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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5
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0answers
36 views

Carmichael numbers of form $m^3+1$ and Ramanujan's $1729$

While researching for a post on tetranacci pseudoprimes I came across a list of Carmichael numbers, $$C_n = 561,\, 1105,\, 1729,\, 2465,\, 2821,\dots$$ Of course, Ramanujan's taxicab number $1729 = ...
1
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0answers
34 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
2
votes
3answers
29 views

Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$.

I need help proving the following: Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$. I know that the smallest $n$ is 8 by ...
1
vote
1answer
21 views

Entropy Calculation and derivation of logarithm

I have probabilities as $$p_1 = 0.4,\ p_2 = 0.3,\ p_3=0.2,\ p_4=0.1$$ hence entropy is given by: $$H(x) = -\big(0.4\cdot \log_2(0.4) + 0.3\cdot \log_2(0.3) + 0.2\cdot \log_2(0.2) + 0.1\cdot ...
4
votes
1answer
103 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
0
votes
1answer
56 views

What are the minimal mathematical prerequisites for starting to study Artificial Intelligence? [on hold]

This question is really broad as there are many aspects to the AI research. However, I just need a bare minimum of prerequisites that will allow me to approach any sub-field of the AI (coupled with ...
1
vote
0answers
53 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
0
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0answers
18 views

Maximize polynomials

Hi guys I need some help. I am reading a paper and I cannot understand something simple. The author has 4 polynomials with a constrain and is trying to find the optimal solution to the problem. ...
12
votes
1answer
165 views

On $1^2+2^2+\dots+24^2 = 70^2$, and $15^3+16^3+\dots+34^3 = 70^3$

It is quite well-known that, $$1^2+2^2+\dots+24^2 = 70^2$$ Not so well-known is, $$15^3+16^3+\dots+34^3 = 70^3$$ The formula for the sum of $m$ consecutive squares starting with $a^2$ is, ...
0
votes
0answers
34 views

If a computer can check 1 million colorings per second, about how long would it take to check all possible three-colorings on 100 vertices?

If we imagine a graph G with 100 vertices, how would we find all possible colorings for G if G(v) = 100? I think that to solve this problem we would start with vertex 1 with 99 edges for the first ...
3
votes
1answer
205 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 ...
-1
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0answers
30 views

fast computation of complete elliptic integral matlab

I'm using matlab to compute complete elliptic integrals of first ($K$) and second kind ($E$). I'm having an issue with computational speed evaluating these integrals using matlab function ...
0
votes
1answer
25 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I don't have a large mathematical background, but I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular ...
2
votes
1answer
52 views

Prove that sets don't intersect

I am trying to solve a computer algorithm problem that boils down to solving the following. I would appreciate some mathematician assistance on the proof. So here goes: Having: Set $S$ - rational ...
1
vote
1answer
19 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
0
votes
0answers
19 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
1
vote
1answer
23 views

Pollard Rho intuition

I have been reading about pollard rho factorization, however their is something I don't understand if we don't use a polynomial that is pick two random numbers and see the gcd(a-b,n) > 1 if it is ...
12
votes
1answer
315 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
1
vote
0answers
51 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
0
votes
1answer
459 views

Calculating eigenvectors and eigenvalues of a 2x2 complex matrix

I've previously asked elsewhere, http://stackoverflow.com/questions/21118820/non-trivial-eigenvectors-of-a-22-matrix-in-code, how to calculate the eigenvectors and eigenvalues of a 2x2 matrix in a ...
4
votes
2answers
485 views

“Opposite” of idempotent operation?

What is the adjective given to a mathematical operation/expression on a variable whose new value can only be described in terms of that variable's existing value? Sequential operation? Example: i = ...
5
votes
0answers
171 views

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. the group $H$ is cyclic, by Ore's theorem), and among them, let $\{ ...
5
votes
1answer
436 views

Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
1
vote
1answer
20 views

Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
2
votes
2answers
56 views

How to calculate the errors of single and double precision

We consider the initial value problem $$\left\{\begin{matrix} y'=y &, 0 \leq t \leq 1 \\ y(0)=1 & \end{matrix}\right.$$ We apply the Euler method with $h=\frac{1}{N}$ and huge number of ...
1
vote
2answers
286 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should approach the ...
46
votes
4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
0
votes
1answer
659 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 ...
2
votes
1answer
29 views

How do I find the sum of first N numbers common to 2 APs?

Here is the question - Certain numbers appear in both arithmetic progressions 17, 21, 25, ... and 16, 21, 26, ... . Find the sum of first 100 numbers appearing in both progressions. The ...
3
votes
1answer
431 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
1
vote
0answers
22 views

numerically solving linear integral equations

I want to solve a 3*3 linear equation system but the equations are integral equations and he coefficients of solutions are to be extracted NUMERICALLY from some other integrals.I do not know how. I ...
0
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0answers
13 views

Norm (modulus) of the derivative of complex function and Newton Method

I am implementing a function that approximates a root of a complex function, say $f(z)$. As we know, at iteration $i$ we ave $$z_i = z_{i-1} - \frac{f(z_{i-1})}{f'(z_{i-1})}$$ The division of ...
1
vote
1answer
45 views

is it possible to find $x$ where $y$ is equal to a whole number in a non iterative fashion

Given the equation $$\frac{635x+326}{637+x} = y$$ where $$x>0$$ Is it possible to find all positive values of $x$ (there is only one) where $x$ is positive and $y$ is a whole number. While I ...
1
vote
2answers
53 views

If negative of negative (--) = positive then why not positive of positive(++)= negative

As per my understanding positive and negative are just indicative of direction of number axes with zero at the center. If that is the case we should apply same laws to both positive and negative ...
-2
votes
1answer
78 views

(x,y) coordinates from gluing together a sequence of right triangles with arbitrary angles [duplicate]

I have been scratching my head all day over this question for one of my assignments. I haven't made any progress and I'm at the point of giving up. Here's what I need help with. Start by gluing ...
1
vote
0answers
20 views

Simpson's rule is not good enough for the best approximation in L2 problem

The problem came from my computation methods (practice) class. It was to write a program which does the following: Original problem statement: We have a [0; 1] segment. Let us divide it into $2^n$ ...
0
votes
0answers
6 views

Fourier Analysis of a p2 continous Galerkin Scheme for the Laplace & Poisson Equation

Background: I am obtaining residual calculations for the 3D Laplace and Poisson Equation using finite element continuous galerkin scheme with lagrange polynomial basis functions for p1, p2, p3 and ...
0
votes
0answers
35 views

does any polyhedral partition admit a convex piecewise quadratic surface defined over?

Given a polyhedral partition, i learnt that there exist some conditions for the existence of a convex piecewise affine surface over this partition for example the following study. ...
0
votes
1answer
16 views

Golden Section Search termination condition

From textbooks I found that the tolerance for Golden Section Search method should be set to $\sqrt{\epsilon}$, where $\epsilon$ - is the machine epsilon. This can be derived from Taylor series. So, in ...
3
votes
3answers
287 views

A bug with the WolframAlpha computational search engine?

I think I may have discovered a bug with WolframAlpha. So I was trying to determine all $x$ such that $$\sum_{i=0}^{5}{x^{-i}} < \frac{13}{12}.$$ WolframAlpha spit out $x > 13$ (see this ...
0
votes
0answers
19 views

Computational verification request

Let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. Note that $\sigma$ is the classical sum-of-divisors function. Previously, I computed for $u$ in the inequality $$\sqrt{3} < ...
0
votes
1answer
29 views

Determining the Change in a variable as a function of change in independent variables

I have an Equation at hand: F = V/P I'd like to find out that for a given number of unit change in F, how many units of change are due to change in V and how many units of change are due to change ...
1
vote
0answers
23 views

Determining an unknown Function

I have an interesting operational situation at hand. I have a dependent variable, let's call it variable Y and a set of independent variables: V, H, N. (relationship is based on my operational ...
0
votes
1answer
32 views

Simple way to find square root of perfect squares

Let me first explain my problem: I am trying to write a program that can generate operations that compare a set of data rather than pulling from a list of possible relations. I have it to the point ...
1
vote
0answers
18 views

Find largest regions bounded by a set of planes

Suppose we are given a set of planes that partition the unit cube into a large number of regions. Is there a computationally efficient way to find the region with the largest volume?
0
votes
0answers
12 views

Automated tool for an algebraic problem

I have a polynomial $f=1+y_1y_2+y_1x_1$ over $GF(2)$. Now variables $y_1,y_2$ are my control but not $x_1$. Hence if put $y_1=1$ and $y_2=1+x_1$, polynomial $f$ will be zero. Similarly I have a ...
0
votes
1answer
41 views

Natural Numbers and $A_x=\{y \in A \mid y \leq x\}$ [closed]

Suppose A is a arbitrary subset of Natural Numbers and $A_x=\{y \in A \mid y \leq x\}$ with respect to $ n \in A \Longleftrightarrow n \in A_n $ and $A_n$ is finte, which of them is True? a) A and ...
10
votes
0answers
101 views

Product of primes mod n

Let $n$ be an odd composite number. I'm trying to compute $$ f(n)=\prod_{n/2<p<n}p\pmod n $$ where $p$ ranges over the primes in the indicated region. Can this be done (significantly) faster ...
2
votes
1answer
82 views

Why does Archimedes Method to calculate Pi decrease in precision after a certain time?

i`m using the following recursive formula to calculate Pi based on Archimedes ideas. $$ S' = \sqrt{2-\sqrt{4-S^2}} $$ The formula gives back the edge length of a Polygon B based on the edge length of ...
1
vote
1answer
75 views

How to obtain a convergent solution iteratively for a linear system of equations?

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} ...