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.

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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 nth 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.

Monday, December 14, 2009

Lecture 7 (Dec 15, 2009)

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.

Merry Hanukka!

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