Questions tagged [computation]
The computation tag has no summary.
42 questions
3
votes
1
answer
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A more general conjecture about palindromic numbers in the comments section of the page of the OEIS sequence A266577
I posed a conjecture which is a generalization of Conjecture on palindromic numbers
My question is to find a proof or a disproof of it.
I have put it in the comments section of the page of the OEIS ...
26
votes
3
answers
3k
views
Background for the Elkies-Klagsbrun curve of rank 29
Elkies and Klagsbrun have recently announced an elliptic curve over Q of rank (at least) 29, improving on the previous record from 2006!
https://web.math.pmf.unizg.hr/~duje/tors/z1.html
It has trivial ...
- 1,347
3
votes
0
answers
115
views
Simplifying sums with CAS or theorem prover
To verify that a certain map is a chain homotopy I could reduce it to an evaluation of
$S = S_1 + S_2$ where
$$
S_1 = \sum_{a=0}^p \sum_{b=0}^{p+1} (-1)^{a+b} n_b \cdot e_a
$$
$$
S_2 = \sum_{b=0}^p \...
- 748
1
vote
0
answers
297
views
Christoph Benzmüller and Gödel's ontological proof?
Are there any notable mathematical or logical issues within Christoph Benzmüller and Bruno Woltzenlogel-Paleo formalized Gödel's ontological proof (pdf) that has been identified by the community?
5
votes
0
answers
188
views
Tutte polynomials of square grid graphs
I'd like to know the Tutte polynomial of the $n$-by-$n$ square grid graph for $n$ between 1 and 16. Criel Merino, in his paper "On the number of tilings of the rectangular board with T-...
- 20.1k
3
votes
1
answer
669
views
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)$ ...
- 2,241
52
votes
8
answers
6k
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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 ...
- 13.7k
5
votes
3
answers
478
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 ...
2
votes
0
answers
126
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, ...
- 6,191
0
votes
0
answers
87
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 ...
5
votes
3
answers
2k
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+...
- 51
1
vote
1
answer
167
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 ...
- 105
11
votes
1
answer
399
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 ...
- 20.1k
3
votes
1
answer
577
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 ...
- 141
1
vote
0
answers
187
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 ...
- 16.1k
4
votes
0
answers
96
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,...
- 519
4
votes
1
answer
185
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 ...
- 111
0
votes
1
answer
871
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 ...
1
vote
1
answer
197
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 ...
- 413
5
votes
0
answers
97
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 ...
2
votes
0
answers
187
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 $C \subset D$, ...
- 6,947
5
votes
1
answer
884
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 $(a(n)...
8
votes
1
answer
533
views
simple conjecture on palindromes in base 10 [closed]
The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \...
1
vote
1
answer
793
views
Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets
Assume $x$ is a variable belongs to $\mathbb R \setminus \{ 0,-1,+1 \}$ and consider for all $i, j \in \mathbb N$,
$$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$
then for all $n \in \mathbb N$ the ...
7
votes
1
answer
556
views
What Turing degree would allow you to "compute" the axioms of ZFC in some countable model of ZFC?
It is established in this post that you there is no computable model of ZFC, yet it can be computed in by a PA-degree oracle machine. Note that when we see "compute a model", we just mean that ...
- 7,158
1
vote
0
answers
157
views
Computer algebra programs for dummies [duplicate]
In the way of my investigations I have encounter the following computational problem: I have a system of 5 algebraic equations and I want to eliminate 4 of them. I also need to do a functional ...
- 1,631
6
votes
1
answer
290
views
Problem on triangles
Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...
user101163
6
votes
1
answer
540
views
Algorithm to compute Matrix Sign Rank?
The (generalised) sign-rank of a (generalised) sign pattern $S\in \{+,-,0\}^{n\times m}$
is the minimum rank of all matrices with the same sign pattern, i.e.
$$
\min\left\{\operatorname{rank}(M)\ :\ M\...
8
votes
2
answers
426
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 ...
- 1,965
7
votes
2
answers
2k
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{-}...
- 153k
19
votes
0
answers
563
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 ...
- 22k
5
votes
0
answers
142
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 ...
- 1,530
1
vote
1
answer
226
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 $s$ ...
- 113
-1
votes
1
answer
121
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-...
- 363
0
votes
0
answers
102
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{...
- 1
3
votes
2
answers
650
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 ...
- 4,160
4
votes
0
answers
140
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} ...
- 20.1k
6
votes
6
answers
541
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
556
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 ...
- 3
95
votes
18
answers
7k
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
2k
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 ...
- 30.9k
3
votes
1
answer
706
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 ...
- 19.9k