How to remove the smallest term from asymptotic expansion?

It is well-known that $e^{-1/x}sim o(x^n)$ as $xto 0^+$ for any $ninmathbb{N}$, thus if I do an asymptotic expansion for a function, say $f=1/(1-x)+e^{-1/x}$ as $xto 0^+$, I expect to receive an answer like $fsim 1+x + x^2+o(x^2)$. Nevertheless both Asymptotic and Series give me results including $e^{-1/x}$, see

In(1):= Asymptotic(1/(1 – x) + E^(-1/x), {x, 0, 2})
Out(1):= 1 + E^(-1/x) + x + x^2
In(2):= Series(1/(1 – x) + E^(-1/x), {x, 0, 2})
Out(2):= E^-(1/x)+O(x)^3+(1+x+x^2+O(x)^3)

My question is how to remove $e^{-1/x}$ in practice.

Thanks in advance!

data structures – k-th smallest element in sliding segment

Consider an array $a(1ldots n)$ and another array $l = a(0)$ (initial value). At each turn we may add next element to array $l$, or remove first element from array $l$. F.e. after first iteration it could be empty or could become $a(0, 1)$. We want to find k-th smallest element at each iteration in array $l$.

First of all if size of $l$ is less than $k$ the answer is ‘No’. Let’s consider more interesting case.

I’ve decided to have two heaps (one min and one max heap).

Max heap contains all k-th smallest elements from $a(l..r)$ and min heap contains elements which are greater than the k-th smallest element. Then answer is head of max-heap (we can take it in O(1)).

But there is a small problem. What if need to consider $a(l+1 .. r)$ (so we need to push left bound). Now of course if $r – l < k$ the answer is ‘No’, but what should we do otherwise? I thought we should do following: if $a(l) > maxheap(0)$ then the answer doesn’t change (because we will delete element greater than k-th smallest element), but what should we do with our heaps? Unfortunately I can’t delete in heap by position (it takes a long time). The best we can do is delete root node in O(log n). How should I affect them?

What is the smallest time/space complexity class for which no sparse language is hard?

For example, whether there exists $mathsf{PSPACE}$-hard sparse language an open problem, as it is not yet known whether polynomial hierarchy collapses.

But is it a solved problem for larger complexity classes like $mathsf{EXP}$ or $mathsf{PR}$? What is the smallest complexity class (that is larger than $mathsf{PSPACE}$) for which it is a solved problem?

arithmetic – Find the smallest insertable number

Let’s say a number $n$ is insertable if for every digit $d$, if we insert $d$ between any two digits of $n$, then the obtained number if a multiple of $d$.
For example, $144$ is not insertable because $1474$ is not divisible by $7$.

The question is the find the smallest insertable positive integer with at least two digits.

It is relatively easy to see that such a number have to be divisible by $2520$. I also ran a script to check all integers below 75,000,000,000 with no success.

Disclaimer. I do not know if such a number do exist.

ag.algebraic geometry – Generalization of: The dimension of a projective $mathbb{F}$-variety equals the smallest codimension of a disjoint linear subspace

Let $mathbb{F}$ be an algebraically closed field. Consider the following definition of the dimension of a (quasi)projective $mathbb{F}$-variety, given in Harris Algebraic Geometry: A First Course:

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It seems nonstandard to take this as the definition of a variety, so I will think of this as a theorem that should be proven from a more conventional definition of dimension, e.g. the Krull dimension.

My question is this: Does Definition 11.2 generalize to structures other than a variety over an algebraically closed field? For example, what if $mathbb{F}=mathbb{R}$? Or $X$ is a manifold embedded in projective space? In these cases, does the dimension of $X$ still agree with the smallest codimension of a disjoint linear subspace?

Note that Definition 11.2 holds when the “irreducible” assumption is dropped, so this small generalization does hold.

Which screen resolution (Largest or smallest) is best for developer hand-offs?

I am currently designing a web app to cater for the following screen resolutions:

  1. 1920 x 1080
  2. 1600 x 864
  3. 1366 x 768

I’m wondering:

  • Which is the default/primary resolution I should design in for developer hand-offs. (It’s a huge web app and I can’t possibly replicate it in all the resolutions.)
  • What trade-offs am I making, or what precaution should I take if I select 1 resolution over another. (I’m currently inclined to design in 1366×768 since it’s easier for developer to imagine scaling up)