Math Problems Vault

Problem 4.111

(i)(ii)

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(i)

For $n=4$ the transform matrix for the Daubechies wavelet is given by

D_4 = \begin{pmatrix} c_0 & c_1 & c_2 & c_3 \\ c_3 & -c_2 & c_1 & -c_0 \\ c_2 & c_3 & c_0 & c_1 \\ c_1 & -c_0 & c_3 & -c_2 \end{pmatrix}, \quad \begin{pmatrix} c_0 \\ c_1 \\ c_2 \\ c_3 \end{pmatrix} = \frac{1}{4\sqrt{2}} \begin{pmatrix} 1+\sqrt{3} \\ 3+\sqrt{3} \\ 3-\sqrt{3} \\ 1-\sqrt{3} \end{pmatrix} .

Is $D_4$ orthogonal? Prove or disprove.

(ii)

For $n=8$ the transform matrix for the Daubechies wavelet is given by

D_8 = \begin{pmatrix} c_0 & c_1 & c_2 & c_3 & 0 & 0 & 0 & 0 \\ c_3 & -c_2 & c_1 & -c_0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_0 & c_1 & c_2 & c_3 & 0 & 0 \\ 0 & 0 & c_3 & -c_2 & c_1 & -c_0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_0 & c_1 & c_2 & c_3 \\ 0 & 0 & 0 & 0 & c_3 & -c_2 & c_1 & -c_0 \\ c_2 & c_3 & 0 & 0 & 0 & 0 & c_0 & c_1 \\ c_1 & -c_0 & 0 & 0 & 0 & 0 & c_3 & -c_2 \end{pmatrix} .

Is $D_8$ orthogonal? Prove or disprove.

Problem 4.112

(i)(ii)(iii)

Consider the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the basis $x, y, h$ and the commutators

[x,h] = -2x, \quad [x,y] = h, \quad [h,y] = -2y .

Let $t \in \mathbb {R}$ .

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(i)

Find $a(t), b(t), c(t)$ such that

\exp (tx)y\exp (-tx) = a(t)x+b(t)y+c(t)h .

(ii)

Find $a(t), b(t), c(t)$ such that

\exp (tx)h\exp (-tx) = a(t)x+b(t)y+c(t)h .

(iii)

Find $a(t), b(t), c(t)$ such that

\exp (ty)h\exp (-ty) = a(t)x+b(t)y+c(t)h .

We consider the representation

h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} .

Problem 4.113

(i)(ii)

A realization of the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ using vector fields is

V_1 = \frac{d}{du}, \quad V_2 = u\frac{d}{du}, \quad V_3 = u^2\frac{d}{du} .

Consider the Riccati equation

\frac{du}{dt} = 1+u+u^2 .

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(i)

Find the solution of the initial value problem using the Lie series

u(t) = \exp (tV)u\vert _{u=u_0}

where

V = V_1+V_2+V_3 = (1+u+u^2)\frac{d}{du} .

(ii)

Find the functions $f(t), g(t), h(t)$ such that

e^{tV} = e^{f(t)d/du}e^{g(t)ud/du}e^{h(t)u^2d/du} .

Problem 4.114

Let $F$ be a field. In the following we have $F = \mathbb {R}$ or $F = \mathbb {C}$ . A Hopf algebra is a vector space $\mathcal{A}$ endowed with five linear operations

\mu : \mathcal{A}\otimes \mathcal{A} \to \mathcal{A}

\eta : F \to \mathcal{A}

\Delta : \mathcal{A} \to \mathcal{A}\otimes \mathcal{A}

\epsilon : \mathcal{A} \to F

\gamma : \mathcal{A} \to \mathcal{A}

which possess the following properties

\mu \cdot (id\otimes \mu ) = \mu \otimes (\mu \otimes id)

\mu \cdot (id\otimes \eta ) = id = \mu \cdot (\eta \otimes id)

(id\otimes \Delta ) \cdot \Delta = (\Delta \otimes id) \cdot \Delta

(\epsilon \otimes id) \cdot \Delta = id = (id\otimes \epsilon ) \cdot \Delta

\mu \cdot (id\otimes \gamma ) \cdot \Delta = \eta \cdot \epsilon = \mu \cdot (\gamma \otimes id) \cdot \Delta

and the co-multiplication $\Delta$ and co-unit $\epsilon$ are $F$ -algebra morphisms, that is, preserves multiplication. Give an example of a Hopf algebra using a finite group $G$ .

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Problem 4.115

Consider the nonlinear wave equation

\frac{\partial ^2 u}{\partial t^2} - \frac{\partial }{\partial x}\left( u\frac{\partial u}{\partial x} \right) = 0 .

Show that this partial differential equation admits the Lie symmetry vector fields

\frac{\partial }{\partial t}, \quad \frac{\partial }{\partial x}, \quad t\frac{\partial }{\partial t}+x\frac{\partial }{\partial x}, \quad t\frac{\partial }{\partial t}-2u\frac{\partial }{\partial u} .

Calculate the commutators of these vector fields and classify the Lie algebra.

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Problem 4.116

Let $\partial := \partial /\partial x$ . The two pseudodifferential operators

L := \partial + \sum _{j=1}^{\infty } u_{j+1}\partial ^{-j}, \quad W := 1+\sum _{j=1}^{\infty } w_j\partial ^{-j}

are called the Lax operator and gauge operator, respectively. With $\partial^{-1}\partial = \partial \partial^{-1} = 1$ the generalized Leibniz rule is satisfied

\partial ^k f\cdot = \sum _{j=0}^{\infty } \binom {k}{j} (\partial ^j f)\partial ^{k-j} .

for any $k \in \mathbb {Z}$ and $L = W\partial W^{-1}$ . The KP hierarchy is given by the Lax equations

\partial _n L = [B_n,L]

where $\partial_n := \partial /\partial t_n$ and $B_n := L_+^n$ is the differential part of $L^n = L_+^n+L_-^n$ with

L_+^n = \sum _{j=0}^{\infty } c_j^n \partial ^j, \quad L_-^n = \sum _{j=-\infty }^{-1} c_j^n \partial ^j .

Derive the first three equations of the KP hierarchy.

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Problem 4.117

Let $\overrightarrow {v}(x,t) = (v_1(x,t),v_2(x,t))^T$ . Consider the linear equations

\frac{\partial \overrightarrow {v}}{\partial x} = \begin{pmatrix} i\left|u\right|^2-2i\zeta ^2 & i\partial u/\partial t+2\zeta u \\ i\partial u^*/\partial t-2\zeta ^*u & -i\left|u\right|^2+2i\zeta ^2 \end{pmatrix} \overrightarrow {v}

\frac{\partial \overrightarrow {v}}{\partial t} = \begin{pmatrix} -i\zeta & u \\ -u^* & i\zeta \end{pmatrix} \overrightarrow {v} .

Show that the compatibility condition of these two equations with $\partial \zeta /\partial t = 0$ provides the unstable nonlinear Schrodinger equation

i\frac{\partial u}{\partial x} + \frac{\partial ^2 u}{\partial t^2} + 2\left|u\right|^2u = 0 .

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Problem 4.118

(i)(ii)(iii)(iv)

Consider the two-dimensional sphere

S_1^2+S_2^2+S_3^2 = S^2

where $S > 0$ is the radius of the sphere. Consider the symplectic structure on this sphere with the symplectic differential two form

\omega := -\frac{1}{2S^2} \sum _{j,k,\ell =1}^{3} \epsilon _{jk\ell }S_j dS_k \wedge dS_\ell

( $\epsilon_{123} = 1$ ) and the Hamilton vector fields

V_{S_j} := \sum _{k,\ell =1}^{3} \epsilon _{jk\ell }S_k\frac{\partial }{\partial S_\ell } .

The Poisson bracket is defined by

[S_j,S_k]_{PB} := -V_{S_j}S_k .

?

(i)

Calculate $[S_j,S_k]_{PB}$ .

(ii)

Calculate $V_{S_j}\rfloor \omega$ .

(iii)

Calculate $V_{S_j}\rfloor V_{S_k}\rfloor \omega$ .

(iv)

Calculate the Lie derivative $L_{V_{S_j}}\omega$ .

Problem 4.119

(i)(ii)

The special unitary group of degree $n$ , denoted $SU(n)$ , is the group of $n \times n$ unitary matrices with determinant 1. The group operation is matrix multiplication. The infinite-dimensional special unitary group $SU(\infty )$ is the limit of unitary groups $SU(n)$ . One can consider elements of $SU(\infty )$ as transformations of the infinite dimensional separable Hilbert space $\ell_2(\mathbb {N})$ . The sequence space $\ell_2(\mathbb {N})$ consists of all infinite sequences $\overrightarrow {z} = (z_1,z_2,\ldots )$ of complex numbers such that

\sum _{j=1}^{\infty } z_j\overline{z}_n = \sum _{j=1}^{\infty } \left|z_j\right|^2 < \infty .

The inner product (scalar product) in the Hilbert space $\ell_2(\mathbb {N})$ is defined by

\left\langle \overrightarrow {z}, \overrightarrow {w} \right\rangle = \left\langle \overline{\overrightarrow {w}}, \overline{\overrightarrow {z}} \right\rangle = \sum _{j=1}^{\infty } z_j\overline{w}_j .

The convergence of this last series follows from the Cauchy-Schwarz inequality. The space $\ell_2(\mathbb {N})$ is complete. This means that each Cauchy series of elements from $\ell_2(\mathbb {N})$ converges to an element of $\ell_2(\mathbb {N})$ . Consider orthonormal sequences of $\overrightarrow {z}_j \in \ell_2(\mathbb {N})$ . This means that if $\overrightarrow {z}_j$ and $\overrightarrow {z}_k$ are two arbitrary distinct elements of orthonormal sequence, then $\left|\overrightarrow {z}_j\right| = \left|\overrightarrow {z}_k\right| = 1$ and $\left\langle \overrightarrow {z}_j, \overrightarrow {z}_k \right\rangle = 0$ . Consider linear operators $A$ in the Hilbert space $\ell_2(\mathbb {N})$ . The linear operator $A$ is a function from the linear subset $D(A)$ of Hilbert space (the domain of the operator) to linear subset $R(A)$ (the range of the operator) that preserves the operations of vector addition and multiplication by scalar.

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(i)

Show that the set of unitary operators in the Hilbert space $\ell_2(\mathbb {N})$ form the group $U(\infty )$ under composition.

(ii)

Let $U$ be the unitary operator in $\ell_2(\mathbb {N})$ that maps

\overrightarrow {u} = \begin{pmatrix} u_1 & u_2 & u_3 & u_4 & u_5 & \ldots & u_{2n} & u_{2n+1} & \ldots \end{pmatrix}^T

onto

A\overrightarrow {u} = \begin{pmatrix} u_2 & u_4 & u_1 & u_6 & u_3 & \ldots & u_{2n+2} & u_{2n-1} & \ldots \end{pmatrix}^T .

Find the spectrum.

Problem 4.120

(i)(ii)

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(i)

Consider the vector space of operators

\left\{ c_{j_1}^\dagger c_{j_2} c_{k_1}^\dagger c_{k_2} \right\}

where $c_j^\dagger , c_j$ are Fermi creation and annihilation operators and $j_1,j_2,k_1,k_2 = 1,2,\ldots ,n$ . Note that depending on $j_1,j_2,k_1,k_2$ the operator $c_{j_1}^\dagger c_{j_2} c_{k_1}^\dagger c_{k_2}$ could be the zero operator. Calculate the commutator

[c_{j_1}^\dagger c_{j_2} c_{k_1}^\dagger c_{k_2}, c_{m_1}^\dagger c_{m_2} c_{n_1}^\dagger c_{n_2}] .

Thus show that we have a Lie algebra for this set of operators. Of course the Jacobi identity must also be checked.

(ii)

Consider the number operator defined by

\widehat{N} := \sum _{j=1}^{n} c_j^\dagger c_j .

Calculate the commutator $[\widehat{N}, c_{j_1}^\dagger c_{j_2} c_{k_1}^\dagger c_{k_2}]$ . Discuss.