# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, November 15, 2016

### Lecture 2 (Nov. 14, 2016).

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 5:07
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## Tangent vectors, vector fields, integration and derivations

Continued discussion of calculus in domains lf $\mathbb R^n$.

• Tangent vector: vector attached to a point, formally a pair $(a,v):\ a\in U\subseteq\mathbb R^n, \ v\in\mathbb R^n$. Tangent space $T_a U=\ \{a\}\times\mathbb R^n$.
• Differential of a smooth map $F: U\to V$ at a point $a\in U$: the linear map from $T_a U$ to $T_b V,\ b=F(a)$.
• Vector field: a smooth map $v(\cdot): a\mapsto v(a)\in T_a U$.  Vector fields as a module $\mathscr X(U)$ over $C^\infty(U)$.
• Special features of $\mathbb R^1\simeq\mathbb R_{\text{field}}$. Special role of functions as maps $f:\ U\to \mathbb R_{\text{field}}$ and curves as maps $\gamma: \mathbb R_{\text{field}}\to U$.
• Integral curves and derivations.
• Algebra of smooth functions $C^\infty(U)$. Contravariant functor $F \mapsto F^*$ which associates with each smooth map $F:U\to V$ a homomorphism of algebras $F^*:C^\infty(V)\to C^\infty(V)$. Composition of maps vs. composition of morphisms.
• Derivation: a $\mathbb R$-linear map $L:C^\infty(U)\to C^\infty(U)$ which satisfies the Leibniz rule $L(fg)=f\cdot Lg+g\cdot Lf$.
• Vector fields as derivations, $v\simeq L_v$. Action of diffeomorphisms on vector fields (push-forward $F_*$).
• Flow map of a vector field: a smooth map $F: \mathbb R\times U\to U$ (caveat: may be undefined for some combinations unless certain precautions are met) such that each curve
$\gamma_a=F|_{\mathbb R\times \{a\}}$ is an integral curve of $v$ at each point $a$. The “deterministic law” $F^t\circ F^s=F^{t+s}\ \forall t,s\in\mathbb R$.
•  One-parametric (commutative) group of self-homomorphisms $A^t=(F^t)^*: C^\infty(U)\to C^\infty(U)$. Consistency: $L=\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0}A^t=\lim_{t\to 0}\frac{A^t-\mathrm{id}}t$ is a derivation (satisfies the Leibniz rule). If $A^t=(F^t)^*$ is associated with the flow map of a vector field $v$, then $L=L_v$.

Update The corrected and amended notes for the first two lectures can be found here. This file replaces the previous version.

## 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.

## Topology of the plane

1. Functions of two variables as maps $f\colon \mathbb R^2\to\mathbb R$. Domains in the plane. Open and closed domains.
2. Convergence of planar points: $\lim_{n\to\infty}(x_n,y_n)=(X,Y)\in\mathbb R^2\iff\lim x_n=X\ \&\ \lim y_n=Y$. Alternative description: any square $Q_\varepsilon(X,Y)=\{|x-X|<\varepsilon,\ |y-Y|<\varepsilon\}$ contains almost all elements of the sequence.
3. Limits of functions: $f\colon U\to\mathbb R$, $Z=(X,Y)\in\mathbb R$. We say that $A=\lim_{(x,y)\to Z}f(x,y)$, if for any sequence of points $\{(x_n,y_n)\}$ converging to $Z$, the sequence $f(x_n,y_n)$ converges to $A$ as $n\to\infty$. We say that $f$ is continuous at $Z$, if $A=f(Z)$.
4. Exercise: $f$ is continuous at $Z$, if the preimage $f^{-1}(J)$, of any interval $J=(A-\delta,A+\delta),\ \delta>0$, contains the intersection $Q_\varepsilon\cap U$ for some sufficiently small $\varepsilon>0$.
5. Exercise: if $f$ is continuous at all points of the rectangle $\{x\in I,\ y\in J\}\subseteq\mathbb R^2$, then for any $y\in J$ the function $f_y\colon I\to\mathbb R$, defined by the formula $f_y(x)=f(x,y)$ is continuous on $I$. Can one exchange the role of $x$ and $y$? Formulate and think about the inverse statement.
6. Exercise: Check that the functions $f(x,y)=x\pm y$ and $g(x,y)=xy$ are continuous. What can be said about the continuity of the function $f(x,y)=y/x$?
7. Exercise: formulate and prove a theorem on continuity of the composite functions.
8. Exercise: Give the definition of a continuous function $f\colon I\to\mathbb R^2$ for $I\subseteq\mathbb R$. Planar curves.  Simple curves. Closed curves.
9. Intermediate value theorem for curves. Connected sets, connected components. Jordan lemma.
10. Rotation of a closed curve around a point. Continuity of the rotation number. Yet another “Intermediate value theorem” for functions of two variables.

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