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Questions tagged [proof-explanation]

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

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Proof of the proposition which says that the column rank of $A$ is equal to the row rank of $A$. (Gilbert Strang's new lecture)

I am watching this new lecture by Gilbert Strang. I have the following question. Let $A = \begin{bmatrix}2&1&3\\3&1&4\\5&7&12\\\end{bmatrix}$. Prof. Strang showed that ...
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0answers
12 views

understanding this proof of the Cramer-Rao bound holding with equality

I've been looking at the following passage of Statistical Signal Processing (1991) by Louis Scharf. It deals with showing when the Cramer-Rao bound holds with equality. In this passage, $\mathbf{C}$ ...
0
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2answers
65 views

Proving $2\cosh 2x+ \sinh x = 5$

I have been sitting on this question for quite some time and I haven't been able to prove this identity. Please anybody who can help me here. I am new to hyperbolics. $$2\cosh 2x+ \sinh x = 5$$ I ...
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1answer
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I don't understand one line in problem to prove $|2^S| = 2^{|S|}$

Problem to prove by induction: If $ S $ is a finite set, then $\vert 2^S \vert = 2^{\vert S \vert}$ Proof: Induction on size of $S$, call it $n$ , $n \ge 0$. Base case: Suppose $n=0$. Now, $|2^S| =...
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1answer
27 views

Explain Proof of Convergence of Matrix when Spectral Radius Less than 1

Theorem: Let $M$ be a square matrix. $M$ is convergent when its spectral radius $\rho(M)$ is less than 1. Proof: Suppose $\rho(M) < 1$. We know that there exists a matrix norm $ \lvert\lvert . \...
1
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1answer
32 views

are $\sum_{i=0}^{n}i$ and $\sum_{i=n}^{0}i$ equivalent?

So here's the ugly history of how I came to ask this question. I was following this proof: and got stuck at this step: $$\sum_{j=0}^{(\log_2n) - 1}\frac{1}{(\log_2n) - j} = \sum_{l = 1}^{\log_2n}\...
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2answers
28 views

monotonic function can only have simple discontinuity

I am self-studying Rudin, Principles of Mathematical Analysis. I am having trouble going through the theorem saying that monotonical functions can only have simple discontinuity, i.e., Suppose $f$ is ...
1
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1answer
24 views

Quadratic reciprocity and decomposition of primes in cyclotomic fields

In Neukirch's Algebraic Number Theory, there is a proof of the quadratic reciprocity which makes use of proposition $10.5$: $$p\text{ is totally split in }\mathbb{Q}(\sqrt{\ell^*})\Leftrightarrow p\...
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1answer
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proof of ED* = (n - 1) / n * D

I've seen this proof in a textbook $ E(D^*)=E(\frac{\sum_{k=1}^{n} (X_k - \bar{X})^2}{n})=E(\frac{\sum_{k=1}^{n} ((X_k - a) - (\bar{X} - a))^2}{n})$ then it simplifies to $\frac{E(\sum_{k=1}^{n} (...
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0answers
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Functional that dispaears on the dual of coset

Theorem(Walter Rudin, Functional Analysis): Let $M$ a closed subspace of a Banach space $X$. b) Let $\pi:X \to X/M$ be the quotient map. Put $Y = X/M$, For each $y^* \in Y^*$, define $$ \tau y^*...
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2answers
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Complex Analysis - Showing function is holomorphic

I have the following problem: Suppose $f$ is holomorphic in a region $U^+$. Define $U^{-}: = \{ z: \overline{z} \in U^+\}$. Prove that $g: U^{-} \rightarrow \mathbb{C} $ given by the formula $g(z) =...
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Tabachnikov Billiards : equality of two angles

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_1^{'}A_1F_2$ and $F_1A_1F_2^{'}$ are equal. Can somebody help me out?
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2answers
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Why can we let $x = 2\cos t\ $ in the solution for the following system of equations

Solve in real number the system of equations $\begin{cases}x^2 = y+2 \\ y^2 = z+2 \\ z^2 = x+2 \end{cases}$ The solution given to me says the following: If we eliminate $y$ and $z$, we obtain a ...
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0answers
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Questions about Hartshorne Proposition II.5.9

My question is about the first part of the proof. Let $X$ be a scheme. For any closed subscheme $Y$ of $X$, the corresponding ideal sheaf $I_Y$ is a quasi-coherent sheaf of ideals. Proof: ...
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2answers
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Please explain the proof of the validity of the Cauchy convergence criterion for real number sequences.

My question pertains to BBFSK, Vol I, Pages 143 and 144. The following appears in the context of developing the real numbers as limits of sequences of rational numbers. It is also easy to prove ...
0
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1answer
45 views

Proof explanation: Every module is the quotient of free module

Let $M$ be a left $R-$ module, it is said (here for example) that $M \approx F(M)/\sim$ The proof is apparently $F(M) \stackrel{\pi}\to M$ where $(0,\dots,1_m,\dots0) \stackrel{\pi}\to m$ and appeal ...
0
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1answer
31 views

LU composition of a tridiagonal matrix

Given a triadiagonal Matrix A with $$A = \begin{bmatrix} d_1 &e1\\ c_2 & d_2 & e_2 \\ & c_3 & \ddots & \ddots \\ & & \ddots & \ddots & e_{n-1} \\ & & &...
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7answers
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Polynomial division: Is this trick obvious?

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+...
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1answer
30 views

In a locally compact 2nd-countable Hausdorff space $E$ there is a sequence of compact subsets $K_n$ with $K_{n-1}⊆\overset∘{K_n}$ and $\bigcup_nK_n=E$

Let $(E,\tau)$ be a locally compact second-countable Hausdorff space. I want to show that there is a $(K_n)_{n\in\mathbb N_0}\subseteq E$ such that $K_n$ is compact and $K_{n-1}\subseteq\overset{\circ}...
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0answers
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Relations between $\Omega_n$'s and $\Omega$ in Weierstrass's theorem

Weierstrass's theorem: Suppose that $f_n(z)$ is analytic in the region $\Omega_n$, and that the sequence $\{f_n(z)\}$ converges to a limit function $f(z)$ in a region $\Omega$, uniformly on every ...
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2answers
17 views

$|z^n|< \epsilon/4$ by taking $n > \log(4/\epsilon) / \log(1/|z|)$

$f_n(z) - z = -2z^{n+1}/(2z^n+1)$. For any given value of $z$ we can make $|z^n|< \epsilon/4$ by taking $n > \log(4/\epsilon) / \log(1/|z|)$. How does taking $n$ greater than that number ...
0
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1answer
61 views

needing help for proofing $\frac{de^x}{dx}=e^x$

could anybody explain why do we proof $\frac{de^x}{dx}=e^x$ in this way "we know $y=e^x=f(x)$ and $(f(y)^-1)' =\frac{1}{f'(x)}$ ,so $\frac{dy}{dx}=\frac{de^x}{dx}=\frac{dln^-1}{dx}=\frac{df(x)^-1}{dx}...
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3answers
35 views

How to find all solutions of $\cos(z) = 0$, where $z\in\mathbb{C}$? [duplicate]

I am stuck on finding all solutions of the equation $\cos(z) = 0$, where $z\in\mathbb{C}$. I found this proof, however, I cannot figure out the logic behind the last few steps. \begin{align} \cos(z) &...
1
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1answer
23 views

Proof of graph theory on dominating sets

Let G be a graph with 3n vertices, with the property that every pair of vertices has codegree at least 1. That is, ∀x∀y∃z such that xz and yz are both edges. Show that G has a dominating set of size n....
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0answers
14 views

Real vector plane with two complex conjugate vectors

I'm reading the proof of this lemma, A vector plane that contains two complex conjugate vector lines, is real. Given proof: Consider two complex conjugate vector lines $\langle w \rangle $ and $\...
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2answers
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Prove: $\inf\big((x,y)\cap \mathbb{Q}\big)=x$ when $x,y\in \mathbb{R}$ and $x<y$

I know that $\mathbb{Q}$ is dense in $\mathbb{R}$. However, I don't know what I am supposed to do. Any impulses?
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1answer
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Question regarding elementary Quotient Algebra Theorem proof

I am having trouble understanding why a proof in my text book proceeds as it does, eventhough the steps involved are quite simple. If possible,I would appreciate some clarifications regarding the ...
0
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0answers
18 views

unique factorisation of non-zero fractional ideals in a Dedekind domain

I'm reading about Dedekind domains from Serre's book, and on Pg. 12, before stating Proposition 7, there are arguments for the proof, which read as follows: If one considers the ideal $a_1 = \...
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0answers
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Doubt in proving sample mean is unbiased estimator of population mean in case of SRSWOR.

I am sharing a proof encountered in William Cochran(1977: page 22): Sampling Techniques.The sample mean $\bar y$ is an unbiased estimate of the population mean $μ$ in a simple random sampling (SRS) ...
0
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1answer
27 views

Definition of smooth functions on a Manifold

Definition. Let $M$ a smooth manifold of dimension $n$. A funciton $f\colon M\to \mathbb{R}$ is said to be smooth at a point $p$ in $M$ if there is a chart $(U,\varphi)$ about $p$ in $M$ such that $f\...
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0answers
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Ideals containing a non-zero element $x$ of a Dedekind domain satisfy the descending chain condition

In Serre's Local Fields, Proposition 6 on Page No. 11, the author claims that the ideals containing $x$, a non-zero element of $A$, which is a Dedekind domain, satisfy the descending chain condition. ...
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Proof in regression model

Im studying for an exam, i don't have the solution, so I hope some of you guys can help me. I have tried a lot but i can't do this proof. Here is the task: Suoppose we have the linear regression ...
2
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2answers
32 views

Proof by induction: $\sum_{j=1}^{n-1}{j^k}<\frac{n^{k+1}}{k+1}$ with $n,k \in \mathbb{N}$ and $n\geq 2$

Proof by induction: $\sum_{j=1}^{n-1}{j^k}<\frac{n^{k+1}}{k+1}$ with $n,k \in \mathbb{N}$ and $n\geq 2$ Inductive step: $$\sum_{j=1}^{n}{j^k}<\frac{(n+1)^{k+1}}{k+1}$$$$\Bigg(\sum_{j=1}^{n-1} j^...
3
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1answer
52 views

heron's formula proof

I have seen an interesting proof of Heron's formula here. It is very simple, but I do not understand one point. The author demands, that the formula should contain factor $(a+b+c)$, because when we ...
0
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3answers
45 views

Why does this square root simplify so much?

I was messing about with a dot product, trying to simplify an expression, when I came across this equality by graphing. Why would these expressions be equal? $\cos(2x)+r^2=\sqrt{r^4+\frac{r^2h^2}{2}\...
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1answer
54 views

$\mathbb{R}\times\mathbb{R}$ $\cong$ $\mathbb{R}^2$ [duplicate]

In my topology class we proved that $\mathbb{R}\times\mathbb{R}$ $\cong$ $\mathbb{R}^2$, when $\mathbb{R}$ and $\mathbb{R}^2$ has the metric topology and $\mathbb{R}\times\mathbb{R}$ has the standard ...
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1answer
62 views

Relationship between tangents and double roots

I am dealing with the proof of the following Theorem, taken from Dale Husemöller's book Elliptic Curves: I have trouble to understand the following underlined section of the proof: Could you please ...
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2answers
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Question on proof: cosets of the alternating group $A_n$

The theorem: If $n \ge 2$, then $|A_n| = \frac{n!}2$. Proof: We will proof that $S_n\backslash A_n$ is the only non-trivial coset of $A_n$ (so aside from $A_n$ itself). Consider an arbitrary, ...
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0answers
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Mean curvature of a fiber in a product manifold

I'm reading this paper and I'm stuck in the lemma $2.1$. There are three points that I can't understand. My first doubt is why the following equality is valid? $$\frac{\partial^2 p^{\alpha}}{\...
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2answers
54 views

Indefinite integral through factorization

So I have $$\int \frac{1}{(x^2+1)^2}dx$$ And the professor does some magic I'm confused. what's with the derivative? I solved the integral via substitution but I'm curious how this works, so I can ...
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3answers
38 views

If we could just use '<' instead of '≤', why are we still using '≤' in many statements?

For example,in " $a\leq b+\epsilon$ if $\forall\epsilon>0$, then $a\leq b$ ",it's impossible to find a case where $a=b+\epsilon$ for very $\epsilon>0$.However,people are still using $\leq$ even ...
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2answers
25 views

Base case for $\sum_{j=1}^{n-1}{j^k}<\frac{n^{k+1}}{k+1}$ with $k\in \mathbb{N}$ and $n\geq 2$

Base case for $\sum_{j=1}^{n-1}{j^k}<\frac{n^{k+1}}{k+1}$ with $k\in \mathbb{N}$ and $n\geq 2$ Do I have to use $k_0=1$ and $n_0=2$? I am a little confused since that's what I can come up with:$$j^...
0
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1answer
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Computing $I=\int_0 ^{2\pi}\frac{dx}{(a+b\cos x)^2}$

Compute the integral using residues: $$I=\int_0 ^{2\pi}\frac{dx}{(a+b\cos x)^2}$$ Resolution(book): $I=2\pi i\sum_{|z_k|<1}res_{z=z_k}f(z)$ where $f(z)=\frac{4z}{i(bz^2+2az+b)^2}\\z_1=\frac{...
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1answer
23 views

Prove a set of vectors $A_i$ perpendicular in pairs is linearly independent

Suppose $A_1, ..., A_r\in F^n$ are non-zero vectors such that $A_i\cdot A_j=0 \space \forall i\neq j$. Let $c_1,...,c_r\in F$ be scalars such that $$\sum_{i=1}^rc_iA_i=0$$ Show that $c_i=0 $ for all $...
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2answers
30 views

Intuition behind lines and points in the projective plane

I've just started learning about projective plane and have trouble understand how to visualize the points and lines in the plane. Specifically, for this lemma below, when its says "all points of $l$ ...
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2answers
27 views

A congruency proof question using circles

This is one of the last questions on a test and I couldn’t get the correct answer. The writing in red is my teacher’s, so my question is how would you prove it with the information my teacher has ...
0
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3answers
44 views

Notation for the product of $n$ rational numbers (proof by induction)

I have a proof I'm trying to solve: For any natural number $n$, the product of $n$ rational numbers is rational. The base case is fairly easy. When $n = 1, 1 =\dfrac{a}{b}.$ $1$ is rational when $a = ...
0
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2answers
25 views

Why is there a one-to-one correspondence between homomorphic images of a group $G$ and normal subgroups of $G$?

I was reading about group theory in Herstein's book (mentioned below) and I came across a couple of propositions that were not clear to me, in the sense that I couldn't quite figure out why they were ...
2
votes
1answer
42 views

Showing for a group $G$ with $a,b,x \in G$ that $ax=bx \implies a=b$

Show that for a group $G$, for $a,b,x \in G, ax=bx \implies a=b$ . I'm not sure why none of the proofs I founds just say something along the lines of: Assume $ax=bx$ holds. Then $axx^{-1}=bxx^{-1}$ $...
0
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1answer
44 views

prove that an equation is bounded by 1

I have an equation that is bounded by 1 and where $s<m$, how can I prove it that it actually bounded by 1. The equation is: $\frac{-m}{s}(1-\frac{s}{m})\ln(1-\frac{s}{m})$ Any help will be ...