# Tagged Questions

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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### A summation involving the ceiling function

I'm trying to find a better method of calculating the sum $$\sum_{k=1}^N\lceil ak\rceil^2$$ where $a$ is an irrational number. So far, my only idea is to somehow use a best rational approximation. ...
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### Two-layer Perceptron for XOR

I'm reading Neural Networks for Pattern Recognition by Christopher M. Bishop. It's for a physics class, but I think the problem is closer to mathematics so I'm asking here instead of PSE. Chapter 4 of ...
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### 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 ...
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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 ...
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### Is there a systematic way of “discovering” an algebra from observations of its universe?

I am faced with the following situation: I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$ These integers go through a series of $N$ possible black boxes that transform them. ...
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### How to find values of x where $a_i x$ are nearly integers? $a_i \in \Bbb R$

I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$ ...
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### 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 ...
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### 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 ...
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### terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $e_1,\cdots,e_n$ and $$\mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n].$$ We define a ...
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### Summation of n terms of a series$1+2+8+64+…$

In one of my problem , I got a series as $1$,$2$,$8$,$64$,$1024$...and so on. can we really get a sum expression for that series$???$ If yes, then what is the expression $f$ $?$ or the sum of $n$ ...
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### I have a finite permutation group and access to a computer algebra system, how can I recognize the structure of the group?

I have a fixed permutation group $G$, and cannot tell which finite group it really is in a "human readable" way. I also have GAP. Is there a step by step computation to give me the structure of this ...
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### How to compute Generalized Group Inverse?

Given a transition matrix $P \in \mathbb{R}^{n \times n}$, i.e. $\sum_j P_{ij} = 1$ and $P_{ij} \geq 0$ for all $i,j$. One can show that there exists a unique group inverse $B$ of $A:= I - P$ which ...
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### Good method for finding roots that *usually* fall within an interval?

I've been using Brent's method to find the roots of a monotonic, nonlinear, non-differentiable function. The roots often fall within a known interval, but Brent's method fails if they occasionally ...
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### 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 ...
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### Is there any efficient progam or software to calculate the fractional chromatic number?

The fractional chromatic number $\chi_f(G)$ is a generation of the chromatic number of a graph $G$. It can be formulated as a linear programming question: Let $\mathcal{I}(G)$ be the set of all ...
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### Computer program to simplify formulas

What is the computer program that attempts to simplify sums of binomial coefficients, factorials, etc.? Possibly Zeilberger wrote it, but I'm unsure. If so, possibly it was talked about in his A=B ...
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### Quantitatively comparing event trains of different lengths for Poissonness

I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I ...
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### LLL and factoring polynomials in $\Bbb Z[x]$

Given a degree $2k$ reducible polynomial $f(x)=\sum_{i=0}^{2k}a_ix^i\in\Bbb Z[x]$ with $gcd(a_{2k},\dots,a_0)=1$ that is known to be of the form $f_1(x)f_2(x)$ with $deg(f_i(x))=\frac{deg(f(x)}{2}=k$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Normalizing an elliptic curve to find integer solutions

I have an elliptic curve $$c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6$$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I ...
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### Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
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### Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
I'm using Sage to solve a problem and would like to use cvxopt to solve a sdp problem. Specifically, I have a list of expressions of the form $$c + \sum_{i,j} a_{i,j} q_{i,j}$$ where each $c$ and all ...