# All Questions

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### Regarding the Poincare Friedrichs inequality

I am working on a two part problem. Part 1 was to prove the Poincare-Friedrichs inequality for n=1: $\int_{0}^{\alpha} |f(t)|^2 dt \le C\int_{0}^{\alpha} |f'(t)|^2 dt$ for some constant $C$. I ...
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I have just started of learning about Manifolds from Milnor's book.I am stuck on following exercise,give me some idea. Let $U$ be an open subset of a manifold $X$. Show that for any $x \in U$ the ...
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### Show that the function is not continuous using the sequence definition.

Consider $f$, given by $f(x,y)$= $xy^2/(x^2+y^4)$ when $(x,y)\ne(0,0)$ and $f(0,0)=0$. Show that $f$ is not continuous at $(0,0)$ using the sequence definition. (which is: a function is continuous if ...
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### Decompose a Real symmetric matrix into only 2 matrices

For a real symmetric matrix $A\in\mathbb{S}^n$, we can decompose it into $A=U\Lambda U^{T}$ by Spectral Theorem. In Matlab, using the command $[L,S,R] = svd(A)$, we obtain $LS R^{T}$ (here, $L$ ...
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### Prove that $\sum_0^n({nCr})^2=2nCn$

Prove that $\sum_{r=0}^n({nCr})^2=2nCn$ I don't know how to prove such probems. Any proof by combinatorics?
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### Exhibiting an isomorphism between two finite fields

So I want to find the isomorphism $\phi$ that takes $F = \mathbb{Z}_3/\langle x^3 - x - 1\rangle$ to $E = \mathbb{Z}_3/\langle x^3 - x + 1\rangle$. I understand that these are both finite fields of ...
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### Does a set $A \subseteq [0,1]$ exist such that $A$ is homeomorphic to $[0,1] \setminus A$?

Does a set $A \subseteq [0,1]$ exist such that $A$ is homeomorphic to $[0,1] \setminus A$? I have no idea how to attack this problem. Any help will be appreciated.
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### Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I ...
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### How to attack this probability question directly?

Below is a example probability question I found in my book. I want to attack this question directly instead of using the complement approach. How do I do that? My book always uses the complement ...
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### A Fundamental Theorem of Calculus

Here is a problem I have been working on recently: Let $f \colon[a,b] \to \mathbb{R}$ be continuous, differentiable on $[a,b]$ except at most for a countable number of points, and $f^{\prime}$ is ...
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241 views

### What are some interesting topics for a differential geometry essay?

As an assignment I have been asked to write an essay on an aspect related to differential geometry. My lecturer has proposed topics such as architecture, navigation and computer graphics. I personally ...
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### Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
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### Let $f$ be a morphism of chain complexes. Show that if $ker(f)$ and $coker(f)$ are acyclic, then $f$ is a quasi-isomorphism.

Let $f$ be a morphism of chain complexes. Show that if $ker(f)$ and $coker(f)$ are acyclic, then $f$ is a quasi-isomorphism. Is the converse true? I am self reader of homology algebra and I stuck in ...
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### Undergrad level presentation on homological algebra and some related topics

I'm a TA of an introductory course about modules, category theory and homological algebra and the students have to do a 2 hour long presentation as a final exam. There's one student who really likes ...
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### Proper map not closed

We call a map proper if the inverse image of quasi-compact sets are quasi-compact sets.(Add the Hausforff axiom if convient) Is there example for such a map not be a closed map? (If the spaces are ...
63 views

### When is the equation $x^2-d^n y^2 = -1$ solvable?

My goal is to prove or disprove that if $x^2-dy^2=-1$ is solvable, then $x^2-d^ny^2 = -1$ is solvable for every odd $n \geq 1$. I do know that the former is solvable if and only if the continued ...
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### Fundamental solution of Poisson equation in the Hyperbolic Plane

If we consider the Poisson's equation $$-\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n,$$ we can construct the fundamental solution $$u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy,$$ where $\Gamma$ is ...
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### $\lim_{x \to \infty} (x+2)-\sqrt{x^2+6x-1}$ [duplicate]

$$\lim_{x \to \infty} (x+2)-\sqrt{x^2+6x-1}$$ I've tried multiplying by conjugate and dividing by $x$ but still get $0$ in the denominator.
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### Principal Ideal Domains with Many Primes of Index 2

By the index of a prime $p$ in a principal ideal domain $R$, I mean the number of cosets of the ideal $(p)$ in $R$ (i.e. $\vert R/(p)\vert$). If I am given a positive integer $m$, I am wondering ...
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### Find a universe for variables x, y, and z for which the statement is true and another universe in which it is false.

Find a universe for variables $x, y$, and $z$ for which the statement $∀x∀y((x ≠ y) → ∀z((z = x) ∨ (z = y)))$ is true and another universe in which it is false. Is there a more ...
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### differentiating $f(x)=\sqrt{x+1}/x$

So I am not sure what I need to do to differentiate this problem. Do I use a combination of chain rule and product rule, and if so what would it look like? Thanks for the help
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### Epsilon-delta proof with $x$ and $y$ defined

I am stuck with the following problem. The question is as follows: prove that for all $x$ in $[0,2]$, there exists $y$ in $[0,2]$ such that the function $f(x,y)=0$. The function $f$ is defined as ...
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### A question about the proof of general lifting theorem in Munkres topology p.479 [duplicate]

after the lifting $\tilde f$ of $f$ is defined, the book goes on to prove its continuity. And in that part, the book says that replacing $U$ by a smaller neighborhood, $V_0$ can be contained in $N$. ...
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### Face post of a subcomplex complement

Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\simeq$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that ...
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### Prove that $\gcd (n^3-1,n+1)=1$ for all even $n$.

Prove that if $n$ is even, then $$\gcd(n^3-1,n+1)=1.$$ I really don't have a clue with this one. Any help would be appreciated.
294 views

### What is wrong with this proof that 3 is less than 1?

What is wrong with this proof? Theorem. 3 is less than 1. Proof. Every number is either less than 1 or greater than 1 or equals 1. Let $c$ be an arbitrary number. Therefore, it is less than 1 or ...
335 views

### Sketch the graph and Determine the domain and range of $h(x)=3+e^{-2x}$.

How do I even start on this? How do I sketch the graph and find the domain and range? I am really lost on how to do this problem! Please walk me through this question!
541 views

### Proof by induction (power rule of the derivative)

Using the differentiation formulas $\displaystyle\frac{d}{dx}x=1$ and $\displaystyle\frac{d}{dx}(fg)=f\frac{dg}{dx}+g \frac{df}{dx}$, prove that $$\frac{d}{dx} x^n=nx^{n-1}$$ for all natural number ...
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### A Limit of a function involving big O notation.

Fix $t$, I was wondering am I allowed to compute the following limit? $$\lim_{n\rightarrow \infty}n \left(\frac{-t^2}{2n} + O(\frac{1}{n}) \right)$$ I'm quite confused about such O-notation.. is ...
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### Vector Space Axioms

I am having trouble understanding vector space axioms seen here (http://imgur.com/qKhgAXu). As an example say we define our potential vector space to be the set of all pairs of real numbers of the ...
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### Summation of Binomial Theorem

The binomial theorem formula: $$\sum\limits_{k=0}^{n} {n \choose k} = \sum\limits_{k=0}^{n}\frac{n!}{k!(n-k)!} = \sum\limits_{k=0}^{n}\frac{n(n-1)(n-2) \cdots (n-k+1)}{k(k-1) \cdots 2\cdot1}.$$ I am ...
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### If $AB+BA=0$, then $A^2B^3=B^2A^3$?

If I have a matrix $A$ and $B$ such that $AB+BA=0$ is it true that $A^2B^3=B^2A^3$? I think that it is false.
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### Unit vectors in $\Bbb R^n$

Suppose that $x$ and $y$ are unit vectors in $\Bbb R^n$. Show that if $\left\Vert{{x+y}\over2}\right\Vert=1$ , then $x =y$. Please enlighten me for this problem! Thanks in advance!
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### Solving homogeneous quaternary quadratic Diophantine equation

Given the equation $w^2+x^2+y^2+z^2=wx+wy+wz+xy+xz+yz$, how does one systematically enumerate all non-negative integer solutions $\{w,x,y,z\}$?
### Distribution of family's disposable income, given pdf, find $F(y)$
Problem: In a certain country, the distribution of a family's disposable income, $Y$, is described by the pdf $$f(y) = ye^{-y}$$ for $y \geq 0$. Find the $F(y)$. Attempt: In the book there it ...
Say I have to place N points in $\mathbf{R}^2$ inside a circle of radius $R$. I want to position them so as to maximize the sum of nearest distances i.e. solve the following problem \begin{align} ...