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

learn more… | top users | synonyms

2
votes
2answers
41 views

Estimate the number of ants in a colony

A friend of mine gave me this weird problem I cannot solve. To estimate the number of ants in a colony an entomologist draws 5500 ants randomly from the colony, labels them with a radioactive isotope ...
0
votes
2answers
41 views

How to apply the transitive law when there is a $\le$

The transitive law states that: For real numbers $a$, $b$ and $c$: $a<b \text{ and } b<c \Rightarrow a<c$ I am not sure how to apply it in the following cases ($x \in \mathbb{R}$): $a ...
1
vote
1answer
34 views

How to prove that $F(x,y)=(f(x)h(y),g(y))$ is a diffeomorphism?

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be given by $F(x,y)=(f(x)h(y),g(y))$, where $h:\mathbb{R}\to\mathbb{R}$ is a diferentiable function and $f,g:\mathbb{R}\to\mathbb{R}$ are diffeomorphisms. ...
2
votes
4answers
460 views

What is the truth table for demorgan's law?

From Demorgan's law: $(A \cup B)^c = A^c \cap B^c$ I constructed the truth table as follows: $$\begin{array}{cccccc|cc} x\in A & x \in B & x \notin A & x \notin B & x \in A^c ...
4
votes
1answer
45 views

How to show that $\varphi(x,y)=(x+f(y),f(x)+y)$ is bijective?

Let $f:\mathbb{R}\to\mathbb{R}$ be a $C^1$ function such that $|f'(t)|\leq k<1$ for all $t\in \mathbb{R}$. Let $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ be the function given by ...
2
votes
1answer
67 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
2
votes
1answer
24 views

Just solution check- Taylor serie around $x_0=2$

I have this solution (I hope it's ok to post picture instead writing in Latex), and I just need someone to check if this is ok precedure for finding serie around point $x_0=2$? -
0
votes
2answers
47 views

If $a$ is a limit point of $f^{-1}(b)$, then the linear mapping $f'(a)$ is not injective.

Let $U\subset\mathbb{R}^m$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function. Suppose that there exists $b\in\mathbb{R}^n$ and $a\in U$ such that $a$ is an accumulation point of ...
9
votes
3answers
644 views

Is there a shorter/faster way to show these two expressions are equal?

I want to know if the two expressions are equivalent: $\frac{1}{2}(k+2)(2a+(k+1)b)$ $\frac{1}{2}(k+1)(2a+kb)+(a+(k+1)b)$ My attempt: First, I decided to start with 2 as 1 looks complicated to me ...
0
votes
3answers
54 views

In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself?

My question is regarding the question posed at the end of the proof. My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false. ...
1
vote
1answer
43 views

Check workings for Strong Induction (Proof by Contradiction)

I want to prove the following: Suppose that $P(n)$ is a statement involving a general positive integer $n$. Then $P(n)$ is true for all positive integers $n$ if: i) $P(1)$ is true, and ...
0
votes
0answers
34 views

Solutions to exercises in Nelson's “An Introduction to Copulas”

I am paving my way through Nelson's "An Introduction to Copulas". The book has exercises (quite good actually), but no solutions. Does anybody have a solution manual for (some of those) exercises? ...
2
votes
3answers
27 views

Help complete proof ('or' statement in conclusion): $a^2 \ge 7a \Rightarrow a\le0 \text { or } a\ge7$

Question Prove that if $a$ is a real number such that $a^2 \ge 7a$ then $a\le0 \text { or } a\ge7$ My Attempt We are given: $a \in \mathbb{R}$ $a^2 \ge 7a$ And need to prove: $a\le 0 \text{ ...
3
votes
1answer
45 views

Question regarding quadratic form exercise in Hoffman Kunze

In the book the quadratic form associated with a bilinear form f is defined as $q(\alpha)=f(\alpha,\alpha)$. Then, if $U$ is a linear operator on $\mathbb R^2$ an operator $U^\dagger$ on the space of ...
0
votes
1answer
35 views

Orthogonalization - need correction

given are $$v_1 = (i, \sqrt{2}i, -i) ,v_2 = (i, 0, -1) ,v_3 = (0, -i, \sqrt{2}i)$$ I need to calculate a orthogonalized basis and then norm then. I am not allowed to immediately use the ...
1
vote
3answers
52 views

Whats wrong with my reasoning here

Find the number of ways of giving $3n$ different toys to Maddy,Jimmy and Tommy so that Maddy and Jimmy together get $2n$ toys. My attemp and flawed reasoning: The number of ways of choosing $2n$ ...
4
votes
1answer
67 views

Help clarify truth of the statement: $n^2-n-2=0 \Leftarrow (n=2 \text{ and } n=-1)$

According to my textbook, the statement $n^2-n-2=0 \Leftarrow (n=2 \text{ and } n=-1)$ is true (full solution was not provided). I am not sure why the statement must be true. My reasoning is as ...
0
votes
0answers
44 views

p--laplacian problem

For $f \in W^{-1,p'}(\Omega)$, $f \geq 0$, we definr $v,$ $v_{\epsilon}$, $g_{\epsilon}$ by $$-\mathrm{div}(|Dv|^{p-2} Dv)=f \quad \mbox{in} \mathcal{D}'(\Omega), \quad v \in W^{1,p}_0(\Omega)$$ with ...
0
votes
1answer
35 views

Proves or counterexamples in retraction and coretractions of modules

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z} $ is not a ...
0
votes
0answers
20 views

Let $ W_1 $ and $ W_2 $ be subspaces of $ V^* $. Prove that $ W_1 = W_2 $ iff $ Ann(W_1) = Ann(W_2) $

Let $ S $ be any subset of $ V^* $ for some finite dimensional space $ V $. Define $ Ann(S) = \lbrace v \in V ~|~ f(v) = 0 $ for all $ f \in S \rbrace $. Let $ W_1 $ and $ W_2 $ be subspaces of $ V^* ...
1
vote
1answer
33 views

Poisson processes and queues

I am trying to understand Poisson processes and queues. I have this exercise: Consider a fuel station with two fuel pumps and one park. Each car that comes to the fuel station when the pumps and the ...
0
votes
0answers
36 views

When can we interchange summation with $L^2$ inner product?

(This question concerns a step in the solution given to Eignvalues of Laplacian operator and Sobolev spaces.) Why can we interchange the sum and the $L^2$ inner product in the following? $$(\sum_n ...
1
vote
1answer
76 views

Help please eigenvalue of Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2\left(\Omega \right)$. Let $\left(\lambda_n\right)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and ...
0
votes
0answers
20 views

eignvalues of laplacian operator and Sobolev spaces -II

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, Let $F=(F_t) \in C^0(I,L^2(\Omega))$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od the ...
12
votes
5answers
272 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
1
vote
2answers
60 views

I still forget concepts even after answering numerous math problems

Note: this is particularly aimed at high-school/entry level college problems When I'm learning a new topic: 1) I read the theory given in the textbook at the start of each topic 2) proceed to read ...
0
votes
0answers
49 views

I want the solution of Bjork Arbitrage Theory in Continuous Time

I am working on Financial Mathematics and also work on Bjork Arbitrage Theory in Continuous Time sometimes I have problem in the Exercises. Does any body know about solution of this book ?
0
votes
0answers
21 views

Explanation for a simple comparison

Ok, Yesterday I started to learn how to solve problems with comparisons, but I couldn't understand one thing of the "solve algotithm". Here is a part from a solve from a simple example problem ...
0
votes
1answer
20 views

Convergent sequences equal to limit, but also not equal to limit

Analysis question: Give an example of, or show that there is no such example of a single convergent sequence (An) with limit l for which An = l for infinitely many values of n, and An does not equal ...
-2
votes
1answer
44 views

Infinite set which is bounded above but not bounded below [closed]

Analysis question: Give an example, or show that there is no example, of an infinite set of real numbers which is bounded above but not bounded below. Support answer with a clear and concise ...
2
votes
1answer
38 views

Is convex set in an ordered set necessarily interval or ray? Munkres 16. 7

The is Problem 7 in Section 16 (page 92) of Munkres' Topology. The problem reads as follows. Let $X$ be an ordered set. If $Y$ is a proper subset of $X$ that is convex in $X$, does it follow ...
1
vote
3answers
124 views

Does anyone know which textbook this question is from? $\displaystyle \int_0^x \frac{\sin t}{t+1}dt > 0$ for all $x>0$

So I'm currently doing integration problems preparing for final exam, and I would like to practice problems like the following: Prove that $\displaystyle \int_0^x \frac{\sin t}{t+1}dt > 0 \ \ $ ...
1
vote
1answer
62 views

Are these quotient spaces homeomorphic to a cylinder and to the Möbius Strip?

Consider for $[0,1]\times [0,2]$ the function $f:\{0\}\times [0,1]\to \{1\}\times [0,1]$ given by $f(0,x)=(1,x)$. Prove that the quotient space given by this $f$ is homeomorphic to the cylinder and ...
2
votes
1answer
51 views

Finite but unbounded set?

I have come across an exercise sheet for my analysis class with the following instructions In each of the cases below, either (i) give an example with the properties indicated or (ii) show that there ...
0
votes
0answers
40 views

Representation of a meromorphic function

I am facing the following question: Let $f(z)$ be a meromorphic function on $\mathbb{C}$ with simple pole at $a_1,a_2,\cdots$ such that $0<|a_1|\leq |a_2|\leq |a_3|\leq\cdots$ and ...
4
votes
1answer
73 views

Is it worthy to buy Kaczor's “Problems in Mathematical Analysis” three volumes?

I'm looking for a problem book in early math analysis, proof based, one single variable calculus problems,limit,continuity, derivative,integral,Taylor theorem, power series, convergence, divergence, ...
1
vote
1answer
39 views

$f^2$ and $f^3$ are holomorphic implies $f$ is holomorphic.

Suppose $f$ is a continuous complex valued function on a domain $\Omega$. Suppose $f^2$ and $f^3$ are holomorphic in $\Omega$. Show that $f$ is also holomorphic in $\Omega$. Assume $f=u+iv$. I see ...
0
votes
0answers
32 views

An exercise using Schwarz Lemma

I am facing the following problem: Let $\alpha$, $\beta\in D$ which is the unit disc in $\mathbb{C}$. Let $F$ be the space of all analytic functions on $D$ s.t $\sup_D |f|\leq 1$ and ...
1
vote
1answer
47 views

convolution -questions

I'm lost, can you help me please. How compute the product convolution between two distributions $T$ and $S$? (we suppose that $T * S$ exist)? How we compute the product convolution between an ...
2
votes
1answer
30 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
1
vote
1answer
52 views

equation in Sobolev space

i have this exercice: Let $f\in L^2(\mathbb{R}^n)$. 1- Prouve that the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admeit a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prouve the ...
2
votes
2answers
102 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
0answers
44 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
45 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
votes
0answers
41 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
88 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
votes
0answers
30 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} ...
0
votes
1answer
55 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
32 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 ...
3
votes
0answers
55 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 ...