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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2
votes
1answer
85 views

How do computers generate primes so quickly?

From what I understand, when a computer encrypts a file using an encryption standard like RSA, one of the steps is to generate two large primes, and multiply them together. I have created RSA keys on ...
6
votes
5answers
280 views

Computing as many digits as possible of $\sqrt{2}$ with a pen and a paper in 5 minutes

You have to compute as many digits as possible of $\sqrt{2}$ with a pen and a paper (an eraser if you're lucky...) in 5 minutes. What will you do? What is your justification for doing it? The ...
0
votes
0answers
23 views

Unknown variable in formula - binomial coefficient? [duplicate]

I'm currently researching a filter and don't quite understand one of the equations used there, since it contains a variable I don't know how to calculate: $$(1)\,\,a^{m, k}_s = \frac{c^{k, m}_s}{m^k}$$...
0
votes
0answers
37 views

deconvolution of exp($x^2$)

I would like to know whether we can get the function of type exp($x^2$) by convoluting any functions. That is which function convolution gives exp($x^2$). Thanks in advance
1
vote
1answer
33 views

computing a fixed interest rate

I've been struggling for hours now with understanding a Topcoder problem, Autoloan , but i cannot grasp the way of computing it from a mathematical point of view. The excerpt goes as follows: A ...
0
votes
0answers
22 views

Analytical expressions for the orthogonalization of a specific set of vectors

I would like to know whether analytical or closed-form expressions could be obtained for the orthogonalization of a set of vectors in the following setting. Let $x_t$ be a vector indexed as a time ...
0
votes
1answer
24 views

Calculate height for rows of rectangles within a given width

I have an array of rectangles, all of the same height, but with different widths. Imagine they are on a single line with a uniform gap between them as shown below... |XXX| |X| |XXXXX| |XXX| |...
0
votes
2answers
90 views

Can car traffic be managed by mathematical formula? [closed]

How car traffic is managed? Is it managed by mathematical algorithm? Or by human(operator)? If it's by operator, can it be managed mathematically? Or is it by physics? By what theories/formula? ...
2
votes
1answer
38 views

Approximate recursively defined error in fixed point iteration

Problem: With an initial guess of $x_0$, the fixed point iteration is given by $$x_{k+1} = e^{-x_k}, \mbox{ for } k=0,1,2,...$$ If $x^*$ is the exact solution, then the approximation error is $$e_n = ...
1
vote
2answers
83 views

Is there any sort algorithm quicker than Quicksort given a random array of integers?

How can we proof (mathematically) that any complexity of sorting algorithm that sorts a random array of integers is no better than $O(n\log n)$?
1
vote
0answers
20 views

Is there a way to delineate the parameter of highest influence in a system of differential equations?

So I have a system of nonlinear ordinary differential equations dependent on parameters. These equations can traditionally be solved numerically with robust methods and the solution is well defined. ...
9
votes
0answers
108 views

Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$

Bellard's "exotic" pi formula has the form, $$a\pi+b = \sum_{n=1}^\infty \dfrac{P(n)}{{\displaystyle \binom{mn}{2n}2^{n-1}}}$$ where $a,b,m$ are integers and he uses $m=7$. However, it seems there ...
0
votes
0answers
27 views

FFT differential equations

Given a generical differential equation what is the procedure to solve it using fft command. Can anyone explain me how to do it? For example: $$\frac{d^2y}{dt^2}+10\cdot \frac{d\:y}{dt}=-5\cdot t^2+...
-1
votes
1answer
36 views

What is transpose multiplier and forward multiplier?

For linear system X = A*s, we define the forward and transpose multiplies Af and At as follows: Af = @(s) A*s; At = @(s) A'*s; I want to know what is forward ...
1
vote
0answers
60 views

Simplifying the Generalized Eigenvalue Problem

Let $\Sigma_1$, $\Sigma_2$ be symmetric positive-definite real $n\times n$ matrices. We want to solve the generalized eigenvalue problem $$ \Sigma_1V=\Lambda\Sigma_2V, $$ where $\Lambda$ is the ...
1
vote
0answers
46 views

Using Orthogonal Collocation to solve Coupled Ordinary Differential Equations

I am trying to solve six first order coupled ODE's, two of these are associated with a heat balance of a catalyst pellet, and four are mass balances. I have been trying to solve these equations using ...
3
votes
1answer
72 views

Prime numbers distribution theorem

I'm trying to understand Gauss' theorem: $$ \frac{\pi(x) }{x/\ln x} \to 1 $$ for large $x$. I've taken the list of first 1000 prime numbers from Utah university site, saved them to file ...
1
vote
0answers
71 views

periodic boundary conditions and the FEM

I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as $$ M = \int{NN^T}dL, $$ where $N$ is the finite element linear ...
1
vote
1answer
42 views

How to make Poisson voronoi diagram

I am facing a problem as follows : I want to make poisson voronoi diagram & for this I have to appropriately choose some generating points. Sources on internet(WIKI) refers that these points have ...
1
vote
1answer
30 views

Calculate pairwise cosine distance only returning the lower triangular matrix

I have a matrix, where each row is a feature vector. I would like to find out the pairwise cosine distance between all of these feature vectors. The cosine value between all rows in a matrix could be ...
1
vote
2answers
45 views

Special properties in the direct solving of sparse symmetric linear systems

In the area of computational solving of large sparse linear systems, some solvers specialize only on symmetric sparse matrices, be it positive definite or indefinite as compared to general (non-...
1
vote
1answer
48 views

Computational complexity of the algorithm

Make an analysis of the computational complexity of the algorithm below, where it is given by the number of elementary operations that the algorithm performs (assignment is not considered). Where $A\...
1
vote
1answer
58 views

inverse of $AQ^{-1}A'$

Suppose that $A$ is a $m\times n$ full row rank sparse matrix, and $Q$ is an $n\times n$ symmetric positive definite sparse matrix with $m<n$. Besides, $m$ is about $10^5$, and $n$ is about $10^6$. ...
7
votes
2answers
153 views

How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
1
vote
0answers
110 views

Cholesky decomposition and rotation matrix inverse

I implemented three methods for inversion of a matrix, all are classic. I wanted to test for the most generalized method, while taking efficiency into account. For Cholesky decomposition, which is ...
4
votes
1answer
42 views

Compute finite series

The problem is to count the sum of the finite series $$\sum_{k=0}^{k_0} \frac{a_k}{b_k}$$ I need to count this series in binary with some precision, that would output $n$ correct binary digits after ...
0
votes
0answers
50 views

Algorithm for finding zero of an odd function from n-sphere -> R^n

There is a well-known Borsuk-Ulam theorem stating that each continuous mapping $f : S^n \rightarrow \mathbb{R}^n$ that is odd in sence of $f(v) = -f(-v)$ for each $v \in S^n$ (where $-v$ denotes the ...
0
votes
1answer
37 views

Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, x^(k-1),...
1
vote
0answers
32 views

Why steepest descent gives a wrong direction search?

I have to minimize the function $ƒ(x_1,x_2)=(x_1-1)^2+x_2^3-x_1x_2$. The initial point is $[1,1]^T$. The gradient of this function is $∇ƒ(x_1,x_2)=[2(x_1-1)-x_2,3x_2^2-x1]$. This gradient evaluated ...
7
votes
1answer
111 views

How to find this number, which is probably a very big prime or a product of big primes?

Let $\mathcal{N}(n)$ be the next prime greater than $n$. Which is the smallest natural number $n>0\;$ such that: $\mathcal N(2\cdot 3\cdot 5\cdot 7\cdot 11\cdot n)−2\cdot 3\cdot 5\cdot 7\cdot ...
0
votes
1answer
72 views

Consider the recursively defined language, L2

Consider the recursively defined language, $L_2$ i) $x \cap L_2$ and $y \in L_2$ ii) if $w \in L_2$, then so is $wxw \in L_2$ Find all strings in L_2 with length less than $7$ characters (...
1
vote
1answer
17 views

Logistic Regression - malty classification

I want to understand why the probability of P(D|p) is presented as a product of mentioned probabilities. I read a lot of texts, but everywhere the explanations are full of terminologies to confuse the ...
7
votes
1answer
179 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and $g^n=g\...
7
votes
1answer
118 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
vote
0answers
76 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
35 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
43 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 \log_2(0....
0
votes
0answers
55 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 ...
2
votes
1answer
60 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
303 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
1answer
99 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
64 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 ...
1
vote
0answers
177 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 ...
4
votes
1answer
139 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 = 149,...
1
vote
1answer
100 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 ...
48
votes
4answers
4k 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, ...
3
votes
1answer
173 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 ...
1
vote
0answers
27 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
votes
0answers
40 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 ...
0
votes
2answers
83 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 ...