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

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48 views

Proof on Limits

I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says $x=0$. If $x=0$, it should cancel everything out. The derivative of $0$ is $0$ so will ...
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0answers
18 views

The kernel of the transpose of the differentiation operator - Solution check

I tried to solve the following problem and I'd like some feedback on my solution: Let $n$ be a positive integer and let $V$ be $P_n(\Bbb R)$the space of all polynomials functions over the field of ...
2
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1answer
35 views

Describe the equivalence classes generated by T

Suppose $S = \{(x,y) \in \mathbb{R}^2\mid y = x + 1\text{ and } 0 < x < 2\}$. Question Describe the equivalence relation T on the real line that is the intersection of all equivalence ...
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1answer
21 views

Perfectly Normal is hereditary

The definitions I'm working with: $(X, T )$ is called perfectly normal if whenever $C$ and $D$ are disjoint, nonempty, closed subsets of $X$, there exists a continuous function $f : X \rightarrow [...
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3answers
34 views

Proving $-1$ is the infimum of $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$

Prove: $\inf\{E\}=-1$ for $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$ Let assume the contrary , there is $x\in E$ such that $\frac{2}{n}+(-1)^n\leq-1$. Because we are looking at negative ...
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1answer
41 views

Finding nilpotent elements in a quotient ring.

Which are nilpotent elements of $\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)$? I tried to decompose in this way: $$\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)\cong\mathbb{Q}[x]/(x^2)\times\mathbb{...
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1answer
42 views

Evaluate $\sum^{100}_{k=1}[{1 \over k}-{1\over k+1}]$

Evaluate $$\sum^{100}_{k=1}[{1 \over k}-{1\over k+1}]$$ I've tried: $$\sum^{100}_{k=1}[{1 \over k}-{1\over k+1}] \\= \sum^{100}_{k=1} k^{-1}-(k+1)^{-1} \\= {2 \over n(n+1)}-({n(n+1)\over 2}+1)^{-1} ...
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3answers
20 views

$a\equiv b\pmod{n}\iff a/x\equiv b/x\pmod{n/\gcd(x,n)}$ for integers $a,b,x~(x\neq 0)$ and $n\in\Bbb Z^+$?

I'm trying to prove/disprove the following: If $a,b,x$ be three integers (where $x\neq 0$) such that $x\mid a,b$ and $n$ be a positive integer, then the following congruence holds: $$a\equiv ...
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2answers
36 views

Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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2answers
39 views

Could someone please verify whether or not this is a book error?

Below is a short extract for which I believe there may be an error: I think that equations $(22.3)$ and $(22.4)$ have been written out wrongly, they should be ...
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13 views

A question about a Jacobian in a proof of Neyman's factorization theorem

everyone. Would you be so kind and explain me the role of a Jacobian in the proof below (Picture attached, source wikipedia: https://en.wikipedia.org/wiki/Sufficient_statistic)? My knowledge is that ...
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0answers
20 views

set equalities proofs

I'm teaching my self topology with the aid of a book, the problem i'm on ask to prove the following: Let $X$ be a topological space and $B$ be a subset of $X$. Prove the following set equalities.(...
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0answers
26 views

Verify if this is correct idea of continuous and homeomorphism

This is a follow up to a previous question. My main motivation is to understand these defintions: Definition of homeomorphism: If $X$ and $Y$ are topological spaces, a homeomorphism from $X$ to $...
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0answers
24 views

Game theory: how is law of large number applied here?

This is a claim rephrased and lifted from from Herbert Gintis' book "Game Theory Evolving" Pg187 Consider an evolutionary game with $n$ pure strategies $i = \{1, \ldots, n\}$, and time periods $t ...
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1answer
52 views

How do I show that $E-\gamma=\lim_{j\to \infty}\sum_{n=1}^{j}n\left({1\over 2^n-1}-{1\over 2^n}+\cdots-{1\over 2^{n+1}-2}\right)$

Given the Erdos-Borwein's constant $E=\sum_{n\geq 1}\frac{1}{2^n-1}$ and the Euler-Mascheroni constant $\gamma=0.5772156...=\sum_{n\geq 1}\left[\frac{1}{n}-\log\left(1+\frac{1}{n}\right)\right]$ how ...
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1answer
58 views

Continuity of $F(x,y)=|x-y|$

Suppose that $F:\mathbb{R}^2\to \mathbb{R}$ defined by $F(x,y)=|x-y|$. Prove using $\epsilon-\delta$ that $F(x,y)$ is continuous. Let $(x_0,y_0)\in \mathbb{R}^2$. We have to show that for any $\...
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4answers
45 views

Evaluate $\sum^{n}_{i=1}(10^{i+1}-10^i)$

Evaluate $$\sum^{n}_{i=1}(10^{i+1}-10^i)$$ Here's what I did $$\sum^{n}_{i=1}(10^{i+1}-10^i) \\ = 10(\sum^n_{i=1} 1^{i+1}-1^i) \\ = -10(\sum^n_{i=1} 1^i-1^{i+1}) \\= -10(1^n-1^{n+1}) \\= 10^{n+1}-...
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1answer
18 views

A $3$-chain is a monotonic subsequence of $3$ integers. Show that any sequence of $5$ distinct integers will contain a $3$-chain

Define a $3$-chain to be a (not necessarily contiguous) subsequence of three integers, which is either monotonically increasing or monotonically decreasing. We will show here that any sequence of five ...
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0answers
12 views

Round up/Round down to Decimal and Significant

(1) 30.9955 is round up to 2 decimal places. 31.00 (2) 30.9955 is round down to 3 significant digits. 30.9000 Is this right?
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1answer
20 views

$A\subseteq R$ is an upper bounded set that contains at least two items/numbers. If $x < \sup{A}$, then $\sup{\left(A\setminus\{x\}\right)}=\sup A$

Prove: $A\subseteq \mathbb{R}$ is an upper bounded set that contains at least two items/numbers. If $x < \sup{A}$, then $\sup{\left(A\setminus\{x\}\right)}=\sup A$. My attempt: Since $A$ is upper ...
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3answers
54 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
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1answer
27 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
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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} : ...
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3answers
118 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 ...
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1answer
41 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
27 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....
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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
45 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
12 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
40 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 ...
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1answer
35 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
26 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? ...
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1answer
17 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 $. ...
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2answers
51 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 $$...
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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 $...
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1answer
40 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 ...
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0answers
24 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=...
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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 ...
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26 views

Convergence of a big sum indexed over $\mathbb{Z}^3$

For a fixed vector $r_j$ consider the function on $\mathbb{R}^3$ defined by the series $$f(r) = \sum_{\substack{n,m,k \in \mathbb{Z} \\ (n,m,k) \neq 0}} \frac{1}{n^2+m^2+k^2}e^{2\pi i(n,m,k) \cdot (r-...
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0answers
10 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. ...
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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
44 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 ...
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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 $n\ne0$, $(b^m)^{1/n}>1$...
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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 ...
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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 ...