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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Euler product of $L(s, \chi)$

Eq. 10.20 in the book Multiplicative number theory I: Classical theory by Hugh Montgomery, Robert C. Vaughan states that $$L(s, \chi) = L(s, \chi^*) \Pi_{p|q}(1- \dfrac{\chi^*(p)}{p^s}),$$ where χ is ...
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Picking $\varepsilon$ to prove open interval in $\mathbb{R}$ is an open set in $(\mathbb{R} , \mathcal{T})$ [closed]

In Abbott's Understanding Analysis, example $3.2.2$, he gives a short proof that open interval $(c,d)$ in $\mathbb{R}$ is an open set in $(\mathbb{R} , \mathcal{T})$ where $\mathcal{T}$ is the usual ...
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Main idea of the proof of the Galois solvability criterion

For an oral exam I want to learn the proof of the Galois solvability criterion (Galois 1831). I have already seen the proof online, I tried to understand it but it is incredibly long. Could someone ...
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Confused by this proof in Jech's set theory

In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) ...
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Regularity of weak solutions with $C^2$ boundary - Evans

I'm reading the proof of Theorem 4 from section 6.3 of Evans - Partial Differential Equations. I'm looking for an explanation about the following claim Previously, I already proved that $u_k\in H^1(V)...
matdlara's user avatar
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Why is this differential injective? Lee Smooth Manifolds Proposition 5.3

I'm trying to understand the following line in the proof to Proposition 5.3 in john M. Lee's 'Introduction to Smooth Manifolds': "Because the projection $\pi_M : M \times N \rightarrow M$ ...
Keshav Balwant Deoskar's user avatar
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Logic behind contrapositive proofs that involves De Morgan's Laws

Suppose $a,b\in\mathbb{Z}$. If both $ab$ and $a+b$ are even, then both $a$ and $b$ are even Proof by contrapositive. Propositions: $P$: $ab$ is even $Q$: $a+b$ is even $R$: $a$ is even $S$: $b$ is ...
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Why we have to proof both $Q$ and $R$ in $P\implies (Q\lor R)$

I'm studying proofs trying to use logic before starting with the proof. A direct proof can be written as $P\implies Q$, by forcing $P$ to be true, we have to force $Q$ to be true so the statement ...
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Change of coordinates in vector fields on Heisenberg group

In the book Geometric Analysis on the Heisenberg Group and Its Generalizations proposition 1.3 says: Under the change of coordinates $$ y_1 = x_1 , \, y_2 = x_2, \, \tau = 4t -2x_1x_2,$$ the vector ...
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Show that $\dfrac{d^{m-1}}{ds^{m-1}} \Big(\dfrac{\zeta'(s)}{\zeta(s)} \Big) =- \sum_{|\gamma -t|<1} \dfrac{1}{(s-\rho)^m} + \mathcal{O} (\log t)$.

Eq. 2.1 in Levinson's paper states $$\dfrac{1}{(s-1)^m} - \sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m} - \sum_{\rho} \dfrac{1}{(s-\rho)^m} = - \sum_{|\gamma -t|<1} \dfrac{1}{(s-\rho)^m} + \mathcal{O} (\...
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Calculating $\dfrac{d^{m-1}}{ds^{m-1}} \Big(\dfrac{\zeta'(s)}{\zeta(s)} \Big)$

Eq. 1.4 in Levinson's paper states $$\dfrac{(-1)^m}{m!} \dfrac{d^{m-1}}{ds^{m-1}} \Big(\dfrac{\zeta'(s)}{\zeta(s)} \Big) = \dfrac{1}{(s-1)^m} - \sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m} - \sum_{\rho} \...
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Why is it that: "If polynomials have integer coefficients, the roots of those polynomials will be a divisor (factor) of the constant term"?

So, in my textbook, I came across a theorem/axiom (?) that states: Given a polynomial $P(x)$ with integer coefficients, its roots, if they exist, are divisors of the independent/constant term of the ...
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Doubt in proof of Prop 2.2.2 in Kesavan's Functional Analysis

Given a normed linear space $V$ (with norm $\lVert \rVert_{V}$) and a closed subspace $W$, Kesavan defines a norm $\lVert \rVert_{V/W}$ on the quotient space $V/W$ as $\lVert x+W \rVert_{V/W} = \inf_{...
Ajin Shaji Jose's user avatar
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Theorem 3.18 from the book by D. S. Jones, The theory of generalised functions

I'm reading D.S.Jones' book, The theory of generalized functions, and in particular I'm studying theorem 3.18 on page 84, in the proof of which I don't understand a statement he makes. Before ...
Nameless's user avatar
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Understanding the proof of Theorem 10.1 in Montgomery & Vaughan's Multiplicative Number Theory

In the last step of the proof of Theorem 10.1 in the book Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan I couldn't understand what exactly "turn the ...
Ali's user avatar
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Let $A\subseteq \Bbb R^n$, then $(\overline{A^c})^c=\mathring A$

Let $A\subseteq \Bbb R^n$, then: $A^{c\bar{}c}=\mathring A$ Notation: As an example $(\bar{A^c})^c=A^{c\bar{}c}$ Dem: We know that $A\subseteq \bar A$, now if we take the complement on both sides: $A^{...
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How to prove the implication $(\neg q \rightarrow (\neg p \lor \neg q))$?

I am working on a proof for the statement: Suppose $x, y \in \mathbb{R}$. If $5|x$ and $5|y$, then $5|xy$ Let's denote the propositions as follows: $P: 5|x$ $R: 5|y$ $Q: 5|xy$ Logically, this is ...
Alexis SM's user avatar
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Use of intermediate value theorem to show that $p = (\cos 2\pi x, \sin 2 \pi x)$ is a covering map for $S^1$.

I am not completely following the argument given in Munkres' topology to show that $p = (\cos 2\pi x, \sin 2 \pi x)$ is a covering map for $S^1$ is a covering map for $S^1$. Here is the outline of the ...
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Step in proof for uniqueness of invariant distribution in continuous-time Markov chain

For an irreducible continuous-time Markov chain, the invariant distribution is unique if it exists. If for some $t_0 > 0$ there is a probability $\boldsymbol{\pi} = (\pi(i))_{i \in S}$ such that $$ ...
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Extending a consistent set of sentences to a complete, witnessing, consistent set of sentences

I'm attempting to unpack the following paragraph from a proof of the Model Existence Theorem (p.44 on this set of notes): Suppose we have a consistent $S$ in the language $L = L(Ω, Π)$. Extend $S$ to ...
Sam's user avatar
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Is there any way to tackle such questions? [closed]

ISI-2023-UG(B) Question (a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta=X Y+Y Z+Z X$ and $\...
SquïdÆir's user avatar
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"Converse" to Chinese Remainder Theorem

There are lots of posts on MSE and the web titled "converse to CRT" but this is not the same. The following is from "Multiplicative number theory I: Classical theory" by Hugh L. ...
Ali's user avatar
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Possible issue with proposed solution to finding the Taylor Polynomial of $f(x)=\frac{e^x-1}{x}$ (a problem in Spivak's Calculus Chapter 20)

I wanted to confirm whether or not I am correctly identifying a mistake in the author-provided solution to the following problem (located within Spivak's Calculus 4th Ed.): Let $f$ be defined by: $\...
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A question about the proof that differentiability implies continuity

When proving that a function f differentiable on an interval I⊆R is necessarily also continuous on that interval, we use the fact that (as x→c)lim(f(x)-f(c)) is equal to zero, but why are we allowed ...
Rinchichi's user avatar
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Understanding the proof of Theorem 9.4 in Montgomery & Vaughan's Multiplicative Number Theory

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: I understand everything in the proof except for: How existence of $c$ such ...
Ali's user avatar
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Understanding the proof of Lemma 9.3 in Montgomery & Vaughan's Multiplicative Number Theory

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: I understand everything in the proof except for: How $m' \equiv n'$ (mod $...
Ali's user avatar
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Trying to write a Taylor series for a matrix exponential using Euler's formula

From my lecture notes I have that: The Lie algebra is introduced as a set of matrices $\theta$ that gave rise to elements $M$ of the corresponding matrix group through $M=e^{i\theta}$. However, just ...
digital's user avatar
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proof of $\pi_1(S^n)=0$ if $n\geq2$ [closed]

I am trying to understand the hatcher's proof of $\pi_1(S^n)=0$ if $n\geq2$. but it is challenging for me to understand each step. We can express $S^n$ as the union of two open sets $A_1$ and $A_2$ ...
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On the probleme des menages solution on Titu Andreescu book

The probleme des menages consists of the following: How many ways can $n$ married couples si at a round table in such a way that there is one man between every two women and no man is seated next to ...
H4z3's user avatar
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Prove there is no rational number q such that q^2 = 2. [duplicate]

I am given the following hint: Express $ q $ as a quotient of integers $ m/n $ where $ m,n $ are mutually prime, and show that $ m^2 = 2n^2 $ leads to a contradiction. Proof solution: $2$ divides the ...
nico's user avatar
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Why does a holomorphic degree $1$ map imply that the Jacobian is strictly positive?

I am currently reading "Lectures on Harmonic Maps" by R. Schoen and S. T. Yau and have problems understanding one step in the proof of a Corollary on p. 13. Corollary: Suppose $N = S^2$, ...
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Prove uniqueness and existence of $\theta$ such that $\theta\left(x,0\right)=x$ and $\theta\left(x,y^{\prime}\right)=\theta\left(x,y\right)^{\prime}.$

The domain of consideration is the set of whole numbers, $\mathbb{N}_0$. The following theorem (see facsimile below) appears in The Number System, by H.A. Thurston: 3. Theorem: There is just one ...
Steven Thomas Hatton's user avatar
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Clarification of the proof of the form $\partial_{1,0}$ in Restrictions on harmonic maps of surfaces by J. Eells and J. C. Wood

I am currently reading "Restrictions on harmonic maps of surfaces" by J. Eells and J. C. Wood and have trouble understanding the proof of the lemma on p. 265. The paper states: Let $X$ and $...
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Where does 2.3.5 come from and how to prove it?

I was looking over this proof in Atkinson’s Numerical Analysis 2nd Edition and I am having a hard time seeing where 2.3.5 is coming from. What is Atkinson trying to show with 2.3.5 and the line below? ...
Dr. J's user avatar
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On a step on a complex inequality with summation.

To make it short I have a doubt in the last step of a complex number inequality, the problem is the next one Let complex numbers $z_1,z_2,z_3,...,z_n$ all modulus $1$ and $z_1+z_2+z_3+...+z_{2012}=0$ ...
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A tournament is acyclic if and only if it has no triangles

A tournament is a directed graph where between any two distinct vertices there is either the edge (u,v) or the edge (v,u) (one of them only). I have not come across a proper explanation on why the ...
Yavuz Bozkurt's user avatar
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Corollary 1.17 in Montgomery & Vaughan's Multiplicative Number Theory

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: The proof is very much ambiguous to me. For example: 1- The claim $\int_1^...
Ali's user avatar
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For $a,b>0$: If $a^2<b^2$, then $a<b$.

I'm trying to answer the following problem: (a) For $a,b>0$: If $a^2<b^2$, then $a<b$. I found a different answer but I'd like to understand The answer/hints in the back: The corollary 1.3 ...
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Connection between variation of a function and its weighted integral

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: My question is that how Eq. D.10 is derived from the previous one? (The ...
Ali's user avatar
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Evaluating $|E_K(x)| = s(x) - \sum_{k=1}^K \dfrac{\sin 2 \pi k x}{\pi k}$ on $0<x \le 1/2$

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: I could understand the full proof but the last paragraph, i.e. to show ...
Ali's user avatar
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Show that $\lim_{K \to \infty} \sum_{k=-K}^{K} \hat{f}(k) e(k \alpha) = \dfrac{f(\alpha^+)+f(\alpha^-)}{2}.$

I am studying Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan. The following is the beginning of Appendix D: I could not understand the last sentence (the ...
Ali's user avatar
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Proving $e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu}+\left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2}$ [duplicate]

I'm trying to show that $$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$ for an absolute constant $C &...
stoic-santiago's user avatar
4 votes
1 answer
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Problems differentiating four-vectors to find the equation of motion using Euler-Lagrange equations.

This post directly follows this question and is very similar in nature: Consider the following Lagrangian: $$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\...
Sirius Black's user avatar
1 vote
1 answer
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The last part of Step 5 in the proof of change of variables theorem (Lemma 19.1) in "Analysis on Manifolds" by James R. Munkres.

I am reading "Analysis on Manifolds" by James R. Munkres. I cannot understand the last part of Step 5 in the proof of change of variables theorem (Lemma 19.1) https://archive.org/details/...
佐武五郎's user avatar
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How $|S(x_n, \xi_n) - S(x'_n, \xi'_n)|$ being arbitrary small implies existence of Stieltjes-Riemann Integral?

I am studying Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan (E-book). In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous ...
Ali's user avatar
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Infinite statements from finite axioms

I want to know if a given finite subset of axioms of PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite ...
jason's user avatar
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is arithmetic finitely consistent? [duplicate]

Let's take PA1( First order axioms of peano arithmetic ) for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest ...
jason's user avatar
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-1 votes
1 answer
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In writing a proof how are we certain whether some part of a goal in the proof can be ignored or not? [duplicate]

I ask this question because I probably meet the same obstacle when writing other proofs. In writing a proof My example would be $(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$ The goal is $(0\le\sqrt{...
Stats Cruncher's user avatar
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1 answer
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Proof that if $card\,A \le card\,B$ and $card\,B \le card\,A$, then $card\,A = card\,B$

The question defines the '$\le$' operation on the cardinality of arbitrary sets A, and B as follows: Let $\mathcal F$ be defined as the family of all injections from subsets of A onto B. Since $\...
Vector's user avatar
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On the universal property of Tensor Product

I was readying the construction of the Tensor Product that was made on the book Introduction to Smooth Manifolds, from John M. Lee. In the proposition 12.7 he proved that his construction satisfied ...
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