# Questions tagged [computation]

The computation tag has no usage guidance.

37
questions

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### Is there a "natural" interpretation of the power function for octonions and for sedenions?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ ...

51
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8
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### Is there a fast way to check if a matrix has any small eigenvalues?

I have hundreds of millions of symmetric 0/1-matrices of moderate size (say 20x20 to 30x30) which (obviously) have real eigenvalues.
I wish to extract from this list the tiny number of matrices that ...

4
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3
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285
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### How to recover integer part from known fractional root part?

Suppose you have $r=n+f$ where $n\in\mathbb{N}$ and $f\in (0,1)$. I know that $r^2$ is an integer and I can also get as many digits of $f$ as I like, is there a way to recover the value of $n$?
Thank ...

2
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0
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84
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### Quantum groups as bialgebra cohomology classes

My question below is about how to view the quantum group $U_q(\mathfrak{g})$ as a bialgebra cohomology class.
Background: If $A$ is a bialgebra, Gerstenhaber and Schack in Bialgebra cohomology, ...

0
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0
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56
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### NC0 randomness vs. non-uniformity

In
Ajtai and Ben-Or. A theorem on probabilistic constant depth
Computations. STOC '84, 1984
Ajtai and Ben-Or show a non-uniform derandomization of BPAC0.
Is there a similar relation known for ...

5
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3
answers

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### How to speed up the process for calculating the Groebner basis?

I am currently trying to get the Groebner basis for 9 equations with 12 variables:
$
a_1^2+b_1^2+c_1^2+d_1^2-48.73=0\\
a_2^2+b_2^2+c_2^2+d_2^2-50.53=0\\
a_3^2+b_3^2+c_3^2+d_3^2-40.69=0\\
a_1a_2+b_1b_2+...

1
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1
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113
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### Problem NP-completeness on a specific graph class

Consider the class of simple connected n/2-regular graphs, n even. Are the maximum clique problem and/or maximum independent set problem NP-complete on such graphs? Is there any known result which ...

11
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1
answer

300
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### Software for recognizing algebraic or D-finite formal power series

I have a formal power series in one variable that I think might be algebraic (or perhaps just D-finite). Is there software that could help me explore this?
By way of comparison, there’s a very simple ...

3
votes

1
answer

431
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### Is there an equivalent of the incompleteness theorems/halting problem in category theory?

Taking the doctrine of computational trinitarianism ( https://ncatlab.org/nlab/show/computational+trinitarianism ), if one understands the incompleteness theorems as the "logic" version, and ...

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0
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107
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### Quick ways to compute transition matrices for classical symmetric function bases

I am trying to implement quick algorithms for computing the transition matrices
involving the monomial, power-sum, elementary, complete homogeneous and Schur polynomials.
There are several relations ...

4
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0
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### Amortized complexity of P

Let $P$ be the class of all polynomial time computable functions from $\{0,1\}^*\rightarrow \{0,1\}$. For any $f\in P$, define function $f^A:\mathbb{N}\rightarrow \{0,1\}^*$ by
$$f^A(n)=(f(x_1),\cdots,...

4
votes

1
answer

128
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### Algorithm to construct basis for Kac-Moody algebra

Suppose I have a Kac-Moody algebra (maybe even Borcherds-Kac-Moody) $\mathfrak{g}$ with symmetric cartan matrix $A$. Let the simple roots be $e_{\alpha_i}$ for $i = 1, \ldots n$.
I know there is ...

0
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1
answer

808
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### Conjecture that relates matrix systems with some specific functions as solution sets

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...

1
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1
answer

155
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### Errors in Waksman's Solution to Cellular Automaton Firing Squad Problem?

Recently, a student and I have been working through Waksman's paper ``An Optimum Solution to the Firing Squad Synchronization Problem.'' The paper claims that for any value of $n$, the proposed ...

5
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0
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77
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### Numerical and computational approaches to limit cycle theory

I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach.
I would like to ask ...

3
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0
answers

162
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### curve blow ups of toric Fano $3$-folds

Suppose $X$ is a smooth toric Fano $3$-fold, and $D$ is a torus invariant divisor corresponding to a face of the polytope associated to $X$. I would like to search for (smooth) curves $C \subset D$, ...

4
votes

1
answer

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### Conjecture on palindromic numbers

The conjecture is as follows:
Let $n\in\mathbb{N}\setminus\{1\}$. Define $a(n)=2^n+1$ and the set:
$$S(n) = \{ (a(n)^m+1)/2\ :\ m\in \mathbb{N}_0\}.$$
Then for all $c\in\mathbb{N}$, the number $(a(n)...

8
votes

1
answer

479
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### simple conjecture on palindromes in base 10 [closed]

The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \...

1
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1
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715
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### Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets

Assume $x$ is a variable belongs to $\mathbb R \setminus \{ 0,-1,+1 \}$ and consider for all $i, j \in \mathbb N$,
$$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$
then for all $n \in \mathbb N$ the ...

7
votes

1
answer

414
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### What Turing degree would allow you to "compute" the axioms of ZFC in some countable model of ZFC?

It is established in this post that you there is no computable model of ZFC, yet it can be computed in by a PA-degree oracle machine. Note that when we see "compute a model", we just mean that ...

0
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0
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150
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### Computer algebra programs for dummies [duplicate]

In the way of my investigations I have encounter the following computational problem: I have a system of 5 algebraic equations and I want to eliminate 4 of them. I also need to do a functional ...

6
votes

1
answer

248
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### Problem on triangles

Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...

6
votes

1
answer

490
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### Algorithm to compute Matrix Sign Rank?

The (generalised) sign-rank of a (generalised) sign pattern $S\in \{+,-,0\}^{n\times m}$
is the minimum rank of all matrices with the same sign pattern, i.e.
$$
\min\left\{\operatorname{rank}(M)\ :\ M\...

8
votes

2
answers

364
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### Curves embedding in plane

Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve can be embedded inside the other one, here embedding ...

6
votes

2
answers

1k
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### Complexity of Turing Machine behavior

If one looks at the code for a Turing Machine (TM) with
$q$ states and, let's say, $2$ symbols, they all look
pretty much the same:
A list of $5$-tuples:
$$
< state, symbol{-}read, symbol{-}to{-}...

19
votes

0
answers

505
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### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all
$x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...

5
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0
answers

109
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### Efficiently computing all equivariant maps between two $GL_n$-representations

This is sort of a strange question; if it's not appropriate for MathOverflow I apologize in advance.
I'm in a situation where I'd like to be able to give a computer two $GL_n$ representations $V$ and ...

1
vote

1
answer

201
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### Polygonal Mersenne numbers [closed]

I posted the same question on Math SE since this one got put on hold.
Link to Math SE question:Polygonal Mersenne numbers
Polygonal numbers are of the form $\cfrac {n^2(s-2)-n(s-4)}{2}$, where $s$ ...

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votes

1
answer

92
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### How to generate computational data in graph theory?

For a given number of nodes how many non-isomorphic graphs are available? Might be this is an open problem. For less number of vertices some computational statistics available.
I want to get all non-...

0
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0
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### Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...

3
votes

2
answers

512
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### efficiently checking that a field extension is Galois

Let $K \subset L$ be an algebraic extension of fields finitely presented over a prime field or over an algebraically closed field. Is there an efficient procedure to check that $L/K$ is Galois? To ...

4
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0
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### Weak randomness relative to finite-state machines

Is there a nice example of a sequence that looks random to any predictor whose predictions use a finite-state machine?
More precisely, consider a finite-state machine $M$ with input alphabet {0,1} ...

6
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6
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452
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### Procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that ...

0
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3
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### System of quadratic equations with 18 unknown

So I want to solve for a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r which satisfy the following system of equations: ( I only need positive integer (or 0) solution)
a g + c h + b i + g j + i ...

85
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17
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### Important open problems that have already been reduced to a finite but infeasible amount of computation

Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer."
Some questions (e.g....

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3
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1k
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### What to do when your research runs into a computationally challenging problem?

Occasionally, but more frequently lately, I would like to perform some hard computations. As an example, yesterday the following question came up:
What is the projective dimension of the edge ideal ...

3
votes

1
answer

624
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### Efficiently computing with pullbacks and pushouts

Often when computing in category theory, one has to show that some square is cartesian. Depending on the number of maps involved, and their arrangement, it's somewhat difficult to write down exactly ...