Use this tag for questions asking about "problem books", "exercise books", and their solutions.

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2
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1answer
477 views

Stochastic Calc

(a) Consider the process $$ \mathrm d\sqrt{v} = (\alpha - \beta\sqrt{v})\mathrm dt + \delta \mathrm dW $$ Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that $$ \mathrm dv = ...
2
votes
2answers
66 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
1
vote
1answer
21 views

Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
1
vote
0answers
14 views

$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I ...
3
votes
0answers
36 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
1
vote
1answer
24 views

Generate some random matrix with given rank

Very often for creating new exercises (I teach basic matrix algebra), I need to a find a matrix $A$ such that: it has integer coefficients, not too big (in order to avoid big numbers computations) ...
0
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0answers
20 views

use Fatou theorem to prouve an convergence

let $u_n$ an sequence uniformaly bounded in $H^1_0(\Omega)$, then, $u_n$ converge weakly to $u$ in $H^1_0$, and strongly in $L^2(\Omega)$ and a.e $x \in \Omega$. Let $g(x,u)$ an Carathedory function ...
3
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2answers
137 views

Nice exercises on resultants

I would like to ask if some one knows a source (a book, or lecture notes ect) that contains several nice exercises on resultants of polynomials (it would be nice if there were some solutions as well ...
1
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0answers
29 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
3
votes
2answers
25 views

How is this exactly equal to $N_1+N_2+\dots+N_r$?

There are $N$ boxes, each containing at most $r$ balls. If the number of boxes containing at least $i$ balls is $N_i$ for $i=1,2,\dots,r,$ then the total number of balls contained in these $N$ ...
0
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0answers
56 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
0
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0answers
31 views

Application of Schauder fixed point theorem

Let $\Omega$ an open bounded in $\mathbb{R}^n$, and let $A(x,u)$ an Caratheodory function, bounded and coercive, i.e., $$|A(x,u)|\leq \beta$$ $$A(x,u) \geq \alpha I,\quad \alpha > 0$$ and let ...
2
votes
2answers
73 views

Lebesgue measurable homework problem

Let $X \subseteq \mathbb{R}$. A subset $E \subseteq \mathbb{R}$ is called a hull of $X$ if $E$ is measurable $X \subseteq E$ If $F$ is any measurable set such that $X \subseteq F$, then $E$\ $F$ is ...
0
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0answers
28 views

Find variational formulation

We have the problem in $\Omega=\mathbb{R}^2_+$ $$ \begin{cases} & -a_{11} \dfrac{\partial^2 u}{\partial x^2} - (a_{12} + a_{21}) \dfrac{\partial^2 u}{\partial x \partial y} - a_{22} ...
3
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0answers
53 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
0
votes
1answer
44 views

differential system-bounded solution

We consider the differential system $$Y'=AY,\quad \mbox{in} [0,+\infty[, Y(0)=Y_0$$ where $A$ is $n \times n$ matrix diagonalisable, $Y_0 \in \mathbb{R}^n$ and $Y \in \mathbb{R}$ What's the suffisant ...
1
vote
0answers
29 views

Prove for i.i.d $X$ and $Y$, $Pr[|X-Y|\leq 2] \leq 3*Pr[|X-Y|\leq 1]$

Given $X$ and $Y$ two independent identically distributed random variables. Prove: $Pr[|X-Y|\leq 2] \leq 3*Pr[|X-Y|\leq 1]$ This is an exercise from The Probabilistic Method 3rd Edition with a STAR ...
2
votes
2answers
57 views

Density in $H^1_0$

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$. For $h \neq 0$ we define $D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$ such as $\forall \varphi \in ...
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0answers
18 views

Exercises for the text “Introduction to Holomorphic Functions of Several Complex Variables”

I would like to know where can I find good exercises and problems that fits with the way Rubert C. Gunning develops his theory on his "Introduction to Holomorphic Functions of Several Complex ...
3
votes
3answers
56 views

Number of solutions of $x+y+z=10$

The number of different solutions $(x,y,z)$ of the equation $x+y+z=10$ where each of $x, y$ and $z$ is a positive integer is $36$. How to derive this answer? I know that $x, y$ and $z$ have to ...
5
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0answers
232 views

ODE Laplace Transform an impulse bring oscillating system to rest

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
0
votes
3answers
69 views

If the sum $\sum_{x=1}^{100}x!$ is divided by $36$, how to find the remainder?

If the sum $$\sum_{x=1}^{100}x!$$ is divided by $36$, the remainder is $9$. But how is it? THIS said me that problem is $9\mod 36$, but how did we get it?
3
votes
3answers
173 views

What does “order matters” regarding permutations refer to?

I psychoanalyze EVERYTHING and permutations/combinations are frustrating me. Sorry for posting so many questions lately but I really appreciate all of the help! Ok so I know the permutation formula: ...
2
votes
1answer
201 views

Phragmen-Lindelöf theorem, question from Conway, chapter VI

Page 141, Question 3: Let $G=\{z:|\operatorname{Im} z| < \pi/2\}$ and suppose $f:G\rightarrow C$ and $\limsup|f(z)| \leq M$ on $w$ in the boundary of $G$. Also, suppose $A < \infty$ and $a ...
2
votes
1answer
40 views

Describe measurable functions

I have the following exersice: Describe the Borel-measurable functions $f:X\to\mathbb{R}$, where $X$ has the $\sigma$-algebra of subsets $A\subseteq X$ such that $A$ is countable or $X\setminus A$ is ...
1
vote
1answer
57 views

Irreducibility of $\frac{X^{p^n}-1}{X^{p^{n-1}}-1}$.

Exercise: Let $p$ be a prime number. Then, the polynomial \begin{equation} \frac{X^{p^n}-1}{X^{p^{n-1}}-1} \end{equation} is irreducible over $\mathbb Z[X]$, for any integer $n \geq 1$. I'm able to ...
0
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0answers
32 views

Contour integration!Help

I have to integrate a function following the route from the point $(0,0,0)$ to $(1,1,1)$ which consists of the 2 curves $C=(t,t^2,0)$ and $K=(1,1,t)$ $0\leq t\leq 1$ .Is it right to take the 2 ...
1
vote
1answer
31 views

Integration Exercise.Help!

I have to integrate the function F(x,y)=x+y on the line segment x=t , y=1-t , z=0 from (0,1,0) to (1,0,0) .So what i did is think the line segment as a vector function(curve) σ(t)=(t,1-t,0) with ...
1
vote
2answers
35 views

minimize $\log_a x+1$ subject to $0\leq a\leq x$

My book says "If $0\leq a \leq x$, then minimum value of $\log_a x+\log_x x$ is $2$." But Wolfram|Alpha says that it isn't! Why is so?
2
votes
1answer
288 views

Finding a conformal map from semi disk to upper half plane.

Find a conformal mapping $f$ of semi-disk$S=\{z: \vert z\vert \lt 1, Im z\gt 0\}$ onto the upper plane. Again I used composition of conformal map. First of all, let's define a conformal map ...
1
vote
1answer
79 views

Is “locally of finite type” affine-local on the source?

Hartshorne exercise II.3.3(c) asks the reader to prove that if $f:X\to Y$ is finite type, then for any $\mathrm{Spec}A\subset Y$ and $\mathrm{Spec}B\subset X$ with $\mathrm{Spec}B\subset f^{-1} ...
0
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0answers
82 views

Convolution of piecewise function

I would like to compute the convolution of piece wise function Following is the piecewise function $$ C_a(t) = \begin{cases}0& t\leq t_d\\ ...
2
votes
3answers
51 views

$2^n$ choose something

Let $m$ be a positive integer, and let $n=2^m$. Prove that the numbers $$ \binom{n}{1}, \binom{n}{2}, \dots , \binom{n}{n-1} $$ are all even. -Source: ASMP sample problems Counting Strategies number ...
1
vote
2answers
39 views

Which way to go?

Given an m x n grid, How many ways are there to go from upper left corner to the lower right one? You can only move right and down, not up or left. Numerical solution are: m=2, n=2, solution=6 ...
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0answers
178 views

Solution manual to Larson's “Problem Solving through Problems”

I am working through Larson's "Problem Solving through Problems" (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) but many of the problems have neither solutions nor sources included. Does ...
3
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2answers
40 views

Commuting two pullbacks

I have stumbled upon some interesting exercise whilst reading the "Category Theory for Scientists" book. Below is the universal property of fiber products: By using the universal property, I can ...
2
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2answers
78 views

Is the sequence $ \frac{1}{10^n} $ convergent?

I must prove that $ f: \Bbb{N} \to \Bbb{R}; n \to\frac{1}{10^n} $ is a convergent sequence. I thought: If $f$ is convergent then $\exists L \in \Bbb{R}(\forall \epsilon >0 (\exists m \in ...
6
votes
1answer
183 views

Maximal ideals in the algebra of continuously differentiable functions on [0,1]

This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
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0answers
73 views

Vakil's exercise 5.5.#

I'd like to check that my solution to the following exercise of Vakil's FOAG is correct. I am bothered that we have to use (B) in it, and I want to make sure that I used it in the end for the correct ...
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0answers
15 views

Finite p-primary group question

Problem is rotman's Intrdouction to theory of group, Exercise 6.8. Let G be a direct sum of b cyclic groups of order $p^{m}$. If $n< m$, then $p^{n}G/p^{n+1}G $is elementary and ...
2
votes
1answer
120 views

Exercice on a differential form

Let $\omega$ be a $q$-form on $\mathbb{R}^2$ and let $Z_{\mathbb{R}^2}(dx_1)=\{p \in \mathbb{R}^2 \colon (dx_1)_{|p}=0\}$ $Z_{S^1}(dx_1)=\{p \in S^1 \colon (dx1_{|S^1})_{|p}=0\}$ where ...
1
vote
1answer
51 views

If $A$ is reduced, Spec $A$ has no embedded points

I've partly solved the following exercise of Vakil's FOAG, but I am not sure I got the last part right. Could some take a look? 5.5.C. EXERCISE (ASSUMING (A)). Show that if $A$ is reduced, Spec ...
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0answers
28 views

Exercises on nets

well I'm learning nets with Munkres, but I'd like to do more exercises than those in this book. Any web site or reference would be welcome. Thanks in advance
2
votes
2answers
55 views

Area with double integral.

$f:\mathbb{R}^2\to\mathbb{R}, f(x,y)=(x^2+y^2)^2-8(x^2-y^2).$ Find the critical points of $f$ and the area of $X=\{(x,y):f(x,y)\leq 0\}$. To find the critical points I just have to find ...
0
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2answers
84 views

bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
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0answers
91 views

Tricky calculus exercise (Now finding a tangent space)

Let $f(x,y,z)=z^3-g(x,y)z+h(x,y)$, with $g,h:\mathbb{R}^2\to \mathbb{R}$ of class $C^1$ and $g(x,y)>0\; \forall (x,y)\in\mathbb{R}^2$. Consider $S=\{(a,b,c)\in\mathbb{R}^3:f(a,b,c)=0\}$, ...
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0answers
32 views

How to solve this second order PDE $(\nabla\cdot k \nabla F+F=0 )$?

I am trying to follow an example of finding a fixed solution of PDE using method of manufactured solutions (p. 58). At some point there is an equation $$\nabla\cdot k \nabla F+F=0 $$ where ...
13
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2answers
306 views

How to deal with exercises with no solutions given?

Probably most people will acknowledge the importance of doing exercises when reading a mathematical textbook. Here I am talking about a textbook of similar level as those ones listed in GTM. However, ...
0
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0answers
38 views

Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
3
votes
2answers
3k views

Transpose of block matrix

I'm attempting to prove that $$ \left[ \begin{array}{c c} A & B \\ C & D \\ \end{array} \right]^\top = \left[ \begin{array}{c c} A^\top & C^\top \\ B^\top & D^\top \\ \end{array} ...