## real analysis – Show that a subsequence is a subset of the original sequence

My main question is the 4th point, but I hope you can clarify some things for me along the way.

The definition of a sequence says that a function $$a: mathbb {N} to S$$ is a sequence on a set $$S$$, denoted $$(a_n)$$.

1. Can i freely restrict the area of ​​function $$a$$ and still call it a sequence? In particular, a) is it valid to define $$a_n$$ on a finite subset of $$mathbb {N}$$ b) on an infinite subset of $$mathbb {N}$$?

Let's say that at each term in the sequence $$(a_n) _ {n in mathbb {N}}$$, I have to define a new sequence from there. I first define a sequence $$(m_k)$$ who maps $${k in mathbb {N}: k geq n } to mathbb {N}, forall n in mathbb {N}$$ with $$m_k (assuming affirmative on question 1b because the domain is an infinite subset of $$mathbb {N}$$). Then, like $$(a_ {m_k})$$ is the composition of the sequence $$(a_n)$$ and the increasing sequence $$(m_k)$$, by definition $$(a_ {m_k})$$ is a subsequence of $$(a_n)$$.

1. Is it a good way to show that these new sequences $$(a_ {m_k})$$ are subsequences?

A set of points defined for the sequence $$(a_n)$$ East $$left {a_n: n in mathbb {N} right }$$.

1. How do you define a set of points for a subsequence $$(a_ {m_k})$$? East $$forall n in mathbb {N}, left {a_ {m_k}: k geq n right }$$ well?

Assuming yes on question 3, the main question is:

1. How to prove this $$forall n in mathbb {N}, left {a_ {m_k}: k geq n right } subseteq left {a_n: n in mathbb {N} right }$$? In other words, that the set of points defined for a subsequence is a subset of the set of points in the original sequence. I think i should take a term off $$left {a_ {m_k}: k geq n right }$$ and deduce that its also in the whole $$left {a_n: n in mathbb {N} right }$$, but I don't know how to do it rigorously.

To give you context on my questions, I want to show that $$sup { left {a_n: n in mathbb {N} right }} geq sup { left {a_ {k}: k geq n right }}$$. Given that $$(a_n)$$ is bounded, defines $$left {a_n: n in mathbb {N} right }$$ and $$left {a_ {k}: k geq n right }$$ are delimited. Having $$left {a_ {k}: k geq n right } subseteq left {a_n: n in mathbb {N} right }$$ I would prove the supremums as in the question Prove the supremum of a subset is smaller than the supremum of the whole.

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## Longest word in the dictionary that is a subsequence of a given string. Javascript

https://jsfiddle.net/dhtmlkitchen/fymuc3ve/62/
const sampleString = "infatuated";

``````const dictionary = Object.freeze((
"fate",
"effete"
));
``````
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## data structures – Check if each prefix sum of the subsequence is \$> = 0 \$ and the sum is \$ = 0 \$

Consider the sequences whose elements are $$-1.0.1$$.
Subequence $$A (i … j)$$ East $$good$$ if sum of its elements $$= 0$$.
Example: for sequence $$1.10, -1, -1.1$$ subsequence $$1.0, -1, -1.1$$ East $$good$$.

The consequence $$A (i … j)$$ East $$supergood$$ if it is good and each sum of its prefix is $$> = 0$$
Example: for sequence $$1.10, -1, -1.1$$ subsequence $$1.0, -1, -1.1$$ is not $$supergood$$ But $$1.0, -1$$ East $$supergood$$.

Now, I want to have a dynamic data structure that allows Mi to effectively perform these operations:

• Insert (S, x, i) – inserts $$x$$ in $$S$$ sure $$i$$& # 39; e position
• Delete (S, i) – Delete $$i$$& # 39; e element
• isSuperGood (S, i, j) – checks if the subsequence $$i$$, $$j$$ is super good

The solution could be an AVL tree with a sum of elements in the left and right subtrees. It is easy to update and allows us to check if the subsequence is good in O (log (n)):

1. Find the node i (say v) O (log (n))
2. Find the node j (say u) O (log (n))
3. Check if v.val + vl.lsum – u.lsum + u.val == 0 O (1)

But if it is a question of checking the super good condition, I do not see how.

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## Longest palindromic subsequence / substring and dynamic programming

The longest palindromic subsequence the problem can be solved using dynamic programming as it is recursive and has overlapping sub-problems, as described in https://www.geeksforgeeks.org/longest-palindromic-subsequence-dp-12/.

On the other hand, AFAIK there is no dynamic programming solution for the longest palindromic substring problem. The DP solution presented at https://www.geeksforgeeks.org/longest-palindrome-substring-set-1/ does not use DP for the original LP substring problem, but for the substring -problem of finding if a substring is palindromic. The global solution consists in comparing the length of all the palindromic substrings, just like in the naive solution; the only difference is that DP is used to speed up the search process if each substring is palindromic, which reduces the time complexity from O (n ^ 3) to O (n ^ 2).

Why is there a DP solution for the LP subsequence problem, but not for the substring problem? Is it because the former can be expressed by recursion and the latter cannot?

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## Dynamic Programming – Find the "Best" Sub-Sequence The Longest Common

I write a program that calculates and displays the diffs. I have implemented the Meyers algorithm which calculates the LCS between 2 subsequences (`seq1` and `seq2`) its output is one of the possible LCS and a partition of `seq1` and `seq2`whose projection is `lcs`.

I want to improve it so that the displayed LCS minimizes the number of breaks; to do this, I implemented a function `f(lcs, seq)`:

• `seq` is a sequence of characters
• `lcs` is a subsequence of `seq`
• the output is a partition of `seq` `p0, p1, p2, ... pn` such as
• no more `p0 + p2 +` or `p1 + p3 + ...` is `lcs`
• and `n` is minimal

I did it using some kind of BFS, at every step, finding the next item not covered in `lcs` in `seq` and eagerly enlarge the common part.

The resulting algorithm is quite slow: on a typical input, about 3 times slower than the Myers algorithm, which seems to calculate something much more complex. See the code here: https://github.com/mookid/diffr/commit/best-lcs/diffr-lib/src/best_projection.rs

Am I missing a better algorithm?

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## c ++ – Why does this longer-escalation sub-sequence problem not work properly?

I have defined recursively to work case by case. I'm getting wrong answers using large enough test data, but it works well for smaller cases.

``````  #include
#include
#include
int LIS(100000);
void LS (int *arr , int n)
{
if (n == 0)
{
LIS(0) = 1;
return;
}
if (LIS(n))
{
return;
}
int i = 0;
int max = 0;
while (i < n)
{
if (arr(i) < arr(n))
{
LS(arr,i);
if (LIS(i) + 1 > max)
{
max = 1 + LIS(i);
}
}
++i;
}
LIS(n) = max;

}
int main()
{
int n;
std::cin >> n;
int arr(n);
for(int i = 0 ; i < n ; ++i) std::cin >> arr(i);
LS(arr,n-1);
std::sort (LIS , LIS+n);
std::cout << "n" << LIS(n-1) << "n";
}
``````

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## calculation – Let a_n be a sequence and assume that sup (a_n) = ∞. Prove that there is a subsequence b_k = a_nk such that b_k tends to.

Can someone help me?
I do not know where to start even these questions

Let a be a sequence and suppose that there exists a subsequence bk = ank such that bk tends to ∞. Prove that sup (an) =.

Let a be a sequence and suppose that supan =. Prove that there is a subsequence bk = ank such that bk tends to

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## real analysis – Convergent subsequence of \$ sin (n ^ 2) \$

What can we say about convergent subsequences of $$sin (n ^ 2)$$ whose existence is guaranteed by the Bolzano-Weierstrass Theorem?

Can we, as a corollary, affirm that for all $$epsilon gt 0$$ it exists $$m, n in mathrm {N}$$ such as $$| sin (n) – sin (m) | lt epsilon$$ ?

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## Grammar without context of the concatenation of an S string and an inverted S subsequence

Instead of trying to remove some symbols from $$x$$ in order to obtain $$y$$, let's try to add some symbols to $$y$$ so that we will get a "supersquence" of $$y ^ R$$, who will be $$x$$. Deletion is difficult to achieve because the activity of context-free grammar is to generate.

Starting from $$#$$, we will either add the same symbol on both sides, or add a symbol only on the left side. Hence the following simple grammar.

$$S to # mid aSa mid bSb mid aS mid bS$$

Exercise. find a grammar without context that generates the language:

$$L_ {a, b} = {xy mid x text {with some} a text {is identical to} y text {with some} b text {inserted. } x, y in {a, b } ^ * }$$.

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## algorithms – maximum GCD per pair of an even subsequence

I'm trying to solve a particular problem, where I've given an array of integers and an integer X and I have to choose a subsequence of the array whose gcd pair is maximal given that the Sub-sequence size should not exceed 2 * X.

So, for example, my table is (5, 7, 10, 8, 3, 4) and X is 2:
the answer would be 9 since GCD (5, 10) + GCD (8, 4) = 9

The only approach I could find was this: at each step, I will find the maximum calculation of the pairwise value of the numbers, then remove the 2 numbers from the array.
Let's continue to add it to a sum (that I have to go out), under the constraints that I do not take more than 2 * X numbers.

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