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Questions tagged [proof-explanation]

For posts seeking explanation or clarification of a specific step in a proof. "Please explain this proof" is off topic (too broad, missing context). Instead, the question must identify precisely which step in the proof requires explanation, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.

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Doubt in a proposition from Fulton's Algebraic Curves, section 5.5 (some criteria for Noether's condition)

I was reading section on Max Noether’s Fundamental Theorem in Fulton's Algebraic Curves and came across the following proposition which gives some sufficient criteria for Noether's condition to hold (...
Ajin Shaji Jose's user avatar
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1 answer
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Differences in meaning for notations $\alpha_i$ versus $\alpha(i)$ and meaning of $\beta(ij)$ for denoting axioms in monomials

The following are partly taken from Malik and Sen's Fundamentals of Abstract Algebra Background First we note that we can reconstruct the monomial $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ from $...
Seth's user avatar
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Proof Clarification: $f$ is convex iff it's convex when restricted to every line intersecting its domain

About this answer: Proof: A function is convex iff it is convex when restricted to any line .. I need clarification about the proof the user @gerw gave. I managed to understand and perform the proof ...
J.N.'s user avatar
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Proof involving integrals, binomial coefficients and Legendre polynomials

I'm currently working on a research paper involving L-moments statistics. This is the first time for me working on that subject. I stumbled upon this article mentioning a very important equality, ...
Thomas SALAÜN's user avatar
1 vote
0 answers
35 views

Why $m \leq n$ in this proof of Steinitz exchange lemma?

I am trying to understand this proof of Steinitz exchange lemma on Wikipedia. Statement: Let $U$ and $W$ be finite subsets of a vector space $V$. If $U$ is a set of linearly independent vectors, and $...
Tomas's user avatar
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Questions on a proof on $p$-constrained groups

Theorem: Let $G$ be a group and $p \in \pi(G)$. Furthermore, suppose that \begin{equation}\label{eq_p-constrained} C_{G/O_{p'}(G)}(O_p(G/{O_{p'}(G)})) \leq O_p(G/{O_{p'}(G)}). \end{equation} If $P$ ...
Stippinator's user avatar
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0 answers
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Trouble understanding the proof of the theorem: If $f\in \mathfrak{a}$, then $m_i\in \mathfrak{a}$ for each $i$,

The following are from Froberg's Introduction to Grobner bases, Malik anad Sen's Fundamentals of Abstract Algebra Background Lemma: Let $I$ be a momomial ideal and $f\in K[x_1,\ldots,x_{n-1},x_n]$. ...
Seth's user avatar
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1 answer
31 views

Proving $\lim_{y\to 0} y^2 \ln|yx^2|=0$ using sequences

I have the function $$f(x, y) = y^2 \ln|yx^2|$$ and I want to prove that $f(x, 0)$ goes to zero but using sequences. SO I thought about this: I choose $b_n = \frac{1}{n}$, and in general any $b_n$ ...
J.N.'s user avatar
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0 answers
23 views

Understanding the Inequality Involving the Probability of Algorithmic Error and Expectation

I am trying to understand the following two inequalities involving a random symmetric matrix $\Theta$ and an algorithm $B$: $ P(B, \Theta) \geq 0.5 \cdot \mathbb{E}\left[\mathbb{I}_{K,L \text{ is not ...
Alan Bakar's user avatar
1 vote
1 answer
46 views

Applying the linearity of $f$ in the proof of Proposition 1.5 in Brezis

This question references the proof that Brezis give that the hyperplane $H = [f = \alpha]$ is closed iff $f$ is continuous. Here, $f$ is a linear functional (not necessarily continuous) that does not ...
Eliot 's user avatar
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0 answers
24 views

Onsager conjecture and properties of Besov Spaces

I am currently studying the result in Peter Constantin 1, Weinan E, and E. S. Titi, Onsager’s Conjecture on the Energy Conservation for Solutions of Euler’s Equation link:https://web.math.princeton....
Radoslav Habarda's user avatar
-3 votes
0 answers
21 views

Lyapunov Inequality [closed]

Lyapunov's Inequality states that: For a random variable $X$ and numbers $0<r<s<\infty$, $E(|X|^r)^{1/r} \leq E(|X|^s)^{1/s}$. I am working through a proof of the central limit theorem using ...
Alex's user avatar
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When $\sup_{x\in\mathcal{X},z\in\mathcal{Z}}|g(z,x)|<t$ how to derive $E_x[f^2(z,x)]-E_x[g^2(z,x)]\geq E_x[\mathbb{1}_{|f(x)|>2t}(f^2(z,x)-g^2(z,x))]$

This is a simplified version of the original question (a step in a proof): $$ \begin{align} &\text{when} \sup_{x \in \mathcal{X}, z \in \mathcal{Z}}\left|\mathbb{E}[c(z ; y) \mid x]-\tilde{h}(z, ...
Sakura Luna's user avatar
3 votes
1 answer
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Surreal numbers by Knuth, the "bad numbers" proof.

This question is about the "Bad numbers" proof (Chapter 4, pages 24-25). This is a proof by contradiction for sets of three numbers, so that: $x \leq y,\ and\ y \leq z,\ then\ x \leq z$ Now,...
Ashesh's user avatar
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-4 votes
1 answer
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proof: If $a$ and $b$ are coprime if and only if $GCD(a,b)=1$ [closed]

In my college textbook, I came across this theorem: My first problem: Then, according to the fundamental theorem of algebra, $a$ and $b$ can be written unambiguously as the product of prime numbers ...
Omar's user avatar
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0 answers
38 views
+50

Self-Organizing maps: why input vectors (x) are dependent on steps (t)?

Based on the paper Essentials of the Self-Organizing maps, I rephrase paragraph 4.1. ->The original, stepwise recursive SOM algorithm: In the mathematical framework $\{\mathbf{x}(t)\}$ represents ...
Nauel's user avatar
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0 answers
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How did Euclid's proof of the isosceles triangle theorem (*pons asinorum*) differ from the High School geometry proof?

Consider the first theorem proved in a High School geometry class. Theorem. The opposite angles of an isosceles triangle are congruent. Proof. Construct an angle bisector through the vertex of the ...
Fomalhaut's user avatar
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0 answers
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Theorem 7.48 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: Lebesgue's Criterion for Riemann Integrability [closed]

Here is Theorem 7.48 (Lebesgue's Criterion for Riemann Integrability) in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition: Let $f$ be defined and ...
Saaqib Mahmood's user avatar
1 vote
1 answer
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Regarding the proof of Primitve element theorem (Theorem 3.3.2) in Murty's Problems in Algebraic Number Theory

I am reading the following proof I understand that we can choose such $\lambda$ so that $\beta$ is only common root of $\phi$ and $g(x)$ but my question is that why the $gcd$ ($\phi$(x),$g(x)$) needs ...
Ritwik Ghosh's user avatar
2 votes
3 answers
56 views

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

I am reading the proof of the following result: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. If $b\in\mathbb{R}$, then $$ V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,...
Shenron's user avatar
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1 answer
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Axler Theorem 5.33: arguing that $U \cap V = \{0\}$. [duplicate]

I'm having trouble understanding a key step in Axler's proof of Theorem 5.33. The theorem states: Let $F = \mathbb{R}$, $V$ a finite-dimensional vector space, and $T: V \to V$ a linear map. Let $b,c \...
Cardinality's user avatar
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2 votes
2 answers
36 views

Explanation of an inequality in proof of nested intervals property

I am having trouble understand part of a proof of Nested intervals property: Let $F$ be an ordered field with the monotone sequences property. Let $I_1\supseteq I_2 \supseteq \cdots$ be closed bounded ...
Tomas's user avatar
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5 votes
1 answer
75 views

$C[a,b]$ is dense in $L^p([a,b])$ from the fact that $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$

We know that the following theorem holds. Theorem. The space $C_c(\mathbb{R}^n)$ of the continuous functions with compact support is dense in $L^p(\mathbb{R}^n)$ for $p\in [1,\infty).$ I don't ...
Jack J.'s user avatar
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1 vote
2 answers
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Question to a solution to an invariance problem from Arthur Engel's book

I have a question with the answer Engel provides to the following problem: Three integers $a, b, c$ are written on a blackboard. Then one of the integers is erased and replaced by the sum of the other ...
H4z3's user avatar
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0 votes
0 answers
44 views

Question about the use of Green's theorem in Do Carmo's proof of the local Gauss-Bonnet theorem.

This is Do Carmo's statement and proof of the local Gauss-Bonnet theorem in "Differential Geometry of Curves and Surfaces": My question is: how can he assert that $$\sum_{i=0}^k \int_{s_i}^{...
Trisztan's user avatar
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5 votes
0 answers
124 views

Are creation and annihilation operators special?

In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written $$O=\sum_{...
Lourenco Entrudo's user avatar
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0 answers
58 views

Defining the completion of a group can be done only using Cauchy sequences

Let $G$ be a group, in Atiyah & MacDonald's Commutative Algebra it says that Assume for simplicity that $0\in G$ has a countable fundamental system of neighborhoods. The completion $\hat G$ of $G$...
ephe's user avatar
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1 vote
0 answers
36 views

Proof of Thompsons $A \times B$-lemma

(Auxiliary lemma) Let $G$ be a $\pi$-group and $a$ a $\pi'$-element acting on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = 1$. Hey guys, I am having a ...
Stippinator's user avatar
1 vote
2 answers
48 views

A question in one step of proving Van Der Corput's Difference Theorem

In proving Van Der Corput's Difference Theorem, there's a lemma: Suppose $\left\{ z_{n} \right\}$ is a bounded complex-valued sequence. Then if for all $d \in \mathbb{N}$ , we have \begin{align*} \lim\...
M_k's user avatar
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1 vote
1 answer
83 views

Theorem 3.6.4 (Separable space) Kreyszig.

Kreyszig , in "Introductory Functional Analysis with Applications" , has this Theorem 3.6.4 (with Proof) concerning Separable spaces. Theorem. Let $H$ be a Hilbert space. If $H$ contains an ...
Jack J.'s user avatar
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1 vote
2 answers
107 views

Proposition 8 Corollary 1, Section 5.7 of Hungerford’s Algebra

Corollary 1.9. Let $E$ and $F$ each be extension fields of $K$ and let $u\in E$ and $v\in F$ be algebraic over $K$. Then $u$ and $v$ are roots of the same irreducible polynomial $f \in K[x]$ if and ...
user264745's user avatar
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0 votes
1 answer
56 views

Help with the proof of this statement

Let $ f : [a, b] \rightarrow \mathbb{R} $ be a bounded function. Then for every $\epsilon > 0$, there exists $\delta > 0$such that for any partition $D$ of the interval $[a, b]$ with norm $\nu(D)...
Binky McSquigglebottom's user avatar
2 votes
1 answer
39 views

Axler Theorem 5.33: Understanding assumption WLOG

Theorem 5.33 in Axler's book is ($\mathcal{L}(V)$ denotes the set of linear map $V \to V$): Suppose $\mathbf{F} = \mathbf{R}$ and $V$ is finite-dimensional. Suppose also $T \in \mathcal{L}(V)$ and $b,...
Cardinality's user avatar
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0 votes
0 answers
27 views

Projection into a Hilbert space with respect to an orthonormal sequence.

Let $H$ be an Hilbert space and let $(e_k)_{k\in\mathbb{N}}$ be an orthonormal sequence in $H$. We define $$V:=\overline{\text{span}(e_k)}$$ I must prove that for all $x\in H$ the projection onto $V$ ...
FoxMath's user avatar
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0 answers
65 views

Proof by contradiction, where is the contradiction?

Example. Let $F=\{E_1,E_2,\ldots,E_s\}$ be a family of subsets with $r$ elements of some set $X$. Show that if the intersection of any $r+1$ (not necessarily distinct) sets in $F$ is nonempty, then ...
Shoe__gazer's user avatar
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0 answers
39 views

Understanding the proof of Perron–Frobenius theorem

I reviewed the proof for Perron–Frobenius' theorem as stated in this article. In the proof, they define Q to be a positive orthant ( $Q:= \{ x \in \mathbb{R}^n: x \geq 0, x \neq 0 \}$) For an ...
malaiyur-mambattiyan's user avatar
1 vote
1 answer
64 views

Using contradiction to validate an argument

In general, when we want to establish the validity of the argument $(p_1 \land p_2 \land ... p_n ) \rightarrow q$, we can establish the validity of the logically equivalent argument $(p_1 \land p_2 \...
user avatar
2 votes
0 answers
53 views

Proof explanation: how to see that $(u_n)$ is bounded above by $v_0$

I am new to this and I am trying understand this proof of Monotone-sequences property $⇒$ least upper bound property. Let $A$ be a non-empty set that's bounded above. Pick $u_0, v_0$ such that $u_0$ ...
Tomas's user avatar
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0 answers
23 views

question equality with norms in real vector space

Suppose $x,\bar{x}$ and $y$ are elements of $\mathbb{R}^n$. A is a set of vectors of $\mathbb{R}^n$. In my textbook I found the following reasoning: $\sum_{x \in A}||x-y||^2 - \sum_{x \in A}||x - \bar{...
user33's user avatar
  • 141
1 vote
0 answers
27 views

Centered Subgaussian Variables have better Properties

I am trying to understand the following proof: Main Confusion: In particular, I am having a very hard time understanding the chain of inequalities in the proof for (3)': I think the first equality is ...
Partial T's user avatar
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0 votes
1 answer
17 views

Understanding the proof for Properties of Subgaussian Variables

Here are the definitions, statements and the proof that I am stuck on: I am stuck on the last part of the proof where the author claims that setting $C = e$ automatically guaranties that (1) holds ...
Partial T's user avatar
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2 votes
2 answers
89 views

Why this simple volume problem in Multivariate Calculus seems to have an anomaly?

Find the volume of the solid inside the cylinder $x^2+y^2-2ay = 0$ and between the plane $z = 0$ and the cone $x^2+y^2 = z^2$. I tried solving this problem as follows: Equation of the cylinder $x^2+(y-...
Thomas Finley's user avatar
4 votes
2 answers
99 views

Trying to prove monotone-sequences property ⇒ Archimedean property

Monotone-sequences property ⇒ Archimedean property Today I just started learning this and having trouble understanding parts of the proof. Sorry if these are easy to some, I just couldn't find this ...
Tomas's user avatar
  • 95
6 votes
3 answers
186 views

Prove that any sequence of five distinct integers must contain a 3-chain

This task is from MIT OpenCourse Mathematics for CS 2010 course, problem set 2, exercise 1(d). I am aware that this question has already been asked several times previously on this platform. Yet, the ...
Lina's user avatar
  • 86
0 votes
1 answer
44 views

For a square matrix $A$, if $\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$ * q ($\lambda$)

For a square matrix $A$, if $\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$ * q ($\lambda$) for some polynomial q. My approach: So we know that A has an ...
brodar's user avatar
  • 157
3 votes
0 answers
37 views

Norm surjective for unramifeid extension of local fields $L/K$

I have a question about the argument used here in proof of Proposition 4.2.5.: For any finite unramified extension $L/K$ of local fields, the map $\text{Norm}_{L/K}: O_L^* \to O_K^*$ is surjective. ...
user267839's user avatar
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0 votes
2 answers
42 views

Axler Theorem 5.18: $\text{null}(T)$ and $\text{range}(T)$ are invariant under $T$.

I am trying to understood Axler's proof of Theorem 5.18. It states that: if $T$ is a linear operator from $V$ to $V$ and $p$ is a polynomial with coefficients in the field $F$, then $\text{null} p(T)$ ...
Cardinality's user avatar
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0 votes
1 answer
90 views

Proposition 12, Section 5.6 of Hungerford’s Algebra

Let $F$ be a finite dimensional extension field of $K$ and $N$ a normal extension field of $K$ containing $F$. The number of distinct $K$-monomorphisms $F\to N$ is precisely $[F : K]_s$, the separable ...
user264745's user avatar
  • 4,239
1 vote
0 answers
38 views

Axler Theorem 5.17, part (b)

I am trying to understand the proof of part (b) of Theorem 5.17 in Axler's Linear Algebra Done Right. He cites part (a) in his proof of (b), so I've written out the full theorem statement below. $\...
Cardinality's user avatar
  • 1,243
0 votes
0 answers
20 views

Proof that the extremum of a function is a critical point [duplicate]

I'm following a proof given by Spivak in his textbook "Calculus" and one of the steps seems slightly unjustified to me. Firstly, here's the theorem: If $f$ is a function defined on $(a,b)$ ...
Aryaan's user avatar
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