Tagged Questions

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

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Exercise 3.3.25 of Karatzas and Shreve

This is the Exercise 3.25 of Karatzas and Shreve on page 163 Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted process ...
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A simpler solution to a limit question?

Okay I saw this limit question in Thomas' Calculus 12th Edition: $\lim \limits_{x \to 0} \frac{\tan3x}{\sin8x}$ The answer is $\frac{3}{8}$. I was able to get the correct answer using this ...
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How many natural numbers less than $10^8$ are there, whose sum of digits equals $7$?

How many natural numbers less than $10^8$ are there, whose sum of digits equals $7$? I got it here.But is there any more effecient and easier way to solve than the link shows?
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Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
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Question from Folland real analysis 6.38

I have been staring at this for hours. I cannot figure out how to prove the following from Folland, problem 6.38. Show that: $$f \in L^p \iff \sum_{k=-\infty}^{+\infty}2^{kp}\lambda_f(2^k)<\infty$$ ...
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Decomposition of polynomials

It is a very simple question but I'm stuck in decomposing this: $x^3+2x^2-2$. I can't find the $x-c$ (Ruffini's rule) form that can enable me to decompose it. Is it possible to decompose? If I can ...
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If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen.

If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen. This is exercise 3.6P of Vakil. I can see that a union of connected components is closed. This is ...
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Classify all invertible meromorphic functions

I am studying complex analysis. And I see a topic, namely meremorphic functions But I cannot find any enough information about this function. So I have insufficient knowledge about this topic. ...
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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 ...
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Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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Proving Fernbahnhof theorem (FHHF) using only concepts of abelian categories

Statement: let $F$ be a right exact functor. Describe a map $FH \rightarrow HF$. (from Vakil's notes 1.6H) attempt of a solution: Let $K$ be the kernel of $FA^i \rightarrow FA^{i+1}$. Since F is ...
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real analysis — lebesgue integration

Let $f$ be integrable over $(-\infty, \infty)$. We know that for each $t$, $\int_{-\infty}^\infty f(x) dx = \int_{-\infty}^\infty f(x+t) dx$. Let $g$ be a bounded measurable function on $\mathbb{R}$. ...
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The Stiefel-Whitney classes of Cartesian product

I am reading the book of characteristic classes by Milnor-Stasheff, and I have a problem with the exercise 4-A: Show that the Stiefel-Whitney classes of a Cartesian product are given by ...
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Exercise about the function field of an (irreducible) affine variety

Look at the following definition: I have problems to solve part iii) and iv) of the following exercise taken from Gathmann's notes: In particular for the part iii) given a class ...
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A topological group question about generators

If $G$ is connected topological group and $e \in V$, $V$ is open. Then prove that $V$ is a set of generators for $G$.
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exercise on the closed subspaces of an Hilbert spaces

I have a question regarding exercise 3.1.13 of "Analysis Now" by Pedersen volume 118 of the Springer GTM. The exercise aim to show that any closed subspace $X$ of $L^2([0,1])\cap L^{\infty}([0,1])]$ ...
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List of (pre-graduate level) exercises

I am about to get my undergraduate degree in (pure) mathematics, but I feel like I'm ill prepared to go through a graduate program. This is why I'm looking for texts like this one ...
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Turning an isometric embedding into a homeomorphism

While in studying functional analysis, there is a part of a homework problem from Rudin's Functional Analysis that asks to show that the isometric embedding $\phi: X \rightarrow X^{**}$ is a ...
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Uniform integrability of sequence of function [closed]

Suppose $μ$ is a finite measure, and assume that for some $\epsilon>0$, $$\sup∫\left|f_n\right|^{1+ε}d\mu<\infty.$$ Prove that $\left\{f_n\right\}$ is uniformly integrable. Help me please.... ...
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A square integrable martingale has orthogonal increments

I am really stuck with the following exercise: $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\ge1},P)$ a filtered probability space. Let $(X_n )_{n\ge1}$ be a sequence of square-integrable random ...
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Prove the indication in Integration

Any body can help me to prove this problem in simple function? Let if $g_n \geq 0$, $g_n \rightarrow g$ and $\int g_n d\mu \leq E \leq \infty$ then $\int g_n d\mu \leq E$ with $(S, \Sigma, \mu)$ ...
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differential equation and sketching of solution graph

I am trying to solve an exercise taken from a book. It is no homework, but i could lie to you so you probably dont care about this much. The exercise: Consider the following differential equation ...
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Coefficient power series problem #49

What is the coefficient of $x^n$ in the power series form of $(1-2x)^{1/3}$? This problem is taken from bona chapter 4, third edition.