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

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1answer
59 views

Proof Verification of Cayley Hamilton

I am wondering about how this proof I am doing for the cayley hamilton is and if it is fully valid. I am also interested in any suggestions, better options or things I should note. Or if I am on right ...
7
votes
4answers
294 views

Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
2
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3answers
51 views

Divisibility Rule for 9

I'm working through an elementary number theory course right now and I think I've come up with a proof here but wanted some feedback on my logic. Question: If the sum of the digits in base 10 is ...
0
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1answer
25 views

Checking region of validity for a standard Laplace transform

In these notes, http://www.math.psu.edu/papikian/Kreh.pdf, Theorem 2.14 it states that $$\mathcal{L}[J_0](s)=\frac{1}{\sqrt{1+s^2}} $$ which I suppose is equivalent to $$ ...
2
votes
1answer
29 views

Inner product property in a convex set.

Question: Let $C \subset \mathbb R^n$ be a convex set. If $x \in \mathbb R^{n}$ and $\overline{x} \in C$ are such that $$\|x - \overline{x}\| = d(x,C)$$ Show that $$\langle x - \overline{x}, y - ...
0
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4answers
52 views

How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
-1
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3answers
46 views

Verify this identity: $1-\frac{\sin^2x}{1−\cos x}=\cos x$ [closed]

This doesn't look like a though one but still i'm having a rough time with it, i guess i'm just doing something wrong but i can't find what i'm doing wrong. I got as close as $1-\cos x$ but i can't ...
0
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1answer
25 views

Application of differential equation (wave equation)

I am not 100% sure what is "steady-state solution w(x)". I tried to solve the question by first making equation (1) equals to zero; then integrate it twice with respect to x and substituted u=0,x=0 ...
0
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4answers
57 views

Show that a group with $p^k$ elements where $p$ is prime has a subgroup of order $p$

Proof- Pick an element $a \in G, \, a\not= e$ Now order $(a) = p^t$ for some $1 \leq t \leq k$ $\,$(by Lagrange) If I could show that $t \not= k$ so $G$ is not cyclic, I could use an inductive ...
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0answers
26 views

There exists a descending chain of symmetry groups from a formal language string down to its smallest grammar.

Background. Let $\tau \in G_i$ be a permutation in the symmetry group of the smallest grammar $g_i$. Then $\tau$ permutes each set of positioned (within $g_i$) symbols $\{x_1^{(1)}, x_1^{(2)}, \dots, ...
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0answers
29 views

Proof that set of functions with derivative zero at a given point is meager in space of strictly increasing twice differentiable functions.

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^2[0,1], f \textrm{ strictly increasing} \}$. I equip $X$ with the topology of uniform convergence. Define the set $A$ as: $$A =\{ f \in X \; | \; ...
9
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1answer
324 views

Separation in compact spaces

There was recently a question that I cannot find about separation in compact spaces. The answer to that question was no for trivial reasons. Motivated by that, let me ask a less trivial version of ...
5
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3answers
46 views

Proving $\sum\limits_{k=1}^{2n} {(-1)^k \cdot k^2}=(2n+1)\cdot n$ for all $n\geq 1$ by induction

How prove the following equality: $a_n$:=$\sum\limits_{k=1}^{2n} {(-1)^k \cdot k^2}=(2n+1)\cdot n$ $1$.presumption: $(-1)^1 \cdot 1^2+(-1)^2+2^2=(2 \cdot 1+1) \cdot 1=3$ that seems legit ...
1
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2answers
57 views

Proof that the derivative of a function $f$ and $g$ are equivalent $\forall x \in$ the domain of $f(x)$ and $g(x)$

Set $ g(x) = \left\{ \begin{array}{lr} \frac{1}{x} & : x > 0 \\ \frac{1}{x} + 1 & : x < 0 ...
2
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2answers
57 views

${\rm sup}\ A\cap B = {\rm min}\ \{ {\rm sup} (A), {\rm sup}(B) \} $

Let $A,B\subseteq \mathbb{R}$ be a non-empty intervals and bounded from above. If $A\cap B\neq \emptyset $ prove that it is bounded from above and that $Sup(A\cap B)=min\{sup(A),Sup(B)\}$ ...
1
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2answers
25 views

Can someone point me in the direction of why $\sum\limits_{n=0}^{T-1}A(1+R)^n = A\frac{(1+R)^T-1}{R}$

Before you ask, this isn't a homework question, I am just curious. I was trying to derive an expression for compound interest with evenly spaced deposits. I reached the point: $F = I(1+R)^T + ...
3
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1answer
32 views

Rational Exponents

I'm just checking to see if have the correct answer because my teacher didn't give us an answer key and i like to know that I have done one question properly before doing the rest. Evaluate. ...
0
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4answers
70 views

Is $f\colon\mathbb{Z}\to\mathbb{Z}, f(x)=x^2$ injective? Surjective?

I would say no: $\text{Suppose } f(a)=f(b) \text{ then } a^2=b^2 \implies \pm a = \pm b \implies -a=b$. Or simply by counterexample: $f(-1)=f(1)$ Further, I would say it does not map $\mathbb{Z}$ ...
3
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0answers
32 views

Tensor product of flat modules - proof verification

Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras. Let $M$ be a flat $B$-module and $N$ a flat $C$-module. I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module. ...
3
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0answers
33 views

My proof that sum of convergent sequences converges to sum of limits

Does my proof appear correct? Also, do you like the notation? $\textbf{Theorem.}$ If $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ are convergent real sequences, then $$ \lim_{n \to ...
1
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1answer
22 views

When doest it make sense to talk about globally well posedness for the ODE $y'=y$?

My Naive Question: How to formulate the following question, rigorous and then to find answer:"There exists a unique continuous solution to the ODE $y'=y$ for all times? Note that the solution is ...
0
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0answers
9 views

Linear Programming Duality with Big M

I wanted to check of my proof for the following is correct. I am least sure of step 3. Given a linear program $LP1$. $$\text{minimize}\left\{\sum_{i\in I}c_iy_i\right\}\\ \text{subject to, }\\ ...
11
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1answer
77 views

Valid proof of Young's Inequality?

Part of an exercise to prove Holder's inequality in Rudin involves proving Young's Inequality... That is, given $\frac{1}{p}+\frac{1}{q} = 1$, prove $$ab \leq \frac{a^p}{p} + \frac{b^q}{q}.$$ ...
0
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0answers
17 views

The radius of convergence of $\sum_{0}^{\infty} a_{n}x^{n}$ where $0 < \alpha \leq |a_{n}| \leq \beta$ for $n \geq 0$

Since $|a_{n}||x|^{n} \leq \beta|x|^{n},$ since $\beta^{1/n}|x| = |x| + o(1)\big|_{n \to \infty}$, and since $\sum \beta |x|^{n}$ converges if $|x| < 1,$ it follows that $\sum|a_{n}||x|^{n}$ ...
0
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1answer
28 views

Proving that HK is a subgroup when K is normal

$HK = \{hk: h \in H\text{ and }k \in K\}$ I need to first prove first $e \in HK$. Since $e \in H$ and $e \in K$. Hence we have $e \cdot e = e \in HK$. Suppose $hk, h'k' \in HK$. $hk \cdot ...
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2answers
71 views

Prove if, $2^n - 1$ is prime, then $n$ is prime. [duplicate]

Prove, when $n$ is a positive integer, if $2^n - 1$ is prime, then $n$ is prime. I did read some sort of proving on the web, but I could not understand it... Any help? And if possible, could the ...
1
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0answers
26 views

(Proof Verification) Prove There Is A Hamiltonian Cycle for Every $n$ dimensional hypercube where $n\geq2$

Prove There Is A Hamiltonian Cycle for Every $n$ dimensional hypercube where $n\geq2$ My book gave a very fancy proof by induction, but to me it seems obvious that if we simply follow the standard ...
3
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1answer
23 views

In $\mathbb{Z}$, let m~n iff m-n is a multiple of 10.

Prove that each of the following is an equivalence relation on the indicated set. Then describe the partition associated with that equivalence relation. In $\mathbb{Z}$, let m~n iff m-n is a ...
2
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1answer
27 views

Proof that $a,r,s$ are odd and $b$ is even

I was trying to do this proof where: Assume $a,b,r,s$ are relatively prime, and that $$a^2+b^2=r^2$$ and $$a^2-b^2=s^2$$ Prove that $a,r,s$ are odd and $b$ is even. So I started off by saying that ...
1
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1answer
24 views

Proof that Mersenne numbers with a composite exponent are also composite

I'm following the book The Haskell Road to Logic, Maths, and Programming, and I am unsure of one of my proofs for one of the exercises. It is to be proven that a number of the form $M_n = 2^n -1$ is ...
4
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1answer
55 views

Clarification of proof of Theorem 4.14 in Baby Rudin

We are proving that $f : X \to Y$, $X$ compact $\Rightarrow f(X)$ is compact. We reach a step saying $$X \subset f^{-1}(V_{\alpha_1}) \cup \dots \cup f^{-1}(V_{\alpha_n})$$ Rudin says that, since ...
0
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1answer
55 views

If $A$ has no max and $B$ is finite, then $\sup(A)=\sup(A\setminus B)$

Let $A\subset \mathbb{R}$ be non-empty and bounded from above, and assume it does not have a maximum. Let $B$ be a finite set of real numbers. Prove: $\sup(A)=\sup(A\setminus B)$ ...
6
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1answer
92 views

Solving $xyt = 1000$

How many nonegative integer solutions (triples), $(x, y, t)$ exist for: $$xyt = 1000$$ I found the prime factorization being, $$1000 = 2^3 \cdot 5^3$$ Let $x = 2^{a} \cdot 3^{b}$, let $y = 2^{c} ...
1
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2answers
50 views

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$. The Mean Value Theorem states: a function $f$ which is continuous on the closed ...
1
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2answers
40 views

Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function.

Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function. Set $f(x)$ to be some function such that $f'(x) > 0 \implies$ f(x) is ...
0
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2answers
59 views

Suppose that $n \in \mathbb{Z}$. Prove that if $n^2 + 1$ is a perfect square, then $n$ is even.

This is a homework problem that I cannot figure out. I have figured out that if $n^2 + 1$ is a perfect square it can be written as such: $n^2 + 1 = k^2$. and if $n$ is even it can be written as ...
0
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3answers
35 views

Prove that a number $u$ is $\sup S$ given certain properties.

Problem Let $S$ be a nonempty subset of $\mathbb{R}$, and let $u$ be a number with the following properties: for each positive integer $n$, the number $u - \frac{1}{n}$ is not an upper bound of ...
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3answers
37 views

Mathematical Induction Proof Question dealing with integers

How would you use mathematical induction to prove that $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots + n \cdot (n + 1) \cdot (n + 2) = \frac{n(n + 1)(n + 2)(n + 3)}{4}$ I tried proving the base ...
1
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1answer
10 views

Finding an equation of a hyperplane given 2 lines

So the book I'm using for my Intro. to Analysis class is An introduction to Analysis by William R. Wade, and I'm stuck on a problem and can't quite match my answer up with the one given in the book. ...
0
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1answer
24 views

condition for a supremum

let $A\subseteq \mathbb{R}$ be a non-empty set and $s\in \mathbb{R}$ and upper bound of $A$. So $s$ is the supremum of $A$ $\iff$ $\forall \epsilon>0$ there is $x\in A$ so $s-\epsilon<x\leq s$. ...
1
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1answer
39 views

Countinuity of the identity map between different topologies

Could you please verify this (rather simple) proof? I'm a bit new to this kind of reasoning. The question is: Let $\mathcal{T}_1$, $\mathcal{T}_2$ be two topologies on some set $X$, when is the ...
2
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1answer
98 views

Proof that this specific function is measurable

Bounty Edit: Considering the nature of the problem at hand (i.e. proving that a specific function is measurable), I think this can be an easy but relevant problem. In particular, it is relevant to ...
0
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0answers
30 views

Prove that Hall's Condition is necessary for complete matching

Here's what I have so far: Hall's Algorithm states that given $n$ girls and $n$ boys a complete matching between the two groups is possible iff any of the three conditions are satisfied: There is ...
1
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1answer
25 views

Two definitions of attaching are equivalent.

Suppose $X,Y$ are topological spaces, $A\subset X$ and $$ f: A\to Y $$ is a quotient map, that is, a surjective continuous map with $U\subset Y$ open if and only if $f^{-1}(U)\subset A$ is open. We ...
1
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1answer
22 views

Show that for a prime $p$, the polynomial $x^p+a$ in $\mathbb Z_p[x]$ is not irreducible for any $a \in \mathbb Z_p$

Here's my attempt: $\mathbb Z_p[x]$ is an integral domain with characteristic $p$ for a prime $p$. Let $\alpha$ be a root of $x^p+a$. So $a=-\alpha^p$ Then the polynomial becomes ...
0
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1answer
39 views

Prove $\sigma(2^{k-1})=2^k-1$

Is there any way to prove this? rather than just plugging in numbers? It's related to Mersenne Primes for anyone interested. I only wanna know the proof to the above statement. Thank you.
2
votes
1answer
59 views

Roots of $z^5 (z − 2) = w $ in Unit disk

The Q is following : Prove that for each w in the unit disc $D(0, 1)$, the equation $z^5 (z − 2) = w $ has exactly five solutions in the unit disc counted with multiplicity. My Approach : let $f(z) ...
0
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1answer
115 views

Is my $1+1+1+1+1…=-\frac{1}{2}$ proof correct?

Let $x = 1+1+1+1+1+1 ...$ Let $y=1-1+1-1+1-1 . . .$ First, let's find the value of $y$. The partial sums of $y$ are $s_n=(1,0,1,0,1,0,...)$ If you take the means of the partial sums, you will get ...
3
votes
1answer
45 views

Proof review- Every sequence in $\mathbb{R}$ has monotone subsequence

I would like to know if my proof is correct. I'm worried that I may have broken some rules of constructive proofs (e.g. providing a construction with infinite steps). Also, please excuse my abuse of ...
2
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0answers
94 views

upper lebesgue sum with a new partition

Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable.Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is ...