All questions

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Fundamental group of $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$

Calculate the fundamental group of the complement in $\mathbb{R}^3$ of $$\{ (x,y,z) \ | \ y = 0 , \ x^{2} + z^{2} = 1\} \cup \{ (x,y,z) \ | \ z = 0 , \ (x-1)^{2} + y^{2} = 1\}.$$ Note: this space is $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$.
2022-07-25 17:47:21
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why function over just a relation?

Is there a the real world instance revealing information that creates a relationship is better than one that creates simply a relationship? what the real world circumstance encourages the added problem on relationship so we have functions? Thanks.
2022-07-25 17:47:21
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Interpretation of a Category theory question

A problem I'm attempting says Let $p:A \to B$ be a map of sets and $p^*: \mathcal{P}B \to \mathcal{P}A$ be the induced map of power sets sending $X \subseteq B$ to $p^*(X) = \{a \in A: p(a) \in X\}$. Exhibit left and right adjoints to $p^*$ but I can't quite work out what it's saying the functor is: namely, are we $(i)$ defining $p^*$ to be ...
2022-07-25 17:47:21
0

how was this angle found?

This is a remedy from a publication entailing trusses (in statics), I do not recognize just how they located the angle $\theta$. What is their method? This is the offered trouble: (the offered sizes vary a little bit in between the trouble in guide and also the remedy guidebook)
2022-07-25 17:47:21
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How can I normalize a percent to a value while still deriving results from the percent?

My mathematics abilities are corroded (at ideal) and also I was asking yourself if I can select individuals is below minds on attempting to identify just how to approach what I'm doing. My trouble is a little bit domain name details so I located a proxy trouble that matches flawlessly and also with any luck is less complex to clarify (simply exc...
2022-07-25 17:47:17
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If $H$ is a subgroup of $\mathbb Q$ then $\mathbb Q/H$ is infinite

I'm trying to work out this question: Prove that if $H$ is a proper subgroup of $\mathbb{Q}$ then $\mathbb{Q}/H$ is infinite, but each of its elements have finite order. I thought, for the first part, that I could assume for contradiction that $\mathbb{Q}/H$ is finite of order $n$, then for all $\dfrac{a}{b}\in\mathbb{Q}$, $\dfrac{a^n}{b^n}$ is ...
2022-07-25 17:47:17
0

Is the Lebesgue integral the completion of integrals on step functions?

Lierre gave a very helpful insight at answer 5 on about what Riemann integrals are. My question relates to whether this can be extended to Lebesgue integrals. Lierre pointed out that Riemman integrals can be seen as the natural extension of the 'obvious' linear form on characteristic (or 'indicatrix') functions on real line intervals. There i...
2022-07-25 17:47:17
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Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the excellent $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime perfect? If so, what is its elevation? I'm stuck attempting to show that $f$ is irreducible.
2022-07-25 17:47:17
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properties of ideals in $K[x_1,\ldots ,x_n]$

I'm trying to convince myself of the 4 following facts: If $X \subseteq Y \subseteq A^n_{k}$ then $I(Y) \subseteq I(X)$ If $J \subseteq K[x_1,\ldots,x_n]$ is an ideal then $J \subseteq I(V(J))$ If $X \subseteq A^n_{k}$ then $X \subseteq V(I(X))$ $X=V(I(X))$ if and only if $X$ is an algebraic set My attemps to prove them: 1) Let $f \in I(...
2022-07-25 17:47:17
0

Card probability problem

Possible Duplicate: ¢ I located the adhering to trouble in Rosen is Discrete Mathematics and also Its Applications 6th ed. : There are 3 cards in a box. Both sides of one card are black, both sides of one card are red, and also the 3rd card has one black side and also one red side. We select a card randomly and also observe just one side. ...
2022-07-25 17:47:14
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Dealing with Tychonoff's Theorem.

Here are my few questions that I encountered while going through Tychonoff's theorem in . a) First of all, so far I was thinking that Heine Borel definition of compactness implies sequential compactness but not the other way around ( although i am failing to find some examples to appreciate it). But what wikipedia says is that " but NEITHER...
2022-07-25 17:47:13
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Why integrals with respect to different variables aren't equal?

I have a function $y=x^2+1$, the integral from $-1$ to $2$ is $\int_{-1}^{2}(x^2+1)dx = 6$. The function $x=\sqrt{y-1}$ is the same as the above function. The integral would be from $0$ to $(2)^2+1=5$. So I thought that $\int_{0}^{5}(\sqrt{y-1})dy$ would be equal to the first one. But it turns out that it does not. The integral of the second fun...
2022-07-25 17:47:13
0

Free abelian group $F$ has a subgroup of index $n$?

Suppose that we have a free abelian group $F$. How can it be proved that $F$ has a subgroup of index $n$ which $n≥1$? Honestly, according to the Theorems, I just know that if we take $X$ as a base for $F$, then $$ F= \bigoplus_{\alpha \in X} \mathbb Z_\alpha \ $$ in which for all $ \alpha \in X$; $\mathbb Z_\alpha \ $ is a copy of $ \mathbb Z $....
2022-07-25 17:47:13
2

Closed form of the sequence $a_{n+1}=a_n^2+1$

If $$a_{n+1}=a_n^2+1,$$ with first $a_1=\frac{1}{2}$. Just how to address this series trouble, i.e., how to stand for $a_n$ in shut kind?
2022-07-25 17:47:13
1

Turing reduction

I'm finding out algorithm concept. Research inquiry is: Are $A$ and also $B$ feasible to make sure that $A\not\le_{tt}B$ (difficult to lower making use of tt), yet $A\le_T B$. Yet I can not consider any kind of instance.
2022-07-25 17:47:10
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Integral equation $\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}}f(R \cos(x)) d x = 1$

Can we confirm that there does not exist a function $f$, which pleases this formula for all $R>0$: $$\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}} f(R \cos(x))\, dx= 1.$$
2022-07-25 17:47:10
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Field Extension problem beyond $\mathbb C$

There are great deals of areas in between $\mathbb C$ and also Meromorphic Functions on $\mathbb C$. As an example set of "All Even Meromorphic Functions on $\mathbb C$" is a subfield in between $\mathbb C$ and also Meromorphic Functions on $\mathbb C$. Inquiry: How to classify such subfields? I have no suggestion whether someone res...
2022-07-25 17:47:10
2

Problem connecting Topology and Algebra via Analysis

Let $C(X):=$ Set of all complex/real valued continuous functions. If $X$ is compact then all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$. Is it true that: If all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$ then $X$ is compact.
2022-07-25 17:47:10
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On Constructions by Marked Straightedge and Compass

Pierpont proved that a regular $n$-gon is constructible by (singly) marked straightedge and compass if and only if $n = k \, p_1 \cdots p_{s}$, where $k = 2^{a_1} 3^{a_2}$ for $a_i \geq 0$ and $p_i = 2^{b_1} 3^{b_2} + 1 > 3$ is prime with $b_i \geq 0$. It has been known since the time of Archimedes that a marked straightedge allows for an...
2022-07-25 17:47:10
1

Composing covers with epis

I am beginner of sheaf-theory and beg your pardon for this maybe silly question. Let $\mathcal{C}$ be a Grothendieck site and $T$ the category of sheaves on $\mathcal{C}$ and let $f:X\rightarrow Y$ be an epic morphism in $T$ into a representable sheaf $Y$. I have a general lack of understanding how such epic morphisms look like and this leads to...
2022-07-25 17:47:06