3
votes
1answer
115 views

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 ...
5
votes
1answer
81 views

Exercise about tangent space

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 ...
0
votes
3answers
122 views

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 ...
0
votes
0answers
51 views

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, $L$ ...
2
votes
1answer
444 views

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?
4
votes
1answer
89 views

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 ...
22
votes
1answer
361 views

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.
0
votes
0answers
92 views

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 ...
1
vote
2answers
47 views

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 ...
5
votes
1answer
198 views

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 ...
1
vote
2answers
39 views

How do I solve this recurrence relation?

Given a recursive relation $$a_n = \begin{cases} (1 - 2b_n)a_{n-1} + b_n, & n > 1 \\ \frac{1}{2}, & n =1 \end{cases} $$, how can I expression $a_n$ in term of $b_i, i \in \{1, 2, \dots ...
1
vote
3answers
69 views

Show $\ell_\infty (M)$ is a Banach Space

I'm working on problems from Carothers' Real Analysis. The following problem is in the section on completions. Given any metric space $(M,d)$, check that $\ell_\infty(M)$ is a Banach space. ...
2
votes
2answers
52 views

Suppose that $L(\alpha):L:K$ and that $[K(\alpha):K]$ and $[L:K]$ are relatively prime.

Show that the minimal polynomial of $\alpha$ over $L$ has its coefficients in $K$. I tried an approach but I got stuck: We have that the following field extensions: $L(\alpha)/L$ and $L/K$ and we ...
0
votes
1answer
25 views

Number of elements in a subgroup of two permutation groups

Let $A=\left\{\beta \in S_{5}\ |\ \beta(1)=1,\beta(3)=3 \right\}.$ How many elements are in $A$? How many elements are in $A$ when $S_{5}$ is replaced by $A_n$ for $n\geq 4$. My thought for the first ...
1
vote
1answer
41 views

Using Cauchy's Theorem on Contour Integral

I need to solve $\int_\gamma (1-e^z)^{-1}$ if $\gamma (t) = 2i + e^{it}$. I would assume Cauchy's Integral theorem applies here, where $\gamma$ is a closed path on a convex open set. I'm having ...
1
vote
2answers
61 views

heat equation with fourier series

Original PDE $$T_t=\alpha T_{xx}$$ I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. Given ...
1
vote
1answer
67 views

Very confused with interpolating polynomials

I have a problem from my homework that I completely botched, and no matter what I do I end up with the wrong answer. Here's the problem: For a given function $f(x)$ let $x_0 = 0, x_1=0.6, x_2 = ...
1
vote
1answer
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 ...
1
vote
2answers
241 views

How to find transformation matrix which converts matrix to simple standard form

I have a matrix A$$ \left( \begin{array}{ccc} 0 & 1 \\ a^2 & 0\\ \end{array} \right) $$ Using eigen values, I convert it into simple standard form B: $$\left( \begin{array}{ccc} a & 0 \\ ...
1
vote
1answer
96 views

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
21
votes
6answers
3k views

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 ...
2
votes
1answer
119 views

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 ...
2
votes
3answers
180 views

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 ...
1
vote
1answer
88 views

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 ...
1
vote
1answer
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 ...
2
votes
0answers
62 views

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 ...
0
votes
2answers
44 views

$\lim_{x \to \infty} (x+2)-\sqrt{x^2+6x-1}$ [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.
4
votes
1answer
68 views

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 ...
1
vote
1answer
121 views

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 ...
1
vote
4answers
41 views

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
1
vote
2answers
50 views

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 ...
2
votes
0answers
57 views

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 $N$. ...
1
vote
0answers
55 views

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 ...
0
votes
3answers
28 views

How to compute expected value

How do I solve the expected value of this problem, if I have already calculated the pmf? Let $X$ be a random variable with cumulative distribution function given below: $$F_X(x) = ...
1
vote
1answer
37 views

Integral solution to a simple equation

consider the following equation: $r^4 + 100 s^4 = q^2$ It has one positive integer solution $(r,s,q) = (6,4,164)$. Is there any general solution (positive integer solution) to it, similar to the ...
16
votes
1answer
364 views

Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
0
votes
1answer
59 views

Suppose that $V ⊂\mathbb R^n$ is a subspace. How to show that $V$ is a normal subgroup of $\mathbb R^n$?

Suppose that $V ⊂\mathbb R^n$ is a subspace. that $V$ is a normal subgroup of $\mathbb R^n$. A bit confused on how to approach this.
0
votes
1answer
35 views

Differentiable L-1 Regularization

In machine learning we are often faced with optimization problems where we want to minimize some energy function using L1 regularization over some of the parameters, e.g.: $$ E(a,w) = [\text{sum of ...
3
votes
4answers
62 views

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.
0
votes
3answers
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 ...
1
vote
2answers
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!
2
votes
1answer
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 ...
1
vote
3answers
132 views

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 ...
0
votes
1answer
83 views

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 ...
1
vote
2answers
235 views

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 ...
1
vote
2answers
66 views

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.
2
votes
4answers
52 views

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!
0
votes
2answers
201 views

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 $\{w,x,y,z\}$?
1
vote
0answers
136 views

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 ...
4
votes
1answer
99 views

Optimal Placement of Points inside a Set

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} ...

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