parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/272665 | 4 | The usual conjugate gradient type algorithms for iteratively finding the inverse of a matrix applied to a vector, $x = A^{-1} y$, works by minimizing $\|Ax - y\|^2$ where $\| \cdot \|$ is the $L^2$-norm. The stopping criterion is usually $\|Ax - y\| < \varepsilon \|y\|$ with some small $\varepsilon$.
Is there an effi... | https://mathoverflow.net/users/41312 | Iterative matrix inversion with $L^\infty$ norm | One approach is to solve the optimization problem:
\begin{equation\*}
\min\_x\quad \|Ax-y\|\_\infty.
\end{equation\*}
This is a nonsmooth optimization problem, but is amenable to a variety of scalable optimization techniques, for instance, [Nesterov's smooth minimization of non-smooth functions](https://link.springer.c... | 4 | https://mathoverflow.net/users/8430 | 272667 | 122,012 |
https://mathoverflow.net/questions/272668 | 11 | Let $T$ denote the unit sphere in complex plane. $\mathbb{Z}/2$ acts by complex conjugation on $T$. There is an induced action on $BT$. The cohomology of the homotopy orbit space of this action is
$$
H^\*(BT//(\mathbb{Z}/2),\mathbb{F}\_2) = H^\*(B\mathbb{Z}/2,\mathbb{F}\_2)\otimes H^\*(BT,\mathbb{F}\_2)=\mathbb{F}\_... | https://mathoverflow.net/users/95770 | Conjugation action on the classifying space of circle | It is the interesting one, with $Sq^1(x) = tx$ and the remainder determined by the axioms of Steenrod operations. This may be seen as the homotopy orbits is a model for BO(2), where one knows the Steenrod operations. (Here I am assuming the choice of class $x$ which pulls back to zero under the section; under the equiv... | 14 | https://mathoverflow.net/users/318 | 272675 | 122,014 |
https://mathoverflow.net/questions/272670 | 5 | Let $G$ be a finite group, and $g,h\in G$ be such that $G = \langle g,h\rangle = \langle g,hgh^{-1}\rangle$ - that is, $g,h$ generate $G$, and so does $g,hgh^{-1}$.
Let $n := |G|$, $r := |g|$, and $e := |ghg^{-1}h^{-1}|$ ("$|.|$" denotes "order")
I was doing some computations with covers of curves, and it seems tha... | https://mathoverflow.net/users/88840 | curious relation between orders of generators of a finite group | Denote $f=hgh^{-1}$.
1) If $r\geqslant 4$, then $n\geqslant 4$ and either $e=1$, $r=n$ or $e\geqslant 2>\frac{n}{2+n/2}\geqslant \frac{n}{2+n-2n/r}$.
2) If $r=3$, then either $e\geqslant 3\geqslant \frac{n}{2+n/3}=\frac{n}{2+n-2n/3}$ or $e=2$, then we have $f^3=g^3=1$, $(fg)^2=1$. It follows that the group generate... | 5 | https://mathoverflow.net/users/4312 | 272677 | 122,015 |
https://mathoverflow.net/questions/272679 | 2 | A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest gets all of what is left.)
Let $s(N)$ be the maximum of all shares of pie if we have $N$ guests. What is the value o... | https://mathoverflow.net/users/8628 | Limit of biggest share of the pie | This limit is infinite. Guest number $k$ gets the quotient $x\_k=\frac kN(1-\frac1N)(1-\frac2N)\dots (1-\frac{k-1}N)$, for $k$ about $\sqrt{N}$ this behaves like $\frac{1}{\sqrt{N}}$, cince the product is bounded by below (actually it tends to $e^{-1/2}$). It is easy to see that $k$ corresponding to the maximal quotien... | 2 | https://mathoverflow.net/users/4312 | 272680 | 122,017 |
https://mathoverflow.net/questions/272684 | 8 | Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix.
Let $f(x\_i,y\_i,z\_i)\geq 0$, $x\_i\geq 0$, $z\_i\geq 0$ for $i\in\{1,2\}$.
(**Edit:** i.e. the matrix is positive semidefinite).
Then
$$
f(x\_1+x\_2,y\_1+y\_2,z\_1+z\_2)\g... | https://mathoverflow.net/users/11100 | A direct proof of a property of symmetric 2x2-determinants | Here is a direct proof: Simple computations show that
$$f\left(x\_1+x\_2,y\_1+y\_2,z\_1+z\_2\right) = f\left(x\_1,y\_1,z\_1\right) + f\left(x\_2,y\_2,z\_2\right) + \left(x\_1z\_2+x\_2z\_1-2y\_1y\_2\right).$$
It thus remains to check that all three addends on the right hand side are nonnegative. For $f\left(x\_1,y\_... | 3 | https://mathoverflow.net/users/2530 | 272689 | 122,021 |
https://mathoverflow.net/questions/272662 | 5 | Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ the base of $\Phi = \Phi(G,T)$ corresponding to $\Phi^+$.
A subset $\Psi$ of $\Phi^+$ is called **closed** if whenever $\a... | https://mathoverflow.net/users/38145 | Reference Request: Derived group of $\mathscr R_u(B)$ | I think the answer is yes.
We can assume that $G$ is semisimple. As you have seen already, and which is is clear from the commutation relations, we have $[U, U] \leq \prod\_{\alpha \in \Phi^+ - \Delta} U\_\alpha$. I guess you could show equality by using the Chevalley commutator relations, although you need to be car... | 3 | https://mathoverflow.net/users/10146 | 272695 | 122,022 |
https://mathoverflow.net/questions/272691 | 3 | Jacobi elliptic function $\mathrm{sn}$ is defined as
$$\operatorname{sn}(u,k)=x\Leftrightarrow u=\int\_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$
and Weierstrass sigma function $\sigma$ is defined as
$$ \sigma(u)=u\prod\_{0\neq\ell\in{L\_z}} \left(1-\frac u \ell\right)e^{\frac u \ell + \frac{1}{2} \left(\frac u \... | https://mathoverflow.net/users/111417 | Jacobi and Weierstrass elliptic function | Weierstrass $\sigma$-function is entire function, it has no poles. Jacobi elliptic sine function (in general case) is an elliptic function, so it has poles. [For each path $\gamma$ connecting $0$ and $\infty$ (in general case and for $k=1$) the integral
$$u\_\gamma=\int\_\gamma \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}$$
is ... | 2 | https://mathoverflow.net/users/5712 | 272696 | 122,023 |
https://mathoverflow.net/questions/243015 | 14 | Let $H$ be a Hilbert space, and
let $A\_1,A\_2,A\_3\subset B(H)$ be three commuting von Neumann algebras.
We write $\odot$ for the algebraic tensor product,
and $\bar\otimes$ for the spatial tensor product of von Neumann algebras.
Suppose that for every $i,j\in\{1,2,3\}$, the map $A\_i\odot A\_j\to B(H)$ extends... | https://mathoverflow.net/users/5690 | On spatial tensor products of von Neumann algebras | No. Consider the case where $G$ is an ICC group, $H=\ell\_2(G\times G)$, $A\_1=(\lambda\otimes\lambda)(G)''$, $A\_2=(1\otimes\rho)(G)''$, and $A\_3=(\rho\otimes1)(G)''$. By the Fell absorption principle $(g,h)\leftrightarrow (g,gh)$, one has $A\_1\vee A\_2 \cong (\lambda\otimes1)(G)''\vee(1\otimes\rho)(G)''\cong A\_1\m... | 6 | https://mathoverflow.net/users/7591 | 272705 | 122,026 |
https://mathoverflow.net/questions/272479 | 3 | I am getting myself acquainted to superalgebras. One often comes across odd polynomial rings of the form
$$k\langle x\_i \rangle\_{i\in I} / (x\_ix\_j -(-1)^{|x\_i||x\_j|} x\_jx\_i)$$
for some index set $I$, where the relation divided out sometimes is required for all choices of $i, j$ (which implies $x\_i^2=0$ if ... | https://mathoverflow.net/users/86062 | Generators of odd polynomial superalgbras | Why should one consider both cases? Just as Lie algebras are "built out of copies of $\mathfrak{sl}(2)$", (basic classical) Lie superalgebras are essentially built out of copies of three elementary algebras: $\mathfrak{sl}(2)$, $\mathfrak{sl}(1|1)$, and $\mathfrak{osp}(1|2)$. (This essentially corresponds to the three ... | 2 | https://mathoverflow.net/users/805 | 272707 | 122,027 |
https://mathoverflow.net/questions/272709 | 7 | Let $H\_d:=\mathbb{C}\_d[x,y,z]$ denote the space of homogeneous degree $d$ polynomials in $x$, $y$, $z$ with complex coefficients. I'd like to show that every $f\in H\_3$ can be represented as
$$
f=\det\begin{pmatrix}q\_1 & q\_2\\ \ell\_1 & \ell\_2\end{pmatrix}, q\_i\in H\_2, \ell\_i\in H\_1, i=1,2.
$$
This is differe... | https://mathoverflow.net/users/11100 | 2x2-determinantal representations of cubic curves | Let $(\ell\_1,\ell\_2)$ be the ideal of a point in $V(f)$, so $f \in (\ell\_1,\ell\_2)$.
| 9 | https://mathoverflow.net/users/88133 | 272711 | 122,028 |
https://mathoverflow.net/questions/272694 | 12 | [Lambek & Scott: Introduction to higher order categorical logic](http://dl.acm.org/citation.cfm?id=7517)
says that
>
> "For example, categorists may be unhappy when we treat categories as special kinds of deductive systems and logicians may be unhappy when we insist that deductive systems need not be freely gene... | https://mathoverflow.net/users/39422 | Category theorists stance on deductive systems | This sentence reads to me like "we will treat Xs as if they were special kinds of of Ys, which will make both X-theorists and Y-theorists unhappy because it is not true". Reminds me of the old joke when a patient says "Doctor, it hurts when I do this" and the doctor says "Then don't do that!"
I think a more modern th... | 14 | https://mathoverflow.net/users/49 | 272720 | 122,030 |
https://mathoverflow.net/questions/218666 | 4 | Let $(P,H)$ be a [Euclidean Hartogs figure](https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem) in $\mathbb{C}^n$, and
let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there is a holomorphic map $F:P\to\mathbb{C}^n$ such that $F\eq... | https://mathoverflow.net/users/40409 | Hartogs's extension theorem | I think what you would like is actually true:
Let $P'=F(P), H'=F(H)=f(H)\subseteq \mathbb C^n$. Consider the holomorphic map $g=f^{-1}:H'\to H$. Since $f$ is injective, $g$ cannot have essential singularities on $P'$, so we may consider it as a meromorphic map $g:P'\to P$.
Now let $x\in P$ a point such that $g=(g\... | 4 | https://mathoverflow.net/users/10076 | 272727 | 122,033 |
https://mathoverflow.net/questions/271376 | 15 | Numerical calculation suggests that for prime $p\ge 5$,
\begin{align\*}
\sum\_{k=1}^{\frac{p-1}{2}}\frac{(-1)^k}{k}\sum\_{i=\lfloor k/2\rfloor +1}^k\frac{1}{2i-1}\equiv 0\pmod{p}.
\end{align\*}
>
> **Question.** How can we prove this congruence?
>
>
>
| https://mathoverflow.net/users/6104 | How to prove $\sum_{k=1}^{\frac{p-1}{2}}\frac{(-1)^k}{k}\sum_{i=\lfloor k/2\rfloor +1}^k\frac{1}{2i-1}\equiv 0\pmod{p}$? | It turned out the OP's question required a good amount of work and a joint effort (with Roberto Tauraso). However, "*the margin here is too small to contain the proof*", so to say.
Instead, readers can find the resolution [at this link](https://www.math.temple.edu/~tewodros/Harmonic_Sumsvnew.pdf). In addition, we pro... | 11 | https://mathoverflow.net/users/66131 | 272741 | 122,037 |
https://mathoverflow.net/questions/272729 | 0 | Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.
Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. Is $ux$ necessarily in $m^k\setminus m^{k+1}$?
The case I care about is when $R$ is an affine semigroup ring over $\Bb... | https://mathoverflow.net/users/36720 | Adic filtration and integral closure | In general, this can be wrong. Consider $R=\mathbb{R}[[x,y]]/(x^2+y^4)$ (or $\mathbb{R}[x,y]\_{(x,y)}/(x^2+y^4)$). Then $u=x/y^2$ is in the integral closure $S$ and it is a unit. $y^2\in\mathfrak{m}^2$, $uy^2=x\in\mathfrak{m}$, but not in its square.
| 3 | https://mathoverflow.net/users/9502 | 272750 | 122,039 |
https://mathoverflow.net/questions/272715 | 16 | $\DeclareMathOperator\cd{cd}$A question that I have already posted in the *Mathematics* section, but which seems to be too delicate for that section (see [here](https://math.stackexchange.com/questions/2320997/cohomological-dimension-of-direct-product) and [here](https://math.stackexchange.com/questions/2325286/cohomol... | https://mathoverflow.net/users/13326 | Cohomological dimension of $G \times G$ | $\DeclareMathOperator\cd{cd}\newcommand{\FP}{\mathrm{FP}}$Another class of groups for which $\cd(G\times G)=2 \cd(G)$, containing duality groups, is the class of groups of type $\FP\_{\infty}$.
If $G$ is of type $\FP\_{\infty}$ that is the trivial module has a projective resolution that is finitely generated in each ... | 8 | https://mathoverflow.net/users/15934 | 272755 | 122,041 |
https://mathoverflow.net/questions/272752 | 4 | Let $G$ be a reductive group acting on an affine variety $X$. For simplicity, one may assume $G=SL\_n$ or $G=U\_n$ and assume the field is $\mathbb C$. Given this one can show $\mathbb C[X]^G$ is finitely generated algebra.
>
> **Question:**
>
>
> (1) Each of the orbit closures $\overline{Gx}$ is a finite union o... | https://mathoverflow.net/users/69190 | The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts | The answer to (2) is affirmative: an affine $G$-variety with a dense open orbit (like $\overline{Gx}$) contains a unique closed orbit. The reason for that is that any two closed are orbits are separated by a $G$-invariant. The closed orbit is clearly the unique orbit of minimal dimension since it is contained in the cl... | 6 | https://mathoverflow.net/users/89948 | 272778 | 122,051 |
https://mathoverflow.net/questions/272776 | 10 | Often, TQFTs are defined in *families*, parametrised by some algebraic data. For example, the Turaev-Viro-Barrett-Westbury TQFTs are parametrised by spherical fusion categories, the Crane-Yetter TQFTs are parametrised by ribbon fusion categories, and the $n$-dimensional Dijkgraaf-Witten theory is parametrised by a fini... | https://mathoverflow.net/users/13767 | Which manifolds are sensitive to the cocycle in the Dijkgraaf-Witten model? | The example $G = \mathbb Z/2$ and $M = \mathbb{RP}^3$ works.
The inclusion $\mathbb Z/2\to\{\pm 1\}\subset\mathrm U(1)$ induces an isomorphism $H^3(B\mathbb Z/2, \mathbb Z/2)\to H^3(B\mathbb Z/2, \mathrm U(1))$, so we can pull the cocycles back to $\mathbb Z/2$ cohomology and evaluate on the $\mathbb Z/2$ fundamental... | 9 | https://mathoverflow.net/users/97265 | 272784 | 122,052 |
https://mathoverflow.net/questions/272599 | 4 | If I consider a Liouville equations in the unit disk $D \setminus\{0\} \subset \mathbb{R}^2$ with singularity at $x=0$,
$$\Delta u= e^{2u}$$
If I define the order of $u$ at origin is defined to be
$$\lim\_{r \rightarrow 0}\frac{\max\limits\_{|x|=r} u}{\log(1/r)}=\alpha$$
It is known that $u$ has the expression
\begin{... | https://mathoverflow.net/users/87922 | The order of the solution of Liouville equations at singularity | The short answer is that $\alpha>1$ cannot happen. Here is why:
For clarity, I will use $z=x+iy$ as the (complex) domain variable, rather than $x$, and let $D^\*= D\setminus\{0\}$ denote the punctured unit disk in the complex plane. The equation $\Delta u = e^{2u}$ is the condition that the metric $g = e^{2u}(\mathrm... | 3 | https://mathoverflow.net/users/13972 | 272790 | 122,054 |
https://mathoverflow.net/questions/272743 | 1 | Let $n\_1, n\_2 \geq 1$ be known integer constants.
Suppose that we have the following system of $n$ polynomial inequalities
for which we know that there exists a feasible solution $(p\_1, p\_2) \in (0,1)^2$. Furthermore, we assume that each $\epsilon\_i$ is much smaller than the corresponding $\mu\_i$ ($\epsilon\_i$... | https://mathoverflow.net/users/102057 | Omitting constraints of polynomial system | Here are some thoughts:
* The system $x+2y=0.75,x^2+2y^2=0.25$ Is the intersection of a line and an ellipse with two exact solutions, One near $(0.45,0.15)$ and the other near $(0.05,0.35).$ A system of your type could have two solution regions. It almost certainly would if $n\_1=n\_2=1$ and hence if they are nearly ... | 1 | https://mathoverflow.net/users/8008 | 272792 | 122,055 |
https://mathoverflow.net/questions/272787 | 11 | Let $(\rho\_{f,\lambda})\_\lambda$ be the system of Deligne's $\ell$-adic representations attached to a modular newform $f$ (where $\lambda$ runs over the finite places of the number field $K$ generated by the Fourier coefficients of $f$), and take a character $\chi\colon\mathrm G\_{\mathbb Q}\rightarrow\overline{\math... | https://mathoverflow.net/users/33820 | "Extra Euler factors" in one definition of the L-function of a twist of a modular form | Let me translate this into a problem purely about automorphic forms:
Take a newform $f \in \mathcal{S}\_k^{\ast}(q,\chi)$, and a primitive Dirichlet character $\psi$ modulo $q'$. Then there exists a newform $f \otimes \psi$ of weight $k$, level dividing $q {q'}^2$, and nebentypus induced by the primitive character in... | 13 | https://mathoverflow.net/users/3803 | 272802 | 122,058 |
https://mathoverflow.net/questions/272807 | 3 | Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably infinite many distinct point masses. For any $P \in \mathcal{C}$ there exists a set $F\_P \in \Sigma^{\infty}$ of full $P^{\i... | https://mathoverflow.net/users/111460 | Measurability of a particular set generated by discrete probability measures | Define $\bar f\_n(x)=\limsup\_{N\to\infty} \frac 1N\#\{j\le N\colon x\_j=x\_n\}$,
$\underline f\_n(x)=\liminf\_{N\to\infty} \frac 1N\#\{j\le N\colon x\_j=x\_n\}$, $g\_n(x)=\prod\_{j<n}\mathbf 1\_{\{x\_j\ne x\_n\}}$.
Then define $F\_n(x)=g\_n(x)\cdot \mathbf 1\_{\bar f\_n(x)=\underline f\_n(x)}\cdot \bar f\_n(x)$.
Thi... | 1 | https://mathoverflow.net/users/11054 | 272817 | 122,065 |
https://mathoverflow.net/questions/272823 | 3 | I am teaching a summer qualifying exam class, and a student was looking up practice questions online. I cannot figure one of them out.
I am trying to solve this problem: Let $M$ be an $n$-dimensional manifold embedded in $\mathbb{R}^{n + 1}$. Then almost every hyperplane in $\mathbb{R}^{n + 1}$ is not tangent to $M$ ... | https://mathoverflow.net/users/40570 | Almost All Hyperplanes are Not Tangent | This should be a little easier than using Sard's theorem:
The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measu... | 6 | https://mathoverflow.net/users/21123 | 272839 | 122,070 |
https://mathoverflow.net/questions/271965 | 2 | In a poset, whenever the meets and joins below exist, their universal properties induce a containment $$(A\vee B)\wedge (A\vee C)\geq A\vee(B\wedge C).$$ This is an instance of *co*distributivity. In a lattice distributivity is equivalent to codistributivity, but this requires the above identity to hold for several ord... | https://mathoverflow.net/users/69037 | Generalizing a codistributive property of sufficiently disjoint normal subgroups to protomodular categories | Any regular protomodular category is also a Mal'tsev category, and thus its lattice of equivalence relation is modular, which means that the following identity holds for all equivalence relations $R,S,T$ :
$$(S\vee R)\wedge T\leq S\vee (R\wedge (S\vee T)).\label{1}\tag{1}$$
Now given three relations $A,B,C$ such that $... | 3 | https://mathoverflow.net/users/111486 | 272847 | 122,072 |
https://mathoverflow.net/questions/272837 | 1 | Is there a useful property for a concrete category $(\mathcal A, U : \mathcal A \to \mathcal X)$ such that in this case $U$ is fibre-complete iff $U$ is topological? (kind of like what "flat" is too "left exact).
The motivation is the following: given a finitely complete category $\mathcal X$ we can define the concr... | https://mathoverflow.net/users/78650 | Generalization of topological functor which need not be fibre-complete | I assume that you mean "topological" in [this sense](https://ncatlab.org/nlab/show/topological+concrete+category), probably with the strict condition that $U(T)=X$ and $g=id$. In that case, an almost-answer is:
>
> $U$ is a bifibration in preorders.
>
>
>
A topological functor certainly has this property. Conv... | 2 | https://mathoverflow.net/users/49 | 272853 | 122,073 |
https://mathoverflow.net/questions/272851 | 4 | Let us consider the moduli space of genus one curves with an effective divisor of degree d; this space is birational to the quotient of the usual moduli space of genus one curves with $d$ marked points $M\_{1,d}$ by the symmetric group $S\_d$ action which permutes the marked points.
I'd like to figure out whether $M\... | https://mathoverflow.net/users/111491 | Unirationality of moduli spaces of marked elliptic curves | Every degree $d$ divisor is a section of a line bundle of degree $d$, which we can write as $\mathcal O(d P)$ for one of $d^2$ possible base points $P$.
So the moduli space of genus $1$ curves with a degree $d$ divisor can be covered by the moduli space of genus one curves with a marked point $P$ and a nonzero secti... | 6 | https://mathoverflow.net/users/18060 | 272856 | 122,074 |
https://mathoverflow.net/questions/270468 | 2 | A ring is called clean if each element of which is a sum of a unit and an idempotent. Recently, I asked a [question](https://mathoverflow.net/questions/267367/a-boolean-quotient-ring-of-a-prime-ring) about a prime ring $R$ in which $R/Soc(R\_R)$ is Boolean where the right socle $Soc(R\_R)$ is non-evident.
My questio... | https://mathoverflow.net/users/48889 | A prime ring which is or is not clean |
>
> So, if any element of $R$ which is a unit in $E$ is also a unit in $R$, then $R$ would be clean too.
>
>
>
Well, that is easy to prove, right? Let $1+x\in R$ be a unit in $E$ ($x\in S$ of course.) Then there exists $y\in E$ such that $(1+x)y=1$. Then $y=1-xy\in R$ since $xy\in S$.
| 1 | https://mathoverflow.net/users/19965 | 272860 | 122,075 |
https://mathoverflow.net/questions/272710 | 19 | Let $\kappa$ be an inaccessible cardinal and let $M \subseteq V\_{\kappa}$ be an inner model of $V\_{\kappa}$, i.e., a transitive model of $\mathsf{ZF}$ containing all the ordinals up to $\kappa$.
My question is whether such a model is always a rank-truncation of an inner model of $\mathbf V$ (defined using $M$ as p... | https://mathoverflow.net/users/100285 | Is every transitive ZF-model of inaccessible height a truncation of an inner model? | **Theorem**: Let $\kappa$ be strongly inaccessible in $V$, such that $V \models ZFC$. If $M\models ZF$, then $L(M) \cap V\_\kappa = M$.
**Proof:** Let us prove by induction on $\alpha < \kappa$ that $L(M) \cap V\_\alpha = M \cap V\_\alpha$.
Let $x \in L(M) \cap V\_{\alpha + 1}$, so $x \subseteq M \cap V\_\alpha$. L... | 10 | https://mathoverflow.net/users/41953 | 272862 | 122,076 |
https://mathoverflow.net/questions/272867 | 4 | Let $X$ be a smooth projective surface, $\mathcal{E}$ a stable torsion free Higgs sheaf of degree 0. Consider the following short exact sequence:
$$0\rightarrow \mathcal{E}\rightarrow \mathcal{E}^{\*\*}\rightarrow \mathcal{S}\rightarrow 0$$ where $\mathcal{E}^{\*\*}$ is the double dual of $\mathcal{E}$. Then $\mathcal{... | https://mathoverflow.net/users/63996 | Second Chern class of Stable Higgs sheaf | Q1: in fact, for a skyscraper sheaf $\mathcal{S}$ on a surface one has $c\_2(\mathcal{S})=- \mathrm{length}(\mathcal{S})$: this follows for instance from example 15.3.1 in Fulton's *Intersection theory*.
Q2: It depends what you call "easy"... It is a nontrivial computation due originally to Lübke, and extended to the... | 5 | https://mathoverflow.net/users/40297 | 272871 | 122,077 |
https://mathoverflow.net/questions/272886 | 2 | Let $X$ be an integral noetherian finite type scheme over an algebraically closed field $k$. Let $X'\to X$ be its normalization. Is the induced homomorphism of etale fundamental groups $\pi\_1(X')\to\pi\_1(X)$ injective?
If so, does this correspond to the fact that the functor Fet(X) $\to $ Fet(X') is fully faithful... | https://mathoverflow.net/users/111506 | Does the fundamental group of the normalization of a scheme inject into the fundamental group of the scheme | Both questions have negative answers. Begin with $\mathbb{P}^4$. Let $L\subset \mathbb{P}^4$, resp. $\Pi\subset \mathbb{P}^4$, be a line, resp. $2$-plane, and assume that $L$ and $\Pi$ are disjoint. Let $C\subset \Pi$ be a smooth projective curve of genus $g>0$. Let $f:C\to L$ be a finite morphism. Let $X$ be the union... | 5 | https://mathoverflow.net/users/13265 | 272889 | 122,084 |
https://mathoverflow.net/questions/272868 | 23 | Let $d \geq 2$ be an integer and $\xi=\exp(\frac{2\pi i}{d})$. I am trying to compute the determinant of the matrix
$$
(\xi^{ij}-1)\_{1 \leq i, j \leq d-1}.
$$ Let me call it $\Delta(d)$. For small values of $d$ I get:
$\Delta(2)=-2$
$\Delta(3)=-3\sqrt{3}i$
$\Delta(4)=-16i$
But I can't seem to fin... | https://mathoverflow.net/users/111500 | Computing a determinant involving roots of unity | Using the earlier responses and comments, I confirm the formula suggested by Neil Strickland:
$$\Delta(d)=d^{d/2}i^{m(d)}\qquad\text{with}\qquad m(d): = 1 + d(7-d)/2\in\mathbb{Z}.$$
Consider the $d\times d$ Vandermonde matrix
$$\Phi(d):=(\xi^{ij})\_{0\leq i,j \leq d-1}.$$
Subtracting the first column from each other co... | 34 | https://mathoverflow.net/users/11919 | 272892 | 122,087 |
https://mathoverflow.net/questions/272895 | 17 | Assume $\Omega$ is an open set in $\mathbb R^3$
such that the intersection of $\Omega$ with any horizontal plane is simply connected.
>
> Can you prove that $\Omega$ is simply connected?
>
>
>
(Note that by the definition, simply connected set can not be empty.)
**Comments.**
* The proof given by Tom Good... | https://mathoverflow.net/users/1441 | Simply connected slices | Yes, I think so. Let's show that every compact set $K\subset \Omega$ is contained in some compact contractible subset of $\Omega$. We use the fact that in a simply connected open subset of the plane every compact set is contained in some compact contractible set.
Denote by $P\_t$ the plane $\mathbb R^2\times t$, and ... | 15 | https://mathoverflow.net/users/6666 | 272899 | 122,090 |
https://mathoverflow.net/questions/272897 | 7 | Let $D$ be the set of *definable ordinals*. An ordinal *s* is definable if there is a predicate $p$ (in the language of (first-order) set theory), such that $p(x) \iff x=s$ for all $x$. This is definitely a set (not a proper class), since the list of all syntactically valid predicates is countable, and $D$ is $\le$ to ... | https://mathoverflow.net/users/65915 | What two ordinals are these (based on definable ordinals)? | In this answer, let me assume as you indicated in the comments that you are working in a second-order set theory with a truth-predicate for first-order truth. Such a theory goes strictly beyond ZFC in consistency strength, but it is provable in Kelley-Morse set theory and indeed, merely in GBC+$\text{ETR}\_\omega$, whi... | 9 | https://mathoverflow.net/users/1946 | 272900 | 122,091 |
https://mathoverflow.net/questions/272883 | 3 | Consider a Brownian sheet $W(t\_1,\ldots,t\_d)$ for some $d\in\mathbb N$. Given some process $X:\mathbb (0,\infty)^d\to\mathbb R$, and time values $T\_1,\ldots,T\_d$, is there a theory similar to that of Itô calculus that makes sense of/studies stochastic integrals of the form
$$\int\_{[0,T\_1]\times\cdots\times[0,T\_d... | https://mathoverflow.net/users/50406 | Itô integral with respect to Brownian sheet | See [Differentiation formulas for stochastic integrals in the plane](http://www.sciencedirect.com/science/article/pii/0304414978900315 "Differentiation formulas for stochastic integrals in the plane") by Wong and Zakai (and references therein) for such an Itô-style calculus in two dimensions.
| 3 | https://mathoverflow.net/users/106304 | 272902 | 122,093 |
https://mathoverflow.net/questions/272885 | 9 | Restudying Marty Isaacs' book *Finite Group Theory*, Chapter 5 - Transfer, I thought of the following by working through some easy examples and I am wondering if it is true. Suppose $G$ is finite and metacyclic in the sense that $G'$ and $G/G'$ are both cyclic. Let $P \in Syl\_p(G)$, where $p$ is the smallest prime div... | https://mathoverflow.net/users/11629 | A metacyclic group $G$ and its Sylow $p$-subgroup, where $p$ is the smallest prime dividing the order of $G$ | Here is my comment expanded slightly. By factoring out Sylow $q$-subgroups of $G'$ for $q \ne p$, we can assume that $G'$ is a $p$-group, so $G' \cap P = G'$. Hence you have shown (using the fact that $p$ is the smallest prime dividing $|G|$) that $G = PC\_G(G')$.
Let $Q$ be a Sylow $p$-complement of $C\_G(G')$. So $... | 4 | https://mathoverflow.net/users/35840 | 272906 | 122,094 |
https://mathoverflow.net/questions/272904 | 0 | A trivalent graph with $2k$ nodes has maximum diameter $2k-1$, where the diameter is defined by the maximum distance between two nodes. So, my question is the following:
What is the lower bound on the diameter of a trivalent graph with $2k$ nodes?
| https://mathoverflow.net/users/91214 | Lower bound on diameter of trivalent graphs | If diameter equals $d$, the total number $n$ of vertices does not exceed $1+3+3\cdot 2+\dots+3\cdot 2^{d-1}=3\cdot 2^d-2$, this gives you a lower estimate on $d$ which is less or more sharp.
| 3 | https://mathoverflow.net/users/4312 | 272907 | 122,095 |
https://mathoverflow.net/questions/272822 | 9 | This is my latest attempt to simplify an old problem of mine so much that the simplified problem can actually be answered.
Starting with the generating function for unrestricted partitions:
$$(1+x+x^2+x^3+\ldots)(1+x^2+x^4+x^6+\ldots)(1+x^3+x^6+x^9+\ldots)\ldots$$
Change some of the plus signs in the leftmost exp... | https://mathoverflow.net/users/40145 | A simplified version of an old problem about the generating function for unrestricted partitions | **REVISED**
No.
Let $$A=(1+x^2+x^4+x^6+\ldots)(1+x^3+x^6+x^9+\ldots)\ldots$$ and call a polynomial $f=1+\sum\_1^da\_ix^i$ *feasible* if all the $a\_i \in \{{-1,1\}}$ and all the coefficients of $fA$ up to that of $x^d$ are in $\{{-1,0,1\}}.$ Finally, call $f$ *maximal* if it is feasible but neither of $f+x^{d+1}$ n... | 3 | https://mathoverflow.net/users/8008 | 272910 | 122,098 |
https://mathoverflow.net/questions/272903 | 7 | This is probably well known but I'll appreciate pointers to references: Is there any model where for a singular cardinal $\kappa$ of cofinality $\omega$, Very Weak Square holds at $\kappa$ but every scale on $\kappa$ has stationarily many bad points of cofinality $\omega\_1$?
Definitions: Very Weak Square at $\kappa... | https://mathoverflow.net/users/23835 | Very weak square and good points | Though Very Weak Square at $\kappa$ does not in general imply $\mathrm{AP}\_\kappa$, it *does* imply that $\kappa^+ \cap \mathrm{cof}(\omega\_1)$ is in the approachability ideal on $\kappa^+$. In fact, the Very Weak Square sequence $\langle C\_\alpha \mid \alpha < \kappa^+ \rangle$ is itself a witness to this, provided... | 8 | https://mathoverflow.net/users/26002 | 272913 | 122,099 |
https://mathoverflow.net/questions/272894 | 2 | Can finite dimensional algebras (over a field $K$) A with $D(A) \otimes\_A D(A) \cong A$ as $A$-bimodules be somehow classified? I think taking for A a selfinjective algebra with nakayama permutation of order at most two should work. Are there nonselfinjective examples? Here $D(A)=Hom\_K(A,K)$ is the dual of the regula... | https://mathoverflow.net/users/61949 | $D(A) \otimes D(A)= A$? | If $A\cong D(A)^{\otimes i}$as $A$-bimodules, for some $i\geq1$, then $-\otimes\_AD(A)$ is a self-equivalence of the module category, and so takes projectives to projectives. But it takes $A$ to $A\otimes\_AD(A)\cong D(A)$, so $A$ must be self-injective.
| 5 | https://mathoverflow.net/users/22989 | 272916 | 122,100 |
https://mathoverflow.net/questions/272878 | 8 | Let $A = \bigoplus\_{n = 0}^\infty A\_n$ be a graded algebra over a field $k$ that is *locally finite*: each $A\_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \bigoplus\_{n \in \mathbb{Z}} P\_n$ is *graded-projective* if there is a graded free module $F = \bigoplus A(l\_i)$ (for... | https://mathoverflow.net/users/778 | Graded and projective (but not bounded below) module that is not graded-projective? | The answer to your question is no. More generally, the properties of being graded-projective or projective are equivalent in settings far more general than yours. See C. Nastasescu, F. Van Oystaeyen, *Graded Ring Theory,* North-Holland Mathematical Library 28, 1982.
| 6 | https://mathoverflow.net/users/11025 | 272918 | 122,101 |
https://mathoverflow.net/questions/95196 | 5 | Please consider the Stratonovich stochastic differential equation (SDE)
$$
dX = b(X)\circ dB
$$
where $B$ is standard Brownian motion and $X(0)=X\_0$. This corresponds to the Ito (SDE)
$$
dX = \frac{1}{2} b(X) b'(X) dt + b(X) dB.
$$
I would like a reference showing (or even just stating) that trajectories of this equ... | https://mathoverflow.net/users/7949 | Time reversibility of Stratonovich Diffusion: Reference Request | To understand what this "time reversibility" means, it is instructive to replace the stochastic differential equations by the corresponding equations for the time dependence of the probability distribution $p(x,t)$. This is sufficient for the comparison of two-time correlations, in this case between $t=0$ and $t'>0$ (f... | 2 | https://mathoverflow.net/users/11260 | 272922 | 122,103 |
https://mathoverflow.net/questions/272848 | 14 | Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices and $k>0$ real. Then $A^k$ is well-defined and experimentally, we have $$\det(A^k+BABA^{-1})\geqslant \det(A^k+BA^{-1}BA),$$or equivalently $$\det(A^k+A^{-1}BAB)\geqslant \det(A^k+ABA^{-1}B).$$ Note that this would imply the inequality $\color{green}{\... | https://mathoverflow.net/users/29783 | A simple but curious determinantal inequality | Let $C = A^{-(k+1)/2} B A B A^{-(k+1)/2}$ and $D = A^{-(k-1)/2} B A^{-1} B A^{-(k-1)/2}$ .
We have to show that $det(I+C) \ge det(I+D)$ .
Now my goal is to apply equation (5.21) in Ando, Majorizations and Inequalities in Matrix Theory, <http://ac.els-cdn.com/0024379594903417/1-s2.0-0024379594903417-main.pdf?_tid=1a... | 3 | https://mathoverflow.net/users/17261 | 272926 | 122,106 |
https://mathoverflow.net/questions/269392 | 10 | From what I've read, in 1975, Jacques Sesiano in his Ph.D. thesis managed to translate four 'new' books from Diophantus' Arithmetica. This brings the total number of books of Arithmetica salvaged to 10 distinct books, if we include the 6 from Heath's Greek translation.
Does anyone know if any of the 6 Greek version o... | https://mathoverflow.net/users/108567 | Other Arabic translations of the Arithmetica | As Sesiano writes in his book, the first three books that once existed in Arabic translation (by Qusta ibn Luqa) are lost. But al-Karaji quoted extensively from Diophantus Book III (and gives almost all of Book IV)
in his work *Fahri*, which proves that the Books I-III must have been known
in Arabic times.
The situa... | 4 | https://mathoverflow.net/users/3503 | 272937 | 122,109 |
https://mathoverflow.net/questions/272920 | 1 | Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ and a graded homomorphism $\phi : R \to S$ , then there exists a graded homomorphism $\bar \phi : R[X]\to S$ such that $... | https://mathoverflow.net/users/nan | Providing a grading for the polynomial ring over a commutative unital graded ring | This answer is only about the case where $G$ is a (commutative) *group.* I have not thought about more general monoids yet.
As observed by Remy in his comment, one has to choose $X$ to be *homogeneous of some fixed degree* and $a$ to be *homogeneous* as well. Then, this is indeed possible for polynomial algebras with... | 0 | https://mathoverflow.net/users/11025 | 272948 | 122,115 |
https://mathoverflow.net/questions/272945 | 2 | **Problem**
Given any positive integer $n$, what is the maximum number of edges in an even-cycle-free graph with $n$ vertices?
Is the above problem an unsolved problem in extremal graph theory? Are there any results about this problem?
| https://mathoverflow.net/users/58096 | The maximum number of edges in an even-cycle-free graph with $n$ vertices | The answer is $\lfloor \frac{3}{2}(n-1)\rfloor$. First note that if $G$ is $2$-connected and even-cycle-free, then $G$ must just be an odd cycle. To see this, consider an [ear-decomposition](https://en.wikipedia.org/wiki/Ear_decomposition) of $G$. If $G$ is not just a cycle, then $G$ contains a cycle and an ear. Howeve... | 11 | https://mathoverflow.net/users/2233 | 272952 | 122,118 |
https://mathoverflow.net/questions/272969 | 6 | Are there some introductory and self-contained books or lecture notes for Mori-program? It would be much better to have enough examples.
| https://mathoverflow.net/users/109262 | Reference for Mori program | Two good introductions to the Mori program and the geometry of higher dimensional varieties are
*Kenji Matsuki*, Introduction to the Mori program, Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478 p. (2002). [ZBL0988.14007](https://zbmath.org/?q=an:0988.14007).
*Olivier Debarre*, Higher-dime... | 6 | https://mathoverflow.net/users/7460 | 272983 | 122,130 |
https://mathoverflow.net/questions/272973 | 3 | By the recent works of Mok, and Kaletha, Shin, White, James, I know that there is a notion of tempered $L$-parameter, square integrable $L$-parameter and generic $L$-parameter of unitary groups.
However, it seems that there is no notion of supercuspidal $L$-parameter corresponding the packet of supercuspidal represen... | https://mathoverflow.net/users/29422 | Is there a definition of supercupidal parameter in the Local Langland correspondence? | Not really. This is because you get two kinds of L-packets which see supercuspidal representations: there are packets consisting purely of supercuspidals (which correspond to what should probably be called "regular" discrete parameters, at least when your group has connected centre), but when your group is non-split th... | 4 | https://mathoverflow.net/users/60848 | 272993 | 122,132 |
https://mathoverflow.net/questions/272986 | 3 | Background of this question is that I recently stumbled over the problem of deforming polygons in area-preserving way, i.e. modifying the angles between adjacent edges while preserving edge-lengths, topological equivalence to a circle and size of enclosed area.
>
> **Questions:**
>
>
> * Has that problem ... | https://mathoverflow.net/users/31310 | Work on "Churning Polygons" | If I understand your question correctly, you're asking about (signed) area-preserving deformations of planar polygonal linkages. One place to start reading about polygonal linkages is [this chapter by Connelly and Demaine](http://www.csun.edu/~ctoth/Handbook/chap9.pdf). You might be interested about questions about the... | 5 | https://mathoverflow.net/users/353 | 273000 | 122,136 |
https://mathoverflow.net/questions/271846 | 15 | In his *"Some topological properties..." (1955)*, Klee gave a construction (simple and beautiful) of an isotopy $h\_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ in the coordinate $\mathbb{R}^n$-subspace to any other homeomorphic compact $K'$ set in this subspace.
The idea is... | https://mathoverflow.net/users/1441 | Klee's trick --- more applications | This answer is a correction to my comment above. In [these notes](http://math.iupui.edu/%7Edramras/M-Cohen-Klee-trick-notes.pdf) from Marshall Cohen's course (from Spring 2001), he uses the Klee trick to give a simple proof that if $f, g: X \to S^n$ are non-surjective embeddings with $X$ compact, then the integral homo... | 9 | https://mathoverflow.net/users/4042 | 273003 | 122,137 |
https://mathoverflow.net/questions/273011 | 5 | I am encountering the following situation which is similar to the Abhyankar's
higher dimensional conjecture on étale fundamental groups, but with much
stronger assumptions:
Let $S$ be a finitely generated subring of $\mathbb{C}$, let $X$ be
a smooth affine variety over $S$, and let $G$ be a finite group such that the... | https://mathoverflow.net/users/103126 | A weak version of high dimensional Abhyankar's conjecture | Here's a sketch of a possible way: Since $p$ is large enough, we can assume that $G$ is of order prime to $p$. If $X$ was projective, then the prime-to-$p$ completions of $\pi\_1(X\_{\mathbb{C}})$ and $\pi\_1(X\_{\bar k})$ would be isomorphic by the results of SGA1, and we would be done. To "reduce" to this case, we mi... | 7 | https://mathoverflow.net/users/3847 | 273015 | 122,141 |
https://mathoverflow.net/questions/272928 | 9 | Let $a(n)$ and $b(n)$ be define by the following;
$E\_6/E\_4 = 1 - 744q + 159768q^2 - 36866976q^3 + 8507424792q^4 - 1963211493744q^5 + \cdots = \Sigma a(n)q^n,$
$E\_8/E\_6 = 1 + 984q + 574488q^2 + 307081056q^3 + 164453203992q^4 + 88062998451984q^5 + \cdots = \Sigma b(n)q^n.$
(Please see A288261( <https://oeis.org/A... | https://mathoverflow.net/users/69834 | Properties of coefficients in expansion of $E_6/E_4$ and $E_8/E_6$ | The congruence is true for all $m,n$, and **Manyama**'s calculations for
$n \leq 500$ are more than enough to prove it. Similar congruences
hold for the coefficients of other quotients of modular forms; e.g. the
$q^n$ and $q^{5n}$ coefficients of $E\_6/E\_8$ are also congruent $\bmod 3000$
for all $n$, while those of $... | 14 | https://mathoverflow.net/users/14830 | 273022 | 122,143 |
https://mathoverflow.net/questions/272692 | 14 | Consider a collection of unit vectors $v\_1, \ldots, v\_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum\_{i\neq j}|\langle v\_i,v\_j\rangle|.$$
Clearly, if $n=d$, the minimum is attained by taking $v\_i=e\_i$. Could it be that for $n>d$ in order to minimiz... | https://mathoverflow.net/users/24494 | The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$ | It is quite likely. At least, the proof for the case $d\mid n$ is easy. First of all, the restriction $i\ne j$ does not matter: adding $n$ ones changes nothing in the problem. Now notice that $|\langle v\_i,v\_j\rangle|\ge \langle v\_i,v\_j\rangle^2=\langle V\_i,V\_j\rangle$ where $V\_i=v\_i\otimes v\_i$. Now, $\langle... | 12 | https://mathoverflow.net/users/1131 | 273028 | 122,146 |
https://mathoverflow.net/questions/273018 | 8 | The [Fibonacci polynomials](https://en.wikipedia.org/wiki/Fibonacci_polynomials) are defined recursively by $F\_0(x)=0, F\_1(x)=1$ and $F\_n(x)=xF\_{n-1}(x)+F\_{n-2}(x)$, for $n\geq2$.
While computing certain integrals, I observe the following (numerically) which prompted me to ask:
>
> **Question.** For $n, k\in... | https://mathoverflow.net/users/66131 | integral transform of Fibonacci polynomials is integral | The integral of each individual monomial will be integral. First we have the identity
$$F\_n(x)=\sum\_{i=0}^{\lfloor(n-1)/2\rfloor}{n-i-1\choose i}x^{n-2i-1},$$
so my claim is that
$$\binom{n-i-1}{i}\int\_0^1 (k+nz)^{n-2i-1}dz=\binom{n-i-1}{i}\cdot\frac{(k+n)^{n-2i}-k^{n-2i}}{n(n-2i)} \in \mathbb Z.$$
By the binomial t... | 14 | https://mathoverflow.net/users/2384 | 273035 | 122,150 |
https://mathoverflow.net/questions/252555 | 2 | A function $f:\mathbb{R} \rightarrow \mathbb{C}$ is called almost periodic if it is the uniform limit of trigonometric polynomials. One can show that for almost periodic $f$ the following pointwise limit (called the mean value of $f$) exists
$$ M(f) = \lim\_{x\rightarrow \infty} \frac{1}{x} \int\_0^x f(t)dt.$$
We call ... | https://mathoverflow.net/users/91098 | Dominated convergence for quasi-periodic functions | The proposition is indeed true. To see this one uses an alternative characterization of quasi-periodic functions. Let $\omega\_1, \dots, \omega\_m\in \mathbb{R}$ a $\mathbb{Z}$-basis of the frequency module of $f$ and set $\omega= (\omega\_1, \dots, \omega\_m)$. By the work of Harald Bohr, we have that for every $f\_n\... | 1 | https://mathoverflow.net/users/91098 | 273048 | 122,154 |
https://mathoverflow.net/questions/273054 | 1 | (This question is a follow-up on an [older one](https://mathoverflow.net/questions/272679/limit-of-biggest-share-of-the-pie).)
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the l... | https://mathoverflow.net/users/8628 | Share of fortunate people in some pie splitting setting | The $p$-th guest receives a share of
$$s\_{p,N}=p N^{-p-1} (N-p+1) \frac{\Gamma(N+1)}{\Gamma(N-p+2)}$$
For large $N$ I substitute the asymptotic expansion of the Gamma function,
$$\Gamma(z)\approx e^{-z}z^z(2\pi/z)^{1/2},$$
to arrive at the approximation
$$s\_{p,N}\approx p(N-p+1) e^{1-p} \sqrt{\frac{1}{N+1}}\sqrt{\f... | 2 | https://mathoverflow.net/users/11260 | 273057 | 122,156 |
https://mathoverflow.net/questions/273053 | 2 | Let $\emptyset \subsetneq S \subsetneq \{1,\cdots,n\}$ be a set with cardinality $s$, and $g\in\mathbb{R}^n$ be a vector such that
$$\sum\_{\{i,j\}\subseteq \{1,\cdots,n\}}{(g\_i-g\_j)^2} = s(n-s).$$
>
> **Question.** Is this true?
> $$\sum\_{\{i,j\}\subseteq \partial S \\(g\_i-g\_j)^2\le1}{(g\_i-g\_j)^2} \ge \f... | https://mathoverflow.net/users/13639 | Lower Bound for $\sum_{\{i,j\}\subseteq \partial S \\ (g_i-g_j)^2 \le1}{(g_i-g_j)^2}$ | No. Take $n=3$, $S=\{3\}$, $g\_3=0$, $g\_2=a=1/1000$, $g\_1>1$ satisfies $f(g\_1)=(g\_1-a)^2+g\_1^2+a^2=2$, such $g\_1$ exists since $f(1)<2$, $f(2)>2$.
| 5 | https://mathoverflow.net/users/4312 | 273069 | 122,160 |
https://mathoverflow.net/questions/273068 | 2 | The paper [ON THE RAPID COMPUTATION OF VARIOUS POLYLOGARITHMIC CONSTANTS](http://www.davidhbailey.com/dhbpapers/digits.pdf) by Bailey, Borwein, and Plouffe gives sophisticated calculations which I will not summarize. But they remark
>
> We are interested in computing in polynomially logarithmic space and
> polynom... | https://mathoverflow.net/users/38783 | What is known about space and time complexity of division and base change? | The paper is outdated. Division was proved to be computable in logarithmic space (and a fortiori SC) by Chiu, Davida, and Litow [1]. In fact, they proved it to be in logspace-uniform $\mathrm{NC^1}$; this was improved by Hesse, Allender, and Barrington [2], who proved it is in fully uniform $\mathrm{TC^0}$. The same re... | 5 | https://mathoverflow.net/users/12705 | 273076 | 122,165 |
https://mathoverflow.net/questions/143901 | 3 | Shannon capacity $\Theta(G)$ of pentagon is achieved at $2$-fold strong product of the pentagon.
It is also known that the Lov\'asz theta $\vartheta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$.
**Is it known that $\Theta(G)^m\neq\alpha(G^{\boxtimes m})$... | https://mathoverflow.net/users/10035 | Capacity of Cycle Graphs | Currently, there are no known examples of graphs where the Shannon Capacity is realized in a dimension other than $1$ or $2$.
However, there are cases (Such as $C\_5$ union a point) where the Shannon Capacity is not achieved in any dimension, and it is believed by many that this is the case for $C\_7$.
| 1 | https://mathoverflow.net/users/12176 | 273081 | 122,168 |
https://mathoverflow.net/questions/273085 | 24 | An $(\infty,1)$-topos according to Lurie is defined as (accessible) left exact localization of a presheaf $(\infty,1)$-category $\text{P}(\mathcal C)$.
Those $(\infty,1)$-topoi $\text{Sh}(\mathcal C)$ arrising from a site $C$ correspond precisely to *topological* left exact localizations of $\text{P}(\mathcal C)$.
Wh... | https://mathoverflow.net/users/76299 | Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology | Marc's examples are good ones, but let me add two more (which are closely related to each other):
1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let $\mathcal{X}$ be the $\infty$-category
of functors from $\mathcal{C}$ to $\mathcal{S}$ which preserve small filtered c... | 20 | https://mathoverflow.net/users/7721 | 273099 | 122,175 |
https://mathoverflow.net/questions/273103 | 2 | Consider the matrix
$$
M=\begin{bmatrix}
\alpha\_1 & \alpha\_2 & \dots & \alpha\_m \\
\alpha\_1^q & \alpha\_2^q & \dots & \alpha\_m^q \\
\alpha\_1^{q^2} & \alpha\_2^{q^2} & \dots & \alpha\_m^{q^2} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha\_1^{q^{m-1}} & \alpha\_2^{q^{m-1}} & \dots & \alpha\_m^{q^{m-1}} \\
\end{... | https://mathoverflow.net/users/111631 | Is the kernel of a Moore Matrix in $\mathbb F_{q^n}$ determined by the linear dependencies of the generators over $\mathbb{F}_q$? | Consider the vector $(\alpha\_1,\ldots,\alpha\_m)$ as the $\mathbf{F}\_q$-linear transformation $T:(\mathbf{F}\_q)^m\to \mathbf{F}\_{q^n}$ given by
$$ (\beta\_1,\ldots,\beta\_m) \mapsto \sum \beta\_i\alpha\_i. $$
Then $\ker(M)$ is the $\mathbf{F}\_{q^n}$-span of $\ker(T)$ in $(\mathbf{F}\_{q^n})^m$.
To see this, note ... | 1 | https://mathoverflow.net/users/89334 | 273108 | 122,177 |
https://mathoverflow.net/questions/273117 | 2 | Let us define a truth predicate that allows one to substitute in variables. For example, for a 3-ary predicate $p$ and sets $A$, $B$, and $C$, then $\text{true}(\ulcorner p \urcorner, A, B, C)$ (where $\ulcorner \cdot \urcorner$ is some sort of Godel numbering) is true if and only if $p(A, B, C)$ is true.
In particul... | https://mathoverflow.net/users/65915 | Is ZFC plus a truth predicate capable of variable substitution consistent? | The usual conception of truth predicate, or satisfaction class, is formulated in the way that you describe, allowing arguments. It doesn't handle only sentences (no free variables), but formulas, with free variables that can be evaluated as one likes.
Namely, a *truth predicate* is a class $T$ consisting of pairs $\... | 5 | https://mathoverflow.net/users/1946 | 273121 | 122,183 |
https://mathoverflow.net/questions/273120 | 4 | I want to know whether there are ways to use algebraic methods for solving graph theory problems (graph coloring problems). For example, is it possible to prove the four-color theorem purely with algebraic or geometric methods?
| https://mathoverflow.net/users/90655 | How to translate a graph coloring problem to algebraic or geometric language and solve it? | See Sebastian A. Csar, Rik Sengupta, and Warut Suksompong, [On a subposet of the Tamari lattice](https://arxiv.org/pdf/1108.5690.pdf);
"Relying on work of Whitney in [13], Kauffman reformulated the Four Color Theorem using the vector cross product in [5]. More recently, in [ 1], Cooper, Rowland and Zeilberger transf... | 5 | https://mathoverflow.net/users/3684 | 273122 | 122,184 |
https://mathoverflow.net/questions/272995 | 2 | If $\mathcal{D}$ is a locally-small category, then a functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ has a right-adjoint if and only if for each object $d$ of $D$, the presheaf $$\mathcal{C}^{\mathrm{op}}\xrightarrow[c\,\mapsto\, \mathcal{D}(F(c),d) ]{}\mathsf{Set}$$ is representable.
* Is there a usual technical ... | https://mathoverflow.net/users/108556 | Technical term for representing object of a presheaf determined by a left-adjoint? | As requested, I'll turn my comments into an answer. There were two questions, the first being what we call the representing object if a presheaf $c \mapsto \mathcal{D}(F c, d)$ is representable, and the second being whether the "the" in "the representing object" is justified.
Based on some other recent discussions, ... | 4 | https://mathoverflow.net/users/2926 | 273123 | 122,185 |
https://mathoverflow.net/questions/273135 | 3 | Let $X$ be a smooth projective variety over a field $k$.
It is well-known that the sum of an ample divisor and an effective divisor is big (in fact this can be taken to be the definition of a big divisor). I'm looking for a weakening of this.
>
> Is the sum of a big divisor and a pseudo-effective divisor itself a... | https://mathoverflow.net/users/5101 | Sum of a big divisor and a pseudo-effective divisor | Let's say the big divisor is $B$ and the pseudoeffective one is $F$. Write $B = A + E$ with $A$ ample and $E$ effective (by Kodaira). Then $B+F = A+E+F = A/2 + E + (F + A/2)$. $F+A/2$ is effective since $F$ is psef, and so we've written $B+F$ as ample plus effective, so it's big.
| 6 | https://mathoverflow.net/users/nan | 273136 | 122,187 |
https://mathoverflow.net/questions/273132 | 19 | I am interested in polynomials with few terms ("short polynomials", "fewnomials") in ideals. A simple to state question is
Given an ideal $I\subset k[x\_1,\dots,x\_n]$, what is the shortest polynomial in $I$?
There are answers for "Does an ideal contain a monomial/binomial?", but the general question seems to be ha... | https://mathoverflow.net/users/5495 | What is the shortest polynomial divisible by $(x-1)(y-1)(x^2y-1)$ | A simpler answer is given by $x^4y^2 - x^4y - x^2y^2 + x^2 + y - 1=(x+1)f$, where $f$ is the generator of the ideal.
| 20 | https://mathoverflow.net/users/18739 | 273151 | 122,195 |
https://mathoverflow.net/questions/273107 | 9 | Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y\_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = aX^2+bXY+cY^2$ of discriminant $b^2-4ac = -D$ (thus $\mathrm{gcd}(a,b,c) = 1$, $-a < b \leq a \leq c$). Those are the ordi... | https://mathoverflow.net/users/26522 | The mean value of $y \log{y}$ over the ordinates of the CM points | Let $g : \Gamma \backslash \mathbb{H} \to \mathbb{C}$ be any bounded continuous function. Duke's theorem states that
\[\frac{1}{h(D)} \sum\_{A \in \mathrm{Cl}\_K} g(z\_A) = \frac{1}{\mathrm{vol}(\Gamma \backslash \mathbb{H})} \int\_{\Gamma \backslash \mathbb{H}} g(z) \, d\mu(z) + o\_g(1)\]
as $D \to -\infty$ through ne... | 10 | https://mathoverflow.net/users/3803 | 273152 | 122,196 |
https://mathoverflow.net/questions/273112 | 8 | Consider the sequence defined by $a\_0=a\_1=1$ and $a\_n=2a\_{n-1}-3a\_{n-2}$ for $n\geq 2$. This is the sequence <https://oeis.org/A087455>.
I would like to prove that $|a\_n|>100$ when $n>10$. How can we do it? (Also, is there an explicit non-trivial lower bound for the sequence $|a\_n|$?)
The first few terms:
... | https://mathoverflow.net/users/111635 | Growth of a linear recurrent sequence | First note that $\frac{1+i\sqrt{2}}{1-i\sqrt{2}}\in\mathbb{Q}(\sqrt{-2})$ is not a root of unity, because it does not equal $\pm 1$. Therefore Baker's famous theorem shows that, for some effectively computable constant $c>0$,
$$\left|\left(\frac{1+i\sqrt{2}}{1-i\sqrt{2}}\right)^m-1\right|>m^{-c},\qquad m\geq 2.\tag{$\*... | 10 | https://mathoverflow.net/users/11919 | 273153 | 122,197 |
https://mathoverflow.net/questions/273156 | 6 | Let $M$ be oriented manifold: this happens if and only if $w\_1(M)=0$ (the first Stiefel Whitney class, being an element in $H^1(M,\mathbb{Z}\_2)$. There is a result that if $M$ is three dimensional then from the vanishing of $w\_1$ automatically follow that also $w\_2=0$. This should be rather easy consequence of prop... | https://mathoverflow.net/users/24078 | Vanishing of $w_2$ for orientable 3-manifolds | From the relation $Sq(v)=w$ we get that $v\_2=w\_2+w\_1^2$. More specifically By Thm 11.14 (which is just an extended form of the above equality) in M-S you have $w\_1=v\_1$ and $w\_2=Sq^1(v\_1)+v\_2=v\_1^2+v\_2$. Plug them together you obtain my first claim.
Recall we have a nice characterisation of the (second) Wu... | 9 | https://mathoverflow.net/users/48216 | 273157 | 122,200 |
https://mathoverflow.net/questions/270862 | 3 | For a **generically** finite morphism $f:X\rightarrow Y$ of smooth projective surfaces over $\mathbb{C}$. Fix any integer $g$, denote $\mathcal{A}$ to be the set of smooth irreducible curve $C$ in $Y$ with geometric genus $g(C)\leq g$.
**Q:** Does the set of geometric genus $\{g(E)| E$ is an integral curve in $f^{-1}... | https://mathoverflow.net/users/63996 | geometric genus of curves and generically finite morphism of surfaces | Does this work?
Take both $X$ and $Y$ to be $\mathbb{P}^1\times\mathbb{P}^1$, with the map $X\rightarrow Y$ corresponding to the product of a degree $2$ map $\mathbb{P}^1\rightarrow\mathbb{P}^1$ and the identity. Now for any $d$, $\mathcal{A}$ contains the generic curve of degree $(1,d)$ (which is automatically ratio... | 3 | https://mathoverflow.net/users/51424 | 273162 | 122,202 |
https://mathoverflow.net/questions/273142 | 1 | Let $S$ be an integral scheme and $X \to S$ be a smooth scheme of finite type over $S$. Let $\mathcal{E}$ be a coherent sheaf on $X$, and $\eta$ be the generic point of $S$. Assume that restriction $\mathcal{E}|\_{X\_\eta}$ is torsion free. Is it true that there exists an open subset $U$ of $S$ such that $\mathcal{E}|\... | https://mathoverflow.net/users/nan | Is relative torsion freeness an open condition? | I am just posting my comment as an answer. Since $f:X\to S$ is smooth of finite type, in particular it is of finite presentation. By limit theorems, after replacing $S$ by the open in an open affine covering, there exists a Cartesian diagram, $$\begin{array}{ccc} X & \xrightarrow{f} & S \\ u~\downarrow & & \downarrow~v... | 0 | https://mathoverflow.net/users/13265 | 273172 | 122,206 |
https://mathoverflow.net/questions/273185 | 6 | Is very well known that the only algebraic curve which admits no deformations is the projective line.
>
> **Q.** What are "rigid" smooth algebraic surfaces? Is there a sensible classification?
>
>
>
| https://mathoverflow.net/users/9833 | Algebraic surfaces with no deformations | There are several different notions of "rigidity" (local rigidity, global rigidity, infinitesimal rigidity, étale rigidity and strong rigidity) and it is possible to provide examples for each of them.
This topic is discussed in the paper by I. Bauer and F. Catanese *[On rigid compact complex surfaces and manifolds](h... | 10 | https://mathoverflow.net/users/7460 | 273190 | 122,218 |
https://mathoverflow.net/questions/273197 | 10 | Let $M$ be a smooth manifold and let $Q$ be an arbitrary $C^\infty(M)$-module. $Q$ is called geometric if $$\bigcap\_{p\in M}\mu\_pQ=0,$$ where $\mu\_p$ is an ideal in $C^\infty(M)$ of functions vanishing at point $p.$
From the definition, $q\in\mu\_pQ$ iff $q=\sum\_{i=1}^Nf\_iq\_i$ for some $f\_i$'s from $\mu\_p$ a... | https://mathoverflow.net/users/62635 | An example of a non-geometric $C^\infty(M)$-module | Let $M$ be the unit circle in $\mathbb C$, and consider the algebra homomorphism $C^\infty(M)\to M\_2(\mathbb R)$ given by $$ f\mapsto \begin{bmatrix} f(1) & \frac{df}{d\theta}(1) \\ 0 & f(1)\end{bmatrix}$$ (where $\theta$ is the angular coordinate on $M$). This homomorphism makes $\mathbb R^2$ into a $C^\infty(M)$-mod... | 12 | https://mathoverflow.net/users/85913 | 273198 | 122,220 |
https://mathoverflow.net/questions/273195 | 3 | I'm not a mathematician, so please forgive my obvious naivety.
I'm interested in generating a representation in vector space of a two dimensional armature, or set of rigid linked elements, with each element articulated by rotation about a point defined on the previous (acyclic) element, about an axis orthogonal to th... | https://mathoverflow.net/users/111672 | modeling mechanical systems with Lie algebras | Probably your best bet is to read through Chapter 2 of [A Mathematical Introduction to Robotic Manipulation](http://www.cds.caltech.edu/~murray/mlswiki/index.php/Main_Page) by Murray et al. (free download at link). You seem to referring to "exponential coordinates" on SO(3) (see for example equation (2.16) on P.30). Al... | 5 | https://mathoverflow.net/users/17945 | 273205 | 122,224 |
https://mathoverflow.net/questions/273164 | 4 | This question will fade from very specific to very generic.
Consider an $\lambda$-accessible category $\mathcal{K}$ and let's call $\text{Pres}\_{\lambda} \mathcal{K}$ its $\lambda$-presentables.
Is the following statement true?
>
> Given a category with directed colimits $ \mathcal{C}$ and a functor $F: \text{Pr... | https://mathoverflow.net/users/104432 | Extending functors defined on dense subcategories | $\mathcal K$ is a free cocompletion of its $\lambda$-presentables under $\lambda$-filtered colimits. Thus $\bar{F}$ exists and preserves $\lambda$-filtered colimits.
| 5 | https://mathoverflow.net/users/73388 | 273206 | 122,225 |
https://mathoverflow.net/questions/273215 | 8 | Let $G$ be a finite group, and let $g,h\in G$ be two elements of the same order. Does there exist a larger finite group $G'\ge G$ such that $g,h$ are conjugate in $G'$?
If not, what is known about groups which have this property?
| https://mathoverflow.net/users/88840 | Given two elements $g,h$ in a finite group $G$ of the same order, does there exist a finite group $G'$ containing $G$ where they are conjugate? | [Two subgroups are isomorphic iff they are conjugate in some larger finite group.](https://groupprops.subwiki.org/wiki/Isomorphic_iff_potentially_conjugate_in_finite)
If you have two subgroups $A,B$ of a finite group $G$ and $\sigma: A\to B$ is an isomorphism. Then there exists a group $G'$ containing $G$ and an elem... | 14 | https://mathoverflow.net/users/2384 | 273217 | 122,228 |
https://mathoverflow.net/questions/273224 | 4 | This question began as [Why are procyclic subgroups of Galois groups of number fields free profinite?](https://mathoverflow.net/questions/273088/why-are-procyclic-subgroups-of-galois-groups-of-number-fields-free-profinite), which fizzled out, but which garnered some helpful comments from YCor.
Let $K$ be a field, tak... | https://mathoverflow.net/users/70019 | What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$? | $\hat{\mathbb Z}$ occurs with probability one.
It is easy to define a homomorphism from the Galois group of a number field to $\hat{\mathbb Z}$. To do this, just define a homomorphism to $\mathbb Z\_p$ for each $p$. It is sufficient to define such homomorphisms for $\mathbb Q$, as the restriction to the Galois group ... | 4 | https://mathoverflow.net/users/18060 | 273226 | 122,233 |
https://mathoverflow.net/questions/265756 | 5 | Does there exist a subset $A\subseteq\mathbb{R}^n$, for some $n$, and numbers $h<m$, such that the Hausdorff dimension $\dim A=h$, while for every cover $A\_i$, $A\subseteq\bigcup\_{i=1}^\infty A\_i$ there exists $i$ such that the lower Minkowski dimension $\underline{\dim}\_MA\_i>m$?
| https://mathoverflow.net/users/16678 | Is there a gap between the Hausdorff and the lower Minkowski dimensions? | I believe what you are asking is whether the Hausdorff dimension can be strictly less than the lower packing dimension (also called the lower modified box dimension). I'm pretty sure the strongest formulation, $h=0$ and $m=n,$ is possible, but I don't know of a reference. However, I think examples showing that $h < m$ ... | 6 | https://mathoverflow.net/users/15780 | 273234 | 122,235 |
https://mathoverflow.net/questions/273237 | 1 | This doubt is born because I am reading an article in this [link in pag 12](https://arxiv.org/pdf/1505.05116.pdf) in order to use these ideas to prove another problem that raised me. My doubt is following:
Let $(\Xi,\mathcal{E})$ be a measurable space and $\mathbb{Q}\_{i}$ measures in this space and $\xi\_{i}\in\Xi$ ... | https://mathoverflow.net/users/109184 | Let $\mathcal{M}(\Xi)$ set of all probability destributions on $\Xi$. Supremum over $\mathcal{M}(\Xi)$ is equal to sup over Dirac distributions | The attempt you made to prove inequality $ \leq $ is correct. The other desiguladad is consequence of the following general fact:
>
> For every fixed function $g$ and every probability distribution $\mu$,
> $$\int\_Xgd\mu\leqslant\sup\_X g\cdot\mu(X)=\sup\_Xg$$
>
>
>
| 1 | https://mathoverflow.net/users/nan | 273245 | 122,239 |
https://mathoverflow.net/questions/273250 | -2 | can anyone show me how
$$\displaystyle\frac{4}{R}\displaystyle\Pi\_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum\_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\right)^2\left(1+2 \sum\_{n=1}^{\infty}\frac{1}{R^{2n^{2}}}\right)^{-2} $$
The above identity is obtained by Komatu [page ... | https://mathoverflow.net/users/111697 | Expression for infinite product | First of all, the original identity has an exponent of $4$ on the left hand side that you have missed. Moreover it is not correct as stated, by simply comparing constant terms of both sides. However it is just a simple typo probably, because the following is true
$$\frac{4}{R}\prod\_{n=1}^{\infty} \left(\frac{1+R^{-4n}... | 4 | https://mathoverflow.net/users/2384 | 273253 | 122,241 |
https://mathoverflow.net/questions/273077 | 3 | Let $X\_t$ be a fixed cadlag semi-martingale and $J\_n$ be a fixed sequence of functions from $\mathbb{R}^d$ to $\mathbb{R}$ which are twice continuously differentiable. If $J\_n$ converge pointwise to the function $J$ which itself is a.e. differentiable on $\mathbb{R}^d$ (wrt to the Lebesgue measure-zero sets) and *no... | https://mathoverflow.net/users/36886 | Convergence of function of stochastic processes | According the definition of Skorokhod topology, for the $J\_n(X\_t)\to J(X\_t)$ in Skorokhod topology,
it suffices to suppose that
$$ \lim\_{n\to\infty}\sup\_{\|x\|\le a}|J\_n(x)-J(x)|=0, \qquad \forall a>0.$$
i.e. $J\_n $ convergence to $J$ uniformly in every compact set.
| 3 | https://mathoverflow.net/users/103256 | 273265 | 122,247 |
https://mathoverflow.net/questions/273270 | -4 | Is there currently an algorithm that given the number of vertices / nodes `n`, and the number of edges per node `l`, output all graph edges?
If not, is there an algorithm that can tell if such graph is possible?
In both cases, this is an undirected, loop-free graph.
| https://mathoverflow.net/users/111712 | Is there an algorithm to generate graph edges given amount of vertices and edges per node? | From the comments, you want every node to have the same weight. This is of course only possible if $n \geq \ell + 1$. Now assume we have such parameters. Pick $\ell+1$ vertices and form a complete graph on them. Continue, until you only have $k \leq \ell$ vertices left. Now there are two cases:
1. $\ell - k$ is odd. ... | 3 | https://mathoverflow.net/users/109932 | 273276 | 122,250 |
https://mathoverflow.net/questions/273232 | 6 | I have a linear algebraic question about an arbitrary correlation matrix. If
$$
\lambda\_{\min}
\begin{pmatrix}
1 & \rho\_1 & \rho\_2
\\
\rho\_1 & 1 & \rho\_3
\\
\rho\_2 & \rho\_3 & 1
\end{pmatrix}
\ge
\alpha
$$
does the following hold?
$$
(\rho\_1,\rho\_3)
\begin{pmatrix}
1 & \rho\_2
\\
\rho\_2 & 1
\end{pmatrix}... | https://mathoverflow.net/users/88033 | A correlation matrix problem | If $\rho\_1=\rho\_3=0$ it is obvious, so we assume $\rho\_1^2+\rho\_3^2>0$.
First note that
$$ A= \begin{pmatrix} 1 & \rho\_2 \\ \rho\_2 & 1 \end{pmatrix}^{-1}=\frac{1}{\rho\_2^2-1}\begin{pmatrix} -1 & \rho\_2 \\ \rho\_2 & -1 \end{pmatrix}$$ with eigenvalues $\lambda\_1=\frac{1}{1-\rho\_2}$ and $\lambda\_2=\frac{1}{1+... | 4 | https://mathoverflow.net/users/54403 | 273281 | 122,253 |
https://mathoverflow.net/questions/273283 | 3 | **Problem.** Is there an infinite set $I\subset 3^\omega$ such that for any infinite subset $J\subset I$ there exists $n\in\omega$ such that $\{x(n):x\in J\}=3$?
Here $3^\omega$ is the set of functions from $\omega$ to $3=\{0,1,2\}$.
| https://mathoverflow.net/users/61536 | An infinite subset in $3^\omega$ with "large" projections? | **Claim:**
For every integer $k>1$ there exists a subset $I\subseteq k^\omega$ of cardinality $\frak{c}$ such that for every subset $J\subseteq I$ such that $|J|=k$ there exists $n\in\omega$ such that $|\{x(n)\mid x\in J\}|=k$.
**Proof:**
By Theorem 7.7 of Jech "Set Theory" there exists an independent family $\mathca... | 6 | https://mathoverflow.net/users/16678 | 273287 | 122,254 |
https://mathoverflow.net/questions/273243 | 6 | Is there an $F\_{\sigma}$-set (countable union of closed subsets of plane) $S \subseteq \mathbb{R}^2$ that meets every circle at 3 points?
| https://mathoverflow.net/users/2689 | F-sigma subset of plane meeting every circle at 3 points | There is not. If there were an $F\_\sigma$ set meeting every circle in three points, then a suitable Moebius transformation would turn it into an $F\_\sigma$ set meeting every line in three points. Bouhjar, Dijkstra, and Maudlin proved that no set meeting every line in exactly three points can be $F\_\sigma$, extending... | 5 | https://mathoverflow.net/users/70618 | 273289 | 122,255 |
https://mathoverflow.net/questions/273296 | 5 | Does there exist a continuous bilinear form $\mathcal{B}$ on $\mathcal{S}(\mathbb{R})\times \mathcal{S}(\mathbb{R})$ such that
\begin{equation}
\mathcal{B}(\varphi\_1, \varphi\_2) =\int\_{\mathbb{R}\times\mathbb{R}} \frac{\varphi\_1(x) \varphi\_2(y)}{\lvert x - y \rvert} \mathrm{d}x\mathrm{d}y
\end{equation}
for every ... | https://mathoverflow.net/users/39261 | No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions? | Yes it is possible. Let $K\in S'(\mathbb{R}^2)$ be the distribution acting on test functions $f(x,y)$ by
$$
K(f)=\int\_{|x-y|\ge 1} \frac{f(x,y)}{|x-y|}\ dx\ dy
\ + \int\_{|x-y|< 1} \frac{f(x,y)-f(x,x)}{|x-y|}\ dx\ dy\ .
$$
What is impossible is to do this extension to the diagonal while preserving the degree -1 homoge... | 8 | https://mathoverflow.net/users/7410 | 273301 | 122,256 |
https://mathoverflow.net/questions/273291 | 4 | This question is a follow-up of [Extending functors defined on dense subcategories](https://mathoverflow.net/q/273164/24563).
>
> Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
> $\mathcal{K}$ is called a *Yoneda object* if the functor
> $\mathcal{K}(X,-):\mathcal{K} \to \text{Set}$ is coli... | https://mathoverflow.net/users/24563 | About the Yoneda objects of a locally presentable category | The usual name for "Yoneda objects" is "[tiny](https://ncatlab.org/nlab/show/tiny+object)" or "small-projective".
1. In general the tiny objects in a presheaf category are the *retracts* of representables. In particular, if $A$ is Cauchy-complete, then every tiny object is representable.
2. Yes. In fact, instead of l... | 7 | https://mathoverflow.net/users/49 | 273304 | 122,257 |
https://mathoverflow.net/questions/273302 | 11 | Does anybody have any ideas how to solve the equation $x=\phi(x)+\phi(x+1)-1$, where $x$ is a natural number and $\phi$ is Euler's totient function?
I failed even to figure out whether this equation has finite number of solutions or infinite.
Any help will be appreciated.
| https://mathoverflow.net/users/110691 | Equation $x=\phi(x)+\phi(x+1)-1$ | This is [OEIS Sequence A067798](https://oeis.org/A067798).
Nothing else seems to be known about it; at any rate OEIS gives no
references to the literature, only a link to
[a list of further such $x$](https://oeis.org/A067798/b067798.txt)
from Giovanni Resta that extends it from the 43rd solution, $722015$,
to the 76... | 18 | https://mathoverflow.net/users/14830 | 273305 | 122,258 |
https://mathoverflow.net/questions/273221 | 5 |
>
> Adjoint functor theorems are theorems stating that under certain conditions a functor that preserves limits is a right adjoint, and a functor that preserves colimits is a left adjoint. (*from [the nLab](https://ncatlab.org/nlab/show/adjoint+functor+theorem)*.)
>
>
>
Now, the condition that a functor is a lef... | https://mathoverflow.net/users/7952 | Adjoint functor theorems on 2-categories | There is actually such a theorem in the original Street-Walters paper on Yoneda structures (p372): any colimit-preserving functor from a [total object](https://ncatlab.org/nlab/show/total+category) to an admissible object has a right adjoint. (This is a "special" adjoint functor theorem, i.e. it imposes "reasonability"... | 2 | https://mathoverflow.net/users/49 | 273306 | 122,259 |
https://mathoverflow.net/questions/273282 | 7 | It is easy to show that if $f\colon[0,1]\to\mathbb R$ and $|f|\leq A$ and $|f''|\leq B$ then~$|f'|\leq 4A+B$. Indeed, by Taylor formula with remainder $f(x)=f(c)+(x-c)f'(c)+\frac12(x-c)^2f''(d)$ where $d$ is between $x$ and $c$. Therefore $f'(c)=\frac{f(x)-f(c)-\frac12(x-c)^2f''(d)}{x-c}$. Now let $c$ be a maximum of $... | https://mathoverflow.net/users/28128 | Is there a bound for Lipschitz constant in terms of second differences? | Smoothifying by convolution as Pietro Majer suggests is pretty ok, but if you prefer more direct argument, you may use a standard
**Lemma.** If a bounded function $f$: $[0,1]\to \mathbb{R}$ satisfies $f(\frac{x+y}2)\leqslant \frac{f(x)+f(y)}2$, then $f$ is convex.
**Proof.** At first, we prove that $f$ is continuo... | 9 | https://mathoverflow.net/users/4312 | 273307 | 122,260 |
https://mathoverflow.net/questions/273284 | 3 | Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, non-degenerate complex surface $X\_n \subset \mathbb{P}^{N}$ with the following properties:
1. for all $p \in X\_n$ there exists a $1$-dimensional family $\mathcal{C}\_p$ of rational normal curves of degree $n$ containing $p$;
2. any tw... | https://mathoverflow.net/users/7460 | Surface in $\mathbb{P}^N$ covered by rational normal curves | I am writing up my comments as an answer. Let $f:\widetilde{X}\to X$ be a minimal desingularization of $X=X\_n$. For a general member $\mathcal{C}$ of the family of curves $\mathcal{C}\_{p,q}$, let $\widetilde{\mathcal{C}}$ be the strict transform of $\mathcal{C}$ in $\widetilde{X}$.
**Lemma.** The surface $\widetild... | 3 | https://mathoverflow.net/users/13265 | 273314 | 122,266 |
https://mathoverflow.net/questions/273235 | 7 | I am learning formal PDE theory for my research and I am currently struggling to have a basic understanding of the operations involved in completing a (say, linear) PDE system to an involutive one (Cartan-Kuranishi procedure) in terms of the partial differential operators involved.
First, let us put some context in o... | https://mathoverflow.net/users/11211 | A very basic question about projections in formal PDE theory | Your conjecture is almost correct. But consider the extreme case when your equation has no integrability conditions, so that $\operatorname{coker}(\sigma(\rho\_{q+1} P)) = 0$, whence $\rho^{(1)}\_{q+1}(\Phi\_P) = 0$, which definitely doesn't give you the the equation $\mathcal{R}^{(1)}\_{k+q}$ that you wanted. What you... | 4 | https://mathoverflow.net/users/2622 | 273320 | 122,268 |
https://mathoverflow.net/questions/273031 | 7 | Let $W$ be any non-empty set. We call a subset $A$ of $\mathscr{P}(W)$ a 3-fold filter on $W$ if the following hold:
(i) $W\in A$ and $\varnothing\notin A$;
(ii) If $a\in A$ and $a\subset b$ then $b\in A$;
(iii) If $a,b,c\in A$ then $(a\cap b)\cup(b\cap c)\cup(c\cap a)\in A$.
Now let $A,B$ be two 3-fold filters... | https://mathoverflow.net/users/101817 | A question about the generalization of filters | First, a bit of notation: if $a\_1,\dots,a\_n\subseteq W$, let $\chi\_{a\_1,\dots,a\_n}\colon W\to\{0,1\}^n$ denote the “$n$-fold characteristic function”
$$\chi\_{a\_1,\dots,a\_n}(u)=([u\in a\_1],\dots,[u\in a\_n]),\qquad u\in W,$$
where $[\ ]$ is the Iverson bracket. For a Boolean function $f\colon\{0,1\}^n\to\{0,1\}... | 5 | https://mathoverflow.net/users/12705 | 273330 | 122,273 |
https://mathoverflow.net/questions/273315 | 7 | In a couple of web pages, I see that Legendre's constant is defined to be $\lim\_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, [here](https://en.wikipedia.org/wiki/File:Legendre%27s_constant.svg) and [here](https://proofwiki.org/wiki/Definition:Legendre%27s_Constant)).
Actually the first uses $\lim\_{n \to \in... | https://mathoverflow.net/users/57425 | Legendre's Constant | **1.** The prime number theorem in the form
$$\pi(n)=\mathrm{li}(n)+O(ne^{-c\sqrt{\log n}})$$
combined with the approximation
$$\mathrm{li}(n)=\frac{n}{\log n}+\frac{n}{\log^2 n}+O\left(\frac{n}{\log^3 n}\right)$$
shows that
$$\pi(n)-\frac{n}{\log n}=\frac{n}{\log^2 n}+O\left(\frac{n}{\log^3 n}\right).$$
So the left ha... | 10 | https://mathoverflow.net/users/11919 | 273337 | 122,276 |
https://mathoverflow.net/questions/273230 | 3 | I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ask some smoothness to the boundary of the region $\ U$ where the DP is posed. The following lemma enables to prove tha... | https://mathoverflow.net/users/86048 | Exercise on a hitting time for a Brownian Motion | Here's one way of proving your first claim.
Let $z$ be the center of the ball. Choose an $\epsilon < 1/4$ and a finite set of points $\{x\_1, \dots, x\_n\} \in B\_{3/4}(z)$ such that for **every** cone $C$ with angle $\alpha$ and vertex at $z$, there is some $x\_i$ such that $B\_\epsilon(x\_i) \subset C$. (This is po... | 4 | https://mathoverflow.net/users/4832 | 273339 | 122,277 |
https://mathoverflow.net/questions/273318 | 0 | Let $1\leq \ell \leq \binom{k}{2}$. It seems that there should exist some set $\mathcal{P}$ of $kd-\ell+1$ points in $\mathbb{R}^d$ which do not lie on the union of any $k$ hyperplanes $H\_1,\ldots, H\_k$, exactly $\ell$ of which are orthogonal. Indeed, this should be generic.
If $\ell=1$, this is easily seen: put t... | https://mathoverflow.net/users/53199 | 7 Points in $\mathbb{R}^3$ that don't lie on the union of 3 orthogonal hyperplanes | Of course, fedja's measure theoretic argument works. Here is another way to see that $7$ general points in $\mathbb{R}^3$ don't lie on $3$ mutually orthogonal planes. For $7$ generic points we will have
1. No $3$ on a line and no $4$ on a hyperplane.
2. For any two triples of points, the hyperplanes they span will n... | 4 | https://mathoverflow.net/users/297 | 273340 | 122,278 |
https://mathoverflow.net/questions/269362 | 14 | Cluster algebras are closely related to totally positivity in algebraic groups and canonical bases in quantum groups.
Is there some relation between cluster algebras and crystal graphs? Can the crystal graph on page 19 of [the book](http://sporadic.stanford.edu/bump/wmd5book.pdf) be described using cluster variables ... | https://mathoverflow.net/users/11877 | Is there some relation between cluster algebras and crystal graphs? | Yes, there are many relations between cluster algebras and crystal graphs. I am by no means an expert on these things, but let me mention one connection. Cluster algebras were originally discovered in the study of totally positive matrices and total positivity. This theory is ultimately interested in Lusztig's canonica... | 8 | https://mathoverflow.net/users/62154 | 273346 | 122,282 |
https://mathoverflow.net/questions/273350 | 1 | I want to study the space $Y$ of all un-ordered $d$-tuples of points on $\mathbb P^1$. By considering the space $V\_d$ of homogeneous polynomials of degree $d$ in two variables, one may identify $Y$ with $\mathbb P(V\_d)\equiv \mathbb P^d$. There is a natural $SL\_2$ action on $\mathbb P(V\_d)\equiv Y$ via change of co... | https://mathoverflow.net/users/69190 | un-ordered distinct $n$-tuples of points on $\mathbb P^1$ | The quotient is actually isomorphic to $\mathbb{P}^1$.
You can find a proof in the notes by G. Ottaviani and J. Vallès [*Moduli of vector bundles and group actions*](https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwiU86yRjuXUAhWFWRQKHTbBDysQFggiMAA&url=http%3A%2F%2Fweb.math.unifi.it%2Fusers%... | 3 | https://mathoverflow.net/users/7460 | 273358 | 122,285 |
https://mathoverflow.net/questions/273254 | 8 | In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle over the base manifold $M$, which we assume to have dimension $n$. Given a bundle map $$\mathscr{L}:J^kE\rightarrow\wedge^n... | https://mathoverflow.net/users/11211 | In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear? | Igor gave a coordinate independent definition of quasilinear *equations* which the Euler-Lagrange equations satisfy. Still missing is a definition of quasilinear differential *operator*, which the Euler-Lagrange operator satisfies.
You already observed that
>
> $M\times\_M V^\circledast E$ is generally *not* a v... | 2 | https://mathoverflow.net/users/745 | 273377 | 122,289 |
https://mathoverflow.net/questions/273381 | 3 | In Hanamura's paper Mixed Motives and Algebraic Cycles III
<http://intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0001/MRL-1999-0006-0001-a005.pdf>
He proved that if assume Grothendieck's standard conjecture, Murre's conjecture and vanishing conjecture, there is a $t$ structure on Voevodsky's categor... | https://mathoverflow.net/users/87910 | Clarification on Hanamura's work on $t$ structure of triangulated category of mixed motives | 1) I believe that the (Murre's) vanishing needed for Hanamura's argument is stronger than the BS conjecture.
2) There are certain standard conditions ensuring that a triangulated category is equivalent to the derived category of the heart of a t-structure on it. Yet I don't known much about them and they were not tre... | 5 | https://mathoverflow.net/users/2191 | 273383 | 122,291 |
https://mathoverflow.net/questions/273348 | 11 | There is a simple characterization of shift-invariant closed subspaces of $l^2$: for any measurable subset $S$ of $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$, the set of elements of $l^2$ whose Fourier transform is supported on $S$ is a shift-invariant closed subspace, and every such subspace has this form. This is pretty... | https://mathoverflow.net/users/23141 | Shift invariant subspaces of $l^1$ | This is not a full answer, as my memory / personal bookshelf is not good enough. But it should give some hints.
As the comments suggest, we can reduce this to a question about the Fourier algebra $A(\mathbb T)$. Indeed, the steps are:
* We can turn $\ell^1(\mathbb Z)$ into a commutative Banach algebra for the convo... | 5 | https://mathoverflow.net/users/406 | 273385 | 122,293 |
https://mathoverflow.net/questions/273390 | 5 | Let $f$ be a rational function with $j$ zeros and $k$ poles, all of which reside in the closed unit disk (excepting of course the zeros or poles at $\infty$ when $j\neq k$). What is the smallest number $R>0$ such that all the (finite) critical points of $f$ must lie in the closed disk centered at the origin with radius... | https://mathoverflow.net/users/35158 | Upper bound on the critical points of a rational function with zeros and poles on the disk | Your conjecture is correct. We can assume that the largest critical point occurs at a positive $z=R>0$. If we denote the zeros and poles by $a\_n$ and $b\_n$, respectively, then the condition for a critical point is
$$
\sum \frac{1}{R-a\_n} = \sum \frac{1}{R-b\_n} .
$$
In particular, the real parts of both sides must b... | 7 | https://mathoverflow.net/users/48839 | 273391 | 122,296 |
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