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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9
votes
3answers
335 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
4
votes
4answers
99 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
1
vote
1answer
24 views

When is ${n \choose k} > (n-k)(k+1) + (n-k-1)k$?

I have two algorithms that output the same result for an input value of a non-negative integer k and a list of n elements, where $1 \leq k \leq n$. However, the two algorithms are very different in ...
1
vote
0answers
25 views

Trying to learn about kernel PCA but cannot understand some math.

I'm trying to learn about kernel PCA by reading through the paper of it's creators (I assume) "Nonlinear Component Analysis as a Kernel Eigenvalue Problem", Bernhard Schölkopf, Alexander Smola, Klaus-...
1
vote
1answer
86 views

How can I optimise a scalar function over a matrix?

I need to optimise the following scalar function with respect to a matrix $S$. $$ f(S) = \boldsymbol{y}^{T}\boldsymbol{X}w - \boldsymbol{1}_{n}^{T} \exp \left\{ \boldsymbol{X}w + \frac{1}{2} \...
2
votes
2answers
82 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for $e$...
2
votes
3answers
73 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
0
votes
1answer
37 views

How to obtain fair competition between two teams

Consider a class with $4$ students having min goals as $\big\{ 1, 3, 4, 5 \big\} $ and max goals as $\big\{ 2, 5, 8, 6 \big\}$. Find the best way to divide the class in such a way that the match is ...
0
votes
0answers
22 views

Finite element boundary conditions

I have a boundary condition given by $\mathbf{n}\cdot \nabla m=\phi$, where $n$ is a vector normal to a surface, $m$ is a physical quantity (say mass) and $\phi$ is a constant. The boundary condition ...
0
votes
1answer
23 views

How can I implement Newton-Raphson's method with a function of one vector and one matrix?

I have a function $f(\mathbf{u}, \Sigma)$ where $\mathbf{u}$ is a $p \times 1$ vector and $\Sigma$ is a $p \times p$ real symmetric matrix (positive semi-definite). I somehow successfully computed ...
0
votes
2answers
81 views

Converting Java Code to Mathematic formula

I have algorithm , but I don't know how to convert it to mathematic formula. ...
0
votes
0answers
13 views

bio heat equation modification

I have the bio heat equation as described .here And the solution to it is, But to this I am trying to include the effect from exercise intensity as well. So the modified bio heat equation is ...
2
votes
0answers
18 views

Given a set of integers $S$, what is the maximum integer that is a product of one or more integers from $S$ not exceeding $X$?

Given a set of integers $S$, which will contain no more than $100$ integers. Now, what would be the fastest approach to find $M$ which is a product of one or more integers from $S$ (and multiple usage ...
0
votes
0answers
30 views

summation of even and odd numbers

In my video processing algoritm, I do some processing even and odd frames seperately. F = E(x) + O(x) where F is the video, E and O contains its even and odd ...
9
votes
1answer
150 views

How do I develop numerical routines for the evaluation of my own special functions?

This question has been cross-posted to ComputationalScience.SE here. When performing computational work, I often come across a univariate function, defined in terms of an integral or differential ...
0
votes
0answers
40 views

How to solve the Allen-Cahn equation with finite element method?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...
1
vote
0answers
26 views

Integrating Wishart density

I have several points $\textbf{s} = s_1,...,s_n$ which follow Wishart distribution. In one of my problem, I have to integrate this Wishart pdf over a ball of radius $r$ at origin in $\mathbf{R}^2$ i.e....
0
votes
0answers
20 views

Obtaining the weak form of Allen-Cahn equation

\begin{equation} \frac{\partial\phi(\mathbf{x},t)}{\partial t}=g(\mathbf{x})(\varepsilon^{2}\Delta\phi-F^{'}(\phi)),\ \ \ \mathbf{x}\in \Omega,t>0\ \ (*) \end{equation} \begin{equation} \frac{\...
0
votes
0answers
11 views

A computational problem with implicit function

Given a function $$ H(x) = \sum_{i=0}^{\infty}a_i x^i $$ where $x_i\in[0,1]$ and $a_i \ge0$. Also, $$ F(r) = \sum_{i=0}^{\infty}a_i G(r)^i $$ G(r) is a cdf function on $[r_1,r_2]$ so that $G(r_1)=0,...
15
votes
3answers
226 views

Plotting $\left(1+\frac{1}{x^n}\right)^{x^n}$.

When I plot the following function, the graph behaves strangely: $$f(x) = \left(1+\frac{1}{x^{16}}\right)^{x^{16}}$$ While $\lim_{x\to +\infty} f(x) = e$ the graph starts to fade at $x \approx 6$. ...
0
votes
0answers
34 views

How to calculate derivative with respect to time for Optical Flow

Suppose we have 2 images in motion for detecting the object in movement according to Lucas and Kanade [u, v] = inv(H)*[dxdt, dydt] where H is the Hessian for partial derivatives for image x and y ...
1
vote
0answers
18 views

Efficiently compute the Determinant of a Banded Matrix

So I've got a large (~ 2 million x 2 million) positive semi-definite, banded, square matrix that I need to find the determinant of. What is the correct way to efficiently compute the determinant of a ...
2
votes
2answers
48 views

evaluating limit using binomial series

I am trying to evaluate the following limit by using binomial series $(1+x)^{1\over 2}=\left ({1/2}\over n\right) x^n$ $$\lim_{x\to 0}{{(1+x)^{1\over 2}-1-{1\over 2}x +{1\over 8}x^2}\over {x^3+x^5}}$$...
1
vote
3answers
72 views

solve differential equation $f'(x)=f(x)$

I want to solve the differential equation $f'(x)=f(x)$ using power series of the form $$f(x)=\sum_{n=0}^{\infty}{c_nx^n}$$ From my previous knowledge I know that the solution is $f(x)=c_0e^x$ I can ...
1
vote
1answer
51 views

explain convergent using power series

I want to use the power series of $(x+1)^{1/2}$ to explain why $$\sum_{n=1}^{\infty}\left({\sqrt{1+{1\over n}}-{1\over n}}\right).$$ converges. I was able to get the expansion of the series using ...
1
vote
1answer
32 views

Are there any errors in the computation of the earth surface area [closed]

The surface area of earth might be completed using the formula A=4pe *r^2. For the surface area of a sphere of radius r. Are there any errors in the computation of the earth surface area using the ...
0
votes
1answer
23 views

test for convergence or divergence

So I am looking at the following series: $$\sum_{n=1}^{\infty}{{ln(n)+n^p+r^n+n!}\over{n^n-n!-r^n-n^p-ln(n)}}$$ Before testing, I wanted to look at some series that I can compare this to but haven'...
1
vote
1answer
47 views

equivalence of theory of reals and Rationals

Present a sentence φ that is in theory of reals but not in thoery of Rationals Following up from this question what is the approach to show that both the theories are equivalent Th(R, 0, 1, +, ≤) ...
1
vote
1answer
35 views

How to find the closest bounded rational approximation to a rational number?

Say I have a rational number $a/b$ and I want to find its closest rational approximation $x/y$ where $$x_- \leq x \leq x_+$$ $$y_- \leq y \leq y_+$$ for some constants $x_\pm$, $y_\pm$. How can I ...
1
vote
1answer
73 views

Particle swarm optimization

I don't know where to start. Like, I don't know how to plug the info into the algorithm. Show two iterations of particle swarm optimization (neighborhood approach) method. Mathematically show two ...
0
votes
2answers
56 views

growth rate of n! versus $r^n$

How do you show that $\lim_{n\to \infty} {n!\over {r^n}} $ approaches $\infty$? the growth rate of $r^n$ is slower than $n!$, so the latter one is increasing faster, but how do you show the above ...
1
vote
4answers
1k views

Newton form vs. Lagrange form for interpolating polynomials

I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other? It seems to me, that the computational cost of the two are ...
1
vote
0answers
17 views

I need help with this Standard Uniform random variable problem

Click here for problem image I need help on how to do b,c and d. Can I please have an idea of how to start it because i have no idea how to do this.
0
votes
0answers
58 views

The solution of Allen-Cahn equation?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...
1
vote
0answers
38 views

for what value of p does the integral converge

For what value of p does $\int_0 ^1 x^p dx$ converge? So I used improper integral value for this, looking at the result $$\lim_{p\to 0+} {1\over p+1}(x^{p+1}-p^{p+1})$$ *** I made a mistake, I got ...
1
vote
0answers
63 views

Find the different values of this integral when all paths of integration are possible

This is the question.. I only know how to do the question from infinity to infinity.. enter image description here Find the different values of this integral when all paths of integration are ...
0
votes
1answer
40 views

An unusual limit

I a reading an article in neural coding. It cintains a limit which I cannot understand. (It remindes the benomial sum but it didn't work for me) I am talking about $(22) \Rightarrow (23)$ in the ...
1
vote
1answer
184 views

Is there a word for an “infinite algorithm”?

According to Knuth's notes (see Slide 3), an algorithm, by definition, satisfies the following five properties: Finiteness: Terminates after a finite number of steps. Definiteness: Each step is ...
1
vote
0answers
46 views

Constructing triangulations algorithmically

I am developing a Python package for computations in algebraic topology (namely: cohomology and Massey products on manifolds). Basically all the stuff I'm doing requires an explicit triangulation of ...
2
votes
0answers
50 views

A recursive problem with GAP concerning lists and an iterator loop

I have the following question concerning a list algorithm in GAP: Let $L_1$ be a non-empty list with certain objects as entries. I wrote a program and called it helping_program_1. The Input for ...
1
vote
0answers
25 views

Algorithm detect simple curves using Voronoi diagram or Delaunay triangulation?

I wonder if there is algorithm/method to determine if closed (or even non closed) curve is simple or not, using the mathematics from the field of computational geometry? Especially I wonder if exist ...
0
votes
1answer
23 views

How can I find an approximation for y(1) using MATLAB without ODE solvers? (Euler's Method & Matrices)

Task: Given \begin{align*} y(t+h)&\approx \underbrace{\left(\begin{matrix}1-h & 5h & h\\3h & 1-h & 0\\0 & -th & 1+h\end{matrix}\right)}_{F(t,h)}y(t)+\underbrace{\left(\...
1
vote
1answer
43 views

Is there a good way to compute this integral?

Sometimes questions like: "How many digits does 2015! have?" become quite trendy. The most reasonable approach would probably use Stirling's formula. However doing this in the same way over and over ...
8
votes
1answer
560 views

Is there a way to calculate the definite integral of inverse of a 5th degree polynomial?

I want to calculate the definite integral of inverse of a 5th degree polynomial. The problem is that the inverse of the polynomial cannot be calculated (by using Matlab). However without calculating ...
0
votes
1answer
34 views

Find RHS from LHS.

$A=\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right]<==>\left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ ...
0
votes
0answers
43 views

Affine Matrix for Computer Graphics

I was given this assignment task which I have no idea where to start with so I'm hoping that someone can help me with this? Student A: Matrix 4x4 (0.989, 0, -0.148, 0, 0.018, 0.992, 0.121, 0, 0....
0
votes
0answers
140 views

Finding The Radix of A Quadratic Equation

I have found previous solutions to finding the radix of a quadratic equation, where both of the provided roots return the same radix or base. However, unless I am some type of arithmetic error of ...
0
votes
1answer
22 views

Question related to the security of RSA method

I learned about the RSA method, where if B wants to send a message $M$, say $0 \leq M <n = pq$ to A with public key $(n,e)$, then B sends $M'= M^e (mod \ n)$. Then A can decode this message using ...
0
votes
0answers
16 views

adding a vertex and edges which do not change lengths of shortest paths (ACM ICPC problem)

I am trying to figure out how to solve the problem "Farm and Factory" from the ICPC archives. For the sake of making this question self-contained, I've reproduced the problem below: All hail ...
0
votes
0answers
78 views

What is meant by “deterministic error”?

I am reading a scientific paper in computer science and I have found the term of deterministic error. I googled to find any meaning to this notion but I did not find anything. So is there anyone to ...