This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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1answer
14 views

Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
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1answer
24 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
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0answers
15 views

Fichtenholz's limit proof for a sequence

I'm reading calculus written by G.M Fichtenholz. And i can't understand his proof. The proof is: $n>2$ There is a sequence $x_n = \frac{n^2 -n + 2}{3n^2+2n-4}$ $x_n - \frac13 = \frac{5n - 10}{3( ...
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1answer
20 views

Question regarding an algebraic manipulation in GFology

How does the author arrive at the last equality in the first line, i.e.$$\text{why is } [x^k]\frac{1}{1-y(1+x)} = \frac{1}{1-y}[x^k]\frac{1}{1-\left(\frac{y}{1-y}\right)x} \text{?}$$
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2answers
56 views

set theory proof explanation

The following lemma is taken from the book 'Introduction to Set Theory' by Hrbacek and Jech. chapter $6$ normal form Can anyone explain to me why the first sentence holds ( the existence of ...
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0answers
22 views

Multilinear Function proof in Spivak?

Note that $$\wedge^n (V)$$ denotes the set of all alternating multilinear functions and $\mathfrak{I}^n(R^n)$ denotes the set of all multilinear function. I don't know what the actual symbol ...
2
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1answer
27 views

Multilinear algebra and matrices

Given $\wedge^k(V)$ an alternating multilinear space and $T : V \to W$ a linear map, then we have $$v_1 \wedge \dots \wedge v_k \in\wedge^k(V).$$ Define $$\wedge^k(T)(v_1\wedge\dots\wedge v_k) = ...
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1answer
34 views

Understanding the proof that $c_0$ is a closed subspace of $\ell^\infty$

The problem is given: source Let $c_0$ be a space of real sequences $x = \{x_n\}_{n = 1}^\infty=0$ converging to $0$. Let $\ell^\infty$ be a set of real sequences $w = \{w_k\}^\infty_{k=0}$ ...
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1answer
70 views

A limit theorem in Rudin. Please elaborate?

Theorem: Let $\{ p_n \} \in X $. If $E \subset X$ and $p$ is a limit point of $E$, then there is a sequence $\{ p_n \}$ such that $p = \lim p_n $ The proof goes like this proof: For each ...
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0answers
35 views

Transfinite Induction (Proof Explanation)

The theorem above is extracted from the book 'Introduction to Set Theory' by Hrbacek and Jech. Questions: $1$)I don't understand the successor case. When $\alpha_2=\beta+1$, why suddenly $W_2$ ...
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1answer
33 views

Help understanding example in Engel's *Problem Solving Strategies*

I've spent a lot of time trying to follow the chain of reasoning, but to no avail. I lose track of how it works at the "Adding (1) and (2)" part. Could someone help me understand this, please?
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1answer
44 views

Proposition 0.16 in Hatcher's AT

In the proof of the quoted proposition, it is mentioned that $D^n \times I$ retracts onto $D^n \times \left\{0\right\} \cup \partial D^n \times I$ and an example is given in a figure with $n=2$, which ...
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1answer
55 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
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1answer
36 views

Proposition 1A.1 in Hatcher's Algebraic Topology

In the proof of the quoted Proposition, we have a connected graph $X$ and a sequence of subgraphs $X_0 \subset X_1 \subset \cdots$ such that $\cup_i X_i$ is both open and closed. Then Hatcher deduces ...
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3answers
49 views

How to determine value from willingness to pay?

I use the British pounds symbol instead of dollars because $ conflicts with Mathjax. Source: p 296, The Legal Analyst, Ward Farnsworth "... one time in a thousand we do lose the film; if you’re ...
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1answer
48 views

Proposition I.6.8 in Hartshorne

In the context of the quoted proposition arises the following question. Let $X$ be an abstract nonsingular curve (as defined in p. 42), $P \in X$ and let $\phi: X-P \rightarrow \mathbb{P}^n$ a ...
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1answer
37 views

Proposition I.6.7 in Hartshorne

Proposition (I.6.7,HAG): Every nonsingular quasi-projective curve $Y$ is isomorphic to an abstract nonsingular curve. The first paragraph of the proof establishes a bijective map $\phi: Y \rightarrow ...
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1answer
56 views

Hartshorne Theorem I.5.3

This question concerns a reduction argument that occurs in the proof of Theorem I.5.3 in Hartshorne. In particular, let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Then by (4.9) $Y$ ...
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1answer
43 views

Weibel “Introduction to homological algebra” Main Theorem 4.4.16

I can't understand the proof of Main Theorem 4.4.16 from Weibel's book "An Introduction to homological algebra". The Theorem states Let $R$ be a local noetherian commutative ring, then $R$ is ...
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1answer
37 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
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1answer
26 views

Question in Fulton and Harris regarding induced representation.

I'm confused by the following paragraph: I don't see why $g\cdot W$ depends only on the left coset $gH$. What does he mean precisely by that? Why is it true that $gh\cdot W = g\cdot(h\cdot W) = ...
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1answer
50 views

Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
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5answers
75 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
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1answer
27 views

The topology of the restriction of a metric is the restriction of the topology.

I'm reading this proof of the following claim. Let $(X,d)$ be a metric space and $(Y,d')$ a subspace of $X$. If $(X,T)$ is the topology induced by $d$ and $(Y,T')$ the typology induced by $d'$, ...
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1answer
161 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
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1answer
35 views

Explain what the teacher did, convergence of improper integral

I'd like someone to explain what the teacher did, because I'm not sure I understand. Basically, the question is for which values of $p$, does the integral $$\int_{1}^{\infty} \frac{dt}{t \log ...
2
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1answer
57 views

Groups product formula's proof

I'm trying to understand the proof of the following theorem: Theorem 2.20 (Product Formula). If $S$ and $T$ are subgroups of a finite group $G$, then $$|ST|\, |S \cap T| = |S|\,|T|.$$ Remark. ...
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2answers
36 views

Uniqueness of Initial Object, or why must a morphism from an object to itself be the identity?

The definition of an initial object in a category $\mathscr{C}$ is defined as an object that only has one map going to each object in $\mathscr{C}$. A basic result supposedly says that any two initial ...
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0answers
37 views

Noetherian rings/Hilbert's Basis Theorem

So I'm studying the proof of Hilbert's Basis Theorem - we've shown that $λ(I)$ is an ideal of $R$ and and then it says "Since R is Noetherian, we have $λ(I) = \sum\limits_{i=1}^k s_iR$ for some $s_1, ...
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1answer
43 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
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1answer
51 views

Proving that every set in the Borel $\sigma$-algebra on $\mathbb{R}$ is Lebesgue-Stieltjes Measurable

I am reading through Richard Bass's Real Analysis and on page 25 we have the following proposition, using $\alpha$ is a Stieltjes function and $m^*$ is the Lebesgue-Stieltjes outer measure on ...
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2answers
40 views

All homomorphisms from a simple ring to a non-zero ring are injective

Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective. Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker ...
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1answer
50 views

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$?

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$ ? And why does $p_i^{\alpha_i} \mid \text{ord}_n(a)$ ? I see that ...
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3answers
111 views

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$.

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$, where $n$ has a primitive root. I have proven the following theorem ...
2
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1answer
49 views

Confused About Step in Proof of Divergence of $\sum \frac{1}{p}$

I was going through the number theory text by Ireland and Rosen, and was following the proof of the divergence of the sum of reciprocal primes. But I came across a step unclear to me. The proof so ...
2
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1answer
45 views

Same characteristic polynomial $\iff$ same eigenvalues?

This proves: Similar matrices have the same characteristic polynomial. (Lay P277 Theorem 4) I prefer http://math.stackexchange.com/a/8407/53259, but this proves that they have the same eigenvalues. ...
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1answer
73 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
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2answers
48 views

How to Complete Sketch of a function of two variables $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ ? [Stewart P930 Question 14.7.4]

For $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ $\implies$ $\partial_x f = 3 - 3x^2, \partial_y f = -4y + 4y^3$. Set both equations to 0 $\implies x = \pm $1 and $y = 0, \pm 1$. $1.$ To determine the ...
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1answer
81 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
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1answer
30 views

2 x 2 Phase Portrait for 2 x 2 Linear System with Real Coefficients. Boyce, p395, Figure 7.5.4a

The given general solution for some linear system is $ x= c_{2} \mathbf{ x^{(1)} }(t) + c_{2} \mathbf{ x^{(2)} }(t) = c_{1}\ \left(\begin{array}{l} 1\\ \sqrt{2} \end{array}\right)\ e^{-t}+ ...
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3answers
63 views

How does this prove: ALL the eigenvalues of a triangular matrix = ALL of its diagonal entries? [Lay P269 Theorem 5.1.1]

For simplicity, consider the $3\times 3$ case. If $A$ is upper triangular, then $ A-\lambda I=\left\{\begin{array}{lll} a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & 0 ...
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0answers
21 views

Explanation for a simple comparison

Ok, Yesterday I started to learn how to solve problems with comparisons, but I couldn't understand one thing of the "solve algotithm". Here is a part from a solve from a simple example problem ...
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2answers
201 views

Proof of Cayley-Hamilton Theorem for Diagonalisable Matrices [Lay P326 Ch 5 Sup Q7]

Proof for Diagonal Matrices from Page 2 of 7: Let $A \in M_{n}(C)$ be diagonal, to wit, $A _{ii}=\lambda_{i}$. Then $ p_{A}(t) = \det(tI-A)= \det \begin{bmatrix} t - \lambda_1 & ~ & ~ \\ ...
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2answers
297 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
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0answers
18 views

Which is the starting and ending basis? - Matrix of a linear transformation [Lay P294 Q 5.4.28]

Denote some arbitrary linear transformation as $L.$ When a question asks "to find a matrix of $L$ with respect to S and T", does this denote $[L]_{T \leftarrow S}$ or $[L]_{S \leftarrow T}$ ? How can ...
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2answers
39 views

Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
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1answer
38 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
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1answer
34 views

Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. [Lay P160 Ch 2 Sup Q4]

Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To verify this, compute $ (I \color{orangered}{-A} ...
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2answers
25 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
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2answers
64 views

Ground Plan - Prove Fermat-Euclid's Totient Theorem with Lagrange's Theorem

If $\gcd(a,n) = 1$, then $a^{\phi(n)}\equiv 1\pmod n$. Here's a three-step proof. An integer a is invertible means there's some $a^{-1}$ such that $aa^{-1}\equiv 1 \pmod n$. By cause of Jones p84 ...