Tag: analysis

calculus – What are the steps in breaking down the exponent in this limit analysis?

I’m trying to understand the reasoning in the following step of a limit analysis:

$$lim_{n to infty} nleft(left(1- frac{1+c}{frac{n}{ln(n)}} right)^{frac{n}{ln (n)}}right)^{(n-1)ln n/n} = lim_{n to infty} ne^{-((1+c)ln(n))}$$

I understand the “inner” part; $lim_{n to infty} (1-frac{1+c}{frac{n}{ln (n)}})^{frac{n}{ln n}} = e^{-(1+c)}.$ And I sort of see that outer exponent $((n-1)ln n) /n = (ln n) – (ln n / n)$ and the second part goes to 0, but it’s not clear to me what rules actually justify “bringing the limit to the exponent”. What are the actual steps involved in deducing this limit?

More generally, these types of asymptotic analysis show up in comp sci all the time and I feel there is a bag of tricks that I am missing.

fa.functional analysis – Decomposition of a function into right-sided and left-sided function

Here I define a distribution $fin D’$ to be right-sided if supp $fsubseteq (0,infty)$ and defnote it by $f_+$ and if the supp $fsubseteq (-infty,0)$ it is called left-sided and denoted by $f_-$.

Now, it is claimed that if $f$ is locally integrable function on $mathbb{R}$, then there is a unique decomposition $f=f_++f_-$ where $f_+$ is right-sided locally integrable function and $f_-$ is left sided locally integrable function.

For an example:

If I have $A(omega)=frac{1}{omega^2+9}$
then I can find a decomposition $A_+(omega)=frac{i}{6(omega+3i)}$ and $A_-(omega)=frac{-i}{6(omega-3i)}$ by inspection.

But, How do I find such a decomposition for function like
$e^{-a x}theta(-x)$ where $theta$ is Heaviside step function? Is there a general process to find such a decomposition?

Complex Analysis: Decomposing function to right-sided and left sides function

Here I define a distribution $fin D’$ to be right-sided if supp $fsubseteq (0,infty)$ and defnote it by $f_+$ and if the supp $fsubseteq (-infty,0)$ it is called left-sided and denoted by $f_-$.

Now, it is claimed that if $f$ is locally inferable function on $mathbb{R}$, then there is a unique decomposition $f=f_++f_-$ where $f_+$ is right-sided locally integrable function and $f_-$ is left sided locally integrable function.

For an example:

If I have $A(omega)=frac{1}{omega^2+9}$
then I can find a decomposition $A_+(omega)=frac{i}{6(omega+3i)}$ and $A_-(omega)=frac{-i}{6(omega+3i)}$ by inspection.

But, How do I find such a decomposition for function like
$e^{-a x}theta(-x)$ where $theta$ is Heaviside step function? Is there a general process to find such a decomposition?

functional analysis – construction of a function and Lebesgue measure

I’ve spent weeks with an interesting problem in my head, the problem would say something like Build, if possible, a continuous and unbounded function f ∈ L1 ((0, ∞)). In case the above is possible, say if it is possible to make the construction so that the exact value of ∫(between 0 and ∞)fdx. Someone dares to solve it

fa.functional analysis – Bochner integral in a Fréchet space

I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $mathcal{P}$ of norms – not just seminorms – and on this space I have a Borel probability measure $nu$. Now, I would like to see whether it is possible to make sense of the integral

begin{equation}
int_V x , mathrm{d} nu left( x right)
end{equation}

within $V$. The measure is such that I actually can prove that the integral exists as a Bochner integral in some Banach space completions of $V$ with respect to some of the norms in $mathcal{P}$. But I am far away from being able to prove this for all of these norms.

Is there perhaps a different way to define Bochner integrals in Fréchet spaces? I only know the usual Banach space setting and it seems to not be enough in this case.

analysis – The Denseness of Q

I am answering the question:

Consider $a, b ∈ R$ where $a < b$. Use Denseness of $mathbb Q$ to show there are infinitely many rationals between $a$ and $b$.

I have chosen to answer this thing using induction.
I know $P_1$ is a true assertion since by the denseness of $mathbb Q$ there exists a rational, $r_1$ such that $a<r_1<b$. I can then assume $P_n$ is true and that there are $n$ distinct rationals between $a$ and $b$ of the form

$a<r_n<r_{n-1}<r_{n-2}<cdots<r_2<r_1<b$

This is where I’m stuck but I know I want to use the denseness of $mathbb Q$ again to say since $a<r_n$, I can find a rational $r_{n+1}$. At the same time, I don’t know what it is about $mathbb Q$ that allows me to say it.

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functional analysis – How is this property equivalent to the Reiter Property?

We have the Reiter Property $(R_2)$ for an action of a group G on a set X:
For any $epsilon>0$, any finite subset $S$ of G, there exists $phiin{ell^2(X)}$ such that $|sphi-phi|_{ell^2}<epsilon{|phi|_{ell^2}}$ for all $sin{S}$.
I am trying to show this is equivalent to the alternative property $(R_2)’$: for any $epsilon>0$, any finite subset $S$ of G, there exists $phiin{ell^2(X)}$ such that $$left|frac{1}{|S|}sum_{sin{S}}{sphi}right|_{ell^2}>(1-epsilon)|phi|_{ell^2}$$
but I am completely stuck.
I have tried using some uniform convexity since $ell^2$ has an inner product, but can only get anything out of it when $|S|=2$, I have heard from someone else that this is related to adjoint operators, so I have tried defining $T:ell^2(X)rightarrowell^2(X)$ by $T(phi)=frac{1}{|S|}sum_{sin{S}}{sphi}$ and can deduce that it has norm 1 and, if we extend S to also contain the inverses of all its elements, is self adjoint, but I can’t see how this could be helpful to solve the problem.

Many thanks.

runtime analysis – How to analyse the worst-case time complexity of this algorithm(a mix of Bubble Sort and Merge Sort)?

Suppose I have a sorting algorithm that sorts a list integers. When the input size(the number of elements) $n$ is odd, it sorts using Bubble Sort and for even $n$ it uses Merge Sort. How do we perform the worst-case time complexity analysis for this algorithm?

The context in which this question came about is when I was going through the analysis of MAX-HEAPIFY algorithm given in CLRS(3rd edition) on page 154. In the worst-case analysis, the author had assumed some arbitrary input size $n$ and then concluded that the worst case occurs when the bottom-most level of the heap is exactly half full. This threw me off since in various texts and articles, $n$ is assumed to be fixed when performing the worst case analysis(and even for best or average cases for that matter) and that the number of elements at the bottom-most level of a heap of $n$ nodes is fixed. In that light, I concocted this algorithm so as to have the worst case dependent on $n$.

My intuition tells me that the worst case time complexity for this algorithm is $mathcal O(n^2)$ since that’s the worst case runtime for Bubble Sort. But I want to know the precise mathematical formulation of the worst-case time complexity analysis for any algorithm. Any insight would be much appreciated.

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