for the Rolle theorem over the complex numbers. As a corollary, rather sharp bounds for the number of complex roots of quasipolynomials $\sum_k p_k(z)\mathrm e^{\lambda_k z}$, $\lambda_k\in\mathbb C,\ p_k\in\mathbb C[z]$ in complex domains are obtained.
2. D. Novikov, S. Yakovenko, Integral curvatures, oscillation and rotation of smooth curves around affine subspaces. Contains the proof of the Main theorem cited below, with a slightly worse weights attached to the integral curvatures.
3. D. Nadler, S. Yakovenko, Oscillation and boundary curvature of holomorphic curves in $\mathbb C^n$. A complex analytic version of the Main theorem with improved estimates.