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Exercise 10.11 | Let \( \delta > 0 \) be given. Consider an interest rate swap paying a fixed interest rate \( K \) and receiving backset LIBOR \( L\left( {{T}_{j - 1},{T}_{j - 1}}\right) \) on a principal of \( 1 \) at each of the payment dates \( {T}_{j} = {\delta j}, j = 1,2,\ldots, n + 1 \) . Show that the value of the swap is
\[
{\delta K}\mathop{\sum }\limits_{{j = 1}}^{{n + 1}}B\left( {0,{T}_{j}}\right) - \delta \mathop{\sum }\limits_{{j = 1}}^{{n + 1}}B\left( {0,{T}_{j}}\right) L\left( {0,{T}_{j - 1}}\right) . \tag{10.7.22}
\] | No |
Exercise 2.22 | Exercise 2.22 Let
\[
A = \left\lbrack \begin{matrix} 1 & 1 & - 1 & - 1 \\ 0 & \varepsilon & 0 & 0 \\ 0 & 0 & \varepsilon & 0 \\ 1 & 0 & 0 & 1 \end{matrix}\right\rbrack ,\;b = \left\lbrack \begin{array}{l} 0 \\ 1 \\ 1 \\ 2 \end{array}\right\rbrack .
\]
The solution of the linear system \( {Ax} = b \) is \( x = {\left\lbrack 1,{\varepsilon }^{-1},{\varepsilon }^{-1},1\right\rbrack }^{T} \) .
(a) Show that this system is well-conditioned but badly scaled, by computing the condition number \( {\kappa }_{C}\left( A\right) = {\begin{Vmatrix}{\left| A\right| }^{-1}\left| A\right| \end{Vmatrix}}_{\infty } \) and the scaling quantity \( \sigma \left( {A, x}\right) \) (see Exercise 2.21). What do you expect from Gaussian elimination when \( \varepsilon \) is substituted by the relative machine precision eps?
(b) Solve the system by a Gaussian elimination program with column pivoting for \( \varepsilon = \) eps. How big is the computed backward error \( \widehat{\eta } \) ?
(c) Check yourself that one single refinement step delivers a stable result. | No |
Exercise 10.2 | Consider a market with short term interest rate \( {\left( {r}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) and two zero-coupon bonds \( P\left( {t,{T}_{1}}\right), P\left( {t,{T}_{2}}\right) \) with maturities \( {T}_{1} = \delta \) and \( {T}_{2} = {2\delta } \), where \( P\left( {t,{T}_{i}}\right) \) is modeled according to
\[
\frac{{dP}\left( {t,{T}_{i}}\right) }{P\left( {t,{T}_{i}}\right) } = {r}_{t}{dt} + {\zeta }_{i}\left( t\right) d{B}_{t},\;i = 1,2.
\]
Consider also the forward LIBOR \( L\left( {t,{T}_{1},{T}_{2}}\right) \) defined by
\[
L\left( {t,{T}_{1},{T}_{2}}\right) = \frac{1}{\delta }\left( {\frac{P\left( {t,{T}_{1}}\right) }{P\left( {t,{T}_{2}}\right) } - 1}\right) ,\;0 \leq t \leq {T}_{1},
\]
and assume that \( L\left( {t,{T}_{1},{T}_{2}}\right) \) is modeled in the BGM model as
\[
\frac{{dL}\left( {t,{T}_{1},{T}_{2}}\right) }{L\left( {t,{T}_{1},{T}_{2}}\right) } = {\gamma d}{B}_{t}^{\left( 2\right) },\;0 \leq t \leq {T}_{1}, \tag{10.25}
\]
where \( \gamma \) is a deterministic constant, and
\[
{B}_{t}^{\left( 2\right) } = {B}_{t} - {\int }_{0}^{t}{\zeta }_{2}\left( s\right) {ds}
\]
is a standard Brownian motion under the forward measure \( {\mathbb{P}}_{2} \) defined by
\[
\frac{d{\mathbb{P}}_{2}}{d\mathbb{P}} = \exp \left( {{\int }_{0}^{{T}_{2}}{\zeta }_{2}\left( s\right) d{B}_{s} - \frac{1}{2}{\int }_{0}^{{T}_{2}}{\left| {\zeta }_{2}\left( s\right) \right| }^{2}{ds}}\right) .
\]
(1) Compute \( L\left( {t,{T}_{1},{T}_{2}}\right) \) by solving Equation (10.25).
(2) Compute the price at time \( t \) :
\[
P\left( {t,{T}_{2}}\right) {\mathbb{E}}_{2}\left\lbrack {{\left( L\left( {T}_{1},{T}_{1},{T}_{2}\right) - \kappa \right) }^{ + } \mid {\mathcal{F}}_{t}}\right\rbrack ,\;0 \leq t \leq {T}_{1},
\]
of the caplet with strike \( \kappa \), where \( {\mathbb{E}}_{2} \) denotes the expectation under the forward measure \( {\mathbb{P}}_{2} \) . | Yes |
Exercise 2.7 | Show that any power of an expanding map is still an expanding map. | No |
Exercise 10 | Exercise 10. Let \( f : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be a function of class \( {C}^{2} \) and \( x = {e}^{r}\cos t \) , \( y = {e}^{r}\sin t \)
(i) Compute \( \frac{{\partial }^{2}f}{\partial {r}^{2}},\frac{{\partial }^{2}f}{\partial r\partial t} \) and \( \frac{{\partial }^{2}}{\partial {t}^{2}} \) ;
(ii) Prove that \( \frac{{\partial }^{2}f}{\partial {r}^{2}} + \frac{{\partial }^{2}f}{\partial {t}^{2}} = {e}^{2r}\left( {\frac{{\partial }^{2}f}{\partial {x}^{2}} + \frac{{\partial }^{2}f}{\partial {y}^{2}}}\right) \) . | No |
Exercise 8.5 | \[
{\mathbb{P}}_{0}\{ B\left( {0,\tau }\right) \cap A \neq \varnothing \} \leq \frac{{C}_{a}}{1 - {e}^{-1/2}}\mathbb{P}\{ B\left\lbrack {0,1}\right\rbrack \cap A \neq \varnothing \} ,
\]
where \( {C}_{a} \leq {e}^{{\left| a\right| }^{2}} + {\mathbb{P}}_{0}\left\{ {\left| {B\left( \frac{1}{2}\right) }\right| > a{\} }^{-1}}\right. \) . To this end, let \( H\left( I\right) = {\mathbb{P}}_{0}\{ B\left( I\right) \cap A \neq \varnothing \} \) ,
where \( I \) is an interval. Then \( H \) satisfies \( H\left\lbrack {t, t + \frac{1}{2}}\right\rbrack \leq {C}_{a}H\left\lbrack {\frac{1}{2},1}\right\rbrack \) for \( t \geq \frac{1}{2} \) . Hence, we can conclude that
\[
\mathbb{E}H\left\lbrack {0,\tau }\right\rbrack \leq H\left\lbrack {0,1}\right\rbrack + \mathop{\sum }\limits_{{j = 2}}^{\infty }{e}^{-j/2}H\left\lbrack {\frac{j}{2},\frac{j + 1}{2}}\right\rbrack \leq {C}_{a}\mathop{\sum }\limits_{{j = 0}}^{\infty }{e}^{-j/2}H\left\lbrack {0,1}\right\rbrack ,
\]
which is the required statement. | No |
Exercise 7.2.7 | Find \( \int {\left( 5{t}^{2} + {10}t + 3\right) }^{3}\left( {{5t} + 5}\right) {dt} \) . | No |
Exercise 19.1 | Exercise 19.1. Use Figure 19.2 to give another proof of (19.1). (Hint: express \( \left| {AC}\right| \) in terms of \( z \) and note that the two shaded triangles are similar.) | No |
Exercise 1.1.3 | Exercise 1.1.3 You have a system of \( k \) equations in two variables, \( k \geq 2 \) . Explain the geometric significance of
(a) No solution.
(b) A unique solution.
(c) An infinite number of solutions. | No |
Exercise 8.3.3 | Exercise 8.3.3. Check the orthonormality of the characters of the irreducible representations of \( {S}_{3} \) and \( {S}_{4} \) . The characters are collected in Table 8.1. | No |
Exercise 1 | Exercise 1. Prove that \( \parallel \cdot {\parallel }_{\infty } \) is indeed a norm on \( {c}_{0}^{\mathbb{K}}\left( I\right) \) . | No |
Exercise 7.17 | Consider a model which consists of a charged complex scalar field interacting with an Abelian gauge field. The classical Lagrangian is
\[
L\left\lbrack {\varphi ,{A}_{\mu }}\right\rbrack = - \frac{1}{2}{\left( {D}_{\mu }\varphi \right) }^{ * }{D}_{\mu }\varphi - \frac{\lambda }{4}{\left( {\left| \varphi \right| }^{2} - {\mu }^{2}\right) }^{2} - \frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }, \tag{7.147}
\]
where \( {F}_{\mu \nu } = {\partial }_{\mu }{A}_{\nu } - {\partial }_{\nu }{A}_{\mu } \) and \( {D}_{\mu } = {\partial }_{\mu } - {ie}{A}_{\mu } \) . The theory is invariant with respect to local \( U\left( 1\right) \) gauge transformations. The classical potential has a continuous family of minima at \( \left| \varphi \right| = \mu \) . Model (7.147) can be used to illustrate the Higgs mechanism; the gauge group is spontaneously broken in the vacuum state because the gauge field acquires a mass \( {m}_{v}^{2} = {e}^{2}{\mu }^{2} \) when \( \left| \varphi \right| = \mu \) .
Calculate the Coleman-Weinberg potential for model (7.147) in the regime when \( {e}^{2} \gg \lambda \) . Show that in the ground state quantum corrections result in appearance of a new minimum where the symmetry is restored. | No |
Exercise 12.2 | Exercise 12.2.
(a) Let \( c \in \mathbf{R} \) be a constant. Use Lagrange multipliers to generate a list of candidate points to be extrema of
\[
h\left( {x, y, z}\right) = \sqrt{\frac{{x}^{2} + {y}^{2} + {z}^{2}}{3}}
\]
on the plane \( x + y + z = {3c} \) . (Hint: explain why squaring a non-negative function doesn’t affect where it achieves its maximal and minimal values.)
(b) The facts that \( h\left( {x, y, z}\right) \) in (a) is non-negative on all inputs (so it is "bounded below") and grows large when \( \parallel \left( {x, y, z}\right) \parallel \) grows large can be used to show that \( h\left( {x, y, z}\right) \) must have a global minimum on the given plane. (You may accept this variant of the Extreme Value Theorem from single-variable calculus; if you are interested, such arguments are taught in Math 115 and Math 171.) Use this and your result from part (a) to find the minimum value of \( h\left( {x, y, z}\right) \) on the plane \( x + y + z = {3c}. \)
(c) Explain why your result from part (b) implies the inequality
\[
\sqrt{\frac{{x}^{2} + {y}^{2} + {z}^{2}}{3}} \geq \frac{x + y + z}{3}
\]
for all \( x, y, z \in \mathbf{R} \) . (Hint: for any given \( \mathbf{v} = \left( {x, y, z}\right) \), define \( c = \left( {1/3}\right) \left( {x + y + z}\right) \) so \( \mathbf{v} \) lies in the constraint plane in the preceding discussion, and compare \( h\left( \mathbf{v}\right) \) to the minimal value of \( h \) on the entire plane using your answer in (b).) The left side is known as the "root mean square" or "quadratic mean," while the right side is the usual or "arithmetic" mean. Both come up often in statistics. | Yes |
Exercise 23 | Exercise 23 (Recession functions) | No |
Exercise 2.23 | Exercise 2.23. Show that if \( A \in {\mathbb{C}}^{n \times n} \) is an invertible triangular matrix with entries \( {a}_{ij} \in \mathbb{C} \) for \( i, j = 1,\ldots, n \), then \( {a}_{ii} \neq 0 \) for \( i = 1,\ldots, n \) . [HINT: Use Theorem 2.4 to show that if the claim is true for \( n = k \), then it is also true for \( n = k + 1 \) .] | No |
Exercise 13.4 | Verify that laplace correctly computes the Laplace Transforms of the functions heaviside \( \left( {t - 2}\right) \) and \( \operatorname{dirac}\left( {t - 3}\right) \) . | No |
Exercise 19.10 | Exercise 19.10. Consider a two-dimensional system where \( \operatorname{tr}\left( A\right) = 0 \) and \( \det \left( A\right) > 0 \) .
a. Given those conditions, explain why \( {\lambda }_{1} + {\lambda }_{2} = 0 \) and \( {\lambda }_{1} \cdot {\lambda }_{2} > 0 \) .
b. What does \( {\lambda }_{1} + {\lambda }_{2} = 0 \) tell you about the relationship between \( {\lambda }_{1} \) and \( {\lambda }_{2} \) ?
c. What does \( {\lambda }_{1} \cdot {\lambda }_{2} > 0 \) tell you about the relationship between \( {\lambda }_{1} \) and \( {\lambda }_{2} \) ?
d. Look back to your previous two responses. First explain why \( {\lambda }_{1} \) and \( {\lambda }_{2} \) must be imaginary eigenvalues (in other words, not real values). Then explain why \( {\lambda }_{1,2} = \pm {bi} \) .
e. Given these constraints, what would the phase plane for this system be?
f. Create a linear two-dimensional system where \( \operatorname{tr}\left( A\right) = 0 \) and \( \det \left( A\right) > 0 \) . Show your system and the phase plane. | No |
Exercise 1.23 | Exercise 1.23 (Boolean Group)
Let \( M \) be a set.
a. If \( X, Y, Z \subseteq M \), then
\[
X \smallsetminus \left( {\left( {Y \smallsetminus Z}\right) \cup \left( {Z \smallsetminus Y}\right) }\right) = \left( {X \smallsetminus \left( {Y \cup Z}\right) }\right) \cup \left( {X \cap Y \cap Z}\right)
\]
and
\[
\left( {\left( {X \smallsetminus Y}\right) \cup \left( {Y \smallsetminus X}\right) }\right) \smallsetminus Z = \left( {X \smallsetminus \left( {Y \cup Z}\right) }\right) \cup \left( {Y \smallsetminus \left( {X \cup Z}\right) }\right) .
\]
b. We define on the power set \( G = \mathcal{P}\left( M\right) = \{ A \mid A \subseteq M\} \) of \( M \) a binary operation
Operation by
\[
A + B \mathrel{\text{:=}} \left( {A \smallsetminus B}\right) \cup \left( {B \smallsetminus A}\right) = \left( {A \cup B}\right) \smallsetminus \left( {A \cap B}\right)
\]
for \( A, B \in G \) . Show that \( \left( {G, + }\right) \) is an abelian group. | No |
Exercise 8.5.3 | Exercise 8.5.3. Modify the birth and death rates and study the behavior of the population over time (you will need to re-initialize the population each time you specify new birth and death rates). | No |
Exercise 4.15 | Recall that \( U\left( 1\right) \) is the group of \( 1 \times 1 \) unitary matrices. Show that this is just the set of complex numbers \( z \) with \( \left| z\right| = 1 \), and that \( U\left( 1\right) \) is isomorphic to \( {SO}\left( 2\right) \). | No |
Exercise 6.13 | Exercise 6.13 (c) Find the expected number of customers seen in the system by the first arrival after time \( {n\delta } \) . Note: The purpose of this exercise is to make you cautious about the meaning of 'the state seen by a random arrival'. | Yes |
Exercise 2.2 | Use Strategy 2.1 to express the following cycles in \( {S}_{7} \) as composites of transpositions.
(a) \( \left( \begin{array}{lllll} 1 & 5 & 2 & 7 & 3 \end{array}\right) \) (b) \( \left( \begin{array}{llllll} 2 & 3 & 7 & 5 & 4 & 6 \end{array}\right) \) (c) \( \left( \begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \end{array}\right) \) | No |
Exercise 4.6.10 | [Exercise 4.6.10 (Kuratowski). Prove that the axioms for topology can be rephrased in terms of the closure. In other words, a topology on \( X \) may be defined as an operation \( A \mapsto \bar{A} \) on subsets of \( X \) satisfying
- \( \bar{\varnothing } = \varnothing \) .
- \( \overline{\{ x\} } = \{ x\} \) .
- \( \overline{\bar{A}} = \bar{A} \) .
- \( \overline{A \cup B} = \bar{A} \cup \bar{B} \) .] | No |
Exercise 9.20 | Compute the variance of the decision alternatives for the decision in Example 9.5. Plot risk profiles and cumulative risk profiles for the decision alternatives. Discuss whether you find the variance or the risk profiles more helpful in determining the risk inherent in each alternative. | No |
Exercise 8.6 | [Exercise 8.6. Verify the claims made in subsection 8.3 .2 about the ranks of the matrices \( {A}_{n} \) for \( n \leq 5 \) .] | Yes |
Exercise 4 | Suppose Properties P1., P2. and P3. hold. State and prove the dual of Property \( {P3} \) . | No |
Exercise 7.1.19 | Exercise 7.1.19. Suppose a topology is regular. Is a finer topology also regular? What about a coarser topology? | No |
Exercise 1.12 | In Example 1.6.6, we began with a standard normal random variable \( X \) on a probability space \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \) and defined the random variable \( Y = X + \theta \), where \( \theta \) is a constant. We also defined \( Z = {e}^{-{\theta X} - \frac{1}{2}{\theta }^{2}} \) and used \( Z \) as the Radon-Nikodým dcrivativc to construct the probability measure \( \widetilde{\mathbb{P}} \) by the formula (1.6.9):
\[
\widetilde{\mathbb{P}}\left( A\right) = {\int }_{A}Z\left( \omega \right) d\mathbb{P}\left( \omega \right) \text{ for all }A \in \mathcal{F}.
\]
Under \( \widetilde{\mathbb{P}} \), the random variable \( Y \) was shown to be standard normal.
We now have a standard normal random variable \( Y \) on the probability space \( \left( {\Omega ,\mathcal{F},\widetilde{\mathbb{P}}}\right) \), and \( X \) is related to \( Y \) by \( X = Y - \theta \) . By what we have just stated, with \( X \) replaced by \( Y \) and \( \theta \) replaced by \( - \theta \), we could define \( \widehat{Z} = {e}^{{\theta Y} - \frac{1}{2}{\theta }^{2}} \) and then use \( \widehat{Z} \) as a Radon-Nikodým derivative to construct a probability measure \( \widehat{\mathbb{P}} \) by the formula
\[
\widehat{\mathbb{P}}\left( A\right) = {\int }_{A}\widehat{Z}\left( \omega \right) d\widetilde{\mathbb{P}}\left( \omega \right) \text{ for all }A \in \mathcal{F},
\]
so that, under \( \widehat{\mathbb{P}} \), the random variable \( X \) is standard normal. Show that \( \widehat{Z} = \frac{1}{Z} \) and \( \widehat{\mathbb{P}} = \mathbb{P} \) . | No |
Exercise 6.31 | Exercise 6.31 Let \( {\Lambda }^{3} = \mathbb{N} \times \mathbb{N} \times 3\mathbb{Z} \) and \( f : {\Lambda }^{3} \rightarrow \mathbb{R} \) be defined as
\[
f\left( t\right) = {t}_{1}{t}_{2}{t}_{3},\;t = \left( {{t}_{1},{t}_{2},{t}_{3}}\right) \in {\Lambda }^{3}.
\]
Find 1. \( {f}^{\sigma }\left( t\right) \) ,
2. \( {f}_{1}^{{\sigma }_{1}}\left( t\right) \) ,
3. \( {f}_{2}^{{\sigma }_{2}}\left( t\right) \) ,
4. \( {f}_{3}^{{\sigma }_{3}}\left( t\right) \) ,
5. \( {f}_{12}^{{\sigma }_{1}{\sigma }_{2}}\left( t\right) \) ,
6. \( {f}_{13}^{{\sigma }_{1}{\sigma }_{3}}\left( t\right) \) ,
7. \( {f}_{23}^{{\sigma }_{2}{\sigma }_{3}}\left( t\right) \) ,
8. \( g\left( t\right) = {f}^{\sigma }\left( t\right) + {f}_{1}^{{\sigma }_{1}}\left( t\right) \) .
Solution \( 1.{t}_{1}{t}_{2}{t}_{3} + 3{t}_{1}{t}_{2} + {t}_{1}{t}_{3} + {t}_{2}{t}_{3} + 3{t}_{1} + 3{t}_{2} + {t}_{3} + 3, \)
2. \( {t}_{1}{t}_{2}{t}_{3} + {t}_{2}{t}_{3} \) ,
3. \( {t}_{1}{t}_{2}{t}_{3} + {t}_{1}{t}_{3} \) ,
4. \( {t}_{1}{t}_{2}{t}_{3} + 3{t}_{1}{t}_{2} \) ,
5. \( {t}_{1}{t}_{2}{t}_{3} + {t}_{1}{t}_{3} + {t}_{2}{t}_{3} + {t}_{3} \) ,
6. \( {t}_{1}{t}_{2}{t}_{3} + 3{t}_{1}{t}_{2} + {t}_{2}{t}_{3} + 3{t}_{2} \) ,
7. \( {t}_{1}{t}_{2}{t}_{3} + 3{t}_{1}{t}_{2} + {t}_{1}{t}_{3} + 3{t}_{1} \) ,
8. \( 2{t}_{1}{t}_{2}{t}_{3} + 3{t}_{1}{t}_{2} + {t}_{1}{t}_{3} + 2{t}_{2}{t}_{3} + 3{t}_{1} + 3{t}_{2} + {t}_{3} + 3 \) . | Yes |
Exercise 23.10 | Exercise 23.10. For any \( n \geq 1 \) we have defined the scalar-valued dot product \( \mathbf{v} \cdot \mathbf{w} \) for any \( n \) -vectors
\( \mathbf{v} \) and \( \mathbf{w} \) . In the case \( n = 3 \) there is another type of "product" that is vector-valued: for \( \mathbf{v} = \left\lbrack \begin{array}{l} {v}_{1} \\ {v}_{2} \\ {v}_{3} \end{array}\right\rbrack \) and
\( \mathbf{w} = \left\lbrack \begin{array}{l} {w}_{1} \\ {w}_{2} \\ {w}_{3} \end{array}\right\rbrack \) the cross product \( \mathbf{v} \times \mathbf{w} \in {\mathbf{R}}^{3} \) is defined to be
\[
\mathbf{v} \times \mathbf{w} = \left\lbrack \begin{array}{l} {v}_{2}{w}_{3} - {v}_{3}{w}_{2} \\ {v}_{3}{w}_{1} - {v}_{1}{w}_{3} \\ {v}_{1}{w}_{2} - {v}_{2}{w}_{1} \end{array}\right\rbrack = \det \left\lbrack \begin{array}{ll} {v}_{2} & {v}_{3} \\ {w}_{2} & {w}_{3} \end{array}\right\rbrack {\mathbf{e}}_{1} - \det \left\lbrack \begin{array}{ll} {v}_{1} & {v}_{3} \\ {w}_{1} & {w}_{3} \end{array}\right\rbrack {\mathbf{e}}_{2} + \det \left\lbrack \begin{array}{ll} {v}_{1} & {v}_{2} \\ {w}_{1} & {w}_{2} \end{array}\right\rbrack {\mathbf{e}}_{3}
\]
(note the minus sign in front of the second determinant on the right). This concept is very specific to the case \( n = 3 \), and arises in a variety of important physics and engineering applications. General details on the cross product are given in Appendix F.
(a) Verify algebraically that \( \mathbf{w} \times \mathbf{v} = - \left( {\mathbf{v} \times \mathbf{w}}\right) \) ("anti-commutative"), and \( \mathbf{v} \times \mathbf{v} = \mathbf{0} \) for every \( \mathbf{v} \) (!).
(b) For \( \mathbf{v} = \left\lbrack \begin{matrix} 2 \\ - 1 \\ 3 \end{matrix}\right\rbrack ,\mathbf{w} = \left\lbrack \begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right\rbrack ,\mathbf{u} = \left\lbrack \begin{matrix} 4 \\ 3 \\ - 2 \end{matrix}\right\rbrack \), use the description via \( 2 \times 2 \) determinants to verify:
\( \mathbf{v} \times \mathbf{w} = \left\lbrack \begin{matrix} - 9 \\ - 3 \\ 5 \end{matrix}\right\rbrack ,\mathbf{w} \times \mathbf{u} = \left\lbrack \begin{matrix} - {13} \\ {14} \\ - 5 \end{matrix}\right\rbrack ,\left( {\mathbf{v} \times \mathbf{w}}\right) \times \mathbf{u} = \left\lbrack \begin{matrix} - 9 \\ 2 \\ - {15} \end{matrix}\right\rbrack \), and \( \mathbf{v} \times \left( {\mathbf{w} \times \mathbf{u}}\right) = \left\lbrack \begin{matrix} - {37} \\ - {29} \\ {15} \end{matrix}\right\rbrack \) . (The latter
two are not equal, illustrating that the cross product is not associative: parentheses matter!)
(c) For a general scalar \( c \) verify algebraically that \( \left( {c\mathbf{v}}\right) \times \mathbf{w} = c\left( {\mathbf{v} \times \mathbf{w}}\right) \), and for a general third vector \( {\mathbf{v}}^{\prime } \) verify algebraically that \( \left( {\mathbf{v} + {\mathbf{v}}^{\prime }}\right) \times \mathbf{w} = \mathbf{v} \times \mathbf{w} + {\mathbf{v}}^{\prime } \times \mathbf{w} \) (distributivity over vector addition, which is the reason this operation deserves to be called a "product").
(d) For linearly independent \( \mathbf{v} \) and \( \mathbf{w} \) making an angle \( \theta \in \left( {0,\pi }\right) \), the vector \( \mathbf{v} \times \mathbf{w} \) is perpendicular to \( \mathbf{v} \) and \( \mathbf{w} \) with magnitude \( \parallel \mathbf{v}\parallel \parallel \mathbf{w}\parallel \) sin \( \left( \theta \right) \) . Verify these orthogonality and magnitude properties for the specific 3-vectors \( \mathbf{v} \) and \( \mathbf{w} \) in (b). (Hint on the magnitude aspect: \( \sin \left( \theta \right) = \sqrt{1 - {\cos }^{2}\left( \theta \right) } \) since \( \sin \left( \theta \right) > 0 \) for \( 0 < \theta < \pi \), and \( \cos \left( \theta \right) \) can be computed via a dot product.) | No |
Exercise 6.10 | Exercise 6.10. Prove or disprove the following statements:
1. In the Smale horseshoe, the periodic points of period odd are dense.
2. In the Smale horseshoe, the periodic points of period prime are dense.
3. In the Smale horseshoe, the periodic points of period at least 100 are dense. | No |
Exercise 8.5.10 | Exercise 8.5.10. Suppose \( A, B \), and \( {AB} \) are symmetric. Show that \( A \) and \( B \) are simultaneously diagonalizable. Is \( {BA} \) symmetric? | No |
Exercise 3.8.2 | Show that if condition 4 is satisfied, then conditions (3.8.4) and (3.8.5) hold. | No |
Exercise 1.1.4 | Exercise 1.1.4 Show that \( \left\{ {c}_{\alpha }\right\} \) is summable if and only if \( \left\{ \left| {c}_{\alpha }\right| \right\} \) is summable; show also that \( \left\{ {c}_{\alpha }\right\} \) is summable if and only if
\[
\left\{ {\left| {\mathop{\sum }\limits_{{\alpha \in A}}{c}_{\alpha }}\right| : A \in F\left( I\right) }\right\}
\]
is bounded. | No |
Exercise 9.10 | Exercise 9.10. A dog’s weight \( W \) (pounds) changes over \( D \) days according to the following function:
\[
W = f\left( {D,{p}_{1},{p}_{2}}\right) = \frac{{p}_{1}}{1 + {e}^{{2.462} - {p}_{2}D}}, \tag{9.9}
\]
where \( {p}_{1} \) and \( {p}_{2} \) are parameters.
a. This function can be used to describe the data wilson. Make a scatterplot with the wilson data. What is the long term weight of the dog?
b. Generate a contour plot for the likelihood function for these data. What are the values of \( {p}_{1} \) and \( {p}_{2} \) that optimize the likelihood? You may assume that \( {p}_{1} \) and \( {p}_{2} \) are both positive.
c. With your values of \( {p}_{1} \) and \( {p}_{2} \) add the function \( W \) to your scatterplot and compare the fitted curve to the data. | Yes |
Exercise 3.31 | Exercise 3.31. (Continuation of Exercise 3.27) Consider matrices of the form
\[
\left( \begin{matrix} p & 1 - p & a \\ q & 1 - q & b \\ 0 & 0 & c \end{matrix}\right) ,
\]
where \( 0 < p, q < 1, a \) and \( b \) are real, and \( c = \pm 1 \) . | No |
Exercise 3.3.15 | b) This matrix equals its own conjugate transpose:
\[
{\left\lbrack \begin{matrix} 0 & 2 + {3i} \\ 2 - {3i} & 4 \end{matrix}\right\rbrack }^{ * } = \left\lbrack \begin{matrix} 0 & 2 + {3i} \\ 2 - {3i} & 4 \end{matrix}\right\rbrack .
\] | No |
Exercise 8.28 | Exercise 8.28 Let \( \mathbb{T} = ( - \infty ,0\rbrack \cup \mathbb{N} \), where \( ( - \infty ,0\rbrack \) is the real line interval. Find \( l\left( \Gamma \right) \), where
\[
\Gamma = \left\{ \begin{array}{l} {x}_{1} = {t}^{3} \\ {x}_{2} = {t}^{2},\;t \in \left\lbrack {-1,0}\right\rbrack \cup \{ 1,2,3\} . \end{array}\right.
\]
Solution \( \frac{1}{27}\left( {8 - {13}\sqrt{13}}\right) + \sqrt{2} + \sqrt{58} + \sqrt{386} \) . | Yes |
Exercise 7.2.5 | Exercise 7.2.5 Let \( X \) be a spectral domain and let \( L \) be its lattice of compact open subsets. Prove that \( \mathcal{J}{\left( L\right) }^{\text{op }} \) is isomorphic to \( \mathrm{K}\left( X\right) \) . Hint. You can describe an isomorphism directly: Send \( p \in \mathrm{K}\left( X\right) \) to the join-prime element \( \uparrow p \) of \( L \) . | No |
Exercise 2.7 | Exercise 2.7. Let \( \{ B\left( t\right) : t \geq 0\} \) be a standard Brownian motion on the line, and \( T \) be a stopping time with \( \mathbb{E}\left\lbrack T\right\rbrack < \infty \) . Define an increasing sequence of stopping times by \( {T}_{1} = T \) and \( {T}_{n} = T\left( {B}_{n}\right) + {T}_{n - 1} \) where the stopping time \( T\left( {B}_{n}\right) \) is the same function as \( T \), but associated with the Brownian motion \( \left\{ {{B}_{n}\left( t\right) : t \geq 0}\right\} \) given by
\[
{B}_{n}\left( t\right) = B\left( {t + {T}_{n - 1}}\right) - B\left( {T}_{n - 1}\right) .
\]
(a) Show that, almost surely,
\[
\mathop{\lim }\limits_{{n \uparrow \infty }}\frac{B\left( {T}_{n}\right) }{n} = 0
\]
(b) Show that \( B\left( T\right) \) is integrable.
(c) Show that, almost surely,
\[
\mathop{\lim }\limits_{{n \uparrow \infty }}\frac{B\left( {T}_{n}\right) }{n} = \mathbb{E}\left\lbrack {B\left( T\right) }\right\rbrack
\]
Combining (a) and (c) implies that \( \mathbb{E}\left\lbrack {B\left( T\right) }\right\rbrack = 0 \), which is Wald’s lemma. | No |
Exercise 2.5 | Exercise 2.5 Imagine two ways other than changing the size of the points (as in Section 2.7.2) to introduce a third variable in the plot. | No |
Exercise 2.6.9 | Exercise 2.6.9. Suppose \( E{X}_{i} = 0 \) . Show that if \( \epsilon > 0 \) then
\[
\mathop{\liminf }\limits_{{n \rightarrow \infty }}P\left( {{S}_{n} \geq {na}}\right) /{nP}\left( {{X}_{1} \geq n\left( {a + \epsilon }\right) }\right) \geq 1
\]
Hint: Let \( {F}_{n} = \left\{ {{X}_{i} \geq n\left( {a + \epsilon }\right) }\right. \) for exactly one \( \left. {i \leq n}\right\} \) . | No |
Exercise 3.1 | Exercise 3.1. Prove the theorem via a direct verification of the Anscombe condition (3.2).
For the law of large numbers it was sufficient that \( N\left( t\right) \overset{a.s.}{ \rightarrow } + \infty \) as \( t \rightarrow \infty \) . That this is not enough for a "random-sum central limit theorem" can be seen as follows. | No |
Exercise 10.3 | Exercise 10.3 Find a rectangular block (not a cube) and label the sides. Determine values of \( {a}_{1},{a}_{2},\ldots ,{a}_{6} \) that represent your prior probability concerning each side coming up when you throw the block.
1. What is your probability of each side coming up on the first throw?
2. Throw the block 20 times. Compute your probability of each side coming up on the next throw. | No |
Exercise 2.6.1 | Exercise 2.6.1. Compute the topological entropy of an expanding endomorphism \( {E}_{m} : {S}^{1} \rightarrow {S}^{1} \) . | Yes |
Exercise 1.3.11 | Exercise 1.3.11. ([28], Proposition 3.4) Let \( M \) be an \( R \) -module, and \( S = \) \( \{ I \subseteq R \mid I = \operatorname{ann}\left( m\right) \), some \( m \in M\} \) . Prove that a maximal element of \( S \) is prime. \( \diamond \) | No |
Exercise 4.4.5 | Let \( {A}_{t} = t - {T}_{N\left( t\right) - 1} \) be the "age" at time \( t \), i.e., the amount of time since the last renewal. If we fix \( x > 0 \) then \( H\left( t\right) = P\left( {{A}_{t} > x}\right) \) satisfies the renewal equation
\[
H\left( t\right) = \left( {1 - F\left( t\right) }\right) \cdot {1}_{\left( x,\infty \right) }\left( t\right) + {\int }_{0}^{t}H\left( {t - s}\right) {dF}\left( s\right)
\]
so \( P\left( {{A}_{t} > x}\right) \rightarrow \frac{1}{\mu }{\int }_{\left( x,\infty \right) }\left( {1 - F\left( t\right) }\right) {dt} \), which is the limit distribution for the residual
lifetime \( {B}_{t} = {T}_{N\left( t\right) } - t \) . | No |
Exercise 7.1.4 | Exercise 7.1.4. By taking the product of two of three topologies \( {\mathbb{R}}_{ \leftrightarrow },{\mathbb{R}}_{ \rightarrow },{\mathbb{R}}_{ \leftarrow } \), we get three topologies on \( {\mathbb{R}}^{2} \) . Which subspaces are Hausdorff?
1. \( \{ \left( {x, y}\right) : x + y \in \mathbb{Z}\} \) . 2. \( \{ \left( {x, y}\right) : {xy} \in \mathbb{Z}\} \) . 3. \( \left\{ {\left( {x, y}\right) : {x}^{2} + {y}^{2} \leq 1}\right\} \) . | No |
Exercise 4.4.32 | Exercise 4.4.32 Show that \( {\int }_{0}^{t}\operatorname{sgn}\left( {B\left( s\right) }\right) {dB}\left( s\right) \) is a Brownian motion. | No |
Exercise 6.18 | Show that if \( \Lambda \) is a hyperbolic set for a flow \( \Phi \), then the stable and unstable subspaces \( {E}^{s}\left( x\right) \) and \( {E}^{u}\left( x\right) \) vary continuously with \( x \in \Lambda \) . | No |
Exercise 8.7.3 | Exercise 8.7.3. Model the problem of finding a nontrivial factor of a given integer as a nonlinear integer optimization problem of the form (8.1). Then explain why the algorithm of this chapter does not imply a polynomial-time algorithm for factoring. | No |
Exercise 7.1.3 | Exercise 7.1.3. Which subspaces of the line with two origins in Example 5.5.2 are Hausdorff? | No |
Exercise 10 | Exercise 10 (Tangents to graphs) | No |
Exercise 6.8.10 | Let \( {V}_{n} \) be an armap (not necessarily smooth or simple) with \( \theta < 1 \) and \( E{\log }^{ + }\left| {\xi }_{n}\right| < \infty \) . Show that \( \mathop{\sum }\limits_{{m \geq 0}}{\theta }^{m}{\xi }_{m} \) converges a.s. and defines a stationary distribution for \( {V}_{n} \) . | No |
Exercise 5.7 | Exercise 5.7. (i) Suppose a multidimensional market model as described in Section 5.4.2 has an arbitrage. In other words, suppose there is a portfolio value process satisfying \( {X}_{1}\left( 0\right) = 0 \) and
\[
\mathbb{P}\left\{ {{X}_{1}\left( T\right) \geq 0}\right\} = 1,\;\mathbb{P}\left\{ {{X}_{1}\left( T\right) > 0}\right\} > 0,
\]
\( \left( {5.4.23}\right) \)
for some positive \( T \) . Show that if \( {X}_{2}\left( 0\right) \) is positive, then there exists a portfolio value process \( {X}_{2}\left( t\right) \) starting at \( {X}_{2}\left( 0\right) \) and satisfying
\[
\mathbb{P}\left\{ {{X}_{2}\left( T\right) \geq \frac{{X}_{2}\left( 0\right) }{D\left( T\right) }}\right\} = 1,\;\mathbb{P}\left\{ {{X}_{2}\left( T\right) > \frac{{X}_{2}\left( 0\right) }{D\left( T\right) }}\right\} > 0.
\]
\( \left( {5.4.24}\right) \)
(ii) Show that if a multidimensional market model has a portfolio value process \( {X}_{2}\left( t\right) \) such that \( {X}_{2}\left( 0\right) \) is positive and (5.4.24) holds, then the model has a portfolio value process \( {X}_{1}\left( 0\right) \) such that \( {X}_{1}\left( 0\right) = 0 \) and (5.4.23) holds. | No |
Exercise 9.30 | Exercise 9.30 Check this, and explicitly describe the (co)equalizers in the categories Set, \( \mathcal{T}{op},\mathcal{A}b,{\mathcal{{Mod}}}_{K}, R \) - \( \mathcal{M}{od},\mathcal{M}{od} \) - \( R,\mathcal{G}{rp},\mathcal{C}{mr} \) .
Intuitively, the existence of equalizers allows one to define "subobjects" by means of equations, whereas the coequalizers allow one to define "quotient objects" by imposing relations. For example, the (co) kernel of a homomorphism of abelian groups \( f : A \rightarrow B \) can be described as the (co)equalizer of \( f \) and the zero homomorphism in the category \( \mathcal{A}b \) . | No |
Exercise 1.16 | Exercise 1.16 Let \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence in a complete metric space \( \left( {\mathcal{X}, d}\right) \) such that \( \mathop{\sum }\limits_{{n \in \mathbb{N}}}d\left( {{x}_{n},{x}_{n + 1}}\right) < + \infty \) . Show that \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) converges and that this is no longer true if we merely assume that \( \mathop{\sum }\limits_{{n \in \mathbb{N}}}{d}^{2}\left( {{x}_{n},{x}_{n + 1}}\right) < + \infty \) . | No |
Exercise 1.5 | For following nonlinear ODEs, find a particular solution:
(1) \( {x}^{2}{y}^{\prime \prime } - {\left( {y}^{\prime }\right) }^{2} + {2y} = 0 \) ,
(2) \( x{y}^{\prime \prime \prime } + 3{y}^{\prime \prime } = x{e}^{-{y}^{\prime }} \) ,
(3) \( {x}^{2}{y}^{\prime \prime } - 2{\left( {y}^{\prime }\right) }^{3} + {6y} = 0 \) ,
(4) \( {y}^{\prime \prime } + \frac{2}{x}{y}^{\prime } = {y}^{m}, m \neq 3 \) ,
(5) \( {y}^{\prime \prime \prime } - \frac{15}{{x}^{2}}{y}^{\prime } = 3{y}^{2} \) . | No |
Exercise 11.7.2 | Use the last results to find that the eigenvalues of matrix \( A \), defined by (11.7.22), are expressed by
\[
{\alpha }_{ik} = {\beta }_{i} + 2\cos \left( {{k\pi }/{n}_{y}}\right) = - 2\left( {1 + {\sigma }^{2}}\right)
\]
\[
+ 2{\sigma }^{2}\cos \left( {{i\pi }/{n}_{x}}\right) + 2\cos \left( {{k\pi }/{n}_{y}}\right), i = 0,\ldots ,{n}_{x}, i = 0,\ldots ,{n}_{y}. \tag{11.7.27}
\]
Deduce that \( A \) is singular. | No |
Exercise 15.2.3 | Exercise 15.2.3. Interpret this combinatorially, in terms of the number of partitions of \( m \) into unequal parts. | No |
Exercise 2.11 | Exercise 2.11. The purpose of this exercise is to familiarize yourself with the transformations of the pushforward operation applied to the bootstrap function b. Let \( v \in \mathcal{P}\left( \mathbb{R}\right) \) be a probability distribution and let \( Z \) be a random variable with distribution \( v \) . Let \( r = 1/2,\gamma = 1/3 \), and let \( R \) be a Bernoulli(1/4) random variable independent of \( Z \) . For each of the following probability distributions:
(i) \( v = {\delta }_{1} \) ;
(ii) \( v = 1/2{\delta }_{-1} + 1/2{\delta }_{1} \) ;
(iii) \( v = \mathcal{N}\left( {2,1}\right) \) ,
express the probability distributions produced by the following operations:
(i) \( {\left( {\mathrm{b}}_{r,1}\right) }_{\# }v = \mathcal{D}\left( {r + Z}\right) \) ;
(ii) \( {\left( {\mathrm{b}}_{0,\gamma }\right) }_{\# }v = \mathcal{D}\left( {\gamma Z}\right) \) ;
(iii) \( {\left( {\mathrm{b}}_{r,\gamma }\right) }_{\# }v = \mathcal{D}\left( {r + {\gamma Z}}\right) \) ; and
(iv) \( \mathbb{E}\left\lbrack {{\left( {\mathrm{b}}_{R,\gamma }\right) }_{\# }v}\right\rbrack = \mathcal{D}\left( {R + {\gamma Z}}\right) \) .\( \bigtriangleup \) | No |
Exercise 3.9.17 | Exercise 3.9.17 Show that the three angles coming together at \( r \) are \( \alpha \mathrel{\text{:=}} \arccos - 3/{10} \) and twice \( \beta \mathrel{\text{:=}} \arccos - \sqrt{7/{20}} \) . Furthermore, show that \( \alpha + {2\beta } = {2\pi } \) | No |
Exercise 1.36 | Exercise 1.36. An element \( a \) of a topological group \( G \) is compact if \( \overline{\left\{ a,{a}^{2},\ldots \right\} } \) is compact. Consider the general linear group \( {GL}\left( {2,\mathbf{C}}\right) \) (the set of nonsingular complex \( 2 \times 2 \) matrices). Let \( {z}_{n} = {e}^{\frac{2\pi i}{n}} \) for \( n = 2,3\ldots \) Show that \( \left( \begin{matrix} {z}_{n} & 1 \\ 0 & 1 \end{matrix}\right) \) generates a finite subgroup of \( {GL}\left( {2,\mathbf{C}}\right) \) . Thus, it is a compact element. Also show that the set of compact elements of \( {GL}\left( {2,\mathbf{C}}\right) \) is not closed. | No |
Exercise 2.9 | Exercise 2.9. Let the spheres \( {S}^{1},{S}^{3} \) and the Lie groups \( \mathbf{{SO}}\left( n\right) \) , \( \mathbf{O}\left( n\right) ,\mathbf{{SU}}\left( n\right) ,\mathbf{U}\left( n\right) \) be equipped with their standard differentiable structures introduced above. Use Proposition 2.21 to prove the following diffeomorphisms
\[
{S}^{1} \cong \mathbf{{SO}}\left( 2\right) ,\;{S}^{3} \cong \mathbf{{SU}}\left( 2\right) ,
\]
\[
\mathbf{{SO}}\left( n\right) \times \mathbf{O}\left( 1\right) \cong \mathbf{O}\left( n\right) ,\;\mathbf{{SU}}\left( n\right) \times \mathbf{U}\left( 1\right) \cong \mathbf{U}\left( n\right) .
\] | No |
Exercise 1.3.5 | Let \( S \mathrel{\text{:=}} \{ \alpha = x + {y\omega } \mid 0 \leq y < x\} \subset \mathbb{Z}\left\lbrack \omega \right\rbrack \smallsetminus \{ 0\} \) . Show that for every element \( \alpha \in \mathbb{Z}\left\lbrack \omega \right\rbrack ,\alpha \neq 0 \), there exists a unique associate element \( {\alpha }^{\prime } \in S \) , \( \alpha \sim {\alpha }^{\prime } \) . Deduce that \( \alpha \) has a factorization
\[
\alpha = \epsilon \cdot {\pi }_{1} \cdot \ldots \cdot {\pi }_{r}
\]
with prime elements \( {\pi }_{i} \in S \) and a unit \( \epsilon \), and that this factorization is unique up to a permutation of the \( {\pi }_{i} \) . | No |
Exercise 10.10 | Exercise 10.10. (i) Use the ordinary differential equations (6.5.8) and (6.5.9) satisfied by the functions \( A\left( {t, T}\right) \) and \( C\left( {t, T}\right) \) in the one-factor Hull-White model to show that this model satisfies the HJM no-arbitrage condition (10.3.27).
(ii) Use the ordinary differential equations (6.5.14) and (6.5.15) satisfied by the functions \( A\left( {t, T}\right) \) and \( C\left( {t, T}\right) \) in the one-factor Cox-Ingersoll-Ross model to show that this model satisfies the HJM no-arbitrage condition (10.3.27). | No |
Exercise 7.2 | Exercise 7.2. Let \( \mathcal{C} \) be an abelian category with enough injective and such that \( \operatorname{dh}\left( \mathcal{C}\right) \leq 1 \) . Let \( F : \mathcal{C} \rightarrow {\mathcal{C}}^{\prime } \) be a left exact functor and let \( X \in {\mathrm{D}}^{ + }\left( \mathcal{C}\right) \) .
(i) Construct an isomorphism \( {H}^{k}\left( {{RF}\left( X\right) }\right) \simeq F\left( {{H}^{k}\left( X\right) }\right) \oplus {R}^{1}F\left( {{H}^{k - 1}\left( X\right) }\right) \) .
(ii) Recall that \( \operatorname{dh}\left( {\operatorname{Mod}\left( \mathbb{Z}\right) }\right) = 1 \) . Let \( X \in {\mathrm{D}}^{ - }\left( \mathbb{Z}\right) \), and let \( M \in \operatorname{Mod}\left( \mathbb{Z}\right) \) . Deduce the isomorphism
\[
{H}^{k}\left( {X\overset{\mathrm{L}}{ \otimes }M}\right) \simeq \left( {{H}^{k}\left( X\right) \otimes M}\right) \oplus {\operatorname{Tor}}_{\mathbb{Z}}^{-1}\left( {{H}^{k + 1}\left( X\right), M}\right) .
\] | No |
Exercise 6.1 | Exercise 6.1. Consider the stochastic differential equation
\[
{dX}\left( u\right) = \left( {a\left( u\right) + b\left( u\right) X\left( u\right) }\right) {du} + \left( {\gamma \left( u\right) + \sigma \left( u\right) X\left( u\right) }\right) {dW}\left( u\right) ,
\]
\( \left( {6.2.4}\right) \)
where \( W\left( u\right) \) is a Brownian motion relative to a filtration \( \mathcal{F}\left( u\right), u \geq 0 \), and we allow \( a\left( u\right), b\left( u\right) ,\gamma \left( u\right) \), and \( \sigma \left( u\right) \) to be processes adapted to this filtration. Fix an initial time \( t \geq 0 \) and an initial position \( x \in \mathbb{R} \) . Define
\[
Z\left( u\right) = \exp \left\{ {{\int }_{t}^{u}\sigma \left( v\right) {dW}\left( v\right) + {\int }_{t}^{u}\left( {b\left( v\right) - \frac{1}{2}{\sigma }^{2}\left( v\right) }\right) {dv}}\right\} ,
\]
\[
Y\left( u\right) = x + {\int }_{t}^{u}\frac{a\left( v\right) - \sigma \left( v\right) \gamma \left( v\right) }{Z\left( v\right) }{dv} + {\int }_{t}^{u}\frac{\gamma \left( v\right) }{Z\left( v\right) }{dW}\left( v\right) .
\]
(i) Show that \( Z\left( t\right) = 1 \) and
\[
{dZ}\left( u\right) = b\left( u\right) Z\left( u\right) {du} + \sigma \left( u\right) Z\left( u\right) {dW}\left( u\right), u \geq t.
\]
(ii) By its very definition, \( Y\left( u\right) \) satisfies \( Y\left( t\right) = x \) and
\[
{dY}\left( u\right) = \frac{a\left( u\right) - \sigma \left( u\right) \gamma \left( u\right) }{Z\left( u\right) }{du} + \frac{\gamma \left( u\right) }{Z\left( u\right) }{dW}\left( u\right), u \geq t.
\]
Show that \( X\left( u\right) = Y\left( u\right) Z\left( u\right) \) solves the stochastic differential equation (6.2.4) and satisfies the initial condition \( X\left( t\right) = x \) . | No |
Exercise 1.4.30 | Exercise 1.4.30 Consider the system \( - {5x} + {2y} - z = 0 \) and \( - {5x} - {2y} - z = 0 \) . Both equations equal zero and so \( - {5x} + {2y} - z = - {5x} - {2y} - z \) which is equivalent to \( y = 0 \) . Does it follow that \( x \) and \( z \) can equal anything? Notice that when \( x = 1, z = - 4 \), and \( y = 0 \) are plugged in to the equations, the equations do not equal 0 . Why? | No |
Exercise 6.8 | [Let \( R \) be a ring.
(i) Prove that \( M \in \operatorname{Mod}\left( R\right) \) is of finite presentation in the sense of Definition 6.3.3 if and only if it is of finite presentation in the classical sense (see Examples 1.2.4 (iv)), that is, if there exists an exact sequence \( {R}^{\oplus {n}_{1}} \rightarrow {R}^{\oplus {n}_{0}} \rightarrow \) \( M \rightarrow 0 \) .
(ii) Prove that any \( R \) -module \( M \) is a small filtrant inductive limit of modules of finite presentation. (Hint: consider the full subcategory of \( {\left( \operatorname{Mod}\left( A\right) \right) }_{M} \) consisting of modules of finite presentation and prove it is essentially small and filtrant.)
(iii) Deduce that the functor \( {J\rho } \) defined in Diagram (6.3.1) induces an equivalence \( {J\rho } : \operatorname{Ind}\left( {{\operatorname{Mod}}^{\mathrm{{fp}}}\left( R\right) }\right) \overset{ \sim }{ \rightarrow }\operatorname{Mod}\left( R\right) \) .] | No |
Exercise 11.18 | Exercise 11.18. By an \( {\mathrm{{FO}}}^{k} \) theory we mean a maximally consistent set of \( {\mathrm{{FO}}}^{k} \) sentences. Define the \( k \) -size of an \( {\mathrm{{FO}}}^{k} \) theory \( T \) as the number of different \( {\mathrm{{FO}}}^{k} \) - types realized by finite models of \( T \) . Prove that there is no recursive bound on the size of the smallest model of an \( {\mathrm{{FO}}}^{k} \) theory in terms of its \( k \) -size. That is, for every \( k \) there is a vocabulary \( {\sigma }_{k} \) such that is no recursive function \( f \) with the property that every \( {\mathrm{{FO}}}^{k} \) theory \( T \) in vocabulary \( {\sigma }_{k} \) has a model of size at most \( f\left( n\right) \), where \( n \) is the \( k \) -size of \( T \) . | No |
Exercise 4.18 | Exercise 4.18. Let \( \mu \) be a probability measure on \( d \times d \) real matrices such that \( {\mu }^{m} \) \{the zero matrix\} is positive for some positive integer \( m \) . Show that \( {\mu }^{n} \) converges weakly to the unit mass at the zero matrix. Does this mean \( \mu \{ 0\} > 0 \) ? If not, give an example. | No |
Exercise 7.5 | Exercise 7.5 (Black-Scholes-Merton equation for lookback option). We wish to verify by direct computation that the function \( v\left( {t, x, y}\right) \) of (7.4.35) satisfies the Black-Scholes-Merton equation (7.4.6). As we saw in Subsection 7.4.3, this is equivalent to showing that the function \( u \) defined by (7.4.36) satisfies the Black-Scholes-Merton equation (7.4.18). We verify that \( u\left( {t, z}\right) \) satisfies (7.4.18) in the following steps. Let \( 0 \leq t < T \) be given, and define \( \tau = T - t \)
(i) Use (7.8.1) to compute \( {u}_{t}\left( {t, z}\right) \), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that
\[
{u}_{t}\left( {t, z}\right) = r{e}^{-{r\tau }}N\left( {-{\delta }_{ - }\left( {\tau, z}\right) }\right) - \frac{1}{2}{\sigma }^{2}{e}^{-{r\tau }}{z}^{1 - \frac{2r}{{\sigma }^{2}}}N\left( {-{\delta }_{ - }\left( {\tau ,{z}^{-1}}\right) }\right)
\]
\[
- \frac{\sigma z}{\sqrt{\tau }}{N}^{\prime }\left( {{\delta }_{ + }\left( {\tau, z}\right) }\right) \tag{7.8.18}
\]
(ii) Use (7.8.2) to compute \( {u}_{z}\left( {t, z}\right) \), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that
\[
{u}_{z}\left( {t, z}\right) = \left( {1 + \frac{{\sigma }^{2}}{2r}}\right) N\left( {{\delta }_{ + }\left( {\tau, z}\right) }\right)
\]
\[
+ \left( {1 - \frac{{\sigma }^{2}}{2r}}\right) {e}^{-{r\tau }}{z}^{-\frac{2r}{{\sigma }^{2}}}N\left( {-{\delta }_{ - }\left( {\tau ,{z}^{-1}}\right) }\right) - 1. \tag{7.8.19}
\]
(iii) Use (7.8.19) and (7.8.2) to compute \( {u}_{z}\left( {t, z}\right) \), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that
\[
{u}_{zz}\left( {t, z}\right) = \left( {1 - \frac{2r}{{\sigma }^{2}}}\right) {e}^{-{r\tau }}{z}^{-\frac{2r}{{\sigma }^{2}} - 1}N\left( {-{\delta }_{ - }\left( {\tau ,{z}^{-1}}\right) }\right) + \frac{2}{{z\sigma }\sqrt{\tau }}{N}^{\prime }\left( {{\delta }_{ + }\left( {\tau, z}\right) }\right) . \tag{7.8.20}
\]
(iv) Verify that \( u\left( {t, z}\right) \) satisfies the Black-Scholes-Merton equation (7.4.18).
(v) Verify that \( u\left( {t, z}\right) \) satisfies the boundary condition (7.4.20). | No |
Exercise 7.27 | Let \( X, Y \) be \( \mathrm{L}\left( {d, d}\right) \) -valued semimartingales. Show that
\[
{X}_{t}{Y}_{t} = {X}_{0}{Y}_{0} + {\int }_{0 + }^{t}{X}_{s - }d{Y}_{s} + {\int }_{0 + }^{t}\left( {d{X}_{s}}\right) {Y}_{s - } + {\left\lbrack X, Y\right\rbrack }_{t}. \tag{7.6.1}
\]
The relation (7.6.1) is the matrix analogue of the integration by parts formula (4.6.7).
Recall our terminology: we say that a \( \mathrm{L}\left( {d, d}\right) \) -valued process \( h \) is \( {L}_{0}\left( d\right) \) -valued if
\[
\mathrm{P}\left( {{h}_{t} \in {\mathbb{L}}_{0}\left( d\right) \forall t \geq 0}\right) = 1.
\] | No |
Exercise 10.6.3 | Exercise 10.6.3. Complete the following:
i.) Substitute (10.26) into (10.23) to show that \( \forall m \in \mathbb{N},{z}_{m}\left( t\right) \) satisfies the IVP
\[
\left\{ \begin{array}{l} {z}_{m}^{\prime }\left( t\right) + \frac{{m}^{2}{\pi }^{2}}{{a}^{2}}{z}_{m}\left( t\right) = {f}_{m}\left( t\right), t > 0, \\ z\left( 0\right) = {z}_{m}^{0}. \end{array}\right. \tag{10.27}
\]
ii.) Use the variation of parameters technique to show that the solution of (10.27) is given
by
\[
{z}_{m}\left( t\right) = {z}_{m}^{0}{e}^{-\left( \frac{{m}^{2}{\pi }^{2}}{{a}^{2}}\right) t} + {\int }_{0}^{t}{e}^{-\left( \frac{{m}^{2}{\pi }^{2}}{{a}^{2}}\right) \left( {t - s}\right) }{f}_{m}\left( s\right) {ds}, t > 0. \tag{10.28}
\]
iii.) Use (10.22) and (10.25) in (10.28) to show that the solution of (10.27) can be simplified to
\[
z\left( {x, t}\right) = {e}^{At}\left( {z}_{0}\right) \left\lbrack x\right\rbrack
\]
\( \left( {10.29}\right) \)
\[
+ {\int }_{0}^{t}{\int }_{0}^{a}\frac{2}{a}\mathop{\sum }\limits_{{m = 1}}^{\infty }{e}^{-\left( \frac{{m}^{2}{\pi }^{2}}{{a}^{2}}\right) \left( {t - s}\right) }\sin \left( {\frac{m\pi }{a}w}\right) \sin \left( {\frac{m\pi }{a}x}\right) f\left( {w, s}\right) {dwds}.
\]
iv.) Finally, use (10.29) to further express the solution of (10.23) in the form
\[
z\left( {\cdot, t}\right) = {e}^{At}\left\lbrack {z}_{0}\right\rbrack \left\lbrack \cdot \right\rbrack + {\int }_{0}^{t}{e}^{A\left( {t - s}\right) }f\left( {s, \cdot }\right) {ds}, t > 0. \tag{10.30}
\] | No |
Exercise 2.11 | Exercise 2.11. Let \( \left( {P}_{\theta }\right) \) be a regular family.
1. Show that the \( {KL} \) -divergence \( \mathcal{K}\left( {\theta ,{\theta }^{\prime }}\right) \) satisfies for any \( \theta ,{\theta }^{\prime } \) :
(a)
\[
{\left. \mathcal{K}\left( \theta ,{\theta }^{\prime }\right) \right| }_{{\theta }^{\prime } = \theta } = 0 \tag{b}
\]
\[
{\left. \frac{d}{d{\theta }^{\prime }}\mathcal{K}\left( \theta ,{\theta }^{\prime }\right) \right| }_{{\theta }^{\prime } = \theta } = 0
\]
(c)
\[
{\left. \frac{{d}^{2}}{d{\theta }^{\prime 2}}\mathcal{K}\left( \theta ,{\theta }^{\prime }\right) \right| }_{{\theta }^{\prime } = \theta } = I\left( \theta \right) .
\]
2. Show that in a small neighborhood of \( \theta \), the KL-divergence can be approximated by
\[
\mathcal{K}\left( {\theta ,{\theta }^{\prime }}\right) \approx I\left( \theta \right) {\left| {\theta }^{\prime } - \theta \right| }^{2}/2
\]
1. Note that
\[
\mathcal{K}\left( {\theta ,{\theta }^{\prime }}\right) = {\mathbb{E}}_{\theta }\log p\left( {x,\theta }\right) - {\mathbb{E}}_{\theta }\log p\left( {x,{\theta }^{\prime }}\right)
\]
(a) First item is trivial.
(b)
\[
\frac{d}{d{\theta }^{\prime }}\mathcal{K}\left( {\theta ,{\theta }^{\prime }}\right) = - \frac{d}{d{\theta }^{\prime }}{\mathbb{E}}_{\theta }\log p\left( {x,{\theta }^{\prime }}\right)
\]
\[
= - \frac{d}{d{\theta }^{\prime }}\int \log p\left( {x,{\theta }^{\prime }}\right) p\left( {x,\theta }\right) {dx}
\]
\[
= - \int \frac{{p}_{{\theta }^{\prime }}^{\prime }\left( {x,{\theta }^{\prime }}\right) }{p\left( {x,{\theta }^{\prime }}\right) }p\left( {x,\theta }\right) {dx}
\]
where \( {p}_{{\theta }^{\prime }}^{\prime }\left( {x,{\theta }^{\prime }}\right) \overset{\text{ def }}{ = }\frac{d}{d{\theta }^{\prime }}p\left( {x,{\theta }^{\prime }}\right) \) . Substitution \( {\theta }^{\prime } = \theta \) gives
\[
{\left. \frac{d}{d{\theta }^{\prime }}\mathcal{K}\left( \theta ,{\theta }^{\prime }\right) \right| }_{{\theta }^{\prime } = \theta } = - {\left. \int \frac{d}{d{\theta }^{\prime }}\left\{ p\left( x,{\theta }^{\prime }\right) \right\} dx\right| }_{{\theta }^{\prime } = \theta }
\]
\[
= - {\left. \frac{d}{d{\theta }^{\prime }}\int p\left( x,{\theta }^{\prime }\right) dx\right| }_{{\theta }^{\prime } = \theta } = 0.
\]
(c)
\[
\frac{{d}^{2}}{d{\theta }^{\prime 2}}\mathcal{K}\left( {\theta ,{\theta }^{\prime }}\right) = - \int \frac{d}{d{\theta }^{\prime }}\left\{ \frac{{p}_{{\theta }^{\prime }}^{\prime }\left( {x,{\theta }^{\prime }}\right) }{p\left( {x,{\theta }^{\prime }}\right) }\right\} p\left( {x,\theta }\right) {dx}
\]
\[
= - \int \left\lbrack \frac{{p}_{{\theta }^{\prime }}^{\prime \prime }\left( {x,{\theta }^{\prime }}\right) p\left( {x,{\theta }^{\prime }}\right) - {\left\{ {p}_{{\theta }^{\prime }}^{\prime }\left( x,{\theta }^{\prime }\right) \right\} }^{2}}{{\left\{ p\left( x,{\theta }^{\prime }\right) \right\} }^{2}}\right\rbrack p\left( {x,\theta }\right) {dx}.
\]
Substitution \( {\theta }^{\prime } = \theta \) yields
\[
{\left. \frac{{d}^{2}}{d{\theta }^{\prime 2}}\mathcal{K}\left( \theta ,{\theta }^{\prime }\right) \right| }_{{\theta }^{\prime } = \theta } = \underset{{\left. \frac{{d}^{2}}{d{\theta }^{\prime 2}}\int p\left( x,{\theta }^{\prime }\right) dx\right| }_{{\theta }^{\prime } = \theta } = 0}{\underbrace{{\left. \int {p}_{{\theta }^{\prime }}^{\prime \prime }\left( x,{\theta }^{\prime }\right) dx\right| }_{{\theta }^{\prime } = \theta }}} + \underset{ = I\left( \theta \right) }{\underbrace{\int \frac{{\left\{ {p}_{\theta }^{\prime }\left( x,\theta \right) \right\} }^{2}}{p\left( {x,\theta }\right) }{dx}}} = I\left( \theta \right) .
\]
2. The required representation directly follows from the Taylor expansion at the point \( {\theta }^{\prime } = \theta \) . | No |
Exercise 3.2 | [As we have seen in Chapter 2, many quantum control systems have a bilinear structure
\[
\dot{X} = {AX} + \mathop{\sum }\limits_{{k = 1}}^{m}{B}_{k}X{u}_{k}
\]
Assume that the set of the possible values for the controls contains a neighborhood of the origin in \( {\mathbf{R}}^{k} \) . Show that the dynamical Lie algebra coincides with the one generated by \( A,{B}_{1},\ldots ,{B}_{m} \).] | No |
Exercise 11.12.5 | Consider the following collection of points in \( {\mathbb{R}}^{2} \) :
\[
\left\{ {\left\lbrack \begin{array}{r} 4 \\ - 3 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} {10} \\ - 9 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} 4 \\ - 7 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} - 2 \\ 3 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} {10} \\ - 3 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} 4 \\ - 3 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} 5 \\ - 3 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} 4 \\ 1 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} - 2 \\ - 3 \end{array}\right\rbrack ,\left\lbrack \begin{array}{r} 3 \\ - 3 \end{array}\right\rbrack }\right\} .
\]
Compute the centroid, and then find the 1-dimensional affine subspace that best approximates this collection of points. What is the total squared distance of the points to the subspace? | Yes |
Exercise 5.6 | Exercise 5.6 Prove that a map \( F : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is locally Lipschitz continuous if and only if \( F \) is Lipschitz on bounded sets. (Hint: Start by considering the segment \( {sY} + \left( {1 - s}\right) X \) with \( s \in \left\lbrack {0,1}\right\rbrack \) .) | No |
Exercise 8.4.1 | Exercise 8.4.1. Let \( F \) be a homogeneous polynomial in \( \mathbb{C}\left\lbrack {{X}_{0},\ldots ,{X}_{n}}\right\rbrack \), and let \( I \) be the homogeneous ideal generated by \( {X}_{0}F,{X}_{1}F,\ldots ,{X}_{n}F \) . Show that
\[
\bar{Z}\left( F\right) = \bar{Z}\left( I\right)
\]
as subsets of \( {\mathbb{P}}^{n} \) . | No |
Exercise 3.1 | Exercise 3.1. According to Definition 3.3.3(iii), for \( 0 \leq t < u \), the Brownian motion increment \( W\left( u\right) - W\left( t\right) \) is independent of the \( \sigma \) -algebra \( \mathcal{F}\left( t\right) \) . Use this property and property (i) of that definition to show that, for \( 0 \leq t < {u}_{1} < {u}_{2} \) , the increment \( W\left( {u}_{2}\right) - W\left( {u}_{1}\right) \) is also independent of \( \mathcal{F}\left( t\right) \) . | No |
Exercise 8.12 | Exercise 8.12. Give an example to show that a discrete vector field need not stabilize at every simplex. | No |
Exercise 1 | Exercise 1 (Basic example of linear regression).
a. Consider the following data points: \( \left( {{x}_{1} = - 2,{y}_{1} = 1}\right) ,\left( {{x}_{2} = 0,{y}_{2} = 2}\right) ,\left( {{x}_{3} = 1,{y}_{3} = - 1}\right) \) . Solve the corresponding linear regression problem, that is, find the best coefficients \( a, b \in \mathbb{R} \) minimizing
\[
\mathop{\sum }\limits_{{i = 1}}^{3}{\left| a{x}_{i} + b - {y}_{i}\right| }^{2}
\] | Yes |
Exercise 5.9 | Exercise 5.9. (a) Suppose that \( \mathbf{U} \) is a finite dimensional real Euclidean space and \( Q \in \operatorname{Sym}\left( \mathbf{U}\right) \) is a positive definite symmetric bilinear form. Prove that there exists a unique positive operator
\[
T : \mathbf{U} \rightarrow \mathbf{U}
\]
such that
\[
Q\left( {\mathbf{u},\mathbf{v}}\right) = \langle T\mathbf{u}, T\mathbf{v}\rangle ,\;\forall \mathbf{u},\mathbf{v} \in \mathbf{U}.
\] | No |
Exercise 2.8.14 | Exercise 2.8.14. Compute a finite free resolution of the ideal generated by the \( 2 \times 2 \) minors of the matrix
\[
\left( \begin{array}{llll} {x}_{0} & {x}_{1} & {x}_{2} & {x}_{3} \\ {x}_{1} & {x}_{2} & {x}_{3} & {x}_{4} \end{array}\right) .
\] | No |
Exercise 4.7.1 | Exercise 4.7.1 This exercise is about the collection of elements \( {\left\{ {a}_{i},{b}_{i}\right\} }_{i \in \mathbb{N}} \) that we construct for an element \( a \in \mathcal{P}\left( X\right) \) in the proof of Lemma 4.81.
(a) Show that \( {a}_{0} \supseteq {b}_{0} \supseteq {a}_{1} \supseteq \cdots \supseteq {b}_{n} \) for all \( n \in \mathbb{N} \) .
(b) Show that
\[
\mathop{\bigcup }\limits_{{i = 0}}^{n}\left( {{a}_{i} - {b}_{i}}\right) = {a}_{0} - \left( {{b}_{0} - \left( {{a}_{1} - \cdots \left( {{a}_{n} - {b}_{n}}\right) \ldots }\right) }\right) .
\]
(c) For an element \( a \in \mathcal{P}\left( X\right) \), we call \( {a}_{0} \supseteq {b}_{0} \supseteq {a}_{1} \supseteq \cdots \supseteq {b}_{n} \) such that \( {a}_{i},{b}_{i} \in \mathcal{U}\left( X\right) \) and
\[
a = \mathop{\bigcup }\limits_{{n = 0}}^{h}\left( {{a}_{n} - {b}_{n}}\right)
\]
a difference chain for \( a \) and we order difference chains for \( a \) by coordinate-wise inclusion (if one is shorter than the other, then we consider it extended with empty sets). Show that if \( a \) has a difference chain then it has a least such and it is the one we define in the proof of Lemma 4.81. | No |
Exercise 8.16 | [Exercise 8.16 Let \( f : X \rightarrow X \) be a measurable map preserving a measure \( \mu \) on \( X \) with \( \mu \left( X\right) = 1 \) . Show that if \( \xi \) is a partition of \( X \), then \( {h}_{\mu }\left( {f,\xi }\right) \leq \log \operatorname{card}\xi \) .] | No |
Exercise 7.1 | Exercise 7.1. (Exercise 6.1 continued).
(1) Derive a stochastic differential equation satisfied by \( t \mapsto P\left( {t, T}\right) \) .
(2) Derive a stochastic differential equation satisfied by \( t \mapsto \) \( {e}^{-{\int }_{0}^{t}{r}_{s}{ds}}P\left( {t, T}\right) . \)
(3) Express the conditional expectation
\[
{\mathbb{E}}_{\mathbb{P}}\left\lbrack {\left. \frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}\right| \;{\mathcal{F}}_{t}}\right\rbrack
\]
in terms of \( P\left( {t, T}\right), P\left( {0, T}\right) \) and \( {e}^{-{\int }_{0}^{t}{r}_{s}{ds}},0 \leq t \leq T \) .
(4) Find a stochastic differential equation satisfied by
\[
t \mapsto {\mathbb{E}}_{\mathbb{P}}\left\lbrack {\left. \frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}\right| \;{\mathcal{F}}_{t}}\right\rbrack
\]
(5) Compute the density \( d\widetilde{\mathbb{P}}/d\mathbb{P} \) of the forward measure with respect to \( \mathbb{P} \) by solving the stochastic differential equation of question 4 .
(6) Using the Girsanov Theorem 2.1, compute the dynamics of \( {r}_{t} \) under the forward measure.
(7) Compute the price
\( {\mathbb{E}}_{\mathbb{P}}\left\lbrack {{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}{\left( P\left( T, S\right) - K\right) }^{ + } \mid {\mathcal{F}}_{t}}\right\rbrack = P\left( {t, T}\right) {\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {{\left( P\left( T, S\right) - K\right) }^{ + } \mid {\mathcal{F}}_{t}}\right\rbrack \)
of a bond call option at time \( t \geq 0 \) . | Yes |
Exercise 8.17 | Compute the metric entropy of the expanding map \( {E}_{m} : {S}^{1} \rightarrow {S}^{1} \) with respect to the \( {E}_{m} \) -invariant measure \( \mu \) defined by (8.5). | No |
Exercise 9.4.8 | Exercise 9.4.8. This exercise is an \( n \) -dimensional version of Exercise 9.4.7 Because of the similarities, we will be less verbose. Let \( k \) be a field and \( p \) a polynomial in \( A = k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) of degree \( d \geq 2 \) . Assume \( p \) is not a square. Let \( p = {p}_{1}^{{e}_{1}}\cdots {p}_{s}^{{e}_{s}} \) be the unique factorization of \( p \) into irreducibles. Write \( {e}_{i} = 2{q}_{i} + {r}_{i} \), where \( 0 \leq {r}_{i} < 2 \) . Set \( r = {p}_{1}^{{r}_{1}}\cdots {p}_{s}^{{r}_{s}} \) and \( q = {p}_{1}^{{q}_{1}}\cdots {p}_{s}^{{q}_{s}} \) . Then \( p = r{q}^{2} \) . Let \( X \) be the affine hypersurface in \( {\mathbb{A}}_{k}^{n + 1} \) defined by \( {z}^{2} - r{q}^{2} = 0 \) . The affine coordinate ring of \( X \) is \( \mathcal{O}\left( X\right) = A\left\lbrack z\right\rbrack /\left( {{z}^{2} - r{q}^{2}}\right) \) . Let \( \widetilde{X} \) be the affine hypersurface in \( {\mathbb{A}}_{k}^{n + 1} \) defined by \( {w}^{2} - r = 0 \) . The affine coordinate ring of \( \widetilde{X} \) is \( \mathcal{O}\left( X\right) = A\left\lbrack w\right\rbrack /\left( {{w}^{2} - r}\right) \) . Define an \( A \) -algebra homomorphism \( \phi : \mathcal{O}\left( X\right) \rightarrow \mathcal{O}\left( \widetilde{X}\right) \) by \( \alpha \mapsto \alpha \) for \( \alpha \in A \) and \( z \mapsto {wq} \) . Identify both rings with subrings of the quotient field of \( \mathcal{O}\left( X\right) \) and show that \( \mathcal{O}\left( \widetilde{X}\right) \) is the integral closure of \( \mathcal{O}\left( X\right) \) . Show that the conductor ideal from \( \mathcal{O}\left( \widetilde{X}\right) \) to \( \mathcal{O}\left( X\right) \) is \( \left( {z, q}\right) \subseteq \mathcal{O}\left( X\right) \) . As an ideal in \( \mathcal{O}\left( \widetilde{X}\right) \), the conductor is the principal ideal \( \left( q\right) \) . | No |
Exercise 3 | [Exercise 3 (Monotonicity). Explain why logistic regression does not work well when \( p\left( x\right) \) is not monotone.] | No |
Exercise 6.3.3 | Exercise 6.3.3. Show that every nonempty subset of a linearly independent set is linearly independent. | No |
Exercise 1.6 | Exercise 1.6
(a) Linear, space invariant.
(b) Nonlinear, this is an affine transformation, but space invariant.
(c) Nonlinear, space invariant.
(d) Linear, not space invariant.
(e) Linear, space invariant. | No |
Exercise 7.2 | Exercise 7.2 It is important to realize that we cannot take just any DAG and expect a joint distribution to equal the product of its conditional distributions in the DAG. This is only true if the Markov condition is satisfied. You will illustrate that this is the case in this exercise. Consider the joint probability distribution \( P \) in Example 7.1.
1. Show that probability distribution \( P \) satisfies the Markov condition with the DAG in Figure 7.29 (a) and that \( P \) is equal to the product of its conditional distributions in that DAG.
2. Show that probability distribution \( P \) satisfies the Markov condition with the DAG in Figure 7.29 (b) and that \( P \) is equal to the product of its conditional distributions in that DAG.
Show that probability distribution \( P \) does not satisfy the Markov condition with the DAG in Figure 7.29 (c) and that \( P \) is not equal to the product of its conditional distributions in that DAG. | No |
Exercise 3 | Exercise 3. Let \( \Phi : G{L}_{N}\left( \mathbb{C}\right) \rightarrow \mathcal{U}\left( N\right) \) be the map which takes an invertible complex matrix \( A \) and applies the Gram-Schmidt procedure to the columns of \( A \) to obtain a unitary matrix. Show that for any \( U \in \mathcal{U}\left( N\right) \), we have \( \Phi \left( {UA}\right) = {U\Phi }\left( A\right) \). | No |
Exercise 2.11 | Exercise 2.11 Regular representation of \( {\mathfrak{S}}_{3} \) .
Decompose the regular representation of \( {\mathfrak{S}}_{3} \) into a direct sum of irreducible representations.
Find a basis of each one-dimensional invariant subspace and a projection onto the support of the representation \( {2\rho } \), where \( \rho \) is the irreducible representation of dimension 2 . | No |
Exercise 14.5 | [Suppose
\[
\mathbf{A}\left( t\right) \mathrel{\text{:=}} \mathbf{D} + t\left( {\mathbf{A} - \mathbf{D}}\right) ,\;\mathbf{D} \mathrel{\text{:=}} \operatorname{diag}\left( {{a}_{11},\ldots ,{a}_{nn}}\right) ,\;t \in \mathbb{R}.
\]
\( 0 \leq {t}_{1} < {t}_{2} \leq 1 \) and that \( \mu \) is an eigenvalue of \( \mathbf{A}\left( {t}_{2}\right) \) . Show, using Theorem 14.2 with \( \mathbf{A} = \mathbf{A}\left( {t}_{1}\right) \) and \( \mathbf{E} = \mathbf{A}\left( {t}_{2}\right) - \mathbf{A}\left( {t}_{1}\right) \), that \( \mathbf{A}\left( {t}_{1}\right) \) has an eigenvalue \( \lambda \) such that
\[
\left| {\lambda - \mu }\right| \leq C{\left( {t}_{2} - {t}_{1}\right) }^{1/n},\text{ where }C \leq 2\left( {\parallel \mathbf{D}{\parallel }_{2} + \parallel \mathbf{A} - \mathbf{D}{\parallel }_{2}}\right) .
\]
Thus, as a function of \( t \), every eigenvalue of \( \mathbf{A}\left( t\right) \) is a continuous function of \( t \) .] | No |
Exercise 4.3.1 | For each matrix \( A \), find the products \( \left( {-2}\right) A,{0A} \), and \( {3A} \) .
(a) \( A = \left\lbrack \begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right\rbrack \)
(b) \( A = \left\lbrack \begin{array}{rr} - 2 & 3 \\ 0 & 2 \end{array}\right\rbrack \)
(c) \( A = \left\lbrack \begin{array}{rrr} 0 & 1 & 2 \\ 1 & - 1 & 3 \\ 4 & 2 & 0 \end{array}\right\rbrack \) | Yes |
Exercise 1.4.6 | Exercise 1.4.6. Let \( \mathcal{H} = {l}_{2} \) and \( {e}_{k} = \left( {{e}_{k1},{e}_{k2},\ldots }\right) \) with
\[
{e}_{kj} = \left\{ \begin{array}{l} 1\text{ if }j = k \\ 0\text{ if }j \neq k, \end{array}\right.
\]
\( j, k \geq 0 \) . Prove that \( {\left\{ {e}_{k}\right\} }_{k = 1}^{\infty } \) converges weakly in \( {l}_{2} \) but does not converge strongly. | No |
Exercise 1.5.2 | Exercise 1.5.2 Reduce each of the matrices from Exercise 1.4.15 to reduced echelon form. | No |
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