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

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1answer
20 views

Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values

I'm talking specifically of population SD, where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ I have a hunch that $s$ is maximized for a certain mean $\bar{x}$ when the values in ...
0
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1answer
31 views

Would the following series of implications be logically correct?

Let $a$ and $b$ be positive integers, and let $f$ be a generic function satisfying $f(1) = 1$, and taking on only positive integer values. Suppose that I have the following propositions: $$\bf{A} : ...
1
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3answers
123 views

$1+1=2$…but Why? [duplicate]

A study that was carried on recently showed that even babies at the age of few months know that $1+1=2$. My question is : is this a fact that can be proved, or is it a just a postulate as those in ...
4
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1answer
56 views

Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
1
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1answer
29 views

Real Analysis, Folland Corollary 2.19 Integration of Nonnegative functions

Corollary 2.19 - If $\{f_n\}\subset L^+$, $f\in L^+$, and $f_n\rightarrow f$ a.e., then $\int f \leq \liminf\int f_n$. Proof - We have that $\{f_n\}\subset L^+$, $f\in L^+$ and $f_n\rightarrow f$ a....
1
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1answer
23 views

Show excluded point topology is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_4=\{U \...
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2answers
23 views

Show particular point topology, is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_3=\{U \...
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2answers
46 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
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0answers
15 views

Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
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0answers
31 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
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3answers
41 views

Is this a valid existence proof for: “there exists a unique real number solution to the equation $x^3 + x^2 - 1 = 0$ between $x = 2/3$ and $x = 1$”

I was wondering if this was a valid existence proof for the following: "there exists a unique real number solution to the equation $x^3 + x^2 - 1 = 0$ between $x = 2/3$ and $x = 1$" Proof: Assume ...
2
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1answer
37 views

Show “countable complement topology” is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set. Show that $\mathscr{T}_2=\{U \subseteq X : U = \emptyset $ or $ X\...
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3answers
27 views

Proof by contradiction for: Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$

I am kind of stuck on a practice problem relating to proof by contradiction that goes as follows: "Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$" For the ...
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4answers
49 views

Prove $3\mathbb{Z}+1=\{6\mathbb{Z}+1\}\cup\{6\mathbb{Z}+4\}$

I was wondering if someone could confirm I have proven the following equality correctly. Also, for part II should I have let $n\in \mathbb{Z}$ as opposed to $n\in 6\mathbb{Z}+1$ or was I correct? ...
3
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1answer
19 views

Prove that $\text{Dom } (S\circ R) ⊆ \text{Dom }R $

Let $R$ be a relation from $A$ to $B$ and $S$ be a relation from $B$ to $C$. Suppose, $x \in \text{Dom }(S\circ R)$. Then, it follows that there $\exists y \in C$ such that $(x,y) \in S\circ R $. ...
4
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2answers
52 views

Real Analysis, 2.18 (Fatou's Lemma) Integration of Nonnegative functions

2.18 Fatou's Lemma - If $\{f_n\}$ is any sequence in $L^+$, then $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) \leq \lim_{n\rightarrow \infty}\inf\int f_n$$ Attempted proof - We know that $$...
1
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0answers
17 views

Sum over square divisors is multiplicative proof verification

I would like someone to verify my proof of the following claim, which I have been using to solve some problems about proving series identities in Ch. 11 of Apostol's analytic number theory text. Let $...
1
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1answer
41 views

Real Analysis, Folland Theorem 2.14 (Monotone Convergence Theorem)

Theorem 2.14 (MCT) - If $\{f_n\}$ is a sequence in $L^+$ such that $f_{n}\leq f_{n+1}$ for all $n$, and $f = \lim_{n\rightarrow \infty}f_n (=\sup_n f_n)$, then $\int f = \lim_{n\rightarrow \infty}\int ...
2
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0answers
28 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
0
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1answer
46 views

Is it my error or the term “normal” has multiple meanings?

I use a definition of normal quasi-uniform spaces from this article. Now I have proved (I do not present the proof because it uses "funcoids" which can be read about only in my manuscripts.) that ...
0
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0answers
11 views

I need input and help understanding how the formula for x arises in a cycloid that is parameterized with theta with the cusp at the origin

Disclaimer: I attempted to answer some of it by using my own deductions. I would feedback on that. The book gives the formulas for how x arises but my problem is understanding how the formulas arose. ...
0
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1answer
29 views

Every completely regular space is regular

The definitions I'm working with: $(X,\mathcal{T})$ is said to be completely regular if for every $x \in X$ and every closed set $C \subseteq X$ not containing x can be separated by a continuous ...
0
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1answer
45 views

Fibonacci Sequence: Prove $f_1+f_3+\dots+f_{2n-1}=f_{2n}$ by Induction.

I believe the majority of my proof is correct I'm just not certain about the base case if any one can explain how to do that base case or fix any error I made I would greatly appreciate it. Recall ...
3
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4answers
36 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
0
votes
1answer
30 views

Contrapositive, Negation, and Converse of statements

I am having trouble with the wording of these statements particularly the negation statement. Is that the best way to put it or could you provide a better alternative? Also for the converse proof ...
1
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1answer
15 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...
2
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4answers
50 views

Antiderrivative of ${d^2 y \over dx^2} = 1-x^2$

At any point $(x,y)$ on a curve, ${d^2 y \over dx^2} = 1-x^2$, and an equation of the tangent line to the curve at the point $(1,1)$ is $y=2-x$. Find an equation of the curve. This is what I've done ...
0
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0answers
19 views

Showing $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ is a Hilbert space

Let $I$ be an open interval in $\mathbb{R}$. We define $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ with the scalar product of the Sobolev space $H^1(I)$, i.e. $(u,v)=(u,v)_{L^2(I)}+(u',v')_{L^...
13
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5answers
203 views

I want to show that $\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$

I want to show that $$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$$ Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$ Let see (substitution of $y=x^2$)...
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0answers
43 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...
0
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0answers
27 views

Eliminate the parameter to find a cartesian equation for the curves

For the first part I am just unsure as to how the book has a different answer than mine. The book has the answer $y = \frac{3}{4} x - \frac{1}{4}$ but given the functions $x(t) = 3 - 4t$ and $y(t) = ...
1
vote
1answer
36 views

Arithmetic progressions form infinite basis on $\mathbb{Z}$

Let $B(a,b) = \{ax+b: a,b \in \mathbb{Z}, a \neq 0, x \in \mathbb{Z}\}$ be a so called arithmetic progression I am required to show that that $\mathcal{B} = \{B(a,b) | a,b \in \mathbb{Z}\}$ is a ...
2
votes
2answers
38 views

Construction of field extension for $[E:\mathbb F_{11}]=3$

Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1 $ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...
2
votes
1answer
31 views

Proof by contradidction that the mean of a set cannot be greater than the greatest value in that set.

I want to prove that given a set of values $x_1, x_2, ..., x_n$, the mean of those values cannot be greater than the greatest of those values. Let the mean $\frac{x_1 + x_2 +... + x_n}{n} = a$ ...
0
votes
0answers
48 views

Unusual integration of 1/cx [duplicate]

Consider an integral: $$\int_2^3 \frac{1}{cx} dx$$ where $c$ is a constant So we can take that out of the integral, so $$\int_2^3 \frac{1}{cx} dx = \frac{1}{c} \int_2^3 \frac{1}{x} dx $$ all is ...
2
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5answers
149 views

Prove: if $|x-1|<\frac{1}{10}$ so $\frac{|x^2-1|}{|x+3|}<\frac{1}{13}$

Prove: $$|x-1|<\frac{1}{10} \rightarrow \frac{|x^2-1|}{|x+3|}<\frac{1}{13}$$ $$|x-1|<\frac{1}{10}$$ $$ -\frac{1}{10}<x-1<\frac{1}{10}$$ $$ \frac{19}{10}<x+1<\frac{21}{10}$$ $...
3
votes
1answer
54 views

Let $g_n(x)=[x\in (r_n,1]]$. Show that $G(x)=\sum_{n\ge 1} g_n(x)/2^n$ is Riemann-integrable

I want to check if my proof about $G(x)$ is Riemann-Integrable in $[0,1]$ is correct. Let $\{r_n\}$ an enumeration of the rationals in $[0,1]$, and $g_n(x)=[x\in (r_n,1]]$. Show that $G(x)=\sum_{n\...
0
votes
1answer
40 views

Show that if $X$ is bounded above, then there exists $y \in a + X$ such that $y$ is an upper bound of $X$

"Let $X \subset \mathbb{R}$ be nonempty and $a > 0$. Define $$a + X = \{a+x: x \in X\}$$ Show that if $X$ is bounded above, then there exists $y \in a + X$ such that $y$ is an upper bound of $X$"...
3
votes
4answers
93 views

$a^n$ even implies $a$ even

I've tried to prove that $(\forall a,n>0 \in \mathbb{N}),(a^n \text{ even} \implies a \text{ even})$, can someone tell me whether my proof is sound? Lemma 1: $a \text{ even} \implies a^2 \text{ ...
1
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1answer
39 views

Real Analysis, Folland Problem 2.1.5 Measurable Functions

Problem 2.1.5 - If $X = A\cup B$ where $A,B\in M$, a function $f$ on $X$ is measurable if and only if $f$ is measurable on $A$ and on $B$. Proof - Suppose, $X = A\cup B$ where $A,B\in M$ and we have ...
2
votes
1answer
40 views

Is this proof of convergence in probability correct?

${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad $ in probability show that $$\lim_{n\to \infty} \min_{...
1
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0answers
26 views

Function check-up exercise

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
2
votes
0answers
33 views

Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
2
votes
2answers
69 views

Are these subsets homeomorphic?

Are the two subsets of the Euclidean Plane $[0,1]\times[0,1)$ and $[0,1)\times[0,1)$ homeomorphic or not? My attempt: We need to find a bijective function $f$ from $[0,1]$ to $[0,1)$ such that $f$ ...
0
votes
1answer
39 views

$\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
0
votes
1answer
41 views

Existence and Uniqueness of an ordinal

$\underline{\mathbf{Problem:}}$ If $\alpha < \beta $ then $\exists$ a unique $\gamma$ st. $\alpha+\gamma=\beta$, where '$+$' denotes ordinal addition. If someone would be so kind to check the ...
0
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1answer
31 views

Prove that any finitely generated submodule of $R^+$ (the field of quotients) is free of rank $1$

I am working on the following problem: Let $R$ be a principal ideal domain and $R^+$ the field of quotients. Then $R^+$ is an $R$-module. Prove that any finitely generated submodule of $R^+$ is a ...
2
votes
2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
0
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2answers
25 views

Metric space where each continuous function has IVP is connected

The question: Let $X$ be a space such that every continuous function $f:X\rightarrow\mathbb{R} $ does have the following property: if $a<c<b$, $f(x) =a$, and $f(y) =b$, then there exists $z\in ...