For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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1answer
19 views

Integrality conditions and proof by double counting.

Theorem $\mathbf{3.4.}$ In a block design of type $2-(v,k,\lambda)$ every element lies in precisely $r$ blocks, where $$r(k-1)=\lambda(v-1)\textit{ and }bk=vr\;.$$ The letter $r$ stands for ...
2
votes
0answers
25 views

convergence in $L^p$ implies convergence in measure

I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in $L^p$ in measure, where $1\le p \le \infty$. Here is my attempt for $p>1$ - Let $\varepsilon>0$ and define ...
3
votes
3answers
55 views

Baby Rudin Exercise 4.2

Can someone check my proof? If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $$f(\overline{E}) \subset \overline{f(E)} $$ for every set $E\subset X$. ...
0
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1answer
12 views

Linear Second order Differential operator proof questions

I have 3 proof questions from my book that I have tried and I would like to see if my solutions are valid and/or there is a simpler way to prove them. Firstly, the notation $ker(L)$ means all $f$ ...
3
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1answer
58 views

When is $x^2 - 75 y^2 = 0$ in $\mathbb{Z}_p$ solvable?

Exercise: For which prime numbers does the equation $x^2 - 75 y^2 = 0$ have non-trivial solution in the $p$-adic integers $\mathbb{Z}_p$? For $p\neq 5$, the non-trivial solvability of the ...
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vote
2answers
37 views

Relevance of prime being divisble by $4k+1$ in proof that 'There are infinitely many primes of the shape $4k+3$'

Show that there are infinitely many primes of the shape $4k+3$ Proof: $1)$ Suppose that there are only finitely many such primes, say $p_1,...p_n$. $2)$ Consider the integer $Q=4p_1...p_n-1$ $3)$ ...
1
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1answer
40 views

Find the group of conformal automorphisms of $U=\lbrace z\in \mathbb{C}: \vert z-1\vert>1\rbrace$

Well $\phi$ is an automorphism of $U$ $\iff$ $1/ \phi$ is an automorphism of $U^C=\lbrace z\in \mathbb{C}:\vert z-1\vert<1\rbrace$ $\iff$ $1/\phi -1$ is an automorphism of the unit disc $\iff$ ...
0
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0answers
15 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
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0answers
12 views

$\mathcal{Z}$-transform of differential equations $y(n+2)-3y(n+1)-10y(n)=(-2)^n$

Is defined function: $$y(n+2)-3y(n+1)-10y(n)=(-2)^n$$ with conditions: $$y(0)=0, y(1)=0 $$ And my solution is (Z-transform): $$\mathcal{Z}\{y(n+2)\}=z^2Y(z)-0z^2-2z=z^2Y(z)-2z$$ ...
1
vote
1answer
48 views

There is no equivariant map $f:S^2 \to S^1$

To fix some notation, let $n \geq 2$ and let $p:S^n \to P^n$ be the canonical double cover. Let $\gamma:I \to S^n$ be a lift of a representative of a nontrivial element in $\pi_1(P_n) \cong ...
1
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0answers
27 views

Series Proof Question [duplicate]

By considering the partial sums for S, that is Sn =1+2+3+···n show that the infinite series S does not converge. However in this video http://www.numberphile.com/videos/analytical_continuation1.html ...
0
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1answer
32 views

Show that if $a, b$ and $m$ are integers such that $m \geq 2$ and $a \equiv b \pmod{m}$, then $\gcd(a, m) = \gcd(b, m)$

Problem 1 (#3.5.32). Show that if $a, b$, and $m$ are integers such that $m \geq 2$ and $a \equiv b \pmod {m}$, then $\gcd(a, m) = gcd(b, m)$. Proof. Let $d = \gcd(a, m)$ Then $d \mid a$ and $d ...
1
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0answers
80 views

Proof that a limit is true

In my assignment I have to prove the following limit, by definition: $$\lim _{x \to 2}\sqrt{3x-2}=2 $$ I have made some calculations trying to prove it, and I've made some way but I'm afraid I'm ...
2
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0answers
48 views

Dimensions of quotient rings of $K[x,y]$

I have tried to solve the following problem and would be very grateful if someone could check my answer. Let $K$ be an algebraically closed field with $\mathrm{char}(K)=0$. I wish to compute ...
0
votes
1answer
57 views

What is wrong with this proof of a number theory competition problem?

Let $a$ and $b$ be positive integers. Suppose $a^n+n| b^n+n$ for any positive integer $n$, prove that $a=b$. My trial: Clearly $b\geq a$, write $b=a+d$, we must show that $d=0$. Now by assumption and ...
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0answers
9 views

Moments bounds VS Chernoff bounds

I have to prove that, when bounding tail probabilities of a nonnegative random variable, the moments method is always better than the classical Chernoff method. In mathematical language, I have to ...
2
votes
1answer
28 views

Does this reasoning work?

Consider the following system of ODEs. $$ \theta'=r\\ r'=1-r^2 $$ On the unit circle, $\theta'=1$, and $r'=0$ Now consider the system $$ \theta'=1\\ r'=0 $$ The solution curves to this system are ...
1
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2answers
24 views

Expected Value and Variance of transformed Random variable

I am trying to find the expected value and variance of $Y_i=\ln(X_i)$ for $X$ is uniformly distributed between $1$ and $3$. I believe that $E(Y_i)=(\ln3)/2$ and $\operatorname{Var}(x)=(\ln3)^2/12$. ...
0
votes
1answer
36 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
1
vote
1answer
27 views

trouble in getting triangle inequality

Let $l_{2}$ be the set of all infinite sequences , $ (x_{n})$ such that $\sum_{n=1}^ {\infty} x_{n}$ converges. Define $$d(x,y)= \sqrt{\sum_{n=1}^{\infty} (x_{n}-y_{n})^{2}}$$ for each $x=(x_{n})$ ...
0
votes
1answer
48 views

Show that there exists a unique function with a certain property

I'm trying to prove the following theorem: "Let $~f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}~$ be a function, and let $~c~$ be a natural number. Show that there exists a unique function $~a: ...
3
votes
1answer
48 views

My answer to this combi problem doesn't match the answer in the book (Problem-Solving Strategies)

[Problems 31 and 32 from Arthur Engel's Problem-Solving Strategies.] Let $n$ children be seated in a line. How many ways can they change their places if they may only move by one place at most? ...
1
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1answer
32 views

Existence of a Map

I just wanted to check to be sure I was correct as the more I stare at my proof the more I doubt myself. Suppose you have a commutative diagram of $R$-modules with exact rows: $$ ...
2
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0answers
43 views

Unifrom Convergence of series of product of two sequences

Suppose {$f_n$}, {$g_n$} defined on $E$ and, (a) $\Sigma f_n$ has uniformly bounded partial sums; (b) $g_n \to 0$ uniformly on $E$ (c) $g_1(x)\geq g_2(x)\geq g_3(x)\geq ...$ for every $x \in E$. ...
0
votes
1answer
61 views

Are these two definitions of prime numbers equal?

In Coq for instance, prime numbers are defined ${n\ is\ prime} \doteq \forall a\in \mathbb{N}: a|n \rightarrow (a=1 \vee a=n)$ ...
0
votes
0answers
31 views

multivaraible chain rule proof

I wanted to prove the multivaraible chain rule; I had to prove that $df \large \frac {({x(t)},{y(t)})}{dt} = \frac{∂ f}{∂ x}\cdot\frac{dx}{dt} + \frac{∂ f}{∂ y}\cdot\frac{dy}{dt}$ So, I took the ...
0
votes
2answers
52 views

Galois Group of an Extension

Question: Determine the isomorphism type of $ \mathrm{Gal}\,(\mathbb{Q}(\sqrt[8]{2},i)/\mathbb{Q}(i)) $. $\\$ This amounts to finding isomorphisms that send $\mathbb{Q}(\sqrt[8]{2},i)$ to ...
1
vote
1answer
37 views

Help for the proof of Lemma for pull-backs

I am learning category theory from the book by Steve Awodey, trying to complete all the proofs, and I got stuck at one. Lemma: Given the diagram above, if the square at the right and the ...
3
votes
1answer
27 views

Are there no polynomials in $\mathbb{C[x]}:f^2 − Xf = −X^2 + 1$?

Are there no polynomials in $\mathbb{C[x]}:f^2 − Xf = −X^2 + 1$? What I did: $$ f^2 − Xf = −X^2 + 1 \iff f^2=Xf-X^2+1 $$ $\deg(f)=n \rightarrow \deg(f^2)=2n$, $\deg(Xf)=n+1$ and $\deg(-X^2+1)$=2 So ...
2
votes
1answer
39 views

$x\in X$ LCS, $f\in X^\ast$ s.t. $f(x)=1$, $f|_Y=0$

Let $X$ be a locally convex space (topology induced by a family of seminorms $P$ which separates points) and $Y\subset X$ a closed subspace. Assume $x\in X\setminus Y$. Show that there exists a $f\in ...
2
votes
2answers
25 views

Volume of a cube with integration.

Say we wanted to derive the formula for the volume of a cube with integration. Each "slice" of the cube has area $x^2$, with "width" $dx$. Integrate from $0$ to $x$, and I believe you would get the ...
3
votes
3answers
88 views

Matrix solving equation.

The number of real solutions of equation $$\begin{vmatrix}x^2-12&-18&-5\\10&x^2+2&1\\-2&12&x^2\end{vmatrix}=0$$ is? Well I wanted to do something like this: ...
0
votes
1answer
33 views

Proving function is measurable

Define $f : [0, 1] → \Bbb R$ by $f(x) = 0$ if $x$ is rational,$1/(d^{1/2})$ if $x$ is irrational and $x = 0.0 . . . 0d . . . $, where $d$ is the first nonzero digit in the decimal expansion of $x$. ...
4
votes
1answer
64 views

Location of the foci of a hyperbola as the value of $a$ becomes increasingly smaller than the value of $b$

"What happens to the location of the foci of a hyperbola as the value of $a$ becomes increasingly smaller than the value of $b$?" I assumed that the hyperbola was in the form ...
1
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0answers
8 views

Is this a sufficient proof that if $M$ and $N$ are midpoints of $AD$ and $BC$ respectively, then $\vec{MN} = \frac{1}{2}(\vec{AB} + \vec{DC})$?

Proof: It can be proved that $\vec{MN} = \frac{1}{2}(\vec{MC} + \vec{MB})$, so it would suffice to show that $\vec{AB} + \vec{DC} = \vec{MC} + \vec{MB}$. $\vec{AB} + \vec{DC} = \vec{MC} + \vec{MB} ...
2
votes
1answer
38 views

Marginal Densities

I just have a few questions about joint density and marginal density questions. Q1: Joint Distribution $f_1=2x+4y$ on triangle with vertices $(0,0), (0,1),(1,0)$. Sketch the region and compute ...
5
votes
3answers
74 views

Proving a little tough trigonometric identity

Show that $$\frac{1+\sin A}{\cos A}+\frac{\cos B}{1-\sin B}=\frac{2\sin A-2\sin B}{\sin(A-B)+\cos A-\cos B}$$ How do I get the $A-B$ term in the denominator? Is RHS to LHS easier? Thanks.
4
votes
0answers
33 views

subset of a compact set in $\mathbb{R}$ with nonempty interior has positive outer measure

Let $A\subset I=[a,b] \subset \mathbb{R}$, $a < b$ such that Int$(A) \neq \emptyset$. Show that $A$ has positive outer measure. What I have so far: Since Int$(A) \subseteq A$, by the ...
4
votes
8answers
150 views

Proving that $5^n-1$ is divisible by $4$ for $n\geq 0$ by induction

I hope this is not counted as a duplicate, as I would like to know if my proof is valid: $P(n): 5^n - 1$ is divisible by $4$ for $n \ge 0$. Base Step: $P(0): 5^0-1 = 1-1 = 0 = 0\times 4$. Induction ...
0
votes
1answer
17 views

If f is a continuous map from X to Y with X limit point compact, does it follow that f(X) is limit point compact?

I'm looking for proof verification/help for the title question. Here is what I have now: Let $f:X \rightarrow Y$ be continous with $X$ limit point compact. Let $V$ be an infinite subset of $f(X)$. ...
0
votes
2answers
19 views

Showing $u_1, u_2, u_3$ is basis

Let $\{v_1, v_2, v_3\}$ be a basis for a vector space $V$. I want to show that $\{u1, u2, u3\}$ is also a basis where $u1 = v1, u2 = v1 + v2$ and $u3 = v1 + v2 + v3$ I wanted to use the standard ...
3
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1answer
43 views

Problem understanding this specific proof that $\sqrt{2}$ is irrational.

The proof (taken from http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm#proof): "To prove that there is no rational number whose square is 2, suppose there were. Then we could ...
0
votes
1answer
21 views

Would it be correct to say that NOT(P OR Q) is (NOT P AND NOT Q)?

I seem to think it is true as $$ x \notin (A \cup B)$$ $$\implies x \notin A \text{ and }x\notin B$$ $$\implies x \in A^C \text{ and }x\in B^C$$ I have deduced this via a venn diagram, and ...
6
votes
5answers
865 views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
12
votes
3answers
64 views

Prove that $4x^2-8xy+5y^2\geq0$ - is this a valid proof?

I need to prove that $4x^2-8xy+5y^2\geq0$ holds for every real numbers $x, y$. First I start with another inequality, i.e. $4x^2-8xy+4y^2\geq0$, which clearly holds as it can be factorized into ...
1
vote
1answer
36 views

Induction Clarification

I had this problem: Is it always necessary to go from n to (n + 1) or from (n - 1) to n in the inductive hypothesis? Is the "direction" always important? Here is my solution to one such proof, which ...
2
votes
1answer
39 views

Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
1
vote
1answer
13 views

U uniform on [-1,1] - Find density of U^2

Let $U$ be uniformly distributed on $[-1,1]$. Find the denstiy of $U^2$. I would start with $F_{U^2}(u)$=$P(U^2\le u)$=$P(-\sqrt{u}\le U\le\sqrt{u})$ for $u\ge 0$. Since it is uniformly distributed ...
1
vote
0answers
18 views

Proving that a union of a countable and an uncountable set is equivalent to the uncountable set (proof check)

Let $A$, $B$ be sets with $|B|=\aleph _0$ and $|A|>\aleph _0$, Prove that $|A\cup B| = |A|$ I've already seen somewhere here (though can't seem to find it now) a proof using the fact that ...
1
vote
1answer
34 views

Find the curve which together with $\gamma$ encloses the greatest area.

I'm working through Gelfand & Fomin's Calculus of Variations by myself, and could use the guidance of someone familiar with the subject. The problem I'm on now is the following: "Given two points ...