complexity theory – Padding in proof of space hierarchy theorems

Suppose that we consider instead the language
$$ L = { langle M rangle : text{$M$ does not accept $langle M rangle$ in space $f(langle M rangle)$} }. $$
We want to show that $L notin mathsf{SPACE}(o(f(n))$, that is, that if $M$ uses space $o(f(n))$ then $L(M) neq L$. This should be the case since $$langle M rangle in L Leftrightarrow langle M rangle notin L(M).$$
But is this really true? According to the definition of $L$, $langle M rangle in L$ iff $M$ does not accept $langle M rangle$ in space $f(langle M rangle)$. It could be that $langle M rangle in L$ and $M$ accepts $langle M rangle$ using more than $f(langle M rangle)$ space. The latter could actually happen, since we are only guaranteed that $M$ uses space $g(n)$ for some function $g(n) = o(f(n))$, which does not preclude $g(|langle M rangle|) > f(|langle M rangle|)$ at the particular value $|langle M rangle|$.

Adding the padding fixes this issue: it cannot be that $g(|(langle M rangle, 10^k)|) > f(|(langle M rangle, 10^k)|)$ for all $k$, since this would contradict $g(n) = o(f(n))$.

real analysis – Wrong inequality in simple proof Cauchy $implies$ Boundedness.

These notes from Oxford University contain an apparently very simple proof that Cauchy sequences (real or complex) imply boundedness.

I understand the Cauchy condition $|a_m – a_n| < epsilon$, and that the proof assigns $epsilon=1$ as an arbitrary value.

Question: I can’t understand how the following inequality is derived from the triangle inequality:

$$|a_m| leq 1 + |a_N|$$

My Attempt: I have tried using the reverse triangle inequality with no success:

$$|a_m| – |a_N| leq |a_n -a_N| < 1$$

And so,

$$|a_m| < 1 + |a_N|$$

Here the inequality is $<$ and not $leq$ as per the reproduced notes.

For convenience, the proof is reproduced below.

enter image description here

probability theory – Donsker and Varadhan inequality proof without absolute continuity assumption

I’ve been attempting to understand the proof of the Donsker-Varadhan dual form of the Kullback-Liebler divergence, as defined by
operatorname{KL}(mu | lambda)
= begin{cases}
int_X logleft(frac{dmu}{dlambda}right) , dmu, & text{if $mu ll lambda$ and $logleft(frac{dmu}{dlambda}right) in L^1(mu)$,} \
infty, & text{otherwise.}

with Donsker-Varadhan dual form
operatorname{KL}(mu | lambda)
= sup_{Phi in mathcal{C}} left(int_X Phi , dmu – logint_X exp(Phi) , dlambdaright).

Many of the steps in the proof are helpfully outlined here: Reconciling Donsker-Varadhan definition of KL divergence with the “usual” definition, and I can follow along readily.

However, a crucial first step is establishing that (for any function $Phi$)
operatorname{KL}(mu|lambda)ge left{int Phi dmu-logint e^{Phi}dlambdaright},$$

said to be an immediate consequence of Jensen’s inequality. I can prove this easily in the case when $mu ll lambda$ and $lambda ll mu$:

$$ operatorname{KL}(mu|lambda) – int Phi dmu = int left( -logleft(frac{e^{Phi}}{dmu / dlambda}right) right) dmu ge -log int frac{e^{Phi}}{dmu / dlambda} dmu = -logintexp(Phi)dlambda.$$
However, this last step appears to crucially rely on the existence of $dlambda/dmu$ and thus that $lambda ll mu$, which isn’t assumed by the overall theorem. Where I have been able to find proofs of the above in the machine learning literature, this assumption seems to be implicitly made, but I don’t believe it is necessary and it is very restrictive.

My question is: how can we prove ref{ineq} without assuming $lambda ll mu$?

proof writing – If $X$ is compact then prove that $X$ is complete and totally bounded.

I tried to do it in a way different from my textbook:

Let $(X,d)$ be a compact metric space then it is totally bounded.(I have been Ble to prove this).

My doubt lies in the following part that I have tried to prove:

Let $X$ be a compact metric space then $X$ is totally bounded.We choose a cauchy sequence ${x_n}$ in $(X,d)$ .

Let $A$ be a subset of $(X,d)$ such that $A={x_n}$.Then $A$ will be totally bounded too as $A subset (X,d)$.

Let for an $epsilon > 0$ there exist points ${x_1′,cdots x_n’}$ such that $B_d(x_k’,epsilon)$ contains infinitely many points of $A$.Then as ${x_n}$ is a cauchy sequence so we can conlcude that ${x_n}$ converges to $x_k’$. Also as $(X,d)$ is a metric space so ${x_n}$ can converge to only one points and we have shown that $x_k’$ is a limit point of ${x_n}$.

Another proof for interior of ${(x,y) mid y=0} cup {(x,y) mid x>0 text{and} y neq 0}$

Consider the set $$C= A cup B ={(x,y) mid y=0} cup {(x,y) mid x>0 text{and} y neq 0}$$ I want to prove $text{Int } B = text{Int } C$.
My attempt:

Clearly $text{Int } B subset text{Int } C$ as $B subset C$. To prove the converse inclusion, let $(x,y) in text{Int } C$, then there must exists a $r>0$ such that $B_d((x,y), r) subset C = A cup B$.

Actually I want to show that $B_d((x,y), r) cap A = emptyset$, which implies that $B_d((x,y), r) subset B$.

But I can not procced more. Please help me.

functions – Help with Strong Induction proof

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real analysis – Help understanding the Proof of Cauchy’s criterior for uniform convergence of sequence of functions

Before asking the question I would like to say that I filtered it into mathstackexchange before, and that every answer to a similar question didn’t really convinced me, so please don’t just write “go see this answer” and leave. That being said, my question is: i’ve seen two types of proofs for this theorem: the one that uses a limit argument at a certain point, for which I don’t understand why are we allowed to use that, since the limit in question is a pointwise limit, and so one needs to fix X in order to make it work, but then claims that the obtained inequality holds for every X in the set for which the sequence {f_n(X)} is uniformly Cauchy. Why can this be done? To me it feels like it’s not that rigourous. Is there any other epsilon-N proof which I can understand (I saw a couple of them on this site but I couldn’t figure out why they worked, since they have a similar flaw of the proof above: they use an argument that to me is justified only when X is fixed, while the X is clearly not fixed. Or, if they fix it, then they claim that it works for every X on the set for which f_n is uniformly Cauchy)

proof techniques – Mutual Friends in a Network?

I always seem to have trouble finding a formal way to analyze this (be through proofs or whatever).

The problem statement is as such:

If A and B are friends, and B and C are friends, then A and C are friends too.

In a simple network like the following, this makes complete sense:

1 — 2 — 3

We can see that 1 and 2 are friends, and 2 and 3 are friends. It follows from the problem statement that 1 and 3 must be friends too. This is the most generic case for a problem like this.

Where I get confused is in a following network:

1 — 2 — 3 — 4

We can see that 1 and 2 are friends, and 2 and 3 are friends; therefore, 1 and 3 must be friends. Also, since 2 and 3 are friends, and 3 and 4 are friends; therefore, 2 and 4 must also be friends.

Since 1 and 2 are already friends, would it follow from our conclusion of the last sentence (that 2 and 4 are friends) that 1 and 4 must also be friends?

Moving forward into a bigger picture, any group of connected nodes would also all be friends?

What’s the best way to analyze this?

computer science – Pumping Lemma proof for this language: L={a^ib^jc^k∣k=i∗j}

i have troubles to show that this language is not context free with the pumping lemma.

As a word I chose: a^mb^mc^(m^2)

I solved all the cases but one, which is:
“vxy contains b’s and c’s”

I came up with the following approach to show that the resulting word can not be in the language:

p * (p+|v|) != p*p + |y|

this is equal to:

p * |v| != |y|

Because p must >= 1 and |vxy| > 0 and |vxy| <= p the resulting word is not in the language.
Is this assumption correct?

Proof of limit of $A(P_n) – A(p_n) = 0$ for exhaustion method

This is the last question on an exhaustion method problem set.

I need to show
$$ lim_{n rightarrow infty} n(sin(frac{2pi}{n}) – tan(frac{2pi}{n}) ) = 0$$

Without using L’ Hospital’s rule…
Any pointers on where to start?