Datasets:

uq-project
/

uq

question_id string	site string	title string	body string	link string	tags list	votes int64	creation_date timestamp[us]	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

End of preview. Expand in Data Studio

Dataset Card for UQ

Dataset Summary

UQ is a collection of 500 popular unsolved questions from StackExchange sites covering Science, Technology, Culture & Recreation, Life & Arts, and Business.

Dataset Structure

Data Instances

An example looks as follows:

{
  'question_id': '5',
  'site': 'crypto',
  'title': 'Finding $x$ such that $g^x\\bmod p&lt;p/k$?',
  'body': "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 $x$ that fits, with expected cost $O(k)$ modular multiplications in $\\mathbb Z_p^*$. Is there better?

Assume $p$ is an $l$-bit prime; $q$ is an $n$-bit prime with $q$ dividing $p-1$; and $g=h^{(p-1)/q}$ for some (random?) $h$ in $\\mathbb Z_p^*$. For concrete values, assume $l\\approx1024$, $n\\approx160$, $k\\approx2^{64}$.

Update: I'm expecting the problem to be hard, and thus that the more difficult problem of exhibiting $x$ with $g^x\\bmod p$ in a range of width $p/k$ must be hard.\
\",
  'link': 'https://crypto.stackexchange.com/questions/48503/finding-x-such-that-gx-bmod-pp-k',
  'tags': ['discrete-logarithm', 'group-theory'],
  'votes': 19,
  'creation_date': datetime.datetime(2017, 6, 20, 8, 57, 11),
  'comments': ['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 discrete logarithm problem in the multi-target setting, with a potentially very high target number depending on $k$. For a $k\\approx \\sqrt{p}$ index calculus seems like the best choice. Once you have one solution, getting more is relatively fast. Unsure if there's something better.", '@SamuelNeves: yes; the only constraint on $g^x\\bmod p$ is that it is small (about $\\log_2(k)$-bit smaller than $p$ is).', 'Do you get to choose what the $g^x \\mod p$ value is?', '@puzzlepalace: thank you for finding a hole; plugged it.', 'Choose $x = 0$ ? Runs in $\\Theta(1)$  ;)'],
  'comment_count': 8,
  'category': 'Technology',
  'diamond': 0,
}

Data Fields

The data fields are the same among all splits:

question_id: unique question id (1-500)
site: name of the StackExchange site
title: question title
body: question body
votes: StackExchange vote count of the question
creation_date: date of question creation on StackExchange
comments: StackExchange comments below the question
comment_count: Number of StackExchange comments
category: StackExchange category, one of {'Culture & Recreation', 'Science', 'Technology', 'Business', 'Life & Arts'}
diamond: Whether this question is included in UQ Diamond Subset

Data Splits

Category	Site	# Questions
Technology	Stack Overflow	21
	Mathematica	8
	Cryptography	5
	Retrocomputing	4
	Quantum Computing	4
	Space Exploration	3
	Unix & Linux	2
	TeX - LaTeX	2
	Code Golf	1
	Signal Processing	1
	Information Security	1
	Subtotal	52
Science	Math Overflow	200
	Mathematics	108
	Theoretical Computer Science	41
	Computer Science	12
	Cross Validated	9
	Physics	6
	Chemistry	4
	History of Science and Mathematics	3
	Linguistics	2
	Proof Assistants	2
	Artificial Intelligence	1
	Economics	1
	Bioacoustics	1
	Biology	1
	Medical Sciences	1
	Matter Modeling	1
	Operations Research	1
	Computational Science	1
	Subtotal	395
Culture & Recreation	Puzzling	8
	History	5
	Mythology & Folklore	2
	Role-playing Games	1
	Subtotal	16
Life & Arts	Science Fiction & Fantasy	35
Business	Quantitative Finance	2
Total	-	500

Sample Usage

You can load the data from 🤗 uq-project/uq via:

# pip install -q datasets
from datasets import load_dataset
dataset = load_dataset("uq-project/uq", split="test")

Additional Information

Licensing Information

The dataset is licensed under CC BY SA.

Citation Information

@misc{nie2025uqassessinglanguagemodels,
      title={UQ: Assessing Language Models on Unsolved Questions}, 
      author={Fan Nie and Ken Ziyu Liu and Zihao Wang and Rui Sun and Wei Liu and Weijia Shi and Huaxiu Yao and Linjun Zhang and Andrew Y. Ng and James Zou and Sanmi Koyejo and Yejin Choi and Percy Liang and Niklas Muennighoff},
      year={2025},
      eprint={2508.17580},
      archivePrefix={arXiv},
      primaryClass={cs.CL},
      url={https://arxiv.org/abs/2508.17580}, 
}

Downloads last month: 63

Paper for uq-project/uq

UQ: Assessing Language Models on Unsolved Questions

Paper • 2508.17580 • Published Aug 25, 2025 • 15