- Consider the language \(L\) consisting of all words in alphabet a,b,c having no subwords aaa. Find the generating function

$$

f(x)=\sum |L_n| z^n,

$$

where \(L_n\) is the set of words of lengths \(n\) in \(L\). Estimate how fast \(\#

#### About yuliy

## problems for review

## midterm 2

\(\def\pd{\partial}

\def\Real{\mathbb{R}}

\)

- Consider the closed cycle \(\gamma:[0,1]\to\Real^2\) shown on the left (we assume that \(\gamma(0)=\gamma(1)\) is the dot on the positive \(x\)-axis). Define the functions

\[

I_-(t)=\int_0^t \left(\frac{(x+1)dy -ydx}{(x+1)^2+y^2}.\dot{\gamma}(t)\right)dt,\quad I_+(t)=\int_0^t \left(\frac{(x-1)dy -ydx}{(x-1)^2+y^2}.\dot{\gamma}(t)\right)dt,

\]

given as partial integrals over the

## Stokes and relatives

**The (general) Stokes Theorem:**this unique theorem puts

Green’s/Gauss/Stokes Theorems (from intermediate calculus)

under one umbrella. Let \(C\) be a polyhedron with boundary \(\partial C\). Then

$$ {\int_{\partial C} \omega = \int_C d\omega }$$

**Example:**integrate the form \(\omega =

## solutions for problems on vector analysis and differential forms

**Exercise:**

Assume that 4 vectors in \(\mathbb{R}^3\) satisfy \(A+B+C+D\)=0.

Simplify

$$ A\times B-B\times C+C \times D-D\times A$$

** Solution: ** By the anti-symmetry \(-B\times C = C\times B\)

and \(-D\times A = A\times D\) we can rewrite the expression as

$$ …

## March 28

$$ \omega(x)= \sum c_I(x) dx_I $$

where the multi-index

\( I=\{1\leq i_1\leq i_2 \leq, \ldots, \leq i_k \leq n\} \).

Each basis element

\(dx_I:= dx_{i_1} \wedge

## Exterior Differential forms

\(\def\pd{\partial}

\def\Real{\mathbb{R}}

\)

- We define a differential \(k\)-form as

$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$

an exterior form with coefficients depending on the positions.(We will be using sometimes simplifying notation \(\xi_i = dx_i\) or \(\eta_i=dy_i\).)

**Pullback maps:**Let

## Exterior Forms

\(\def\Real{\mathbb{R}}

\)

- To generalize the notion of 1-forms, we need to develop some algebraic apparatus. It is called
*Exterior Calculus.* -
Consider a vector space \(V\cong \mathbb{R}^n\).

We say that the functional

$$ \omega:\underbrace{V\times \cdots \times V}_{k \text{ times}}\to \mathbb{R} $$

## Calculus of Exterior and Differential Forms: Motivation

\(\def\pd{\partial}\def\Real{\mathbb{R}}

\)

- Let \(\Omega \subset \mathbb{R}^n\) be some Euclidean domain, and consider

a curve \( \gamma:\underbrace{[0,1]}_{I} \to \Omega \) given by

$$\gamma(t) = \begin{pmatrix} \gamma_1(t), & \ldots &, \gamma_n(t) \end{pmatrix}. $$Given a real-valued multivariate function \(f:\Omega\to \mathbb{R}\) we’ll try to

## 3.14 Vector Analysis And Classical Identities in 3D

\(\def\pd{\partial}\def\Real{\mathbb{R}}

\)

- There are two main types of products between vectors in \(\mathbb{R}^3\):

The inner/scalar/dot product

$$ A\cdot B = A_x+B_x + A_yB_y + A_z Bz \in \mathbb{R} $$

is commutative, distributive, and homogenenous.

The vector (cross) product:

$$ A\times

## 2.28 operators in Hermitian spaces and their spectra.

\(\def\Real{\mathbb{R}}

\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}

\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}

\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}

\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}

\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}

\def\tr{\mathrm{tr\,}}

\def\braket#1#2{\langle #1|#2\rangle}

\)

**1.**

Given a bilinear form \(Q(\cdot,\cdot)\), or, equivalently, a mapping \(Q:U\to U^*\), one can bake easily *new* bilinear forms from linear operators \(U\to U\): just take \(Q_A(u,v):=Q(u,Av)\).

This opens …