Use this tag for questions asking about "problem books", "exercise books", and their solutions.
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52 views
Analyticity of a function in $x$ and $y$, without employing the Cauchy-Riemann eqns
Exercise from Saff & Snider's Complex Analysis:
How to determine the analyticity of this function, without using the Cauchy-Riemann equations? I tried to work from first principles (taking the ...
1
vote
0answers
23 views
Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?
Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow,
...
2
votes
1answer
48 views
Ergodic theory question about the support of a measure.
I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:
...
2
votes
1answer
47 views
Problem books in ODE
I'm studying Ordinary differential equations right now in the level of Hartman's book.
I've never seen problem books in ODE in this level even if you consider it without solutions.
I would like to ...
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votes
1answer
24 views
Length of life of a fire detector
The length of life of a flame detector is exponentially distributed with paramater $\lambda=0.1/year$. Die number of events which activate the flame detector in an interval with length $t$ (heat, ...
1
vote
1answer
52 views
exercise: limit orthonormal sequence, “Banach Space Theory”
I have an exercise from "Banach Space Theory":
Suppose $\{x^k\}_{k=1}^\infty$ is an orthonormal sequence in $l_2$, where $x^k:=(x_i^k)$. Show that $\lim_{k\rightarrow \infty} x_i^k =0 \ \forall_{i\in ...
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0answers
66 views
Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.
Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$
is the Fourier transform of a function in $L^1(R^d,m)$.
[Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
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4answers
64 views
The matrix has rank $n$ if and only if $A$ is nonsingular and $B = A^{-1}$.
Let $A$ and $B$ be $n \times n$ matrices with real entries. Show that the
matrix $$M = \left( \begin{matrix} A&I\\ I&B \end{matrix} \right)$$
has rank $n$ if and only if $A$ is nonsingular
...
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votes
0answers
61 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 ...
2
votes
1answer
47 views
Extension of a Bounded Operator on $L^p$ to $L^r$
Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
4
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1answer
36 views
$L^{p}$ functions from Rudin Exercises 3.5
I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
3
votes
1answer
55 views
Simple module is isomorphic to R/M where M is a maximal ideal
In Michael Artin's Algebra textbook page 484 Chapter 12 Exercise 1.6:
A module is called simple if it is not the zero module and if it has no proper submodule.
(a) Prove that any simple module is ...
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0answers
35 views
Unbounded self- adjoint and von Neumann algebra
I am reading Conway's Functional Analysis. Here is one exercise problem.I don't know how to show the following fact. For unbounded self-adjoint $T$ in Hilbert space $H$
1) $T$ commutes with its Borel ...
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0answers
25 views
Solvable Lie algebra with codimension 1 ideal
There is an exercise in Humphreys "Any nilpotent Lie algebra contains a codimension 1 ideal".
The proof I am thinking of is the following.
Suppose the Lie algebra $L$ is non-satisfies $L\neq[L,L]$. ...
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votes
2answers
33 views
Linear Algebra : find the kernel of this transformation.
Q. I think I find the kernel but several... which is correct? Seems like depending on which variable I put as kernel, I can get several kernels. Correct?
T is the transformation from $\mathbb{R}^2$ ...
2
votes
1answer
88 views
question 9 - chap 5 evans PDE
The question is :
Integrate by parts to prove :
$$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$
for $ 2 \leq p < \infty$ ...
1
vote
1answer
48 views
How do I determine a formula for a given trig function?
Assume that 0 < x < pi/2 and sin(x) = z
a.) Find a formula that gives the value of sin(x/2) in terms of z
b.) Corroborate the validity of the formula for these values of x:
pi/4
pi/3
pi/6
...
2
votes
1answer
58 views
Getting an acute angle for an obtuse angle using law of Sines.
I have done this problem over and over again. I even looked up tutorials on how to properly use law of sines. It's rather embarrassing that I'm struggling so much wish this simple trigonometric stuff.
...
6
votes
4answers
125 views
Show that if $T_1$, $T_2$ are normal operators that commutes then $T_1+T_2$ and $T_1T_2$ are normal.
Let $V$ be a finite dimensional inner-product space, and suppose that $T_1$, $T_2$ are normal operators on $V$ that commutes. How to show that $T_1+T_2$ and $T_1T_2$ are then normal?
It is clear if ...
2
votes
1answer
34 views
Abstract integral - Borel measures - $L^p$ spaces
Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
1
vote
1answer
40 views
Weak convergence-exercice
Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$
Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
4
votes
1answer
79 views
Matrix Norm set #2
As a complement of the question
Matrix Norm set
and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions:
(3) ...
5
votes
1answer
42 views
Substitution problem
My question is something I've been thinking about for some time now.
Q: Why is it possible to make substitutions or change in variables ?
I mean, how do I know which substitutions are allowed ?
For ...
1
vote
1answer
52 views
Prove there is a Borel measure u such that $u[x,y) = a(y) - a(x)$
If anyone has a solution to the following exercise, I would be grateful. Thanks.
Let $\alpha$ be continuous and increasing on $[a,b]$. Prove that there exists a unique Borel measure $\mu$ on ...
1
vote
1answer
32 views
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3answers
88 views
Homework - Prove that a given set is a group
I have a homework question and I don't know how to approach this exercise.
The exercise is the following:
Let's suppose $G$ is a set with binary function * defined for its members, which is:
...
2
votes
0answers
91 views
Question on groups of order $pq$
Let $G$ be a group of order $pq$, $p>q$ and $p$, $q$ are primes. Then prove that
If $q\mid p-1$ then there exists a non abelian group of order $pq$.
Any two non-abelian groups of ...
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votes
2answers
32 views
How to determine the conditional expectation $\mathbb{E}[x^2\mid y]$
If $[x,y]^{T}$ is a two dimensional Gaussian random variable with zero mean and
\begin{equation}
\mathbb{E}[x,y]^{T}[x,y]=\begin{bmatrix}
\sigma_x^2 & r_{xy}\\
r_{yx}& \sigma_y^2\\
...
2
votes
0answers
35 views
Composition of a subharmonic function and a conformal mapping
this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
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votes
0answers
51 views
K3-surface is not the blow-up of any other smooth complex surface?
Good evening,
I'm stuck in the following exercise in Huybrechts, Complex Geometry, chapter 2, page 103.
Let $X$ be a K3 surface, i.e. X is a compact complex surface with $K_X \cong \mathcal{O}_X$ ...
2
votes
2answers
69 views
Show $\int_X f d\nu = \int_X fgd\mu$ if $\nu(E)=\int_E g d\mu$ .
$f$ and $g$ are both non-negative functions where the integral of non-negative function is defined as the supremum over all simple functions dominated by the non-negative function.
Would going ...
3
votes
1answer
68 views
Prove the inequality?
Let $f$ be an analytic function in the unit disc without zeros satisfying $|f|\leqq 1$. Prove that
$$
\sup_{|z\leqq{1/5}|}|f(z)|^2\leqq \inf_{|z|\leqq{1/7}}|f(z)|
$$
Help me please. These questions ...
3
votes
0answers
39 views
How to show $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
How can I construct a random variable $X$ such that: $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
3
votes
2answers
52 views
Let $\{X_n\}$ be i.i.d integrable r.v.s, show that $\frac{1}{n}\max_{1\leq j\leq n}|X_j|\to 0 \quad \mbox{a.e.}$
This problem is an exercise in Probability theory,independence,interchangeable, martingale(Chow), exercise 4.1.10.
Let $\{X_n,n\geq 1\}$ be independent identical distributed integrable random ...
4
votes
1answer
67 views
algebraic geometry exercise: infinite subset is dense
A hypersurface $C \subset \mathbb{A}^2$ we call a plane curve.
Show that any infinite subset of an irreducible plane curve $C \subset \mathbb{A}^2$ is dense in $C.$
Note. We call hypersurface the ...
2
votes
1answer
75 views
Math Analysis - Problem dealing with bounded variation
Let $f\colon[0,1] \rightarrow \mathbb R$ (all real numbers) be defined by
$\displaystyle f(x) = x \sin \left(\frac{\pi}{2x}\right)$ if $0 \lt x \le 1$ and $f(x) = 0$ if $x=0$.
Determine ...
5
votes
1answer
97 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, ...
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vote
2answers
154 views
Question 2.1 of Bartle's Elements of Integration
The problem 2.1 of Bartle's Elements of Integration says:
Give an exemple of a function $f$ on $X$ to $\mathbb{R}$ which is not
$\boldsymbol{X}$-mensurable, but is such that the function $|f|$ ...
10
votes
1answer
121 views
Question on a homomorphism of a set G.
I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19
Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
2
votes
1answer
152 views
Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.
I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
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votes
1answer
50 views
Finding an algebra of smooth functions on a manifold with a given product.
I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
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votes
0answers
64 views
Stokes' Theorem Example
I am reading Wade's Introduction to Analysis. One of the exercises is to show that
$$
\int_{\partial M}\sum_{k=1}^n \, dx_1dx_2\cdots \hat{dx_i}\cdots dx_n
$$
is equal to the volume of $M$ if $n$ is ...
3
votes
3answers
198 views
Analytic function f constant if $f(z) = 0$ or $f'(z) = 0$ for all $z$.
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be analytic and suppose that for all $z \in \mathbb{C}$, at least one of $f(z)$ and $f'(z)$ is equal to 0. Proof that $f$ is constant.
Any ideas? Thanks.
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votes
2answers
60 views
Question from Folland on modes of convergence
I have been reading through Folland, and I am having a hard time answering the following question. Any help will be much appreciated.
Suppose $\lvert f_n \rvert \leq g \in L^1$ and $f_n \rightarrow ...
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votes
2answers
118 views
$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence
Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence.
$C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$.
And, the ...
1
vote
2answers
68 views
Stuck at a differential equation, particular solution
The problem is:
$y'' + 4y' + 3y = (4x-2).e^{-3x}$
with conditions $y(0)=2$ & $y'(0)=0$
I first find the characteristic polynomial $p(r) = (r+3)(r+1)$ which gives me the homogeneous solution $yh ...
2
votes
1answer
85 views
Real Analysis Qual Problem 2
This shouldn't be a hard problem, but I am stuck on it. I just need to prove the statement or come up with a counterexample. Any help will be appreciated.
Let $f: [0, 1] \rightarrow [0, \infty)$ be ...
2
votes
1answer
163 views
Real Analysis Qualifying Exam Problem
I think this should be an easy question, and I believe the answer should be in the positive, but I am not sure how to start. I would appreciate some help. Thank you.
Suppose that $f_j$ is a ...
2
votes
1answer
83 views
A necessary and sufficient condition for ergodicity
Let $(X,\mathcal B,\mu)$ be a probability space and $T\colon X\rightarrow X$ be measure preserving. I need to prove that $T$ is ergodic if and only if the following property holds:
If $f\colon ...
5
votes
6answers
340 views
List of problem books in undergraduate and graduate mathematics
I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
