Probability theory – Complexity / hardness of a generalization of a problem of inclusion / exclusion

For each element of $ S $, count the number of times that he is a member of $ A_i $, $ 1 the i the n $. If this number is $ m $we add 1 to the count. The final count is what we wanted.

The algorithm above takes $ O (tn) $ time if the calculation cost to check the membership is $ O (1) $. It takes $ O (ttn) $ time if the cost in calculation to check the membership is proportional to the size of the set, which is at most $ t $.

So, you can see that the problem is in $ P $. This has nothing to do with this general principle of inclusion – exclusion.

Theory of types – How to reverse a stream using corecursion

Here is the definition of the codata stream:

Stream codata where
hd: Stream -> A
tl: Stream -> Stream

To simplify, I guess I only have one bit stream. Now I would like to define a function flip: Stream -> Stream , which reverses the order of flow elements.

To define: hd (flip S) = head (S) is quite simple.

However, I do not see how we can define tl (flip S)) correctly. Does anyone have an idea, if that is possible and if so, how?

graph theory – monotone Fotler vector

Let $ u_0 geq u_1 geq cdots geq u_ {n-1} $ be positive numbers and define a matrix $ n times n $ by $ M_ {i, j} = u _ { left | i-j right |} $ for everyone $ i, j $.

Let $ L = D – M $ to be the Laplacian matrix of $ M $, or $ D $ is a diagonal matrix such that $ D_ {ii} = sum_j M_ {i, j} $ for everyone $ i $. Yes $ lambda_1 = 0 leq lambda_2 leq cdots leq lambda_n $ are the eigenvalues ​​of $ L $. Then we can show that $ lambda_2 $ has a monotone clean vector, called Fiedler vector.

(for proof see eg https://arxiv.org/abs/1411.0210).

If we put $ L = I – D ^ {- 1/2} MD ^ {- 1/2} $ is it possible to get the same result? It is that there is a monotone clean vector $ v $ corresponding to the second smallest eigenvalue of $ L & # 39;?

Number Theory – Is it true that $ sum_ {k = 1} ^ infty frac { binom {2k} k ^ 2} {k16 ^ k} (H_ {2k} -H_k) = frac23 sum_ {k = 0} ^ infty frac { binom {2k} k ^ 2H_ {2k}} {(2k + 1) 16 ^ k} $?

Six years ago, I conjectured the identity
$$ sum_ {k = 1} ^ infty frac { binom {2k} k ^ 2} {k16 ^ k} (H_ {2k} -H_k) = frac23 sum_ {k = 0} ^ infty frac { binom {2k} k ^ 2H_ {2k}} {(2k + 1) 16 ^ k}, tag {$ * $} $$
or $ H_n $ denotes the harmonic number $ sum_ {k = 1} ^ n frac1k $. As the two series converge slowly, I lack compelling numerical data to support $ (*) $.

Question. Is the identity $ (*) $ true? Can we check further? If that is true, how to prove it?

Your comments are welcome!

Game Theory: Using a Convex Hull Algorithm to Map Pareto Results

I started to study the concept of Pareto efficiency in game theory. The definition I know well is:

Strategy Profile $ mathbf {s} $ Pareto dominates the strategy $ mathbf {s} & # 39; $ if for all $ i in mathcal {N} $, $ u_i ( mathbf {s}) geq u_i ( mathbf {s} $) $and there are $ j in mathcal {N} $ For who $ u_j ( mathbf {s})> u_j ( mathbf {s} & # 39;) $. Strategy Profile $ mathbf {s} $ Is Pareto optimal, or strictly effective, if there is no other strategy profile? $ mathbf {s} & # 39; in S $ that Pareto dominates $ mathbf {s} $.

I am interested in finite games of normal form, for example, the $ n $prisoner's dilemma. Clearly, for $ n = $ 2 we have three Pareto results and it's not too difficult to get them.

But my concern is about a lot of players and non-constant utilities. How can we find the results of Pareto efficiency? If you know of an article that does that, please share it with me.

Is there a more effective and efficient way to calculate the effectiveness of different results of a game? Perhaps the price of anarchy?

I would appreciate any help or hint. Thank you.

Complexity Theory – Why Every DNF Formula for $ (x_ {1} vee y_ {1}) wedge (x_ {2} vee y_ {2}) wedge ldots wedge (x_ {n} vee y_ {n}) $ have at least $ 2 ^ {n} $ terms?

Why each DNF formula for $ (x_ {1} vee y_ {1}) wedge (x_ {2} vee y_ {2}) wedge dots wedge (x_ {n} vee y_ {n}) $ have at least $ 2 ^ {n} $ terms?

This statement can be found on the Wikipedia page of the DNF form here: https://en.wikipedia.org/wiki/Disjunctive_normal_form

I can not understand why that's true though. I saw an explanation saying that we can distribute the ET and OR, but I was wondering if there was a better way to understand that. I do not know too much logic, and I would appreciate it if anyone could help me, please.

Agalgic Geometry – The p-adic Hodge Theory for Singular Projective Varieties

In the p-adic Hodge theory, there are comparison theorems linking, for example, the crystalline cohomology of the special fiber of a smooth clean family to the étale cohomology of the rigid analytical generic fiber.

The so-called rigid cohomology extends the crystalline cohomology to the case of not necessarily smooth schemes. For separate finite type schemes, rigid cohomology groups satisfy reasonable finiteness properties (as opposed to crystalline cohomology). It may not be totally unreasonable to ask the following question: if rigid cohomology is considered, does p-adic Hodge theory make sense for families of singular special fiber schemes? ?

Complexity Theory – NP Integrity of the PROCESS PLANNING PROBLEM

I have to prove the NP-Completeness of the following problem:

PROBLEM OF PROCESS PLANNING

Entry: Jobs $ P_1, …, P_n $ with duration $ t_1, .., t_n $ and slots $ l_i, r_i $ for each job, which means that the job must start and end in the interval $[l_i,r_i]$. There is only one processor and it can only do one job.
Result: Is there a schedule so that each job can be done on time?

For example:

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The answer is yes, because there is a timetable for all jobs.

My idea is a reduction of Partition Problem to PSP.

So, given a partition instance $ a_1, a_2, .., a_n $ with $ sum_ {i = 0} ^ n a_i = 2A. $ Is there an index-set $ I subseteq {1, .., n } $ such as $ sum_ {i in I} a_i = A $?

So I have a partition instance $ a_1, …, a_n $ are jobs $ J_1, .., J_n $ with times $ a_1, …, a_n $ and $ l_i = 0, r_i = 2A + 1 $ for each work 1 to n.

We add a new job $ J_ {n + 1} $ with $ t_ {n + 1} = 1, l_ {n + 1} = A, r_ {n + 1} = A + 1 $.

Is this construction correct?
So, the idea is that I force the last job to be between A and A + 1, so I have a separation of 2 and it only works if there are two sets that have A as the sum.

Ag.Algenic Geometry – Applications of the Integral P-Adic Hodge Theory

What are some geometric applications of the integral p-adic Hodge theory (as opposed to the rational p-adic Hodge theory)? I know some applications to the question of interesting reductions of varieties (for example this or that).

It seems that this question is not a duplicate since the Berthelot et al paper mentioned there uses the rational p-adic Hodge theory.

group theory – quasi homomorphism from whole numbers to real numbers

Reference: lemma 5.3 of "Groups acting on the circle" of Etiyenne Ghys.

Let $ f: mathbb {Z} rightarrow mathbb {R} $ to be a quasi homomorphism, that is to say $ | f (a + b) -f (a) -f (b) | D $ $ forall $ $ a $ and $ b $ in $ mathbb {Z} $ ($ mathbb {R} $ and $ mathbb {Z} $ here are considered as additive groups and so you see the plus sign). I have to prove that there is a unique number $ tau $ $ epsilon $ $ mathbb {R} $ such as $ f (n) -n tau $ is delimited.

I have separately shown the uniqueness part and found limits that work in the following restrictive cases:
(i) If $ n $ is positive, I have delimited it above by $ f (0) $ using $ tau = f (1) + D $.
(ii) if $ n $ is positive, I have limited it below by $ f (0) $ using $ tau = f (1) -D $.
(iii) if $ n $ is negative, I have limited it above by $ f (0) $ using $ tau = f (-1) + D $.
(iv) If $ n $ is negative, I have limited it below by $ f (0) $ using $ tau = f (-1) -D $. I understand that I've always used $ f (0) $ to link them, but that's only what I can see since I separate $ n $ as $ n $ times the generator $ 1 $ $ epsilon $ $ mathbb {Z} $. Please, help me connect this universally by using a single $ tau $.