**2**

votes

**2**answers

698 views

### viewing the second fundamental form as a tensor

Dear all,
Thank you for your time reading this post. I am a student in computer science so this viewpoint of the second fundamental form may be interesting to you.
I would like to understand the ...

**3**

votes

**2**answers

1k views

### finding numbers at k hamming distance

Guys,
I have N < 2^n randomly generated n-bit numbers stored in a file the lookup for which is expensive. Given a number Y, I have to search for a number in the file that is at most k hamming ...

**8**

votes

**5**answers

416 views

### Syntactically capturing complexity classes

Primitive recursive functions are syntactically constructible in the sense that from a set of "axioms" we can build every function in the set $PR$. This basicly means that we can build a machine that ...

**1**

vote

**1**answer

252 views

### Can a polynomial size CFG describe the finite language \{$w \pi(w)$ : $\pi(w)$ is fixed string permutation, $|w|=n$ is fixed\} over alphabet \{0,1\}?

Can a polynomial size Context free grammar describe the finite language {$w \pi(w)$ : $\pi(w)$ is fixed string permutation, $|w|=n$ is fixed} over alphabet of {0,1}?
One case this is possible is when ...

**4**

votes

**1**answer

467 views

### What fails when using call/cc as realizer of the Peirce formula

Define the axiom constants $p_{A,B}^{((A\rightarrow B)\rightarrow A)\rightarrow A}$ as realizers of the Peirce formula, and $f_A^{\bot\rightarrow A}$ as realizers of the Ex Falso Quodlibet. Then ...

**0**

votes

**0**answers

380 views

### Applications of the property of Kendall-Mann numbers

I am looking for an application of the Kendall-Mann sequence (KM) which uses the property $M(n+1)/M(n) = n - 1/2 + O(1/n)$ ($n \to \infty$) in science ( computer science ( sorting), physics, biology ...

**28**

votes

**17**answers

6k views

### Computer Science for Mathematicians

This is a big-list community question, so I'm sorry in advance if it is deemed too soft but I haven't seen anything similar yet.
I've seen computer scienctists post questions looking to learn things ...

**1**

vote

**2**answers

734 views

### Verifying a sequence that converges to pi [closed]

A computer program ouputs the digits of $\pi$ by evaluating the recurrence relation
$a_{n+1} = a_n + sin \ a_n$
with $a_0 = \frac{6}{5}$
Does the sequence actually converge or is this just ...

**5**

votes

**1**answer

387 views

### What is the pathwidth of the 3D-grid (mesh or lattice) with sidelength k?

This question is now also on http://cstheory.stackexchange.com/questions/4081/what-is-the-pathwidth-of-the-3d-grid-mesh-or-lattice-with-sidelength-k, where a discussion started, and one reference ...

**4**

votes

**2**answers

581 views

### Coloring edges on a graph s.t. the set of edges for any two vertices have no more than 'k' colors in common

Please imagine the case where one has a planar graph, $G$, with a set of $|V|$ vertices, $(v_1, ..., v_{|V|}) \in V$, and $|E|$ edges, $(e_1, ..., e_{|E|}) \in E$. Now, provided a total of $N$ ...

**8**

votes

**3**answers

2k views

### n-dimensional voronoi diagram

Hi, I need to compute the voronoi diagram of a set of points in $R^n$.
I'm quite unschooled on the topic, could someone point me to the right references so that I can
a) understand the theory behind ...

**5**

votes

**1**answer

353 views

### Are innermost reductions perpetual in untyped $\lambda$-calculus?

Background
In the untyped lambda calculus, a term may contain many redexes, and
different choices about which one to reduce may produce wildly
different results (e.g. $(\lambda x.y)((\lambda ...

**2**

votes

**3**answers

317 views

### Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?

I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...

**15**

votes

**7**answers

1k views

### Between mu- and primitive recursion

It is well known that primitive recursion is not powerful enough
to express all functions, Ackermann function being probably the best
known example.
Now, in the logic courses (that I have had look ...

**5**

votes

**1**answer

333 views

### Drawing graphs on circles

Please consider the following problem:
Given: a simple graph (without self-loops and without multiple edges) $G$ on $n$ vertices.
Task: place equidistantly the vertices of $G$ on a circle of unit ...

**5**

votes

**1**answer

3k views

### Meaning of \Subset notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...

**28**

votes

**1**answer

3k views

### An edge partitioning problem on cubic graphs

Hello everyone,
I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...

**6**

votes

**1**answer

424 views

### post correspondence problem variant

Is there an algorithm which takes as input two lists of words $v_1,...,v_n$ and $w_1,...,w_n$ over an alphabet $X$ and decides if there is an infinite sequence $(k_i)$ where $1 \leq k_i \leq n$ for ...

**4**

votes

**3**answers

986 views

### Optimal packing of spheres tangent to a central sphere

Please consider a central, ordinary 2-sphere $S_1$, of some radius $r_1$, and a second ordinary sphere, $S_2$, of radius $r_2$, where $r_2 \leq r_1$.
My question concerns optimal values for the ...

**0**

votes

**3**answers

297 views

### boolean functions and averaging / counting

Hey guys,
I have a slightly imprecise question. I would like say something about a whole set of binary strings evaluated by a binary function by just looking at some type of average. The easiest ...

**1**

vote

**2**answers

776 views

### What is the right citation for the power iteration method to find eigenvalues?

What is the right citation for the power iteration method to find eigenvalues, if I want to cite the method in a paper? I've seen some Google PageRank references in this context. But Brin and Page ...

**2**

votes

**4**answers

743 views

### Enumerative algorithm through inclusion-exclusion

Hello everybody !
I wondered, without really knowing where to search, whether there was a "smart" way to enumerate/iterate over all the elements of a set which can be counted by inclusion-exclusion. ...

**3**

votes

**1**answer

472 views

### Approximating an integral representation of the Number Partition Problem

One can write out an integral whose solution gives the number of solutions to the NP-Complete Number Partition Problem and I'm wondering if anyone has an suggestions or ideas on who to solve or ...

**2**

votes

**0**answers

257 views

### Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...

**2**

votes

**0**answers

183 views

### Constructing hard inputs for the complement of bounded halting

If there is always a hard input for the complement of bounded halting, can that input be constructed?
More precisely, suppose that
for any deterministic TM $M$ accepting
$$
...

**8**

votes

**1**answer

852 views

### Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice

Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l_1, ..., l_N) \in L$ and total lattice points $P = \prod{l_i}$, for $i = 1, ..., N$. At each ...

**1**

vote

**2**answers

955 views

### post correspondence problem

I have read a couple of proofs for the undecidability of the post correspondence problem, but neither reference gave a concrete example of two lists of words over a fixed alphabet such that the ...

**1**

vote

**2**answers

814 views

### Practical use of probability amplification for randomized algorithms

Normally a 2-sided error randomized algorithm will have some constant error $\varepsilon < 1/2$. We know that we can replace the error term for any inverse polynomial. And the inverse polynomial ...

**2**

votes

**1**answer

291 views

### Approximating a recursively-defined function

Let $$f(k) := \frac{2k-1}{k}\bigl(1-\sum\limits_{i\lt k}\frac{i\ f(i)}{k+i-1}\bigr)$$ for $k\in\mathbb{N}^{+}$.
So $f(1) = 1$, $f(2) = 3/4$, $f(3) = 35/72$, etc.
(This function arises when ...

**4**

votes

**3**answers

886 views

### Complete problems for randomized complexity classes

It is believed that $BPP$ has no complete problems. Even for $BPP^O$ for a suitable oracle $O$ it is believed not to have complete problems, unless P=BPP. I wonder if the class MA (the randomized ...

**10**

votes

**3**answers

640 views

### Complete Extensions of First Order Logic (or Language)

Lindstrom's theorem states that any extension of FOL more expressible than FOL fails to have either compactness or Lowenheim-Skolem. When I first read Lindstrom's theorem my first reaction was: "Does ...

**1**

vote

**0**answers

366 views

### Minimizing quadratic form over permutations

Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...

**6**

votes

**2**answers

882 views

### Bijective proof of weak form of Stirling's approximation

There are short and sweet proofs of various forms of Stirling's approximation. But even the sweetest among them don't instill the same conviction in the reader as a direct bijective proof.
Computer ...

**2**

votes

**2**answers

331 views

### Efficient computation of AB^-1 for matrices

Hi there,
Sorry if this has already been asked before. I tried googling for it, but perhaps I could not find the right words to search for. My question is: Which is the fastest way to compute ...

**28**

votes

**3**answers

4k views

### Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \leq k } ...

**6**

votes

**5**answers

2k views

### Aren't “oracle machines” unsound concepts?

From Wikipedia (bold emphasis at the end is mine):
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as ...

**14**

votes

**2**answers

1k views

### Structure theorems for Turing-decidable languages?

Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such ...

**1**

vote

**1**answer

417 views

### Number of subset sums

Let $\mathbb{F}_q$ be a finite field with characteristic $p$ and $p < q$ (i.e. not a prime field). Let $D\subseteq \mathbb{F}_q$ be a some set with $|D|=n$. Find a non-empty subset ...

**3**

votes

**4**answers

1k views

### Does an “efficient” random number generator exist?

Given some number $n$ and a seed number $s$<$n$, I want a random number generator (RNG) that will go through all integers `<$n$ before coming back to $s$. The resulting random number must be ...

**4**

votes

**7**answers

1k views

### How to generate a net on a 8-dimensional sphere

Using Matlab, how to generate a net of 3^10 points that are evenly located (or distributed) on the 8-dimensional unit sphere?
Thanks for any helpful answers!

**8**

votes

**4**answers

503 views

### What is the relationship between “translation” and time complexity?

Consider the problem of deciding a language $L$; for concreteness, say that this is the graph isomorphism problem. That is, $L$ consists of pairs of graphs $(G, H)$ such that $G\simeq H$. Now the ...

**13**

votes

**3**answers

1k views

### What is the history of the Y-combinator?

Inspired by the comments to this question, I wonder if someone can explain the history of the fixed point combinator (often called the Y combinator) in lambda calculus.
Where did it first appear? ...

**24**

votes

**7**answers

5k views

### Problems known to be in both NP and coNP, but not known to be in P

One such problem I know is integer factorization.
What are other interesting cases?

**2**

votes

**2**answers

187 views

### Indexing schemes of binary sequences

I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight.
Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to ...

**3**

votes

**0**answers

306 views

### Inversion density: Have you seen this concept?

Let n > 1 be an integer.
Let A be an array, indexed from 1 to n, of n values
A(i) coming from the finite set {0,1}.
(More generally, the values can come from any
totally ordered set, but I only need ...

**0**

votes

**1**answer

171 views

### Building optimal rewriting rules.

Please give me some pointers where I can learn more about the following problem:
I have two alphabets A and B. A have a dictionary which contains words in A together with their translation in B (ie. ...

**1**

vote

**1**answer

322 views

### Algorithm for generating a size k error-correcting code on n bits

I want to generate a code on n bits for k different inputs that I want to classify. The main requirement of this code is the error-correcting criteria: that the minimum pairwise distance between any ...

**5**

votes

**3**answers

754 views

### Software for Tree-Decompositions

Does anybody know about software that exactly calculates the tree-width of a given graph and outputs a tree-decomposition? I am only interested in tree-decompositions of reasonbly small graphs, but ...

**1**

vote

**1**answer

183 views

### $\omega$-monoids

Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of ...

**32**

votes

**9**answers

3k views

### What is the shortest program for which halting is unknown?

In short, my question is:
What is the shortest computer program for which it is not known whether or not the program halts?
Of course, this depends on the description language; I also have the ...