How do I calculate the probability of an event x with probability P(x) happening y times, given n trials?

For instance, let’s say each time I play the lottery, I have a 1% chance of winning, and I play 100 times in my life. How do I calculate the probability that I end up winning exactly thrice?

Also, as a related question, how do I determine the average total number of wins I could expect to see in my lifetime?

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CSDIH: About the computation of the action of an ideal over a Montgomery curve [Post quantum cryptography]

Let p be a prime. Given a Montgomery elliptic curve E0: y^2=x^3+Ax^2+x over Fp, we can consider Cl(O) and Ell(O), where O=End_p(E0). Cl(O) is the class group action and Ell(O) is the set of all elliptic curves over Fp with End_p=O.

This setup allows for an action of Cl(O) on Ell(O) as follows:

I*E=E/H_I

Where H_I = { Points of E/F_p | f(P)=0 for all f in I } and E/H_I is the unique curve for which there’s a (unique) isogeny from E to E/H_I with kernel = H_I

I am trying to understand the computation of an ideal I in Cl(O) on a Montgomery curve: y^2=x^3+Ax^2+x, i.e. how do we compute the curve E/H_I. In Martingale’s slides (one of the authors of the original paper on CSIDH) we have an example which I will show with images:

https://i.redd.it/qedap8mt0fb21.png

https://i.redd.it/yzha83yu0fb21.png

I guess that taking b=1 we have a Montgomery curve. What I’d like to know is, why is it an advantage to have P in E(F_p)? Why does it favour the computation of Velu’s formula? Here I present a proposition from Joost Renes’ Computing Isogenies between Montgomery Curves Using the Action of (0, 0):

https://i.redd.it/7wx9o1nw0fb21.png

If we take a point P of order ℓ , then <P> is already in E(alg.closure(K)) (as the hypothesis require), so, how does it make the computation easier if P is defined over F_p?

There’s another site: https://www.esat.kuleuven.be/cosic/csidh-post-quantum-key-exchange-using-isogeny-based-group-actions/ which states:

https://i.redd.it/vbzoe3ix0fb21.png

Here again, “the situation is particularly Velú-friendly”. I can’t see why. I do see that “all the computations could be done over F_p”, but, how having all the computations over Fp does make it easier as opposed to having P in Fp2 or the algebraic closure of Fp? Would it be difficult for the Velú formulas to compute the curve if the points weren’t defined over F_p? And why would it be that difficult?

From my understanding, if a curve is supersingular, all points are defined over Fp or Fp^2, but I don’t know how to relate this fact to what I am not understanding. I think I have all the pieces there but I can’t see it. Any help will be highly appreciated.

Thanks!

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Simple Questions – January 18, 2019

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than “what is the answer to this problem?”. For example, here are some kinds of questions that we’d like to see in this thread: Can someone explain the concept of maпifolds to me? What are the applications of Represeпtation Theory? What’s a good starter book for Numerical Aпalysis? What can I do to prepare for college/grad school/getting a job? Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. submitted by /u/AutoModerator [link] [comments]

How do Compression Algorithms fit with Shannon’s formula?

Shannon discovered how to measure in the information in a message:

I = -k Sum p ln(p)

where p is the frequency of each symbol in the message.

Then we can choose how many different symbols we can use to represent that information, either just 2, like 1 and 0, or 3 or 10 or whatever, and using Shannon’s formula we can know how many of those symbols we need to encode the whole message.

So far so good, but then, how does that explain compression algorithms?, specifically noiseless compression, how come we can reduce the amount of symbols and have the same amount of information?.

Does that mean that Shannon’s formula is wrong?, is it a special case?, if so, how can we modify it to know exactly into how many symbols we can compress something?

submitted by /u/Frigorifico
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Finding a bijective mapping between [0,1]^n and a manifold

I have a CS background and have taken only undergrad math courses, so the terminology I’m using may be incorrect. I’m interested in the following problem:

I have a set of k differentiable functions f_i : R^n -> R. There is a non-empty set of points x in R^n for which f_i(x) = 0 for all i. Lets denote this set as M. It holds that k < n. The set M forms one “blob”. By blob I mean that for any point in that set, there’s a path to an other point that does not lead outside of M (as several have pointed out, the correct terminology is path connectedness). This set M is what I call (probably incorrectly) the manifold.

If possible, how can one find a differentiable bijective mapping between [0,1]^n and M?
If it isn’t possible generally, is it possible under certain constraints on M (or f_i) and if so, how?
If it isn’t possible even under certain constraints, is it possible to approximate such a mapping?

Ideally, small changes in the domain of the bijective mapping should lead to small changes in its image.

I’d appreciate any solution, tips or push in the correct direction.

Thanks!

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Is it possible to analyticly determine the time of collision for two rotating bodies?

Hi,

I recently started writing a small program that simulates discs traveling in a plane and bumping into each other. While fun I feel it would be cooler to add more objects. But I am not sure if its possible to solve all of the equations with a closed formula. So I am hoping someone here can inform me of the answer. Please be kind 🙂

  1. Is it possible to find time of collision between two rotating line segments?
  2. Is it possible to find the time of collision between two rotating line segments whose center of rotation is also rotating?

If it is, could you please point me in the right direction for solving these? So far all I’ve managed to solve are collisions between non-rotating bodies.

If there is no such formula, is there a fast stable algorithm for finding the time of collision?

If neither are possible, please be so kind as to tell me that as well.

Eitherway, thank you for reading this far, I hope you will have a good day!

submitted by /u/tobbe2064
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Topics

Does anyone have topic for a person which is familiar w/ calculus (would say between 1-2 if u squeeze it into a college course), linear algebra, exponential equations, scratched analysis and knows “everything“ you need to know about complex numbers and has used quaternions in 1-2 calculations? I’m interrested in number theory, machine learning (math but also programming) and new discoveries. I dislike/hate statistics and topology.

submitted by /u/candlelightener
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Why is the Root Mean Squared of electric signals the same formula for euclidean distances?

If 2 things have the same mathematical form is because in some way they are the same, right?, and you can look for formulas with pi in places you wouldn’t expect it and try to find the “hidden circle”.

But here the Root Mean Squared, specifically of waveform combinations is exactly the same as an euclidean distance in a space with as many dimensions as waveforms, which is fascinating, I’m trying to find the “hidden distance” but I just can’t find it.

submitted by /u/Frigorifico
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