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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Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process with volatility $\sigma _t$ (that is, $dX_t = \sigma…
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Intermediate value for derivative, Apostol text

In the textbook, Mathematical Analysis of Apostol, there is the intermediate value theorem for derivative as shown below, Now after the theorem there is the note that this theorem is also true if one or both of the one sided derivatives at the…
Khoa ta
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Great Circles in $SU_{2}$

So I am working on the proof that all great circles in $SU_{2}$ (circles of radius 1) are a coset of a longitude, and I am unsure what a great circle looks like in matrix form. Clearly any point on the 3-sphere takes the form $$\begin{bmatrix}…
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Proofs in linear algebra

I'm pretty awful at proving linear algebra proofs, I just don't understand how you know what to do or where the information comes from. I have some sample questions below of what I mean, I have no idea how I'm supposed to prove them. Any help would…
Emily
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I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where $x$ and $y$ are distinct and $x$ is not among the free variables…
alim
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Proving $\lim_{x\to 1}\frac{2+4x}{3}=2$ using the $\epsilon$ -$\delta$ definition of a limit

I'm looking for a verification of my $\epsilon-\delta$ proof of a limit example, if my proof is not completely mathematically rigorous, please tear it apart. Required to Prove: $$\lim_{x \to 1} \ \ \frac{2+4x}{3} = 2$$ Epsilon-Delta Definition $$…
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How to find number of vertices in a given graph?

I am studying on the graphs where eccentricity of every vertex is same. If $G$ is such graph where eccentricity is $r$ for every vertex and for a vertex $x$ if there exists at least two vertices such that $d(x,y) = d(x,z) =r$, then how to find…
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Logic Behind Epsilon-Delta Proofs (Single-Variable Calculus)

Most of what I am asking is based off this (fairly popular) article I've read here : https://bobobobo.wordpress.com/2008/01/20/how-to-do-epsilon-delta-proofs-1st-year-calculus/, but most lecturers, use this same process to tackle epsilon-delta…
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Integrating over a sequence of sets $A_n$ with $\mu(A_n)\to0$

I am going through the proof of the following. Let $(X,\mu)$ be a measure space and $f\colon X\to\overline{\mathbb R}$ be a measurable function with finite integral. If $A_1,A_2,\dots$ are $\mu$-measurable and $\lim_{n\to\infty}\mu(A_n)=0$, then…
Ryuky
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Using a combinatorial argument

I am having some difficulty with this problem: Use a combinatorial argument to show that $$\binom{m + n}{r} = \binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r - 1} + \dots + \binom{m}{r}\binom{n}{0}$$ My book shows how to derive an identity, but…
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Show that $H_n$ is almost always a non-terminating decimal.

The following theorem and its proof is not so clear at some points: Theorem. $H_n$ is almost always a non-terminating decimal. Proof. We have that $H_1 = 1$, $H_2 = 1.5$ and $H_6 = 2.45$ and of course, since $H_n$ is always a fraction, its decimal…
user200918
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Can somebody explain me this proof on Kasch book (Modules and rings)?

I have a question about one step in the proof of Proposition 13.2.6: If $R_R$ is injective and ${_R}R$ is noetherian then $R_R$ is a cogenerator and $R$ is artinian on both sides. In the proof of this Proposition Kasch deduces that: $Rad(R)$ is…
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Proof that $\|S_N\|_p < \infty $ is equivalent to $\|S_N f - f\|_p \to 0$ as $N \to \infty$

I am having difficulties with the proof of proposition 1.9 in the book "Classical and multilinear harmonic analysis, Vol. 1" by C. Muscalu and W. Schlag. The following statements are equivalent for any $1 \leq p \leq \infty$: $(i)$ for every $f \in…
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Proof of a Four-Pole Tower of Hanoi

Four-Pole Tower of Hanoi: Suppose that the Tower of Hanoi problem has four poles in a row instead of three. Disks can be transferred one by one from one pole to any other pole, but at no time may a larger disk be placed on top of a smaller disk. Let…
Vinicius
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Proof with induction for a Tower of Hanoi with Adjacency Requirement

Tower of Hanoi with Adjacency Requirement: Suppose that in addition to the requirement that they never move a larger disk on top of a smaller one, the person who move the disks of the Tower of Hanoi are also allowed only to move disks one by one…
Vinicius
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