## SOLVE SYSTEM OF EQUATION SOLVE BY EULER MARUYAUMA OR MILSTEIN APPROXIMATION

APPROXIMATION OF EULER MARUYAUMA AND MILSTEIN METHOD. HOW DO IS SOLVE SYSTEM OF EQUATIONS IN ITO PROCESS METHOD OF EULER MARUYAUM AND MILSTEIN APPROXIMATION BELOW PICTURE IS GIVEN…

## Euler Angles of Any Great Circle from hit.point of a sphere

How do you get the Euler Angles of Euler Angles of Any Great Circle that surrounds the raycast hit.point of a sphere?

## Project Euler #3 with Python

I tried to solve some Euler’s problems, this is the solution I found for problem #3 with Python. The problem asks:

The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ?

Can you, please, suggests me improvements?

def get_largest_prime2(value):     start = time.clock() #used to measure the execution time     divisors = []  #this list will be filled with all number's divisors     i = 1     while value != 1:          while value%i == 0: #check if the value can be divided by i and, if so, iterate the process.             value = (value // i)             divisors.append(i)             if i == 1 : break #it's needed to avoid an infinity loop         if i >= 3: i += 2 #when i is bigger than 3, search divisors among odd numbers( this reduces the search field)         else: i += 1       print( time.clock()-start)     return max(divisors) #returns the max prime divisors 

## Elixir solution for Project Euler #2

I have already done many of the Project Euler questions in other languages before, but whenever learning a new language, I like to do the problems again in that language.

Here is my elixir version of

Find the sum of all fibbonaci numbers below 4,000,000.

stream = Stream.unfold({0,1}, fn {a, b} -> {a, {b, a + b}} end) Enum.reduce_while(stream, 0, &(     cond do         &1 < 4000000 and rem(&1, 2) == 0 ->             {:cont, &2 + &1}         &1 < 4000000 ->             {:cont, &2}         true ->             {:halt, &2}     end )) 

Can anyone spot a way to make my code fit the elixir paradigm more? Are there things I could improve?

## Euler characteristics in the rank one case

Suppose $$E$$ is an elliptic curve over a number field with good ordinary reduction at the primes above a fixed odd prime $$p$$. We are interested in the Iwasawa theory over the cyclotomic $$\mathbb{Z}_p$$ extension under the additional assumption that $$E$$ has a point of infinite order and also that $$L(E,s)$$ has a simple zero at $$s=1$$.

Is there a definition of a modified Euler characteristic for the Selmer group over the cyclotomic extension which can be related to the derivative $$L'(E,1)$$ in the framework of the BSD conjecture? Can someone give me some precise references.

## bare hand integration of euler class of blow up of a complex surface at a point

let X be a product of two Riemann surface of genus $$\geq1$$ and p be a point of X. If X’ is the blow up of X at p. Let E denote the exceptional divisor I am aware that the the second Chern number of X’ since it’s the euler characteristic. I would to verify this by integration of the second chern class. Up to now I try as follows:

Let W be a tubular neighborhood of the exceptional divisor E. Let N be the normal bundle of E $$c_2(X’) = p^*(c_2(X))+c_2(TN)=p^*(c_2(X))+ c_1(TE)c_1(N)$$ Now I am not sure how to proceed. $$\int_{X’}p^*(c_2(X))+ c_1(TE)c_1(N)$$ This reduces to $$\int_{X’}c_1(TE)c_1(N)$$ which I tend to understand as the intersection number. But usually one performs this computation in $$\mathbb{C}P^n$$, which is not the case here. Could anybody give me some help? Thanks.

## Project Euler #7 10001st prime in C++

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

How to optimize this code?

#include <iostream>  bool is_prime(int num) {     for (int i = 2; i <= num/2; ++i)     {         if (num % i == 0)         {             return false;         }     }     return true; }  int main() {     int count = 2;     for (int i = 5; ; i = i+2)     {         if (is_prime(i))         {             count++;         }         if (count == 10001)         {             std::cout << i;             return 0;         }     } } 

## Project Euler #1 : Finding the sum of multiples of 3 and 5 under a given number

I wrote the following code in c++14 for hackerRank Project Euler challenge 1.

First I came up with a naive solution by iteration and passed most test cases except 2 and 3 due to timeout as solution is O(N) and test cases 2 and 3 have large input.

The below code is my second attempt at a solution. ALthough I pass the sample test case 0 and custom test case successfully, while attempting to submit I fail at all hidden cases except case 5. I can’t seem to figure out, how do I proceed further. Is my code logic flawed and failing at edge cases that I dont seem to figure out or there is some other issue with the code.

As I am just starting out, please enlighten me with standardised way of doing things.

# include <iostream>  long T; long N;  /* Naive Solution 2 test cases failed due to timeout time complexity O(N)  int Sum_multiple_3and5 (int n) {  int sum = 0;  for (int j = 1; j < N; j++)  {    if ((j % 3 == 0) || (j % 5 == 0))    {      sum = sum + j;    }  }  return sum ; } */  long Sum_multiple_3and5(long n)  { long Given_No = n; long Kfor3 = (Given_No - 1) / 3; //std::cout << Kfor3 << "\n"; long Kfor5 = (Given_No - 1) / 5; //std::cout << Kfor5 << "\n"; long SKfor3 = (((Kfor3) * (3 + (3 * Kfor3))) /2); //std::cout << SKfor3 << "\n"; long SKfor5 = (((Kfor5) * (5 + (5 * Kfor5))) /2); //std::cout << SKfor5 << "\n"; long KforExtraSum = (Given_No - 1) / 15; //std::cout << KforExtraSum << "\n"; long SKforExtraSum = (KforExtraSum /2) * (15 + (15 * KforExtraSum)); //std::cout << SKforExtraSum << "\n"; long sum = ((SKfor3 + SKfor5) - SKforExtraSum); //std::cout << sum << "\n"; return sum; }  /* using sum of sequence formula to sum multiples of 3  to multiple of 5 and subtract from the total the  sum of multiples of 15 */  int main()  { std::cin >> T;  for ( long i = 0; i<T;i++) {  std::cin >> N; std::cout << Sum_multiple_3and5(N) << std::endl; } return 0; }     

## Casson invariant and Euler characteristic

A slogan I frequently hear is: “the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere”. Is there a single paper/reference that essentially states this as a result? It has been difficult to locate precise definitions. In addition, a sketch of how this works would be very helpful.

## Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $$\chi(t,s)$$ is a curve in $$\mathbb R^3$$, be derived from the Euler equation $$\partial_t \omega + v\cdot \nabla \omega = \omega \cdot \nabla v, \quad \operatorname{div} v = 0,$$ where $$v:\mathbb R \times \mathbb R^3 \to \mathbb R^3$$, and $$\omega = \operatorname{curl}(v):\mathbb R \times \mathbb R^3 \to \mathbb R^3.$$

I’ve asked a more general question at Survey on the vortex filament equation.