This tag concerns computational problems central to mathematical and scientific computating. The scope includes algorithms, numerical analysis, optimization, and linear algebra, computational topology, computational geometry, symbolic methods, and inverse problems.

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1answer
729 views

Minimizing mean squared error

I want to find a $d$ that minimizes the value of the expression below. I think the first step is to find the derivative w.r.t. $d$ (is that correct? If not, what is the first step?). If so, I'm having ...
6
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3answers
265 views

Mathematical Limitations of Computer Experiments

One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
1
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1answer
76 views

Discrete numerical derivative with respect to d/d(n*x)

How can I generate a stencil for a d/d(n*x) operator? I am writing a program that needs a method to calculate line derivatives in an image. If we want to calculate the simplest forward derivative ...
-1
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1answer
113 views

Qubit state finding [closed]

Suppose we have two qubits in the state $x|00\rangle+y|11\rangle $. What is the resulting state of the second qubit in that case? Use and to denote and respectively.
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2answers
156 views

Can product of all pairwise sums be computed faster than the naive method?

Let $S$ be a set of integers. $|S|=n$. Can we find the product $\prod_{a,b\in S} a+b$ faster than naively add all pairs then multiply them one by one? By faster, I mean use less than $O(n^2)$ ...
1
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1answer
72 views

How to find a close form expression in terms generating functions for the triple summation

Given $$\sum\limits_{i=0}^\infty a_i z^i=A(z)$$ and $$\sum\limits_{i=1}^\infty b_i z^i=B(z)$$ and $$\sum\limits_{i=0}^\infty c_i z^i=C(z)$$ Find $\sum\limits_{i=1}^\infty\sum\limits_{j=0}^\infty a_j ...
8
votes
9answers
10k views

Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
2
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1answer
67 views

Amicable numbers

Def: a pair natural numbers $a$, $b$, $a\ne b$ are an Amicable pair if $\sum_{d|a,a\ne d}d = b$ and $\sum_{d|b, b\ne d}d = a$. Ok. So I'm trying to optimize a calculation for finding the number of ...
3
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3answers
773 views

Understanding recursive definitions of a language.

I am having difficulty understanding the recursive definition of a language. The problem asked how to write this non recursively. But I want to understand just how a recursive definition of a ...
0
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1answer
43 views

Probability : Dividing a list into 2 classes

I have a list of integer numbers ($n$). I am dividing it into two parts $n_1$ (smaller) and $n_2$ (bigger) such that the length of $n_1 \ge a*n$; $a$ is positive and $a \lt 0.5$. What is the ...
5
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1answer
1k views

A search for integers which can be written as a sum of two squares in multiple ways

As part of a number theory hobby project, I'm looking for a computational way to enumerate all integers $n$ which can be written as a sum of two integer squares in three or more ways. The range of ...
7
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1answer
156 views

Fractional part of exp(x)

I have a real number $x$ (for concreteness, say $10^4<x<10^6$) and would like to find $e^x-\lfloor e^x\rfloor$ to reasonable precision (10-20 decimal places). What is the most efficient method? ...
2
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0answers
334 views

How to convert a hologram into an image?

Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
2
votes
2answers
107 views

Number of ways to move 1 or more elements from one list to the previous list until one list remains

Given N elements, divided into at most N groups, which are then labeled 1 thru N, move all of the elements into the group labeled 1. By moving 1 to all of the elements, in group i to i-1. This means ...
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2answers
2k views

Questions that can be solved using Excel.

I recently started to realize that Excel is a powerful tool that can solve many problems. What interesting mathematics problems are there can be solved using excel? I am looking for a set of ...
1
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1answer
172 views

Where does the input x in Turing Machine subroutines come from in solving reductions to undecidable problems?

I'm taking an introduction to computation theory class and we went over the chapter on undecidable problems and proving undecidability through reductions. I can't seem to grasp some of the simplest ...
2
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1answer
133 views

Oracle turing machine

I am learning computational complexity and this is a question of my assignment that I have issues trying to solve/understand. An oracle Turing Machine M with oracle A is a Turing Machine with an ...
2
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1answer
343 views

Complement of NP-Complete

If a language L is NP-complete, with respect to polynomial time reducibility, does L ≤ co-L in polynomial time?
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0answers
25 views

Model to reward adaptative communication

I'm currently working in different simulations focused on the study of artificial evolution of communications in robots, and I have stumbled against the problem of its mathematic formulation. The ...
0
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2answers
127 views

floating point binary arithmetic

Prove that the decimal number $\displaystyle \frac{1}{5}$ cannot be represented by a finite expansion in the binary system.
1
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2answers
250 views

Finite representation in the binary $\implies$ finite representation in the decimal system

Any number that has a finite representation in the binary system have a finite representation in the decimal system. Why?
1
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1answer
156 views

Simplified way of finding a complex number raised to another complex number

This question here has the answer but I'm still in school and I don't understand any of it. I'm writing a computer program that takes a complex number a + ib and raises it to c + id and I need to ...
1
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1answer
451 views

Exponential Probability Monte Carlo simulation

I need to write a Matlab program to estimate the quantity $\theta = \mathrm{Pr}(X < 1)$, where $X$ is an exponential random variable with mean $1$. I am doing this for multiple monte carlo ...
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0answers
217 views

Power sums, fast algorithm

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
2
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1answer
65 views

Is regularity is preserved under reversal?

When talking about languages and regular languages. Can I say that reversal preserved regularity since if the language L is regular, we can generate it by right linear grammar. Therefore, the ...
1
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2answers
135 views

Solve right linear set equations

Trying to solve this set of equations. I'm feeling like I'm making it so complicated. Of course + is union. Am I on the right track? A = 0B + 1D B = 0C + 1A C = 0A + 1B + λ D = OD + 1C + λ A ...
1
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1answer
47 views

Show Two f.s.a. machines accept different input

I'm trying to solve it for two hours already. I know it somehow related to the pumping lemma Let $M_1 = \langle Q_1,S,f_1,s_1,F_1\rangle$ and $M_2 = \langle Q_2,S,f_2,s_2,F_2\rangle$ be two machines, ...
1
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1answer
1k views

Show two finite state machines are equivalent

Suppose $M_1 = \langle Q_1,S,R,f_1,g_1\rangle$ and $M_2 = \langle Q_2,S,R,f_2,g_2\rangle$ are two strongly connected machines. I need to show that $M_1 \equiv M_2$ iff there exist a state $p \in Q_1$ ...
3
votes
1answer
784 views

Computational Complexity of Modular Exponentiation

The following was posted from a lecture: "($a^n \bmod N$) has a runtime complexity of $\mathcal{O}(n*|a|*|N|)$ using the brute force method. $Z_1 = a \bmod N$ $Z_2 = (aZ_1) \bmod N$ $Z_3 = (aZ_2) ...
2
votes
1answer
1k views

radial basis function and neural networks

actually i need a simple explanation consider it for dummies about what is Radial basis function are?and what is the relation between radial basis function and neural networks ?and is there's any ...
2
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2answers
128 views

Approximating $\pi$ in Binary

I am interested in creating a Java program that generates digits of $\pi$ (in Binary though). To be clear, the number I'm looking for begins: $11.00100100 \dots$ I am unsure of the most efficient way ...
2
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0answers
446 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
1
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1answer
211 views

Is all mathematics computable?

No matter how good a computer is, it will never compute the whole sequence of PI, but we can approximate it to arbitrary degeree. We can also implement programs that can do calculus and linear ...
0
votes
1answer
128 views

Finding if two machines can be equivalent

I have this problem: Consider the following machines M1 and M2. M1 has initial state A and the initial state of M2 is unspecified. Can the machines be made equivalent by the correct choice of ...
1
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0answers
219 views

Confusion related to state equivalence of finite state machines

I have this confusion if there are two states of a machine p and q. Let x be an input string such that length of x = k, g be the output function and let g(p,x) and g(q,x) be the output when the input ...
0
votes
1answer
777 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
3
votes
1answer
213 views

What is the sixth Martin quadruple $\sqrt[n]{x_1^k+x_2^k+x_3^k+x_4^k} =\text{Integer}$ for $k=1,2,3$?

Define a Martin quadruple {a,b,c,d} as a solution in non-zero integers to the system, $a+b+c+d = x^2$ $a^2+b^2+c^2+d^2 = y^2$ $a^3+b^3+c^3+d^3 = z^3$ It can be shown that there are an infinite ...
3
votes
0answers
360 views

“Green Globs” question

When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are ...
2
votes
1answer
107 views

Sieve higher powers with logarithmic optimization

I am factoring number $N = 90283$ using quadratic sieve. Bound is $B = 44$. I find factor base to be $\{2, 3, 7, 17, 23, 29, 37, 41\}$. I have $50$ element sieving interval: $\{318, 921, 1526, ...
14
votes
1answer
2k views

Quadratic sieve algorithm

I am stuck with the sieving stage of Quadratic Sieve algorithm. I've read lots of papers to this point but I can't find any guidlines how to choose sieving interval or how sieving is actually done ...
3
votes
4answers
256 views

Prime factorization, Composite integers.

Describe how to find a prime factor of 1742399 using at most 441 integer divisions and one square root. So far I have only square rooted 1742399 to get 1319.9996. I have also tried to find a prime ...
2
votes
2answers
244 views

Given two sets of vectors, how do I find a change of basis that will convert one set to another?

Given two sets of dimension $n$ vectors $\lbrace v_1 , v_2 , \ldots , v_m \rbrace$, $\lbrace u_1, u_2, \ldots , u_m \rbrace$, where $m > n$, is there a computational method (in particular, using ...
3
votes
2answers
3k views

How to handle big powers on big numbers e.g. $n^{915937897123891}$

I'm struggling with the way to calculate an expression like $n^{915937897123891}$ where $n$ could be really any number between 1 and the power itself. I'm trying to program (C#) this and therefor ...
2
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1answer
357 views

How to compute efficiently the norm of a cyclotomic integer

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\alpha \in A$. How ...
2
votes
1answer
292 views

Using sympy im Python to substitute values into a 10x10 matrix

'm using the symbolic package sympy to store a 10X10 antisymmetric matrix in terms of 10 variables. and then at every iteration step, i substitute numerical values into the entries of the matrix. ...
4
votes
0answers
528 views

Obtain a contradiction

Motivation : The motivation is to show that the equation $x^{2b}.x^{2a} +(3-x^{2b}) x^{a} + (1-s^2)=0 $ has no solutions in integers for any values of $x,b,a,s$ ( choosen as per the constraints ...
4
votes
1answer
182 views

Hilbert curve scheduling not theoretically well defined?

In high-dimensional task scheduling, it is common to use a Hilbert-curve ordering. Given a set of points $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$ the goal is to linearly order them such that points ...
0
votes
1answer
116 views

Help understanding a $3n+1$ problem programming project

I was going through this website where I found 3n + 1 problem. I was not able to understand what the output should be. I ...
2
votes
1answer
214 views

Asymptotic expansion of a special integral

I need an asymptotic expansion of J(n) $J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$ Can anybody help to find the asymptotic analytically or at ...
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2answers
259 views

Computational efficiency of Machin-like formulae

From what I have read, it appears that the most efficient methods of calculating $ \pi $ are Machin-like formulae. And it is known that certain formulas are more efficient than others. Are there any ...