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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
201 | mathoverflow | Is there a model category describing shape theory? | Is there a nice model structure on some category of topological spaces compatible with [shape theory](https://en.wikipedia.org/wiki/Shape_theory_\(mathematics\))? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, the [Polish Circle](https://en.wikipedia.org/wiki/Shape_theo... | https://mathoverflow.net/questions/345396/is-there-a-model-category-describing-shape-theory | [
"at.algebraic-topology",
"sheaf-theory",
"model-categories",
"shape-theory"
] | 18 | 2019-11-06T08:52:29 | [
"@DenisNardin Thanks for the suggestion. I see the point now and did the edit.",
"@SebastianGoette Well, it is sort of the same definition, only you need to generalize it to a notion of the shape of a topos rather than just of topological spaces (you might have heard it under the name \"étale homotopy type\").The... | 10 | Science | 0 |
202 | mathoverflow | Are simplicial finite CW complexes and simplicial finite simplicial sets equivalent? | **Edit** Originally the question was whether an arbitrary diagram of finite CW complexes can be approximated by a diagram of finite simplicial sets. In view of Tyler's comment, this was clearly asking for too much. I restricted the question from arbitrary diagrams to simplicial diagrams.
* * *
It is well-known that f... | https://mathoverflow.net/questions/338660/are-simplicial-finite-cw-complexes-and-simplicial-finite-simplicial-sets-equival | [
"at.algebraic-topology",
"homotopy-theory"
] | 18 | 2019-08-19T00:48:40 | [
"The point of the original question is that a model does not support every morphism. Tyler's example of a functor from $B\\mathbb N$ is just the choice of an object and a morphism. If you allow Kan complexes or finite CW complexes, then they are cofibrant-fibrant and thus they support arbitrary morphisms. In the pu... | 9 | Science | 0 |
203 | mathoverflow | What is the logical complexity of the Hodge conjecture? | The Hodge conjecture seems to me the most mysterious among the Millennium problems (and many others). In particular, I am not sure about its logical complexity. It is not difficult to see that the conjecture is equivalent to a $\Pi^0_2$ statement if restricted to varieties over $\overline{\mathbb{Q}}$. (Basically, it i... | https://mathoverflow.net/questions/289745/what-is-the-logical-complexity-of-the-hodge-conjecture | [
"ag.algebraic-geometry",
"lo.logic"
] | 18 | 2018-01-02T03:07:33 | [
"@Libli See en.wikipedia.org/wiki/Arithmetical_hierarchy",
"what are $\\Pi_{2}^0$ and $\\Pi_{3}^0$?"
] | 2 | Science | 0 |
204 | mathoverflow | Is this Variation of the Continuum Hypothesis Inconsistent with ZFC or ZF? | It is a well-known fact that the Generalized Continuum Hypothesis is undecidable from ZFC. For similar sentences $\phi$, this is simply equivalent to ZFC having a model $M$ for which $M\models\phi$.
I wanted to see if there was any way to make all $\beth$-numbers limits. At first, this seemed dubious, but now, after c... | https://mathoverflow.net/questions/283443/is-this-variation-of-the-continuum-hypothesis-inconsistent-with-zfc-or-zf | [
"lo.logic",
"set-theory",
"continuum-hypothesis"
] | 18 | 2017-10-13T21:31:39 | [
"I mainly mentioned NCH because it seems like this is a \"variant\" of NCH (where cardinal exponentiation speeds up)",
"Just as a reminder, I would like to mention that despite its tempting appearance and intuitive background from natural numbers, the Natural Continuum Hypothesis (NCH) is inconsistent with ZFC du... | 5 | Science | 0 |
205 | mathoverflow | Is the universality of the surreal number line a weak global choice principle? | I'd like to consider the principle asserting that the surreal number line is universal for all class linear orders, or in other words, that every linear order (including proper-class-sized) linear orders, order-embeds into the surreal number line.
This principle is the class-analogue of the corresponding result for co... | https://mathoverflow.net/questions/227849/is-the-universality-of-the-surreal-number-line-a-weak-global-choice-principle | [
"lo.logic",
"set-theory",
"axiom-of-choice",
"linear-orders",
"surreal-numbers"
] | 18 | 2016-01-06T22:01:00 | [
"Or to put it the other way round: the fact that a class-sized linear order A is embeddable in No apparently does not give a way to Ord-enumerate A. If one wants to somehow use the property `all class-sized linear orders are embeddable in No', then again, we trivially know that all class-sized linear orders isomorp... | 12 | Science | 0 |
206 | mathoverflow | Does the "holomorphic spheres-to-continuous spheres" forgetful function respect the mixed Hodge structures on homotopy groups? | For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was extended to (possibly singular) quasi-projective varieties independently by Hain, by Deligne, and by Navarro Aznar. For ~~any~... | https://mathoverflow.net/questions/233979/does-the-holomorphic-spheres-to-continuous-spheres-forgetful-function-respect | [
"ag.algebraic-geometry",
"homotopy-theory",
"hodge-structure"
] | 18 | 2016-03-18T08:22:50 | [
"@AknazarKazhymurat No, unfortunately I have not found an answer to this question.",
". . . This reduces to the smooth, projective case. In that case, the Sullivan minimal model of the De Rham complex agrees with the Sullivan minimal model of the De Rham cohomology (with zero differentials). Since the Sullivan ... | 12 | Science | 0 |
207 | mathoverflow | Deforming a basis of a polynomial ring | The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\\{S_\lambda\\}$, with structure constants $c_{\lambda\mu}^\nu$ called Littlewood-Richardson numbers.
I have been studying a c... | https://mathoverflow.net/questions/199537/deforming-a-basis-of-a-polynomial-ring | [
"co.combinatorics",
"ac.commutative-algebra",
"symmetric-functions",
"algebraic-combinatorics",
"schur-functions"
] | 18 | 2015-03-09T18:40:56 | [
"I think $a=0$ is triangular one way, $b=0$ triangular the other way. The one-box rule is here: mathoverflow.net/questions/88569/…",
"Is it possible that the deformation is upper-triangular (i.e. that $S_\\lambda(a,b) = S_\\lambda + \\textrm{lower order terms}$)? How complicated are the structure constants when $... | 2 | Science | 0 |
208 | mathoverflow | An algebraic strengthening of the Saturation Conjecture | The Saturation Conjecture (proved by Knutson-Tao) asserts that $c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq 0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ is a positive integer. Let $\mathfrak{o}$ be a (commutative) discrete valuation ring with a finite residue field $k=\math... | https://mathoverflow.net/questions/212368/an-algebraic-strengthening-of-the-saturation-conjecture | [
"co.combinatorics",
"symmetric-functions"
] | 18 | 2015-07-26T11:57:23 | [
"@DavidSpeyer You are right about the definition of Hall algebra. I have fixed this.",
"A note on terminology (and then I will think about this interesting question). I usually understood the Hall algebra to refer to the ring whose elements were formal sums of isomorphism classes of $\\mathfrak{o}$-modules, and w... | 3 | Science | 0 |
209 | mathoverflow | Smooth curves on smooth varieties | Let $X$ be a smooth, proper algebraic variety over a field $k$, of positive dimension.
Is it true that $X$ contains a smooth Zariski-closed curve?
If it is projective, this is true by Bertini. But is it true in general?
| https://mathoverflow.net/questions/112524/smooth-curves-on-smooth-varieties | [
"ag.algebraic-geometry"
] | 18 | 2012-11-15T13:44:16 | [
"@algori : yes, indeed ! The answer is positive if $X$ is separably rationally connected (over an algebraically closed field). Indeed, if $X$ is of dimension $\\leq 2$, it is projective so there is no problem. And if it is of dimension $\\geq 3$ it contains (lots of) smooth rational curves by Theorem IV 3.9 in Kol... | 4 | Science | 0 |
210 | mathoverflow | An "exercise" on von Neumann algebra tensor product | The following problem appears to be an easy exercise on von Neumann algebra tensor products, but since I've been failing to find a rigorous proof, I'd like to make sure it's not that trivial. Suppose $M$ and $N$ are von Neuamann factors such that $L^\infty[0,1] \mathbin{\bar\otimes} M \cong L^\infty[0,1] \mathbin{\bar\... | https://mathoverflow.net/questions/270650/an-exercise-on-von-neumann-algebra-tensor-product | [
"oa.operator-algebras",
"von-neumann-algebras"
] | 18 | 2017-05-25T03:32:20 | [
"Obviously when $M$ or $N$ are Type $\\mathrm{I}$ it can done in a different way and the result is well-known, but I'm unsure how to proceed if the factors in question are Type $\\mathrm{II}_\\infty$ or Type $\\mathrm{III}$, even assuming separability.",
"This is obviously achievable if $M$ and $N$ are $\\mathrm{... | 3 | Science | 0 |
211 | mathoverflow | What is the Hochschild cohomology of the Fukaya-Seidel category? | Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, \mathbb{C})$, at least at the level of vector spaces (the product structure should be the quantum cohomology on $H^\bul... | https://mathoverflow.net/questions/218232/what-is-the-hochschild-cohomology-of-the-fukaya-seidel-category | [
"sg.symplectic-geometry",
"mirror-symmetry",
"hochschild-cohomology",
"fukaya-category"
] | 18 | 2015-09-13T14:06:08 | [
"Another important paper which is related to your question is Seidel's symplectic homology as a Hochschild homology. If you denote by $\\mathscr{B}$ the Fukaya category of the fiber and $\\mathscr{A}$ the Fukaya-Seidel category, then $HH_\\ast(\\mathscr{A}\\oplus t\\mathscr{B}[[t]])$ gives the symplectic homology o... | 3 | Science | 0 |
212 | mathoverflow | A cohomology class associated with a complex representation of a group | $\newcommand\CC{\mathbb C}\newcommand\ZZ{\mathbb Z}\newcommand\ad{\mathsf{ad}}\newcommand\Ext{\operatorname{Ext}}$ Suppose that $G$ is a finite group and that it acts on a finite dimensional complex vector space $V$.
For each $g\in G$ we may consider the subspace $V_g$ of $V$ spanned by the eigenvectors of $g$ corres... | https://mathoverflow.net/questions/232090/a-cohomology-class-associated-with-a-complex-representation-of-a-group | [
"rt.representation-theory",
"group-cohomology"
] | 18 | 2016-02-24T21:43:54 | [
"(Here LBG is the free loop space on BG) The first Chern class of this descended bundle is an element of $H^2(LBG,\\mathbb Z)$, which is a direct sum of cohomologies of centralizers. Is that class the same as the one described? That would be an answer to \"what is c(V)?\".",
"@Andreas, well, I know that the class... | 9 | Science | 0 |
213 | mathoverflow | Cohomological characterization of CM curves | In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable tensoring the decomposition of the algebraic $H_{\rm dR}^{1}(E,K)$ in eigenspaces for the natural $F^\times$-action coincide... | https://mathoverflow.net/questions/13681/cohomological-characterization-of-cm-curves | [
"elliptic-curves",
"cohomology",
"nt.number-theory"
] | 18 | 2010-02-01T06:36:19 | [
"Ok, I guess I was a bit lazy myself too..... :-) $K$ is a finite extension of $\\Bbb Q$ over which the complex multiplications are defined, that is $F\\subseteq K$. The comparison between algebraic $H^1$ and the Rham over $\\Bbb C$ is allowed by choosing (and fixing) an embedding $K\\rightarrow{\\Bbb C}$.",
"Wh... | 2 | Science | 0 |
214 | mathoverflow | Lipschitz constant of a homotopy | Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole.
A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family of maps $h_x\colon M\to M$, $x\in \mathbb S^n$ such that
* $h_1=\mathop{\rm id}_M$
* For any Riemannian metric $g$ on $M$ and any family of m... | https://mathoverflow.net/questions/99276/lipschitz-constant-of-a-homotopy | [
"dg.differential-geometry",
"mg.metric-geometry",
"homotopy-theory",
"at.algebraic-topology"
] | 18 | 2012-06-11T00:57:45 | [
"If it were me, I'd try to start : Let $\\Sigma$ be an exotic $n$-sphere. By conjugacy through the round sphere, this carries a very-transitive action by homeomorphisms of $S O (n+1)$; this action cannot be by diffeomorphisms, or else one could use it to give $\\Sigma$ an $S O (n+1)$-equivariant Riemann metric... ... | 4 | Science | 0 |
215 | mathoverflow | local equivalence of loop group representations | Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the **_local loop group_** $$ L_IG := \\{\gamma\in LG \ | \ \forall z\not\in I \ \gamma(z)=e\\} $$ which is a subgroup of $LG$. Let $k\ge 1$ be an i... | https://mathoverflow.net/questions/66859/local-equivalence-of-loop-group-representations | [
"loop-spaces",
"conformal-field-theory",
"von-neumann-algebras",
"rt.representation-theory",
"reference-request"
] | 18 | 2011-06-03T16:35:03 | [
"After having a further look at his preprint, it is now my opinion that Antony Wassermann does not provide a complete proof of the desired claim. So the question is open.",
"Many thanks to the person who sent me Wassermann's preprint.",
"There seems to be a proof of Claim 2 in some unpublished notes, namely in ... | 4 | Science | 0 |
216 | mathoverflow | Steenrod algebra at a prime power | Let $n=p^k$ be a prime power.
When $k=1$, the algebra of stable operations in mod $p$ cohomology is the [Steenrod algebra](http://en.wikipedia.org/wiki/Steenrod_algebra) $\mathcal{A}_p$. It has a nice description in terms of generators and relations. Its dual (as a Hopf algebra) is also well understood by work of Miln... | https://mathoverflow.net/questions/106602/steenrod-algebra-at-a-prime-power | [
"at.algebraic-topology",
"cohomology",
"steenrod-algebra"
] | 18 | 2012-09-07T05:54:20 | [
"@elidiot: I don't know anything about mod p^k Dyer-Lashof operations, but that's probably just my own ignorance. It seems worthy of an MO question for sure!",
"I know that this is another question but as it is related I'm taking the liberty to ask it here: has anything been written about mod p^k Dyer Lashof oper... | 11 | Science | 0 |
217 | mathoverflow | Elliptic $\infty$-line bundles over $B G$ | Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" ([pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf)) states the equivalence of two maps
$$ B G \longrightarrow B \mathrm{GL}_1(A) $$
for $A$ an $E_\infty$-ring carrying an oriented derived elliptic curve $E \to \mathrm{Spec}(A)$.
The theorem is s... | https://mathoverflow.net/questions/185721/elliptic-infty-line-bundles-over-b-g | [
"at.algebraic-topology",
"elliptic-curves",
"cohomology",
"derived-algebraic-geometry"
] | 18 | 2014-10-29T10:11:41 | [] | 0 | Science | 0 |
218 | mathoverflow | Computation of low weight Siegel modular forms | We have these huge tables of elliptic curves, which were generated by computing modular forms of weight $2$ and level $\Gamma_0(N)$ as N increased.
For abelian surfaces over $\mathbb{Q}$ we have very little as far as I know. The Langlands philosophy suggests that every abelian surface should be attached to a Siegel mo... | https://mathoverflow.net/questions/22949/computation-of-low-weight-siegel-modular-forms | [
"computational-number-theory",
"siegel-modular-forms",
"nt.number-theory",
"trace-formula"
] | 18 | 2010-04-28T23:46:38 | [
"@David: in which case my answer to your aside is then simply \"yes\".",
"@Kevin: I am aware of the low-dimensional coincidences SO(2,1)=PGL(2) and Spin(2,3)=Sp(4). I suppose n > 2 was implicit in my query. :)",
"@David: there is a low-degree coincidence: the dual group of GSp_4 is GSp_4 (think about the Dynki... | 10 | Science | 0 |
219 | mathoverflow | Quotients of residually finite groups by amenable normal subgroups | My questions are:
> Is there any group, which cannot be written as the quotient of a residually finite group by an amenable normal subgroup? Is it possible for large classes of groups?
and
> Is there a group, such that every extension by an amenable group is split?
| https://mathoverflow.net/questions/40991/quotients-of-residually-finite-groups-by-amenable-normal-subgroups | [
"gr.group-theory"
] | 18 | 2010-10-03T23:19:30 | [
"@Andreas: Me too! Can you send me the text? This sounds very interesting!",
"Have you imagined a way to answer the first question considering only the one-relator groups?",
"@Andreas: Could you please send me a copy of this paper, also? I am very interested in this.",
"@Andreas: Can you send me the text? I s... | 12 | Science | 0 |
220 | mathoverflow | What is known about module categories over general monoidal categories? | All of the literature I have seen on module categories over monoidal categories has been in the rigid $k$-linear semisimple case, more or less in the spirit of Ostrik's paper,
* Ostrik, V. _Module categories, weak Hopf algebras and modular invariants_ , Transformation Groups **8** (2003) 177–206, doi:[10.1007/s00031... | https://mathoverflow.net/questions/6775/what-is-known-about-module-categories-over-general-monoidal-categories | [
"ct.category-theory",
"monoidal-categories"
] | 18 | 2009-11-24T23:08:19 | [
"The \"module categories are categories of modules\" yoga is probably very special to the finite rigid setting. We tried to be careful about what exactly was needed for this here:arxiv:1406.4204. An interesting example is the following (non-rigid) tensor category C. It is finite, semisimple and has two simple objec... | 4 | Science | 0 |
221 | mathoverflow | Almost complex 4-manifolds with a "holomorphic" vector field | **Main question.** What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
_The following sub question is rewritten thanks to the comment of Robert Bryant:_
Is it true that if $(M,J)$ admits a vector field that preserves $J$ ... | https://mathoverflow.net/questions/9518/almost-complex-4-manifolds-with-a-holomorphic-vector-field | [
"4-manifolds",
"ds.dynamical-systems",
"gt.geometric-topology",
"complex-geometry"
] | 18 | 2009-12-21T16:14:00 | [
"Robert, thanks for this remark, you are right that this second phrase makes no sense. I rewrote so that I it makes sense.",
"@Dmitri: Perhaps I don't understand your terminology, but it seems to me that $S^4$ admits a smooth $S^1$ action, but it doesn't admit any almost complex structure, let alone one with a (... | 8 | Science | 0 |
222 | mathoverflow | $G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian? | During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you :
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian ?
| https://mathoverflow.net/questions/273976/g-a-group-with-p-a-prime-number-and-g-2p-1-is-it-abelian | [
"nt.number-theory",
"gr.group-theory",
"finite-groups"
] | 18 | 2017-07-08T05:27:26 | [
"@YemonChoi Well, I would wait 500 years for the math community to find a complex 600-page solution, like for Fermat.",
"If you are claiming that you can answer this question, then you can either add an answer below, or write up your work as a preprint in the normal academic way, or write it in a blog post and l... | 11 | Science | 0 |
223 | mathoverflow | Symmetries of local systems on the punctured sphere | Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex representations of $\pi_1(X)=F_{n-1}$), $\text{LocSys}(X)$, has many natural autoequivalences.
For example, if $f: S^2\to S^... | https://mathoverflow.net/questions/446416/symmetries-of-local-systems-on-the-punctured-sphere | [
"ag.algebraic-geometry",
"rt.representation-theory",
"braid-groups",
"local-systems"
] | 17 | 2023-05-08T12:40:43 | [] | 0 | Science | 0 |
224 | mathoverflow | Does the Ackermann function count something? | Let $\mathrm{FinSet}$ be the category of finite sets.
A _finite set structure_ is a faithful functor $F\colon C\to \mathrm{FinSet}$ such that, for any $n\geq 1$, there are only finitely many isomorphism classes of objects $F$ maps to $\\{1, \dots, n\\}$.
> **Question.** Is there a natural finite set structure realizi... | https://mathoverflow.net/questions/392387/does-the-ackermann-function-count-something | [
"co.combinatorics",
"ct.category-theory"
] | 17 | 2021-05-10T07:20:52 | [
"So, I do not know what the Ackermann function counts, but the TREE function does count something explicit and combinatorial, and the Ackermann function is tiny in comparison with TREE. mathoverflow.net/questions/93828/how-large-is-tree3",
"The inverse Ackermann function appears in some algorithms, e.g. using a ... | 15 | Science | 0 |
225 | mathoverflow | Picture of Lambert's proof that $\pi$ is irrational? | With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of [the irrationality of square roots](https://www.cut-the-knot.org/proofs/GraphicalSqRoots.shtml) and even of the [irrationality of $e$](https://arxiv.org/pdf/0704.1282.pdf). However, I am not a... | https://mathoverflow.net/questions/376318/picture-of-lamberts-proof-that-pi-is-irrational | [
"nt.number-theory",
"continued-fractions",
"transcendental-number-theory",
"irrational-numbers",
"alternative-proof"
] | 17 | 2020-11-12T11:09:09 | [
"I don't have a specific suggestion, but of possible related interest is a collection of references I assembled 3 years ago in my answer to Irrationality of $\\pi^2$ and $\\pi^3$.",
"Correction: The above formula should have $\\tan(x^{-1})$ on the right hand side. I also think there might be a sign error somewher... | 3 | Science | 0 |
226 | mathoverflow | When is the determinant an $8$-th power? | I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (non-trivially) as the power of another polynomial. Let me give some examples where such a phenomenon appears, before askin... | https://mathoverflow.net/questions/278981/when-is-the-determinant-an-8-th-power | [
"ag.algebraic-geometry",
"rt.representation-theory",
"matrices",
"ra.rings-and-algebras",
"octonions"
] | 17 | 2017-08-17T13:13:33 | [
"But I may be wrong and I would be delighted to learn that there is such an embedding!",
"@DenisSerre : Yes sure, I tried a lot of stuff related to the Jordan algebra theory. In fact I am now able to prove that there is an embedding of $\\mathcal{H}_{3}(\\mathbb{O})$ in $\\mathbb{R}^{24}$ such that $\\mathrm{det}... | 9 | Science | 0 |
227 | mathoverflow | Kan's simplicial formula for the Whitehead product | In his article on _Simplicial Homotopy Theory_ (Advances in Math., 6, (1971), 107 –209) Curtis quotes a formula (on page 197) for the Whitehead and Samelson products in a simplicial group $G$. The formula for the Samelson product $\langle x,y\rangle$ of elements $x\in \pi_p(G)$ and $y\in \pi_q(G)$ is in terms of an ord... | https://mathoverflow.net/questions/296479/kans-simplicial-formula-for-the-whitehead-product | [
"at.algebraic-topology",
"homotopy-theory",
"simplicial-stuff"
] | 17 | 2018-03-29T00:19:56 | [] | 0 | Science | 0 |
228 | mathoverflow | Almost monochromatic point sets | There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as [Van der Waerden](https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem), [Hales-Jewett](https://en.wikipedia.org/wiki/Hales%E2%80%93Jewett_theorem) or [Gallai's theorem](https://arxiv.org/abs/1411.1038), the ... | https://mathoverflow.net/questions/300604/almost-monochromatic-point-sets | [
"nt.number-theory",
"co.combinatorics",
"mg.metric-geometry",
"ramsey-theory",
"polymath16"
] | 17 | 2018-05-19T13:50:40 | [
"@fedja How can I cite your comment?",
"@fedja I've just discovered that essentially the same problem was posed by some dude called Erdős and his pals in '75, and they made the same conjecture, so you might want to spell out your argument for their sake; see Conjecture 4 in old.renyi.hu/~p_erdos/1975-12.pdf.",
... | 8 | Science | 0 |
229 | mathoverflow | On manifolds which do not admit (smooth) actions of finite groups | **Question** : Assume a smooth manifold $M$ does not admit any effective smooth group actions of finite groups $G \neq 1$, does it follow that $M$ also admits no _continuous_ effective group actions of finite groups $G \neq 1$?
Manifolds which do not admit actions of finite groups are interesting because if $M$ is com... | https://mathoverflow.net/questions/253815/on-manifolds-which-do-not-admit-smooth-actions-of-finite-groups | [
"at.algebraic-topology",
"differential-topology",
"smooth-manifolds",
"group-actions"
] | 17 | 2016-11-03T01:45:57 | [
"The linked article of V. Puppe seems to say \"manifold\" for \"(connected) closed manifold\". Is the same implicit assumption made in the question?",
"Restatement of the question using group theory language: If $\\mathrm{Diff}^\\infty(M)$ is torsion-free, does it follow that $\\mathrm{Homeo}(M)$ is torsion-free?... | 2 | Science | 0 |
230 | mathoverflow | Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first? | The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a _vector_ bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\rightarrow M$$ of the vector bundles of $E$-valued totally symmetric rank-$k$ covariant tensors for all $k=0,\ldots,r$ (here... | https://mathoverflow.net/questions/240329/jets-of-sections-of-vector-bundles-expressed-by-symmetrized-iterated-covariant-d | [
"reference-request",
"dg.differential-geometry",
"ho.history-overview",
"jets"
] | 17 | 2016-06-02T22:30:00 | [
"Conversely, given $P$ as defined in the question above, it is clear that $[P,f]=$ commutator of $P$ and (multiplication by) a smooth function $f$ on $M$ is a linear differential operator of order $k-1$. (Remark: one has to add to your definition the \"initial condition\" that linear differential operators of order... | 8 | Science | 0 |
231 | mathoverflow | Trying to reconcile two facts about the Appell-Lerch sum learned from Polishchuk and Zwegers | One of the key characters in the [thesis](https://arxiv.org/abs/0807.4834) of Zwegers is the modular correction $\tilde\mu(u,v;\tau)=\mu(u,v;\tau)+\frac i2R(u-v;\tau)$ of the Lerch sum $\mu(u,v;\tau)=\frac{e^{\pi i u}}{\vartheta(v;\tau)}\sum_{n\in\mathbb Z}(-1)^n\frac{e^{\pi i(n^2+n)\tau}e^{2\pi inv}}{1-e^{2\pi in\tau}... | https://mathoverflow.net/questions/266199/trying-to-reconcile-two-facts-about-the-appell-lerch-sum-learned-from-polishchuk | [
"ag.algebraic-geometry",
"elliptic-curves",
"modular-forms",
"vector-bundles"
] | 17 | 2017-04-02T11:59:10 | [] | 0 | Science | 0 |
232 | mathoverflow | Are there two non-equivalent exotic structures on $\mathbb{R}^4$ coming from topologically slice, non-slice knots? | For a knot $K \subset S^3$, which is topologically slice but not slice (in a smooth way), there's a four manifold $\mathbb{R}^4_K$, homeomorphic but not diffeomorphic to standard euclidean $\mathbb{R}^4$.
There are ways to find such knots, particularly knots with the Alexander polynomial equal to 1, are topologically... | https://mathoverflow.net/questions/277368/are-there-two-non-equivalent-exotic-structures-on-mathbbr4-coming-from-top | [
"differential-topology",
"knot-theory",
"4-manifolds",
"heegaard-floer-homology",
"khovanov-homology"
] | 17 | 2017-07-27T03:27:14 | [] | 0 | Science | 0 |
233 | mathoverflow | Need explicit formula for certain "$q$-numbers" involving gcd's | The question is motivated by yet another possible approach to a combinatorial problem formulated previously in ["Special" meanders](https://mathoverflow.net/q/146802/41291). I'm not giving details of the connection as I believe the question is sufficiently motivated by itself.
I've got a vector space with basis $\left... | https://mathoverflow.net/questions/210448/need-explicit-formula-for-certain-q-numbers-involving-gcds | [
"nt.number-theory",
"co.combinatorics",
"orthogonal-matrices"
] | 17 | 2015-06-30T00:37:31 | [
"@OfirGorodetsky Thanks for the link! Reading that - seems an interesting idea to give it a combinatorial interpretation in terms of the Möbius function of some poset. In fact this might be important for the original problem which is of combinatorial nature.",
"@alpoge But this is great! I was sticking to $e_0$ s... | 13 | Science | 0 |
234 | mathoverflow | Bunnity of multilinear maps | Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it _bunnity_ after myself:-)) If it already has a name, it be nice to know it. References are particularly appreciated.
Let $V$ be a $d$-dimensional vector space over a field $K$, $f:V^{\otimes n}... | https://mathoverflow.net/questions/206211/bunnity-of-multilinear-maps | [
"linear-algebra",
"multilinear-algebra"
] | 17 | 2015-05-10T07:36:12 | [
"The dimension of the null-space: think of $f$ as a map $V\\rightarrow V^\\ast$ and take the dimension of its kernel. Similarly, the usual multinullity is defined for any $n$: use position $i$ to write $f$ as $V\\rightarrow (V^{\\otimes (n-1)})^\\ast$ and take the dimension of its kernel.",
"Could you remind what... | 5 | Science | 0 |
235 | mathoverflow | Question about combinatorics on words | Let $\\{a_1,a_2,...,a_n\\}$ be an alphabet and let $\\{u_1,...,u_n\\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$ some prefix of the word $\phi^m(a_1)$ coincides with some suffix of the word $\phi^k(a_2)$?
This is related to a ... | https://mathoverflow.net/questions/250290/question-about-combinatorics-on-words | [
"co.combinatorics",
"gr.group-theory",
"semigroups-and-monoids",
"combinatorics-on-words"
] | 17 | 2016-09-19T20:08:39 | [
"@domotorp The equality version is decidable, by Theorem 4 of [1]. Unfortunately, this doesn't seem very helpful for the original question, because it goes via saying that if the substitution homomorphism is not injective, then it factors through a smaller alphabet and we are happy by induction. [1] A. Ehrenfeucht,... | 6 | Science | 0 |
236 | mathoverflow | The topos for forcing in computability theory | My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this [nlab article on forcing](http://ncatlab.org/nlab/show/forcing) describes forcing as a "the topos of sheaves on a suitable site."
My question concerns forcing in computability theory, for example as described in Ch... | https://mathoverflow.net/questions/195794/the-topos-for-forcing-in-computability-theory | [
"ag.algebraic-geometry",
"ct.category-theory",
"lo.logic",
"computability-theory",
"topos-theory"
] | 17 | 2015-02-05T19:55:21 | [
"Have you, since asking the question two years ago, learned about a topos-theoretic treatment of forcing in computability theory?",
"@BjørnKjos-Hanssen, I would be happy to know the answer in that case. (The application I have in mind is a bit more general, but I think knowing what is going on in the case of a c... | 3 | Science | 0 |
237 | mathoverflow | Elements of finite fields with many powers of trace zero | Let $p$ be an odd prime number, $n>1$ be an integer, and $\mathrm{tr}$ be the trace map of the field extension $\mathrm{GF}(p^{2n})/\mathrm{GF}(p)$. For which pair $(p,n)$ does there exists $x\in\mathrm{GF}(p^{2n})$ such that $x^{2(1+p^n)}=1$ and $\mathrm{tr}(x^{1+p^j})=0$ for $j=0,1,\dots,n-1$?
After computing the gc... | https://mathoverflow.net/questions/257733/elements-of-finite-fields-with-many-powers-of-trace-zero | [
"nt.number-theory",
"galois-theory",
"finite-fields"
] | 17 | 2016-12-20T21:45:08 | [] | 0 | Science | 0 |
238 | mathoverflow | Groups generated by 3 involutions | Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everyth... | https://mathoverflow.net/questions/112527/groups-generated-by-3-involutions | [
"permutation-groups",
"computational-group-theory",
"gr.group-theory",
"nt.number-theory",
"computability-theory"
] | 17 | 2012-11-15T14:14:42 | [] | 0 | Science | 0 |
239 | mathoverflow | Is there a n/2 version of the Erdős-Hanani conjecture? | This question comes out of REU research from this past summer. Unfortunately weeks of thought led to only trivial observations and the conclusion that the problem is quite hard.
Fix $k,t$. Let $F$ be a set of $k$-subsets of $[n] := \\{1,\ldots,n\\}$ of minimal cardinality such that $F$ covers all $t$-subsets of $[n]$ ... | https://mathoverflow.net/questions/121961/is-there-a-n-2-version-of-the-erd%c5%91s-hanani-conjecture | [
"co.combinatorics"
] | 17 | 2013-02-15T17:02:33 | [
"@Gerhard Paseman: it looks like the repository has $n < 100$, $k \\leq 25$ and $t \\leq 8$. With such small values it is not possible to detect the presence or absence of a log factor.",
"Some structural results on this part of the boolean lattice can be found with the search phrases \"middle levels\" and \"bool... | 6 | Science | 0 |
240 | mathoverflow | Katz--Mazur for abelian varieties | Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]$ classifying abelian varieties with $N$-level structure.
> Is there a finite proper map $A_g(N)\to A_g$ over $\mathbb... | https://mathoverflow.net/questions/152986/katz-mazur-for-abelian-varieties | [
"ag.algebraic-geometry",
"nt.number-theory",
"moduli-spaces",
"abelian-varieties",
"characteristic-p"
] | 17 | 2013-12-28T11:48:13 | [
"Dear Daniel Litt: Thanks for tracking it down; that is indeed the right paper. It is also worth noting (in view of the question posed) that this paper provides a sense in which mere normalization in higher level (which doesn't have positive-dimensional fibers) is the \"wrong\" thing to consider, suggesting that ... | 4 | Science | 0 |
241 | mathoverflow | Souslin trees and weakly compact cardinals | In [Souslin trees on the first inaccessible cardinal](https://mathoverflow.net/questions/143530/souslin-trees-on-the-first-inaccessible-cardinal) it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like to ask an apparently simp... | https://mathoverflow.net/questions/161868/souslin-trees-and-weakly-compact-cardinals | [
"lo.logic",
"set-theory",
"forcing",
"large-cardinals"
] | 17 | 2014-03-30T04:28:37 | [
"Its about your comment above (since in the situation of your comment $S$ is non reflecting at every $\\alpha$, as witnessed by $\\text{acc }C_\\alpha$).",
"Your comment is about my above comment for a weaker sufficient condition?",
"@Mohammad: you don't actually need a $\\square (\\kappa )$ sequence, all you n... | 8 | Science | 0 |
242 | mathoverflow | Maximum automorphism group for a 3-connected cubic graph | The following arose as a side issue in a project on graph reconstruction.
**Problem:** Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a good upper bound on $a(n)$.
There is a [paper of Opstall and Veliche](http://arxiv.org/abs/math/0608645) that finds t... | https://mathoverflow.net/questions/136929/maximum-automorphism-group-for-a-3-connected-cubic-graph | [
"co.combinatorics",
"gr.group-theory",
"graph-theory",
"finite-groups",
"automorphism-groups"
] | 17 | 2013-07-16T18:29:15 | [
"In the cubic vertex-transitive case and n twice an odd number, it immediately follows from the same paper that you get a polynomial rather than exponential upper bound. For example Corollary 4 yields that, for large enough n, we have |G|<n^2. This is not best possible, but is not far off, at least for some values ... | 4 | Science | 0 |
243 | mathoverflow | Actions on ℍⁿ generated by torsion elements | Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, I need a cocompact properly discontinuous isometric action $\Gamma\curvearrowright\mathbb{H}^n$ such that $\mathbb{H}^n... | https://mathoverflow.net/questions/102644/actions-on-%e2%84%8d%e2%81%bf-generated-by-torsion-elements | [
"group-actions",
"hyperbolic-geometry",
"algebraic-groups",
"open-problems"
] | 17 | 2012-07-19T04:21:58 | [
"@ Anton: I mean he computed the reflection subgroup for $n$ such that the subgroup is finitely generated. The full group is then generated by the reflection subgroup together with the symmetries of its fundamental domain, so in some sense he computed $O(n,1;Z)$. All that I'm pointing out is that these are actually... | 4 | Science | 0 |
244 | mathoverflow | Does a symplectic group act on a tensor power of a spin representation? | $\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$. Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ which commutes with the action of $\Spin(2k+1)$?
I am not asking for a duality. That is, I do not require that each isoty... | https://mathoverflow.net/questions/57244/does-a-symplectic-group-act-on-a-tensor-power-of-a-spin-representation | [
"co.combinatorics",
"rt.representation-theory",
"lie-groups",
"invariant-theory"
] | 17 | 2011-03-03T03:38:18 | [
"(For an explanation of a similar, well-known numerical \"coincidence,\" but for $SL_n$ and $GL_m$, using Howe duality, see: sbseminar.wordpress.com/2007/08/10/the-ubiquity-of-howe-duality/…)",
"As mentioned in the bounty message, I would be interested in any algebraic explanation of the numerical coincidence of ... | 2 | Science | 0 |
245 | mathoverflow | Antichains of Cardinals in ZF Without Choice... | With the Axiom of Choice, the cardinals form a nice linearly ordered "set". In the absence of the Axiom of Choice, the cardinals form a partially ordered "set". Broadly, I am wondering what properties these posets can have.
A specific question I am interested in is the following.
**Is there, for each (infinite) subse... | https://mathoverflow.net/questions/82972/antichains-of-cardinals-in-zf-without-choice | [
"lo.logic",
"set-theory",
"axiom-of-choice"
] | 17 | 2011-12-08T07:09:36 | [
"Joel: My motivation for requiring infinite $S$ was the initial context: How complicated can the theory of cardinals in a model of $ZF$ (in the language of posets) be? That said, now my interest goes beyond the initial context, so I am curious about sets such as $\\{1,2\\}$. Thanks for elaborating! <p> For comp... | 11 | Science | 0 |
246 | mathoverflow | "extended TQFT" versus "TQFT with defects" | There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a d-dimensional TFT is a symmetric monoidal functor
\begin{equation*}\text{Bord}_d\rightarrow\mathcal{C}\end{equation*}
... | https://mathoverflow.net/questions/184454/extended-tqft-versus-tqft-with-defects | [
"mp.mathematical-physics",
"higher-category-theory",
"quantum-field-theory",
"extended-tqft"
] | 17 | 2014-10-14T19:07:01 | [
"One approach to this is discussed in detail at arxiv.org/abs/1009.5025 . See sections 2 and 6, and the remark about defects near the start of section 6.7.",
"Perhaps the following question needs to be answered first: a defect theory (an object of the k-category of defects that decorates a k-fold) is itself a TF... | 6 | Science | 0 |
247 | mathoverflow | The Grothendieck Ring of Higher Stacks | The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the _cut & paste_ (or scissor) relations, which say that $[X] = [U] + [Y]$, for any closed $Y \subset X$ and where $U$ is the open complement.
Now, we can equally speak of a Grothendieck ring o... | https://mathoverflow.net/questions/121984/the-grothendieck-ring-of-higher-stacks | [
"ag.algebraic-geometry",
"stacks"
] | 17 | 2013-02-16T05:10:36 | [
"are there any interesting phenomena you expect to happen? or just for fun?"
] | 1 | Science | 0 |
248 | mathoverflow | Vector Bundles on the Moduli Stack of Elliptic Curves | As is well known, there is classification of line bundles on the moduli stack of elliptic curves over a nearly arbitrary base scheme in the paper _The Picard group of $M_{1,1}$_ by Fulton and Olsson: every line bundle is isomorphic to a tensor power of the line bundle of differentials $\omega$ and $\omega^{12}$ is triv... | https://mathoverflow.net/questions/20824/vector-bundles-on-the-moduli-stack-of-elliptic-curves | [
"ag.algebraic-geometry"
] | 17 | 2010-04-09T03:06:40 | [
"The fact that vector bundles on $\\mathcal M_{1,1}$ split as sums of line bundles in characteristic larger than 3 can also be seen by the classical description of $\\mathcal M_{1,1}$ as an open substack of the weighted projective stack $\\mathbb P(4,6)$, coming from the Weierstrass form. Any locally free sheaf on ... | 8 | Science | 0 |
249 | mathoverflow | Combinatorial identity involving the Coxeter numbers of root systems | The setup is:
$R$ = irreducible (reduced) root system;
$D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$;
$\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$;
$\alpha_k$ = $k^{th}$ simple root ($1\le k\le r$); $\alpha_0$ = -(highest root);
label $n_k$ ($1\le k\le n$) = multiplicit... | https://mathoverflow.net/questions/38992/combinatorial-identity-involving-the-coxeter-numbers-of-root-systems | [
"co.combinatorics",
"rt.representation-theory",
"lie-algebras",
"root-systems",
"weyl-group"
] | 17 | 2010-09-16T09:47:55 | [
"@Jeff: One small nit-pick about your formulation is that the Coxeter number $h$ only depends on the Weyl group, not on $D$; for example, Lie types $B_r, C_r$ yield the same $h$. Coxeter originally found the concept in the framework of finite real groups generated by reflections, which is independent of Lie theo... | 3 | Science | 0 |
250 | mathoverflow | How many Hecke operators span the Hecke algebra? | This is a generalisation of my [earlier question](https://mathoverflow.net/questions/42809/how-many-hecke-operators-span-the-level-1-hecke-algebra) about generators for the level 1 Hecke algebra.
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, and $k \ge 1$ an integer. There's an explicit b... | https://mathoverflow.net/questions/42815/how-many-hecke-operators-span-the-hecke-algebra | [
"nt.number-theory",
"modular-forms"
] | 17 | 2010-10-19T11:30:31 | [
"Joel: see e.g. wstein.org/books/modform/modform/newforms.html#sturm-s-theorem.",
"David, do you know a good reference for sturm bounds?",
"A data point: for $\\Gamma_0(p)$ in weight 4, $p$ prime, the Sturm bound $S$ is $(p + 1)/3$, but the Eisenstein series $E_4(pz)$ has $a_n = 0$ for $1 \\le n < p$, so $S'$... | 7 | Science | 0 |
251 | mathoverflow | monomorphisms and epimorphisms of local rings | I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
* Every monomorphism is injective.
Proof: Let $R \to S$ be a monomorphism and $a,b \in R$ mapped to the same elemen... | https://mathoverflow.net/questions/24066/monomorphisms-and-epimorphisms-of-local-rings | [
"ac.commutative-algebra",
"ct.category-theory"
] | 17 | 2010-05-10T00:17:01 | [
"Oh yes, this is the example. I'll take a look at Paul Balmer's explanation.",
"The link to Lazard's article seems to be broken; is this (mathoverflow.net/questions/19282/…) the counterexample you were referring to? If so, I don't see why $C$ maps to $D$ either.",
"@Martin: here's proof of non-existence when G... | 7 | Science | 0 |
252 | mathoverflow | Does every connected set that is not a line segment cross some dyadic square? | A _dyadic square_ is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ _crosses_ a square $S$ if there exists a connected subset of $A \cap S$ which intersects two opposite sides of the square $S$. Clearly, the 45 degree line $\\{ (\pi + t, t) :... | https://mathoverflow.net/questions/120415/does-every-connected-set-that-is-not-a-line-segment-cross-some-dyadic-square | [
"geometry",
"discrete-geometry",
"euclidean-geometry",
"gt.geometric-topology",
"mg.metric-geometry"
] | 17 | 2013-01-31T06:07:14 | [
"Since you @mirko don't mind rephrasing-reposting the question I'd repost as a dimension/measure-theoretical question on curves in the plane, someone will spit some result at you really quick if you're convinced it's true. But I think it's false, you should be able to get good control of limiting behavior with a r... | 12 | Science | 0 |
253 | mathoverflow | Rational equivalence of smooth closed manifolds | All spaces below will be assumed simply connected. A continuous map is a _rational equivalence_ if it induces an isomorphism of the rational homology groups. Two spaces are _rationally equivalent_ if they can be connected by a sequence of rational equivalences (not necessarily all going in the same direction). Now supp... | https://mathoverflow.net/questions/429394/rational-equivalence-of-smooth-closed-manifolds | [
"at.algebraic-topology",
"gt.geometric-topology",
"smooth-manifolds",
"rational-homotopy-theory"
] | 16 | 2022-08-29T13:31:35 | [
"To add to Denis T's comment, if the rational homotopy type in question is formal, then $X$ admits endomorphisms of arbitrarily divisible degree (see Infinitesimal Computations in Topology Theorem 12.2, and F. Manin's \"Positive weights and self-maps\" for a detailed discussion, Corollary 1.1 arxiv.org/abs/2108.021... | 6 | Science | 0 |
254 | mathoverflow | Transcendence of sum of reciprocals of factorials | For $A \subseteq \mathbb{N}$, define $\displaystyle x_A = \sum_{n \in A} \frac{1}{n!}$. It is easy to see that for every infinite $A$, $x_A$ is irrational.
**Question** : Is there an infinite $A \subseteq \mathbb{N}$ for which $\displaystyle x_A = \sum_{n \in A} \frac{1}{n!}$ is algebraic?
| https://mathoverflow.net/questions/419523/transcendence-of-sum-of-reciprocals-of-factorials | [
"nt.number-theory",
"analytic-number-theory",
"transcendental-number-theory"
] | 16 | 2022-04-02T22:22:00 | [
"At least we can prove that if $x_{A}$ is an algebraic number then $A$ is not a cofinite subset of $\\mathbb{N}$. :)",
"For other results along these lines, see Kumar and Vance and the references therein. I'm not familiar with them all but at first glance they all require faster growth rates than $n!$. And for an... | 4 | Science | 0 |
255 | mathoverflow | Is "Escherian metamorphosis" always possible? | $\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$_This is a tweaked version of a question which was asked and bountied[at MSE](https://math.stackexchange.com/q/4555946/28111); that question s... | https://mathoverflow.net/questions/433479/is-escherian-metamorphosis-always-possible | [
"mg.metric-geometry",
"euclidean-geometry",
"plane-geometry",
"tiling"
] | 16 | 2022-10-29T10:38:49 | [
"@MattF. Metamorphosis II has a portion that goes squares-lizards-hexagons, I believe.",
"Which of the Escher images do you find most suggestive for an answer about unit squares and unit hexagons?",
"Interesting question!"
] | 3 | Science | 0 |
256 | math | A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals? | **Background.** If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'm aware of uses quite a bit of commutative algebra and non-trivial theorems such as Krull's intersection theorem. It... | https://math.stackexchange.com/questions/358423 | [
"ring-theory",
"commutative-algebra",
"noetherian",
"krull-dimension",
"dimension-theory-algebra"
] | 873 | 2013-04-11T08:19:32 | [
"@Alexey Back then I removed the remark about first-order because of your comment. But I have just added this back, because we can just change the language to make it first-order. I have added an explanation for this.",
"Why this question has no answer or answers were deleted?",
"Currently 18 answers, all delet... | 0 | Science | 1 |
257 | math | Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not? | Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ is connected.
Question: Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with th... | https://math.stackexchange.com/questions/952466 | [
"general-topology",
"metric-spaces",
"examples-counterexamples",
"connectedness"
] | 632 | 2014-09-30T05:18:04 | [
"@HelloWorld: thank you for the reminder. The cash prize is now posted on X.",
"Will there be a cash price in twelve days? ;)",
"Two dozen answers, all deleted.",
"@Carlyle: nothing immediate comes to mind. You can have highly discontinuous functions whose graph still remain connected.",
"How does this defi... | 0 | Science | 1 |
258 | math | Does every ring of integers sit inside a ring of integers that has a power basis? | Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form
$$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$
with $a_i \in \mathbb{Q}$.
However, the ring of integers, $\mathcal{O}_K$, of $K$ need not have a basis over $\mathbb{Z}... | https://math.stackexchange.com/questions/1754860 | [
"abstract-algebra",
"number-theory",
"ring-theory",
"algebraic-number-theory"
] | 185 | 2016-04-22T16:54:33 | [
"The difficulty in finding a general approach to construct such an extension arises from the fact that the structure of the ring of integers can be quite complicated, and power bases may not always exist for every number field.",
"@Jacob Wakem, if you take a field extension, let's say $\\mathbb{Q}(\\sqrt{2})/\\ma... | 0 | Science | 1 |
259 | math | If polynomials are almost surjective over a field, is the field algebraically closed? | Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. That is, for all but finitely many $a\in K$, $f(x)-a$ has a root.
Clearly polynomials are almost surjective over any f... | https://math.stackexchange.com/questions/1792464 | [
"abstract-algebra",
"polynomials",
"field-theory"
] | 142 | 2016-05-19T19:06:55 | [
"I guess the problem with the original reasoning is that it relies on the fact that the image of $\\mathbb{C}$ has to be simply connected, but this is hard to use for abitrary fields, the only topology you could use would be zariski but knowing zariski would give you info about the algebraic closeness. Maybe studyi... | 0 | Science | 1 |
260 | math | Probability for an $n\times n$ matrix to have only real eigenvalues | Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues?
The answer cannot be $0$ or $1$, since the set of matrices with distinct real eigenvalues is open, and also the set with distinct, but not all real, eigen... | https://math.stackexchange.com/questions/3770846 | [
"linear-algebra",
"probability",
"matrices",
"eigenvalues-eigenvectors",
"random-matrices"
] | 135 | 2020-07-27T03:59:33 | [
"@EXodd Ok. Then we can try this way for $n\\geq 4$ case. Let $f(x)=det(xI−A)$ be the characteristic polynomial. Let $a_1 >⋯>a_{n−1}$ be roots of $f′(x)$. Then the roots of f are all real if $f(a_1)<0,f(a_3)<0,…$ and $f(a2)>0,f(a4)>0,…$. For $n=4$, this seem to give the explicit algebraic relation. For $n>4$, I am ... | 0 | Science | 1 |
261 | math | A question about divisibility of sum of two consecutive primes | I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question:
Find the least natural number $k$ so that there will be only a finite
number of $2$ consecutive primes whose sum is divisible by $k$.
Although I c... | https://math.stackexchange.com/questions/527495 | [
"number-theory",
"prime-numbers",
"prime-gaps"
] | 122 | 2013-10-15T11:52:59 | [
"This is equivalent to: Find the least natural number $k$ so that there is no pair of $2$ consecutive primes whose sum is divisible by $k$. Let say $k$ is the least natural number such there is only a finite number of pair of $2$ consecutive primes whose sum is divisible by $k$, and that $p+q=kn$ is the largest of ... | 0 | Science | 1 |
262 | math | What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism? | Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with a unit perimeter?
A reasonable first guess would be the regular tetrahedron of side length $1/6$, with volume $\left(\frac16\right)^3\cdot\frac1{6\sqrt{2}}=\frac{\sqrt{2}}{2592}\approx 0.0005456$. Howe... | https://math.stackexchange.com/questions/4044670 | [
"geometry",
"optimization",
"volume",
"calculus-of-variations",
"polyhedra"
] | 112 | 2021-03-01T09:12:52 | [
"The term \"skeleton\" in the title should be avoided because this term is used in Mathematical Morphology for something else, rather different from the meaning given here.",
"A good starting point would be to consider Gauss's Shoelace formula for volume en.wikipedia.org/wiki/Shoelace_formula#Generalization . Eve... | 0 | Science | 1 |
263 | math | Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way? |
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there exists a positive real number $\epsilon (w) > 0$, such that any finite gr... | https://math.stackexchange.com/questions/3070332 | [
"combinatorics",
"group-theory",
"finite-groups",
"conjectures",
"universal-algebra"
] | 94 | 2019-01-11T12:51:53 | [
"Which group words w admit an ε(w) > 0 such that: A finite group G satisfies w = e globally ⟺ P_w(G) > 1 - ε(w)?",
"@Z. A. K. (Well, technically I proved them during my PhD, but I wrote them up earlier this year)",
"I proved analogous results about the equations $xy^2=y^2x$, $xy^3=y^3x$ and $xy=yx^{-1}$ (and in... | 0 | Science | 1 |
264 | math | Complete, Finitely Axiomatizable, Theory with 3 Countable Models | Does there exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models?
A few relevant comments:
There is a classical example of a complete theory with exacly $3$ models. This theory is not finitely axiomatizable (For the trivial reason that the language is infinite).... | https://math.stackexchange.com/questions/913049 | [
"logic",
"model-theory"
] | 92 | 2014-08-29T06:01:33 | [
"and by Scott's theorem a complete $\\mathcal{L}_{\\omega_1,\\omega}$-theory has at most one countable model up to isomorphism (as long as the language is countable, anyways); actually, Scott showed that for every countable structure $\\mathcal{A}$ (in a countable language), there is a single $\\mathcal{L}_{\\omega... | 0 | Science | 1 |
265 | math | Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles? | The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a known regular way to do this? Just playing and not having any specific algorithm, I got as far as the picture below, which s... | https://math.stackexchange.com/questions/1164035 | [
"sequences-and-series",
"visualization",
"egyptian-fractions"
] | 85 | 2015-02-24T15:08:28 | [
"This question is basically identical to one posted on MathOverflow. I am leaving the question here as a reference for interested readers in the future.",
"And that's popularity....",
"I ❤ this question !!!!!!!!!",
"Nice problem. But there seems to be no method of arrangeing it analytically.",
"@alex.jorda... | 0 | Science | 0 |
266 | math | Does the $32$-inator exist? | Background
It is common popular-math knowledge that as we extend the real numbers to complex numbers, quaternions, octonions, sedenions, $32$-nions, etc. using the Cayley-Dickson construction, we lose algebraic properties at each step such as commutativity, associativity, alternativity, etc.
We can express this phenome... | https://math.stackexchange.com/questions/4498328 | [
"abstract-algebra",
"recreational-mathematics",
"multilinear-algebra",
"octonions",
"sedenions"
] | 84 | 2022-07-22T14:58:59 | [
"In regards to conjugation. One way to view it is as an invertible operator on a vector of reals, that can only change the signs and the order of elements but not the magnitudes of the elements themselves. For an $n$ vector there are $2^n n!$ such conjugation operators. That gives a lot of options to work with.",
... | 0 | Science | 0 |
267 | math | Is there a "ping-pong lemma proof" that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2? | Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime.
In the book, by Pierre de la Harpe, Topics in Geometric Group Theory section $\textrm{II.B.40}$, as a research problem, it asks to fin... | https://math.stackexchange.com/questions/1295967 | [
"group-theory",
"reference-request",
"group-actions",
"geometric-group-theory"
] | 80 | 2015-05-23T15:06:23 | [
"I don't expect that linearity would be easier than free. For instance, the subgroup $H$ generated by $f,g$ and in addition $h:x\\mapsto 2x$ is not linear because of the relations $hfh^{-1}=f^2$, $ghg^{-1}=h^3$. Indeed the last implies that $g^nhg^{-n}f^{\\pm 1}g^nh^{-1}g^{-n}=f^{\\pm 2^{(3^n)}}$. If these are comp... | 0 | Science | 0 |
268 | math | Dedekind Sum Congruences | For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} \mathfrak{s}( \tfrac{bc}{d}, \tfrac{a}{b^{\prime} c^{\prime}}) + b^{\prime} \mathfrak{s}( \tfrac{ac}{d}, \tfrac{b}{a^{\prim... | https://math.stackexchange.com/questions/61026 | [
"number-theory",
"reference-request"
] | 73 | 2011-08-31T13:39:55 | [] | 0 | Science | 0 |
269 | math | Determinant of a matrix that contains the first $n^2$ primes. | Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix
$$
\left(\begin{matrix}
p_1 & p_2 & \cdots & p_n \\
p_{n+1} & p_{n+2} & \cdots & p_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\cdots & \cdots & \cdots & p_{n^2}
\end{matrix}
\right)
$$
we can take the determinant.... | https://math.stackexchange.com/questions/2355787 | [
"linear-algebra",
"number-theory",
"prime-numbers",
"determinant"
] | 69 | 2017-07-11T23:55:08 | [
"Is it possible to prove this by showing that inverse of given matrix always exists as there always exists trivial solutions to Ax = 0 equation?. Trivial solutions are only possible as the rows of the matrix will always be independent, as each element of matix is a distinct prime number.",
"Has anyone thought of ... | 0 | Science | 0 |
270 | math | Does the average primeness of natural numbers tend to zero? | Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click here for a somewhat related question.
A number is either prime or composite, hence primality is a binary concept. Inst... | https://math.stackexchange.com/questions/3176228 | [
"number-theory",
"limits",
"statistics",
"prime-numbers",
"natural-numbers"
] | 62 | 2019-04-05T10:59:06 | [
"If $f(n)=1$ iff $n$ is a prime then I would have thought $f(n)$ cannot uniquely identify $n$ (i.e. $f(n)$ is not $1-1$ injective), since it cannot when $n$ is a prime.",
"@daniel One of the reasons why I invented this definition was because even if two numbers have the same number of divisors or the same number ... | 0 | Science | 0 |
271 | math | On the equivalence relation $(a,b) \sim (c,d)\iff a+b=c+d$ | Setup
Let $A=\{a_1<a_2<\cdots<a_p\}$ and $B=\{b_1<b_2<\cdots<b_q\}$ be two finite sets of real numbers. Define an equivalence relation $\sim_{A,B}$ on $R_{p,q}=\{1,\dots,p\}\times\{1,\dots,q\}$ by setting $$(i,j)\sim_{A,B}(k,l)\iff a_i+b_j=a_k+b_l$$ and further define a total order on the set of equivalence classes $R_... | https://math.stackexchange.com/questions/4103564 | [
"combinatorics",
"discrete-mathematics",
"equivalence-relations",
"additive-combinatorics",
"configuration-space"
] | 56 | 2021-04-15T10:36:32 | [
"Sets where you have at most one pair from each equivalence class are called Sidon sets and there's huge theory behind them. Might be worth a shot to look into that",
"@SamuelMuldoon Thanks! I'm using Goodnotes on ipad",
"@OlivierBégassat The diagrams having a graph-paper/grid background are very pretty. What c... | 0 | Science | 0 |
272 | math | Does there exist a polynomial $P(x,y)$ which detects all non-squares? | Problem. Does there exist a two-variable polynomial $P(x, y)$ with integer coefficients such that a positive integer $n$ is not a perfect square if and only if there is a pair $(x, y)$ of positive integers such that $P(x, y)=n$?
Context. The answer is positive for polynomials in 3 variables! This appeared as a problem ... | https://math.stackexchange.com/questions/4438559 | [
"number-theory",
"polynomials",
"contest-math"
] | 49 | 2022-04-28T11:58:00 | [
"@Sil Great suggestion! Just posted the problem on MathOverFlow at this link.",
"Interesting problem, you might want to consider asking on MathOverflow as it was not resolved here even after bounty.",
"In fact, the polynomial $P=p*q=[(p-1)/2 + (p+1)/2]*[(q-1)/2 + (q+1)/2]$ can represent non-squares and non-prim... | 0 | Science | 0 |
273 | math | The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms | PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms.
It suffices to show that the terms of the sequence
$$\,b_n=\mathrm{e}^n\,\mathrm{mod}\, 2,\,\,\,n\in\mathbb N,$$
are dense in $[0,2]$.
Unfortunately, Weyl's Theorem does not look helpful in ... | https://math.stackexchange.com/questions/2621776 | [
"real-analysis",
"sequences-and-series",
"uniform-distribution"
] | 46 | 2018-01-25T23:17:44 | [
"For any $a\\in\\mathbb N$, $\\lfloor a^n\\rfloor$ will have the same parity as $a$. For $0 < a < 1$ we always have $\\lfloor a^n\\rfloor = 0$. But for any other values of $a$ I don't see obvious way to prove or disprove the property from the problem. Maybe it has nothing to do specifically with $e$ and a proof wou... | 0 | Science | 0 |
274 | math | Is every finite list of integers coprime to $n$ congruent $\pmod n$ to a list of consecutive primes? | For example the list $(2, 1, 2, 1)$ is congruent $\pmod 3$ to the consecutive primes $(5, 7, 11, 13)$. But how about the list $(1,1,1,1,1,1,1,1,2,3,4,3,2,3,1) \mod 5$?
More generally, we are given some integer $n \geq 2$ and a finite list of integers that are coprime to and less than $n$. Is it always possible to produ... | https://math.stackexchange.com/questions/2194973 | [
"number-theory",
"elementary-number-theory",
"prime-numbers",
"prime-gaps"
] | 40 | 2017-03-20T05:52:33 | [
"For those seeking Shiu's paper, here is the citation: Shiu, D.K.L. (2000), Strings of Congruent Primes. Journal of the London Mathematical Society, 61: 359-373. doi.org/10.1112/S0024610799007863",
"Very interesting question. Just as an observation, if we took $n$ to be a prime and a finite list of values less th... | 0 | Science | 0 |
275 | math | Are these generalizations known in the literature? | By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(2k)}{(a-2k)!}\ln^{a-2k}(z)\tag{b}$$
I managed to find:
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+yx^2}dx=\frac{\left(\frac{\pi}... | https://math.stackexchange.com/questions/4397045 | [
"integration",
"sequences-and-series",
"reference-request",
"harmonic-numbers",
"polylogarithm"
] | 40 | 2022-03-05T22:37:13 | [
"Very impressive and nice work !",
"Marvelous job. A lot of work.",
"Thanks for great effort in exploring to such an amazing depth!",
"@AliShadhar Thank you so much for your comment. I meant significance in a broad sense... Have similar identities been applied in other areas of math (to give better approximat... | 0 | Science | 0 |
276 | math | If $K\cong K(X)$ then must $K$ be a field of rational functions in infinitely many variables? | If $k$ is any field, then the field $K=k(X_0,X_1,\dots)$ of rational functions in infinitely many variables satisfies $K(X)\cong K$ (by mapping $X$ to $X_0$ and $X_n$ to $X_{n+1}$). My question is, does the converse hold? That is:
Suppose $K$ is a field such that $K\cong K(X)$. Must there exist a field $k$ such tha... | https://math.stackexchange.com/questions/2856060 | [
"abstract-algebra",
"field-theory"
] | 37 | 2018-07-18T15:05:43 | [
"@JeremyRickard Nevermind, those are the same thing. And I see now why my idéa doesn't work.",
"@JeremyRickard I thought that $K(X)$ denoted $\\text{Frac} (K[X])$, not field extension.",
"@Kilian I don't see why $K$ would be isomorphic to $K(X)$. $\\mathbb{R}((X_0,X_1,\\dots))$ is not generated as a field exten... | 0 | Science | 0 |
277 | math | Constructing an infinite chain of subfields of 'hyper' algebraic numbers? | This has now been cross posted to MO.
Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn from $F$.
That is, we let $S_F$ denote
$$\bigg \{x \in \mathbb{R}: 0=\sum_{i=1}^n{a_i x^{e_i}}: e_i \in F \text{ distinc... | https://math.stackexchange.com/questions/3014759 | [
"real-analysis",
"field-theory",
"transcendental-numbers"
] | 36 | 2018-11-26T10:55:38 | [
"@JackM, $x=0$ satisfies this and is algebraic. I am not sure I understand your meaning... But if you take $x=y^6$ then we can replace your equation with $y^3+y^2=0$ and we can always manage to do this type of change when the exponents are rational.",
"Why is $S_\\mathbb Q$ equal to the set of algebraic real num... | 0 | Science | 0 |
278 | math | An iterative logarithmic transformation of a power series | Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion:
$$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$
Then, at each step we apply the following transformation:
$$f_{n+1}(x)=x^{-1}\log\left(\frac{f_n(x)}{f_n(0)}\right).\tag2$$
A few initial i... | https://math.stackexchange.com/questions/4338032 | [
"limits",
"logarithms",
"power-series",
"closed-form",
"lambert-w"
] | 36 | 2021-12-19T19:33:11 | [
"He also gives the first terms of the next power series in the sequence. Then, he writes the first ever recorded insights on the formula for iteration of power series $f \\circ \\ldots \\circ f(x)$, which then insipired Schröder in his 1872 paper, in which he first considered, what now call, the \"Schröder equation... | 0 | Science | 0 |
279 | math | Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations | Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$
Questions:
1. Do either $S$ or its elements have an accepted name?
2. Where can I learn more about the set $S$ and it elements... | https://math.stackexchange.com/questions/1081468 | [
"number-theory",
"reference-request",
"algebraic-number-theory",
"real-numbers",
"transcendental-numbers"
] | 36 | 2014-12-26T06:48:47 | [
"Could we build this set by transfinite recursion and then study its properties (like for the Borel Hierarchy)?",
"@EricWofsey, if S is closed under subtraction it would include $2^\\sqrt{2}-1$, and that seems unlikely.",
"S is a subset of the positive \"elementary-logarithmic\" or \"closed-form\" numbers, as d... | 0 | Science | 0 |
280 | math | Determining the kernel of a Vandermonde-like matrix | The kernel of a Vandermonde matrix can be determined using this formula.
The following type of matrix has a similar structure, and should also have a one-dimensional kernel.
$$V=
\begin{bmatrix}
1 & 1 & 1 & \ldots & 1 \\
x_1 & x_2 & x_3 & \ldots & x_n \\
x_1^2 & x_2^2 & x_3^2 & \ldots & x_n^2 \\
\vdots & \vdots & \v... | https://math.stackexchange.com/questions/407484 | [
"linear-algebra",
"matrices",
"determinant"
] | 35 | 2013-05-31T00:32:53 | [
"Wouldn't there be an additional row of $y_1^2 \\cdots y_n^2$ before the row $y_1^2x_1 \\cdots y_n^2x_n$?",
"Assuming you're in the complex field, what you're asking for is exactly which are the two-variable polynomials in $\\mathbb{C}[x,y]$ with degree $m-1$ in both $x$ and $y$, such that they are zero on a give... | 0 | Science | 0 |
281 | math | Is the power of $2$ in the Euclidean norm related to the fact that the algebraic closure of the reals is $2$-dimensional? | Consider any local field $K$, endowed with its topological field structure. We define the function $| \cdot | : K \to \mathbb{R}_{\ge 0}$ as
$$|x| = \frac{\mu(xS)}{\mu(S)},$$
where $\mu$ is any Haar measure (which exists because a local field is additively a locally compact group), and $S$ is any measurable set of nonz... | https://math.stackexchange.com/questions/4353238 | [
"linear-algebra",
"number-theory",
"measure-theory",
"soft-question",
"local-field"
] | 34 | 2022-01-10T04:38:48 | [
"The question has been posted on MathOverflow.",
"I also suggest asking on MO, as Nolord suggested. You can repost it there and link the questions to each other (for example, with a brief heading saying \"Cross-posted to/from MO/MSE.\").",
"@user76284 Edited, thanks for the suggestion!",
"Minor suggestion: Yo... | 0 | Science | 0 |
282 | math | Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity? | It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of rotating objects can be surprising (see the Dzhanibekov effect); in 4D it could be more surprising.
A 2D or 3D spacecraft (with no reaction wheels or gimbaling etc.) needs at least two thrusters to control its spin. See my answer... | https://math.stackexchange.com/questions/4472850 | [
"ordinary-differential-equations",
"analysis",
"dynamical-systems",
"physics"
] | 34 | 2022-06-14T18:13:22 | [
"Cross-posted to MO: mathoverflow.net/questions/448773/…",
"Oh, I didn't explain what $M$ is (though the linked physics.SE post did explain it). It's the matrix of second moments of mass: $M=\\int\\mathbf r\\,\\mathbf r^T\\,\\rho\\,dV$, where $\\mathbf r$ is position relative to the centre of mass, $\\rho$ is den... | 0 | Science | 0 |
283 | math | Semirings induced by symmetric monoidal categories with finite coproducts | A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite coproducts (this includes an initial object $0$) with the property that $\otimes$ preserves finite coproducts in each variable. ... | https://math.stackexchange.com/questions/813676 | [
"ring-theory",
"category-theory",
"examples-counterexamples",
"monoidal-categories"
] | 33 | 2014-05-29T05:39:16 | [
"@Arducode This does not define a functor for $\\otimes$. Or how do you define the tensor product of morphisms? Also notice that it is required that the tensor product is distributive - this is not guaranteed by arbitrary isomorphisms, they have to be the canonical ones. Therefore I do not think that the skeleton w... | 0 | Science | 0 |
284 | math | Can you obtain $\pi$ using elements of $\mathbb{N}$, and finite number of basic arithmetic operations + exponentiation? | Is it possible to obtain $\pi$ from finite amount of operations
$\{+,-,\cdot,\div,\wedge\}$ on $\mathbb{N}$ (or $\mathbb{Q}$, the answer will still be the same)? Note that the set of all real numbers obtainable this way contains numbers that are not algebraic (for example $2^{2^{1/2}}$ is transcendental).
Bonus: If it ... | https://math.stackexchange.com/questions/2865253 | [
"number-theory",
"pi",
"transcendental-numbers"
] | 33 | 2018-07-28T06:29:14 | [
"As mentioned above, it is probably false, but a disproof is difficult. A hope could be to describe a subfield of $\\Bbb{R}$ containing all elements generated by the method you describe, but not containing $\\pi$. How such a field would look is difficult to imagine though.",
"The bonus question is the same questi... | 0 | Science | 0 |
285 | math | A holomorphic function sending integers (and only integers) to $\{0,1,2,3\}$ | Does there exist a function $f$, holomorphic on the whole complex plane $\mathbb{C}$, such that $f\left(\mathbb{Z}\right)=\{0,1,2,3\}$ and
$\forall z\in\mathbb{C}\ (f(z)\in\{0,1,2,3\}\Rightarrow z\in\mathbb{Z})$?
If yes, is it possible to have an explicit construction?
Note that, for example, $h(z)=\frac{1}{6} \left(9-... | https://math.stackexchange.com/questions/4624439 | [
"complex-analysis",
"galois-theory",
"riemann-surfaces",
"elliptic-functions"
] | 32 | 2023-01-23T11:39:57 | [
"It is already solved on MO",
"Crossposted to MathOverflow: A holomorphic function sending integers (and only integers) to $\\{0,1,2,3\\}$",
"I suggest you post this to MathOverflow, adding a link to this MSE post to let readers know you've crossposted.",
"I'd ask for a Laurent polynomial solution in a separa... | 0 | Science | 0 |
286 | math | A conjecture about Catalan sequence | For $n=2$, consider the free abelian group generated by polynomials corresponding to $\frac{(2 n)!}{2^n n!}=3$ partitions of $2n=4$ vertices into pairs (chords):
$$
(x_1-x_3)(x_2-x_4),(x_1-x_2)(x_3-x_4),(x_1-x_4)(x_2-x_3)
$$
Euler's Identity: $
(x_1-x_3)(x_2-x_4)=(x_1-x_2)(x_3-x_4)+(x_1-x_4)(x_2-x_3)$.
So$$\{(x_1-x_2)... | https://math.stackexchange.com/questions/3792709 | [
"polynomials",
"conjectures",
"catalan-numbers",
"free-abelian-group"
] | 31 | 2020-08-16T03:38:47 | [
"@DietrichBurde The first example you gave only starts failing at the $x^{10}$ term. $132$ and $429$, as well as the two terms that follow $1430$ and $4862$, are Catalan numbers. The second example evaluates to $3$ when $k=3$. Though I do get your point that it's best not to be too confident about integer sequences... | 0 | Science | 0 |
287 | math | How do I find the common invariant subspaces of a span of matrices? | Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices.
Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set of $G$ matrices, and let $\mathcal{S}$ be any linear subspace of $\mathcal{G}$.
Finally, let $\mathcal{I}$ be a linear sub... | https://math.stackexchange.com/questions/1122655 | [
"linear-algebra",
"matrices",
"vector-spaces",
"eigenvalues-eigenvectors"
] | 31 | 2015-01-27T16:06:22 | [
"How about computing products $G_1G_2\\cdots G_k$, $\\forall k \\in \\{1,N\\}$ and multiply a random vector with that. Then put all vectors with some large enough norm into a new matrix and then iterate by the new matrix with the sequence of matrices. Then iterate the procedure. This (I hope) will give something si... | 0 | Science | 0 |
288 | math | Minimal time gossip problem | The gossip problem (telephone problem) is an information dissemination problem
where each of $n$ nodes of a communication network has a unique piece of information
that must be transmitted to all the other nodes using two-way communications
(telephone calls) between the pairs of nodes. Upon a call between the given two... | https://math.stackexchange.com/questions/576326 | [
"combinatorics",
"graph-theory",
"discrete-mathematics"
] | 31 | 2013-11-21T11:25:40 | [
"Yes, this paper is about fault-tolerance of the gossip schemes. I have some ideas, and I want simply find any results about minimum number of calls of a gossip scheme with minimal time $T(n)$ to develop the theory. Presumably it is an open question.",
"It was this one:arxiv.org/abs/1304.5633 but I didn't read th... | 0 | Science | 0 |
289 | math | The limit of $(\sin(n!)+1)^{1/n}$ as n approaches infinity | Calculate the limit
$$
\lim_{n\rightarrow\infty}(\sin(n!)+1)^{1/n}
$$
or prove that the limit does not exist.
This appeared as a problem in my mathematical analysis test, and the answer was that the limit exists and it was $1$. But later the teacher found a mistake in his proof and eventually removed the problem from t... | https://math.stackexchange.com/questions/4577234 | [
"sequences-and-series",
"limits",
"analysis",
"trigonometry",
"factorial"
] | 30 | 2022-11-15T07:22:14 | [
"One way to show that this sequence diverges would be to construct a subsequence which diverges.",
"AFAIK, $1$ will be an accumulation point (limit of a subsequence). But it is much harder to say that $0$ is not another one...",
"Factorial is not defined on not-integers. If the gamma function is used to extend ... | 0 | Science | 0 |
290 | math | Wave equation: predicting geometric dispersion with group theory | Context
The wave equation
$$
\partial_{tt}\psi=v^2\nabla^2 \psi
$$
describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is different in odd vs even number of spatial dimensions, $n$. A point source in odd-$n$ creates a disturbance that propagates... | https://math.stackexchange.com/questions/4278597 | [
"group-theory",
"partial-differential-equations",
"mathematical-physics",
"wave-equation",
"dispersive-pde"
] | 29 | 2021-10-16T12:35:52 | [
"@GiuseppeNegro I had tried studying the Howe-Tan account of the Huygens' principle about five years ago and at the time I too could not handle the representation theory involved (I'm afraid I still can't). I don't really have any useful insights, as such I can't claim any credits, though thank you for acknowledgin... | 0 | Science | 0 |
291 | math | Binomial coefficients modulo a prime | Consider an odd prime $p\equiv1 \pmod {16}$ and set $M=\frac{p-1}{2}$ for notational convenience. Then is there even a single prime $p$ of the above form for which the following congruence holds? $$\binom{M}{M/2}\binom{M}{M/4}\equiv \pm \binom{M}{3M/8}\binom{M}{7M/8}\pmod p ?$$
Here the symbol $\binom{n}{r}$ is a binom... | https://math.stackexchange.com/questions/3411958 | [
"modular-arithmetic",
"binomial-coefficients",
"finite-fields"
] | 29 | 2019-10-27T21:43:24 | [
"@JyrkiLahtonen it's been a while but yes the presence of that last factor is critical. For some more information, this is about superspecial hyperelliptic curves in the family $y^2 =x(x^4-1)(x^4+ax^2+1)$. I was trying to prove that for all primes $p = 1,7 \\mod 8$, and $p>7$, there is a superspecial curve in this ... | 0 | Science | 0 |
292 | math | $f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity | Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^1 d(f(t),w)\,dt$ for every $w\in X$. If such $m_f\in X$ exists then we call $f$ as $D$-integrable with $D$-integral $m_... | https://math.stackexchange.com/questions/49208 | [
"calculus",
"group-theory",
"metric-spaces"
] | 28 | 2011-07-03T07:06:02 | [] | 0 | Science | 0 |
293 | math | Lowest dimensional faithful representation of a finite group | How does one compute the lowest dimensional faithful representation of a finite group?
This question originated in the context of given a finite group $G$: trying to find the lowest dimensional shape whose rotational/reflection symmetries form $G$. (Formally stated as finding the lowest dimensional faithful representat... | https://math.stackexchange.com/questions/1688173 | [
"linear-algebra",
"group-theory",
"finite-groups",
"representation-theory",
"linear-transformations"
] | 28 | 2016-03-08T00:31:03 | [
"For future visitors: some research developments since this question was asked mathoverflow.net/questions/351938/… as well it is interesting to ask even whether there are small irreducible representations, as discussed here: mathoverflow.net/questions/400864/… Finally, note that $j$ in the question statement is cal... | 0 | Science | 0 |
294 | math | Dividing the whole into a minimal amount of parts to equally distribute it between different groups. | Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed in $x_i$ equal piles for any $1\leq i\leq n$.
The question is what is the minimal amount of parts for some $x_1, x_2, .... | https://math.stackexchange.com/questions/1381042 | [
"number-theory",
"elementary-number-theory",
"divisibility"
] | 28 | 2015-08-01T03:46:39 | [
"Note that the same question has been asked at math.stackexchange.com/questions/1383406/…",
"OK, so for $n$ coprime numbers, we can always do it with $1+\\sum(x_i-1)$ pieces, and the question is, under what circumstances can we do it with fewer, and how many fewer. So I wonder whether the $3,4,5$ construction gen... | 0 | Science | 0 |
295 | math | Circles of radius $1, 2, 3, ..., n$ all touch a middle circle. How to make the middle circle as small as possible? | Non-overlapping circles of radius $1, 2, 3, ..., n$ are all externally tangent to a middle circle.
How should we arrange the surrounding circles, in order to minimize the middle circle's radius $R$?
Take $n=10$ for example.
On the left, going anticlockwise the radii are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, and $R\approx... | https://math.stackexchange.com/questions/4619480 | [
"geometry",
"optimization",
"circles",
"discrete-geometry"
] | 27 | 2023-01-16T03:46:59 | [
"I wrote a program that loops over all permutations of 2...10 (excluding the circle of radius 1), and finds an $R$ that solves the equation that you wrote (the sum of the arc-cosines). It confirmed that the optimal permutation is the one that you conjectured: 2, 9, 4, 7, 6, 5, 8, 3, 10, where the radius of the inne... | 0 | Science | 0 |
296 | math | A curious identity on $q$-binomial coefficients | Let's first recall some notations:
The $q$-Pochhammer symbol is defined as
$$(x)_n = (x;q)_n := \prod_{0\leq l\leq n-1}(1-q^l x).$$
The $q$-binomial coefficient (also known as the Gaussian binomial coefficient) is defined as $$\binom{n}{k}_q := \frac{(q)_{n}}{(q)_{n-k}(q)_{k}}.$$
I found the following curious ident... | https://math.stackexchange.com/questions/4434174 | [
"binomial-coefficients",
"q-analogs",
"q-series"
] | 27 | 2022-04-22T23:52:21 | [
"So we begin with a combinatorial interpretation : Consider all $n$-tilings using $k$ green tiles and $n-k$ red tiles, such that every green tile's weight is $q^r$ where $r$ is the number of red tiles preceding this green tile. The weight of a tiling is the product of weights of all green tiles, and the sum over al... | 0 | Science | 0 |
297 | math | What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$? | As in this post, define the ff:
$$K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$
$$K_4(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac14,\tfrac34,1,\,k^2\right)}$$
$$K_6(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac16,\tfrac56,1,\,k^2\right)... | https://math.stackexchange.com/questions/3103805 | [
"integration",
"definite-integrals",
"closed-form",
"hypergeometric-function",
"elliptic-functions"
] | 27 | 2019-02-07T05:32:18 | [
"At one point, I convinced myself that dlmf.nist.gov/15.5 (at least a couple of them) worked for fractional derivatives. If so, then the order might be reduced To 1F1(a,b;k^2), which might be easier to evaluate (and invert). If you're still interested, I might take a stab at that. No warranties implied :)",
"A... | 0 | Science | 0 |
298 | math | Is there a group $G$ for which $\mathrm{Aut}(G) \simeq (\mathbb{R},+)$? | I know the classic theorem that $(\mathbb{Q},+)$ cannot be expressed as an automorphism group, i.e. there is no group $G$ such that $\mathrm{Aut}(G)\simeq (\mathbb{Q},+)$.
Theorem A. If $L$ is a locally cyclic group with no element of order $2$, then $L$ cannot be expressed as an automorphism group.
But how about $(\... | https://math.stackexchange.com/questions/3925067 | [
"group-theory",
"automorphism-group"
] | 27 | 2020-11-27T07:48:46 | [
"Thanks, I did already know that! Same idea to prove that $\\mathbb{Q}$ is not an automorphism group.",
"Partial answer: $G$ cannot be abelian. If $G$ is abelian, then $x\\mapsto x^{-1}$ is an automorphism of $G$ of order $\\leq 2$. $\\mathbb{R}$ has no nonidentity elements of finite order, so $x=x^{-1}$ for all ... | 0 | Science | 0 |
299 | math | If $r\in\mathbb{Q}\setminus\mathbb{Z}$ is it possible that $r^{r^{r^r}}\in \mathbb{Q}$? | It's straightforward to prove that $r^r\notin\mathbb{Q}$, which furthermore allows us to use the Gelfond-Schneider theorem to prove that $r^{r^r}\notin\mathbb{Q}$.
Is it true that $r^{r^{r^r}}\notin\mathbb{Q}$?
It seems like it ought to be true, though my guess would be that this is an open problem. If not, does anyo... | https://math.stackexchange.com/questions/3655352 | [
"reference-request",
"exponentiation",
"irrational-numbers",
"tetration"
] | 27 | 2020-05-02T09:25:14 | [
"en.wikipedia.org/wiki/… - the techniques for the proof by infinite descent here can be generalized in a straightforward way",
"@DarkMalthorp how do u prove $r^r \\not \\in \\mathbb{Q}$?",
"Yes, indeed it is a stronger result :) Of course, I fully expect that $r^{r^{r^r}}$ is also transcendental.",
"@darkmalt... | 0 | Science | 0 |
300 | math | What functions from $\Bbb N\to\Bbb N$ can be made from $+$, $-$, $\times$, $\div$, exponentiation, and $\lfloor\cdot\rfloor$? | What functions from $\Bbb N\to\Bbb N$ can be made from $+$, $-$, $\times$, $\div$, exponentiation, and $\lfloor\cdot\rfloor$? Call this class of functions $\mathcal Flex$ (for "floor and exponentiation").
The $\rm mod$ function $a\operatorname{mod}b$ can be defined by $a-b\lfloor a/b\rfloor$. In addition,
$$\binom nk=\... | https://math.stackexchange.com/questions/4605213 | [
"exponential-function",
"diophantine-equations",
"recursion",
"computability",
"ceiling-and-floor-functions"
] | 26 | 2022-12-24T20:59:34 | [
"Given that we have binomials and modular arithmetic, the function $\\Big\\lfloor 10^n\\frac{16^{f(n)}}{f(n){2f(n)\\choose f(n)}^2} \\pmod{10}\\Big\\rfloor$ probably* gives the $n^\\textrm{th}$ digit of $\\pi$ for sufficiently fast-growing $f(n)$ like $n^{n^{n^n}}$. *Unless pi has an incredibly long string of zero... | 0 | Science | 0 |
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