ID stringlengths 9 15 | Exercise stringlengths 27 6k | judge stringclasses 2
values |
|---|---|---|
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{arr... | Yes |
Problem 7.4 | Problem 7.4. Let \( A \) be an \( N \times N \) matrix with eigenvalues \( {\rho }_{1} \geq {\rho }_{2} \geq \cdots \geq \) \( {\rho }_{N} \) . Consider a \( d \) -dimensional parallelepiped with the vectors \( {\xi }_{0}^{\left( 1\right) },\cdots {\xi }_{0}^{\left( d\right) } \) , as its sides, and let \( {V}_{0}^{\le... | Yes |
Example 8.9 | Example 8.9. We will now solve the wave equation on a disk and compare the fundamental frequency of a circular drum to that of a square drum. We consider the IBVP
\[
\frac{{\partial }^{2}u}{\partial {t}^{2}} - {c}^{2}{\Delta }_{p}u = 0,\left( {r,\theta }\right) \in \Omega, t > 0, \tag{8.49}
\]
\[
u\left( {r,\theta ,0... | Yes |
Example 3.3.11 | Example 3.3.11. Suppose that \( G \) is a group of order 8 that contains an element \( x \) of order 4 . Let \( y \) be another element in \( G \) that is distinct from any power of \( x \) . With these criteria, we know so far that \( G \) contains the distinct elements \( e, x,{x}^{2},{x}^{3}, y \) . The element \( {... | No |
Example 5.5 | [Example 5.5 (See [7, Example 2.7]) Let \( x \) be an indeterminate over \( {Z}_{p}, k \mathrel{\text{:=}} \) \( {Z}_{p}\left( {x}^{p}\right) \), and \( K \mathrel{\text{:=}} L \mathrel{\text{:=}} {Z}_{p}\left( x\right) \) . Then, \( K{ \otimes }_{k}L \) is Noetherian, and therefore it is locally complete intersection ... | No |
Example 3.3 | Example 3.3 Let us prove eqn (3.17) by induction. Because of linearity, it is sufficient to prove this equation for the case where \( {A}_{\left\lbrack p\right\rbrack } \) is a simple \( p \) -vector. First, we slightly change the notation, so that it shall be suitable for this purpose. Let us denote a simple \( p \) -... | No |
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-pr... | 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 \( {T}_{n} = T\left( {B}_{n}\right) + {T}_{n - 1} \) whe... | 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 |
Example 5.6.39 | Example 5.6.39 Consider the second-degree equation
\[
\frac{3}{2}{x}^{2} + {y}^{2} + \frac{3}{2}{z}^{2} - {xz} + x - 1 = 0.
\]
The matrix of the quadratic form associated to this equation is the matrix \( A \) of Example 5.5.19. Its eigenvalues are \( 1,1,2 \) . If we transform the equation using the orthogonal trans... | No |
Problem 4.56 | Problem 4.56 Make a full report with a sketch for each of the following functions. Include diagonal asymptotes if any.
1. \( f\left( x\right) = \frac{{x}^{2} + {2x} + 1}{x - 1} \) 4. \( r\left( x\right) = \frac{{x}^{3}}{{x}^{2} + 2} \)
2. \( g\left( x\right) = \frac{2{x}^{2} + {3x} + 1}{x} \) 5. \( s\left( x\right) =... | No |
Example 5.1 | [Plaintext: MEET ME TODAY
Cipher-text: PHHW PH WRGDB \( \left( {k = 3}\right) \)] | 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 }\rig... | No |
Problem 19 | Problem 19. Prove that if a sequence \( 0 \rightarrow A\xrightarrow[]{\varphi }B\xrightarrow[]{\psi }C \) is exact, then so is the sequence
\[
0 \rightarrow \operatorname{Hom}\left( {G, A}\right) \overset{\widetilde{\varphi }}{ \rightarrow }\operatorname{Hom}\left( {G, B}\right) \overset{\widetilde{\psi }}{ \rightarro... | No |
Example 10.11 | Example 10.11 (Inner faithfulness for groups and Lie algebras). If \( H = \mathbb{k}G \) is a group algebra, then Hopf ideals of \( \mathbb{k}G \) are exactly the ideals of the form \( (g - 1 \mid g \in \) \( N \) ), where \( N \) is a normal subgroup of \( G \) (Exercise 9.3.4). It follows that \( \mathcal{H}I \) is t... | No |
Example 2 | 7 For this joint probability matrix with \( \operatorname{Prob}\left( {{x}_{1},{y}_{2}}\right) = {0.3} \), find \( \operatorname{Prob}\left( {{y}_{2} \mid {x}_{1}}\right) \) and \( \operatorname{Prob}\left( {x}_{1}\right) \).
\[
P = \left\lbrack \begin{array}{ll} {p}_{11} & {p}_{12} \\ {p}_{21} & {p}_{22} \end{array}\... | Yes |
Example 1 | Example 1 (a) If \( n \) is 3 and \( m \) is 16, then \( {16} = 5\left( 3\right) + 1 \) so \( q \) is 5 and \( r \) is 1 .
(b) If \( n \) is 10 and \( m \) is 3, then \( 3 = 0\left( {10}\right) + 3 \) so \( q \) is 0 and \( r \) is 3 .
(c) If \( n \) is 5 and \( m \) is -11, then -11 = -3(5) +4 so \( q \) is -3 and \... | Yes |
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 fol... | No |
Problem 1.275 | Problem 1.275 Let \( X \) be a non-empty set. Let \( \mathcal{M} \) be a \( \sigma \) -algebra in \( X \) . Let \( \mathcal{M} \) be an infinite set. Show that
a. \( X \) is an infinite set,
b. there exists a sequence \( \left\{ {{B}_{1},{B}_{2},\ldots }\right\} \) of members in \( \mathcal{M} \) such that \( \varnot... | No |
Example 2 | 4. The following are ten measurements of \( {\mu }^{\prime } \) and eight measurements of \( \mu \) :
\[
{\mu }^{\prime } : {17.3},{17.1},{18.2},{17.5},{15.8},{16.9},{17.0},{17.5},{17.8},{17.1}
\]
\[
\mu : {3.2},{3.2},{3.9},{3.3},{2.7},{3.4},{4.0},{2.9}\text{.}
\]
Obtain an estimate of \( {\mu }^{\prime } - \mu \) . | Yes |
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... | No |
Example 2.13 | Example 2.13 \( {L}^{2}\left( \left\lbrack {-a, a}\right\rbrack \right) \)
We know from experience in quantum mechanics that all square integrable functions on an interval \( \left\lbrack {-a, a}\right\rbrack \) have an expansion \( {}^{10} \)
\[
f = \mathop{\sum }\limits_{{m = - \infty }}^{\infty }{c}_{m}{e}^{i\frac... | No |
Example 6.1 | Measuring the distance \( d \) between the two points on the plane involves measuring the angle \( \phi \) . This angle is estimated by averaging \( n = 7 \) readings each containing error drawn independently from a single normal distribution with mean zero. The sample standard deviation is found from (6.8) to be \( s ... | No |
Example 5.6 | Example 5.6 Let \( {\mathbb{T}}_{1} = \mathbb{Z},{\mathbb{T}}_{2} = {2}^{{\mathbb{N}}_{0}},\left\lbrack {{a}_{1},{b}_{1}}\right\rbrack = \left\lbrack {-1,1}\right\rbrack ,\left\lbrack {{a}_{2},{b}_{2}}\right\rbrack = \left\lbrack {1,4}\right\rbrack \) . We
consider
\[
I\left( {t}_{2}\right) = {\int }_{-1}^{1}\left( {... | No |
Example 5.2 | 5.2 Prove relations (5.22). | No |
Example 6.2 | The \( {C}^{1} \) path \( \gamma : \mathbb{R} \rightarrow \mathbb{H} \) defined by \( \gamma \left( t\right) = i{e}^{t} \) travels along the geodesic \( \{ z \in \mathbb{H} : \operatorname{Re}z = 0\} \) . Moreover,
\[
{\left| {\gamma }^{\prime }\left( t\right) \right| }_{\gamma \left( t\right) } = \frac{{\left\langle ... | 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 |
Example 9.25 | Example 9.25. Let a signal be a digital image generated by the following 4- dimensional commutative linear representation system
\( \sigma = \left( {\left( {{K}^{4},{F}_{\alpha },{F}_{\beta }}\right) ,{x}^{0}, h}\right) \) with a vector index \( \nu = \left( 4\right) \) ,
where \( {F}_{\alpha } = \left\lbrack \begin{... | Yes |
Example 7.4 | Example 7.4 The stationary covariance function of \( X\left( t\right) \) in Example 7.3 with \( k = 2 \) has the expression
\[
c\left( {t, s}\right) = \frac{2}{\rho \left( {\rho + {2\alpha }}\right) }{e}^{-\rho \left( {t - s}\right) } + \frac{2}{{\rho }^{2} - 4{\alpha }^{2}}\left( {{e}^{-{2\alpha }\left( {t - s}\right... | Yes |
Example 4.22 | Example 4.22. Let \( A \) be the matrix
\( \mathrm{A} = \)
1 2 3
\( \begin{array}{lll} 4 & 5 & 6 \end{array} \)
To create a \( 3 \times 2 \) tiling using \( A \) as a tile, we write \( B = \operatorname{repmat}\left( {A,3,2}\right) \) , which results in \( \mathrm{B} = \)
<table><tr><td>1</td><td>2</td><td>3</td><... | No |
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 | 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) \... | 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... | No |
Example 5.3.8 | Example 5.3.8 It is easy to check that for \( q = 1 \) the formula in Theorem 5.3.7 becomes the formula in Theorem 5.3.2. For \( q = 2 \), the formula in Theorem 5.3.7 becomes
\[
{\left\lbrack n\right\rbrack }_{m + 2}{\widehat{\chi }}_{m,2,{1}^{n - m - 2}}^{\lambda } = \left\lbrack {{c}_{3}^{\lambda }\left( 2\right) -... | Yes |
Example 15 | Example 15. Let \( V \) be the space of all polynomial functions from \( R \) into \( R \) of the form
\[
f\left( x\right) = {c}_{0} + {c}_{1}x + {c}_{2}{x}^{2} + {c}_{3}{x}^{3}
\]
that is, the space of polynomial functions of degree three or less. The differentiation operator \( D \) of Example 2 maps \( V \) into \... | No |
Problem 99 | Problem 99. Prove that \( {o}_{{2j} + 1} = {\beta }^{ * }{o}_{2j} \) . | No |
Example 7.4 | Example 7.4. Consider the finite-state automaton shown in Figure 7.2, whose associated alphabet is \( \{ a, b\} \) . The vertices represented by double circles are the accept states, and the start state is indicated by the letter S. The language for this FSA consists of all words in \( \{ a, b{\} }^{ * } \) that contai... | No |
Example 6 | [Example 6 (Non-elementary substructure). In the language of rings, \( \mathbb{R} \) is a substructure of \( \mathbb{C} \), since it is a subfield. On the other hand, in notation of Example 4, \( \varphi \left( \mathbb{R}\right) = {\mathbb{R}}_{0}^{ + } \) and \( \varphi \left( \mathbb{C}\right) = \mathbb{C} \) so \( \... | No |
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 |
Example 5.4.6 | Example 5.4.6. Consider the operator \( D : {C}^{1}\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \rightarrow {C}^{0}\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \) defined by taking the derivative \( D\left( f\right) = {f}^{\prime } \) . This is not a ring homomorphism. It is true that \(... | No |
Example 1.4.6 | Example 1.4.6
Standard Matrices
of Reflections
---
Find the standard matrices of the linear transformations that reflect through the lines in the direction of the following vectors, and depict these reflections geometrically.
a) \( \mathbf{u} = \left( {0,1}\right) \in {\mathbb{R}}^{2} \), and
b) \( \mathbf{w} = \... | No |
Example 3 | Let \( X \) and \( Y \) have the CDFs
\[
F\left( x\right) = \frac{{x}^{2}}{4},0 < x < 2\text{ and }G\left( x\right) = \exp \left\{ {\frac{1}{2} - \frac{1}{x}}\right\} ,0 < x < 2,
\]
respectively. The \( {\mathrm{{WDRD}}}_{\alpha } \) between \( X \) and \( Y \) is plotted in Fig. 1. This shows that the \( {\mathrm{{W... | No |
Example 5.1 | Example 5.1 Choose primitive polynomial \( p\left( x\right) = {x}^{4} + x + 1 \in {Z}_{2}\left\lbrack x\right\rbrack \) . The nonzero elements in \( F = {Z}_{2}\left\lbrack x\right\rbrack /\left( {p\left( x\right) }\right) \) are listed in the table in Example 4.3. Using this field \( F \), we obtain the following gene... | No |
Example 2.14 | During some construction, a network blackout occurs on Monday with probability 0.7 and on Tuesday with probability 0.5 . Then, does it appear on Monday or Tuesday with probability \( {0.7} + {0.5} = {1.2} \) ? Obviously not, because probability should always be between 0 and 1 ! Probabilities are not additive here beca... | 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 |
Example 2 | We next consider a problem for the set \( G \) in Figure 5.6, but now the reflection direction is the unit vector in the direction \( \left( {2,3}\right) \) . See Figures 5.7a, b. Here, several possibilities are of interest. Clearly, this direction cannot be achieved as a convex combination of the vectors \( \left( {1,... | No |
Example 4.7 | Given \( a > 4 \), consider the map \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) defined by
\[
f\left( x\right) = {ax}\left( {1 - x}\right) .
\]
We have
\[
f\left( \left\lbrack {\frac{1}{a},\frac{1}{2}}\right\rbrack \right) = \left\lbrack {1 - \frac{1}{a},\frac{a}{4}}\right\rbrack \supset \left\l... | No |
Example 5.3.2 | Consider \( X \sim P\left( I\right) \left( {\sigma ,\alpha }\right) \) a classical Pareto distribution. The sequences of absolute Gini indices are
\[
{m}_{n} = \frac{\alpha n\sigma }{{\alpha n} - 1},\;n = 1,2,\ldots
\]
if \( {\alpha n} > 1 \), since the distribution of the minimum is again of the Pareto form, i.e., \... | Yes |
Example 2.3.15 | Example 2.3.15 For the lexicographical ordering \( > \) in the variables \( {x}_{1},{x}_{2} \), the set \( {D}_{ > } \) is plotted in Figure 2.6. The figure also shows the line \( {w\delta } = 0 \), where \( w \) is a weight vector representing \( {lp} \) on all monomials of degree \( \leq 4 \) . For more examples see ... | No |
Example 5.5.1 | Example 5.5.1 What is the distribution of \( W\left( t\right) + W\left( \tau \right) \), where \( t \leq \tau \) ?
Solution. We can write that
\[
W\left( t\right) + W\left( \tau \right) = {2W}\left( t\right) + \left\lbrack {W\left( \tau \right) - W\left( t\right) }\right\rbrack \mathrel{\text{:=}} Y + Z,
\]
where \(... | Yes |
Example 9.28 | Example 9.28 ( \( E \) -related transformation and Concurrency Theorem). We use the construction in the proof of Fact 5.29 to construct an \( E \) -dependency relation and an \( E \) -concurrent production for the sequentially dependent transformations \( A{G}_{1}\overset{{addClass},{m}_{2}}{ \Rightarrow }A{G}_{2}\over... | No |
Example 5 | Example 5 For any real numbers \( {a}_{1},{a}_{2},{a}_{3},{b}_{1},{b}_{2} \), and \( {b}_{3} \), show that
\[
\left| {{a}_{1}{b}_{1} + {a}_{2}{b}_{2} + {a}_{3}{b}_{3}}\right| \leq \sqrt{{a}_{1}^{2} + {a}_{2}^{2} + {a}_{3}^{2}}\sqrt{{b}_{1}^{2} + {b}_{2}^{2} + {b}_{3}^{2}}.
\] | No |
Example 3.8 | Example 3.8. The ends of a uniform string of linear density \( \rho \) and length \( l \) are fixed, and all external forces are neglected. Displace the string from equilibrium by shifting the point \( x = {x}_{0} \) by a distance \( h \) at time \( t = 0 \) and then release it with zero initial speed. Find the displac... | No |
Example 3.18 | The spectral distribution function of the weakly stationary real-valued process in Example 3.14 is \( S\left( v\right) = \alpha + \mathop{\sum }\limits_{{k = 1}}^{n}\left( {{\sigma }_{k}^{2}/2}\right) \left\lbrack {1\left( {v \geq - {v}_{k}}\right) + 1\left( {v \geq {v}_{k}}\right) }\right\rbrack \) , where \( \alpha \... | No |
Example 1.7 | [Use Gauss's lemma to evaluate the Legendre symbol \( \left( \frac{6}{11}\right) \) . By Gauss’s lemma, \( \left( \frac{6}{11}\right) = {\left( -1\right) }^{\omega } \), where \( \omega \) is the number of integers in the set
\[
\{ 1 \cdot 6,2 \cdot 6,3 \cdot 6,4 \cdot 6,5 \cdot 6\}
\]
whose least residues modulo 11 ... | Yes |
Example 3.1.2 | Example 3.1.2
Computing a
Coordinate Vector
in the Range
of a Matrix
---
Find a basis \( B \) of the range of the following matrix \( A \) and then compute the coordinate vector \( {\left\lbrack \mathbf{v}\right\rbrack }_{B} \) of the vector \( \mathbf{v} = \left( {2,1, - 3,1,2}\right) \in \operatorname{range}\le... | Yes |
Example 1 | Let \( A \) fuzzy set and \( R = \left( {Z,+,\text{.}}\right) {bethering} \) of all integers. Define a mapping \( f : A \rightarrow F\left( {{NR}\left( Z\right) }\right) \) where, for any \( a \in A \) and \( x \in Z \) ,
\[
{A}_{f}\left( x\right) = \left\{ \begin{array}{l} 0\text{ if }x\text{ is odd } \\ \frac{1}{a}\... | No |
Example 4.24 | Example 4.24 General Simple Random Walk. Now consider the general simple random walk with the following transition probabilities:
\[
{p}_{i, i + 1} = {p}_{i}\text{ for }i \geq 0,
\]
\[
{p}_{i, i - 1} = {q}_{i} = 1 - {p}_{i}\text{ for }i \geq 1,
\]
\[
{p}_{0,0} = 1 - {p}_{0}
\]
Assume that \( 0 < {p}_{i} < 1 \) for ... | No |
Example 24.9 | Example 24.9 Let \( f \in {\Gamma }_{0}\left( \mathcal{H}\right) \) . Then \( \partial f \) is \( {3}^{ * } \) monotone. | No |
Problem 6.10.5 | Problem 6.10.5. Consider the Lie group
\[
G = \left\{ {\left( \begin{array}{ll} 1 & 0 \\ x & y \end{array}\right) : x, y \in \mathbb{R}, y > 0}\right\} .
\]
(1) Prove that its Lie algebra is \( \mathfrak{g} = \langle y\partial /\partial x, y\partial /\partial y\rangle \) .
(2) Write the left-invariant metric on \( G... | No |
Example 6.5 | \[
Y\left( t\right) = \left\lbrack \begin{array}{l} s + t \\ X\left( t\right) \end{array}\right\rbrack \in {\mathbb{R}}^{2};Y\left( 0\right) = y = \left( {s, x}\right) \tag{6.2.17}
\]
where \( X\left( t\right) = x + B\left( t\right) + {\int }_{0}^{t}{\int }_{\mathbb{R}}z\widetilde{N}\left( {{ds},{dz}}\right) \) and \(... | No |
Example 4.3.1 | Example 4.3.1. Solve the quadratic equation
\[
2{x}^{2} - {5x} + 1 = 0
\]
by completing the square. Divide through by 2 to make the quadratic monic giving
\[
{x}^{2} - \frac{5}{2}x + \frac{1}{2} = 0
\]
We now want to write
\[
{x}^{2} - \frac{5}{2}x
\]
as a perfect square plus a number. We get
\[
{x}^{2} - \frac{... | Yes |
Problem 1 | Problem 1. Does there exist a separable (complete separable, \( \sigma \) -compact, compact) metric space \( X \) such that
\[
{\operatorname{DIM}}_{\mathrm{{PC}}}X = \operatorname{ind}X - 1\text{?} \tag{8}
\]
It is clear that if \( X \) satisfies (8), then \( X \) cannot be a finite-dimensional manifold since \( {\o... | No |
Example 3.66 | Example 3.66 (intersection and product machine-Sakarovitch [12]) The recognizer \( M \) of the strings that contain as substrings both digrams \( {ab} \) and \( {ba} \) is naturally specified through the intersection of languages \( {L}^{\prime } \) and \( {L}^{\prime \prime } \) :
\[
{L}^{\prime } = {\left( a \mid b\... | No |
Example 2 | [The language \( {L}_{{fc} = {lc}} \) (see Example 1) of pictures \( p \) whose first column is equal to the last one, is in UREC. Indeed, we can define a tiling system as done before and this is unambiguous. This because there is only one possible counter-image for the first column of a picture \( p \) and there is a ... | No |
Example 2 | Example 2. We know by CLT theorem that \( {Y}_{n} = \bar{X} \) is \( \mathrm{{AN}}\left( {\mu ,{\sigma }^{2}/n}\right) \) . Suppose \( g\left( \bar{X}\right) = \) \( \bar{X}\left( {1 - \bar{X}}\right) \) where \( \bar{X} \) is the sample mean in random sampling from a population with mean \( \mu \) and variance \( {\si... | No |
Example 6 | Example 6. Let \( F\left( {X, Y}\right) = p\left( X\right) {Y}^{d} + {Y}^{d - 1} + q\left( X\right) {Y}^{2} + r\left( X\right) \), with \( \deg \left( p\right) = \) \( m \geq 1,\deg \left( q\right) = d + m - 1,\deg \left( r\right) = d + m + 1, d \geq 5 \) . We have
\[
\frac{\deg \left( {P}_{1}\right) - \deg \left( {P}... | No |
Exercise 6.8.10 | [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 |
Example 7 | Example 7. Marcinkiewicz-Jackson-de La Vallée-Poussin summation. Let
\[
{\theta }_{8}\left( t\right) = \left\{ \begin{array}{ll} 1 - 3{t}^{2}/2 + 3{\left| t\right| }^{3}/4 & \text{ if }\left| t\right| \leq 1 \\ {\left( 2 - \left| t\right| \right) }^{3}/4 & \text{ if }1 < \left| t\right| \leq 2 \\ 0 & \text{ if }\left|... | No |
Example 4.2 | John Slow is driving from Boston to the New York area, a distance of 180 miles at a constant speed, whose value is uniformly distributed between 30 and 60 miles per hour. What is the PDF of the duration of the trip? Let \( X \) be the speed and let \( Y = g\left( X\right) \) be the trip duration:
\[
g\left( X\right) =... | No |
Example 14.59 | Example 14.59. Let \( G \) be the metrizable compact compact compact connected abelian group \( {S}^{\mathbb{N}} \) and \( X \) an arbitrary subset of \( \mathbb{N} \) . We form \( {H}_{X} = {S}_{a}^{X} \times {S}^{\mathbb{N} \smallsetminus X} \) considered in the obvious way as a subgroup of \( G = {S}^{\mathbb{N}} \)... | 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\righ... | No |
Problem 3 | [Problem 3. Is it true that the cohomotopical dimension \( \pi - \dim X = \dim X \) for every \( X \) ?] | No |
Example 2.6.4 | Example 2.6.4. We determine all proper representations of 1 by \( f = {X}^{2} + {Y}^{2} \) . The discriminant of \( f \) is \( \Delta \left( f\right) = - 4 \) . So we compute all \( \Gamma \) -orbits of forms \( \left( {1, B, C}\right) \) of discriminant -4 . By (2.14) any such \( \Gamma \) -orbit contains a form \( \l... | Yes |
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 mean... | No |
Problem 13.2.18 | Problem 13.2.18 (Optimal stochastic control problem with average cost for a finite stochastic control system with complete observations) Consider the stochastic control system of Definition 13.2.17. The optimal stochastic control problem with average cost is to determine a control law \( {g}^{ * } \in G \) such that
\... | No |
Example 7.13 | Previously, we stated that the exact value of the integral (7.5) is \( {\int }_{0}^{1}{x}^{2}\sqrt{1 - {x}^{2}}\mathrm{\;d}x = \frac{\pi }{16} \) . If we use the substitution \( x = \sin t,0 \leq t \leq \frac{\pi }{2} \), we will see why:
\[
{\int }_{0}^{1}{x}^{2}\sqrt{1 - {x}^{2}}\mathrm{\;d}x = {\int }_{0}^{\frac{\p... | Yes |
Example 1.3 | Here are some examples of functions on finite sets - one which is surjective, and one which is not.
A surjection from \( \{ 1,2,3,4\} \) to \( \{ a, b, c\} \)
![0191b05f-39d9-7158-8173-95092112665e_18_568... | No |
Example 6.2 | Example 6.2. Consider the Laplacian on the unit disk \( \mathbb{D} \subset {\mathbb{R}}^{2} \) . In polar coordinates,
\[
\Delta = \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}}\right) + \frac{1}{{r}^{2}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}.
\]
If we substitute \( \phi \left( ... | No |
Example 1.3 | Let \( k \) be a commutative ring, \( G \) a finite group, and \( A \) a \( G \) - module algebra. Let \( \mathcal{C} = { \oplus }_{\sigma \in G}A{v}_{\sigma } \) be the left free \( A \) -module with basis indexed by \( G \), and let \( {p}_{\sigma } : \mathcal{C} \rightarrow A \) be the projection onto the free compo... | 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 \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) conver... | No |
Example 6.4.1 | 6.4.1 Suppose that \( {\pi \sigma } \in {\mathbb{L}}_{-1\mathrm{{oc}}}^{2,4},{V}_{0} \in {\mathbb{D}}_{1\mathrm{{oc}}}^{1,2},\left( {{\mu }_{t} - {\rho }_{t}}\right) {\pi }_{t} - \frac{{\left( {\pi }_{t}{\sigma }_{t}\right) }^{2}}{2} \in {\mathbb{L}}_{1\mathrm{{oc}}}^{1,2} \) and the value process \( {V}_{t} \) has con... | No |
Example 2.34 | Example 2.34. Any random matrix and the identity are asymptotically free. | 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} = ... | No |
Example 1.19 | Example 1.19 Let \( X = {\mathbb{R}}^{2} \) . One can write any function \( f : S \rightarrow {\mathbb{R}}^{2} \) in terms of component functions \( {fs} = \left( {{xs},{ys}}\right) \), where the components \( {xs} \) and \( {ys} \) are simply given by the composition
![0191b040-d6f7-7f03-83f6-23d666ea31d2_32_314365.j... | No |
Example 4.1.1 | Example 4.1.1 Let \( {\mathcal{F}}_{t} \) be the information available until time \( t \) regarding the evolution of a stock. Assume the price of the stock at time \( t = 0 \) is \( \$ {50} \) per share. The following decisions are stopping times:
(a) Sell the stock when it reaches for the first time the price of \( \... | No |
Problem 6.3 | Problem 6.3. An airport bus deposits 25 passengers at 7 stops. Each passenger is as likely to get off at any stop as at any other, and the passengers act independently of one another. The bus makes a stop only if someone wants to get off. Use Markov chain analysis to calculate the probability mass function of the numbe... | Yes |
Example 5.11 | Example 5.11. Consider the curve defined by \( {y}^{2} = \left( {x - {r}_{1}}\right) \left( {x - {r}_{2}}\right) \left( {x - {r}_{3}}\right) \) . Let’s assume that all three \( {r}_{i} \) are distinct, say \( {y}^{2} = \left( {x + 1}\right) x\left( {x - 1}\right) - \) that is, \( {y}^{2} = {x}^{3} - x \) . Figure 5.3 d... | No |
Problem 4.7 | Problem 4.7. Let \( f : K \rightarrow \mathbb{R} \) be a discrete Morse function with \( \overrightarrow{f} \) a perfect discrete Morse vector. Show that
(i) \( \overrightarrow{f} \) is unique;
(ii) \( \overrightarrow{f} \) is optimal. | No |
Example 7.3.5 | (1) Let \( G = \mathbf{R} \) . Then, as we have already seen (see Example 3.4.10), the dual group is isomorphic as a group to \( \mathbf{R} \) using the map
\[
\left\{ \begin{matrix} \mathbf{R} & \rightarrow & \widehat{G} \\ t & \mapsto & {e}_{t}, \end{matrix}\right.
\]
where \( {e}_{t}\left( x\right) = {e}^{itx} \) ... | No |
Example 1.3.1 | Example 1.3.1. Verify that the function \( u\left( x\right) = {e}^{-x} \) is a solution of the ODE \( {u}^{\prime } = - u \) . Solution: Observe that \( {u}^{\prime }\left( x\right) = - {e}^{-x} \) and that \( - u\left( x\right) = - {e}^{-x} \), for any real number \( x \) . Substituting these into the given ODE result... | No |
Example 2.32 | [Example 2.32. In the case of \( {S}^{1} \), the map \( f\left( z\right) = {z}^{k} \), where we view \( {S}^{1} \) as the unit circle in \( \mathbb{C} \), has degree \( k \) . This is evident in the case \( k = 0 \) since \( f \) is then constant. The case \( k < 0 \) reduces to the case \( k > 0 \) by composing with \... | Yes |
Example 2.68 | [Example 2.68 (Arens algebra \( {L}^{\omega }\left( {0,1}\right) \) )
Let \( \parallel \cdot {\parallel }_{p} \) denote the norm of \( {L}^{p}\left( {0,1}\right) \) with respect to the Lebesgue measure on \( \left\lbrack {0,1}\right\rbrack \) . Let \( p \in \left( {1, + \infty }\right) \) . For \( f, g \in {L}^{p}\lef... | No |
Example 4.26 | Example 4.26 Find the inflection points and concave up and down ranges of
\[
f\left( x\right) = {x}^{3} - {4x}
\]
## Solution:
We need the second derivative:
\[
{f}^{\prime }\left( x\right) = 3{x}^{2} - 4
\]
\[
{f}^{\prime \prime }\left( x\right) = {6x}
\]
Solve \( {6x} = 0 \), and we get that there is an inflect... | No |
Example 15 | Example 15. Let \( \operatorname{Prog} = \left\{ {P\left( {\widehat{{x}_{1}},\widehat{g\left( {x}_{2}\right) }}\right) \leftarrow {P}^{\prime }\left( {\widehat{{x}_{1}},\widehat{{x}_{2}}}\right) .P\left( {\widehat{f\left( {x}_{1}\right) },\widehat{{x}_{2}}}\right) \leftarrow {P}^{\prime \prime }\left( {\widehat{{x}_{1}... | 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)... | No |
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