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https://mathoverflow.net/questions/396256 | 0 | Suppose $f, g: \mathbb R \to \mathbb R$ are continuous, non negative functions with $f \leq g$.
Fix some $T > 0$, and denote by $X^f$ the stochastic integral $\int\_{[0, T]} f(s) \ dW\_s$, where $W\_s$ is a standard Brownian motion. Similarly write $X^g$ for the corresponding integral of $g$.
Write $F^f$ for the cu... | https://mathoverflow.net/users/173490 | A monotonicity formula for the stochastic integral with respect to Brownian motion | $X^f$ and $X^g$ are Gaussian random variables with zero mean and variances $$\mathbb{E}(|X^f|^2) = \int\_0^T|f(t)|^2dt \leq \int\_0^T|g(t)|^2dt = \mathbb{E}(|X^g|^2).$$
Thus, we know the probability density functions of both random variables. Comparison is up to the reader.
| 2 | https://mathoverflow.net/users/164350 | 396262 | 163,643 |
https://mathoverflow.net/questions/396243 | 6 | I am seeking for an Artin $k$-algebra (especially for group algebra) which is infinite-dimensional over some field $k$. It's known that any complex group algebra has trivial Jacobson radical. So I have the following question:
Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$... | https://mathoverflow.net/users/134942 | Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$ is semisimple? | The answer is negative as any semisimple Hopf algebra is finite-dimensional. More generally, the same conclusion is true for all Artinian Hopf algebras. See e.g. [Liu and Zhang - Artinian Hopf algebras are finite dimensional](https://www.ams.org/journals/proc/2007-135-06/S0002-9939-07-08711-4/home.html).
| 8 | https://mathoverflow.net/users/14653 | 396264 | 163,645 |
https://mathoverflow.net/questions/396250 | 5 | For the function $f\_1(x)=\dfrac{ax+b}{a'x+b'},\quad a'\neq0$ , we have $$f\_1'(x)=\dfrac{\begin{vmatrix}{a} && {b} \\ {a'} && {b'}\end{vmatrix}}{(a'x+b')^2}$$
For $f\_2(x)=\dfrac{ax^2+bx+c}{a'x^2+b'x+c'},\quad a'\neq0$, we have
$$f\_2'(x)=\dfrac{{ \begin{vmatrix}{a} && {b} \\ {a'} && {b'}\end{vmatrix} }x^2+2{ \beg... | https://mathoverflow.net/users/171386 | General formulas for derivative of $f_n(x)=\dfrac{ax^n+bx^{n-1}+cx^{n-2}+\cdots}{a'x^n+b'x^{n-1}+c'x^{n-2}+\cdots},\quad a'\neq0$ | You can easily extend this, but for $n\geq 3$ you will end up with more than one term per monomial:
For two functions $f$, $g$ rewrite the quotient rule using a determinant
$$\frac{d}{dx} \frac{f}{g} = \frac{\frac{df}{dx}g-f \frac{dg}{dx}}{g^2} = \frac{\begin{vmatrix} \frac{df}{dx} & f \\ \frac{dg}{dx} & g \end{vma... | 6 | https://mathoverflow.net/users/51695 | 396267 | 163,647 |
https://mathoverflow.net/questions/396209 | 10 | I want to calculate the number of solutions to the quadratic equation $$x\_1^2+\dots+x\_m^2=0$$ where $m$ is odd (a given number) and $x\_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given positive integer $n$.
I guess one can consider the projective variety over the $p$-adic field $\mathbb{Q}\_p$ and count... | https://mathoverflow.net/users/nan | What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$? | $\newcommand\Z{\mathbf{Z}}$Here's a solution for odd $p$ ($p=2$ seems to have specific complications), granted the case $n=1$.
>
> Lemma 1: ($p$ odd) Let $u\_n$ be the number of solutions of $\sum\_{i=1}^mx\_i^2=0$ in $\Z/p^n\Z$ such that $(x\_1,\dots,x\_m)$ is not $(0,\dots,0)$ mod $p$. Then, for $n\ge 1$, $u\_n=p... | 6 | https://mathoverflow.net/users/14094 | 396268 | 163,648 |
https://mathoverflow.net/questions/396234 | 2 | Let $M$ be the space of right continuous functions $\ell: \mathbb R\_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>t]$ for all $\ell \in M$ and $t\ge 0$, where $\tau^{\ell}:=\inf\{t\ge 0: X^{\ell}\_t\le 0\}$ and
$$X^{\ell}\_t:=1+t+\in... | https://mathoverflow.net/users/261243 | On the continuity of map $\Gamma$ | I believe we can. Let $\ell\_n \to \ell$ in your topology, and fix $t\_0 \in \mathbb R\_+$. We show that $\Gamma(\ell\_n)(t\_0) \to \Gamma(\ell)(t\_0)$. We work over the interval $[0, T]$ with $T > t\_0$.
**Step 1:** We first note that $\ell\_n$ converges to $\ell$ in measure.
Indeed, let $\varepsilon > 0$ be arbit... | 5 | https://mathoverflow.net/users/173490 | 396269 | 163,649 |
https://mathoverflow.net/questions/396271 | 6 | Let $\underline{\Omega}\_X^{\bullet}$ denote the Deligne -- Du Bois complex of a normal variety $X$. What kind sigularities satisfy $gr^k\underline{\Omega}\_X^{\bullet}[k] \simeq \Omega\_X^{[k]}$ where $\Omega\_X^{[k]} := j\_\*\Omega^k\_{X^{\operatorname{reg}}}$ and $j\colon X^{\operatorname{reg}} \hookrightarrow X$ is... | https://mathoverflow.net/users/164620 | Is there a generalization of Du Bois singularities? | Good question. Your first question has a positive answer when $X$ has finite quotient singularities, i.e. locally analytically $\mathbb{C}^n/G$, with $G\subset GL\_n(\mathbb{C})$ finite. This was essentially proved in Du Bois' original paper "Complexe de de Rham filtré...", see also Steenbrink "Mixed Hodge structure on... | 8 | https://mathoverflow.net/users/4144 | 396272 | 163,650 |
https://mathoverflow.net/questions/396182 | 4 | Let $\mathbf{C}$ be a closed symmetric monoidal category (I probably need even less than this; the examples I have in mind are simply the category of modules over a commutative ring and the category of sets) and let $\omega$ be an (arbitrary) object in $\mathbf{C}$ which I will term the “dualizing object”. Define a con... | https://mathoverflow.net/users/17064 | The bidualizing monad | It has been explained by Maxime Ramzi in the comments that this monad simply arises from the adjunction $[-,\omega] \vdash [-,\omega]^{\mathrm{op}}$.
As for the name, it's called the double dualization monad. The classical reference is
>
> A. Kock, *On double dualization monads*, Math. Scand. 27 (1970), 151-165, ... | 7 | https://mathoverflow.net/users/2841 | 396278 | 163,651 |
https://mathoverflow.net/questions/382866 | 2 | Let $G$ be a compact group with finite-dimensional, real representations $\phi$ and $\psi$ on $V$ and $W$ respectively. (e.g. $V = \mathbb{R}^m$, $W = \mathbb{R}^n$.)
Is it true that, as is the case for finite groups, the dimension of the intertwiner space of the two representations is equal to the inner product of the... | https://mathoverflow.net/users/102255 | Dimension of intertwiner space: finite-dimensional representations of compact groups | Yes, this formula holds, and such representations exist.
If $\mathbb K$ is either $\mathbb R$ or $\mathbb C$ and $G$ is a compact topological group with $\mathbb K$-representations $V$ and $W$, then $Hom\_{\mathbb K}(V,W)$ is another representation of $G$ satisfying
$$ Hom\_G(V,W) = Hom\_{\mathbb K}(V,W)^G.$$
For any... | 1 | https://mathoverflow.net/users/125523 | 396288 | 163,652 |
https://mathoverflow.net/questions/396119 | 6 | *I'd like to close a gap left open in [an old question of mine](https://mathoverflow.net/questions/285651/undetermined-games-of-overdetermined-type); I've tweaked the terminology to be a bit nicer.*
For a (boldface) pointclass $\Gamma$ and a payoff set $G\subseteq\omega^\omega$, say that $G$ is $\Gamma$-**narrow** if... | https://mathoverflow.net/users/8133 | Are there "very narrow" undetermined games? | Solovay's coded club game is $\Pi^1\_1\vee\Sigma^1\_1$-narrow. So $\mathsf{ZFC}$ proves the existence of a $\Pi^1\_1\vee\Sigma^1\_1$-narrow undetermined game. I don't know about $\Pi^1\_1\wedge\Sigma^1\_1.$
It's just optimizing the analysis in the other answer you linked to, [Undetermined games of "overdetermined" ty... | 1 | https://mathoverflow.net/users/164965 | 396290 | 163,653 |
https://mathoverflow.net/questions/396297 | 11 | My question is about the Hamming Weight (or number of 1's in binary expansion) of $a\_n = \frac{2^{\varphi(3^n)}-1}{3^n}$ [A152007](https://oeis.org/A152007)
For example, $a\_3 = 9709 = (10110111101001)\_2 $ has nine 1's in binary expansion
I guess the answer is $3^{(n-1)}$ but I can't prove it
Is that correct?
... | https://mathoverflow.net/users/36456 | Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$ | It is, indeed, correct. Notice first that $2-(-1)=3$ is divisible by $3$, so by lifting-the-exponent lemma the number
$$
A=\frac{2^{3^{n-1}}-(-1)^{3^{n-1}}}{3^n}=\frac{2^{3^{n-1}}+1}{3^n}
$$
is an integer. Notice also that for $n>0$ it has less than $3^{n-1}$ binary digits. Assume that it has $m$ binary digits. We have... | 22 | https://mathoverflow.net/users/101078 | 396300 | 163,656 |
https://mathoverflow.net/questions/396125 | 2 | Let $M$ be a 3-dimensional complex manifold, and $\Lambda$ a discrete lattice in $\mathbb C^2$. Suppose there is a holomorphic submersion $f:M\to\mathbb{C}^2/\Lambda$ with fibers given by 1-dimensional compact complex manifolds. And these fibers form the leaves of a 1-dimensional holomorphic foliation $\mathcal{W}$.
... | https://mathoverflow.net/users/167284 | Complex fibration over complex torus | **Added.** Thanks to the comment of abx below I understood that there was a big gap in the reasoning, and $M$ doesn't need to be a fiber bundle.
**Example.** I'll construct an example $S$ of a complex surface that admits a submersion to an elliptic curve $E$ but that is not a fiber bundle over $E$. Then $M$ can be ta... | 1 | https://mathoverflow.net/users/943 | 396303 | 163,658 |
https://mathoverflow.net/questions/396310 | 5 | Let $\ n\ $ be an arbitrary natural number ($\ 1\ 2\ \ldots).\ $ Then
* $\ n\ $ is coarse $\ \Leftarrow:\Rightarrow\ $ there exists a prime divisor $p$ of $\ n\ $ such that $\ p^3>n.$;
* $\ n\ $ is a p-cube $\ \Leftarrow:\Rightarrow\ $ the positive cubical root of $\ n\ $ is a prime number;
* $\ n\ $ is fine $\ \Left... | https://mathoverflow.net/users/110389 | Can all three numbers $\ n\ \ n^2-1\ \ n^2+1\ $ be fine (as opposed to coarse)? | $n = 2673$ has largest prime factor $11$ whose cube is $1331$.
$n^2 - 1 = 7144928$ has largest prime factor $191$ whose cube is $6967871$.
$n^2 + 1 = 7144930$ has largest prime factor $61$ whose cube is $226981$.
| 10 | https://mathoverflow.net/users/46140 | 396318 | 163,661 |
https://mathoverflow.net/questions/396071 | 1 | This question is an offshoot of [this closely related MO question](https://mathoverflow.net/questions/393738).
Here, we consider the Diophantine equation
$$m^2 - p^k = 2^r t,$$
where $r \geq 2$ and $\gcd(2,t)=1$.
Furthermore, we place the following restrictions:
$$p \equiv k \equiv 1 \pmod 4$$
$$2^r \neq t$$
$$2^r ... | https://mathoverflow.net/users/10365 | On the Diophantine equation $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$ | I think that
$$m=10^{375}+1,p=5,k=1,r=2,t=25\cdot 10^{748}+5\cdot 10^{374}-1\tag1$$
is a solution.
*Proof* :
When $m=10^{375}+1,p=5,k=1$, we have
$$m^2-p^k=(10^{375}+1)^2-5\equiv 1-5\equiv 4\pmod 8$$
from which $r=2$ and
$$t=\frac{(10^{375}+1)^2-5}{2^r}=25\cdot 10^{748}+5\cdot 10^{374}-1$$
follow.
Now, (1) satisf... | 1 | https://mathoverflow.net/users/34490 | 396321 | 163,663 |
https://mathoverflow.net/questions/396326 | 16 | The following real $2 \times 2$ matrix has determinant $1$:
$$\begin{pmatrix}
\sqrt{1+a^2} & a \\
a & \sqrt{1+a^2}
\end{pmatrix}$$
The natural generalisation of this to a real $2 \times 2$ block matrix would appear to be the following, where $A$ is an $n \times m$ matrix:
$$\begin{pmatrix}
\sqrt{I\_n+AA^T} & A \\... | https://mathoverflow.net/users/119987 | Proof that block matrix has determinant $1$ | Write the SVD of $A$, say $A=PDQ^T$ with $D$ diagonal with non-negative entries and $P\in O(n),Q\in O(m)$. Then $\sqrt{I\_n + AA^T} = P\sqrt{1+D^2}P^T$
and $\sqrt{I\_m+ A^TA} = Q\sqrt{1+D^2}Q^T$. This gives
$$
\begin{pmatrix}
\sqrt{I\_n + AA^T} & A \\ A^T& \sqrt{I\_m+A^TA}
\end{pmatrix}
=
\begin{pmatrix}
P & 0 \\
0 & Q... | 25 | https://mathoverflow.net/users/141760 | 396329 | 163,666 |
https://mathoverflow.net/questions/396314 | 4 | Let $\mathcal{H}$ be a separable Hilbert space and let $TC( \mathcal{H})$, $HS(\mathcal{H})$ be the space of trace-class operators and Hilbert-Schmidt operators on $\mathcal{H}$. Recall that these space are Banach spaces and that $HS(\mathcal{H})$ is a even a Hilbert space. For a Banach space $X$ let $B(X)$ be the spac... | https://mathoverflow.net/users/143779 | Which operators on the trace-class operators extend to operators on Hilbert-Schmidt operators? | No, this space is not closed under involution. Choose $A \in TC$ and $B \in HS\setminus TC$ and consider the rank one operator $T \mapsto \langle T, B\rangle A$ on $HS$. This restricts to a bounded operator on $TC$ but its adjoint doesn't.
| 6 | https://mathoverflow.net/users/23141 | 396332 | 163,668 |
https://mathoverflow.net/questions/396344 | 2 | Let $(\mathbb R^{1+2},\eta)$ be Minkowski with the metric $\eta= -dt^2+(dx^1)^2+(dx^2)^2$. Suppose $\Sigma$ is a smooth timelike hypersurface and denote by $h$ the second fundamental form on $\Sigma$. Assume that
$$ h(N,N) \geq 0 \quad \forall N\in L\_p\Sigma \quad \text{and}\quad p \in \Sigma,$$
where $L\_p\Sigma= \{N... | https://mathoverflow.net/users/50438 | Hyperboloids in Minkowski geometry | By hyperboloid, you seem to mean the one-sheeted surface $\{ |x|^2 - t^2 = 1\}$; this surface is umbilical with the induced metric proportional to the mean curvature, and so $h(N,N) = 0$ for all $N\in L\_p\Sigma$.
Before answering your question, a couple comments:
1. It is meaningless to specify $h(N,N) \geq 0$ wit... | 4 | https://mathoverflow.net/users/3948 | 396349 | 163,672 |
https://mathoverflow.net/questions/330771 | 7 | Let $G$ be a compact Lie group. The classical Peter-Weyl theorem shows that $L^2(G)$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$. This is a powerful statement as it allows to answer questions about functions on $G$ in terms of matrix coefficients of irreducible ... | https://mathoverflow.net/users/17047 | Peter–Weyl theory for vector fields | Let $\mathfrak g \to \mathfrak{X}(G)$ be the inclusion of right-invariant vector fields on $G$ into vector fields on $G$. Then we have an isomorphism $C^\infty(G) \otimes\_{\mathbb R} \mathfrak g \to \mathfrak{X}(G)$ of left $C^\infty(G)$-modules defined by $f \otimes\_{\mathbb R} x \mapsto fx$. It is $G \times G$-equi... | 6 | https://mathoverflow.net/users/125523 | 396360 | 163,677 |
https://mathoverflow.net/questions/396368 | 1 | Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set $P\subseteq\mathcal{C}$, $f|P$ maps surjectively onto $\mathcal{C}$?
Such functions (on $\mathbb{R}$) are called *per... | https://mathoverflow.net/users/16107 | A Borel perfectly everywhere surjective function on the Cantor set | As suggested by the comment above, the answer is no. The relevant fact is that every nonmeager subset of $\mathcal{C}$ with the Baire property (in particular, any Borel set) contains a nonempty closed perfect set.
Suppose that there was a Borel function $f$ with this property. Observe that for every $p\in\mathcal{C}$... | 3 | https://mathoverflow.net/users/16107 | 396371 | 163,682 |
https://mathoverflow.net/questions/192314 | 12 | I'm broadly interested in notions of "generic presentability" - when a given object exists in *every* forcing extension of the universe by some fixed forcing, at least up to the appropriate notion of equivalence. Sometimes this is boring - [per Solovay](https://mathoverflow.net/questions/155915/who-proved-sets-in-every... | https://mathoverflow.net/users/8133 | A new cardinality living in every forcing extension? | Partial answer. There is a $V\models ZFA$ containing a generically presentable cardinal $(\nu,\mathbb{P})$ such that for all $a\in V$ we have $\Vdash\_{\mathbb{P}^2}\nu[G\_0]\not\equiv\check{a}.$ This should be equivalent to your first condition, assuming the first condition can even be stated i.e. the ground model is ... | 1 | https://mathoverflow.net/users/164965 | 396375 | 163,683 |
https://mathoverflow.net/questions/396381 | 1 | Suppose that $A = M\_n(\mathbb{C})$ be the algebra of $n\*n$ matrices over $\mathbb{C}$.
If com(A) = {$B \in M\_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
| https://mathoverflow.net/users/137242 | Dimension of commutant | This is known for a general field by a theorem of Frobenius:
Let $F$ be a field and $V$ a finite dimensional $F$-vector sapce with a linear operator A. When $f\_i(X)$ denote the invariant factors of $A$ (such that $f\_i(X)$ divides $f\_{i+1}(X)$, then the dimension is equal to $\sum\limits\_{i=1}^{k}{(2k-2i+1)deg(f\_... | 6 | https://mathoverflow.net/users/61949 | 396383 | 163,686 |
https://mathoverflow.net/questions/396378 | 2 | *This note is related to*
[Can all three numbers $\ n\ \ n^2-1\ \ n^2+1\ $ be fine (as opposed to coarse)?](https://mathoverflow.net/questions/396310/can-all-three-numbers-n-n2-1-n21-be-fine-as-opposed-to-coarse)
---
Let
$$ m\ n\ \in\ \mathbb N\_{\_{>1}}\ :=\ \{x\in\mathbb Z: x>1\} $$
be arbitrary. Let $\ P(n... | https://mathoverflow.net/users/110389 | Molecularity $\ M(n)$ | For every $u>0$, there exists $n$ such that each of $P(n-1)$, $P(n)$, $P(n-1)$ is less than $n^u$. This was proved by Eggleton and Selfridge (Consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 22 (1976), 1–11). In fact their proof is constructive (see pp. 2-3 of their paper). It follows tha... | 5 | https://mathoverflow.net/users/11919 | 396385 | 163,687 |
https://mathoverflow.net/questions/395939 | 9 | Might there be a good historical reference on Egyptian number theory ($ \sim 2000$ B.C.)? The following online reference by a professor at the UCLA indicates that they were aware of the Pythagorean theorem [1]. This makes me wonder whether Egyptian scientists and engineers might have done fundamental work in number the... | https://mathoverflow.net/users/56328 | Egyptian number theory | According to the chapter on Egyptian mathematics and astronomy in the book by O. Neugebauer, *The Exact Sciences in Antiquity* (1951, 1957), Egyptian mathematics only involved basic arithmetic with positive integers and a restricted class of fractions. So the answer to your question appears to be "no".
| 2 | https://mathoverflow.net/users/106467 | 396388 | 163,690 |
https://mathoverflow.net/questions/396390 | 4 | Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n\_k\}$ of natural numbers with positive lower density, we have that for all $f \in L^1(X)$,
$$\frac{1}{N} \sum\_{i=0}^{N-1} f(T^{n\_i} x) \to \int\_X f d\mu$$
as $n \to ... | https://mathoverflow.net/users/173490 | A sufficient condition for an ergodic system to be weakly mixing | I think $\mathbf{X}$ has to be weakly mixing.
In order to prove it, note that one characterization of weak mixing is that the Koopman operator $U\_Tf:= f\circ T$ does not have any eigenvectors on the space $L^{2}\_{0}$ of mean zero square-integrable functions. Suppose then that there is an eigenvector $f$ such that $... | 3 | https://mathoverflow.net/users/24953 | 396394 | 163,691 |
https://mathoverflow.net/questions/396384 | 1 | Let $B=\left\{ \left(x,y\right)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\right\} $
be the unit ball in $\mathbb{R}^{2}.$
Can we construct a subharmonic
function $f:B\rightarrow\left[-\infty,0\right]$ such that
$$
0<\int\_{\widetilde{B}}\left(1-x^{2}-y^{2}\right)^{-2}dV<\infty,
$$
where $\widetilde{B}=\left\{ \left(x,y\right)\i... | https://mathoverflow.net/users/310212 | A subharmonic function with a growth property | The answer is "yes". Let $E$ be some Jordan region in the unit disk on which
$$\int\_E(1-x^2-y^2)^{-2}dxdy<\infty,$$
and such that $E$ contains $[0,1)$, and the closure of $E$ is contained in the open unit disk, except the point $1$. Let $\phi$ be a conformal map of $E$ onto the right half-plane, such that $\phi(1)=\in... | 1 | https://mathoverflow.net/users/25510 | 396408 | 163,698 |
https://mathoverflow.net/questions/396338 | 5 | For a Banach space $X$ let $S\_X$ denote its unit sphere and let $\mathrm{Iso}\_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\mathrm{Iso}\_0(X)\curvearrowright S\_X$.
When $X=L^p([0,1])$ the group $\mathrm{Iso}\_0(X)$ is Polish, so we can ask ... | https://mathoverflow.net/users/49381 | How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$? | The orbit structure is extremely simple. If $p=2$, there is one orbit (the isometry group of a Hilbert space acts transitively), whereas for $p\neq 2$ there are exactly $2$ orbits: the (classes of) functions that do not vanish on a set of positive measure and its complement, the functions that do vanish on a set of pos... | 5 | https://mathoverflow.net/users/10265 | 396412 | 163,700 |
https://mathoverflow.net/questions/396356 | 2 | We consider $2a$ - periodic smooth solutions for
\begin{eqnarray\*}
-\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a]
\end{eqnarray\*}
We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume that (up to multiplication with a real number) there exists only one odd $2a$ - periodic solution.
Can one say anythin... | https://mathoverflow.net/users/79956 | Even and odd solutions for the Schrödinger equation | Because of the uniqueness of the initial value problem, there can be at most two solutions, i.e., if we have one odd $2a$-periodic solution, then there can be at most one more even $2a$-periodic solution. For example, for $V(x)=-(\pi /a)^2 $, we have the odd solution $\sin \pi x/a $ and the even solution $\cos \pi x/a ... | 6 | https://mathoverflow.net/users/134299 | 396418 | 163,702 |
https://mathoverflow.net/questions/396416 | 2 | Cohomology ring and cup product can be defined on simplicial complex (ie a triangulation of a manifold). Can we define cohomology ring and cup product on a more general complex? In particulate, I am interested in defining cohomology ring and cup product on a complex and its dual complex. The dual of a simplicial comple... | https://mathoverflow.net/users/17787 | Cohomology ring on non-simplicial complex | If $H^\ast(-,R)$ is cohomology with coefficients in a ring, the cup product may be defined purely via the functoriality of $H^\ast(-,-)$ and certain compatibilities of tensor products. There is a map
$$H^\ast(X,A) \otimes H^\ast(Y,B) \to H^\ast(X \times Y, A \otimes B)$$
coming from taking the tensor product of cochain... | 1 | https://mathoverflow.net/users/125523 | 396422 | 163,703 |
https://mathoverflow.net/questions/396414 | 1 | I am reading a paper '**[Periodic Nonlinear Schrodinger Equation and Invariant Measures](https://math.mit.edu/classes/18.158/gibbs-1D.pdf)**' written by **J.Bourgain**. And I am wondering if I can have some help from this website.
My question is an inequality at **(3.18)** of the paper. The inequality is, with $\lamb... | https://mathoverflow.net/users/127918 | An inequality between sum of exponential functions wrt dyadic index | $\newcommand\si\sigma\newcommand\la\lambda$It seems to be (tacitly) assumed in the paper that $p\ge2$ -- see the display between (3.11) and (3.12).
Also, $c$ and $C$ seem to be (tacitly) assumed in the paper to be positive real constants.
Note that
$$M^{-2/p}+M\_0/M\le\si\_M^2\le2M^{-2/p}+2M\_0/M.$$
So,
$$\begin{alig... | 1 | https://mathoverflow.net/users/36721 | 396425 | 163,704 |
https://mathoverflow.net/questions/396413 | 12 | I'm an analyst who needs to use Deligne's Theorem 8.4 in **1**, but I feel lost in the maze of definitions, and I don't trust my geometric intuition here.
>
> Theorem 8.4: Let $Q$ be a polynomial in $n$ variables and of degree $d$ over $\mathbb{F}\_q$, $Q\_d$ the homogeneous part of degree $d$ of $Q$, and $\psi:\ma... | https://mathoverflow.net/users/90189 | Deligne's theorem on exponential sums | Yes, smoothness is equivalent to the gradient being nonzero for every $x \in \overline{\mathbb F}\_q^n \setminus \{0\}$.
I would define smoothness of the hypersurface defined by $Q\_d$ as the condition that for each such $x$, either the gradient or the value of $Q\_d$ at $x$ is nonzero, but as you note, by homogeneit... | 17 | https://mathoverflow.net/users/18060 | 396428 | 163,705 |
https://mathoverflow.net/questions/396411 | 1 | Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w\_1,\ldots,w\_m$ be iid uniformly distributed on the unit-sphere in $\mathbb R^d$. Consider the quartic form
$$
F := \frac{1}{m}\sum\_{j,\ell=1}^m (w\_j^\top w\_\ell)(w\_j^\... | https://mathoverflow.net/users/78539 | Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices | Assume iid $N(0,1)$ entries, assume $C$ diagonal, and focus first on the non-diagonal terms:
$G=\sum\_j \sum\_{l\ne j} w\_j^Tw\_l w\_j^TCw\_l
= \sum\_{j\ne l, ik} w\_{ji}w\_{li} c\_i w\_{jk} w\_{lk}$.
Write this quantity as
$$
\begin{split}
G=\sum\_{j\ne l, i\ne k} w\_{ji}w\_{li} c\_i w\_{jk} w\_{lk}
&+
\sum\_{j\ne l... | 1 | https://mathoverflow.net/users/141760 | 396431 | 163,708 |
https://mathoverflow.net/questions/396427 | 1 | The Erdős–Rado notation $a \rightarrow (b)^c\_d$ is common in partition calculus / combinatorial set theory, as well as its negation $a \not\rightarrow (b)^c\_d$. In that field, is there a standard way to read them out loud?
| https://mathoverflow.net/users/310472 | Pronunciation: the Erdős–Rado partition notation | Community wiki because it is answered over at [MSE](https://math.stackexchange.com/questions/674961/how-to-pronounce-the-partition-relation).
[source](https://books.google.nl/books?id=tCATRdtV_cwC)
| 3 | https://mathoverflow.net/users/11260 | 396432 | 163,709 |
https://mathoverflow.net/questions/393532 | 5 | While studying the so-called [Mittag-Leffler Polynomials](https://mathworld.wolfram.com/Mittag-LefflerPolynomial.html), denoted $M\_n(x)$, I was looking into the sequence $\frac1{n!}M\_n(n)$ which takes the following form
$$a\_n:=\sum\_{k=1}^n\binom{n-1}{k-1}\binom{n}k2^k.$$
>
> **QUESTION 1.** Let $\nu\_2(m)$ deno... | https://mathoverflow.net/users/66131 | Power of $2$ dividing a specialized Mittag-Leffler polynomial | I will confine myself to Question 1 since you mentioned that you know how to do Question 2. Also the case when $n$ is odd is easy, and let us restrict to $n$ being divisible exactly by $2^r$ with $r\ge 1$, and we need to show that the exact power of $2$ dividing $a\_n$ is $3r$. Thus in what follows we may discard any t... | 6 | https://mathoverflow.net/users/38624 | 396433 | 163,710 |
https://mathoverflow.net/questions/396420 | 1 | Let $F(x,y)$ be a real polynomial in two variables of degree $d$. How many roots can $F(x,e^x)$ have? In other words, is there a bound one can place on the number of intersection points of $F(x,y)=0$ and $y=e^x$?
| https://mathoverflow.net/users/nan | Roots of $F(x,e^x)$ | Such a function has at most $\frac{d(d + 3)}{2}$ roots. In fact, let us show by induction on $k$ the following general statement:
Let $p\_0, p\_1, \dots, p\_k$ be polynomials of degree at most $n\_0, n\_1, \dots, n\_k$ respectively, not all of which are 0. Then, the function $f(x) = \sum\_{j = 0}^{k} p\_j (x) e^{j x}... | 4 | https://mathoverflow.net/users/88679 | 396434 | 163,711 |
https://mathoverflow.net/questions/396364 | 5 | I know that the group structure on Hom sets can be recovered from biproducts if they exit. Indeed, if $f, g : A \to B$ are two maps then there is a uniquely defined map $f \oplus g : A \oplus A \to B \oplus B$ and then there are diagonal and codiagonal maps giving
$$ A \to A \oplus A \to B \oplus B \to B $$
so we get a... | https://mathoverflow.net/users/154157 | Does an Ab-enriched category have a unique Ab-enrichment? | As Martin Brandenburg and Maxime Ramzi suggest, it is easy to construct examples on small categories.
For example, a one object Ab-enriched category is exactly a ring. The category corresponds to the monoid (multiplication) of the ring and the Ab-enrichment to the addition law. There are monoids which have multiple a... | 11 | https://mathoverflow.net/users/154157 | 396441 | 163,712 |
https://mathoverflow.net/questions/396426 | 3 | Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?
This is a refinement of the question I asked in: [A Borel perfectly everywhere surjective function on the Cantor set](https://math... | https://mathoverflow.net/users/16107 | A Borel perfectly everywhere dominating family of functions | The answer is no. By Lusin's Theorem there exists $A \subseteq 2^\omega$ closed and of positive measure such that $f \restriction A$ is continuous. Since $A$ cannot be countable it must contain a perfect set $P$. Now $f[P] \subseteq \omega^\omega$ is compact, hence bounded in $\omega^\omega$.
| 6 | https://mathoverflow.net/users/134910 | 396443 | 163,714 |
https://mathoverflow.net/questions/396436 | 9 | Judging by the compact regular case, and more generally the spatial case, regular projectivity of locales, resp. regular injectivity of frames, must have something to do with $\neg p\lor\neg\neg p$ and $\neg(x\land y)\to(\neg x\lor\neg y)$. On the other hand, existence of locales without points shows that the terminal ... | https://mathoverflow.net/users/41291 | What are projective locales / injective frames? | So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps).
The problem is that there exists a family of non-trivial Boolean locales $B\_\kappa$ indexed by infinite cardinal numbers $\kappa$, such that the only ... | 12 | https://mathoverflow.net/users/22131 | 396444 | 163,715 |
https://mathoverflow.net/questions/396220 | 1 | Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}\_{\ge 0}$ with respect to which $F$ is complete. Let $\mathcal{O}\_{F}$ denote the ring of integers of $F$.
>
> First define the product norm $|\cdot|\_{prod}$ on $F\o... | https://mathoverflow.net/users/105386 | Product absolute value in rings of integers | The natural map $i \colon \mathcal O\_F \to F$ is an injective map of $\mathbb F\_p$-vector spaces, hence we can choose a splitting, i.e. an $\mathbb F\_p$-linear map $s \colon F \to \mathcal O\_F$ such that $s \circ i$ is the identity.
We have $|s(x)| \leq |x|$ since if $x \in \mathcal O\_F$ then $|s(x) | = |x|$ and... | 3 | https://mathoverflow.net/users/18060 | 396446 | 163,716 |
https://mathoverflow.net/questions/396126 | 9 | I would have a proof of the following fact; but it's a bit clunky, and am wondering if one can get a more elegant one (and/or improve the constants). I couldn't find this anywhere, and searching properties of the Skellam distribution didn't help much either.
>
> Let $X\sim\operatorname{Poi}(\lambda)$ and $Y\sim\ope... | https://mathoverflow.net/users/37266 | Bounds on the expectation of $|X-Y|$ for $X,Y$ Poisson | $\newcommand{\la}{\lambda}$Let us show that
\begin{equation\*}
E|Z|\ge J(1)\min[c,\sqrt c\,], \tag{1}
\end{equation\*}
where
\begin{equation\*}
Z:=X-Y,\quad c:=\la+\mu,
\end{equation\*}
\begin{equation\*}
J(x):=\frac2\pi\,\int\_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.
\end{equation\*}
Mathematica's comma... | 9 | https://mathoverflow.net/users/36721 | 396447 | 163,717 |
https://mathoverflow.net/questions/396448 | 3 | We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is of the form
$$
\mu\_{|[I]}=\psi(\mu^{+} \times \mu^{-}),$$
where $\psi$ is continuous and positive, and $\mu^{+}$ and $\... | https://mathoverflow.net/users/127839 | Does full shift have the local product structure? | No. Take $\mu$ to be a measure supported on a Sturmian shift corresponding to some irrational rotation $R\_\alpha$. If $\mathcal F^+$ denotes the $\sigma$-algebra generated by the coordinates in $\mathbb Z\_{\ge 0}$ and $\mathcal F^-$ denotes the $\sigma$-algebra generated by coordinates in $\mathbb Z\_{<0}$, then cond... | 4 | https://mathoverflow.net/users/11054 | 396449 | 163,718 |
https://mathoverflow.net/questions/396174 | 8 | $\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$Let $G$ be a compact Lie group and $\lambda\in
H^4(BG;\Z)$. The data $(G, \lambda)$ determine a 3d topological field theory called Chern-Simons theory, except not
quite: there is an obstruction to defining it on general closed, oriented $3$-manifolds called the *... | https://mathoverflow.net/users/97265 | Formula for the anomalies of spin Chern-Simons theories? | This is not a direct answer to your question, but I think it's relevant.
One way of thinking about the anomaly for ordinary (oriented) Chern-Simons theories is that it's the evaluation of the associated 3+1-dimension Crane-Yetter theory (what Freed would call the anomaly theory) on a generator of 4d oriented bordism,... | 8 | https://mathoverflow.net/users/284 | 396453 | 163,720 |
https://mathoverflow.net/questions/396377 | 5 | The following question concerns that without $ZF+DC$, can every function be "a little bit" continuous?
>
> **Question** Is it consistent with $ZF+DC$ that for any function $f:[0,1]\to [0,1]$ and any positive measure set $A$, there is a positive measure closed set $B\subseteq A$ so that $f^{-1}(B)$ is closed?
>
>
... | https://mathoverflow.net/users/14340 | Continuity of real functions | Any Borel $f:[0, 1] \to [0, 1]$ satisfying $f^{-1}[\{y\}]$ is dense in $[0, 1]$ for every $y \in [0, 1]$ is a counterexample. For example, $f(x) = \limsup\_n \frac{x\_1 + x\_2 + \dots + x\_n}{n}$ where $x = 0.x\_1 x\_2 \dots$ is the binary expansion of $x$.
| 7 | https://mathoverflow.net/users/310787 | 396465 | 163,724 |
https://mathoverflow.net/questions/396291 | 2 | So, I asked a similar question at math stackexchange and was directed here. Please let me know if this question is better suited elsewhere.
Let $U$ and $V$ be (infinite-dimensional) Banach spaces. Assume we have a sequence $(u\_n)$ in $U$ converging weakly to $u\_0$ and a nonlinear function $F:U \to V$, where we can ... | https://mathoverflow.net/users/308734 | Weakly continuous function implies some sort of triviality | From the question and from the OP's [related question](https://math.stackexchange.com/q/4180682/793015) on Mathematics StackExchange, I infer that the OP is in general interested in the weak continuity of nonlinear mappings. So here are two general facts the seem to be relevant:
Let $X$, $Y$ be Banach spaces (over th... | 1 | https://mathoverflow.net/users/102946 | 396469 | 163,726 |
https://mathoverflow.net/questions/396424 | 2 | Consider the drifted Brownian motion $X\_t=1+\lambda(t)+W\_t$, where $\lambda: \mathbb R\to [0,\infty)$ with $1\le \lambda'(t)\le 2$ and $(W\_t)\_{\ge 0}$ denotes a Brownian motion. Define the hitting time
$$\tau:=\inf\{t\ge 0: X\_t\le 0\}$$
and further the function $p(t):=\mathbb P[\tau>t]$ for all $t\ge 0$. Can w... | https://mathoverflow.net/users/261243 | Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$ | I claim this is not a complete answer. I wonder whether it can be improved to obtain the Lipschitz continuity.
Define
$$\left(\frac{d\mathbb Q}{d\mathbb P}\right)\_t := \exp\left(-\int\_0^t \lambda'(s)dW\_s-\frac{1}{2}\int\_0^t(\lambda'(s))^2ds\right).$$
Then $B\_t:=W\_t+\lambda(t)$ is a Brownian motion under $\m... | 1 | https://mathoverflow.net/users/261243 | 396484 | 163,730 |
https://mathoverflow.net/questions/396483 | 4 | Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition?
One way to get such a bound would be to give a lower bound for the dimension of a non-trivial representation (over $\mat... | https://mathoverflow.net/users/32210 | Subgroups and representations of finite groups of Lie type | Explicit values for the minimum degree of a primitive permutation representation of a simple group of Lie type can be found in Table 4 of this paper:
*Guest, Simon; Morris, Joy; Praeger, Cheryl E.; Spiga, Pablo*, [**On the maximum orders of elements of finite almost simple groups and primitive permutation groups.**](... | 5 | https://mathoverflow.net/users/801 | 396489 | 163,731 |
https://mathoverflow.net/questions/396488 | 2 | Let $\ell \geq 5$ be a prime. Show that there exists an imaginary quadratic field $K$ with odd fundamental discriminant:
(a) $\ell$ inert in $K$,
(b) $(\ell, h\_{K})=1$.
Remark. Without requiring the discriminant being odd, the existence due to Horie-Onishi.
| https://mathoverflow.net/users/311108 | Imaginary quadratic fields with $\ell$-indivisible class number | Here's an elementary argument. For $\ell < 41$, $K = \mathbb{Q}(\sqrt{-163})$ works. For $\ell = 41$, $K = \mathbb{Q}(\sqrt{-3})$ works. Assume then that $\ell \geq 43$.
Choose an integer $1 \leq n \leq \ell - 1$ so that $\left(\frac{-n}{\ell}\right) = -1$ and let $k \in \{ 0, 1, 2, 3 \}$ be the unique integer so tha... | 2 | https://mathoverflow.net/users/48142 | 396495 | 163,734 |
https://mathoverflow.net/questions/396474 | 2 | The following fraction shows up when trying to show consistency of the OLS slope estimator in a simple linear regression on a log-log scale where the window of observation changes as the sample size $T$ increases and I want to find a constant $c > 0$ such that
$$
\frac{\sum\_{k = 1}^{N(T)} \vert x\_{k, T} - \bar{x}\_{... | https://mathoverflow.net/users/302666 | Asymptotic bound of quotient of absolute and squared deviation from mean | $\newcommand{\ep}{\varepsilon}\newcommand{\num}{\operatorname{num}}\newcommand{\den}{\operatorname{den}}\newcommand{\R}{\mathbb R}$Let
$$x\_k:=x\_{k,T}=\ln(S+k-1),\quad S:=T^\ep,\quad N:=N(T).$$
Let us ensure that $N$ is an integer, assuming, more generally, just that
\begin{equation}
N\sim(m-1)S.
\end{equation}
We ha... | 1 | https://mathoverflow.net/users/36721 | 396497 | 163,735 |
https://mathoverflow.net/questions/396487 | 1 | Apologies if this question is a little vague.
I have seen written in a few places that the space of projective structures on a Riemann surface is an affine space modelled on the space of holomorphic quadratic differentials. Firstly, is this true for real curves and higher dimensional projective structures? Secondly, ... | https://mathoverflow.net/users/163024 | Understanding the space of structures | The space of affine connections is an affine space, since any two affine connections, say with Christoffel symbols $\Gamma^i\_{jk}$ and $\bar\Gamma^i\_{jk}$ differ by the difference of their Christoffel symbols $\Gamma^i\_{jk}-\bar\Gamma^i\_{jk}=a^i\_{jk}$, so the difference determines a tensor $a^i\_{jk}dx^j dx^k\part... | 3 | https://mathoverflow.net/users/13268 | 396501 | 163,737 |
https://mathoverflow.net/questions/396435 | 3 | **Problem summary:** I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in [this paper by Brav and Dyckerhoff](https://arxiv.org/abs/1812.11913)) actually looks like/how to give an alternative identification of the moduli of objects in certain cases.
**Edited to add:** Adding ... | https://mathoverflow.net/users/143797 | Intuition for points of the moduli of objects for a dg-category | [It would be good to mention the original paper of Toën-Vaquié https://arxiv.org/abs/math/0503269 where these moduli spaces of objects are defined, and maybe that of Lowrey for the ``coherent" version https://arxiv.org/abs/1110.5117 ]
Having a continuous right adjoint (in the compactly generated presentable / "large"... | 2 | https://mathoverflow.net/users/582 | 396514 | 163,742 |
https://mathoverflow.net/questions/396513 | 3 | Let $R$ be a ring and let $\mathcal{C}$ be the category of perfect $R$-complexes. Suppose that $$S=\bigoplus\_{i=1}^{\infty}R$$
Let us define $\mathcal{D}$ the smallest thick category generated by $S$.
Clearly $\mathcal{C}$ is a full subcategory of $\mathcal{D}$
The natural embedding $i:\mathcal{C}\rightarrow \ma... | https://mathoverflow.net/users/165456 | Algebraic K-theory of a category containing all perfect complexes | Following Achim's comment, it suffices to show that one can apply the usual Eilenberg swindle argument.
Maybe it's not super clear to the OP that D is closed under countable direct sums of any single object (which is all that one needs for the Eilenberg swindle).
The point is as follows : let $E$ be the subcategory... | 8 | https://mathoverflow.net/users/102343 | 396515 | 163,743 |
https://mathoverflow.net/questions/396452 | 3 | I should preface this question by saying that I strongly suspect the answer is negative, but I couldn't find the counterexample myself.
Say we are working on the unit disc $D \subset \mathbf{R}^n$, where we are given two uniformly elliptic operators with coefficients $A^{ij}$ and $a^{ij}$. (These may be as regular as... | https://mathoverflow.net/users/103792 | Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation? | $U = 2x^2-y^2$ and $u = x^2-2y^2$ solve constant-coefficient elliptic equations, but $U - u = x^2 + y^2$ has an interior minimum and thus cannot solve an elliptic equation.
| 4 | https://mathoverflow.net/users/16659 | 396518 | 163,745 |
https://mathoverflow.net/questions/396274 | 1 | Let $M$ be a type III$\_1$ factor. Suppose $\rho$ is a normal state on $M$, given any $c\in [0,2]$, can we find a normal state $\rho'$ on $M$ such that $\|\rho-\rho'\|=c$? Or can we find a sequence of normal states $\rho\_n$ such that $\|\rho-\rho\_n\|\to c$ as $n\to \infty$?
| https://mathoverflow.net/users/153196 | Normal states on a type III$_1$ factor | I don't think you can get $c = 2$, but you can do this for any $c \in [0,2)$. Given $\epsilon > 0$, find a projection $p \in M$ with $\rho(p) < \epsilon$ and then find $\rho'$ supported on $p$. Then $$(\rho - \rho')(1 - 2p) = 1 - 2\rho(p) - 1 + 2 = 2(1 - \rho(p)),$$ showing that $\|\rho - \rho'\|$ can be arbitrarily cl... | 3 | https://mathoverflow.net/users/23141 | 396523 | 163,747 |
https://mathoverflow.net/questions/396521 | 7 | In [arxiv:0909.3140](https://arxiv.org/abs/0909.3140) the $G$-extensions of a fusion category $\mathcal{D}$ are classified via monoidal 2-functors $G \to \underline{\underline{BrPic}}(\mathcal{D})$. A crucial part of the classification is Theorem 7.7 that shows that equivalence classes of such functors are in 1-1 corre... | https://mathoverflow.net/users/122206 | Why are $G$-Extensions of fusion categories rigid, when constructed via monoidal 2-functors $G \to \underline{\underline{BrPic}}(\mathcal{D})$? | Great question! Yes, this is missing from the original paper and is certainly needed for the main result. It is true though. There's two different proofs currently in the literature, [Deshpande and Mukhopadhyay](https://arxiv.org/pdf/1909.10799.pdf) Corollary 2.11 and [Davydov and Nikshych](https://arxiv.org/pdf/2006.0... | 7 | https://mathoverflow.net/users/22 | 396529 | 163,749 |
https://mathoverflow.net/questions/396505 | 2 | I apologize in advance if this question is not up to the level of research level questions on Math overflow. I am a complete outsider to invariant theory/representation theory and would like someone more knowledgeable than me to direct me to where I should read.
Let $Sym^k(\mathbb{R}^n)$ be the vector space of symmet... | https://mathoverflow.net/users/32135 | $O(n)$ Polynomial invariant of symmetric tensors | Just in case this wasn't obvious from Abdemalek Abdesselam's answer, the following is a well-known fact. The algebra of invariant polynomials $\mathcal{A}^{O(n)}$ consists of linear combinations of all possible total contractions of any number of copies of the symmetric tensors from $Sym^k(\mathbb{R}^n)$. These contrac... | 4 | https://mathoverflow.net/users/2622 | 396530 | 163,750 |
https://mathoverflow.net/questions/396519 | 3 | *This was [asked at MSE](https://math.stackexchange.com/questions/4182108/how-complicated-must-a-real-with-an-easy-total-computability-problem-be-to-com) without success. Granted, a bounty is still ongoing there, but it doesn't look like it will be answered.*
---
For (noncomputable) $A\subseteq\omega$ let $\tilde... | https://mathoverflow.net/users/8133 | Is there a $\Delta^0_2$ real with "easy total computability problem"? | It seems to me that if $G$ is 1-generic and recursive in $0'$ then $\tilde{G}$ is Boolean $\Sigma^0\_2$, which is sufficient to conclude that $\tilde{G}$ is not $\Sigma^0\_3(G)$-complete.
First, show that if (1) there is a condition $p$ in $G$ such that there is no $\varphi\_e$-splitting pair of conditions extending ... | 5 | https://mathoverflow.net/users/31026 | 396540 | 163,752 |
https://mathoverflow.net/questions/396423 | 5 | Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 1)$ a velocity field such that every solution $(x\_t)\_{t\geq 0}$ of $(d/dt)x\_t=v(x\_t)$ is periodic. Denote, for a non-stationary point $x\in\mathbb{R}^n$ (meaning $v(x)\neq0$), by $T(x)$ the period of such a solution $(x\_t)\_{t\geq0}$ such that $x(0)=x$.
... | https://mathoverflow.net/users/159940 | Continuity of the period for a periodic dynamical system | The answer is trivially negative already for $n=3$: Start with the flow following along the "long" lines of a thickened (say with disc-shaped cut) Möbius strip. You can imagine this like a thick “rope” bent to a loop, making half a rotation on its way, and the flow follows the fibres of the rope (with a fixed speed).
... | 4 | https://mathoverflow.net/users/165275 | 396548 | 163,757 |
https://mathoverflow.net/questions/396537 | 1 | Let $k$ be a field of characteristic 0 and let $\varphi:\mathfrak{g}\rightarrow\mathfrak{f}$ and $\psi:\mathfrak{h}\rightarrow\mathfrak{f}$ be maps of Lie algebras. Is there a reference showing that the pullback (in the category of Lie $k$-algebras) of $\mathfrak{g}$ and $\mathfrak{h}$ along these maps is given by the ... | https://mathoverflow.net/users/226648 | Pullback of Lie algebras | A reference for pullbacks for modules over a ring: Proposition 5.11, p.222 in "An Introduction to Homological Algebra" by Rotman. For Lie $k$-algebras (or just $k$-algebras), you could refer to this and say the same construction/proof works.
| 1 | https://mathoverflow.net/users/38068 | 396559 | 163,762 |
https://mathoverflow.net/questions/396563 | 10 | I am working on a article in poset theory. In that article, I am defining a subposet of a poset. The definition is following:
Let $P$ be a finite poset. A subposet $P'$ of $P$ is called **closed under covering** if for every $x,y \in P'$ with $x\lessdot y$ in $P'$, we have $x\lessdot y$ in $P$. Here, $x \lessdot y$ m... | https://mathoverflow.net/users/311595 | A definition in poset theory | I recall seeing in various sources the terminology "cover preserving embedding" and "cover preserving subposet". Googling it now (<https://www.google.com/search?q=poset+%22cover+preserving%22>) brings some 4000 results, many of which are research articles (with some repetitions - I am not implying there are 4000 distin... | 10 | https://mathoverflow.net/users/1306 | 396566 | 163,765 |
https://mathoverflow.net/questions/347835 | 10 | * A space $X$ is said to be *sequential* if whenever $A \subset X$ is not closed then $A$ contains a sequence converging to a point outside of $A$.
* A space $X$ is said to have *countable tightness* if for every non-closed set $A \subset X$ and every point $x \in \overline{A} \setminus A$ there is a countable subset $... | https://mathoverflow.net/users/11647 | A variant of the Moore-Mrowka problem | This question was answered in the negative by Alan Dow and Istvan Juhász in a recent preprint.
[On the cardinality of separable pseudoradial spaces.](https://webpages.uncc.edu/%7Eadow/mioduszewskiDec4.pdf)
| 5 | https://mathoverflow.net/users/11647 | 396572 | 163,768 |
https://mathoverflow.net/questions/396511 | 3 | In the comments section of [this question](https://mathoverflow.net/questions/301882/on-functors-preserving-monoid-objects#comment879728_301882) there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the question, which... | https://mathoverflow.net/users/144957 | Functors that preserve monoids | A simple class of counterexamples can be found among [thin](https://ncatlab.org/nlab/show/thin+category) [small](https://ncatlab.org/nlab/show/small+category) [strict monoidal categories](https://ncatlab.org/nlab/show/strict+monoidal+category), i.e. preordered monoids. These are just sets equipped with a preorder and a... | 6 | https://mathoverflow.net/users/2841 | 396577 | 163,770 |
https://mathoverflow.net/questions/396590 | 2 | I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int\_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper *Uber die konvexe Hulle von n zufaillig gewahlten Punkten* they use the relation $$ \varPhi(p) = 1 - \frac{e^{\frac{-p^2}{2}}}{\sqrt{2\pi}p\... | https://mathoverflow.net/users/311932 | Approximation of $\Phi (p)$ | Equality
$$ \varPhi(p) = 1 - \frac{e^{\frac{-p^2}{2}}}{\sqrt{2\pi}p\left(1 + \frac{\theta\_p}{p^2}\right)} $$
for some $\theta\_p\in(0,1)$ can be rewritten as
$$r\_2(p):=\frac1{p+1/p}<r(p):=\frac{1-\varPhi(p)}{e^{-p^2/2}/\sqrt{2\pi}}<r\_1(p):=\frac1p,\tag{1}$$
which actually holds for all real $p>0$ and is a special ca... | 1 | https://mathoverflow.net/users/36721 | 396597 | 163,775 |
https://mathoverflow.net/questions/396595 | 6 | In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive integer $n$.
I am not the one that created this inequality. So the my question is where references for this inequality ... | https://mathoverflow.net/users/161614 | Need a reference for a trigonometric inequality | According to the last sentence on page 16 of [this paper](https://www.dcce.ibilce.unesp.br/%7Edimitrov/papers/main.pdf), this inequality was proved by
W. H. Young, On certain series of Fourier, Proc. London Math. Soc. (2) 11
(1912), 357–366.
| 9 | https://mathoverflow.net/users/36721 | 396599 | 163,776 |
https://mathoverflow.net/questions/394266 | 4 | Let $p:E\to B$ and $p':E'\to B$ be fibrations. It is [well known](https://en.wikipedia.org/wiki/Fiber-homotopy_equivalence) that if $f:E\to E'$ a fibrewise map that is also a homotopy equivalence, then it is a fibrewise homotopy equivalence.
What about the more general situation of fibrations $p:E\to B$ and $p':E'\to... | https://mathoverflow.net/users/8103 | fibre-preserving homotopy equivalence | The answer to the question as asked is no: a fibre-preserving map of fibrations in which the maps of total and base spaces are homotopy equivalences is neccessarily a fibre-preserving homotopy equivalence (also known as a *homotopy equivalence of fibrations*). A reference was supplied in the comments by Gustavo Granja,... | 1 | https://mathoverflow.net/users/8103 | 396607 | 163,779 |
https://mathoverflow.net/questions/396547 | 2 | Let $(M,\omega,H)$ be a Hamiltonian system and assume that $\gamma$ is a periodic orbit on a regular energy hypersurface. Then the regular orbit cylinder theorem (see for example *Abraham/Marsden: Foundations of Mechanics* Theorem 8.2.2 or the book by *Hofer/Zehnder* on Hamiltonian Dynamics Proposition 2 on page 110) s... | https://mathoverflow.net/users/98139 | On the existence of regular orbit cylinders | The two multipliers correspond to the exponential growth rate of perturbations 1). Along the periodic orbit, and 2). Normal to energy hypersurface.
Unless there is additional symmetry in the system, two multipliers = 1 is the generic situation. A common example of symmetry is integral of motion of Hamiltonian system ... | 1 | https://mathoverflow.net/users/230017 | 396618 | 163,781 |
https://mathoverflow.net/questions/396613 | 2 | Non-extendable 2D TQFTs correspond to finite dimensional Frobenius algebras [1].
How about 3D TQFTs? While the answer is clear for the extended ones (e.g. (3,2,1) TQFTs almost correspond to modular tensor categories [2]), I have not seen any discussion for (3,2) TQFTs.
More precisely, can one classify the functors
... | https://mathoverflow.net/users/124549 | Non-extendable 3D TQFTs | Check out Andras Juhasz' paper: <https://arxiv.org/pdf/1408.0668.pdf>
Specifically, Theorem 1.10:
>
> There is an equivalence between the symmetric monoidal category
> of (2+1)-dimensional TQFTs and the category of J-Algebras.
>
>
>
J-Algebras are somewhat cumbersome to define.
They are graded vector spaces ... | 7 | https://mathoverflow.net/users/5690 | 396621 | 163,782 |
https://mathoverflow.net/questions/396622 | 4 | Let $X$ be a CAT(0) space, $x \in X$ and $\xi$ be a point in the boundary at infinity of $X$. Denote by $\rho(t)$ the geodesic ray (parameterized by arc length) starting from $x$ and asymptotic to $\xi$. Let $y$ be a point close to $x$ and let $\overline{xy}$ denote the geodesic segment connecting $x$ and $y$. I would ... | https://mathoverflow.net/users/127739 | Angles and Busemann function in CAT(0) | The answer is "no" even in the hyperbolic plane $\mathbb{H}^2$.
Consider the horocycle about $\xi$ through $x$: this is the set of points such that $b\_{\xi}(z)=0$. Let $\gamma$ be the geodesic through $x$ tangent to the horocycle, and take $y$ to be any point on $\gamma$ (except $x$ itself). Then the angle between $... | 5 | https://mathoverflow.net/users/1463 | 396636 | 163,787 |
https://mathoverflow.net/questions/396633 | 0 | Suppose $X, Y$ are $L^1$ random variables, and $X\_t$ and $Y\_t$ are real valued stochastic processes with $X\_t, Y\_t \in L^1$ for all $t$ such that the following convergences hold:
i) $X\_t \to X$, $Y\_t \to Y$ a.s and in $L^1$.
ii) $E(X\_t| Y\_t) \to E(X)$ a.s.
iii) $E(Y\_t|X\_t) \to E(Y)$ a.s.
Does it follo... | https://mathoverflow.net/users/173490 | Independence of limits of asymptotically independent processes | The title mentions asymptotic independence, but I don't see any sort of asymptotic independence condition here. For random variables $X$ and $Y$, the conditions $E(X|Y)=E(X)$ a.s. and $E(Y|X)=E(Y)$ a.s. don't imply that $X$ and $Y$ are independent.
For your question, you can take $(X,Y)$ to be uniformly distributed o... | 0 | https://mathoverflow.net/users/5784 | 396639 | 163,788 |
https://mathoverflow.net/questions/396600 | 8 | Let $\alpha$ be an exterior product of a harmonic and a parallel form on a Riemannian manifold. Then $\alpha$ is known to be harmonic. I have heard that this is an old result due to R. Bott, but I could never find a reference. I would be very grateful for any pointers to the early literature.
| https://mathoverflow.net/users/3377 | reference to a theorem about a product of harmonic and parallel forms | One place where (a generalization of) the desired result is stated explicitly is in a 1973 paper by J. H. Sampson, [On a theorem of Chern](https://www.jstor.org/stable/1996588). Sampson gives a simplified proof of Chern's main result in his 1957 paper *On a generalization of Kähler geometry* (Algebraic geometry and top... | 8 | https://mathoverflow.net/users/13972 | 396647 | 163,790 |
https://mathoverflow.net/questions/396644 | 1 | Fix $n\geq 2$ and let $\mathbb{H}^{n}=\mathbb{R}\_{+}\times \mathbb{S}^{n-1}$ be the hyperbolic space be defined as a Riemannian manifold equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}rd\theta^{2},$$ where $dr^{2}$ is the standard metric on $\mathbb{R}\_{+}$ and $d\theta^{2}$ is the standard metric on the sph... | https://mathoverflow.net/users/163368 | Isoperimetric inequality for exterior domains on $\mathbb{H}^{n}$ | I might be missing something, but if you require $\Omega$ to be precompact in $\mathbb{H}^n \setminus K$, then $K$ makes no difference for the purpose of determining the measures and you can use the original inequality. The same if you only require pre-compactness in $\mathbb{H}^n$ but include the measure of $\partial ... | 1 | https://mathoverflow.net/users/51695 | 396649 | 163,792 |
https://mathoverflow.net/questions/396656 | 0 | Given a sequence $u\_k\in W^{1,p}(B\_1)\cap C^{\alpha}(B\_1)$ such that $\|u\_k\|\_{C^{\alpha}(B\_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u\_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B\_1)$}
$$
and we also have
$$
u\_k \rightarrow u\_0\;\;\mbox{in $C^{\alpha}(B\_1)$}.
$$
Can I imply from the abov... | https://mathoverflow.net/users/68370 | Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$? | I don't think so. Define a sequence of 'zig-zag' functions $(f\_n)$ on $[0,1]$ as follows. Let $f\_n$ be $\frac{2}{n}$-periodic with $f\_n(x) = x$ when $x \in [0,\frac{1}{n}]$ and $f\_n(x) = \frac{2}{n}- x$ when $x \in [\frac{1}{n},\frac{2}{n}]$. Every function is Lipschitz with $\lvert f\_n \rvert\_\infty \leq 1/n$ an... | 2 | https://mathoverflow.net/users/103792 | 396661 | 163,797 |
https://mathoverflow.net/questions/396662 | 3 | The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the title is the opposite of the one in the body. Fortunately, [@LaurentMoret-Bailly's answer](https://mathoverflow.net/a/39666... | https://mathoverflow.net/users/2383 | Can non-geometrically reduced reduced subschemes happen for reductive groups? | No, it does not follow that $G\_\mathrm{red}$ is geometrically reduced (thus the answer to the title question is yes).
Let $p=\mathrm{char}(k)>0$ and let $H=\mathbb{G}\_a \rtimes \mathbb{G}\_m$ be the group of affine transformations of $\mathbb{A}^1\_k$.
Let $(x,y)\in H$ act on $\mathbb{A}^1\_k$ by the "twisted" ac... | 4 | https://mathoverflow.net/users/7666 | 396667 | 163,800 |
https://mathoverflow.net/questions/313825 | 19 | Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k\_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim\_{t \to 0} t \log k\_{t}(x,y) = -\frac{1}{4}d^{2}(x,y)$. Since the heat kernel $k\_{t}(x,y)$ can be expressed in terms of eigenfunctions and eige... | https://mathoverflow.net/users/13341 | Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice? | Here is a sketch of an idea of how to show that the set $\mathcal{E}(g)\subset C^\infty(M)$ of *all* the eigenfunctions of the metric $g$ on a compact manifold $M$ determines $g$ up to a constant multiple. (Note that I do not assume that the corresponding eigenvalues are given, which would be easier.) Clearly, this is ... | 4 | https://mathoverflow.net/users/13972 | 396678 | 163,802 |
https://mathoverflow.net/questions/228359 | 23 | Let $\mathbb{F}$ be a field. The Tits building for $\text{SL}\_n(\mathbb{F})$, denoted $T\_n(\mathbb{F})$, is the simplicial complex whose $k$-simplices are flags
$$0 \subsetneq V\_0 \subsetneq \cdots \subsetneq V\_k \subsetneq \mathbb{F}^n.$$
The space $T\_n(\mathbb{F})$ is $(n-2)$-dimensional, and the Solomon-Tits th... | https://mathoverflow.net/users/317 | Is the Steinberg representation always irreducible? | Andrew Snowden and I managed to finally answer this question in our paper "The Steinberg representation is irreducible", available [here](https://arxiv.org/abs/2107.00794). As you might guess from the title, we prove that the Steinberg representation over an infinite field is always irreducible. In fact, we prove somet... | 9 | https://mathoverflow.net/users/317 | 396684 | 163,804 |
https://mathoverflow.net/questions/396280 | 1 | In the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks in page $1009$ the authors claim that a map $w:D^2\rightarrow S^2$ with $w|\_{\partial D^2}\subset L$, where $L$ is the equator, and such that $\mu(w)=2$, where $\mu$ is the maslov index of the map, cannot be multiply covered and hen... | https://mathoverflow.net/users/nan | Maslov index equal to $2$ implies that the disk is not multiply covered | First, I assume you want $w$ to be holomorphic (or else it isn't true).
If the target is really $S^2$ with boundary on the equator $L$ then this should be easy to prove directly. Take a point $p$ not on $L$. We can define an intersection number between $w$ and $p$ using the intersection pairing between $H\_2(S^2,L)$ ... | 0 | https://mathoverflow.net/users/10839 | 396697 | 163,808 |
https://mathoverflow.net/questions/396682 | 0 | Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $D\sim\_{\mathbb{Q}} aC+bF$ (numerical equivalence) where $a,b\in\mathbb{Q}$.
Is it true then that $mD$ must be an integra... | https://mathoverflow.net/users/14514 | Pseudoeffective divisors on surfaces | If I understand the question correctly, then the Picard number of $S$ is 2 and hence the existence of the curve $F$ implies that the cone of effective curves is closed, so every pseudoeffective divisor is actually effective.
**Added to answer Friedrich's question in the comments:** Let $C\subseteq S$ be a(n effective... | 2 | https://mathoverflow.net/users/10076 | 396701 | 163,810 |
https://mathoverflow.net/questions/396715 | 1 | Let $k$ be a field; $X$ a projective scheme over $k$; $\mathcal A$ an ample $\mathcal O\_X$-module. Sometimes we want to apply induction on dimension by finding an effective Cartier divisor $D$ such that $\mathcal O\_X(D) \cong \mathcal A$. However, the problem is, whether $D$ always exists, i.e., a regular section of ... | https://mathoverflow.net/users/129738 | Effective Cartier divisors corresponding to a line bundle | If $\mathcal A$ is generated by global sections and $k$ is infinite then a regular section must exist. For each irreducible component of $X$, global sections vanishing on that component form a positive-codimension subspace of global sections. For non-reduced $X$, global sections vanishing on the induced reduced subsche... | 2 | https://mathoverflow.net/users/18060 | 396719 | 163,812 |
https://mathoverflow.net/questions/396596 | 12 | In searching for a counterexample in homological stability, I came across the following question:
>
> Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ has infinite Bass [stable rank](https://encyclopediaofmath.org/wiki/Stable_rank)?
>
>
>
| https://mathoverflow.net/users/157284 | Group ring with infinite stable rank | **Yes**, the integral group ring $\mathbb{Z}[F\_2]$ of the free group $F\_2$ on two generators has **infinite** stable rank.
This can be deduced from [1, Corollary 3.6]:
>
> There exists a cyclic $\mathbb{Z}[F\_2]$-module $M$ with the following property.
> For every $N \ge 1$, there exists an epimorphism $\theta\... | 7 | https://mathoverflow.net/users/84349 | 396729 | 163,814 |
https://mathoverflow.net/questions/396727 | 6 | I am trying to understand the relation between Wigner's $3j$-symbols (or Clebsch-Gordan coefficients) and matrix coefficients of intertwiners. I am new to this topic and need some help to understand it properly.
So, let us start with the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$, which is the complexification of $\ma... | https://mathoverflow.net/users/259525 | Relations between $3j$-symbols and intertwiners | There is a [standard basis](https://en.wikipedia.org/wiki/Angular_momentum_operator#Orbital_angular_momentum_in_spherical_coordinates) for finite-dimensional $SO(3)$ (and hence $\mathfrak{su}(2)$) representations, denoted by $\left| j, m\right\rangle$; the total spin $j\ge 0$ labels the representation, the magnetic qua... | 6 | https://mathoverflow.net/users/2622 | 396737 | 163,816 |
https://mathoverflow.net/questions/396706 | 1 | Suppose we are given a sequence $\phi\_k$ of traces (i.e. functions defined on boundary $\partial B\_1$) such that
$$
\phi\_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B\_1)$}
$$
(one can consider $C^{\alpha}$ convergence if required).
Now we consider affine subspaces Sobolev space $W^{1,p}(B\_1)$
$$
W^{1,p}\_{\p... | https://mathoverflow.net/users/68370 | Characterization on smallest element in affine Sobolev subspace | It seems to depend on the strength assumed of the convergence of the $\phi\_k$. This is a partial answer, where in some parts it is assumed that $p > n - 1$. Under this hypothesis, the convergence does not hold if only $\lvert \phi\_k \rvert\_{L^\infty(\partial B\_1)} \to 0$. However if $\lvert \phi\_k \rvert\_{C^{0,\b... | 2 | https://mathoverflow.net/users/103792 | 396741 | 163,818 |
https://mathoverflow.net/questions/396754 | 2 | Let $X$ be an open complex manifold, e.g., the complement of a simple normal crossing divisor $D$ in a (smooth) projective manifold $M$. Let $T^{1,0}X$ be the holomorphic tangent bundle of $X$. Let $K \subset X$ be a compact set with $U \subset X$ a sufficiently small open neighbourhood of $K$. Let $V$ be a holomorphic... | https://mathoverflow.net/users/nan | Local extension of holomorphic vector fields | By Theorem II.9.5 in Bredon's Sheaf Theory,
for any closed subset $K$ of a paracompact space $X$
and for any sheaf of abelian groups $F$ on $X$,
the canonical map $$\mathop{\rm colim} F(U) \to F(K)$$ is an isomorphism, where $U$ runs over all open neighborhoods of $K$.
In particular, every element of $F(K)$ is the re... | 5 | https://mathoverflow.net/users/402 | 396755 | 163,824 |
https://mathoverflow.net/questions/396751 | 1 | The first chapter of this [paper](http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf) (Gersten’s conjecture and the homology of schemes) defines a "Poincaré duality theory with supports" is a *twisted cohomology* theory satisfying certain properties on page 184.
The second chapter of this paper mentioned some e... | https://mathoverflow.net/users/119770 | Twist cohomology theory of algebraic de Rham cohomology | Nothing in the axioms says that the Tate twist should be nontrivial: it could be that $H^k(X,n)=H^k (X,m) $ for all $n,m $. This will be the case in both algebraic de Rham and Betti cohomology.
If you want the Tate twist in the de Rham theory to be nontrivial, you must keep track of extra structure, for example the c... | 3 | https://mathoverflow.net/users/1310 | 396761 | 163,825 |
https://mathoverflow.net/questions/396735 | 2 | Let $X^n$ and $X$ be stochastic processes defined by
$$X^n\_t=1+\int\_0^tb\_n(s)ds+\int\_0^t\sigma\_n(s)dW\_s \quad\mbox{and}\quad X\_t=1+\int\_0^tb(s)ds+\int\_0^t\sigma(s)dW\_s,$$
where $b\_n, \sigma\_n, b, \sigma$ are uniformly bounded measurable functions s.t.
$$\lim\_{n\to\infty}\sup\_{0\le t\le T}|b\_n(t)-b(... | https://mathoverflow.net/users/261243 | Convergence in law of stopped stochastic processes | I believe convergence in law holds for all $t \geq 0$. The proof proceeds in three steps.
**Step 1:** Note that by the dominated convergence theorem for stochastic integrals (see, for example [Theorem 7](https://almostsuremath.com/2010/01/11/properties-of-the-stochastic-integral/) here), we have that $X\_n$ converges... | 2 | https://mathoverflow.net/users/173490 | 396763 | 163,826 |
https://mathoverflow.net/questions/396757 | 3 | Let $p$ be a prime number and $q=p-1$. I’m trying to prove that the nonzero coefficients $a\_{qk}$ ($k\ge1$) of the power series
$$ \sum\_{k\ge1} a\_{qk} z^{qk} := \left( \sum\_{k\ge0} \frac{z^{qk+1}}{(qk+1)!} \right)^q $$
satisfy the congruence
$$ a\_{qk} \cdot (qk)! \equiv -1 \mod p. $$
*I’ve managed to work out th... | https://mathoverflow.net/users/313592 | On the arithmetic of powers of subseries of the exponential series | Your argument is in fact almost complete. It reduces the problem to checking the normalized coefficient $a\_q$ of $z^q$ is congruent to $-1$ mod $p$, but since $$ f = z + \frac{z^{q+1}}{(q+1)!} + \dots ,$$ we have $$f^q = z^q + \frac{ q z^{2q}}{ (q+1)!} + \dots$$ and so $$a\_q = q! \equiv -1 \mod p.$$
---
Here is... | 2 | https://mathoverflow.net/users/18060 | 396778 | 163,831 |
https://mathoverflow.net/questions/396768 | 5 | $\DeclareMathOperator\SO{SO}$Suppose we have a (continuous) linear action of $\SO(n,\mathbb R)$ on a vector space $\mathbb R^N$. Consider the ring of invariants $A\subset \mathbb R[x\_1,\ldots, x\_N]$, which is an $\mathbb R$-algebra. Is it true that the orbits of the $\SO(n,\mathbb R)$ action are in one-to-one corresp... | https://mathoverflow.net/users/13441 | Invariant theory over $\mathbb R$ | The answer depends on what you mean by "one-to-one correspondence". Is it bijective or just injective? Robert Bryant's (standard) argument shows that $\mathbb R^N/\mathrm{SO}(n)\to \mathrm{AlgHom}\_{\mathbb R}(A,\mathbb R)$ is injective. In general, this map is very far from being surjective, though. In other words, no... | 9 | https://mathoverflow.net/users/89948 | 396780 | 163,832 |
https://mathoverflow.net/questions/396769 | 3 | The last few days I am trying my best to understand a part of a proof from [these](https://www.win.tue.nl/%7Erhofstad/percolation_randomgraphs_rev.pdf) lecture notes on page 14:
[Picture of the relevant part](https://i.stack.imgur.com/0297Xl.png)
The setting is percolation on a regular tree with degree $r$, $C\_{BP... | https://mathoverflow.net/users/313721 | Deriving an asymptotic statement from a recursion | $\newcommand{\ep}{\varepsilon}$Let
\begin{equation\*}
x\_n:=\theta\_n,\quad s:=r-1>1,\quad b:=\frac{s-1}{2s}>0,
\end{equation\*}
\begin{equation\*}
f(x):=1-(1-x/s)^s,
\end{equation\*}
so that
\begin{equation\*}
x\_n=f(x\_{n-1})
\end{equation\*}
for natural $n$, with $x\_0\in[0,1]$. Without loss of generality, $x\_0... | 3 | https://mathoverflow.net/users/36721 | 396788 | 163,834 |
https://mathoverflow.net/questions/396685 | 1 | Let $N\_{n}=(1/n)\_{i=1,j=1}^{n}$ be the $n\times n$-matrix where all the entries are equal.
Suppose $n>0$. Let $\delta\_{n}$ be the least natural number such that $N\_{n}$ can be factored as $N\_{n}=A\_{1}\dots A\_{k}$ for some $k$ and $A\_{1},\dots,A\_{k}$ such that if $1\leq i\leq k$, then $A\_{i}$ is the convex c... | https://mathoverflow.net/users/22277 | Factorizing the doubly stochastic matrix where all entries are equal such that the factors are all convex combinations of few permutation matrices | I claim that $\delta\_{n}=2$ for all $n$.
>
> Lemma: For each $n$, and each vector $[x\_{1},\dots,x\_{n}]^{T}$ with
> $x\_{1}+\dots+x\_{n}=0$, there are matrices $A\_{1},\dots,A\_{k}$ where
> each $A\_{i}$ is the convex combination of $2$ permutation matrices and
> where $A\_{1}\dots A\_{k}[x\_{1},\dots,x\_{n}]^{T}... | 0 | https://mathoverflow.net/users/22277 | 396795 | 163,835 |
https://mathoverflow.net/questions/396797 | 6 | Let $k$ be an algebraically closed field. Let $H\_1, H\_2$ be two smooth hypersurfaces of the same degree $d$ in $P^n\_k$. Let $U\_1,U\_2$ be their complements respectively. Are $U\_1,U\_2$ isomorphic as algebraic varieties?
In $n=1,d=1$ case this is true, because the complement of any point is isomorphic to $A^1$.
... | https://mathoverflow.net/users/177957 | Open complement of hypersurfaces | The answer is no. Perhaps the simplest case is $n=2$, $d=4$. There is a unique double covering $\pi \_i:S\_i\rightarrow \mathbb{P}^2$ branched along $H\_i$. If $U\_1$ and $U\_2$ are isomorphic, $S\_1$ and $S\_2$ are isomorphic; then $H\_1$ and $H\_2$ are isomorphic, because $H\_i$ is the branch locus of the morphism $\... | 9 | https://mathoverflow.net/users/40297 | 396802 | 163,839 |
https://mathoverflow.net/questions/396762 | 5 | Is there a well-defined notion of connection on a measurable bundle of Hilbert spaces?
| https://mathoverflow.net/users/78032 | Connection on a Hilbert bundle | This is an answer to the refined question formulated in the comments:
the base space is the unitary dual of a Lie group $G$.
The definition can be carried out in the setting of stacks in groupoids (or simplicial sets) on the site of cartesian spaces ($\def\R{{\bf R}} \R^n$ with smooth maps, for all $n≥0$).
Specific... | 3 | https://mathoverflow.net/users/402 | 396805 | 163,842 |
https://mathoverflow.net/questions/396794 | 2 | Let's say that a (right) module $M$ is *well complemented* if every non-zero submodule of $M$ has an indecomposable direct summand (by the way, is there a better or more standard name for this property?). For instance, every module of finite [uniform dimension](https://en.wikipedia.org/wiki/Uniform_module#Uniform_dimen... | https://mathoverflow.net/users/16537 | Every non-zero submodule of $R_R$ has an indecomposable direct summand: True when $R$ is von Neumann regular? | The answer is no. Take a compact totally disconnected space $X$ with no isolated points, like the Cantor set. Let $K$ be any field and let $R$ be the ring of locally constant functions $f\colon X\to K$ with pointwise operations. This is a commutative von Neumann regular ring. The idempotents of $R$ are precisely the ch... | 4 | https://mathoverflow.net/users/15934 | 396806 | 163,843 |
https://mathoverflow.net/questions/396214 | 7 | A nonempty subset $D$ of a group $G$ is called
$\bullet$ *decomposable* if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$;
$\bullet$ *product-one* if there exists $n\in\mathbb N$ and pairwise distinct elements $x\_1,\dots,x\_n\in D$ such that $x\_... | https://mathoverflow.net/users/61536 | Product-one sets in non-commutative groups | GAP shows that the groups SmallGroup(27,3), SmallGroup(27,4), SmallGroup(36,11), SmallGroup(39,1) SmallGroup(48,3) do contain many 5-element decomposable sets, which are not product-one. So, the lower bound 5 for the smallest cardinality of a counterexample, obtained by @YCor in his answer, is the best possible.
Belo... | 3 | https://mathoverflow.net/users/61536 | 396817 | 163,849 |
https://mathoverflow.net/questions/396746 | 4 | Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ideal of $R$ is principal?
*Thoughts*: We have $\mathfrak m^2 \subseteq (0:\_R \mathfrak m)$. Since $R$ is Artinian Goren... | https://mathoverflow.net/users/158239 | Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$ | We use the fact that in an Artinian Gorenstein ring, any ideal contains the socle. The assumption tells us that the socle of $A$ is $\mathfrak m^2$, which is principal.
Let $I\neq (0)$ be a non-maximal ideal. If $I=\mathfrak m^2$, we are done. Otherwise, $I$ strictly contains $\mathfrak m^2$. Thus $\mathfrak mI\neq 0... | 4 | https://mathoverflow.net/users/2083 | 396824 | 163,851 |
https://mathoverflow.net/questions/396820 | 0 | Let $n=3^m$ for some positive integer $m$. Let $G\leq S\_n$ be a transitive permutation group on $n$ letters. Denote the largest normal subgroup of $G$ with odd order by $O\_{2'}(G)$. My question is the following:
Does there exist $G$ such that $G/O\_{2'}(G)\cong A\_4$ or $S\_4$ for suitable $m$?
| https://mathoverflow.net/users/134942 | Is there a permutation group satisfying the following property? | Let $\Omega=\{1,2,3,4,5,6,7,8,9\}$ and let $G\leq\mathrm{Sym}(\Omega)$ be the group generated by the following permutations:
* $(1, 2, 9)$
* $(4, 5)(7, 8)$
* $(1, 4, 7)(2, 5, 8)(3, 6, 9)$
* $(3, 6)(4, 7)(5, 8)$
This is an example with $n=9$, with $G/O\_{2'}\cong S\_4$. If you want $G/O\_{2'}\cong A\_4$, remove the ... | 5 | https://mathoverflow.net/users/22377 | 396830 | 163,852 |
https://mathoverflow.net/questions/396615 | 4 | Let $C$ be a class of plane curves, regarded as subsets of $\mathbb{R}^2$ (parametrization won't matter), I'm thinking for example of splines or algebraic subsets. Let $D$ be a class of topological discs bounded by plane curves from a possibly different class, for example circles, ellipses, closed splines.
Now if I t... | https://mathoverflow.net/users/5339 | Classes of curves closed under Minkowsky sum | There are lots of finite dimensional curve families that are closed under Minkowski sum with circular disks. Here's a way to construct examples:
First, choose a finite dimensional space $\mathcal{C}$ of smooth, functions that contains the constants. Then, for any $f\in\mathcal{C}$, consider the curve $X\_f$ defined b... | 3 | https://mathoverflow.net/users/13972 | 396850 | 163,856 |
https://mathoverflow.net/questions/392187 | 3 | I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.
Let's say we have a principal bundle $\mathcal{P}=(P, M, \pi ; G)$ and the isomorphism $T\_{e} L\_{p}: \mathfrak{g} \longrightarrow V\_{p}(... | https://mathoverflow.net/users/209074 | Local coordinates of one form on a principal bundle | Since you cross-posted, I'll cross-answer my reply from MSE. I'M rather new to the MO forum so I could use some reputation...
* I assume by $T\_A$ you mean a set of basis vectors of $\mathfrak{g}$.
* I assume by $\omega$ (without bar) you mean the connection of an associated vector bundle, associated via some represe... | 4 | https://mathoverflow.net/users/276879 | 396853 | 163,858 |
https://mathoverflow.net/questions/396843 | 1 | I am studying the notes of Sorger concerning the moduli problem of principal bundles over curve <https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/005/38005695.pdf> and there is something I don't quite understand. He considers the moduli stack of principal $G$-bundles over a curve $G$, and it is denoted by... | https://mathoverflow.net/users/140062 | Universal principal bundle on stack | Expanding on abx’s comment, let’s suppose $\mathcal{M}\_{G,C}$ was representable as a scheme. Then for any $\mathbb{C}$-scheme $S$ there is a canonical isomorphism
$Hom\_\mathbb{C}(S, \mathcal{M}\_{G,C}) \cong \{\text{Groupoid of principal }G\text{-bundles over }C \times S\}$.
Now take $S = \mathcal{M}\_{G,C}$, and... | 2 | https://mathoverflow.net/users/5513 | 396862 | 163,861 |
https://mathoverflow.net/questions/396856 | 9 | Let $X$ be a smooth complex algebraic variety endowed with a $\mathbb{C}^\*$ action. We assume also to have an antiholomorphic involution $\sigma$ over $X$ such that it anticommutes with the action above i.e $$\sigma(t \cdot x)=\bar{t}\cdot \sigma(x) .$$
Let us also assume that the $\mathbb{C}^\*$-action respects the... | https://mathoverflow.net/users/146464 | Bialynicki-Birula decomposition for real analytic varieties | No, consider the following $\mathbb{C}^\*$-action on $\mathbb{CP}^2$ : $$z.[z\_0:z\_1:z\_{2}] = [z\_0:z.z\_1:z^2.z\_2] ,$$ along with the antiholomorphic map $\sigma ([z\_0:z\_1:z\_2]) = [\bar{z\_{0}}:\bar{z\_{1}}:\bar{z\_2}]$. One map check that $\sigma$ anticommutes with the $\mathbb{C}^\*$-action.
This is the nice... | 6 | https://mathoverflow.net/users/99732 | 396864 | 163,862 |
https://mathoverflow.net/questions/396705 | 2 | Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$.
For $p \geq 1$, let $\lVert \cdot \rVert\_p$ denote the Schatten $p$-norm on the space of compact operators and let $1\_{\Gamma\_n}(x)$ d... | https://mathoverflow.net/users/271621 | Finiteness of Schatten $p$-norm of truncated free resolvent | I found a positive answer to this question in this [paper](https://arxiv.org/abs/1404.2817) of Frank and Sabin: *Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates*, Theorem 12.
| 0 | https://mathoverflow.net/users/271621 | 396865 | 163,863 |
https://mathoverflow.net/questions/396668 | 15 | It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A \to \widehat{\mathbf A}$ exhibits $\widehat{\mathbf A}$ as the cocompletion of $\mathbf A$ under small colimits.
Wher... | https://mathoverflow.net/users/152679 | Original reference for categories of presheaves as free cocompletions of small categories | The earliest reference I can find to the universal property of the presheaf construction is Remark 2.29 of Ulmer's [Properties of Dense and Relative Adjoint Functors](https://core.ac.uk/download/pdf/82101024.pdf) (1968). However, the proof is only lightly sketched, and in the introduction Ulmer states:
>
> As an ap... | 12 | https://mathoverflow.net/users/152679 | 396866 | 163,864 |
https://mathoverflow.net/questions/396855 | 1 | This question seems to be related to Theorem IX.7.28 in [J. Jacod and A. Shiryaev's *Limit theorems for stochastic processes* (2013)](https://www.google.co.kr/books/edition/Limit_Theorems_for_Stochastic_Processes/mSD4CAAAQBAJ?hl=ko&gbpv=1&printsec=frontcover), and it is very important to prove asymptotic properties of ... | https://mathoverflow.net/users/159685 | Convergence of discretized process when its predictable part converges to infinite variation process | Let $h\_i:=W\_{i/n}-W\_{(i-1)/n}$ and $n\_t:=\lfloor nt\rfloor$, so that $t-1/n\le n\_t/n\le t$. Then
$$X^n\_t=\sum\_{i=1}^{n\_t}(h\_{i-1}+h\_i)
=\sum\_{i=1}^{n\_t}h\_{i-1}+\sum\_{i=1}^{n\_t}h\_i=W\_{n\_t/n-1/n}-W\_{-1/n}+W\_{n\_t/n},$$
whence
$$|X^n\_t-2W\_t|\le|W\_{n\_t/n-1/n}-W\_t|+|W\_{-1/n}|+|W\_{n\_t/n}-W\_t|$$
a... | 1 | https://mathoverflow.net/users/36721 | 396868 | 163,865 |
https://mathoverflow.net/questions/396827 | 3 | Consider the symmetric group $S\_n$ under the uniform distribution. For integer $k > 1$, suppose we draw $k$ elements $s\_1, \dots, s\_k$ independently at random. What is the probability that there exists at least one rearrangement of the $s\_i$ that composes to the identity?
| https://mathoverflow.net/users/173490 | Probability that k randomly drawn permutations can be arranged to compose to the identity | Only a partial answer, which however is too long for a comment: Let $p\_{n,k}$ denote the given probability. Then we have
(1) $p\_{n,1} = p\_{n,2} = \frac{1}{n!}$, $p\_{n,3} = \frac{2\cdot n!-p(n)}{(n!)^2}$.
(2) $p\_{1,k}=1$, $p\_{2,k}=\frac{1}{2}$.
(3) $\frac{1}{n!}\leq p\_{n,k}\leq \frac{(k-1)!}{n!}$.
(4) For... | 3 | https://mathoverflow.net/users/65801 | 396880 | 163,869 |
https://mathoverflow.net/questions/396833 | 1 | $$
x\_{n}=\sum^{n-1}\_{i=0} {a\_i x\_{n-1-i}}
$$
where
$$
\sum^{+\infty}\_{i=0} {a\_i}=1,1>a\_i>0,1>x\_i>0
$$
In fact, the specific problem (comes from probability theory) I want to solve is that:
$0<d<0.2$ is a constant.
$E\_n(p) \in C[0,1]$is a function of p, $E\_0(p)=p$, and $E\_n(p)$ can be defined by：
$$
E\_... | https://mathoverflow.net/users/151339 | How to prove the convergence of this kind of sequence? | If $p>1-5d$, then $a\_n(p)=0$ for all $n\ge1$ and hence $E\_n(p)=0$ for all $n\ge1$.
So, without loss of generality $0\le p\le1-5d$. Then, letting $E\_{-1}:=1$, for all $n\ge0$ we get
\begin{equation\*}
E\_n=\sum\_{i=-1}^{n-1}l\_{n-1-i}E\_i,
\end{equation\*}
where $E\_n:=E\_n(p)$ and $l\_n:=l\_n(p)$.
Here is the ... | 1 | https://mathoverflow.net/users/36721 | 396896 | 163,873 |
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