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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/431642 | 43 | Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?
| https://mathoverflow.net/users/148161 | Do rings of smooth functions differ from rings of continuous functions? | No. In both the smooth function ring and the continuous function ring a maximal ideal $\frak m$ consists of the functions vanishing at some point. In the smooth case $\frak m/\frak m^2$ is the cotangent space of the manifold at that point, while in the continuous case $\frak m^2=\frak m$.
| 60 | https://mathoverflow.net/users/6666 | 431644 | 174,744 |
https://mathoverflow.net/questions/431625 | 10 | A finitely generated group $G$ *algebraically fibers* if there is an epimorphism $G\to\mathbb{Z}$ with finitely generated kernel. Since this kernel is finitely generated, we can ask whether \*it\* algebraically fibers. If it does, then we can keep going, etc etc etc. In any examples I can think of, this process always ... | https://mathoverflow.net/users/164670 | Iterated algebraic fibering | Let $G\_1$ be the wreath product of $\mathbb{Z}$ with $\mathbb{Z}$. There is a surjective homomorphism $G\_1\rightarrow \mathbb{Z}$ whose kernel $G\_2$ is isomorphic to $G\_1$. This gives a sequence of groups in which each $G\_i$ is isomorphic to $G\_1$.
If we write $G\_1:=\langle a,b: [a,a^{b^i}]\,\forall i\rangle$... | 5 | https://mathoverflow.net/users/124004 | 431660 | 174,749 |
https://mathoverflow.net/questions/430897 | 9 | A $\mathrm{C}^\*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) subspace (not subalgebra) we have an *operator system* $\mathcal{O}\subset \mathcal{A}$.
(I am aware there are abstrac... | https://mathoverflow.net/users/35482 | Why operator systems? | Collating some comments, it appears that one reason why operator systems are studied is because they are *useful*:
* Uri Bader mentions [a paper of Kalantar & Kennedy](https://arxiv.org/pdf/1405.4359.pdf) which makes extensive use of Hamana's theory for injective envelopes for operator systems equipped with a group a... | 4 | https://mathoverflow.net/users/35482 | 431662 | 174,750 |
https://mathoverflow.net/questions/431665 | 5 | The well-known *matrix determinant lemma* states that for an invertible square matrix $A$ and column vectors $u,v$ one has
$$
\det(A + uv^T) = \det(A)(1 + v^T A^{-1} u).
$$
Is there any analogous formula in case we are adding to $A$ a matrix which is not rank-one, but still has some special structure? The case I have i... | https://mathoverflow.net/users/2192 | Matrix determinant lemma for non-rank-one updates | If $A,B$ are square matrices, then $A\otimes I+I\otimes B$ is commonly known as the Kronecker sum of $A$ and $B$. If $A$ has eigenvalues $\mu\_1,\dots,\mu\_m$ and $B$ has eigenvalues $\nu\_1,\dots,\nu\_n$, then $A\otimes I-I\otimes B$ has eigenvalues
$\mu\_i-\nu\_j$ for $1\leq i\leq m,1\leq j\leq n$. Therefore, $\det(A... | 4 | https://mathoverflow.net/users/22277 | 431679 | 174,753 |
https://mathoverflow.net/questions/431676 | 11 | Let $X=(X\_1,\ldots,X\_n)$ be an iid sequence of random variables, and let $\nu$ be a *uniformly random* integer in the range $1,\ldots,n$. Then $\xi\_\nu$ is a random entry of $X$. Is it always true that after deleting such a random entry from $X$, the remaining sequence is still iid?
Due to the uniform randomness o... | https://mathoverflow.net/users/101180 | Can deleting a random entry from an iid sequence destroy the iid property? | The independence will be then in general lost. E.g., let $X\_1,\dots,X\_n$ be independent random variables each uniformly distributed on $[0,1]$. Let $M:=\max(X\_1,\dots,X\_n)=X\_\nu$, so that $\nu$ is uniformly distributed on $[n]:=\{1,\dots,n\}$. Let $(Y\_1,\dots,Y\_{n-1})$ be the leftover sequence, after the removal... | 19 | https://mathoverflow.net/users/36721 | 431681 | 174,754 |
https://mathoverflow.net/questions/431680 | 3 | Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder <https://arxiv.org/pdf/2207.06477.pdf> Lemma 6.18, $D^b(X)$ contains a categorical ordinary double point subcategory $\mathcal{P}\subset D^b(X)$, which is generated by a $\ma... | https://mathoverflow.net/users/41650 | Semi-orthogonal decomposition for maximally non-factorial Fano threefolds | If you assume that $\mathcal{P}$ and $\mathcal{P'}$ are semiorthogonal, this is true. The easiest way to see this is by looking at the singularity category. If $X$ has one node, (the idempotent completion of) the singularity category of $X$ is the category of $\mathbb{Z}/2$-graded vector spaces, and if
$$
D^b(X) = \lan... | 6 | https://mathoverflow.net/users/4428 | 431692 | 174,757 |
https://mathoverflow.net/questions/431616 | 3 | In the context of Banach space theory, what is the correct terminology: *extremally* disconnected or *extremely* disconnected. Looking through the internet I have met using both [extremely](https://math.stackexchange.com/questions/1808481/proof-that-ck-is-a-grothendieck-space-for-k-an-extremely-disconnected-comp) and [... | https://mathoverflow.net/users/61536 | Extremely disconnected or extremally disconnected? | The correct word is **extremally** (which is not in standard English) and is indeed often mistakenly corrected to the more common "extremely" — possibly, in recent case, with the unfortunate help of automatic correctors. Extremely would mean "very very", while "extremally" purports to mean, as far as I understand "in a... | 4 | https://mathoverflow.net/users/14094 | 431701 | 174,759 |
https://mathoverflow.net/questions/431556 | 20 | I can prove the following result.
>
> **Theorem 1.** Let $f\_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
> that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
> Then $f$ is convex in the
> sense that there is a convex function $F:\mathbb{R}^n\to\mathbb{R}$
> such that $... | https://mathoverflow.net/users/121665 | Convergence of convex functions | It follows from Theorem 10.8 in
R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No.
28. Princeton University Press, Princeton, N.J., 1970.
This theorem essentially says the following: if $f\_n$ are convex functions on an open domain $\Omega\subset\mathbb{R}^n$ that converge pointwise (to a f... | 9 | https://mathoverflow.net/users/140505 | 431703 | 174,760 |
https://mathoverflow.net/questions/431672 | 7 | Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum\_{i=1}^d b\_i(x) \frac{\partial}{\partial x\_i},
\end{align}
where $b\_i(x) \in \mathcal{C}\_b^2(\mathbb{R}^d)$, $i=1,2,\ldots,d$, and initial condition $x \in \mathbb{R}^d$. Ikeda and Watanabe prove the fol... | https://mathoverflow.net/users/474675 | Onsager-Machlup functional when drift is time-dependent | Using the argument of [http://users.sussex.ac.uk/~md326/MAP.pdf](http://users.sussex.ac.uk/%7Emd326/MAP.pdf) or <https://arxiv.org/abs/2209.04523>
We have that if $\mu\_0$ is a centered Gaussian measure then its Onsager-Machlup function is $\operatorname{OM}\_{\mu\_0}(z)=\frac{1}{2}\|z\|\_{\mu\_0}^2$. If $\mu$ is equ... | 3 | https://mathoverflow.net/users/479223 | 431704 | 174,761 |
https://mathoverflow.net/questions/431709 | 0 | I would like to know if there is a method to solve the Problem.
**Problem**:
Maximize the following function: $$f(p\_{1,i},p\_{2,i},\dotsc,p\_{m,i})=\sum\_{i=1}^{n}\begin{bmatrix}p\_{1,i} & p\_{2,i} & \cdots & p\_{m,i} \end{bmatrix}\*\begin{bmatrix}e\_1 \\ e\_2 \\ \vdots \\ e\_m \end{bmatrix} \* c\_i$$
where we... | https://mathoverflow.net/users/492293 | Method for (binary) optimization under constraints | This is the [transportation problem](https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)) in a bipartite network, with a supply of $1$ at each $j$ node and a demand of $t\_i$ at demand node $i$. The problem can be solved via linear programming, a minimum-cost network flow algorithm, or a specialized algor... | 4 | https://mathoverflow.net/users/141766 | 431710 | 174,764 |
https://mathoverflow.net/questions/430844 | 3 | Consider the following NLS:
$$i u\_t + \Delta u- 2 \operatorname{Re} u = F(u),$$
where $F(u):=(u + \bar{u} + |u|^2)u.$
In [Scattering for the Gross–Pitaevskii equation](https://arxiv.org/abs/math/0510080), the authors S. Gustafson, K. Nakanishi, and T.-P. Tsai used a change of variables to get the free evolution ... | https://mathoverflow.net/users/471464 | Change of variables for obtaining a unitary group | Let $u=u\_1+iu\_2$, $v=v\_1+iv\_2$ with $u\_1,u\_2,v\_1,v\_2$ being real valued.
What one would like to do is find a $2\times 2$ matrix of real symmetric operators
$$
\begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
$$
such that if one lets
$$
\begin{pmatrix}v\_1\\ v\_2\end{pmatrix}:=\begin{pmatrix}
A & B\\
C & D
\end{pmatr... | 2 | https://mathoverflow.net/users/7410 | 431713 | 174,765 |
https://mathoverflow.net/questions/429875 | 6 | A very fluffy question in which I'm ignorant of homology/cohomology, grading etc:
The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the wrapped Fukaya category of some symplectic manifold, are proven to be (ring, not sure about BV-algebra?) isomorphisms in... | https://mathoverflow.net/users/490756 | Relation between symplectic (co)homology and Hochschild (co)homology and deformations | First, the closed-open string map is expected to be an isomorphism between BV algebras. In the case when $X$ is a Weinstein manifold, it is known to be an isomorphism of BV algebras, just make sure that it is the wrapped Fukaya category $\mathcal{W}(X)$ that is under consideration.
Assume that $X$ is a smooth affine ... | 3 | https://mathoverflow.net/users/43423 | 431719 | 174,767 |
https://mathoverflow.net/questions/431567 | 7 | Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density Hypothesis" usually means one of the two following unproven conjectures.
**1."Weak" Density Hypothesis:**
Let $\varepsilon>0$ b... | https://mathoverflow.net/users/153908 | What consequences would follow from the density hypothesis? | The "strong" density hypothesis implies that if $\epsilon>0$ fixed and $X$ is large, then the Lebesgue measure of the $x\in[0,X]$ such that the interval $[x,x+x^{\epsilon}]$ does not contain a prime is $o(X)$. (See the bottom of page 131 in Montgomery's "Topics in Multiplicative Number Theory".) For comparison, RH impl... | 8 | https://mathoverflow.net/users/111215 | 431727 | 174,770 |
https://mathoverflow.net/questions/431714 | 1 | Is it possible to find an example of a family $\mathcal{F}$ of $n$ finite distinct non-empty sets, a universe of maximum size $n/4$, with at least $\lfloor \frac{2}{3}{n \choose 2} \rfloor$ unordered couples of sets with at least one element in common between the two sets, and no element belonging to at least $n/2$ set... | https://mathoverflow.net/users/136218 | Existence of a family of sets with some properties | Take a finite projective plane: for the sake of concreteness, the Fano plane $PG(2, 2)$. It has seven points $P$ and seven lines $L$, where each line goes through three points and each pair of lines intersects.
Take a second set $S$ which is disjoint from $P$ and an element $x$ which is not in either. Consider $$\mat... | 0 | https://mathoverflow.net/users/46140 | 431743 | 174,774 |
https://mathoverflow.net/questions/431686 | 0 | Suppose we have a 5 tuple of positive real numbers $(l\_1,l\_2,m\_1,m\_2,m\_3)$, with $m\_i \in (0,\pi)$ for all $i$. Now fix a point $v\_1$ in the hyperbolic plane. Then consider a geodesic of length $l\_1$ starting at $v\_1$. suppose that ends at $v\_2$. At $v\_2$ draw another geodesic of length $l\_2$ which makes an... | https://mathoverflow.net/users/490039 | When a polygonal line become a loop in hyperbolic plane? | Your broken line closes iff the triangles $(v\_1,v\_2,v\_3)$
and $(v\_3,v\_4,v\_5)$ are equal, and $[v\_1,v\_3]$ is their common side. So there are two conditions: one is that $m\_1=m\_3$, by the hyperbolic law of cosines.
To state the second one,
Apply the hyperbolic law of cosines to find the length $x$ of the side... | 1 | https://mathoverflow.net/users/25510 | 431749 | 174,777 |
https://mathoverflow.net/questions/431484 | 5 | Fix $\epsilon>0$.
For all large $N$, does there exist $A\subset [N]:=\{1,\dots,N\}$ such that both $A+A$ and $A^c:=[N]\setminus A$ lack arithmetic progressions of length $N^\epsilon$?
I am aware Rusza used “niveau sets” to create dense subsets of $[N]$ whose sumsets lack progressions of length $\exp(\log^{2/3+o(1)}... | https://mathoverflow.net/users/130484 | Progressions in sumset or complement | Ah, I realized how to construct such $A$.
The idea is to just crudely mimic the construction for the off-diagonal van der Waerden numbers $w(3,k)$.
We fix parameters $D,M,\rho$ which will be optimized later.
We shall sample $M$ points $x\_1,\dots,x\_M$ from the $D$-dimensional torus $\Bbb{T}^D := \Bbb{R}^D/\Bbb{Z... | 2 | https://mathoverflow.net/users/130484 | 431752 | 174,778 |
https://mathoverflow.net/questions/431683 | 12 | This is an obvious continuation of [an MO question](https://mathoverflow.net/q/431642). Let $r,s\in\mathbb N\cup\{\infty\}$ with $r\neq s$, and $M,N$ two connected manifolds of positive dimension (which roots out the trivial case of a single point). I wonder whether the rings $C^r(M)$ and $C^s(N)$ of real-valued contin... | https://mathoverflow.net/users/176381 | Is $C^r(M)$ non-isomorphic to $C^s(N)$ for $r\neq s$ and nontrivial manifolds $M,N$? | If $r\neq s$, then the rings $C\_r(M),C\_s(N)$ cannot be isomorphic nor can they be elementarily equivalent to each other. I claim that for all $r<\infty$, there is a first order formula $\phi$ where $C\_r(M)\models\phi$ if and only if $s=r$.
If $p/q$ is a reduced rational number where $q$ is odd, and $f$ is a real v... | 6 | https://mathoverflow.net/users/22277 | 431757 | 174,780 |
https://mathoverflow.net/questions/431765 | 4 | The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes\_i^n (\det \mathcal{E}\_i)^{⊗ (-1)^i}
\end{equation}
where $\mathcal{E}\_\bullet \to F$ is a locally free resolution of $\mathcal{F}$ (which we can t... | https://mathoverflow.net/users/123448 | Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)? | No.
Working on projective space, consider a composition $$ \mathcal O \to \mathcal O \oplus \mathcal O/\mathcal O(-1) \to \mathcal O $$ where $\mathcal O/\mathcal O(-1)$ is the constant sheaf on a hyperplane, and the maps are just defined to be the identity on $\mathcal O$ and $0$ on $\mathcal O/\mathcal O(-1)$.
Si... | 10 | https://mathoverflow.net/users/18060 | 431767 | 174,783 |
https://mathoverflow.net/questions/431723 | 6 | In Martin-Löf type theory, a weakly Tarski universe is a type $\mathcal{U}$ with a type family $\mathcal{T}(A)$ indexed by terms $A:\mathcal{U}$, which is closed under identity types, dependent products, and dependent sums: there are functions
$$\Sigma\_\mathcal{U}:\prod\_{A:\mathcal{U}} (\mathcal{T}(A) \to \mathcal{U}... | https://mathoverflow.net/users/483446 | Univalence for weakly Tarski universes | Yes, $\mathrm{transport}^\mathcal{T}$ is an equivalence if and only if $\mathrm{idtoequiv}$ is an equivalence. To show this, it is enough to prove that the underlying functions of $\mathrm{transport}^\mathcal{T}(p)$ and $\mathrm{equiv}\_\simeq (\mathrm{idtoequiv}(p))$ are equal for all $p : A =\_\mathcal{U} B$. By path... | 2 | https://mathoverflow.net/users/62782 | 431772 | 174,784 |
https://mathoverflow.net/questions/431776 | 4 | Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions:
$$Tf(x)=\int\_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t}
\int\_{|x-y|^2\leq t}\frac{f(y)}{|x-y|^{\frac{n}{p^{\prime}}}}dy dt,\qquad x\in \mathbb{R}^{n}.$$
I have tried to prove that $... | https://mathoverflow.net/users/116555 | Is this an $L^p-L^{\infty}$ operator? | No this is not true. Take $n=1$, $p=2$ and $$f(y)=\frac{1}{\sqrt {|y|} (|\log|y||)^\alpha}\chi\_{(-1,1)}(y)$$ with $\frac 12 < \alpha <1$. Then $f \in L^2(\mathbb R)$ but the innermost integral diverges at $x=0$ for every $t>0$.
EDIT The same counterexample can be done for general $n$ and $1<p<\infty$ (the case $p=1$... | 5 | https://mathoverflow.net/users/150653 | 431781 | 174,785 |
https://mathoverflow.net/questions/431775 | 3 | Let $\mathcal{O}$ be a Dedekind domain and $K = \mathrm{Frac}(\mathcal{O})$ its field of fractions. Let $E / K$ be an elliptic curve and $\mathcal{E} / \mathcal{O}$ its Neron model and $\mathcal{E}^\circ$ the connected component of the identity (fiberwise).
Then the natural map $H^1\_{\mathrm{fppf}}(\mathcal{O}, \mat... | https://mathoverflow.net/users/154157 | Selmer groups and fppf cohomology | The following paper of Kestutis Cesnavicius answer these and related questions completely: <https://arxiv.org/abs/1301.4724>
| 3 | https://mathoverflow.net/users/110362 | 431791 | 174,789 |
https://mathoverflow.net/questions/431300 | 0 |
>
> Let $ \{a\_k\}\_{i=1}^n $ is a positive sequence. For $ 0<p<\infty $, space $ L^{p,\infty} $ is defined by
> $$
> \left\{f:\|f\|\_{p,\infty}=\inf\left\{C>0:a\_f(\lambda)\leq C/\lambda^p\right\}\right\}
> $$
> where
> $$
> a\_f(\lambda)=|\{x\in\mathbb{R}^n:|f(x)|>\lambda\}|.
> $$
> Then
> $$
> \left\|\sum\_{k=1}^{... | https://mathoverflow.net/users/241460 | How to prove this inequality for the norm $ \|\cdot\|_{1,\infty} $? | For any $ \lambda>0 $, we have
\begin{align}
\left|\left\{x:\left|\sum\_{i=1}^nf\_k\right|>\left(1+\sum\_{i=1}^na\_k\right)\lambda\right\}\right|&\leq\left|\left\{x:\sum\_{i=1}^n|f\_k|>\left(1+\sum\_{i=1}^na\_k\right)\lambda\right\}\right|\\
&\leq\left|\left\{x:\sum\_{i=1}^n(|f\_k|-a\_k\lambda)>\lambda\right\}\right|\\... | 1 | https://mathoverflow.net/users/241460 | 431798 | 174,790 |
https://mathoverflow.net/questions/431777 | 5 | I was reading [this post about the Bell Numbers](https://math.stackexchange.com/a/1527678) where users Lucian and Vladimir Reshetnikov give us Dobiński's formula for the Bell numbers
$$ B(x) = \frac{1}{e} \sum\_{k=1}^{\infty} \frac{k^x}{k!}. $$
Now I was trying to reason about this function on the complex plane. It... | https://mathoverflow.net/users/46536 | Has anyone characterized the zeroes of the Bell numbers? | The function $B(z)$ is an example of an almost periodic function. The zeroes of an almost periodic function that is holomorphic on some strip are also almost periodic, so such a function either has no zeroes at all or infinitely many zeros.
Let $f(z)=\sum\_{k=0}^{\infty}a\_ke^{b\_kz}$ where each $b\_k$ is real and ea... | 3 | https://mathoverflow.net/users/22277 | 431803 | 174,792 |
https://mathoverflow.net/questions/431802 | 3 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$I'm a PhD student in physics working in the broad area of photonic quantum computing. My current project looks at the equivalence of any two $n$-photon $m$-mode Fock states under linear optical evolution.
These two Fock states can be r... | https://mathoverflow.net/users/490056 | Generators of polynomial invariant ring of compact Lie groups | The method/result you are looking for is commonly known under the name "unitary trick" (of Hurwitz and Weyl), - this keyword should bring you a great deal of accessible explanations of all possible levels (since your post does not give a clear idea of your level, it is better if you look through several different sourc... | 3 | https://mathoverflow.net/users/1306 | 431805 | 174,793 |
https://mathoverflow.net/questions/431769 | 7 | I am only a beginner in the field of type theory, and I'm wondering if the community could point me out a few open problems in the field. I have a good background in logic, in particular, proof theory and I have a reasonable knowledge on category theory. While I am interested in homotopy type theory, I'd like to receiv... | https://mathoverflow.net/users/492345 | Open problems in type theory | Thierry Coquand has a list of open problems, based on Vladimir Voevodsky's research on Dependent Type Theory. Scroll to the end of this [presentation.](https://web.archive.org/web/20190518164032/https://eutypes.cs.ru.nl/eutypes_pmwiki/uploads/Meetings/Coquand.pdf)
And I presume you know the HoTT open problem list at ... | 6 | https://mathoverflow.net/users/11260 | 431807 | 174,794 |
https://mathoverflow.net/questions/431804 | 0 | The ratio between the number of unordered couples of sets, with empty intersection between the two sets, and the total number of unordered couples of sets, for a powerset on $n$ elements without the empty set, $\mathcal{P}([n]) \setminus \emptyset$, is:
$$\frac{{n+1 \brace 3}}{{2^n-1 \choose 2}}=\frac{(1 + 3^n - 2^{n... | https://mathoverflow.net/users/136218 | Number of couples of sets with empty intersection in a separating union-closed family of sets | The numerator grows faster than the denominator, so we can do better by making a minimal extension of a previous powerset: $2^{[k]} \cup \{[m] : k+1 \le m \le n \} \setminus \emptyset$ gives $$\frac{(1 + 3^k - 2^{k+1})}{(2^k+n-k-1)(2^n+n-k-2)}$$ and for $n \ge 5$ the optimal value of $k$ is less than $n$ and grows loga... | 1 | https://mathoverflow.net/users/46140 | 431809 | 174,795 |
https://mathoverflow.net/questions/431748 | 8 | Here are some definitions:
A space is *homotopy finite* if it is homotopy equivalent to a finite CW complex.
A space *finitely dominated* if it is a retract of a homotopy finite space.
A space $X$ is a *Poincaré duality space* of dimension $d$ if there exists a pair
$$
({\mathscr L},[X])
$$
consisting of a rank one... | https://mathoverflow.net/users/8032 | Finite domination and Poincaré duality spaces | Corollary 5.4.2 of Wall's article `Poincaré complexes I', *Ann. Math.* **86** (1967) 213-245 gives examples of 4-dimensional Poincaré complexes $X$ with fundamental group of prime order $p\geq 23$ for which the Wall finiteness obstruction $\chi(X)$ is non-zero.
Incidentally, Theorem 1.3 of the same article is very cl... | 9 | https://mathoverflow.net/users/124004 | 431812 | 174,797 |
https://mathoverflow.net/questions/431794 | 4 | Let $u$ be a solution of the heat equation
$$u\_t - \Delta u = 0, \qquad t >0, \ x \in \mathbb T^d$$
and $v$ be a solution of the bi-harmonic heat equation
$$v\_t +\Delta^2 v = 0, \qquad t >0, \ x \in \mathbb T^d$$
with the same initial data $f$.
Is it true that, for every fixed time $T >0$,
$$\|v(T,\cdot)\|\_{L^2} \le... | https://mathoverflow.net/users/110835 | $L^2$ norm for solutions of evolution equations driven by different elliptic operators | Not necessarily. I mean, it depends upon the torus you consider. Notice that in the case of the standard one ${\mathbb T}^d={\mathbb R}^d/{\mathbb Z}^d$, the answer is positive. But if you torus is ${\mathbb R}^d/a{\mathbb Z}^d$, then it is positive if $a\le2\pi$ and negative otherwise. The reason is that both semi-gro... | 4 | https://mathoverflow.net/users/8799 | 431815 | 174,798 |
https://mathoverflow.net/questions/431800 | 3 | For the elliptic equation with non-divergence form
$$
\sum\_{i,j=1}^na\_{ij}(x)\partial\_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1),
$$
where $ \{a\_{ij}(x)\} $ is a matrix-valued function such that for any $ \xi\in\mathbb{R}^n $ and $ x\in B(0,1) $
$$
\mu|\xi|^2\leq\sum\_{i,j=1}^na\_{... | https://mathoverflow.net/users/241460 | Schauder estimates with boundary conditions | The result is true. Let $L=\sum\_{ij}a\_{ij}D\_{ij}$ and consider $$L^{-1}: C^{2+\alpha}(\partial \Omega) \mapsto C^{2+\alpha}(\bar \Omega)$$ with $L^{-1}f=u$ is $Lu=0$ and $u=f$ at the boundary.
$L^{-1}$ is bounded from $C^{2+\alpha}(\partial \Omega)$ to $ C^{2+\alpha}(\bar \Omega)$ by the Schauder theory and from ... | 2 | https://mathoverflow.net/users/150653 | 431835 | 174,804 |
https://mathoverflow.net/questions/431852 | 4 | Lurie's $\infty$-categorical Dold-Kan Correspondence relates simplicial objects and sequential diagrams in a stable $\infty$-category. Is there any reference for an equivalence to a category of homotopy chain complexes? That this equivalence holds is well-known, and I feel comfortable with the argument, but I'd like to... | https://mathoverflow.net/users/28033 | Reference for the equivalence between chain complexes and sequential diagrams in a stable $\infty$-category | I believe <https://arxiv.org/abs/2109.01017> does what you want! The description of coherent chain complexes used there is a bit different than what you suggest, but they look equivalent at first glance.
| 4 | https://mathoverflow.net/users/39747 | 431856 | 174,812 |
https://mathoverflow.net/questions/431842 | 6 | Let $(M, \omega)$ be a holomorphic symplectic manifold of (complex) dimension $2n$. Let $x$ be a point in $M$. My understanding from the discussion and answers to [this MO question](https://mathoverflow.net/q/322447/184) is that there exists a neighborhood $U \subseteq M$ of $x \in M$, and a neighborhood $V \subseteq T... | https://mathoverflow.net/users/184 | Are holomorphic Lagrangians locally graphs? | The answer is 'yes'. Specifically, the holomorphic version of the Darboux-Weinstein theorem holds, just as it does in the smooth category. In particular, if $L\subset M$ is a holomorphic Lagrangian submanifold where $M$ is a holomorphic symplectic complex manifold and $x\in L$ is specified, then there is an open $x$-ne... | 9 | https://mathoverflow.net/users/13972 | 431857 | 174,813 |
https://mathoverflow.net/questions/431859 | 2 | Let $\wp(u) = \frac{1}{u^2} + \sum\limits\_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ be a Weierstrass pe function.
My question is, how can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$
If $αL=L$, the answer is easy, we can replace sum up over $L$ to sum up over $αL$, $\wp... | https://mathoverflow.net/users/144623 | How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$ | $\alpha L$ is of finite index in $L$ so you can write $L = \cup\_{i=1}^n (\alpha L + \lambda\_i)$ for some $\lambda\_i, n$.
If you rewrite the sum defining $\wp(\alpha u)$ as a sum over $\cup\_{i=1}^n (\alpha L + \lambda\_i)$ and rearrange terms (without worrying too much about convergence) you should get something l... | 3 | https://mathoverflow.net/users/2290 | 431879 | 174,819 |
https://mathoverflow.net/questions/431867 | 1 | I'm reading Escobar's The Yamabe Problem On Manifolds With Boundary.
He says
>
> Let $(y\_{1},\cdots,y\_{n})$ be normal coordinates around $0\in \partial M$, such that $\eta(0)=-\frac{\partial}{\partial y\_{n}}$, and second fundamental form of $\partial M$ at 0 has a diagonal form.
>
>
>
Here $M$ is a Rieman... | https://mathoverflow.net/users/148247 | Local geometry of nonumbilic points | You can—at least locally—find a normal coordinate system adapted to any submanifold $N \subset M$. (You can extend $M$ past its boundary to have $N = \partial M$ lie in the interior, but really that is not necessary here.)
Now suppose you have normal coordinates $(y^1,\dots,y^n)$ near a point $0 \in \partial M$, with... | 1 | https://mathoverflow.net/users/103792 | 431882 | 174,821 |
https://mathoverflow.net/questions/431883 | 2 | $\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Seg{Seg}$I apologize in advance for my naïve understanding of real algebraic geometry. I define a *real projective variety* to be a subset of $\mathbb{P}(\real^n)$ that is the ... | https://mathoverflow.net/users/150898 | Is the Segre embedding of two real varieties a real variety? | $\mathrm{Seg}(X\times Y)$ is a real projective variety since the full Segre map is an isomorphism of real algebraic varieties onto its image.
As for your second question, I think the answer is "no". If you take $X=Y=\mathbf{P}(\mathbf R^2)$, the composition $\Pi\circ\mathrm{Seg}$ is nothing but a quotient map for the... | 4 | https://mathoverflow.net/users/85592 | 431889 | 174,822 |
https://mathoverflow.net/questions/431847 | 2 | I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert or Gudermann-like functions.
| https://mathoverflow.net/users/492387 | Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ | $$y=e^{- \epsilon x} \sinh (x)=\sum\_{n=0}^\infty \frac{(1-\epsilon )^n+(-1)^{n+1} (1+\epsilon )^n}{2 n!}\,x^n$$ Now use [series reversion](https://mathworld.wolfram.com/SeriesReversion.html) using the explicit formula for the $n^{\text{th}}$ term as given by Morse and Feshbach.
This will give
$$x=y+\sum\_{n=2}^\inft... | 2 | https://mathoverflow.net/users/42185 | 431892 | 174,824 |
https://mathoverflow.net/questions/431888 | 4 | **Context**
Given a finitary monad $T:\operatorname{gSet}\_n\to\operatorname{gSet}\_n$ we can define categories $\operatorname{Comp}\_k^T$ of $k$-computads for $T$, for any $k=0,\cdots,n+1$. This is nicely explained in Schommer-Pries' [thesis](https://arxiv.org/abs/1112.1000), for example and the original source is t... | https://mathoverflow.net/users/14078 | Is the category of computads for a finitary monad on $n$-globular sets cocomplete? | The answer to $1$ and $2$ are both yes. I don't know if this appears in the literature. The argument you give seems reasonable - I don't completely follow your notation but the general idea is that in the "inductive" definition of computads, you can show that colimits of "k-computads" are computed by taking the colimit... | 6 | https://mathoverflow.net/users/22131 | 431904 | 174,826 |
https://mathoverflow.net/questions/431731 | 1 | Let $M$ be some non-well-founded model of $\sf ZF$, can we have a sequence $(S\_n)\_{n \in \mathbb N}$ of nonempty sets in $M$, where each $S\_n \subset \mathcal P(S\_{n+1})$; and such that there exists a sequence of bijective functions $(f\_n)\_{n \in \mathbb N}: S\_{n+1} \to S\_n$, having : $f\_n(x) = f\_{n+1}[x]$?
... | https://mathoverflow.net/users/95347 | Can we have such an infinite descending sequence of functions with prior ones inside their successors? | (Note that I am assuming the axiom of choice in the metatheory. If you want to ask if this holds without choice, clarify and add the appropriate tag. I suspect that this particular compactness instance should be true even without choice, but I am not an expert, nor did I think much about it.)
This is true by a standa... | 3 | https://mathoverflow.net/users/54415 | 431911 | 174,828 |
https://mathoverflow.net/questions/431900 | 6 | Let $\xi$ be a random variable valued in the space of Schwartz distributions $\mathcal{S}'(\mathbb{R}^d)$.
For any open set $R\subset\mathbb{R}^d$ let $\Sigma(R)$ be the $\sigma$-algebra generated by $\{\xi(f)\,|\,f\in\mathcal{S}(\mathbb{R}^d), \,\textrm{supp}\,f\subset R\}$.
For any $\lambda\in(0,\infty)$ the rand... | https://mathoverflow.net/users/47256 | Abstract characterization of white noise | These assumptions are not sufficient. Take $d=1$ and for simplicity let us work with the antiderivative of $\xi$. You are then asking if a process with independent increments and Brownian scaling is necessarily a Brownian motion. Any $\alpha$-stable Levy process $L$ has independent increments and scaling $(L\_t)\_{t\ge... | 6 | https://mathoverflow.net/users/100941 | 431914 | 174,829 |
https://mathoverflow.net/questions/431795 | 0 | While reading a preprint [Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process](https://arxiv.org/abs/1903.04522) I came across the following SDE in Section 3:
$$d X\_t=d B\_t+\nabla \log P\_{1-t} f\left(X\_t\right) d t$$
where $B\_t$ is the Brownian motion and $P\_t$ den... | https://mathoverflow.net/users/68232 | Change of measure formula for the Föllmer process | One can show it using the Feynman-Kac formula. In particular, from Feynman-Kac, we know that $h(x, t) := \mathbb{E}[u(X\_1) \mid \mathcal{F}\_t ]$ solves the following PDE:
$ \frac{\partial}{\partial t} h(x, t) + \nabla \log P\_{1-t} f(X\_t)^T \nabla\_x h(x, t) + \frac{1}{2} \Delta\_x h(x, t) = 0.$
with the termina... | 2 | https://mathoverflow.net/users/116327 | 431925 | 174,832 |
https://mathoverflow.net/questions/431814 | 2 | We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$?
$$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$
Here $\{x\}$ denotes the fractional part of $x$, $\{x\} = x - [x]$.
**Background**. I encountered this problem when I w... | https://mathoverflow.net/users/492366 | Almost Diophantine approximation | Such $b$ exists for every real $a$.
Define an increasiung sequence of positive integers $n\_1,n\_2,\dots$ as follows. Picj $n\_1$ so that $\{an\_1\}<1/2$. If $n\_i$ has been already defined, choose $n\_{i+1}>n\_i$ such that
$$
\{an\_{i+1}\}\in\left[\{an\_i\},\{an\_i\}+\frac1{2^{n\_i+1}}\right];
$$
such $n\_{i+1}$ cl... | 1 | https://mathoverflow.net/users/17581 | 431933 | 174,834 |
https://mathoverflow.net/questions/431936 | 7 | Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am wondering what is usually done in that respect in mathematics. To formulate a question (answers to any of the three variant... | https://mathoverflow.net/users/40120 | Mathematics of sustainable development and energy sobriety in the classroom | My former colleague David Mond at Warwick [has developed some materials on this issue](https://homepages.warwick.ac.uk/~masbm/climate.html). He has some talks (at school and u/g level) on Climate change and game theory. He also lists some links there, to [a page by John Baez](https://math.ucr.edu/home/baez/what/hong_ko... | 13 | https://mathoverflow.net/users/6107 | 431938 | 174,835 |
https://mathoverflow.net/questions/431924 | 1 | Let $G$ be a finite group, $p$ a prime number, and $k$ an algebraically closed field of characteristic $p$. Then we can consider the cohomological variety of $G$, namely the maximal spectrum $V\_G$ of the graded algebra $H^\bullet(G,k)=\operatorname{Ext}^\bullet\_{k[G]}(k,k)$. By the Quillen stratification, this space ... | https://mathoverflow.net/users/97652 | Cohomological variety in case that Sylow subgroup is elementary abelian | Yes. There is a stronger result too, which predates Quillen's theorem: if $G$ is a finite group whose Sylow $p$-subgroup $P$ is abelian, then the restriction map $H^\*(G;\mathbb{F}\_p)\rightarrow H^\*(P;\mathbb{F}\_p)$ has image equal to the fixed points for the action of the normalizer $N\_G(P)$ on $H^\*(P;\mathbb{F}\... | 3 | https://mathoverflow.net/users/124004 | 431939 | 174,836 |
https://mathoverflow.net/questions/431901 | 4 | Let $D \subset \mathbf{R}^2$ be the unit disc, and $L > 0$. Let $u: D \times (-L,L) \to \mathbf{R}$ satisfy
\begin{equation}
\begin{cases}
\Delta u = 0 \quad \text{ on $D \times (-L,L)$ } \\
\frac{\partial u}{\partial z} = 0 \quad \text{ on $D \times \{ -L , L \}$}.
\end{cases}
\end{equation}
**Question.** Is it poss... | https://mathoverflow.net/users/103792 | Is there a harmonic function with just one singular point? | **Yes**, this is possible. An explicit example is
$$u(x, y, z) = 1 - I\_0\left(\sqrt{x^2 + y^2}\right) \, \cos z$$
when $L = \pi$, and $u\big(\frac{\pi x}{L}, \frac{\pi x}{L}, \frac{\pi x}{L}\big)$ for a general $L$. Here $I\_0$ is the Bessel $I$ function.
| 7 | https://mathoverflow.net/users/108637 | 431940 | 174,837 |
https://mathoverflow.net/questions/431793 | 7 | Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map
$$
J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty
$$
returning the infinite jet at 0, which is a surjection by [Borel's lemma](https://ncatlab.org/nlab/show/Borel%27s+theorem). Here $\math... | https://mathoverflow.net/users/4177 | Is the Borel lemma projection a smooth principal bundle? | **No**, there is not even any $C^1$ (in the Michal−Bastiani, i.e. Keller $C\_c$ sence) map $\mathbb R^\infty=G\sqsupseteq{\rm dom}\,f\to E=C^\infty(\mathbb R)=C^\infty(\mathbb R,\mathbb R)$ with ${\rm dom}\,f$ a zero neighbourhood and $J\circ f={\rm id}$ on ${\rm dom}\,f$. The argument goes as follows. Supposing there ... | 6 | https://mathoverflow.net/users/12643 | 431944 | 174,839 |
https://mathoverflow.net/questions/431958 | 3 | Let $f: [0, 1] \to \mathbb R$ be a measurable function. A function $g: [0, 1] \to \mathbb R$ is said to be a *condensation limit* of $f$ if $g$ is continuous and agrees with $f$ on a dense subset of $[0, 1]$.
Let $k \geq 1$ be an integer, and $f: [0, 1] \to \mathbb R$ be a measurable function whose graph is dense in ... | https://mathoverflow.net/users/173490 | Restriction to dense subset of functions whose graph is dense | **Yes.** This is true.
Proposition: Let $A\subseteq[0,1]\times\mathbb{R}$ be dense. Then for each $k\geq 0$ and $\epsilon>0$ and $g\in C^k([0,1])$, there is some $C^k$ function $f:[0,1]\rightarrow\mathbb{R}$ where $\{x:(x,f(x))\in A\}$ is dense in $[0,1]$ and where $\|f-g\|<\epsilon$.
Proof: We shall construct a se... | 4 | https://mathoverflow.net/users/22277 | 431961 | 174,844 |
https://mathoverflow.net/questions/431970 | 1 | Let $d$ be a large positive integer and fix $r \ge 0$. Set $S := B\_2^n \cap [-r,r]^d$, where $B\_2^d$ is the euclidean unit-ball in $\mathbb R^d$. Finally, let $\omega(S)$ be the *Gaussian width* of $S$, defined by
$$
\omega(S) := \mathbb E \sup\_{x \in S} x^\top z,
$$
where the expectation is over $z \sim N(0,I\_... | https://mathoverflow.net/users/78539 | Gaussian width of intersection of cube and ball in high-dimensional euclidean space | The answer is, up to a constant factor $\omega(S) = \Theta(\min(\sqrt{d}, rd))$.
To see the upper bound, we can use the fact that if $S \subset Q$, then $\omega(S) \leq \omega(Q)$, therefore
$$
\omega(B\_2^n \cap r B\_\infty^n) \leq \min(\omega(B\_2^n), r \omega(B\_\infty^n)) = \min( \mathbb{E}\_{z \sim \mathcal{N}(0... | 2 | https://mathoverflow.net/users/468679 | 431975 | 174,845 |
https://mathoverflow.net/questions/431953 | 6 | Let $u\_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly harmonic (via mean value property). Suppose we denote by $Z\_{f}$ the set of zeros of a function $f$.
My question is, ... | https://mathoverflow.net/users/492517 | Limit of zero sets of harmonic functions | **Yes.**
With an appropriate topology, the function mapping a non-zero harmonic function to its zero set is continuous. The result actually applies to open mappings from a locally compact locally connected space $X$ to $\mathbb{R}$, and harmonic functions are just a particular example of such open mappings.
If $X$ ... | 4 | https://mathoverflow.net/users/22277 | 431979 | 174,847 |
https://mathoverflow.net/questions/431973 | 0 | $H$ is an $n\times m$ matrix with non-negative coefficients and $n < m$. $H'$ is the transpose of $H$.
Are the following statements true?
1. If $\det(HH’) > 0$, the rows of $H$ define the edges of an $n$-dimensional conic polyhedron. No row of $H$ is a linear combination of the other rows of $H$ using non-negative ... | https://mathoverflow.net/users/170274 | $\det(HH’) = 0$ for nonnegative $H$ | 1. Yes, because $H$ must be of rank $n$ (for $n=\operatorname{rk}(HH^\top)\leq\operatorname{rk}(H)$), so none of its rows is a linear combination of the others.
2. No, the matrix $$H = \begin{pmatrix} 1 & 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \end{pmatrix}$$ provides a counter-exa... | 0 | https://mathoverflow.net/users/5018 | 431981 | 174,849 |
https://mathoverflow.net/questions/431982 | 2 | *Question:* If $M$ is a compact smooth finite-dimensional manifold with boundary, is the inclusion of a closed subspace $A \subseteq M$ a cofibration? (I'm specifically interested in the case when $A$ is a smooth submanifold with boundary).
Does the following sketch proof work? *Sketch:* $M$ is homotopy equivalent to... | https://mathoverflow.net/users/83360 | If $M$ is a compact smooth finite-dimensional manifold with boundary, is the inclusion of a closed subspace $A \subseteq M$ a cofibration? | If the closed subset is locally compact and locally contractible then yes, the inclusion is a cofibration. This is surprisingly not very well known, but it follows from the classification of finite dimensional ANR's and the following fact from the answer of Tyrone [here](https://mathoverflow.net/questions/374160/is-the... | 6 | https://mathoverflow.net/users/134512 | 431988 | 174,852 |
https://mathoverflow.net/questions/46239 | 0 | I'm unable to access the following paper: J. Dénes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Inst. Hung. Acad. Sci. 4 (1959) 63-70. Can anyone help me?
| https://mathoverflow.net/users/3356 | Request for a copy of a paper of J. Dénes on permutation factorisations | The *PUBLICATION OF THE MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES* is available online.
<http://real-j.mtak.hu/view/journal/A_Magyar_Tudom=E1nyos_Akad=E9mia_Matematikai_Kutat=F3_Int=E9zet=E9nek_k=F6zlem=E9nyei.html>
You can download this specific paper here: <http://real-j.mtak.hu/510/>
| 3 | https://mathoverflow.net/users/37580 | 432008 | 174,856 |
https://mathoverflow.net/questions/431999 | 2 | Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set
$$
\alpha := \sup\_{(x,y) \in C} ax + b y.
$$
>
> **Question.** In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?
>
>
>
Some special cases
------------... | https://mathoverflow.net/users/78539 | Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$ | We have
$$\alpha = \left\{\begin{array}{ll}
\sqrt{a^2 + b^2} & \mathrm{if} ~ (a^2 + b^2)R^2 \ge a^2 + cb^2, \\[6pt]
R\sqrt{a^2 + b^2/c}& \mathrm{if} ~ a^2c^2 + b^2c \ge (a^2c^2 + b^2)R^2, \\[6pt]
|a|\sqrt{\frac{c-R^2}{c-1}}
+ |b|\sqrt{\frac{R^2 - 1}{c-1}} & \mathrm{otherwise}.
\end{array}
\right.$$
**Proof**:
I... | 2 | https://mathoverflow.net/users/141801 | 432010 | 174,857 |
https://mathoverflow.net/questions/432000 | 0 | The following interesting problem was asked at [Aops](https://artofproblemsolving.com/community/u738252h2922192p26122750) and I wonder if it was based on some research paper:
>
> Let $K$ be a convex body in $\mathbb R^2$, such that the diameter of $K$ is less than $\sqrt2$.
>
> Prove that there is a lattice-poin... | https://mathoverflow.net/users/70464 | Lattice-point-free body diameter | For higher dimension, this does not hold, see the answer by Sergei Ivanov [here](https://mathoverflow.net/q/42774/4312).
For dimension 2, in the closed set $K$ without lattice-point free translate (in other words, such that $K$ has a point in every translate $\mathbb{Z}^2+c$ of the lattice) we may even find two point... | 2 | https://mathoverflow.net/users/4312 | 432029 | 174,863 |
https://mathoverflow.net/questions/432026 | 8 | In an old paper of Glaisher, I find the following formulas:
$$\dfrac{\sin(\pi x)}{\pi x}=1-\dfrac{x^2}{1^2}-\dfrac{x^2(1^2-x^2)}{(1.2)^2}-\dfrac{x^2(1^2-x^2)(2^2-x^2)}{(1.2.3)^2}-\cdots$$
$$\cos(\pi x/2)=1-x^2-\dfrac{x^2(1^2-x^2)}{(1.3)^2}-\dfrac{x^2(1^2-x^2)(3^2-x^2)}{(1.3.5)^2}-\cdots$$
These are trivial since the $n... | https://mathoverflow.net/users/81776 | Trivial (?) product/series expansions for sine and cosine | You can derive at least some of these formulas from expansions of $\sin x\theta$ and $\cos x\theta$ as Taylor series in $\sin\theta$,
$$\sin x\theta=x\sin \theta-\frac{x(x^2-1)}{3!}\sin^3\theta+\frac{x(x^2-1)(x^2-3^2)}{5!}\sin^5\theta+-\cdots$$
$$\cos x\theta=1-\frac{x^2}{2!}\sin^2 \theta+\frac{x^2(x^2-2^2)}{4!}\sin^4\... | 10 | https://mathoverflow.net/users/11260 | 432030 | 174,864 |
https://mathoverflow.net/questions/431833 | 1 | This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete spectra for operators that also have continuous spectrum. We can assume perturbation is compact and hence continuous s... | https://mathoverflow.net/users/30684 | Spectral perturbation theory of discrete spectra in presence of continuous spectrum | For this purpose ("beginning grad student") it would make sense to focus on the case that the discrete eigenvalues appear in an energy range that does not overlap with the continuous spectrum (say, $E>E\_0$). One can then simply use the [formulas](https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)) f... | 1 | https://mathoverflow.net/users/11260 | 432049 | 174,868 |
https://mathoverflow.net/questions/432053 | 2 | I am relatively new to the world of braids/knots so really sorry if this question is simple. However, I am not able to find if there is any theorem/procedure that determines if a closed braid, given its representation in the Artin braid group, is a link or an unlink. Or, any theorem that says this cannot be determined?... | https://mathoverflow.net/users/492606 | Determine if a closed braid is a link/unlink | A braid gives a braid closure. This can be drawn as a knot (or link) diagram. There are then various approaches to solve the unknot (or unlink) recognition problem given a diagram. This begins with work of Haken, and then work of Hass, Lagarias, and Pippenger, and then work of Lackenby.
There is also another line of ... | 4 | https://mathoverflow.net/users/1650 | 432062 | 174,872 |
https://mathoverflow.net/questions/432071 | 4 | $\newcommand\Set{\mathbf{Set}}\newcommand\Ob{\mathbf{Ob}}\newcommand\Hom{\mathbf{Hom}}$Work in a foundation that admits a countable hierarchy of notions of ‘set’, and say that a category is *$n$-small* iff its object and arrow collections are $n$-sets. Denote the category of $n$-sets by $\Set\_n$.
For each $n$-small ... | https://mathoverflow.net/users/92164 | There are no abstract categories | The argument you give in your original post is essentially the proof that *every small category is concrete.*
So if you work in a setting that have enough universes/inaccessible cardinal/notion of smallness so that it is reasonable to only consider "small" categories, then of course all categories are concrete.
How... | 20 | https://mathoverflow.net/users/22131 | 432073 | 174,876 |
https://mathoverflow.net/questions/431974 | 4 | There is a descent spectral sequence computing $\pi\_\*L\_{K(n)}S^0$ with $E\_2$-term
$$E\_2^{s,t}\cong H^s\_c(\mathbb{G}\_n,(E\_n)\_t)$$
It is mentioned in [Barthel-Beaudry](https://arxiv.org/pdf/1901.09004.pdf) (in the description of Figure 3.30) that if $p>2$ and $2(p-1)>n^2$, then there are only nonzero entries whe... | https://mathoverflow.net/users/492531 | On the sparsity of the descent spectral sequence computing homotopy groups of the K(n)-local sphere | This sparsity holds even without the assumptions that $p>2$ and $2(p-1)>n^2$. (Those are used to get a horizontal vanishing line). As in the comments, it comes down to the fact that you can use $\widehat{E(n)}\simeq L\_{K(n)}E(n)$ in place of $E\_n$, and $\widehat{E(n)}$ has homotopy groups concentrated in degrees that... | 10 | https://mathoverflow.net/users/111541 | 432074 | 174,877 |
https://mathoverflow.net/questions/432088 | 10 |
>
> What are your favorite models of the KL-axioms?
>
>
>
The motivation is having some basic models to understand the axiom scheme as presented e.g. in [Synthetic Geometry of Manifolds](https://users-math.au.dk/kock/SGM-final.pdf) by Kock.
In that text he references his other text, [Synthetic Differential Geo... | https://mathoverflow.net/users/92164 | Models of the Kock-Lawvere axioms | Here is what I once worked out; I think I remember seeing this published somewhere too but I forget where.
*Most of the interesting Kock-Lawvere algebra can be captured in a ring*:
$$R\_1= \mathbb{R}[t\_1,t\_2,\dots]/(t\_1^2,t\_2^2,\dots)$$
For example, this satisfies the principle that only zero can annihilate a... | 8 | https://mathoverflow.net/users/nan | 432100 | 174,886 |
https://mathoverflow.net/questions/432106 | 16 | As the title says, for every Lebesgue integrable function $f:\mathbb{R}\to\mathbb{R}$ is there a Riemann integrable function $g:\mathbb{R}\to\mathbb{R}$ such that $f=g$ almost everywhere?
For example, $\chi\_{\mathbb{Q}\cap [0, 1]}$ is a well known example of lebesgue integrable function that is not riemann integrabl... | https://mathoverflow.net/users/481551 | Equivalence between Lebesgue integrable and Riemann integrable functions | Let $A$ be a measurable subset of $[0,1]$ such that both it and its complement have positive measure in every open interval in $[0,1]$ (see [here](https://math.stackexchange.com/questions/57317/construction-of-a-borel-set-with-positive-but-not-full-measure-in-each-interval) for example). Its characteristic function is ... | 29 | https://mathoverflow.net/users/23141 | 432110 | 174,888 |
https://mathoverflow.net/questions/432104 | 3 | Let
$$ n\ := \sum\_{k=0}^L\ a\_k\cdot10^k $$
where $\ L\ $ is a non-negative integer, and $\ a\_k\ $ are decimal digits, and $\ a\_L>0.$
Let $\ n\ $ be called ***mixed*** $\ \Leftarrow:\Rightarrow$
$$ n\ =\ \sum\_{k=0}^L a\_k\cdot\prod\_{k=0}^L a\_k $$
For instance, $\,\ n:=1\,\ $ and $\,\ n:=144\ $ are mixed.
... | https://mathoverflow.net/users/110389 | Diophantine entertainment -- mixed natural numbers | <https://oeis.org/A038369> "Numbers $k$ such that $k$ = (product of digits of $k$) times (sum of digits of $k$)."
"$0, 1, 135, 144$. The list is complete. Proof: One shows that the number of digits is at most $84$ and then it is only necessary to consider numbers of the forms $2^i3^j7^k$ and $3^i5^j7^k$. - David W. W... | 10 | https://mathoverflow.net/users/3684 | 432111 | 174,889 |
https://mathoverflow.net/questions/432091 | 11 | The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is
$$
x^TAx+bx+c=0,
$$
where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with integer entries, $b$ vector, $c$ integer. We can try simplify it by linear substitution $x=Hy$, where $y$ are new variables ... | https://mathoverflow.net/users/89064 | How to describe all integer solutions to $x^2+y^2=3z^2+1$? | Ok, I now was able to solve the equation myself.
If $(x,y,z)$ is any solution, then $(y,x,z)$, $(x,-y,z)$, $(x,y,-z)$, and $(x,3z-2y,2z-y)$ are also solutions. To check the last one, observe that
$$
x^2 + (3z-2y)^2 - 3(2z-y)^2 = x^2 + y^2 - 3z^2 = 1.
$$
All these transformations are invertible: if we apply any of the... | 14 | https://mathoverflow.net/users/89064 | 432118 | 174,890 |
https://mathoverflow.net/questions/432114 | 5 |
>
> Suppose that $\{X\_{ij}\}\_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X\_{11})=0$ and $\mathrm{Var}(X\_{11})=1$, does the following convergence hold:
> $$
> \max\_{1\leqslant j\leqslant n}\biggl\{\frac{1}{n^2}\sum\_{1\leqslant i\neq i'\leqslant n}(X\_{ij}X\_{i'j})\biggr\} \to 0 \qquad ... | https://mathoverflow.net/users/481055 | Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$ | In the Math Stack Exchange post, I gave a proof based on Lemma 2 in Bai and Yin (1993). I will give an alternative proof.
Expressing $\sum\_{1\leqslant i\neq i'\leqslant n}X\_{i,j}X\_{i',j}$ as
$\left(\sum\_{i=1}^n X\_{i,j}\right)^2-\sum\_{i=1}^nX\_{i,j}^2$ and considering dyadic numbers, we are reduced to show that
... | 3 | https://mathoverflow.net/users/17118 | 432135 | 174,895 |
https://mathoverflow.net/questions/432109 | 1 | Suppose that $f$ is a continuous, nonconstant function on $[0,1]$. Fix some $0<a<1$. Is it possible to establish the following inequality
$$ |f(x+h)-f(x)| \leq C \left[ |h|^a + |2f(x)-f(x+h)-f(x-h)| \right] ~~~~~\forall x,x-h,x+h\in I, $$
where $C$ only depends on $f$ and $a$?
$|h|^a$ in the RHS of the inequality is ... | https://mathoverflow.net/users/152618 | Can the second-order difference control the first-order difference for nowhere differentiable functions? | No. E.g., take any $b\in(0,1)$ such that $1-b<a$. Take any strictly decreasing sequence $(x\_n)$ in $[0,1]$ converging to $0$.
Then there clearly exists a continuous function $f$ on $[0,1]$ such that $f(0)=0$ and for all natural $k$ and all $x\in[x\_{2k},x\_{2k-1}]$ we have
$$f(x)=2h\_{2k}^{-b}(x-x\_{2k}),$$
where $h... | 2 | https://mathoverflow.net/users/36721 | 432144 | 174,898 |
https://mathoverflow.net/questions/428693 | 6 | Let $V$ be the real vector space of finitely supported functions $f: \Omega\to \mathbf{R}$ such that $\sum\_\omega f(\omega)=0$, where $\Omega$ is a given uncountable set.
Endow $V$ with the weak topology $\sigma:=\sigma(V,\mathbf{R}^\Omega)$, so that, more explicitly, a net $(f\_i)$ in $V$ is $\sigma$-convergent to ... | https://mathoverflow.net/users/32898 | Weakly sequentially closed convex cone which is not weakly closed | The answer is **negative**, for both questions.
Let $\{Z\_1,Z\_2\}$ be a partition of $\Omega$ such that $|Z\_1|=|Z\_2|$.
Let also $h: Z\_1\to Z\_2$ be a bijection and fix $a \in Z\_1$.
For each nonempty finite subset $B\subseteq Z\_1$, define
$$
\tilde{e}(B):=\frac{1}{|B|^2}\sum\_{b \in B}(e\_b-e\_{h(b)}),
$$
where ... | 0 | https://mathoverflow.net/users/32898 | 432162 | 174,902 |
https://mathoverflow.net/questions/432080 | 6 | Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H\_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H\_v$ is the boundary of the smallest closed half-space with outward normal $v$ that contains $C$). Let $U\subseteq S^{n-1}$ be the set of directions $v$ such that $... | https://mathoverflow.net/users/26809 | For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point? | The support function of $C$, restricted to the unit sphere, is differentiable exactly at directions such that (the relevant) hyperplane normal to that direction has a single contact point with $C$.
Because $C$ is bounded, its support function is Lipschitz. Rademacher's theorem then says it is differentiable almost ev... | 5 | https://mathoverflow.net/users/112954 | 432165 | 174,903 |
https://mathoverflow.net/questions/430285 | 2 | Some time ago, I asked a [question](https://mathoverflow.net/questions/424067/duke-and-schulze-pillot-condition-for-equidistribution) about equidistribution on a paper of Duke and Schulze-Pillot that was usefully answered.
However, on the answer there was a statement that was unimportant for me back then but catch my... | https://mathoverflow.net/users/411616 | Primitive representation of integers by some form on the genus of a quadratic form | The quoted text was written by me. As I corrected myself recently in a comment below the original post: I secretly assumed that $n$ was coprime to $\det(a\_{ij})$. In that case, I believe that modulo $4\det(a\_{ij})$ is sufficient. See Hilfssatz 13 in Siegel: Über die analytische Theorie der quadratischen Formen, Ann. ... | 3 | https://mathoverflow.net/users/11919 | 432174 | 174,906 |
https://mathoverflow.net/questions/432179 | 3 | For $A \in \mathbb R^{m \times n}$ and the induced norms:
$$
\| A \|\_1 = \max\_{x \ne 0} \frac{\|Ax\|\_1}{\|x\|\_1}
$$
$$
\| A \|\_2 = \max\_{x \ne 0} \frac{\|Ax\|\_2}{\|x\|\_2}
$$
... where:
$$
\|x\|\_1 = \sum\_{k=1}^n |x\_k|
$$
$$
\|x\|\_2 = \sqrt{\sum\_{k=1}^n |x\_k|^2}
$$
... does the following inequalit... | https://mathoverflow.net/users/146993 | Is the matrix induced L1-norm greater than the induced L2-norm? | To avoid ambiguity I will write $\lVert\cdot\rVert\_{p\to r}$ for the $\ell\_p$-to-$\ell\_r$-norm. Note that in general, $\lVert A\rVert\_{1\to r} = \max\_{1\leq j\leq n} \lVert (Ae\_j)\rVert\_r$.
Let $A$ be the $n\times n$ matrix whose top row has $1$ in every entry, and all other entries of the matrix are $0$. Then... | 7 | https://mathoverflow.net/users/763 | 432181 | 174,909 |
https://mathoverflow.net/questions/432168 | 2 | For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define
$$
\begin{split}
K\_n(a,t) &:= \inf\_{x \in \mathbb R^n} \|x-a\|\_2 + t\|x\|\_1,\\
M\_n(a,t) &:= \min(\|a\|\_2,t\|a\|\_1),\\
R\_n(a,t) &:= K\_n(a,t)/M\_n(a,t).
\end{split}
$$
**Note.** $K\_n$ defines the Peetre's K-... | https://mathoverflow.net/users/78539 | Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$ | This is to extend Christian Remling's [comment](https://mathoverflow.net/questions/432168/fix-positive-t-construct-a-n-in-mathbb-rn-such-that-inf-x-x-a-n-2#comment1112378_432168) to all real $t>0$, with an explicit lower bound on $K/M$, where $K:=K\_n(a,t)$ and $M:=M\_n(a,t)$.
$\newcommand\norm[1]{\lVert#1\rVert}$The... | 3 | https://mathoverflow.net/users/36721 | 432184 | 174,910 |
https://mathoverflow.net/questions/432186 | 0 | Let $R$ be a henselian DVR with fraction field $K$ and residue field $k$ of characteristic $p>0$. Let $\overline K$ be an algebraic closure of $K$, $\overline R$ the normalization of $R$ in $\overline K$ and $\overline k$ the residue field of $\overline R$. Let $G=\mathrm{Gal}(\overline K/K)$ be the absolute Galois gro... | https://mathoverflow.net/users/125617 | Nearby cycles for schemes with semi-stable reduction | According to this statement, $\mathrm R^q\Psi\Lambda$ is supported on points where $\mathrm R^1\Psi\Lambda $ has rank at least $q$ (by a property of wedge powers) and thus supported on points where $\bigoplus\_{i} \Lambda\_{Y\_i}$ has rank at least $q+1$ (by a property of quotients) and thus on $Y^{(q+1)}$ (since each ... | 3 | https://mathoverflow.net/users/18060 | 432187 | 174,912 |
https://mathoverflow.net/questions/432202 | 6 | When is $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ a Galois extension of $\mathbb{Q}$?
I was motivated by the question that $\mathbb{Q}(\sqrt{5+\sqrt{5}})$ is a Galois extension of $\mathbb{Q}$. Here is a rough sketch of the proof.
First, note that the polynomial
$$f(x)=(x-\sqrt{5+\sqrt{5}}) (x+\sqrt{5+\sqrt{5}})(x-\sqrt{5-\s... | https://mathoverflow.net/users/369754 | When is $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ a Galois extension of $\mathbb{Q}$? | Note that $f$ cannot have an irreducible factor of degree $3$ since it is even, so if $f$ is reducible, it means that the minimal polynomial of $\sqrt{p+\sqrt{p}}$ has degree at most $2$, so your extension is automatically Galois in this case.
Hence we may assume without loss of generality that $f$ is irreducible.
... | 11 | https://mathoverflow.net/users/36683 | 432209 | 174,916 |
https://mathoverflow.net/questions/432208 | 13 | I want to grasp the moving frames method but I find some obstacles. I don't know if this question is suitable for MO, if it is not the case please let me know and I will move it.
I am aware there are other related questions here like [this one](https://mathoverflow.net/questions/337294/moving-frames-method-for-non-m... | https://mathoverflow.net/users/129995 | Moving frames method | I think you might want to read a couple of articles on the moving frame that carefully discuss this issue (and show that it is more subtle than most people realize).
The first is a paper by Mark Green, *The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces* (Duke Math. J. 45... | 16 | https://mathoverflow.net/users/13972 | 432215 | 174,918 |
https://mathoverflow.net/questions/432155 | 1 | We introduce the following functional to study Yamabe problem with boundary.
$$
Q\_g(\varphi)=\frac{\int\_M\left(|\nabla \varphi|\_g^2+\frac{n-2}{4(n-1)} R\_g \varphi^2\right) d v+\frac{n-2}{2} \int\_{\partial M} h\_g \varphi^2 d \sigma}{\left(\int\_M|\varphi|^{2 n /(n-2)} d v\right)^{(n-2) / 2}}
$$
$$
Q(M)=\inf \l... | https://mathoverflow.net/users/148247 | How to calculate the infimum of Yamabe functional on upper hemisphere | The first observation is that $Q\_g(\phi)$ is conformally invariant.
Define
\begin{align\*}
L\_gu & := -\Delta u + \frac{n-2}{4(n-1)}Ru , \\
B\_gu & := \partial\_\nu u + \frac{n-2}{2}Hu ,
\end{align\*}
where $\nu$ is the outward-pointing unit normal and everything is defined with respect to the metric $g$.
On the one... | 1 | https://mathoverflow.net/users/121820 | 432227 | 174,922 |
https://mathoverflow.net/questions/432223 | 7 | Let $X \neq \emptyset$ be a set. We say that ${\cal F} \subseteq {\cal P}(X)$ is a *down-set* if ${\cal F}$ is closed under taking subsets. Whenever $a \in X$, we let ${\cal F}\_a = \{ S \in F : a \in S\}$.
We say ${\cal G} \subseteq {\cal P}(X)$ is *intersecting* if $S \cap T$ is non-empty whenever $S, T \in {\cal G... | https://mathoverflow.net/users/8628 | Counterexample for Chvatal's conjecture in an infinite set | Let $\kappa$ be a cardinal of
countable cofinality that is strictly larger than the continuum.
I will construct a counterexample to (C) on
$X=(\kappa\times \omega)\cup \omega$.
Since $\kappa$ has countable cofinality, there exists
a countable increasing sequence of cardinals
$(\kappa\_i)\_{i\in\omega}$ with limit $\... | 3 | https://mathoverflow.net/users/75735 | 432245 | 174,927 |
https://mathoverflow.net/questions/432195 | 3 | This problem has been asked in [MSE](https://math.stackexchange.com/questions/4548511/an-inequality-equivalent-to-h%c3%b6rmanders-condition-sup-y-in-mathbb-rn-int), but got no answers. I guess that this exam problem may be a small lemma in some research papers, so I post it here on MathOverflow.
>
> Let $K\in L\_{\... | https://mathoverflow.net/users/141451 | An inequality equivalent to Hörmander's condition $\sup_{y\in\mathbb R^n}\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx<\infty$ | Condition \eqref{2} implies \eqref{1} with $5|y|$ instead of $2|y|$. Fix $y$, let $r=|y|$ and $I=\int\_{|x| >5|y| }|K(x-y)-K(x)|\, dx$. Then
$$I \leq \int\_{|x| >5r }|K(x-y)-K(x-z)|\, dx+\int\_{|x| >5r }|K(x-z)-K(x)|\, dx:=I\_1(z)+I\_2(z)
$$
for every $|z| \leq r$. If $K$ is the supremum in \eqref{2}, then $r^{-n} \int... | 3 | https://mathoverflow.net/users/150653 | 432260 | 174,931 |
https://mathoverflow.net/questions/431969 | 3 | Suppose $\{a\_n(t)\}\_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a\_n(0) \leq 1$ for all $n\in \mathbb N$ (ii) $a\_n(t) \approx 1 - \mu2^{-n}$ uniformly in $t$ for $n \gg 1$ (iii) $a'\_n = a^2\_{n+1} - a\_n.$ My goal is to show that $$a\_n(t) \xrightarrow{t \to \inft... | https://mathoverflow.net/users/163454 | Showing convergence of an infinite ODE system | Theorem 2 below offers a sufficient condition for convergence to the nontrivial equilibrium that you are referring.
The results are contingent on the uniqueness of solutions to this infinite-dimensional ODE -- I will include the proofs when possible, as needed, but, and unless I am missing something, they are quite s... | 3 | https://mathoverflow.net/users/138242 | 432268 | 174,932 |
https://mathoverflow.net/questions/432262 | 4 | Let $x\_0=1$ and
$$x\_{k+1} = (1-a\_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x\_k}\right)$$
where $a\_n$ is a known sequence satisfying that $a\_k\in(0,1)$ for all $k$ and $a\_k\to 0$ as $k\to\infty$. How to prove that $x\_k\to 1$ as $k\to\infty$?
---
The difficulty here is that
1. It is not known how fast $a\_k... | https://mathoverflow.net/users/490600 | Convergence of a sequence | If, as you say, $a\_k<0.1$ for all $k$, then we can prove by induction that $x\_k>\frac{3}{4}$ for all $k$, with induction step $x\_{k+1}>0.9\left(\frac{3}{2}-\frac{1}{2\cdot\frac{3}{4}}\right)=\frac{3}{4}$. By a similar induction we get $x\_k\in\big[\frac{3}{4},1\big]\;\forall k$.
So $1-x\_k\in\big[0,\frac{1}{4}\big... | 7 | https://mathoverflow.net/users/172802 | 432270 | 174,933 |
https://mathoverflow.net/questions/432250 | 1 | Fix a Gaussian random matrix $A$ with $E[A\_{ij}]=0$ for $i, j=1,\dots n$ and $E[A\_{ij}^2]=\frac{1}{n}$. Let $v\_1$ be the leading eigenvector of $A$. What is the non-asymptotic upper bound for $v\_1$, that is something like
$$
P(v\_1\cdot u\ge t)\le e^{-\alpha t}
$$
where $u$ is distributed uniformly on the unit sphe... | https://mathoverflow.net/users/168083 | What is the non-asymptotic upper bound for the leading eigenvector of the random matrix? | We know the distribution of $x=v\_1\cdot u$, with $v\_1$ of length $n$ uniformly distributed on the unit $n$-sphere and $u$ an arbitrary unit vector. This distribution is given by
$$P(x)=\frac{\Gamma(n/2)}{\sqrt{\pi}\,\Gamma(n/2-1/2)}(1-x^2)^{n/2-3/2}\,\theta(1-x^2),\;\;n>1,$$
with $\theta(x)$ the unit step function. (... | 1 | https://mathoverflow.net/users/11260 | 432277 | 174,936 |
https://mathoverflow.net/questions/432258 | 2 | For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p\_\omega$-norm of the sequence $v$ by
$$\Vert v\Vert\_{\ell^p\_\omega} := \left(\sum\_{k=1}^\infty \omega\_k^p |v|\_k^p\right)^{1/p} .$$
**My question is:** If $v\in\ell^q$ and $p>q$, under which conditions on $\omega$ will $v\in\ell^p\_... | https://mathoverflow.net/users/75500 | Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm? | The answer to both questions is no.
* Taking $q=1$, $p=2$, $v\_k = k^{-(1+\varepsilon)}$ and $\omega\_k = k^{(1+3/2\varepsilon)/2}$ for some $\varepsilon>0$ provides a counter-example to the first assertion.
* [This answer](https://math.stackexchange.com/a/2296826/303111) provides a counter-example for the second ass... | 2 | https://mathoverflow.net/users/75500 | 432289 | 174,940 |
https://mathoverflow.net/questions/432201 | 5 | Here is what I know. Assume $M\cong G/K$ is an irreducible hermitian symmetric space. Denote the Lie-algebra of $K$ by $\mathfrak{t}$. Proposition 1.2. chapter 3 in
[Wienhard - Bounded cohomology and geometry](https://arxiv.org/abs/math/0501258)
says that for a symmetric space, being hermitian is equivalent to the exis... | https://mathoverflow.net/users/492740 | Can all hermitian symmetric spaces be realised as coadjoint orbits? | This is true. One can use a few facts from Helgason's *Differential Geometry, Lie Groups, and Symmetric Spaces* to show that, indeed, $K = \mathrm{Stab}\_G(Z)$.
Since $M=G/K$ is an irreducible Hermitian symmetric space, one can apply Theorem 6.1 from Chapter VIII of Helgason. The main point is that one can regard $G$... | 8 | https://mathoverflow.net/users/13972 | 432297 | 174,943 |
https://mathoverflow.net/questions/432263 | 3 | We begin by considering the usual general first order linear equation of the form
$$ a\_0 y' + a\_1 y + a\_2 = 0 $$
Where $a\_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone's favorite undergrad ODE class that this has a general solution in terms of integration factors.
$$ y = - e^{-... | https://mathoverflow.net/users/46536 | Is there a theory of "elementary closed form solution" at the operator level for differential equations? | There are several theories which deal with this question.
One is the differential algebra, see, for example the little book
I. Kaplansky, Introduction to differential algebra,
Publ. de l'Institut de Mathématique de l'Université de Nancago, No. V. Actualités Scientifiques et Industrielles No. 1251. Hermann, Paris, 197... | 5 | https://mathoverflow.net/users/25510 | 432300 | 174,946 |
https://mathoverflow.net/questions/428933 | 3 | Let $F$ be a number field, $G$ an $F$-simple affine algebraic group.
Then is the Weil restriction $\operatorname{Res}\_{F/\mathbb{Q}} G$ $\mathbb{Q}$-simple?
I couldn’t find any references.
| https://mathoverflow.net/users/159361 | The Weil restriction of a simple algebraic group | As noted, the answer is yes. Let $G$ be a simple algebraic group (meaning $F$-simple) over $F$.
Assume first that $F$ is Galois over $\mathbb{Q}$. Then $(\mathrm{Res}\_{F/\mathbb{Q}}G)\_{F}\simeq G\_{1}\times\cdots\times G\_{r}$, where the $G\_{i}$ are the conjugates of $G$ under $\mathrm{Gal}(F/\mathbb{Q})$. If $H$ ... | 3 | https://mathoverflow.net/users/492821 | 432303 | 174,947 |
https://mathoverflow.net/questions/432281 | 5 | Suppose $R$ is a semisimple ring and if $L$ is a minimal left ideal. Let $B$ be the direct sum of all minimal left ideals isomorphic to $L$ ($B$ is called a simple component corresponding to $L$). It is a standard result that $B$ is a two-sided ideal. The argument critically uses the fact that $R$ is semisimple.
I am... | https://mathoverflow.net/users/165646 | Simple component that is not a two-sided ideal | I'm not sure why you believe it depends on $R$ being semisimple. The usual argument goes like this and makes no reference to semisimplicity:
If $r\in R$ and $L'\cong L$ where $L'$ is a left ideal of $R$, then by right multiplication, $r$ defines a homomorphism of left $R$ modules $L'\to R$. Because $L'$ is simple, ei... | 12 | https://mathoverflow.net/users/19965 | 432304 | 174,948 |
https://mathoverflow.net/questions/432291 | 5 | This question is related to [Homotopy type theory : how to disprove that $0=\mathrm{succ}(0)$
in the type $\mathbb N$](https://mathoverflow.net/questions/421450/homotopy-type-theory-how-to-disprove-that-0-operatornamesucc0-in-the-ty).
Section 2.13 in [The HoTT Book](https://homotopytypetheory.org/book/) uses "the enc... | https://mathoverflow.net/users/492810 | Homotopy type theory: why are $0:\mathbb N$ and $\mathrm{succ}(0):\mathbb N$ not judgementally equal? | Daniel's answer is correct that the judgmental distinctness of $0$ and $\mathsf{succ}(m)$ is not what justifies a definition by pattern-matching. However, it is still a meaningful question of how to prove that $0$ and $\mathsf{succ}(m)$ are not judgmentally equal. (Stretching terminology a bit, Daniel answered your [X]... | 10 | https://mathoverflow.net/users/49 | 432308 | 174,949 |
https://mathoverflow.net/questions/380787 | 0 | It is well known that if a family of meromorphic functions is not normal (a family is said to be normal if each sequence of functions in the family has a subsequence which converges locally uniformly to a limit function which is either meromorphic or identically $\infty$) on some domain, then the corresponding family o... | https://mathoverflow.net/users/143655 | Is there any non-normal family $\mathcal{F}$ of meromorphic functions on $|z|<1$ whose each zero has multiplicity $2$ but $\mathcal{F'}$ is normal | The answer to this question is negative.
This follows easily from the following result of Chen and Lappan (Adv. Math.,vol. 24, 1996, 517-524):
Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D$ such that each $f\in\mathcal{F}$ has zeros of multiplicity at least $k+1,$ where $k$ is a positive integ... | 1 | https://mathoverflow.net/users/143655 | 432319 | 174,952 |
https://mathoverflow.net/questions/432317 | 8 | Let $K$ be a number field. $O\_K$ be its ring of integers, so $O\_K^\*$ are the units.
We have sequence
$1 \rightarrow O\_K^\* \rightarrow K^\* \rightarrow K^\*/O\_K^\* \rightarrow 1$
Note that $K^\*/O\_K^\*$ is essentially the group of principal fractional ideals.
Does this sequence split for all number fields? ... | https://mathoverflow.net/users/3208 | Does this exact sequence split? | The exact sequence in the original post splits for every number field $K$. To see this, let $P\_K$ be the multiplicative group of nonzero principal fractional ideals of $K$, and let $I\_K$ be the multiplicative group of all nonzero fractional ideals of $K$. Clearly, $P\_K$ is a subgroup of $I\_K$, and it is isomorphic ... | 14 | https://mathoverflow.net/users/11919 | 432321 | 174,953 |
https://mathoverflow.net/questions/432294 | 4 | Let $\alpha(x) : \mathbb{R} \to (0,\infty)$ have bounded variation (BV) and suppose $\inf\_{\mathbb{R}} \alpha > 0$. Consider the second order differential operator
$$H : =-\partial\_x (\alpha(x) \partial\_x) : L^2(\mathbb{R}) \to L^2(\mathbb{R}).$$
It's not too hard to show that
$$ \{u \in L^2(\mathbb{R}) : u, u' \... | https://mathoverflow.net/users/87862 | Is the Sobolev space $H^1(\mathbb{R})$ contained in the domain of $(-\partial_x \alpha(x) \partial_x)^{1/2}$? | We can actually do this directly and my comment above is not that relevant. Let $u\in H^1$ and also assume that $u$ is compactly supported, so $\int u'=0$. Approximate $\alpha u'$ in $L^2$ by $v\_n\in C\_0^{\infty}$. Here we can also insist that $\int v\_n/\alpha=0$. Then also $v\_n/\alpha\to u'$ in $L^2$, so $u\_n(x)=... | 1 | https://mathoverflow.net/users/48839 | 432325 | 174,955 |
https://mathoverflow.net/questions/432312 | 6 | By an (intuitionistic) **propositional formula** $\varphi(x\_1,\ldots,x\_n)$ I mean a formula built up from a (finite) number of variables $x\_1,\ldots,x\_n$ using connectors $\top, \bot, \land, \lor, \Rightarrow$. Given such a formula $\varphi(x\_1,\ldots,x\_n)$, given a Heyting algebra $H$ and elements $u\_1,\ldots,u... | https://mathoverflow.net/users/17064 | Variable elimination for propositional formulas in Heyting algebras | The answer for $\bigwedge\_t$ is no. Perhaps the idea here can be adapted to $\bigvee\_t$.
Consider the propositional formula $t \vee (t \to x)$ in the complete Heyting algebra of open subsets of $\mathbb{R}$.
*Claim.* For any open set $U \subseteq \mathbb{R}$, $\bigwedge\_t t \vee (t \to U)$ is the set $U^\bullet ... | 6 | https://mathoverflow.net/users/83901 | 432328 | 174,957 |
https://mathoverflow.net/questions/421918 | 3 | Given a polytope $P$, what do the points of the secondary polytope correspond to?
I know that the vertices of the secondary polytope correspond to regular triangulations of $P$.
But what do the interior points of the secondary polytope correspond to?
| https://mathoverflow.net/users/5690 | Secondary polytope | As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (<https://doi.org/10.2307/2946575>).
This fiber polytope view provides the answer (or, at least, one answer) to what *each point* in the secondary polytope represents:
... | 4 | https://mathoverflow.net/users/22608 | 432332 | 174,960 |
https://mathoverflow.net/questions/432333 | 3 | This question arose through reading "Interactions between homotopy theory and algebra" ([the first chapter by Goerss and Schemmerhorn](https://arxiv.org/abs/math/0609537)). In particular, I am struggling with the proof of Proposition 4.32, the cofiber sequence of cotangent complexes associated to two composable ring ma... | https://mathoverflow.net/users/170467 | Pushout along weak equivalence gives weakly equivalent object | This is true in general in any *left proper* model category. To be left proper means the pushout of a weak equivalence (for you, $X\to A$) along a cofibration (for you, $X\to Y$) is again a weak equivalence (for you, $Y \to A\otimes\_X Y$).
It is not true that any pushout of a weak equivalence is a weak equivalence (... | 7 | https://mathoverflow.net/users/11540 | 432334 | 174,961 |
https://mathoverflow.net/questions/385116 | 9 | Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
| https://mathoverflow.net/users/398 | Vinogradov-Korobov for Dirichlet L-functions? | My paper <http://arxiv.org/abs/2210.06457> establishes several explicit Vinogradov--Korobov type zero-free regions for Dirichlet $L$-functions. In particular, Theorem 1.1 states the following:
Let $q \geq 3$, and let $\chi\pmod{q}$ be a Dirichlet character. The Dirichlet $L$-function $L(\sigma+it,\chi)$ does not vani... | 7 | https://mathoverflow.net/users/167279 | 432336 | 174,962 |
https://mathoverflow.net/questions/432338 | 4 | So there's an elementary (but in my opinion quite interesting!) result which is that the Laurent series expansion of
$$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \cdots$$
Now the reason that is interesting is because each of those coefficients are equal to $\frac{1}{n!}B\_n$ whe... | https://mathoverflow.net/users/46536 | Is there a recurrence for the coefficients of the Laurent series expansion of $\frac{1}{1-e^{e^x - 1}}$? | $e^{e^x-1}$ is the exponential generating function for [Bell numbers](https://en.wikipedia.org/wiki/Bell_number) ${\cal B}\_n$:
$$e^{e^x-1} = \sum\_{n\geq 0} {\cal B}\_n \frac{x^n}{n!}.$$
Then
$$g(x) := \frac{e^{e^x-1}-1}{x} = \sum\_{n\geq 0} {\cal B}\_{n+1} \frac{x^n}{(n+1)!}.$$
Correspondingly, the coefficient of $... | 11 | https://mathoverflow.net/users/7076 | 432341 | 174,963 |
https://mathoverflow.net/questions/432342 | 5 | Let $E \to \mathbb{C}P^\infty$ be any topological complex vector bundle over the infinite complex projective space. I'm wondering if it makes sense to possibly define a "holomorphic structure" on $E$. This a priori requires a complex structure on $\mathbb{C}P^\infty$, which is also something I don't know whether it exi... | https://mathoverflow.net/users/143629 | Does it make sense to define a holomorphic structure on $\mathbb{C}P^\infty$ and vector bundles over it? | Yes, there is lots of literature on this subject.
However, Tyurin proved that all vector bundles on $CP^\infty$ are
direct sum of line bundles. There are several more recent papers by
Penkov and Tikhomirov about vector bundles on $C P^\infty$ (they treat other infinite-dimensional
manifolds, too).
* A. N. Tjurin (Tyu... | 8 | https://mathoverflow.net/users/3377 | 432346 | 174,964 |
https://mathoverflow.net/questions/432355 | 0 | Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell\_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f\_a(x) := \|x-a\|\_2 + \|x\|$ is strictly convex and has a unique minimizer $x(a)$.
**Question.** Given $a,b \in \mathbb R^n$, can $\|x(a)-x(b)\|\_2$ be bounded in ... | https://mathoverflow.net/users/78539 | Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive? | First of all, the function $f\_a$ is not strictly convex, and hence, you should expect multiple minimizers. As such, non-expansiveness (even in some generalized sense) does not seem likely. Consider the one-dimensional case where
$$f\_a(x) = |x-a| + |x|$$
for which $\operatorname{argmin} f\_a = [0,a]$. So you can selec... | 4 | https://mathoverflow.net/users/9652 | 432356 | 174,967 |
https://mathoverflow.net/questions/432266 | 4 | Suppose $x\_0=x\_1=1$, define $y\_k=x\_k+\frac{1}{2}(x\_k-x\_{k-1})$ and $x\_{k+1}=y\_k-\eta y\_k^3$ where $\eta\in(0,1/8)$. If we know $x\_k\to 0$ as $k\to\infty$. How to show that $x\_k=\Theta(1/\sqrt{k})$?
---
It suffices to show that $x\_k^{-2}=\Theta(k)$. By Taylor expansion, we have
$$x\_{k+1}^{-2} = (y\_k-... | https://mathoverflow.net/users/490600 | Convergence rate of a sequence | Suppose that we already know that $x\_k>0\forall k$. Clearly $(x\_k)$ and $(y\_k)$ are decreasing sequences which converge to $0$. Then we can prove that $b\_k:=\frac{x\_{k}}{x\_{k-1}}\to 1$. To do it note first that $b\_k\in[0,1]\forall k$ and
$$b\_{k+1}=\frac{y\_k-\eta y\_k^3}{x\_k}=\frac{x\_k+\frac{1}{2}(x\_k-x\_{... | 1 | https://mathoverflow.net/users/172802 | 432369 | 174,971 |
https://mathoverflow.net/questions/432365 | 1 | In Proposition 5.7 on page 34 in [lectures on condensed mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) Peter Scholze shows that $\mathbb{Z}[S]^\blacksquare$ is solid. He shows that the two relevant expressions are isomorphic, however, in the definition of solidity it says that the isomorphism h... | https://mathoverflow.net/users/473423 | Isomorphism of RHoms in condensed mathematics | I guess that this basically follows from keeping track of isomorphisms. Seemingly it is a bit harder to trace the following isomorphisms, due to the occurrence of the shift $\require{AMScd}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\iHom{\underline{Hom}}\DeclareMathOperator\fin{fin}\DeclareMathOperator\colim{coli... | 2 | https://mathoverflow.net/users/176381 | 432384 | 174,972 |
https://mathoverflow.net/questions/431872 | 4 | I am looking at (the limitation of) the extension of the singular value decomposition to tensors. I would like to show that there is a tensor $A\_{i,j,k}$ that **cannot** be decomposed in the following singular value decomposition fashion
$$A\_{i,j,k}=\sum\_n \lambda\_n u\_{i,n}v\_{j,n}w\_{k,n} \tag 1\label{1}$$
where ... | https://mathoverflow.net/users/32660 | Singular value decomposition for tensor | [Asher Peres: *Higher Order Schmidt Decomposition*](https://arxiv.org/abs/quant-ph/9504006),
Physics Letters, A 202, No. 1, 16-17 (1995), [MR1337627](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1337627), [Zbl 1020.81540](https://zbmath.org/?q=an%3A1020.81540), gives the necessary and sufficient condition.
| 2 | https://mathoverflow.net/users/32660 | 432386 | 174,973 |
https://mathoverflow.net/questions/432343 | 0 | In [the LPS paper "Ramanujan graphs"](http://www.ma.huji.ac.il/%7Ealexlub/PAPERS/ramanujan%20graphs/ramanujanGraphs.pdf) the adjacency matrix of $X^{p,q}$, for simplicity say that $p,q\equiv1\mod{4}$ and $\left(\frac{p}{q}\right)=1$ (so, nonbipartite) and $n=\lvert X^{p,q}\rvert$, is considered with spectrum $p+1=\lamb... | https://mathoverflow.net/users/159965 | Series analyzed in Lubotzky–Phillips–Sarnak "Ramanujan Graphs" | **1.** Regarding your first question, we have $e^{i\theta\_0}=p^{-1/2}$ as emphasized two lines below (4.11). Therefore,
$$\frac{\sin((k+1)\theta\_0)}{\sin\theta\_0}=
\frac{e^{-i(k+1)\theta\_0}-e^{i(k+1)\theta\_0}}{e^{-i\theta\_0}-e^{i\theta\_0}}
=\frac{p^{(k+1)/2}-p^{-(k+1)/2}}{p^{1/2}-p^{-1/2}},$$
so that
$$\frac{2p^... | 3 | https://mathoverflow.net/users/11919 | 432390 | 174,974 |
https://mathoverflow.net/questions/432387 | 2 | I've been playing around numerically with Haar random $\text{CUE}$ unitary matrices of size $N$ by $N$, with $N$ around $1000$. If I "truncate" the matrix by keeping the upper left $fN$ by $fN$ block for some fixed $f$ independent of $N$, setting all other entries to zero, the resulting matrix is no longer unitary. In ... | https://mathoverflow.net/users/153549 | The singular values of truncated Haar unitaries | Your numerical findings have a simple explanation: Let me denote the upper left block of the $N\times N$ unitary matrix by $R$ and the upper right block by $T$; the matrix $R$ has dimension $fN\times fN$, the upper right block has dimension $fN\times(1-f)N$; unitarity requires that
$$RR^\dagger+TT^\dagger=I,$$
with $\d... | 4 | https://mathoverflow.net/users/11260 | 432391 | 174,975 |
https://mathoverflow.net/questions/416294 | 11 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL\_n(q),\SO\_n(q)$ etc via Deligne-Lusztig induction and so on. If I correctly understood, there's another geometric way to build the characters (at least in the $\GL\_n$ case... | https://mathoverflow.net/users/146464 | Reference for character sheaves over $\mathrm{GL}_n(q)$ | I am not exactly sure what you have read already, but how about this set of notes, from a course given by Victor Ostrik in Luminy in 2010 (notes by Geordie Williamson)?
* [Character sheaves, tensor categories and non-Abelian Fourier transform](http://people.mpim-bonn.mpg.de/geordie/Ostrik.pdf)
It says in $\S~2.1$ t... | 3 | https://mathoverflow.net/users/481162 | 432392 | 174,976 |
https://mathoverflow.net/questions/432389 | 3 | **Notation**:
$L/K$, finite extension of global fields
$K^\times$, unit group of $K$
$L^\times$, units group of $L$
$\mathbb{A}\_L^\times$, ideles of $L$
$N\_{L/K}$, the norm map
The **knot group** of an extension of global fields $L/K$ is defined as the quotient group of 'local norms' by 'global ... | https://mathoverflow.net/users/92433 | Knot group of a field extension | Arnold Scholz was fond of a colorful language in mathematics. I don't think there's any connection to actual knots except perhaps for a faint reference to the Gordian knot, which is difficult to solve without a striking idea.
| 6 | https://mathoverflow.net/users/3503 | 432393 | 174,977 |
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