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

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1answer
20 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.
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2answers
65 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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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 $\mu$ ...
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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 $N$ ...
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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 ...
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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 ...
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2answers
71 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 ...
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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} ...
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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 ...
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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 ...
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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 ...
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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 ...
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3answers
55 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 ...
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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?
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3answers
169 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
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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 ...
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1answer
56 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 ...
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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 ...
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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 ...
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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?
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81 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\\ ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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
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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 ...
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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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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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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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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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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 ...
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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 ...
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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 $U_i$. ...
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1answer
54 views

Hilbert space, orthonormal system, compact set of vecors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
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1answer
13 views

Probability and Statistics Expected Weekly Loss

So I have this exercise (picture below) and I have the mean calculated and the variance of the random variable X, so I'm left with the formula for Loss, should I simplu substitute the X in the E(10X ...
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1answer
20 views

Probability Density Distribution Exercise

So here I have 2 exercises to which I have solutions of mine, but wonder if they are correct, so they are as follows: My solutions are: 1) I used this formula; E(X) = x$_1$p(x$_1$) + ...
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4answers
104 views

Probability Exercise - Contracts

Here I have this exercise which I am not sure of how to approach, it is in the conditional probability section but I cannot see the use in here, I will state the question and then state my intuitive ...
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2answers
81 views

Probability Exercise (Java and C++)

So I have this probability exercise and I'd like to know if it is correct, along with my reasoning, so here is the exercise: In a computer installation, 200 programs are written each week, 120 in ...
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2answers
92 views

$\|.\|_2$ closure of a set which is dense in $L^2[0,2\pi].$

The following is an exercise of Conway's Functional analysis, chapter 1, section 5. Let $L=\{f\in C[0,2\pi]|f(0)=f(2\pi)\}$ and show that $L$ is dense in $L^2[0,2\pi]$.
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1answer
75 views

An integration question.

An help in the following problem: Let $f:[-1,1] \longrightarrow \mathbb{R}$ a $C^1$ function, i.e., continuously differentiable. Suppose that we have $$\int_{-1}^{1} f(x)\;dx = \pi ...
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1answer
82 views

MacLane's CWM Exercise

Categories for the Working Mathematician, S. Mac Lane, 3.2, ex. 1: Let functors $K, K'\colon D\to \mathbf{Set}$ have representations $\left<r,\psi\right>$ and $\left<r', \psi'\right>$, ...
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0answers
156 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If ...
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3answers
42 views

Let $S$ be a subset of $V$ . Identify which of the following statements is true:

Let $V$ be a vector space of dimension $d < \infty$, over $\mathbb{R}$. Let $U$ be a vector subspace of $V$ . Let $S$ be a subset of $V$. Identify which of the following statements is true: ...