Questions tagged [computation]

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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 $\log(x)$ ...
Dieter Kadelka's user avatar
51 votes
8 answers
5k views

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 ...
Gordon Royle's user avatar
  • 12.3k
4 votes
3 answers
286 views

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 ...
ReverseFlowControl's user avatar
2 votes
0 answers
85 views

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, ...
Pulcinella's user avatar
  • 5,506
0 votes
0 answers
56 views

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 ...
user499408's user avatar
5 votes
3 answers
1k views

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+...
Gabriel's user avatar
  • 51
1 vote
1 answer
113 views

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 ...
Valentin Brimkov's user avatar
11 votes
1 answer
300 views

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 ...
James Propp's user avatar
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3 votes
1 answer
439 views

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 ...
Tristan Duquesne's user avatar
1 vote
0 answers
110 views

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 ...
Per Alexandersson's user avatar
4 votes
0 answers
75 views

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,...
Paul's user avatar
  • 509
4 votes
1 answer
128 views

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 ...
Enclitic Sarcool's user avatar
0 votes
1 answer
808 views

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 ...
Ahmad Jamil Ahmad Masad's user avatar
1 vote
1 answer
155 views

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 ...
Andrew Penland's user avatar
5 votes
0 answers
77 views

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 votes
0 answers
162 views

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 $C \subset D$, ...
Nick L's user avatar
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4 votes
1 answer
747 views

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)...
Ahmad Jamil Ahmad Masad's user avatar
8 votes
1 answer
479 views

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 \...
Ahmad Jamil Ahmad Masad's user avatar
1 vote
1 answer
715 views

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 ...
Ahmad Jamil Ahmad Masad's user avatar
7 votes
1 answer
419 views

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 ...
Christopher King's user avatar
0 votes
0 answers
150 views

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 ...
Johnny Cage's user avatar
  • 1,543
6 votes
1 answer
248 views

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 ...
user avatar
6 votes
1 answer
493 views

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\...
Shant Boodaghians's user avatar
8 votes
2 answers
367 views

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 ...
DLIN's user avatar
  • 1,905
6 votes
2 answers
1k views

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{-}...
Joseph O'Rourke's user avatar
19 votes
0 answers
505 views

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 ...
H A Helfgott's user avatar
  • 19.3k
5 votes
0 answers
109 views

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 ...
Nicolas Ford's user avatar
  • 1,520
1 vote
1 answer
201 views

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 $s$ ...
redelectrons's user avatar
-1 votes
1 answer
92 views

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-...
Supriyo's user avatar
  • 343
0 votes
0 answers
80 views

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{...
Turbo's user avatar
  • 13.7k
3 votes
2 answers
525 views

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 ...
Dima Sustretov's user avatar
4 votes
0 answers
122 views

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} ...
James Propp's user avatar
  • 19.4k
6 votes
6 answers
454 views

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 votes
3 answers
475 views

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 ...
Mathfish's user avatar
85 votes
17 answers
6k views

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....
16 votes
3 answers
1k views

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 ...
Hailong Dao's user avatar
  • 30.3k
3 votes
1 answer
625 views

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 ...
Harry Gindi's user avatar
  • 19.4k