question_id string | site string | title string | body string | link string | tags list | votes int64 | creation_date timestamp[s] | comments list | comment_count int64 | category string | diamond int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|
1 | bioacoustics | How do animals adapt to a partially or fully deaf individual of their group? | In eusocial animals for which hearing is important to communicate between individuals, is there any species apart humans for which there are some evidence that the group adapts their acoustic communication to a fully or partially deaf individual of the group?
For instance, by systematically vocalizing louder/closer to... | https://bioacoustics.stackexchange.com/questions/1235/how-do-animals-adapt-to-a-partially-or-fully-deaf-individual-of-their-group | [
"hearing",
"adaptation"
] | 6 | 2022-11-14T07:01:24 | [
"Thanks @DanStowell, done!",
"Did you search for possible literature? If so it would be good to give some indication of that. Also, I think \"eurosocial\" should be \"eusocial\", right?"
] | 2 | Science | 0 |
2 | codegolf | Minimum cost of solving the Eni-Puzzle | You're tasked with writing an algorithm to efficiently estimate cost of solving an [Eni-Puzzle](http://www.enipuzzles.com/buy/eni-braille-puzzle-with-bold-colors) from a scrambled state as follows:
You're given m lists of containing n elements each(representing the rows of the puzzle). The elements are numbers between... | https://codegolf.stackexchange.com/questions/182647/minimum-cost-of-solving-the-eni-puzzle | [
"puzzle-solver",
"fastest-algorithm",
"sliding-puzzle"
] | 6 | 2019-04-03T19:53:28 | [
"@Jonah Because The costs I've outlined here are not the actual number of moves required to make equivalent moves on the puzzle. \"Swapping\" moves have a higher cost. To solve the actual puzzle, you first reach the state I described, or a state that is only one sliding move away. Because my specifications don't al... | 5 | Technology | 0 |
3 | crypto | Share Conversion between Different Finite Fields | Let us have any linear secret sharing scheme (LSSS) that works on some field $Z_{p}$, where p is some prime or a power of a prime e.g., Shamir Secret Sharing, Additive secret Sharing. The problem at hand is simple, for any secret shared value in $Z_{p}$, is it possible to convert it (and its shares) to elements on $Z_{... | https://crypto.stackexchange.com/questions/47554/share-conversion-between-different-finite-fields | [
"secret-sharing",
"multiparty-computation",
"finite-field",
"garbled-circuits",
"function-evaluation"
] | 11 | 2017-05-19T06:58:27 | [
"@Dragos... Btw.... Are you from Bristol's group? that Dragos?",
"I see..Even converting shares from $Z_p \\rightarrow Z_2^{r}$ is a hard problem - at least I have encountered it several times and don't know how to solve it efficiently other than bit decompose and simulate the binary circuit in $Z_p$.",
"@Drago... | 6 | Technology | 0 |
4 | crypto | RSA key such that pi deciphers to your name per RSA-OAEP | Can you efficiently construct an RSA public/private key pair with $8k$-bit public modulus such that $C=\left\lfloor\pi\,2^{8k-2}\right\rfloor$ deciphers per RSA-OAEP to your name as a bytestring in ASCII or UTF-8?
The decryption must be per RSAES-OAEP of [PKCS#1v2.2](http://mpqs.free.fr/h11300-pkcs-1v2-2-rsa-cryptogra... | https://crypto.stackexchange.com/questions/80211/rsa-key-such-that-pi-deciphers-to-your-name-per-rsa-oaep | [
"rsa",
"oaep"
] | 13 | 2020-04-23T04:22:27 | [] | 0 | Technology | 0 |
5 | crypto | Finding $x$ such that $g^x\bmod p<p/k$? | In a Schnorr group as used for DSA, of prime modulus $p$, prime order $q$, generator $g$ (with $p/g$ small), how can we efficiently exhibit an $x$ with $0<x<q$ such that $g^x\bmod p<p/k$, for sizable $k$ but $k\ll\sqrt q$ ?
I see that for small $k$, it is enough to try incremental values of $x$ until finding an $x$ th... | https://crypto.stackexchange.com/questions/48503/finding-x-such-that-gx-bmod-pp-k | [
"discrete-logarithm",
"group-theory"
] | 19 | 2017-06-20T08:57:11 | [
"Can you please provide me with some real values of variables in question instead of me attempting with random values.",
"@pintor: no, I have made no progress nor got more feedback than the above comments.",
"@fgrieu, did you get any more updates on the problem? I'm also curious",
"This seems to reduce to a d... | 8 | Technology | 0 |
6 | crypto | Time-memory tradeoffs in Shor's algorithm | Can a quantum computer with insufficient qubits to factor an integer of a given size make _any_ progress in factoring it? For example, what if a quantum computer is only one qubit short of what is necessary to attack a specific integer? Is it capable of making any progress in factoring it, or would it be just as useles... | https://crypto.stackexchange.com/questions/67910/time-memory-tradeoffs-in-shors-algorithm | [
"factoring",
"quantum-cryptanalysis",
"shors-algorithm"
] | 8 | 2019-03-09T22:20:47 | [
"Interesting question. One problem certainly could occur when you have less qubits than the number of the period $r$ ($a^r \\bmod N$). This would probably mess up the inverse quantum fourier transform which amplifies the correct period (quantum wave interference), thus rendering the whole computation impractical. I... | 3 | Technology | 0 |
7 | crypto | Does the bias in RC4 drop asymptotically further in the keystream? | It's well-known that the RC4 keystream has significant biases that become less prominent later in the keystream. The most severe bias is in the second byte, which has a 128-1 bias towards zero. Other biases remain, and it's typically recommended to drop between 768 and 3072 bytes of the keystream.
Will dropping one mo... | https://crypto.stackexchange.com/questions/72353/does-the-bias-in-rc4-drop-asymptotically-further-in-the-keystream | [
"stream-cipher",
"rc4",
"distinguisher"
] | 7 | 2019-08-03T01:44:25 | [
"@MaartenBodewes I have not yet, but I also haven't done much research into it. There's a lot of academic text on the (in)security of RC4 which I should familiarize myself with.",
"Hey Forest. This question popped up again. Unfortunately I don't know the answer, but maybe you've found one in the mean time?",
"@... | 7 | Technology | 0 |
8 | cstheory | Partial circulant matrices: Is there a non-zero vector $v\in \{-1,0,1\}^n$ such that $Mv=0$? | The following question arose as a side product of some work I have been part of recently. An $m$ by $n$ $(0,1)$-matrix $M$ is partial circulant if it can be formed by taking the first $m$ rows of a [circulant matrix](http://en.wikipedia.org/wiki/Circulant_matrix) and all its entries are either $0$ or $1$. We assume tha... | https://cstheory.stackexchange.com/questions/12060/partial-circulant-matrices-is-there-a-non-zero-vector-v-in-1-0-1-n-such | [
"cc.complexity-theory",
"ds.algorithms",
"matrices"
] | 20 | 2014-11-04T06:46:47 | [
"The circulant case was in fact considered at mathoverflow.net/questions/168474/… . However, it seems that there is fatal bug in one of the answers and the other answer is not a complete solution.",
"would it be more natural to study the circulant case 1st vs the partial circulant? also (idea) the other problem a... | 2 | Science | 1 |
9 | cstheory | Are monotone Boolean functions in P well-approximated by monotone polynomial-size circuits? | **Question 1:** Is it true that for every polynomial $p(n)$ and $\epsilon >0$ there is a polynomial $q(n)$ such that every monotone Boolean function on $n$ variables that can be expressed by a Boolean circuit of size $p(n)$ can be $\epsilon$-approximated by a _monotone_ Boolean circuit of size at most $q(n)$.
A funct... | https://cstheory.stackexchange.com/questions/31473/are-monotone-boolean-functions-in-p-well-approximated-by-monotone-polynomial-siz | [
"cc.complexity-theory",
"circuit-complexity",
"pr.probability"
] | 18 | 2015-05-12T23:58:06 | [
"Dear Gil, now I see the point: I just wrongly interpreted your question! We indeed have that every monotone circuit, which needs only to coincide with b-Clique on an extremely small, but special set of inputs, must be large. You, however, do not specify this subset of \"hard\" inputs, only its ratio $1-\\epsilon$.... | 8 | Science | 0 |
10 | cstheory | Weighted Hamming distance | Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is not necessarily Theoretical Computer Science but I think similar things come up sometimes, for instance in randomness extraction.
Define:
$d(x,y)=$ the Hamming distance between binary strings $x$ and $y$ of leng... | https://cstheory.stackexchange.com/questions/2637/weighted-hamming-distance | [
"co.combinatorics",
"randomness"
] | 20 | 2010-11-01T22:06:48 | [] | 0 | Science | 1 |
11 | cstheory | To what extent MSO = WS1S, when adding relations? | [This question has been asked on MathOverflow with no luck a month ago.]
Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma =\\{a_1, \ldots, a_n\\}$, I define two structures:
$\mathbb{N}(w) = \langle \mathbb{N}, <, Q_{a_1}, \ldots, Q_{a_n} \rangle$,
and the more usual _word model_ :
$\ma... | https://cstheory.stackexchange.com/questions/15/to-what-extent-mso-ws1s-when-adding-relations | [
"lo.logic",
"automata-theory",
"descriptive-complexity"
] | 19 | 2010-08-16T13:32:53 | [
"Your model $\\mathbb{N}(w)$ is very unusual : what is the need of considering a finite word on an infinite underlying structure ? If you forget the letter predicates (which do not change the problem a lot), you are in fact comparing MSO on finite linear orders, and WS1S on the order $\\omega$. This is strange, bec... | 6 | Science | 0 |
12 | cstheory | Descriptive complexity of communication complexity classes | It is well known that some major complexity classes, like P or NP, admit a full logical characterization (e.g NP = existential 2nd order logic by Fagin's theorem). On the other hand, one can also define complexity classes in communication complexity (where P = problems solvable with poly(logN) communication etc. - see ... | https://cstheory.stackexchange.com/questions/9041/descriptive-complexity-of-communication-complexity-classes | [
"cc.complexity-theory",
"lo.logic",
"communication-complexity"
] | 18 | 2011-11-19T14:15:36 | [
"You may want to limit the computational power of the parties, which is not done in the usual communication complexity models."
] | 1 | Science | 0 |
13 | cstheory | Does $EXP\neq ZPP$ imply sub-exponential simulation of BPP or NP? | By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary.
I think this is a proof: if $EXP\neq BPP$ then from [IW98] we get that BPP has such a simulation. Otherwise we have that $EXP=BPP$, which implies $RP=NP$... | https://cstheory.stackexchange.com/questions/430/does-exp-neq-zpp-imply-sub-exponential-simulation-of-bpp-or-np | [
"cc.complexity-theory",
"reference-request",
"complexity-classes",
"conditional-results"
] | 29 | 2010-08-23T13:21:04 | [
"ZPP is closed under complement, and contained in NP. So, $\\mathrm{NP\\subseteq ZPP}$ implies NP = ZPP = coNP, hence PH = NP = ZPP.",
"Why does NP in ZPP imply PH in ZPP?",
"(Just for reference): Identical question on MO: mathoverflow.net/questions/35945. Maybe someone finds the comments there inspiring.",
"... | 4 | Science | 1 |
14 | cstheory | Interesting PCP characterization of classes smaller than P? | The PCP theorem, $\mathsf{NP} = \mathsf{PCP}(\mathsf{log}\, n, 1)$, involves probabilistically checkable proofs with polynomial time verifiers, so the smallest class that can be characterized in this way (that is, $\mathsf{PCP}(0, 0)$) must be $\mathsf{P}$. There are also PCP characterizations of larger complexity clas... | https://cstheory.stackexchange.com/questions/12060/interesting-pcp-characterization-of-classes-smaller-than-p | [
"cc.complexity-theory",
"complexity-classes",
"pcp"
] | 20 | 2012-07-18T11:54:42 | [
"Your comment is contradictory to your question where you count “PCP(0, 0) = P” as a PCP characterization of P.",
"I suppose I meant are there any characterizations that don't follow immediately from the definition, in the same way that PCP(log n, 1) is non-obvious characterization of NP.",
"Why not? If you res... | 3 | Science | 1 |
15 | cstheory | Is Hankelability NP-hard? | I asked this question on [SO](https://stackoverflow.com/questions/29484864/an-algorithm-to-detect-permutations-of-hankel-matrices) on April 7 and added a bounty which has now expired but no poly time solution has been found yet.
I am trying to write code to detect if a matrix is a permutation of a Hankel matrix. Here... | https://cstheory.stackexchange.com/questions/31174/is-hankelability-np-hard | [
"cc.complexity-theory"
] | 28 | 2015-04-16T10:44:55 | [
"Cross-posted now to mathoverflow.net/questions/204294/is-hankelability-np-hard",
"@IgorShinkar I mean first some permutation on the order of the rows and then a possibly different permutation on the order of the columns, then stop.",
"Do you mean apply first some permutation on the rows and then some permutati... | 6 | Science | 1 |
16 | cstheory | Model-checking for three-variable logics and restricted structures | Denote the $k$-variable fragment of logic $L$ by $L^{(k)}$. The model-checking problem for a logic $L$ with respect to a class of structures $C$, denoted $MC(L,C)$, is the decision problem
> $MC(L,C)$
> _Input:_ formula $\phi$ of $L$, structure $S$ from $C$
> _Question:_ does $S$ satisfy $\phi$?
Is there a logi... | https://cstheory.stackexchange.com/questions/2637/model-checking-for-three-variable-logics-and-restricted-structures | [
"cc.complexity-theory",
"co.combinatorics",
"lo.logic",
"descriptive-complexity"
] | 20 | 2010-11-03T11:47:19 | [
"@MichaëlCadilhac alas, no.",
"Have you made any progress on that question, András?"
] | 2 | Science | 1 |
17 | cstheory | In an $m$ by $n$ Boolean matrix, can you find a square block whose four corners are ones in $O(m \cdot n)$ time? | **Decision Problem**
Input: An $m$ by $n$ Boolean matrix $M$.
Decision Question: Does there exist a square block within $M$ such that upper-left corner entry == upper-right corner entry == lower-left corner entry == lower-right corner entry == 1? That is, all four corners of the square block are 1's.
**Cubic Time So... | https://cstheory.stackexchange.com/questions/47588/in-an-m-by-n-boolean-matrix-can-you-find-a-square-block-whose-four-corners | [
"ds.algorithms",
"matrices",
"computational-geometry",
"search-problem",
"boolean-matrix"
] | 18 | 2020-09-18T11:04:38 | [
"It feels somehow like an FFT would help, although I haven't been able to puzzle out details. You're looking for something translation-invariant, and working in frequency space often helps with that sort of thing. Tossing out this comment in case someone else can spot something.",
"@MichaelWehar: ok, let me know ... | 14 | Science | 0 |
18 | cstheory | Problem unsolvable in $2^{o(n)}$ on inputs with $n$ bits, assuming ETH? | If we assume the Exponential-Time Hypothesis, then there is no $2^{o(n)}$ algorithm for $n$-variable 3-SAT, and many other natural problems, such as 3-COLORING on graphs with $n$ vertices. Notice though that, in general, encoding the input for $n$-variable 3-SAT or $n$-vertex 3-COLORING takes something like $O(n\log n)... | https://cstheory.stackexchange.com/questions/16148/problem-unsolvable-in-2on-on-inputs-with-n-bits-assuming-eth | [
"cc.complexity-theory",
"sat",
"planar-graphs",
"succinct"
] | 48 | 2013-01-19T08:24:59 | [
"I believe that it is an open question if we can have a lower bound $2^{\\Omega(n)}$ under ETH, where $n$ is the bit size. We know that proving lower bound $\\Omega(c^n)$ under SETH would disprove SETH.",
"How about the following problem, for some large constant $c>0$? Given the encoding of a non-deterministic T... | 8 | Science | 1 |
19 | cstheory | Sylver Coinage Game | A game in which the players alternately name positive integers that are not sums of previously named integers (with repetitions being allowed). The person who names 1 (so ending the game) is the loser.
The question is: If player 1 names ‘16’, and both players play optimally thereafter, then who wins?
It has been know... | https://cstheory.stackexchange.com/questions/22112/sylver-coinage-game | [
"co.combinatorics",
"combinatorial-game-theory"
] | 18 | 2014-04-14T04:03:06 | [
"@NealYoung The algorithm in Winning Ways only gives whether an opening move is winning or not -- in accordance with Conway's comment, it relies on a theorem saying finitely moves of the form 2^a 3^b win, and I imagine (Conway having told me of that fact before) that this is what he meant when he said that an algor... | 7 | Science | 0 |
20 | cstheory | Complexity of the homomorphism problem parameterized by treewidth | The _homomorphism problem_ $\text{Hom}(\mathcal{G}, \mathcal{H})$ for two classes $\mathcal{G}$ and $\mathcal{H}$ of graphs is defined as follows:
> **Input:** a graph $G$ in $\mathcal{G}$, a graph $H$ in $\mathcal{H}$
>
> **Output:** decide if there is a homomorphism from $G$ to $H$, i.e., a mapping $h$ from the ver... | https://cstheory.stackexchange.com/questions/34877/complexity-of-the-homomorphism-problem-parameterized-by-treewidth | [
"graph-algorithms",
"parameterized-complexity",
"treewidth",
"fixed-parameter-tractable",
"homomorphism"
] | 18 | 2016-06-02T03:37:34 | [
"This question is still open, but one remark: there is an FPT algorithm parameterized by treewidth for the graph isomorphism problem, here: epubs.siam.org/doi/abs/10.1137/… (Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh, \"Fixed-Parameter Tractable Canonization and Isomorphism Test fo... | 1 | Science | 0 |
21 | cstheory | Is Node Multiway Cut NP-complete on planar graphs when all terminals lie on the outer face? | I am interested in the following problem.
**Node Multiway Cut on Planar Graphs with terminals on the outer face**
* Instance: A plane graph G, and integer k, and a set $S \subseteq V(G)$ of terminals which are all incident on the outer face of G.
* Question: Is there a set of vertices $X \subseteq V(G)$ of size a... | https://cstheory.stackexchange.com/questions/8969/is-node-multiway-cut-np-complete-on-planar-graphs-when-all-terminals-lie-on-the | [
"cc.complexity-theory",
"np-hardness",
"planar-graphs"
] | 17 | 2011-11-14T09:06:13 | [
"I know for sure that the algorithm I mentioned works for edge weighted Steiner tree, but one should be able to adapt it to work for vertices.",
"I think that this is equivalent to solving vertex weighted Steiner tree where all terminals on the outer face. The very loose idea would be something like: consider the... | 2 | Science | 0 |
22 | cstheory | Can short-distance connectivity be harder than connectivity? | Has anybody seen the following (or similar) question being considered:
> Can it be **easier** to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of _short_ $s$-$t$ paths?
A bit more formally, the _distance_ -$k$ _connectivity_ problem STCON(n,k) is, given a subgraph of a comple... | https://cstheory.stackexchange.com/questions/31005/can-short-distance-connectivity-be-harder-than-connectivity | [
"cc.complexity-theory",
"graph-algorithms",
"circuit-complexity",
"lower-bounds",
"dynamic-programming"
] | 17 | 2015-04-03T10:00:23 | [] | 0 | Science | 0 |
23 | cstheory | Sequences with sublogarithmic concat and approximate split | Is there a data structure for representing sequences that supports the operations:
* **concat** takes two sequences of size $m$ and $n$ and produces a new sequence of size $m+n$ by joining them in time $o(\lg \min(n,m/n))$ (or $o(\lg \min(m,n/m))$ if $n>m$).
* For some constants $c$ and $N$, **approximate split** ... | https://cstheory.stackexchange.com/questions/5964/sequences-with-sublogarithmic-concat-and-approximate-split | [
"ds.data-structures"
] | 17 | 2011-04-08T22:01:48 | [] | 0 | Science | 0 |
24 | cstheory | Linear-time algorithm to test if clique number equals degeneracy bound? | Given a connected simple graph $G=(V,E)$, let $d$ denote its [degeneracy](https://en.wikipedia.org/wiki/Degeneracy_\(graph_theory\)) and let $\omega$ denote the size of a [maximum clique](https://en.wikipedia.org/wiki/Clique_problem).
A well-known bound on the clique number is $\omega\le d+1$, which is helpful when so... | https://cstheory.stackexchange.com/questions/47947/linear-time-algorithm-to-test-if-clique-number-equals-degeneracy-bound | [
"graph-algorithms",
"parameterized-complexity",
"clique",
"fixed-parameter-tractable"
] | 17 | 2020-11-30T14:49:56 | [
"Maybe you are aware, but this problem recently appeared in codeforces codeforces.com/contest/1439/problem/B. It seems nobody in the discussion mentions anything better than $O(m^{1.5})$.",
"Um...it looks like their article was just published and they list your question as Open Problem 1 (minus the possible conne... | 5 | Science | 0 |
25 | cstheory | Is it possible to boost the error probability of a Consensus protocol over dynamic network? | Consider the binary consensus problem in a synchronous setting over dynamic network (thus, there are $n$ nodes, and some of them are connected by edges that may change round to round). Given a randomized $\delta$-error Monte Carlo protocol for Consensus in this setting, is it possible to use this protocol as a black bo... | https://cstheory.stackexchange.com/questions/32442/is-it-possible-to-boost-the-error-probability-of-a-consensus-protocol-over-dynam | [
"ds.algorithms",
"reductions",
"dc.distributed-comp"
] | 16 | 2015-08-23T23:24:43 | [
"@Peter Also this question is more of a whether it is possible to obtain a better protocol using a black box protocol. For example, in Sum, each node can run the protocol multiple times (either in parallel or in a row) and takes a majority. Using Chernoff's bound, for any $0 < \\delta' < \\delta < \\frac{1}{3}$, on... | 7 | Science | 0 |
26 | cstheory | complexity of checking if a subspace is a Euclidean section of L1 | If $X$ is a linear subspace of ${\mathbb R}^n$, $X$ is high-dimensional, and for every $x\in X$ we have
$(1-\epsilon) \sqrt n ||x||_2 \leq ||x||_1 \leq \sqrt n ||x||_2$
for some small $\epsilon >0$, then we say that $X$ is an almost-Euclidean section of $\ell_1^n$, and (the matrix whose image is) X is useful in compr... | https://cstheory.stackexchange.com/questions/4992/complexity-of-checking-if-a-subspace-is-a-euclidean-section-of-l1 | [
"cc.complexity-theory",
"cg.comp-geom",
"norms",
"compressed-sensing"
] | 17 | 2011-02-17T18:27:44 | [
"This is not an answer to your question. But in the vein of computational problems arising from compressed sensing applications, Koiran and Zouzias have a recent paper on checking whether a matrix satisfies the restricted isoperimetry property (RIP) and related problems.",
"Thanks Sariel, I had inverted the plac... | 3 | Science | 0 |
27 | cstheory | Looking for an operator on polynomials | I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc). I wonder whether someone might have an idea.
I'm looking for an operator T that can be applied to m-variate polynomials over a finit... | https://cstheory.stackexchange.com/questions/10881/looking-for-an-operator-on-polynomials | [
"it.information-theory",
"coding-theory",
"algebra",
"polynomials"
] | 16 | 2012-03-28T08:17:46 | [
"Honestly, I asked this question so long ago, I only barely remember what I needed it for :-)... I believe that the issue is that multiplying by a fixed polynomial is \"trivial\" in the sense that to compute the multiplication at a point, you only need to know the value of F at the point (and nothing about F's inne... | 8 | Science | 0 |
28 | cstheory | When does adding edges decrease the cover time of a graph? | When first learning about random walks on a graph $G$, one may have an intuitive feeling that adding edges to $G$ will decrease its cover time $C(G)$. However, this is not the case. The [path graph](https://en.wikipedia.org/wiki/Path_graph) $P_n$ has cover time $C(P_n) = \Theta(n^2)$ while the [($\frac{n}{2}$, $\frac{n... | https://cstheory.stackexchange.com/questions/33071/when-does-adding-edges-decrease-the-cover-time-of-a-graph | [
"graph-theory",
"pr.probability"
] | 16 | 2015-11-12T18:35:47 | [
"My bad, I confused spanning with induced.",
"@chazisop Those subgraphs are not spanning.",
"Wouldn't the complete graph $K_{n}$ and any of its subgraphs $K_{k}$, $k < n$ be a counterexample for general vertex transitivity?"
] | 3 | Science | 0 |
29 | cstheory | Complexity of approximating the range of a matrix | Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\\{-1,1\\}$, let us define:
$$S_M = |\\{Mx : x \in \\{-1,1\\}^n\\}|.$$
I believe that it is NP-hard to compute $S_M$ exactly, by applying the reductions from [Decide whether a matrix's kernel contains any non-zero vector all of whose entries are -1, 0,... | https://cstheory.stackexchange.com/questions/33676/complexity-of-approximating-the-range-of-a-matrix | [
"cc.complexity-theory",
"approximation-algorithms",
"linear-algebra"
] | 15 | 2016-01-28T02:00:10 | [
"(1) If there is a poly-time $2^{n^{1-\\epsilon}}$-approximation for any $\\epsilon>0$, then there is a poly-time $(1+1/q(n))$-approximation for any polynomial $q$. To see this, given $M$, consider the block-diagonal matrix $M(k)$ formed by $k$ independent copies of $M$ along the diagonal, so $S_M=(S_{M(k)})^{1/k}... | 1 | Science | 0 |
30 | cstheory | Intersecting Complexity Classes with Advice | In [on hiding information from an oracle](http://linkinghub.elsevier.com/retrieve/pii/0022000089900184), the authors (Abadi, Feigenbaum, and Kilian) wrote:
> $(\mathsf{NP/poly} \cap \mathsf{co\text-NP}{/poly})$ ... is **not known** to be equal to $(\mathsf{NP} ∩ \mathsf{co\text-NP}){/poly}$.
They highlighted that in... | https://cstheory.stackexchange.com/questions/5839/intersecting-complexity-classes-with-advice | [
"cc.complexity-theory",
"reference-request",
"complexity-classes",
"advice-and-nonuniformity"
] | 15 | 2011-04-03T03:17:43 | [
"@HenryYuen: I think that a Language $L$ is in $(NP \\cap coNP)/poly$ iff there is a language $K$ in $NP \\cap coNP$ and $a_i$ advice of polynomial length s.t. $x \\in L$ iff $(x, a_{|x|}) \\in K$.",
"@HenryYuen: Oh, I got your point. Strangely, the definition of (NP ∩ coNP)/poly in Complexity Zoo (qwiki.stanford... | 7 | Science | 0 |
31 | cstheory | Intermediate problems between PSPACE and EXPTIME | Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves.
Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, and resisting all efforts to be proved complete for one of these classes ?
Lifted (succinct) versions of problems betwee... | https://cstheory.stackexchange.com/questions/38393/intermediate-problems-between-pspace-and-exptime | [
"cc.complexity-theory",
"complexity-classes",
"exp-time-algorithms",
"pspace"
] | 17 | 2017-06-09T06:50:47 | [
"I think the more correct way of stating the question is asking for candidate problems in ExpTime - PSpace - ExpTime-hard. Note that e.g. in the case of P vs. NP we don't know if Factoring or GI is P-hard."
] | 1 | Science | 0 |
32 | cstheory | Mutual information vs. Product sets | Suppose we have two _dependent_ random variables $X$ and $Y$, each of which is uniform over $\\{0,1\\}^n$, such that their mutual information $I(X;Y)$ is small, say, at most $\sqrt{n}$. Does this imply that there exist large sets $\mathcal{X}, \mathcal{Y} \subset \\{0,1\\}^n$ such that the product set $\mathcal{X} \tim... | https://cstheory.stackexchange.com/questions/10225/mutual-information-vs-product-sets | [
"co.combinatorics",
"it.information-theory"
] | 15 | 2012-02-15T17:49:06 | [
"If $(X,Y)$ is a product of iid trials $(X_i,Y_i)$ then I am inclined to say yes.",
"If my calculation is correct, a standard use of the probabilistic method shows that for sufficiently large N, there exists a bipartite graph on N+N vertices with at least (1/2)N^2 edges that does not contain K_{k,k} as a subgraph... | 3 | Science | 0 |
33 | cstheory | Computability of a "weird" set | The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are [absurdly high](https://plus.google.com/+johncbaez999/posts/Pr8LgYYxvbM). This leads to the following general question: Is the set $C\subseteq {\mathb... | https://cstheory.stackexchange.com/questions/39383/computability-of-a-weird-set | [
"computability"
] | 14 | 2017-10-26T07:49:40 | [
"Cross-posted from mathoverflow.net/questions/278747/is-this-set-computable",
"I'm not an expert, but if you simplify the equation you get a cubic diophantine equation in three variables, and I think it's an open problem if universality can be achieved in such a setting. In every case, quadratic diophantine equat... | 2 | Science | 0 |
34 | cstheory | Can a Penrose tile cellular automaton be Turing-complete? | This question was based on an incorrect premise ... see Colin's comment below. Forget it.
This was inspired by the discussion on [this Math Overflow question](https://mathoverflow.net/questions/45378/undecidability-in-conways-game-of-life/45382). First, I need to define our terms.
In a Penrose tiling, there are gener... | https://cstheory.stackexchange.com/questions/2883/can-a-penrose-tile-cellular-automaton-be-turing-complete | [
"computability",
"machine-models",
"cellular-automata"
] | 15 | 2010-11-11T06:14:56 | [
"Yes, that is true for tilings formed via the deflation process. But an arbitrary planar tiling might have a \"rift\" along a line in the plane, along which the higher level tiles do not match up (like a tiling of the plane by squares in which the upper half plane is shifted right by a fraction of a unit). This is ... | 14 | Science | 0 |
35 | cstheory | NP-Hardness of 4-cycle packing problem in complete bipartite digraph? | A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a complete bipartite graph.
Given a directed complete bipartite graph, we are asked to find the largest set of edge-disjoint 4... | https://cstheory.stackexchange.com/questions/45998/np-hardness-of-4-cycle-packing-problem-in-complete-bipartite-digraph | [
"graph-theory",
"np-hardness"
] | 14 | 2019-12-10T00:40:50 | [] | 0 | Science | 0 |
36 | cstheory | Is there a P-complete language X such that succinct-X is in P? | I came across a paper called "A Note on Succinct Representation of Graphs". It seems that in the discussion section they claim that for any problem $X$ that is $\mathrm{P}$-hard under projections, $\mbox{succinct-X}$ is $\mathrm{EXP}$-hard.
This made me wonder if the theorem fails for more complicated kinds of reducti... | https://cstheory.stackexchange.com/questions/42185/is-there-a-p-complete-language-x-such-that-succinct-x-is-in-p | [
"cc.complexity-theory",
"complexity-classes",
"reductions",
"succinct"
] | 14 | 2019-01-09T22:34:19 | [
"Werent p-projections used in Valiants' original completeness and algebra paper?",
"@EmilJeřábek Yes, I noticed this too. :)",
"Seeing that projections are a restricted class of polylogtime reductions, let me add that what I wrote above about logtime reductions also applies to polylogtime reductions.",
"@Emi... | 13 | Science | 0 |
37 | cstheory | Fourier spectrum of the parity of two monotone Boolean functions | This is a question that I've been pondering, on and off, for a while, and unsuccessfully. I'd be very interested in any insight regarding this conjecture. (Or rather, these conjectures.)
Recall that, given a Boolean function $f\colon \\{-1,1\\}^n \to \\{-1,1\\}$, the Kahn—Kalai—Linial theorem states that $$\max_{i\in[... | https://cstheory.stackexchange.com/questions/37722/fourier-spectrum-of-the-parity-of-two-monotone-boolean-functions | [
"boolean-functions",
"fourier-analysis",
"monotone"
] | 14 | 2017-03-07T06:55:43 | [
"@daniello As far as I can tell, not much, at least in terms of blackbox interpretation of the results. The question above would have applications to weak learning (learning to advantage $1/2+1/\\mathrm{poly}(n)$), while the paper you link shows lower bound on testing (and a relaxation, parameterized testing). As f... | 2 | Science | 0 |
38 | cstheory | Is it possible to find the median with a linear size sorting network? | Is there a sorting network that makes only $O(n)$ comparisons and finds the median?
The AKS sorting network sorts with $O(\log n)$ parallel steps, but here I am only interested in the number of comparisons. The median of medians algorithm finds the median with $O(n)$ comparisons, but it cannot be implemented as a sort... | https://cstheory.stackexchange.com/questions/27765/is-it-possible-to-find-the-median-with-a-linear-size-sorting-network | [
"ds.algorithms",
"sorting",
"sorting-network",
"selection"
] | 14 | 2014-12-09T06:40:04 | [
"Actually the comparator circuit model is a bit different, since you are allowed to repeat inputs and their negations. What you are looking for is a comparator network for majority.",
"@Yuval Your result seems to be the only hit on google for \"comparator circuit complexity\" and even there I cannot see the defin... | 3 | Science | 0 |
39 | cstheory | Exponential-time factorization of polynomials | Let an _explicit_ field be a field for which equality is decidable (in some standard model of computation). I am interested in the factorization of univariate polynomials over an explicit field.
It is known that over some explicit fields, testing irreducibility of polynomials is undecidable [1]. Though, over many fiel... | https://cstheory.stackexchange.com/questions/18502/exponential-time-factorization-of-polynomials | [
"algebraic-complexity",
"polynomials",
"exp-time-algorithms"
] | 14 | 2013-07-30T01:12:45 | [
"Thanks @JoshuaGrochow! I found the paper incredibly easy to read, even though my German lessons are quite old now...",
"Depending on your definition of \"explicit field\", the result you cite as [1] was also proven by van der Waerden much earlier (1930). Amazingly, he proved this before the notion of computabili... | 4 | Science | 0 |
40 | cstheory | Pseudorandom functions in ACC^0? | In the lower bound result by Ryan Williams (Non-uniform $\mathsf{ACC}$ circuit lower bounds), there is a mention of "little evidence that Pseudorandom function generators exist in $\mathsf{ACC}^0$. Is there any development in this regard that might be of interest? (Even old results implying something along these lines ... | https://cstheory.stackexchange.com/questions/10220/pseudorandom-functions-in-acc0 | [
"cc.complexity-theory",
"reference-request",
"cr.crypto-security",
"circuit-complexity",
"natural-proofs"
] | 14 | 2012-02-15T01:26:37 | [
"The following paper doesn't answer your question but does show somewhat simpler candidate PRFs ccs.neu.edu/home/viola/papers/spn.pdf"
] | 1 | Science | 0 |
41 | cstheory | DPLL and Lovász Local Lemma | Let $\varphi$ be a CNF formula. Suppose that each of $\varphi$'s clauses consist of exactly $t$ literals (and, moreover, all literals within one particular clause correspond to different variables). It is well known that if every clause has less than $2^t / e$ clauses that share variables with it, then $\varphi$ is sat... | https://cstheory.stackexchange.com/questions/7720/dpll-and-lov%c3%a1sz-local-lemma | [
"cc.complexity-theory",
"ds.algorithms",
"sat"
] | 14 | 2011-08-13T02:32:29 | [
"@ilyaraz Still hoping you'll post an official answer to this question.",
"@Stasys Actually, Dmitriy Itsykson (who you probably know) convinced me that the answer is negative. I'll elaborate on it soon. Argument is more or less the following: we pick a hard unsatisfiable formula and pad it in order to make it eas... | 3 | Science | 0 |
42 | cstheory | Which monotone DNFs are evasive? | A Boolean function $\phi$ on variables $X$ is _[evasive](https://en.wikipedia.org/wiki/Evasive_Boolean_function)_ if every decision tree for $\phi$ has height $|X|$. In other words, for any strategy that picks variables of $X$ and asks for their value, an adversary can answer the queries such that the strategy needs to... | https://cstheory.stackexchange.com/questions/44278/which-monotone-dnfs-are-evasive | [
"np-hardness",
"boolean-functions"
] | 13 | 2019-07-16T13:28:04 | [
"Since the Alexander dual of the corresponding simplicial complex consists of all the independent sets of $G$, a necessary condition for evasiveness is that the independence polynomial of $G$ vanishes at $-1$.",
"Possibly relevant references for the lazy: A survey of evasiveness: Lower bounds on the decision-tree... | 2 | Science | 0 |
43 | cstheory | Question on Products of Graphs | Let $S_{n}(G)$, $C_{n}(G)$, $T_{n}(G)$ be the $n$-fold Strong Product, Cartesian Product and Tensor Product of a graph $G$ on $|V|$ vertices.
Let the chromatic number ($\chi(G)$) and the independence number ($\alpha(G)$) of $G$ be known through a (possibly exponential time) algorithm. Is it known that the calculation ... | https://cstheory.stackexchange.com/questions/7522/question-on-products-of-graphs | [
"cc.complexity-theory",
"graph-theory",
"graph-colouring"
] | 14 | 2011-07-26T18:45:56 | [
"@Andrew: The first one",
"By $n$-fold strong product do you mean $G\\boxtimes G\\boxtimes \\ldots \\boxtimes G$, or $G\\boxtimes K_n$?",
"Here is Wikipedia's article on Hedetniemi's conjecture.",
"Hi Hsien: Actually you answered part of the question. If you assume the conjecture, then we already have an $o(0... | 6 | Science | 0 |
44 | cstheory | Deterministic context-free languages that can be represented as the word problem of a group | Consider a group $G$. We call $G$ virtually free is it contains a free subgroup of finite index.
If $G$ is finitely generated by some set $X \subseteq G$ one can consider the _word problem_ $W\\!P(G)$ that is the formal language consisting of all words over the alphabet $X \cup X^{-1}$ that evaluate to the unit in $G$... | https://cstheory.stackexchange.com/questions/39954/deterministic-context-free-languages-that-can-be-represented-as-the-word-problem | [
"fl.formal-languages",
"gr.group-theory"
] | 13 | 2018-01-11T04:49:31 | [
"I'm not sure if you're still interested in this, but it is known that such languages are closed under deletion and have universal prefix. This is true for all word problems, not just DCFL. There's a proof here: D. W. Parkes and R. M. Thomas. Groups with context-free reduced word problem. Communications in Algebra,... | 6 | Science | 0 |
45 | cstheory | Complexity to compute the eigenvalue signs of the adjacency matrix | Let $A$ be the $n\times n$ adjacency matrix of a (non-bipartite) graph. Assume that we are given the amplitudes of its eigenvalues, i.e., $|\lambda_1|=a_1,\ldots, |\lambda_n|=a_n$, and we would like to calculate their signs. Is there a faster way of computing the signs of these eigenvalues, other than recomputing the e... | https://cstheory.stackexchange.com/questions/16789/complexity-to-compute-the-eigenvalue-signs-of-the-adjacency-matrix | [
"cc.complexity-theory",
"ds.algorithms",
"linear-algebra",
"spectral-graph-theory"
] | 13 | 2013-03-07T12:38:48 | [
"Thanks, that's exactly the point. I was wondering if the knowledge of the amplitudes made the problem any simpler. I think that this could have interesting implications in problems where eigenvalues are well approximated by the squared degrees of the graph, etc.",
"Nice question! For comparison, the best known r... | 2 | Science | 0 |
46 | cstheory | Name and references for balanced variant of the long code? | The [long code](https://en.wikipedia.org/wiki/Long_code_%28mathematics%29) (arising in PCP theory etc) is an encoding of a set of $k$ values, using binary strings of length $2^k$ (double exponential in the number of bits needed to specify a value), with one coordinate position in the encoding for each of the $2^k$ diff... | https://cstheory.stackexchange.com/questions/19602/name-and-references-for-balanced-variant-of-the-long-code | [
"reference-request",
"coding-theory"
] | 13 | 2013-10-31T16:11:44 | [
"Ok, if it wasn't in the literature before, it is now. See section 6 of arxiv.org/abs/1303.1136 — this code gives the smallest set of vertices to delete from a hypercube graph in order to make all remaining hypercube subgraphs have at most 1/8 of the number of vertices of the original graph."
] | 1 | Science | 0 |
47 | cstheory | Generalizing limit-colimit coincidence to Scott-continuous adjunctions: any uses? | In Abramsky and Jung's 1994 handbook chapter on denotational semantics, after proving that the limit and colimit of expanding sequences exist and coincide, they have the following to say about generalizing this theorem:
> We can generalize Theorem 3.3.7 [the limit-colimit coincidence of expanding sequences in DCPO] [... | https://cstheory.stackexchange.com/questions/2528/generalizing-limit-colimit-coincidence-to-scott-continuous-adjunctions-any-uses | [
"pl.programming-languages",
"semantics",
"denotational-semantics",
"domain-theory"
] | 13 | 2010-10-28T10:07:39 | [] | 0 | Science | 0 |
48 | cstheory | The best known upper bound for two-way probabilistic finite automata with one-counter | It is known that the class of languages recognized by two-way deterministic finite automata with one-counter (2D1CAs) is a proper subset of $ \mathsf{L} $ (deterministic log-space): A 2D1CA can run at most $ O(n^2) $ steps if it halts on a given input, and there is a language in $ \mathsf{L} $, i.e. \begin{equation} \m... | https://cstheory.stackexchange.com/questions/11080/the-best-known-upper-bound-for-two-way-probabilistic-finite-automata-with-one-co | [
"cc.complexity-theory",
"reference-request",
"automata-theory",
"upper-bounds"
] | 14 | 2012-04-13T13:36:31 | [] | 0 | Science | 0 |
49 | history | Are there any references to entombed animals in ancient India? | The 13th century Hindu philosopher Arulnandi Shivacharya wrote a work called the Shiva Jnana Siddhiyar, which among other things contains a refutation of Buddhist philosophy. In [this excerpt](https://i.sstatic.net/KxNh7.jpg), various Buddhist theories of what the ultimate cause of the body is are refuted:
> If you sa... | https://history.stackexchange.com/questions/34768/are-there-any-references-to-entombed-animals-in-ancient-india | [
"india",
"science",
"animals",
"ancient-india",
"folklore"
] | 16 | 2017-01-05T23:06:24 | [
"THe proposed edit seems to replace a question with an answer. I'm confused.",
"Unless someone can say what's available in the article I linked to above, there's not much else that can be said—at least the last time I researched this, I spent a fair bit of time and didn't uncover anything useful, but it's so very... | 4 | Culture & Recreation | 0 |
50 | history | What was the first overland road from Sweden to Finland? | The [Swedish post road](https://en.wikipedia.org/wiki/King%27s_Road_\(Finland\)) from Norway, through Sweden, used the Åland archipelago to pass into Sweden, and this is easily found (evidence of) in the south of Finland to the present day. **When (and where) was the first overland route constructed overland from Swede... | https://history.stackexchange.com/questions/62286/what-was-the-first-overland-road-from-sweden-to-finland | [
"transportation",
"sweden",
"finland"
] | 6 | 2020-12-21T03:13:26 | [
"Carl Linnaeus's 1732 trip to Lapland was recorded in a fascinating book. Of course he was not interested in the speediest or easiest route, so much as the most interesting, but it does give first-hand detailed account of his travels & travel conditions. Wikipedia summary: \"Linnaeus travelled clockwise around ... | 7 | Culture & Recreation | 0 |
51 | history | Which misdeeds of F. Reineke was J.B.B. de Lesseps "obliged to suppress"? | F. Reineke was commandant of Kamchatka from about 1780 to 1784. According to Витер's _История символов как история административного деления государства_ he was responsible for moving the territorial administration from Bolsheretesk to Nizhnekamchatsk. According to J.B.B. de Lesseps, Reineke resigned in 1784 ["for reas... | https://history.stackexchange.com/questions/55995/which-misdeeds-of-f-reineke-was-j-b-b-de-lesseps-obliged-to-suppress | [
"russia",
"18th-century",
"government",
"crime",
"kamchatka"
] | 6 | 2019-12-31T20:47:10 | [
"One more thing: According to this source, in 1786 F. Reineke moves from Kamchatka to Irkutsk and in 1798 retires from the civil service. All this suggests that his record in Kamchatka was just fine. There was some controversy related to his subordinates, as well as some misdeeds in the Russian postal service in Ka... | 7 | Culture & Recreation | 0 |
52 | history | Who owned the ship Ensayo, and what were they doing near Baja California in 1842? | According to Plummer's _The Shogun's Reluctant Ambassadors_ , in 1842, sea drifters from the _Eijū-maru_ were picked up and by the _Ensayo_ , a "Spanish pirate ship" with a Philippine crew. It was "carrying illegal goods between Spain or Mexico and Manila," and "[b]ecause it was on a 'wanted list', it could not enter a... | https://history.stackexchange.com/questions/47476/who-owned-the-ship-ensayo-and-what-were-they-doing-near-baja-california-in-1842 | [
"19th-century",
"spain",
"age-of-sail",
"philippines",
"piracy"
] | 6 | 2018-08-05T23:12:12 | [
"@AlbertoYagos: I concur with T.E.D.'s view.",
"I had \"Vida en México de trece náufragos japoneses\" in my hands, but forgot to look for this point!!",
"@AlbertoYagos - I can't always speak for all our voters, but I for one would consider what you have there acceptable, with perhaps a GoodReads link to the boo... | 6 | Culture & Recreation | 0 |
53 | history | What would a criminal defense attorney charge in late 19th century America? | What would a criminal defense attorney charge on the [American frontier](https://en.wikipedia.org/wiki/American_frontier#The_Postbellum_West) in a murder trial in 1885? I am specifcally interested in what it would have cost three cowpokes to mount a murder defence in Rapid City, Dakota Territory 1885. I know they were ... | https://history.stackexchange.com/questions/68240/what-would-a-criminal-defense-attorney-charge-in-late-19th-century-america | [
"united-states",
"19th-century",
"law",
"old-west"
] | 5 | 2022-02-04T11:16:22 | [
"Would also significantly help if you documented your preliminary research. I imagine that the answer is highly variable - urban setting vs rural, proximity of a a railroad, the presence of skilled attorney vs having to recruit one from elsewhere, Of course, the cost will range from zero (self representation) t... | 2 | Culture & Recreation | 0 |
54 | linguistics | In which non-Sinitic languages do negative clauses retain older constituent order in SVC-derived complex predicates? | Many complex predicates are historically derived from serial verb constructions. This is not only true of the Sinitic family. For example, in Saramaccan (Byrne 1987, as cited in Givón 2009):
(1) a bi-fefi di-wosu kaba.
he TNS-paint the-house finish
He finished painting the house.
... | https://linguistics.stackexchange.com/questions/21555/in-which-non-sinitic-languages-do-negative-clauses-retain-older-constituent-orde | [
"syntax",
"historical-linguistics",
"word-order",
"negation"
] | 7 | 2017-04-06T01:26:05 | [] | 0 | Science | 0 |
55 | linguistics | Are there any languages where the first person cannot be an object? | In some languages, nouns low on the animacy hierarchy, particularly inanimates cannot surface as A, and if a situation arises where they are underlyingly A, some reparative strategy such as a passive must be used. Some theorise that this was the case in early PIE, and a friend of mine mentioned Algonquian and Japhung Q... | https://linguistics.stackexchange.com/questions/27161/are-there-any-languages-where-the-first-person-cannot-be-an-object | [
"syntax",
"grammar",
"list-of-languages",
"grammatical-object",
"person"
] | 7 | 2018-02-16T13:02:04 | [
"Inanimate objects not being able to be agents makes some sense (if your language doesn't use a lot of metaphor), but why would any language prohibit 1st person objects? I can imagine this as a kind of taboo, but not as an actual ungrammaticality."
] | 1 | Science | 0 |
56 | mathoverflow | 2, 3, and 4 (a possible fixed point result ?) | The question below is related to the classical [Browder-Goehde-Kirk fixed point theorem](https://www.researchgate.net/publication/38323269_An_elementary_proof_of_the_fixed_point_theorem_of_Browder_and_Kirk).
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$ be a mapping such that $$\Vert Tx-Ty\... | https://mathoverflow.net/questions/18264/2-3-and-4-a-possible-fixed-point-result | [
"fa.functional-analysis",
"open-problems",
"banach-spaces"
] | 79 | 2010-03-15T06:26:45 | [
"$FP(p,p,\\infty)$ is false for any $p$. We can let $(Tx)_i = x_{i-1}$ for $i<1$ and $(Tx)_0 =(1/2)( 1 - \\sum_i x_i^2)$. Then $||Tx||_{\\ell_2}^2 = (1/4)(1-||x||_{\\ell_2}^2)^2+ x_{\\ell_2}^2 \\leq 1$ since $x_{\\ell_2}\\leq 1$ and, because each coordinate is a $1$-Lipschitz function in the $\\ell_2$ norm, the who... | 13 | Science | 1 |
57 | mathoverflow | Grothendieck's Period Conjecture and the missing p-adic Hodge Theories | Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the extra structure of Weil cohomology theories - grading, cup product, orientation map, and cycle class map. We can consider t... | https://mathoverflow.net/questions/200083/grothendiecks-period-conjecture-and-the-missing-p-adic-hodge-theories | [
"nt.number-theory",
"tannakian-category",
"p-adic-hodge-theory"
] | 61 | 2015-03-15T14:18:19 | [
"I should correct myself: to obtain that \"over a finite field mixed motives are pure\" we must assume that such mixed motives exist (which follows from the Tate conjecture) and that they have a weight filtration respected by Frobenius. Then for each mixed motive $M$ appropriate polynomials in Frobenius are idempot... | 7 | Science | 1 |
58 | mathoverflow | Does every triangle-free graph with maximum degree at most 6 have a 5-colouring? | ## A very specific case of Reed's Conjecture
Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic number, $\Delta$ is the maximum degree, and $\omega$ is the clique number.
When restricted to triangle-free graphs, t... | https://mathoverflow.net/questions/37923/does-every-triangle-free-graph-with-maximum-degree-at-most-6-have-a-5-colouring | [
"graph-theory",
"graph-colorings",
"co.combinatorics"
] | 55 | 2010-09-06T12:58:11 | [
"Another possible tangential attack is to establish the conjecture for the \"warmth\" $w$ of the graph, because Brightwell and Winkler proved that $w \\le \\chi$.",
"Good question; it is known to hold, without the round-up. Reed never published the result in a paper but it's in Graph Colouring and the Probabilist... | 3 | Science | 1 |
59 | mathoverflow | Class function counting solutions of equation in finite group: when is it a virtual character? | Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,x_n)=g \\}|. $$ The question is:
> Can we characterize the words $w$ for which $f_w$ is a virtual character for an... | https://mathoverflow.net/questions/105854/class-function-counting-solutions-of-equation-in-finite-group-when-is-it-a-virt | [
"gr.group-theory",
"rt.representation-theory",
"finite-groups",
"characters"
] | 54 | 2012-08-29T08:38:56 | [
"To add to HJRW's comment a function is a $\\mathbb Q$-linear combination of characters iff it is constant on elements generating conjugate subgroups. In your set up because it is an algebraic integer combination of characters you just need it to be a $\\mathbb Q$-linear combination to be a virtual character. So a... | 2 | Science | 1 |
60 | mathoverflow | How many algebraic closures can a field have? | Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is algebraically closed, then $K\cong\overline F$ via an isomorphism which fixes $F$. We can therefore say that $\overline ... | https://mathoverflow.net/questions/325756/how-many-algebraic-closures-can-a-field-have | [
"set-theory",
"ac.commutative-algebra",
"lo.logic",
"axiom-of-choice",
"field-extensions"
] | 54 | 2019-03-19T02:41:22 | [
"@DavidRoberts: Should, yes. One day...",
"Shouldn't you add an answer about this now, pointing to your own paper?",
"@AsafKaragila thanks!",
"@Holo: arxiv.org/abs/1911.09285",
"@Holo: The one I am putting in arxiv later today. Check Monday's Replacements.",
"@AsafKaragila hi, may I ask which paper of you... | 22 | Science | 1 |
61 | mathoverflow | Atiyah's paper on complex structures on $S^6$ | M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
<https://arxiv.org/abs/1610.09366>
It relies on the topological $K$-theory $KR$ and in particular on the forgetful map from topological complex $K$-theory to $KR$.
Question: what are the nice references to learn ... | https://mathoverflow.net/questions/253577/atiyahs-paper-on-complex-structures-on-s6 | [
"dg.differential-geometry",
"at.algebraic-topology",
"complex-geometry",
"kt.k-theory-and-homology",
"index-theory"
] | 52 | 2016-10-31T13:01:46 | [
"I think question should be edited to remove any implication that it is asking about possible correctness of Atiyah's paper. If the title were to say something about your actual question on KR, that would be good.",
"Regarding the map you wish to understand, I think that the KSp should be seen as coming from one ... | 16 | Science | 0 |
62 | mathoverflow | Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositions: Can we prove there is no Borel decomposition? | This question follows up on a [comment](https://mathoverflow.net/questions/92919/filling-mathbbr3-with-skew-lines#comment238532_92919) I made on Joseph O'Rourke's recent [question](https://mathoverflow.net/questions/92919/filling-mathbbr3-with-skew-lines), one of several questions here on mathoverflow concerning surpri... | https://mathoverflow.net/questions/93601/concerning-proofs-from-the-axiom-of-choice-that-%e2%84%9d%c2%b3-admits-surprising-geometrical | [
"lo.logic",
"set-theory",
"axiom-of-choice",
"mg.metric-geometry",
"descriptive-set-theory"
] | 51 | 2012-04-09T13:32:24 | [
"@LSpice Thanks!",
"Sounds like a good idea. Perhaps one could make such a kind of argument work with determinacy or $\\text{AD}_{\\mathbb{R}}$?",
"I really like these questions and every so often I go back and think about them. For me the obvious strategy to attempt is analytic - under some convenient assumpti... | 17 | Science | 1 |
63 | mathoverflow | Enriched Categories: Ideals/Submodules and algebraic geometry | While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed with the product of ideals
$$I\cdot J:=\langle\\{ab\mid a\in I, b\in J\\}\rangle$$
The internal Hom is given by the ideal... | https://mathoverflow.net/questions/76443/enriched-categories-ideals-submodules-and-algebraic-geometry | [
"ac.commutative-algebra",
"ag.algebraic-geometry",
"ct.category-theory",
"monoidal-categories",
"enriched-category-theory"
] | 48 | 2011-09-26T11:24:59 | [
"\"The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed with the product of ideals\" That's how I'll explain commutative algebra to my son!",
"Very interesting question. $I(A)$ is equivalent to the full subcategory of the comma-category which consists of ... | 2 | Science | 0 |
64 | mathoverflow | Extending a line-arrangement so that the bounded components of its complement are triangles | Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that $\\{L_1,\dots,L_m\\}$ is _triangulating away from infinity_ iff every $R_j$ that is bounded is triangular.
Can every finite line-arrangem... | https://mathoverflow.net/questions/155196/extending-a-line-arrangement-so-that-the-bounded-components-of-its-complement-ar | [
"discrete-geometry",
"plane-geometry",
"hyperplane-arrangements"
] | 47 | 2014-01-20T11:45:57 | [
"the dual version of Silvester-Caley thm in a stronger version shows that a simplicial argangment has to satisfy an optimal incidency condition that cannot hold if one move some lines in a certain neigbourhood while some other will remain fixed. In another hand a set of lines construced with the condition of my fi... | 30 | Science | 0 |
65 | mathoverflow | Homotopy type of TOP(4)/PL(4) | It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(4)\to K({\mathbb Z}/2,3)$ is 5-connected. Is it known whether $\mathrm{TOP}(4)/\mathrm{PL}(4)$ is not equivalent to $K({\ma... | https://mathoverflow.net/questions/106971/homotopy-type-of-top4-pl4 | [
"at.algebraic-topology",
"gt.geometric-topology",
"homotopy-theory"
] | 45 | 2012-09-11T19:38:57 | [
"@Ricardo: yes, that's what I meant.",
"@Peter: Thank you for remarking that $\\mathrm{PL}(n)$ is not defined as a subgroup of $\\mathrm{TOP}(n)$. In fact, by $\\mathrm{TOP}(n)/\\mathrm{PL}(n)$ I meant the homotopy fibre of the map $B \\mathrm{PL}(n)\\to B \\mathrm{TOP}(n)$, which I believe is the usual meaning. ... | 9 | Science | 0 |
66 | mathoverflow | A kaleidoscopic coloring of the plane | > **Problem.** Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\frac12\lambda(D)$?
(I.V.Protasov called such partitions _kaleidoscopic_).
Observe that for the $\ell_1$\- or $\... | https://mathoverflow.net/questions/219860/a-kaleidoscopic-coloring-of-the-plane | [
"real-analysis",
"mg.metric-geometry",
"measure-theory",
"harmonic-analysis",
"geometric-measure-theory"
] | 44 | 2015-10-02T11:55:32 | [
"@AntonPetrunin No, I do not know such a figure. It seems that for any triangle a kaleidoscopic coloring does exist.",
"Do you know any plane figure (maybe nonconvex) without kaleidoscopic colorings? (wrt parallel translations)",
"I seem to see a 3-color kaleidoscope for the perfect hexagonal ball.",
"I'll re... | 5 | Science | 0 |
67 | mathoverflow | What does the theta divisor of a number field know about its arithmetic? | This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) ([link](http://arxiv.org/pdf/math/9802121v3)).
Let me first of all recall that a curve $C$ over a function field can be recovered from its Jac... | https://mathoverflow.net/questions/47920/what-does-the-theta-divisor-of-a-number-field-know-about-its-arithmetic | [
"nt.number-theory",
"arakelov-theory",
"theta-functions",
"reference-request"
] | 41 | 2010-12-01T08:17:15 | [
"Since I think it is useful information to have around, I summarize from memory from the deleted answer and subsequent discussion: the above mentioned paper is the thesis of R. Groenewegen \"Vector bundles and geometry of numbers\", more specifically its second part \"Torelli for number fields\", which explores the... | 2 | Science | 0 |
68 | mathoverflow | Homology of $\mathrm{PGL}_2(F)$ | **Update:** As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on injectivity of $H_i(\mathrm{GL}_n(F))\to H_i(\mathrm{GL}_{n+1}(F))$ up to torsion (in the case $i=4$, $n=3$) together with... | https://mathoverflow.net/questions/298336/homology-of-mathrmpgl-2f | [
"gr.group-theory",
"algebraic-groups",
"kt.k-theory-and-homology",
"group-cohomology",
"algebraic-k-theory"
] | 39 | 2018-04-20T02:58:29 | [
"@Matthias Wendt: Thanks a lot for the Borel-Yang reference! Let's not worry right now about the exact shape of the torsion; I'd be happy enough with the statement rationally (but then my \"evidence\" would be vacuous). About homotopy vs. homology: One reason is that the homotopy of the K-theory space is usually s... | 10 | Science | 0 |
69 | mathoverflow | Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices | In the course of a physics problem ([arXiv:1206.6687](http://arxiv.org/abs/1206.6687)), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and $\bar{U}$ its complex conjugate, on the one hand, and the random orthogonal matrix $O... | https://mathoverflow.net/questions/105512/correspondence-between-eigenvalue-distributions-of-random-unitary-and-random-ort | [
"pr.probability",
"random-matrices",
"haar-measure",
"mp.mathematical-physics",
"linear-algebra"
] | 37 | 2012-08-26T03:42:59 | [
"@student -- the first statement [$U^TU\\in{\\rm COE}(n)$] is correct, but the second statement is not, I think. The eigenvalue distribution of $U\\bar{U}$, with $U\\in{\\rm CUE}(n)$ is not the eigenvalue distribution of ${\\rm CRE}(n)$, it's the distribution of the eigenvalues $\\neq -1$ in ${\\rm CRE}(n+1)$. So t... | 13 | Science | 0 |
70 | mathoverflow | Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$? | In a [recent answer](https://mathoverflow.net/a/282154/17581) Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.
> **QUESTION** Is there any (**added:** strictly increasing) sequence of positive integers $c_1,c_2,c_3,\dots$ satis... | https://mathoverflow.net/questions/282159/is-there-any-positive-integer-sequence-c-n1-fracc-nc-nndn | [
"nt.number-theory",
"integer-sequences",
"recurrences"
] | 37 | 2017-09-27T08:35:39 | [
"@WillSawin: phrased this way, the transformation sort of looks like those found in the theory of cluster algebras, so maybe if there is a positive answer to the question it could be found via the \"Laurent phenomenon.\"",
"If we ignore the division by n, a necessary condition for the integrality of the original ... | 4 | Science | 0 |
71 | mathoverflow | What is the three-dimensional hyperbolic volume of a four-manifold? | Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
We can of course order the handles according to their index. Handles of index 0 and 1 form a connected 4-dimensional h... | https://mathoverflow.net/questions/69819/what-is-the-three-dimensional-hyperbolic-volume-of-a-four-manifold | [
"gt.geometric-topology",
"4-manifolds",
"3-manifolds",
"knot-theory"
] | 37 | 2011-07-08T12:14:14 | [
"I think this might be related to this question: mathoverflow.net/q/19974/1345 In particular, if the 3-dimensional volume is zero, I suspect that the manifold admits an F-structure, and the simplicial volume is zero (and likely minimal volume =0). The point is that the circle action on a Seifert manifold M extends ... | 2 | Science | 0 |
72 | mathoverflow | Two-convexity ⇒ Lefschetz? | Assume that
* $\Omega$ is an open simply connected set in $\mathbb R^n$
* _(two-convexity)_ if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of intersection of $\Omega$ with any 2-plane is simply connected?
**Comments:**
* Note that an open connecte... | https://mathoverflow.net/questions/55788/two-convexity-%e2%87%92-lefschetz | [
"convexity",
"convex-geometry",
"morse-theory",
"tag-removed"
] | 36 | 2011-02-17T14:12:07 | [
"@WillSawin, I guess your question is about last par, starting with \"Let $L_{t_0}$...\". We are in hypereplane, i.e., in $\\mathbb R^3$. Imagine that your parabolic cylinders have exactly one point of intersection. Then the complement is homotoipically equivalent to $\\mathbb{S}^1$.",
"I don't understand the exa... | 4 | Science | 0 |
73 | mathoverflow | Orthogonal vectors with entries from $\{-1,0,1\}$ | Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \\{-1,0,1\\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $n \in \\{1,2\\} \cup \\{4k : k \in \mathbb{N}\\}$?
A few notes are in order:
1. If we do not allow "$0$" entries i... | https://mathoverflow.net/questions/446995/orthogonal-vectors-with-entries-from-1-0-1 | [
"co.combinatorics",
"linear-algebra",
"matrices"
] | 35 | 2023-05-17T12:10:00 | [
"I've computationally verified the conjecture $n=15$ and together with Ilya's and Mikhail's arguments that makes it proved for all $n\\leq 20$. For $n=21$, I can tell for now that if a counterexample exists, it should have at least 6 vectors with exactly 14 nonzero components.",
"@MaxAlekseyev Same argument that ... | 28 | Science | 0 |
74 | mathoverflow | Is there a Mathieu groupoid M_31? | I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might exist.
Indeed, it exists, and the group of permutations of the other points generated by paths in which the hole star... | https://mathoverflow.net/questions/70680/is-there-a-mathieu-groupoid-m-31 | [
"gr.group-theory",
"finite-groups",
"reference-request"
] | 34 | 2011-07-18T14:48:41 | [
"It's good to know. 31 vs. 13 for M12."
] | 1 | Science | 0 |
75 | mathoverflow | Subalgebras of von Neumann algebras | In the late 70s, Cuntz and Behncke had a paper
H. Behncke and J. Cuntz, _Local Completeness of Operator Algebras_ , Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95- 100
the about the following question. Let $A$ be a $C^\star$-algebra and let $B \subset A$ be some $*$-subalgebra.... | https://mathoverflow.net/questions/34692/subalgebras-of-von-neumann-algebras | [
"oa.operator-algebras",
"fa.functional-analysis"
] | 33 | 2010-08-05T14:40:37 | [
"If I understand correctly, you need to add a density hypothesis in your question?"
] | 1 | Science | 0 |
76 | mathoverflow | Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces | Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the type: each finite ultrametrics admits an isomeric embedding into any infinite dimensional Banach space. See Shkarin, S. A. ... | https://mathoverflow.net/questions/221181/isometric-embeddings-of-finite-subsets-of-ell-2-into-infinite-dimensional-ban | [
"mg.metric-geometry",
"banach-spaces",
"metric-embeddings"
] | 33 | 2015-10-17T17:44:32 | [
"The last result I know in this direction is in Petrov, F. V.; Stolyarov, D. M.; Zatitskiy, P. B. On embeddings of finite metric spaces in ln∞. Mathematika 56 (2010), no. 1, 135–139",
"Is much known when $X$ is $\\ell^n_{\\infty}$ for $n<\\#F$?"
] | 2 | Science | 0 |
77 | mathoverflow | Peano Arithmetic and the Field of Rationals | In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is _first order definable_ in $(\Bbb{Q}, +, \cdot$).
It is not hard to see that Robinson's result can be reformulated in the following symmetric form.
**Theor... | https://mathoverflow.net/questions/65664/peano-arithmetic-and-the-field-of-rationals | [
"lo.logic",
"nt.number-theory"
] | 32 | 2011-05-21T11:01:13 | [
"@Samuel Ried: No I have not (sorry for the tardy reply, I have been \"away\" for a long while).",
"@AliEnayat: Have you found an answer to your question yet? I would be very interested to know.",
"SJR: The answer to your first question is positive, i.e., nothing fancier is going on. Regarding the second one: ... | 4 | Science | 0 |
78 | mathoverflow | $f\circ f=g$ revisited | This may be related to [solving $f(f(x))=g(x)$](https://mathoverflow.net/questions/17614). Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\\{g\in C(\mathbb{R}) ; \exists\; f\in C(\mathbb{R})$ s.t. $f\circ f=g \\}$ . Is there some infinite dim... | https://mathoverflow.net/questions/18882/f-circ-f-g-revisited | [
"ca.classical-analysis-and-odes",
"fractional-iteration"
] | 32 | 2010-03-20T14:35:10 | [
"In the previous comment, the full stop was taken as a part of the url - however, the intention was clearly to link to this post: solving $f(f(x))=g(x)$.",
"Unfortunately, not even two vectors in that cone generate a subspace in ${\\mathcal S}$ since ${\\mathcal S}$ contains no decreasing functions (see mathoverf... | 4 | Science | 0 |
79 | mathoverflow | The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126 | I understand from [a helpful earlier MO question](https://mathoverflow.net/questions/101031/kervaire-invariant-why-dimension-126-especially-difficult) that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on dimension 126, to th... | https://mathoverflow.net/questions/397905/the-central-insight-in-the-proof-of-the-existence-of-a-class-of-kervaire-invaria | [
"at.algebraic-topology",
"homotopy-theory",
"stable-homotopy"
] | 31 | 2021-07-19T16:48:31 | [
"Probably you aware by now but just in case, there is also this preprint \"On the Last Kervaire Invariant Problem\" by Weinan Lin, Guozhen Wang, and Zhouli Xu (arxiv.org/abs/2412.10879) which also claims to resolve the final case of dimension 126 in the affirmative.",
"You're most welcome. \"Suitable for MathOve... | 3 | Science | 0 |
80 | mathoverflow | On the definition of regular (non-noetherian, commutative) rings | All rings are commutative with unit. A ring $R$ is called _regular_ if it satisfies
**(Reg)** Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual definition if $R$ is noetherian. In the non-noetherian world, valuation rings are regular (this is the reason why I got in... | https://mathoverflow.net/questions/327031/on-the-definition-of-regular-non-noetherian-commutative-rings | [
"ag.algebraic-geometry",
"ac.commutative-algebra",
"homological-algebra",
"regular-rings",
"valuation-rings"
] | 31 | 2019-04-03T01:38:58 | [
"If I understand correctly, to show that regular Noetherian is regular in this sense, you have to use the fact that for Noetherian rings being proj dimension <n at a point implies it in a neighborhood?",
"@KestutisCesnavicius: Thanks for the reference. Both properties are stable under any localization, and are Za... | 3 | Science | 0 |
81 | mathoverflow | Todd class as an Euler class | Let $X$ be a relatively nice scheme or topological space.
In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. Here $LX$ is the loop space of $X$. They usually call it $e_{reg}$, the regularised Euler class.
**Question:** Is there... | https://mathoverflow.net/questions/379590/todd-class-as-an-euler-class | [
"ag.algebraic-geometry",
"at.algebraic-topology",
"loop-spaces"
] | 31 | 2020-12-23T03:29:19 | [
"You can define $i_*$ for any closed embedding between smooths actually, and $i^*i_*=e$ is still true.",
"@Pulcinella The two questions are indeed related: in finite dimensions in the equivariant setting, after localization both $e(N_i)$ and $i^*$ become invertible, so that one can define $i_*$ from the formula $... | 14 | Science | 0 |
82 | mathoverflow | When are two C*-algebras isomorphic as Banach spaces? | We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its Banach space structure.
Clearly, finite-dimensionality and other cardinality properties can be detected. Thus, to avoi... | https://mathoverflow.net/questions/156621/when-are-two-c-algebras-isomorphic-as-banach-spaces | [
"fa.functional-analysis",
"oa.operator-algebras"
] | 31 | 2014-02-03T15:01:12 | [
"This has been answered many times in the comments but is the pair $c_0$ (the null sequences) and the algebra of continuous functions on the closed unit interval not a simple counter example to your initial question?",
"It is an open problem whether $C^∗_r(F_2)$ is Banach isomorphic to a nuclear $C^*$-algebra (sa... | 17 | Science | 0 |
83 | mathoverflow | "Three great cocycles" in Complex Analysis as cohomology generators | In his [lecture notes](http://www.math.harvard.edu/~ctm/home/text/class/harvard/213a/00/html/course/course.pdf), C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian derivative $$f\mapsto \left(\frac{f'''... | https://mathoverflow.net/questions/231823/three-great-cocycles-in-complex-analysis-as-cohomology-generators | [
"reference-request",
"cv.complex-variables",
"group-cohomology"
] | 31 | 2016-02-22T05:57:23 | [
"I'm confused by the notation, especially $D$ and $\\circle$.",
"See perhaps \"Explicit formulae for cocycles of holomorphic vector fields with values in $ \\lambda$ densities\" by Wagemann (arxiv.org/pdf/math-ph/0003035.pdf).",
"@WillSawin, well, in the case of the circle the space of tensor densities of degre... | 4 | Science | 0 |
84 | mathoverflow | When is a compact topological 4-manifold a CW complex? | Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". Kirby showed that a compact 4-manifold has a handlebody structure if and only if it is smoothable: [1](http://www.manifoldatlas.him.uni-bo... | https://mathoverflow.net/questions/73428/when-is-a-compact-topological-4-manifold-a-cw-complex | [
"at.algebraic-topology",
"open-problems",
"4-manifolds",
"cw-complexes",
"topological-manifolds"
] | 31 | 2011-08-22T11:32:18 | [
"I have good reason to believe that it is an open question! Apologies - I hadn't seen the earlier posting mathoverflow.net/questions/36838/… ",
"I take it this is an open problem? Small note, but your question was effectively asked in another form, here: mathoverflow.net/questions/36838/… I usually think of CW... | 2 | Science | 0 |
85 | mathoverflow | Cohomology of symmetric groups and the integers mod 12 | When $n \ge 4$, the third homology group $H_3(S_n,\mathbb{Z})$ of the symmetric group $S_n$ contains $\mathbb{Z}_{12}$ as a summand. Using the universal coefficient theorem we get $\mathbb{Z}_{12}$ as a summand of the cohomology group $H^3(S_n,\mathbb{Z}_{12})$.
**What's a nice 'formula' for a $\mathbb{Z}_{12}$-value... | https://mathoverflow.net/questions/314626/cohomology-of-symmetric-groups-and-the-integers-mod-12 | [
"gr.group-theory",
"group-cohomology"
] | 30 | 2018-11-05T12:17:19 | [
"@DevSinha - can you explain that class and how it works in more detail, or point me toward an explanation? I wanted a \"formula\", but an explicit geometrical description like this would also be nice.",
"@PhillippLampe - another typo! I'll fix it, thanks.",
"@JohnBaez Why do the second and the third summand... | 27 | Science | 0 |
86 | mathoverflow | Is there an Ehrhart polynomial for Gaussian integers | Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various Gaussian integers. For any Gaussian integer $u+vi$, let $h_P(u+vi)$ be the number of lattice point in $(u+vi) \cdot P$. I woul... | https://mathoverflow.net/questions/165884/is-there-an-ehrhart-polynomial-for-gaussian-integers | [
"nt.number-theory",
"mg.metric-geometry",
"lattices"
] | 30 | 2014-05-11T18:46:47 | [
"Dear David. Sorry for the late question, but, did you eventually found some place (in addition to the articles you mention) where this is explained/worked, hopefully for other number fields?",
"Yeah, right, I didn't think straight. Sorry for noise.",
"I do not see this. A rotation can change the lattice in a q... | 6 | Science | 0 |
87 | mathoverflow | Nontrivial tangent bundle that is diffeomorphic to the trivial bundle | Is there an example of a smooth $n$-manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$?
(This question was inspired by [Trivial fiber bundle](https://mathoverflow.net/questions/58685/trivial-fiber-bundle).)
| https://mathoverflow.net/questions/58793/nontrivial-tangent-bundle-that-is-diffeomorphic-to-the-trivial-bundle | [
"dg.differential-geometry",
"differential-topology"
] | 30 | 2011-03-17T16:25:39 | [
"$T^*G\\cong G\\times \\mathfrak g^*\\cong G\\times \\mathbb R^{n}$",
"@Dylan and @Somnath: Tangent bundles over spheres never give such examples. Moreover, in general, if the homotopy-equivalence $M\\to M$ induced by a diffeomorphism $TM\\to M\\times {\\mathbb R}^n$ is homotopic to a diffeomorphism $M\\to M$, th... | 6 | Science | 0 |
88 | mathoverflow | Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$? | > Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(x_0,x_1,\dots)$. But in this case we have $F \cong F(x)$.
This is related to [this](https://mathoverflow.net/question... | https://mathoverflow.net/questions/258647/is-there-a-field-f-which-is-isomorphic-to-fx-y-but-not-to-fx | [
"ra.rings-and-algebras",
"examples",
"fields"
] | 30 | 2017-01-03T03:25:32 | [
"Oops, I forgot to mention that I've posted a related question here: mathoverflow.net/questions/259117/…",
"I have an example where I know $F\\cong F(X,Y)$ but I don't know whether $F\\cong F(X)$. There are torsion-free abelian groups $G$ such that $G\\cong G\\oplus\\mathbb{Z}\\oplus\\mathbb{Z}$ but $G\\not\\cong... | 3 | Science | 0 |
89 | mathoverflow | The field of fractions of the rational group algebra of a torsion free abelian group | Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
> Are there non-isomorphic torsion free abelian groups $G$ and $H$ such that $\mathbb{Q}(G)\cong\mathbb{Q}(H)$? If so, are ... | https://mathoverflow.net/questions/259117/the-field-of-fractions-of-the-rational-group-algebra-of-a-torsion-free-abelian-g | [
"gr.group-theory",
"ac.commutative-algebra",
"fields",
"abelian-groups"
] | 29 | 2017-01-08T13:30:07 | [
"This question could be related.",
"Lovely question Jeremy. I only wish I could say something useful about it!",
"Just a remark: if $G$ has no primitive element (in the sense that for every $g\\in G$ there exists $n\\ge 2$ such that $n^{-1}g\\in G$) then I guess that $\\pm G$ is characterized within $\\mathbb{Q... | 3 | Science | 0 |
90 | mathoverflow | Linking formulas by Euler, Pólya, Nekrasov-Okounkov | Consider the formal product $$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$ (a) If $z=2$ then on the one hand we get Euler's $$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$ on the other we get Pólya's formula (the "cycle index decomposition", see Stanley's EC2, p.19) $$F(t,x,2)=\sum_{n\geq0}Z(S_n,(... | https://mathoverflow.net/questions/251068/linking-formulas-by-euler-p%c3%b3lya-nekrasov-okounkov | [
"nt.number-theory",
"co.combinatorics",
"rt.representation-theory",
"power-series",
"gauge-theory"
] | 29 | 2016-09-29T21:18:19 | [
"Neither one follows from the other.",
"never mind -- NO formula generalizes the cycle index",
"does Nekrasov-Okounkov formula really fall out of cycle index for permutation groups??"
] | 3 | Science | 0 |
91 | mathoverflow | Non-linear expanders? | Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an _expander family_ if there is a $\delta>0$ such that, on every graph in the family, the discrete Laplacian (or the adjacency matrix) has spectral gap $|\lambda_0-\lambda_1|\geq \delta$. (Assume all graphs in the family hav... | https://mathoverflow.net/questions/168426/non-linear-expanders | [
"nt.number-theory",
"gr.group-theory",
"graph-theory"
] | 29 | 2014-05-28T07:17:01 | [
"Can you say anything on the structure of those $A\\subset{\\mathbb F}_p$ with $|(3A)\\setminus(A+1)|<c|A|$ (or smaller than $|A|^\\delta$ at least)? If so, it might be possible to use this information to show that such $A$ cannot be stable under the map $x\\mapsto x^3$.",
"Yes, as I said it should do something l... | 10 | Science | 0 |
92 | mathoverflow | Can every 3-dimensional convex body be trapped in a tetrahedral cage? | Can every 3-dimensional convex body be trapped in a tetrahedral cage?
Although the question is fairly unambiguous, I give all relevant definitions:
$\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-dimensional _**convex body**_ if $C$ is convex, compact, and has non-empty interior.
$\bullet$ A _**polyhedral cage**_... | https://mathoverflow.net/questions/379560/can-every-3-dimensional-convex-body-be-trapped-in-a-tetrahedral-cage | [
"mg.metric-geometry",
"convex-geometry"
] | 28 | 2020-12-22T20:17:05 | [
"@Anton The half-circle, for example, and many other, related.",
"In 2-dimensional case, one can ask which convex figures can be trapped by three nails. Rectangle cannot be trapped, but I do not see other examples.",
"@DanRomik Romik : Yes, it is quite a bit different.",
"The paper How to cage an egg by Oded... | 6 | Science | 0 |
93 | mathoverflow | Mathieu group $M_{23}$ as an algebraic group via additive polynomials | An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is the group of additive maps of $F$ to itself which permute the set $C$. The restriction to $C$ then is the natural actio... | https://mathoverflow.net/questions/264513/mathieu-group-m-23-as-an-algebraic-group-via-additive-polynomials | [
"ag.algebraic-geometry",
"gr.group-theory",
"polynomials",
"finite-groups"
] | 28 | 2017-03-13T06:48:14 | [
"@DaveBenson Indeed, I had also tried something along this idea. The upper bound of the Galois group of $Y^{23}+XY^3+1$ is achieved by showing that this polynomial divides an additive polynomial of degree $2^{11}$. So actually this additive polynomial has Galois group $M_{23}$, and its groups acts on an $11$-dimens... | 14 | Science | 0 |
94 | mathoverflow | Derivative of Class number of real quadratic fields | Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$, and define the Fekete polynomials $$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$ Define $$ f_N(X) = \frac{F_N(X)}{X^N-1}. $$ Canceling common factors it can be shown that $$ f_N(X) = \frac{H_N(X)}{\Phi_N(X)}, $$ where $\Phi_N$ is the $... | https://mathoverflow.net/questions/114434/derivative-of-class-number-of-real-quadratic-fields | [
"nt.number-theory",
"algebraic-number-theory"
] | 28 | 2012-11-25T10:14:45 | [
"Hi! I think these are generalized Bernoulli numbers $B_{2,\\chi}$ associated the quadratic character $\\chi$ of conductor $\\Delta$. Apart from a power of $2$ perhaps, they are equal to the order of the $\\chi$-part of the cuspidal divisor class group of the modular curve $X_1(\\Delta)$. See the book by Kubert-La... | 1 | Science | 0 |
95 | mathoverflow | derived category of equivariant coherent sheaves and fixed points | The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient field of the representation ring $R(T)$ of $T$.
Is there a similar result for $D_T(Coh X)$ and $D_T(Coh X^T)$, derived... | https://mathoverflow.net/questions/48407/derived-category-of-equivariant-coherent-sheaves-and-fixed-points | [
"kt.k-theory-and-homology",
"ag.algebraic-geometry"
] | 28 | 2010-12-05T16:09:13 | [
"@Ben, you're absolutely right, somehow I missed the salient word \"coherent\" in the question. But even for constructible sheaves, is there a localization statement at the level of derived categories?",
"It doesn't seem like that to me at all. Is there anything about coherent sheaves in GKM?",
"This seems li... | 3 | Science | 0 |
96 | mathoverflow | Computational complexity of topological K-theory | I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with field coefficients. However, it is known that computing the rational homotopy groups of $X$ is at least $NP$-hard (in f... | https://mathoverflow.net/questions/125745/computational-complexity-of-topological-k-theory | [
"kt.k-theory-and-homology",
"at.algebraic-topology",
"computational-complexity"
] | 28 | 2013-03-27T10:15:43 | [
"Just thought I'd add a citation for the fact that rational homotopy groups of simply-connected spaces are NP hard here. In particular, this is not just some $\\pi_1$ issue.",
"Link: jointmathematicsmeetings.org/meetings/national/jmm2012/…",
"Have you looked at Gunnar Carlsson's stuff? They have this software f... | 8 | Science | 0 |
97 | mathoverflow | Finite-dimensional subalgebras of $C^\star$-algebras | Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. Clearly, if $A$ is finite-dimensional, then every element in $A$ (and hence $B$) has finite spectrum. I am asking for the ... | https://mathoverflow.net/questions/35207/finite-dimensional-subalgebras-of-c-star-algebras | [
"oa.operator-algebras",
"c-star-algebras",
"von-neumann-algebras",
"open-problems"
] | 28 | 2010-08-11T02:49:03 | [
"Every finitely generated *-algebra is generated by a finite list of normal elements (the real and imaginary parts of its generators) so the assumption that the $a_i$ are normal is superfluous.",
"@AliTaghavi: Yes, in this case, the spectrum of $A$ is equal to the joint spectrum of $a_1,\\dots,a_n$ (which is cont... | 10 | Science | 0 |
98 | mathoverflow | Can one divide by the cardinal of an amorphous set? | This question arose in a discussion with Peter Doyle.
It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \\{1,\ldots,k\\} \simeq Y \times \\{1,\ldots,k\\}$ then $X \simeq Y,$ where I use $\simeq$ to denote the existence of a bijection between two sets.
Assuming the Axiom of Cho... | https://mathoverflow.net/questions/235798/can-one-divide-by-the-cardinal-of-an-amorphous-set | [
"lo.logic",
"set-theory",
"axiom-of-choice"
] | 28 | 2016-04-09T15:05:50 | [
"@Lorenzo: A hint would imply that this is easy... it's not! Peter Doyle has written up division by three and division by four - math.dartmouth.edu/~doyle",
"Can someone give me a hint on how to prove that in $\\text{ZF}$ we can divide by finite cardinals?",
"I like this question!",
"There is Lovasz's unique ... | 4 | Science | 0 |
99 | mathoverflow | Unital $C^{*}$ algebras whose all elements have path connected spectrum | A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
> **What is an example of a non commutative path connected $C^*$ algebra? What is an example of a simple path connected $C^{*}$ algebra? In particular does $C^{*}_{red} F_{2}$ s... | https://mathoverflow.net/questions/195900/unital-c-algebras-whose-all-elements-have-path-connected-spectrum | [
"fa.functional-analysis",
"oa.operator-algebras",
"operator-theory",
"sp.spectral-theory",
"c-star-algebras"
] | 28 | 2015-02-06T16:24:12 | [
"@JohannesHahn Thanks for your comment. As you pointed out a minimal dynamical system gives a simple C* algebra. But what about the spectrum of its elements(Path connected spectrum)?",
"@HannesThiel Thank you very much for your comment and your attention. The motivation for the second part of my que... | 13 | Science | 0 |
100 | mathoverflow | A question on simultaneous conjugation of permutations | Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$.
Magma says that the answer is yes up to $n\leq 11$ (computed by M. Zieve). The answer is also yes if $a$ and $b$ commute via the argument below. ... | https://mathoverflow.net/questions/267191/a-question-on-simultaneous-conjugation-of-permutations | [
"co.combinatorics",
"gr.group-theory",
"permutations",
"algebraic-combinatorics"
] | 27 | 2017-04-14T04:11:08 | [
"For the record: This is proved in Theorem 4.9 of arXiv:1810.08971v4 by Junyao Pan. (I have not checked the proof.)",
"Consider the polygonal mesh with vertices $\\{1,\\dots,n\\}$, directed edges $i\\to a(i)$ and $i\\to b(i)$ for each $i$, and for each (possibly trivial) cycle $(k\\;\\cdots\\;k+m-1)$ of $[a,b]$ a... | 7 | Science | 0 |
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