Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods.

learn more… | top users | synonyms

0
votes
2answers
36 views

Inverse Percentage.

Sorry for asking this foolish question. Here is the data i have. I purchased the product as $5 and additional fee is 2%. So Here is the total dollor ...
4
votes
0answers
61 views

Finding every $n$ such that $n\times$ ('reverse' number of $n)=m^2$ such as $1584\times 4851={2772}^2$

Let $r(n)$ be the 'reverse' number of $n$ in the decimal system. For example, $r(1234)=4321$. Then, here is my question. Question : Can we find every $n(\in\mathbb N)$, which is not a square ...
3
votes
3answers
77 views

A good way to find $a_{50000}$ where $a_n$ is a number in the form of $2^j\cdot 3^k$

Letting $A=\{2^j\cdot 3^k| j,k \ \text{are non-negative integers} \}$, let us define $a_n$ as the $n$-th element of $A$ in ascending order. We can see $$a_1=1, ...
0
votes
1answer
33 views

computing and algebra

I have a question: Let $f(x)=\sqrt{x^2 +1} - 1$. When $x=10^{-3}$ compute $f(x)$ working to 5 sf. Show algebriacally $f(x)=\frac{x^2}{\sqrt{x^2+1}+1}.$ After desperately rearranging I'm just going ...
0
votes
0answers
111 views

Computation mathematics, sequences and roots

a) For $n=1,2,3..,$ let $I_n = \int_0^1 \frac{x^{n-1}}{2-x} dx$ Writing $x^n = x^{n-1}(2-(2-x))$, show that this sequence of numbers satisfies the recurrence relation: $I_{n+1} = 2I_n - ...
1
vote
0answers
79 views

Lamé's equation - elliptic PDE?

I have a specific PDE in 3-dimensional space + time (the right one). u(x,t) is the unknown function (values are in R^3) and F(x,t) the right hand side, mu and lambda are positive constants. Now ...
1
vote
0answers
25 views

Nonlinear optimization using parallel input/output

I have a system that accepts a vector and returns a function value. The goal is to change the elements of the vector such that the function value is minimized using a derivative-free solver, eg. using ...
1
vote
0answers
43 views

The Command “TzGoGo” in GAP

I am learning GAP and would like to ask one question about a command called "TzGoGo": If $P$ is a finite presentation of a group $G$, then will the eventual result of the command "TzGoGo(P)" be ...
3
votes
1answer
76 views

GAP Responding Time

I am running GAP 4.6.5 on my six-year-old computer and sometimes it takes like forever for GAP to respond to my simple commands. An easy example is as follows: ...
1
vote
1answer
63 views

Need help for a beginner to floating-point arithmetic

I have this question that I need to complete and I literally have zero idea on what to do. I basically need someone to talk me through it and will appreciate all the help I can get. The question ...
1
vote
1answer
45 views

Survey/encyclopaedia/website of mathematical theorems connected

Is there, or is someone creating a survey/encyclopaedia/website of mathematical theorems which connects theorems together with their assumptions (axioms, other theorems, hypotheses etc.)? I'm thinking ...
0
votes
1answer
120 views

Multi layer perceptron activation function

How can you show that the Fourier series approximation of a function (so $f(x)=\sum\limits_{n=0}^{\infty} (a_n cos(nx) + b_n sin(nx))$ can be approximated to arbitrary precision by a feedforward ...
0
votes
0answers
49 views

Urgently Seeking for Help in Calculating the Abelian Invariants of a Group with GAP

I have asked a more complicated question here but I decided to start with an easier version: Let $K_3$ be the group generated by three elements $a$, $b$, $c$ subject to the relation that every simple ...
1
vote
1answer
61 views

Ackermann function and primitive recursiveness

If we define $b_n(m) := a(n,m)$ for all $n$ and $m \in \mathbb{N}$. For which $n$ is the function $b_n$ primitive recursive and for which $n$ it is not a primitive recursive function? Can anyone ...
2
votes
0answers
93 views

Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated ...
4
votes
1answer
128 views

What is the upper bound on the error of a matrix multiplication

When both A and B are n x n upper-triangular matrices, the entries of C = AB are defined as follows: $$ c_{ij} = \begin{cases} \sum _{k=i}^ja_{ik}b_{kj} & 1\leq i\leq j\leq n \\0 & 1\leq j\lt ...
0
votes
1answer
167 views

Simultaneous recursion

I have no idea how to even start proving the following theorem: If $f_0, f_1: \mathbb{N}^r \rightarrow \mathbb{N}$ and $g_0, g_1: \mathbb{N}^{r+3} \rightarrow \mathbb{N}$ are primitive recursive, ...
3
votes
1answer
90 views

Algorithm for Finitely Presented Torsion-Free Nilpotent Groups

I am studying some finitely presented, torsion-free and nilpotent groups $G$ and need to consider the following question: Let $H$ be a subgroup of $G$ and suppose that $H$ is generated by ...
2
votes
3answers
97 views

Websites/Software for Group computation

Anyone knows a website or software that helps to do computations in a group? For example, by inputting generators and relations in the group, can we tell when two particular elements in the group ...
2
votes
1answer
85 views

Is WolframAlpha computing this radical correctly?

Is WolframAlpha computing this radical correctly? $$\sqrt{\frac{1}{1 + {10}^{-375}}}$$ When I double-check again, the inequality: $$\sqrt{\frac{1}{1 + {10}^{-x}}} > 1$$ leads to a ...
3
votes
0answers
114 views

$\pi$, disjunctive numbers, and finite sequences of given length

It is an open problem whether the number $\pi$ is disjunctive in base $10$, i.e., whether every finite sequence appears (at least once) in the base $10$ expansion of $\pi$. Of course, every sequence ...
1
vote
0answers
304 views

Solving large, sparse system of linear equations

I have a system of linear equations as follows: $$(A+I)x=B$$ where $I$ is the $n\times n$ identity matrix, $A$ is a $n\times n$ matrix such that the first and last rows are blank, and, for every ...
5
votes
2answers
794 views

Fast Matlab Code for hypergeometric function $_2F_1$

I am looking for a good numerical algorithm to evaluate the hypergeometric function $_2F_1$ in Matlab (hypergeom in Matlab is very slow). I looked across the ...
0
votes
2answers
45 views

Generating a random number of higher range

I had a discussion with my friend about writing a function (a computer program) which generates 9 values randomly using a random number generator which generates 4 values, i.e I have a PRNG rand(4) ...
3
votes
0answers
47 views

Supremum over unitary group action

Let $A$ and $B$ are two given Hermitian operators on matrix algebra $M_n(\mathbb{C})$. $A$ is positive semi-definite with unit trace. I want to know the general method for calculating the following ...
4
votes
1answer
285 views

Wolfram Alpha error?

I was seeing some equations in WA, and i got with http://www.wolframalpha.com/input/?i=%28k%2B1%29%5E2%3E%3D4%28k-1%29%5E2 Let's manually solve the equation $$(k+1)^2\ge4(k-1)^2$$ ...
10
votes
1answer
459 views

Why are there mathematicians that do not use computers?

I was watching a video on Andrew Wiles and his proof of Fermat's Last Theorem and I quite liked the video, especially the complexity of the proof only to prove a simple concept which can be understood ...
2
votes
1answer
130 views

Is there a computer programm or CAS (maybe GAP?) that can calculate with projective (indecomposable) A-modules (A is a finite dimensional k-algebra)?

I have the following question(s): I have an "Algebra-With-One" $R$ as a subalgebra of a full matrix algebra in GAP. Furthermore, I have 5 primitive orthogonal idempotents $e_1,...,e_5$, which sum up ...
6
votes
1answer
194 views

how to calculate this complementary Bessel function?

I am trying to calculate this complementary Bessel function $$\Psi(a,b,\gamma)=\int_0^\infty\Phi({a\over \sqrt{u}}+b\sqrt{u}){u^{\gamma-1}e^{-u}\over \Gamma(\gamma)}du$$ where $\Phi$ is the standard ...
2
votes
0answers
78 views

Solving a particular system of Diophantine equations in $n$ variables (Frobenius equations)

I have a particular system of linear Diophantine equations in $n$ variables for which I need to find all nonnegative integer solutions. Specifically, they are Frobenius equations, meaning the ...
1
vote
1answer
65 views

Is there a good strategy for computing eigenspace corresponding to $1$ of a matrix with entries of trigonom

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
-2
votes
2answers
75 views

Is there any good strategy for computing null space of a matrix with entries $\cos x$ and $\sin x$?

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
2
votes
1answer
276 views

Numerical Integration over 2D Regions with Discontinuous Functions

I've run into a tricky problem, and I haven't really thought up a good solution. I have to compute many integrals of the form $$\iint\limits_{B((x_k,y_k),\epsilon))} f(x,y) dy dx$$ where ...
0
votes
0answers
34 views

Polytime programming

Given a linear system of the form: $$x_r = a$$ $$x_j = b$$ $$c_1x_1 + c_2x_2 ... c_nx_n = n$$ $$x_1 + x_2 + x_3 ... x_n = k $$ $$0 \leq a,b,x_1, x_2, x_3 ... x_n \leq 1$$ $$k \geq 0$$ How quickly ...
4
votes
4answers
305 views

Computing partition numbers

Today a friend and myself came up with the question of computing partitions of numbers, i.e.: given a number $n$, what is the number $p(n)$ of was of different ways writing $n$ as a sum of non-zero ...
2
votes
2answers
89 views

Uniform grid on a disc

Do there exist any known methods of drawing a uniform grid on a disk ? I am looking for a map that converts a grid on a square to a grid on a disk.
2
votes
1answer
89 views

What does noncomputable really mean?

I believe I understand the definition of a noncomputable problem from an introductory computer science class, but I don't understand what it really means. One of my hypothesis was that a ...
6
votes
2answers
127 views

What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube?

Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random ...
2
votes
0answers
32 views

Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order ...
0
votes
0answers
104 views

Effects of numerical integration stepsize on impulse inputs (e.g., delta function)

Some models of neurons treat synaptic input (from other neurons) as a single impulse, such as the Dirac delta. But doesn't this make the magnitude of that impulse a function of numerical integration ...
0
votes
1answer
32 views

Find all points where the gradient of a high-dimensional function are equal to zero in some domain, numerically

I was wondering if anybody was aware of a numerical method to find all points where the gradient of a high-dimensional function are equal to zero in some domain. Thanks
1
vote
0answers
30 views

Finding k-d sum for all numbers upto maxVal [duplicate]

a number n is a k-d number if either of the following holds true: a) number of digits <=k b) sum of first k digits is equal ...
5
votes
1answer
306 views

cylinder-ray intersections equation

Can You Pleas Help with this one I found an article http://www.mrl.nyu.edu/~dzorin/rendering/lectures/lecture3/lecture3.pdf for Infinite cylinder-ray intersections And I don't know how they develop ...
1
vote
2answers
143 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 ...
0
votes
0answers
58 views

Constrained computational optimization of a functional of a vector valued function.

I am trying to increase the efficiency of a program I have written that must run in real time. I am asking this question in a broad sense, since I'm not sure what tools are available to me. I am ...
1
vote
0answers
13 views

Tools for optimizing asymptotic bounds.

Is there any tool for this task ? Given the asymptotic bound in term of $n$ and other paramaters $t_1,\dots,t_r$, then return the value for each $t_i$ which optimizes the expression in term of $n$, ...
4
votes
3answers
1k views

Derivative of Associated Legendre polynomials at $x = \pm 1$

I'm creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I'm at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the ...
4
votes
1answer
53 views

About parallel time computation

I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
5
votes
2answers
460 views

What free software can I use to solve a system of linear equations containing an unknown?

Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$? Presumably I could do ...
0
votes
0answers
107 views

Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that: ...