parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/425569 | 5 | I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks.
More precisely, on the one hand, we have the construction of Reshetikhin–Turaev as well as Blanchet–Habegger–Masbaum–Vogel of "... | https://mathoverflow.net/users/484855 | Relation between TQFT representations and factorizable sheaves | To sum up: these tangent vectors are always present. In the literature on WZW one usually choose local formal coordinates at the marked point, ie an identification of the neighborhood of those with a chosen formal punctured disc, but this amount to essentially the same thing. The reason is compatibility with gluing: th... | 1 | https://mathoverflow.net/users/13552 | 425676 | 172,813 |
https://mathoverflow.net/questions/425610 | 2 | Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$
\begin{align}
\Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\
u&=0 \text{ on }\partial \Omega
\end{align}
Are there any explicit expressions for a kernel $K$ such that,
$$u(x)=\int\_{\Omega} K(x,y)f(y)dy$$ when $\Omega=\mathbb{R}^n$ or ... | https://mathoverflow.net/users/68232 | Kernel for an equation involving the Ornstein-Uhlenbeck operator | There is a trick that reduces the equation $u\_t=Lu$, $L=\Delta-x \nabla$ to the heat equation $u\_t-\Delta$. It is genuinely parabolic and gives the parabolic kernel in the whole space, from which the elliptic kernel can be deduced by integrating in time.
If $v\_t(t,x)=\Delta v(t,x)-x\nabla v (t,x)$ with $v(0,x)=f(x)$... | 3 | https://mathoverflow.net/users/150653 | 425690 | 172,817 |
https://mathoverflow.net/questions/425652 | 8 | In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, in the natural language, are obviously linked by their opposition, the *souci* or concern that stems from it can only be ... | https://mathoverflow.net/users/138089 | Formalisation of intuitive concepts in the language leading to mathematical progress | An example that is closely related to computation is proof. Mathematicians have been proving theorems ever since Euclid (and presumably even earlier). But it was not until the 20th century that the concept of a proof was formalized to the point where they could be studied as mathematical objects in their own right.
T... | 4 | https://mathoverflow.net/users/3106 | 425692 | 172,818 |
https://mathoverflow.net/questions/425667 | 11 | [It is well known](https://math.stackexchange.com/questions/2965261/homotopy-type-of-the-diffeomorphism-group-of-the-sphere) that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}\_{\partial D^n}(D^n)$ of the orthogonal group and ... | https://mathoverflow.net/users/1708 | Characteristic classes of non-linear sphere bundles | For many values of $n$, the answer to both questions is no. Since the fundamental groups of $BX$ and $B\mathrm{Diff}(S^n)$ are finite for $n\ge5$ (This uses that $\pi\_0\mathrm{Diff}\_\partial(D^n)$ is isomorphic to the group of homotopy $(n+1)$-spheres which is finite by a famous result of Kervaire--Milnor.), their fi... | 11 | https://mathoverflow.net/users/32022 | 425693 | 172,819 |
https://mathoverflow.net/questions/425190 | 0 | Let $S,\Sigma$ in $\mathbb{R}^d$ be finite measure set. The Amrein-Berthier uncertainty principle states that there exists $C=C(S,\Sigma)>0$ such that for all $f\in L^2(\mathbb{R}^d)$, $\int\_{\mathbb{R}^d} |f|^2\leq C \left(\int\_{\mathbb{R}^d\setminus S} |f|^2+\int\_{\mathbb{R}^d\setminus \Sigma} |\widehat{f}|^2\righ... | https://mathoverflow.net/users/107004 | Constant in Amrein-Berthier uncertainty principle | I remembered that this theorem is "essential" only when $|S||\Sigma|\geq1$. If $|S||\Sigma|<1$, one has an easier estimate (for $d=1$)
$$||f||^2\_{L^2}\leq\frac{1}{\sqrt{1-|S||\Sigma|}}\bigg(\int\_{S^c}|f|^2+\int\_{\Sigma^c}|\hat{f}|^2\bigg)$$
So you may see that $C\rightarrow1$ as $|S||\Sigma|$ goes to $0$.
I think th... | 1 | https://mathoverflow.net/users/484991 | 425703 | 172,821 |
https://mathoverflow.net/questions/425714 | 0 | Let $A \subset \mathbb{R}^{n}$ be a closed set. Does there exist an open set $O$ containing $A$, and a smooth function $f : O \to \mathbb{R} $ such that
$f(x) = 0$ for all $x \in A$,
$f(x) > 0$ and $\nabla f(x) \ne 0$ for all $x \in O \setminus A$,
and $f(x) \to \infty$ as $x \to \partial O$ ?
| https://mathoverflow.net/users/153602 | Existence of a particular positive definite and radially unbounded function | Not always possible.
Let $n = 1$. Take $A \subsetneq [0,1]$ be the usual ternary Cantor set.
Every point in $A$ is a limit point.
Let $O$ be any open set containing $A$, then $O$ must contain *some* interval of the form $(a,b)$ with $a,b\in A$. Since $f(a) = f(b) = 0$ is needed, if we want $f(x) > 0$ on $O\setm... | 3 | https://mathoverflow.net/users/3948 | 425715 | 172,824 |
https://mathoverflow.net/questions/425699 | 2 | Let $M\_t$ be a continuous time real valued martingale, and $\mathcal F\_t$ its natural filtration.
Suppose that $\mathcal F\_t \setminus \mathcal F\_s$ is nonempty for all $t > s$.
Let $\mathcal G$ be a sigma algebra, and define the filtration $\mathcal H\_t := F\_t \vee \mathcal G$.
**Question:** Is it true tha... | https://mathoverflow.net/users/173490 | Enlargement of filtration | I think that I have a counterexample. Let $(X,Y)$ be a Brownian motion in $\mathbb{R}^2$. Then $M = \int\_0^\cdot X\_s \mathrm{d}Y\_s$ is a martingale, in the natural filtration of $(X,Y)$, in its own filtration $(\mathcal{F}\_t)\_{t \ge 0}$ and also in $(\mathcal{F}\_t \vee \sigma(X))\_{t \ge 0}$. Yet, $X$ is not inde... | 4 | https://mathoverflow.net/users/169474 | 425723 | 172,826 |
https://mathoverflow.net/questions/425695 | 1 | Let $D\_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f\_n)\_{n\ge 1}\subset D\_+$ be a sequence of continuous functions s.t. $\lim\_{n\to\infty }f\_n(t)$ exits for each $t\in [0,T]$. Denote by $\hat f$ its pointwise limit and by $f$ the right-continuous modification of ... | https://mathoverflow.net/users/nan | Does pointwise convergence yield the convergence under Skorokhod topology? | The answer is no to both questions. E.g., suppose that $T=2$ and
$$f\_n(t)=1(t\le1-\tfrac1n)+n(1-t)1(1-\tfrac1n<t\le1).$$
Then $f\_n(t)$ increases in $n$ to $f(t)=1(t<1)$, and all the other conditions on $f\_n$ hold.
However, by the [definition of the Skorokhod metric](https://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g#S... | 1 | https://mathoverflow.net/users/36721 | 425724 | 172,827 |
https://mathoverflow.net/questions/425585 | 9 | In the quadratic case, it does. Given an irreducible quadratic polynomial $f(x)=ax^2+bx+c$, the discriminant of the quadratic number field $\frac{\mathbb{Q}[x]}{f(x)}$ is $\operatorname{sqf}(d)$ or $4\cdot \operatorname{sqf}(d)$, depending on if $d \equiv 1 \mod 4$, where $d=b^2-4ac$ is the polynomial discriminant of $... | https://mathoverflow.net/users/482554 | Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates? | Another counterexample:
$f\_1(x)=x^3-9x-20, f\_2(x)=x^3-6x-18$, discriminant = $-4\*27\*73$ for both $f\_1$ and $f\_2$.
Both polynomials are irreducible over $\mathbb{Q}$ by the rational root test.
The 2-adic Newton polygons show that $f\_1$ has 2 roots of valuation=0 and 1 root of valuation=2, and $f\_2$ has 3 r... | 3 | https://mathoverflow.net/users/59248 | 425731 | 172,828 |
https://mathoverflow.net/questions/425733 | 1 | Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It seems that there are very few materials in the literature. Is the residue category of $X$ an Enriques category in the sense t... | https://mathoverflow.net/users/41650 | Semi-orthogonal decomposition of Verra threefold | The projection of $X$ to each factor is a conic bundle, therefore there is a decomposition
$$
D^b(X) = \langle D^b(\mathbb{P}^2), D^b(\mathbb{P}^2,\mathcal{B}\_0) \rangle,
$$
where $\mathcal{B}\_0$ is the even part of the Clifford algebra. The second component is indeed an Enriques category.
| 3 | https://mathoverflow.net/users/4428 | 425739 | 172,832 |
https://mathoverflow.net/questions/425738 | 1 | Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^\*: D^b\_{coh}(Y)\to D^b\_{coh}(X)$ and $i\_\*: D^b\_{coh}(X)\to D^b\_{coh}(Y)$. By adjunction, for any $\mathcal{F}\in D^b\_{coh}(X)$ we have a canonical morphism $i^\*i\_\*\mathcal{F}\to \mathcal... | https://mathoverflow.net/users/24965 | How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$? | The cone of $i^\*i\_\*\mathcal{F} \to \mathcal{F}$ is isomorphic to $\mathcal{F} \otimes \mathcal{O}\_X(-X)[2]$.
EDIT. Let me write an argument for a sheaf $F$. Consider the distinguished triangle
$$
i^\*i\_\*F \to F \to F'.
$$
We need to identify $F'$.
Applying $i\_\*$ we obtain
$$
i\_\*i^\*i\_\*F \to i\_\*F \to i\_... | 6 | https://mathoverflow.net/users/4428 | 425740 | 172,833 |
https://mathoverflow.net/questions/425737 | 10 | Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$.
Suppose $K/\mathbb{Q}$ is a finite Galois extension. Cebotarev's density theorem implies that there are infinitely ma... | https://mathoverflow.net/users/485009 | Infinitely many primes that split completely in an arithmetic progression | **Theorem**. *Let $K$ and $L$ be finite Galois extensions of $\mathbf Q$*. Set $F = K\cap L$.
(1) *If $F = \mathbf Q$, then for each conjugacy class $C$ in
${\rm Gal}(L/\mathbf Q)$ there are infinitely many primes that are unramified in $L$ with Frobebius conjugacy class $C$ and split completely in $K$*.
(2) *If $F... | 13 | https://mathoverflow.net/users/3272 | 425742 | 172,834 |
https://mathoverflow.net/questions/425743 | 1 | Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
| https://mathoverflow.net/users/83515 | Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$ | It seems like the lower half of the Alexander polynomial of the pretzel knot $ P(2m+1,2n,2k+1)$ , up to multiplication by $\pm t^{\alpha}$ , is given by $$ \Delta\_{h}(t)= -nt + \sum\_{i=2}^{2m+1}(-1)^{i}( i+2n-1) t^{i} \\ +(2m+2n+1) \sum\_{i=2m+2}^{k+m+2} (-1)^{i} t^{i} $$
for $k\ge m\geq 0,n\ne 0.$ The subindex $h$... | 2 | https://mathoverflow.net/users/83515 | 425744 | 172,835 |
https://mathoverflow.net/questions/425481 | 3 | In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are usually written in books and papers but I do not know how to find them. I tried to see how these laws were found but I co... | https://mathoverflow.net/users/471464 | Hamiltonian, energy, and conservation laws of nonlinear PDEs | There is a great variety of methods to obtain conservation laws of nonlinear evolution equations. In a broad classification one can divide these in symmetry-based approaches ([Noether's theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem) relates a symmetry to a conserved quantity) and direct approaches. From a ... | 1 | https://mathoverflow.net/users/11260 | 425747 | 172,837 |
https://mathoverflow.net/questions/425725 | 0 | [Edit, July 6, 2022: Removed erroneous characterization of Faber polynomials as an Appell sequence.]
[Dress and Siebeneicher](https://www.sciencedirect.com/science/article/pii/0001870889900273?via%3Dihub) in their tale of the Burnside family express an opinion (1.2) that, if I read it correctly, leads me to believe t... | https://mathoverflow.net/users/12178 | A Newton identity and the primes--the Faber partition polynomials and modular arithmetic | From $$\ln[A(x)] = \ln[1 + a\_1 x + a\_2 x^2 + \cdots ] = \sum\_{n \geq 1} -F\_n(a\_1,...,a\_n)\; \frac{x^n}{n}$$ and letting $A'(x) = A(x) - 1$ we have
$$\begin{eqnarray\*}
F\_n(a\_1,...,a\_n) &=& n [x^n] \sum\_{i=1}^{\infty} \frac{(-A'(x))^i}{i} \\
&=& (-a\_1)^n + n [x^n] \sum\_{i=1}^{n-1} \frac{(-A'(x))^i}{i} \\
%&=... | 1 | https://mathoverflow.net/users/46140 | 425753 | 172,840 |
https://mathoverflow.net/questions/425736 | -3 | Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int\_{\xi\_1 +\xi\_2 +...=0} \hat{f}(\xi\_1) \hat{g}(\xi\_2)... d\xi\_1 d\xi\_2...$$
Could you please show me where to find details, statements and proofs about this relationship? Thanks in Advance.
| https://mathoverflow.net/users/471464 | Asking for reference about a relation related to Fourier transform | If $f\_1,\ldots,f\_n$ belong to the Schwarz space, Fourier inversion formula and Fourier-convolution properties yield
\begin{eqnarray\*}
\int\_{\xi\_1+\cdots+\xi\_n=0} \hat{f\_1}(\xi\_1) \cdots \hat{f\_n}(\xi\_n) d\xi\_1 \cdots d\xi\_n
&=& (\hat{f\_1} \* \cdots \* \hat{f\_n})(0) \\
&=& (2\pi)^{-1} \mathcal{F}(\overline... | 4 | https://mathoverflow.net/users/169474 | 425755 | 172,842 |
https://mathoverflow.net/questions/423877 | 6 | A Cartan-Hadamard 3-space $M$ is a complete simply connected 3-dimensional Riemannian manifold with nonpositive sectional curvature. A (smooth) convex surface $\Gamma\subset M$ is an embedded topological sphere with nonnegative second fundamental form $\mathrm{I\!I}$. The total (Gauss-Kronecker) curvature of $\Gamma$ i... | https://mathoverflow.net/users/68969 | Convex surfaces with minimal total curvature in Cartan-Hadamard 3-space | Note that curvature in sectional directions tangent to the surface, say $\Sigma$ vanish.
It seems sufficient to conclude that the usual Peterson–Codazzi equations hold for the surface.
It follows that there is a convex surface $\Sigma'$ in the Euclidian space that is isometric to $\Sigma$; plus they have identical seco... | 2 | https://mathoverflow.net/users/1441 | 425764 | 172,844 |
https://mathoverflow.net/questions/425788 | 13 | I've been trying and failing to find a paper/article/blog post (I think it was a paper) on a particular algebraic structure. The paper describes a structure consisting of something like a constant $0$, an associative binary operation $+$ for which $0$ is the unit, and an $\omega$-ary operation $\Sigma$ that agrees with... | https://mathoverflow.net/users/478205 | Looking for a half-remembered reference on 'magnitude algebras' | @Neil Strickland gave me all I needed in his comment (thank you!). Escardó and Simpson's paper is not what I was looking for, but they do cite it: *A Universal Characterization of $[0,\infty]$*, by Denis Higgs.
For those curious but not curious enough to find a text of the paper: a *magnitude module* is a variety of ... | 11 | https://mathoverflow.net/users/478205 | 425792 | 172,854 |
https://mathoverflow.net/questions/425727 | 21 | One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma\_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-preserving diffeomorphism $f\colon \Sigma\_g \rightarrow \Sigma\_g$ with the following two properties:
1. $f$ has no fi... | https://mathoverflow.net/users/485005 | Fixed-point free diffeomorphisms of surfaces fixing no homology classes | Goodwillie's construction (in genus two) generalises to all higher genus as follows.
Let $P\_n$ be the regular $n$-gon in the plane with vertices at roots of unity. When $n$ is even, we can glue opposite (and thus parallel) sides to obtain an oriented surface $F\_n$. Suppose that $n = 4g + 2$. In this case $F\_n$ has... | 15 | https://mathoverflow.net/users/1650 | 425796 | 172,856 |
https://mathoverflow.net/questions/425801 | 2 | Define $N(F)$ to be the number of monomials of a multi-variable polynomial $F$. For example $N(x^2y+3xy-y^5)=3$.
If $\mathbf{x}=(x\_1,\dots,x\_n)$ and $F\_n(\mathbf{x})=\prod\_{k=1}^n(x\_1+\cdots+x\_k)$ then it's easy to verify that $N(F\_n)=C\_n$ where $C\_n=\frac1{n+1}\binom{2n}n$ are the Catalan numbers.
Conside... | https://mathoverflow.net/users/66131 | Counting monomials and $q$-Catalan polynomials | Yes, bijectively.
Associate to a monomial $\prod\_{i=1}^n x\_i^{e\_i}$ with $\sum\_{i=1}^n e\_i =n$ the path where we walk one step to the right, then up $e\_n$ steps, then one step to the right, then up $e\_{n-1}$ steps, then $\dots$, and finally walk up $e\_1$ steps.
It's easy to see that this gives a bijection b... | 5 | https://mathoverflow.net/users/18060 | 425805 | 172,860 |
https://mathoverflow.net/questions/425799 | 1 | Let $X := \mathbb R^d$, $\lambda^d$ be the $d$-dimensional Lebesgue measure on $X$, and $f:X \to \mathbb R$ convex. Then [there is](https://math.stackexchange.com/questions/4483452/is-it-true-that-if-f-mathbb-rd-to-mathbb-r-is-continuous-then-the-differ) a Borel set $N \subset X$ such that $\lambda^d (N) = 0$ and $f$ i... | https://mathoverflow.net/users/99469 | Let $f$ be convex and $A$ a Borel subset of $\mathbb R^d$ on which $f$ is differentiable. Is the gradient $\nabla f: A \to \mathbb R^d$ measurable? | Yes, since a convex function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ is locally Lipschitz (see e.g. Theorem 6.3.1, pp. 236-239 of the book by L. C. Evans and R. F. Gariepy, *Measure Theory and Fine Properties of Functions*, CRC Press, 1992).
Recall that, as you wrote it yourself in the particular case of convex functio... | 4 | https://mathoverflow.net/users/11211 | 425807 | 172,861 |
https://mathoverflow.net/questions/425557 | 6 | So I've read (for instance in the introduction to R.S de Jong's [thesis](https://www.math.leidenuniv.nl/%7Ejongrsde/publications/thesis.pdf) ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection theory. Most notably we lack a "good" canonical cla... | https://mathoverflow.net/users/152554 | Mordell conjecture over function fields | You can find a good explanation of the proof of the (effective) Mordell conjecture based on such an inequality in:
Parshin, A. N.: Application of ramified coverings in the theory of Diophantine equations. Math. USSR Sbornik 66 (1990), no. 1, 249–264.
The corresponding proof for function fields has been published by... | 2 | https://mathoverflow.net/users/61532 | 425824 | 172,865 |
https://mathoverflow.net/questions/423224 | 2 | It is well known and easy to see (modulo standard basic facts) that any compact 1-dimensional Alexandrov space with curvature bounded from below is isometric either to a circle or to a segment.
I am looking for a precise reference.
| https://mathoverflow.net/users/16183 | Isometric classification of 1-dimensional Alexandrov spaces | See 15.18 in our [Alexandrov geometry: foundations](https://arxiv.org/abs/1903.08539).
| 1 | https://mathoverflow.net/users/1441 | 425825 | 172,866 |
https://mathoverflow.net/questions/425794 | 9 | I would like to know which books I should read to understand the paper ["The local Langlands correspondence for $\mathrm{GL}\_n$ over $p$-adic fields"](https://www.math.uni-bonn.de/people/scholze/LocalLanglands.pdf) written by Peter Scholze.
I only know mathematics subjects at the master's degree level. I also learne... | https://mathoverflow.net/users/481822 | Roadmap to understand the Scholze's proof of the local Langlands correspondence for $\text{GL}_n$ over $p$-adic fields | @user860322 The local Langlands correspondence (LLC) for $\mathrm{GL}\_n$ can be understood (very sloppy) as:
There exists a group for $F$, called the Weil(-Deligne)-Group $W\_F$ (and not only for local fields), that encodes the (admissible) representation theory of $\mathrm{GL}\_n(F)$ for every $n$ in a *very nice* ... | 12 | https://mathoverflow.net/users/484997 | 425829 | 172,868 |
https://mathoverflow.net/questions/124616 | 26 | Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying to understand explicitly why we need to use these?
>
> **Background**
>
>
>
Very roughly, the idea is that on... | https://mathoverflow.net/users/184 | Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle? | I recommend a very nice paper of Sander Kupers, "Three applications of delooping to $h$-principles". In an appendix he discusses and compares various convenient categories of spaces when working with continuous sheaves. However Kupers does not work with quasitopological spaces in the body of the paper. His preferred co... | 5 | https://mathoverflow.net/users/1310 | 425850 | 172,876 |
https://mathoverflow.net/questions/425855 | 1 | Let $D \subseteq \mathbb{R}^n$ be open, connected, bounded. Let $f : D \to \mathbb{R}$ be $C^1$ and assume that $f$ as well as $\partial\_i f$ extend as continuous functions to the closure $\overline{D}$ of $D$ for all $i \in \{1, \ldots, n\}$.
The paper [Functions Differentiable on the Boundaries of Regions](https:/... | https://mathoverflow.net/users/13338 | If function and derivative extend continuously to boundary of domain, can one extend as $C^1$ function? | The failure is essentially local, so the boundedness we can ignore for the purpose of this example.
* Let $D = \{ y < |x|^{1/4}: (x,y)\in \mathbb{R}^2\}$.
* Let $\phi:\mathbb{R}\to\mathbb{R}$ be such that $\phi(y) = 0$ for $y \leq 0$ and $\phi(y) = y^2$ for $y > 0$.
* Let $f:D\to \mathbb{R}$ be given by $f(x,y) = \ma... | 3 | https://mathoverflow.net/users/3948 | 425857 | 172,878 |
https://mathoverflow.net/questions/425836 | 2 | I am looking to replicate, in the fiberwise setting, the result of Spivak/Wall that the fiber homotopy type of the normal sphere bundle of a manifold is preserved under homotopy equivalences.
A particularly slick way to prove this result, is to notice that the mapping cylinder of a homotopy equivalence $M \rightarrow... | https://mathoverflow.net/users/134512 | Preservation of fiberwise normal bundles under fiberwise homotopy equivalences | After looking at the actual construction of the fiberwise embedding of a bundle $M \rightarrow E \rightarrow B$ into $B \times \mathbb{R}^N$, it became clear how to adjust it to embed the mapping cylinder. For convenience, let's assume that $B$ is a finite CW complex.
To obtain a fiberwise embedding of the bundle $p:... | 1 | https://mathoverflow.net/users/134512 | 425869 | 172,886 |
https://mathoverflow.net/questions/425887 | 2 | Let $f:U \to V$ be a flat, quasi-finite, surjective morphism between two affine varieties defined over $\mathbb{C}$. Assume that every closed fiber is reduced. Consider the function $\eta$ that sends a closed point $v \in V$ to the cardinality of the fiber $f^{-1}(v)$ over $v$. Is this function semi-continuous? If so, ... | https://mathoverflow.net/users/38832 | Semi-continuity in quasi-finite morphisms without properness | The function is lower semi-continuous, see EGA4, Prop. 15.5.1.
Note that since you assume the fibres reduced hence (because the characteristic is 0) geometrically reduced, the map $f$ is in fact étale; and closed points have residue field $\mathbb{C}$ which is algebraically closed; hence your $\eta(v)$ is indeed the ... | 7 | https://mathoverflow.net/users/17988 | 425888 | 172,892 |
https://mathoverflow.net/questions/425647 | 1 | I want to compute a integral of a polynomial $f(x, y)$ over a polygon domain $D$ of $n$ sides.
$$
I(f) = \int\_{D} f(x, \ y) \ dx \ dy
$$
The vertex of this polygon are
$$\vec{p}\_{i} = (x\_i, \ y\_i) \ \ \ \ \ \ \ \ \forall \ i = 1, \ 2, \ \cdots , \ n$$
The main aproach is transform this integral over the dom... | https://mathoverflow.net/users/173662 | Double integral in a polygon domain | In the example, I got that
$$
I(1) = \sum\_{i=1}^{n} \dfrac{(x\_{i+1}+x\_{i})(y\_{i+1}-y\_{i})}{2} = \sum\_{i=1}^{n} \dfrac{(x\_{i+1}-x\_{i})(y\_{i+1}+y\_{i})}{2}
$$
And it's indeed correct. If I expand around $j$, I get that
$$
I(1) = \sum\_{i=1}^{j-2} \square + x\_{j}\left(y\_{j+1}-y\_{j-1}\right) + y\_{j}\left... | 1 | https://mathoverflow.net/users/173662 | 425903 | 172,894 |
https://mathoverflow.net/questions/425803 | 7 | Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric realization functor.
Is an analogous statement true for *topological groups*? That is, given a topological group $G$, regarded... | https://mathoverflow.net/users/144250 | Simplicial nerve of a topological group | This is an answer to the edited question.
First, observe that the composition of functors $\def\N{{\rm N}}\def\Sing{{\rm Sing}}\N∘\Sing$ in the main post computes
the homotopy colimit of the simplicial object $[n]↦(\Sing(G))^n$.
This can be seen as follows.
Observe that the functors $\Sing$ and the homotopy coheren... | 3 | https://mathoverflow.net/users/402 | 425910 | 172,896 |
https://mathoverflow.net/questions/425906 | 1 | I am interested in the convergence of the following Euler product:
$$
\prod\_p \frac{1}{1-\chi(p)\cdot p^{-s}}.
$$
The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =3$ and
$\chi(p)=-1$ if $p \bmod 4 =1$. Here $\chi(2)=\pm 1$, the sign does not matter. Also $s=\sigma+ it$ to use the ... | https://mathoverflow.net/users/140356 | Abscissa of convergence for a very specific Dirichlet series / Euler product | What do you mean by the word "orbit"? Please define that term in the body of your question.
You ask where the product over $p$ converges. Although you did not specify it, I assume you mean *product with terms in the order of increasing $p$*. You need to specify the order of the terms if you're dealing with something ... | 7 | https://mathoverflow.net/users/3272 | 425913 | 172,898 |
https://mathoverflow.net/questions/425920 | 20 | Does there exist a pair of finite groups $G$ and $H$ satisfying both of the short exact sequences $1 \rightarrow G \rightarrow H \rightarrow A\_4 \rightarrow 1$ and $1 \rightarrow G \rightarrow H \rightarrow D\_6 \rightarrow 1$? Of course the homomorphisms $G \to H$ in these short exact sequences are not the same.
| https://mathoverflow.net/users/172799 | Pair of short exact sequences of groups | Call two finite groups $Q\_1$ and $Q\_2$ *compatible* if there exists a finite group $G$ with two isomorphic normal subgroups $N\_1$ and $N\_2$ such that $G/N\_i\cong Q\_i$.
One can show the following:
**Proposition:**
If two groups are compatible, then they have subnormal series of the same length with the same fa... | 26 | https://mathoverflow.net/users/22377 | 425924 | 172,901 |
https://mathoverflow.net/questions/425453 | 1 | Fix $\alpha \in (0,1)$ and $\psi\in C^{\infty}\_{c}(\mathbb{R}\to \mathbb{R})$. For a smooth function $\phi\geq 0$ define the integral $$J\_{\alpha}(\phi):=\int \frac{\psi}{\phi^{\alpha}}.$$
If $|\phi^{\prime}|$ is bounded below away from zero, then
$J\_{\alpha}$ exists for all $\alpha<1$. Indeed, one can simply integr... | https://mathoverflow.net/users/116555 | Determine $\alpha \in (0,1)$ such that $J_{\alpha}(\phi):=\int \psi/\phi^{\alpha}$ exists? | $\newcommand\al\alpha\newcommand\R{\mathbb R}$Let us prove the following generalization of your desired statement:
>
> Let $\psi\colon\R\to\R$ be a continuous function with compact support $S$. Let $\phi\colon\R\to\R$ be a nonnegative twice differentiable function with the following property: for any $x\_0\in S$ su... | 1 | https://mathoverflow.net/users/36721 | 425941 | 172,904 |
https://mathoverflow.net/questions/425895 | 1 | I have a question on subfactors of $B(H)$ (the von Neumann algebra of bounded operators on a complex Hilbert space).
Say I have a subfactor $M$ of $B(H)$. Is it true that another subfactor $N \subset B(H)$ which commutes with $M$ and is such that the vN algebra $(M\cup N)''$ generated by $M$ and $N$ is all of $B(H)$,... | https://mathoverflow.net/users/485160 | Commuting and generating subfactors of $ B(H)$ | Let $N \subset M'$ be an *irreducible* inclusion of subfactors, meaning that $N \neq M'$ but the relative commutant is trivial, i.e., $N' \cap M' = \mathbb{C}\cdot I$. There are lots of examples of such things (Google "example of irreducible subfactor"). Then the commutant of the von Neumann algebra generated by $M \cu... | 0 | https://mathoverflow.net/users/23141 | 425942 | 172,905 |
https://mathoverflow.net/questions/425943 | 0 | Let $\{ f\_n \}$ be a sequence of Schwartz functions on $\mathbb{R}^n$ converging to some Schwartz function $f$ in the $L^2(\mathbb{R}^n)$ norm.
Then, it is an elementary fact that we can extract a subsequence converging almost everywhere to $f$.
However, I wonder if we can extract "nicer" subsequences as well. For... | https://mathoverflow.net/users/56524 | $L^2$ convergence of Schwartz functions to a Schwartz function and possibility of extracting a "nicer" subsequence? | To get an example: consider non-negative smooth functions $\chi\_n$ with compact support in the intervals $[0, 1/n]$ with maximum $1$ at some point in this interval. Note that such functions indeed exist and belong to the Schwartz space (being smooth with compact support). Their $L^2$-norm is then at most $1/n$, thus t... | 5 | https://mathoverflow.net/users/12482 | 425946 | 172,906 |
https://mathoverflow.net/questions/425939 | 3 | Consider the following weak version of the Noether–Enriques theorem (field is $\mathbb{C}$):
>
> Let $\varphi:X\rightarrow Z$ be a morphism from a smooth projective surface onto a smooth curve with $F\_z:=\varphi^{-1}(z)\cong\mathbb{P}^1$ for every point $z$. Then there exists a Zariski open neighborhood $U$ of any... | https://mathoverflow.net/users/nan | Noether–Enriques using Tsen's lemma | I will assume that $X$ is proper. Then the generic fiber is a smooth projective curve of genus $0$ over the function field $K = k(Z)$ but any such curve can be embedded as a conic in $\mathbb{P}^2\_K$ using the anticanonical linear series. By Tsen's theorem, the conic has a $K$-point which we can spread out to a $U$-po... | 3 | https://mathoverflow.net/users/12402 | 425949 | 172,907 |
https://mathoverflow.net/questions/425957 | 9 | I wonder whether $\sum\_{k=0}^n \exp(r\_k z)$ has a complex zero for any $n\in \mathbb{Z}\_n^\*,0=r\_0<r\_1<r\_2<\dotsb<r\_n$. I think the answer is affirmative.
| https://mathoverflow.net/users/481550 | Zeros of a complex function | An affirmative answer follows from (9) in [this paper by Ritt](https://www.ams.org/journals/tran/1929-031-04/S0002-9947-1929-1501506-6/S0002-9947-1929-1501506-6.pdf).
| 8 | https://mathoverflow.net/users/11919 | 425959 | 172,910 |
https://mathoverflow.net/questions/424902 | 7 | Let $B\_{0}\supseteq B\_{1}\supseteq\dots\supseteq B\_{\alpha}\supseteq\dots\,\,\left(\alpha<\kappa\right)$ be a descending sequence of complete Boolean algebras, $B\_{\kappa}:=\bigcap\_{\alpha<\kappa}B\_{\alpha}$, $G\_0$ a $V$-generic filter on $B\_0$ and for every $\alpha\leq\kappa$, $G\_{\alpha}:=G\cap B\_{\alpha}$.... | https://mathoverflow.net/users/91680 | Understanding descending intersections of generic extensions | The theorem should state that the separative *quotient* of $B\_{0}/{\sim}$
is isomorphic to $B\_{\kappa}^+$.
*Proof.* Recall that the separative quotient of a poset $P$ is the
unique (up to isomorphism) separative $Q$ such that there is an order preserving $h:P\to Q$ such that $x$ is compatible with $y$ iff $h(x)$
is... | 3 | https://mathoverflow.net/users/91680 | 425961 | 172,911 |
https://mathoverflow.net/questions/425960 | 6 | Last month @godelian, alias Christian Espíndola, in a FOM post has mentioned Joyal's proof of Godel's second incompleteness via the so-called **Arithmetic Universes**, introduced by Joyal around 1973, and further investigated by Vickers, Maietti and Taylor.
After perusing the relatively sparse literature on them, I d... | https://mathoverflow.net/users/15293 | Joyal arithmetic universes and the Box operator | As discussed in the comments, the AU's are all models of the initial one, which is equivalent to the pretopos completion of the syntactic category of PRA (in coherent logic). In particular a set model of PRA cannot be made initial because it could collapse two different arrows in the initial AU (i.e., there could be tw... | 6 | https://mathoverflow.net/users/12976 | 425965 | 172,913 |
https://mathoverflow.net/questions/425005 | 4 | Is it true that for every infinite group $G$ and every $m\in\mathbb{N}$ there are infinite subsets $A\_0,\dots,A\_{m-1}$ such that all the products $a\_0\cdot\dots\cdot a\_{m-1}$ with $a\_i\in A\_i$ are distinct?
(This is true at least when $G$ is either uncountable or contains an infinite abelian subgroup, and for $... | https://mathoverflow.net/users/484402 | Existence of m infinite subsets in an arbitrary group such that all products of one element from each (in order) are distinct | Yes, such infinite subsets $A\_0, \dots, A\_{m-1}$ of an infinite group $G$ always exist. We can show this by wedging the problem into one of two polar opposite cases: abelian and ICC (i.e. every non-trivial element has an infinite conjugacy class).
**Fact.** An infinite group either has an infinite abelian subgroup ... | 4 | https://mathoverflow.net/users/24447 | 425984 | 172,922 |
https://mathoverflow.net/questions/425352 | 1 | When I say that an $r$-sided simple (*i.e.*, not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular $n$-gon. Let $M(r,n)$ stand for the number of different simple $r$-gons that can be inscribed into a regular $n$-gon.... | https://mathoverflow.net/users/481690 | Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon? | $$M(r, n) = \frac{1}{n} \sum\_{d \,\mid\, \gcd(r,n)} \varphi(d) \binom{n/d}{r/d}$$
See [OEIS A047996](http://oeis.org/A047996) for references.
| 2 | https://mathoverflow.net/users/46140 | 425988 | 172,924 |
https://mathoverflow.net/questions/425981 | 5 | In the description of the integral Adams spectral sequence, representations of the following quiver (with relations) arise naturally:
* We have two objects $A, B$,
* we have two arrows $\pi: A \rightarrow B$ and $\delta: B \rightarrow A$ and
* we have a single relation $\delta \circ \pi = 0$.
Does this quiver arise... | https://mathoverflow.net/users/16981 | Quiver with two objects and two arrows composing to zero | This is a Nakayama algebra with Kupisch series [2,3] and the representation theory of Nakayama algebras (a Nakayama algebra is a quiver algebra with admissible relations whose quiver is a linear oriented line or a cycle) is completely understood in nearly all details. (see for example [Farnsteiner - Nakayama algebras: ... | 7 | https://mathoverflow.net/users/61949 | 425989 | 172,925 |
https://mathoverflow.net/questions/425987 | 2 | Consider the second-order ordinary differential system
$$
A y'' + i B y' - C y = 0,
$$
where $A$ and $C$ are real-valued $d\times d$ SPD matrices satisfying
$$
I \le A,C \le 2I,
$$
and $B$ is real-valued and symmetric, having operator 2-norm less than 2 (and $i = \sqrt{-1}$). I would like to show that
1. There are $2... | https://mathoverflow.net/users/98733 | A question about solutions to ordinary differential systems | EDIT 1: I think one needs to assume that $\|B\|<2$, since otherwise as @Iosif Pinelis example in the comments shows, the eigenvalues may be purely imaginary.
EDIT 2: I changed the notation so my answer become clearer. In the original answer, I had the following fact in mind: if $P(\xi)$ is the principle symbol of a l... | 2 | https://mathoverflow.net/users/144247 | 426010 | 172,931 |
https://mathoverflow.net/questions/426003 | 1 | Let $f(x,y)=\frac{10xy-(x+y)+1}{8xy-2(x+y)+5}$ and $g(x,y)=\frac{1}{4}\left[1+\frac{1}{3}(4x-1)(4y-1)\right]$. I want to prove that for any $0.5\le a\le b\le 1$ and $0.7\le c\le d\le 1$, it holds that $g(f(a,b),f(c,d))>f(g(a,c),g(b,d))$. I have checked the result by programming a search algorithm that this seems to be ... | https://mathoverflow.net/users/52871 | An inequality in four variables | $\newcommand{\dif}{\text{dif}}\newcommand{\num}{\text{num}}\newcommand{\den}{\text{den}}\newcommand{\nnt}{\text{nn2}}$We want to show that
\begin{equation}
\dif:=g(f(a,b),f(c,d))-f(g(a,c),g(b,d))>0 \tag{?}\label{?}
\end{equation}
given conditions
\begin{equation}
\tfrac12\le a\le b\le 1,\quad \tfrac7{10}\le c\le d\le... | 2 | https://mathoverflow.net/users/36721 | 426016 | 172,932 |
https://mathoverflow.net/questions/426019 | 2 | Let $A$ be a non-unital $C^\*$-algebra. Let $S\subseteq A^\*$ be a set of continuous functionals that separates the points of $A$. Every element $\omega \in A^\*$ extends uniquely to a strictly continuous functional $\omega \in M(A)^\*$, so we can ask: does $S$ also separate the points of $M(A)$? Concretely, if $m \in ... | https://mathoverflow.net/users/216007 | If $S\subseteq A^*$ is separating, does $S$ also separate $M(A)$? | You're right, it's not true in general. If $A = c\_0$ then the multiplier algebra is the same as the double dual, $l^\infty$. Then we want a linear subspace of $l^1$ which is weak\* dense (so it separates $c\_0$) but not norm dense (so it doesn't separate $l^\infty$). This can be achieved by taking any $\vec{a} \in l^\... | 5 | https://mathoverflow.net/users/23141 | 426025 | 172,935 |
https://mathoverflow.net/questions/426038 | 14 | For nice topological spaces (say Haudorff spaces) $X$ and $Y$, there is a bijection between continuous maps $X\to Y$ and isomorphism classes of geometric morphisms $\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$.
Question: Is there a similar statement for "nice" schemes, i.e., that morphisms of schemes $X\to Y$ correspond bijecti... | https://mathoverflow.net/users/485276 | Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi | Below is a proof that the (pseudo)functor that sends a scheme to its petit étale topos is not fully faithful, for any category of schemes over an algebraically closed base field $k$, assuming that this category contains both $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k[t])$.
*Proof:* The morphisms of schemes
$\mathrm{Spec... | 12 | https://mathoverflow.net/users/37368 | 426046 | 172,938 |
https://mathoverflow.net/questions/414390 | 3 | For a knot $K$, let $J\_N(K)$ denote the $N$th Kashaev invariant of $K$. This [is the same as](https://arxiv.org/abs/math/9905075) the $N$th colored Jones polynomial evaluated at an $N$th root of unity (or $2N$th depending on your conventions for what the variable is).
In the literature on the volume conjecture it is... | https://mathoverflow.net/users/113402 | Closed-form expressions for the Kashaev invariant via surgery | I think the reference I was looking for was [H], which indeed uses presentations of various knots as surgery on the Borromean rings to derive closed forms for their Kashaev invariants.
Skein modules also show up, which I think is where the surgery formula comes from.
In [H] the author cites another paper "to appear" ... | 0 | https://mathoverflow.net/users/113402 | 426055 | 172,941 |
https://mathoverflow.net/questions/425849 | 7 | So this is probably blasphemist to ask and I've resisted asking this for a while. Essentially my question is why are locally symmetric spaces/Shimura varieties the "right" object to study from a number theoretic perspective?
The merit of them, say from a Langlandian perspective, is for sure beyond debate, and certain... | https://mathoverflow.net/users/152554 | Why are Shimura varieties the "right" objects? | This is a long comment. Number theorists have long been interested in modular functions and modular forms, which are functions on the complex upper half plane. Elliptic modular curves are the spaces on which these functions live. An important property of modular curves, which helps explain why they are of interest to n... | 3 | https://mathoverflow.net/users/nan | 426058 | 172,943 |
https://mathoverflow.net/questions/426052 | 2 | How to prove the following inequality $$\forall t>0,\quad\int^\infty\_0 \sin(rt)\frac{r^3}{\sinh^2(r)} dr\leq c \big(te^{-At}\big)$$
for some constants $A>0,c>0$
| https://mathoverflow.net/users/172078 | The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$ | This is to complement the answer by Carlo Beenakker by showing that
$$I(t)\le\pi^4 te^{-\pi t}\tag{1}\label{1}$$
for real $t\ge0$, where $I(t)$ is the integral in question.
Indeed, according to Carlo Beenakker,
$$I(t)=\frac{\pi ^3}{8} \frac{\pi t \cosh \pi t+2\pi t-3 \sinh \pi t}{\sinh^4(\pi t/2)}.$$
This expression ... | 5 | https://mathoverflow.net/users/36721 | 426061 | 172,945 |
https://mathoverflow.net/questions/426043 | 3 | Let $E$ be a nowhere dense subset of $\mathbb{R}\times \mathbb{R}$.
For $x\in \mathbb{R}$, define
$$E\_x=\{ y\in\mathbb{R}\mid (x,y)\in E\}.$$
Let $D$ denote the set of $x$ for which $E\_x$ is NOT nowhere dense in $\mathbb{R}$.
By the Kuratowski-Ulam Theorem, we know that $D$ is of first
cateogory in $\mathbb{R}$. My q... | https://mathoverflow.net/users/122378 | Kuratowski-Ulam Theorem, nowhere dense set in product space | List the rational numbers as $\{q\_n : n = 1, 2, \cdots \}$ and for each $n$, let $X\_n = \{q\_n\} \times [n, n+1]$. Finally, let $X = \cup\_{n = 1}^\infty X\_n$. Then $X$ is nowhere dense in $\mathbb R\times\mathbb R$ and for $x$ a rational number, $E\_x$ is a non-degenerate closed interval, so the set $D$ is the set ... | 6 | https://mathoverflow.net/users/89233 | 426068 | 172,948 |
https://mathoverflow.net/questions/426049 | 1 | I would like to give the following object, $M=S^5 \setminus \sqcup\_{2 \text{ copies}} \text{int}(S^1\times D^4)$, a handle decomposition. It is then to be attached to another manifold. along the two copies of $S^1\times S^3$, so there is no need to start with a $0$-handle. My attempt, below, seems quite complicated (t... | https://mathoverflow.net/users/477214 | An example of handle decomposition on modified $S^5$ | I didn't really follow the details of your description, but here is a handle decomposition along the lines you suggest. Start with $((S^1 \times S^3)\_a \coprod (S^1 \times S^3)\_b) \times I$ where $I= [0,1]$. Add a 1-handle along $((S^1 \times S^3)\_a \coprod (S^1 \times S^3)\_b) \times \{1\}$. Note that by choosing t... | 2 | https://mathoverflow.net/users/3460 | 426086 | 172,951 |
https://mathoverflow.net/questions/426084 | 3 | Let $K$ be a cubic field and let $\mathcal{O}\_K$ be its ring of integers. Does there always exist elements $\alpha, \beta \in \mathcal{O}\_K$ with $\text{Tr}(\alpha) = \text{Tr}(\beta) = 0$ such that $\{1, \alpha, \beta\}$ forms a $\mathbb{Z}$-basis for $\mathcal{O}\_K$? If the answer is generally no, then what about ... | https://mathoverflow.net/users/10898 | Trace-free basis for $\mathcal{O}_K$, $K$ a cubic field | No. There do not always exist such $\alpha$ and $\beta$. If $K$ is a cubic field and such $\alpha$ and $\beta$ exist, then for all $x \in \mathcal{O}\_{K}$, ${\rm Tr}\left(\frac{1}{3} \cdot x\right) \in \mathbb{Z}$ and this implies that $1/3$ is in the inverse different of $\mathcal{O}\_{K}$, and hence that the differe... | 8 | https://mathoverflow.net/users/48142 | 426088 | 172,952 |
https://mathoverflow.net/questions/426075 | 5 | Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=\pi(m)$. It is clear that $G$ acts smoothly on each fiber $M\_b$ for $b\in B$.
Noe fix a $g\in G$. For each $b\in B$ ... | https://mathoverflow.net/users/24965 | Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds? | I think the answer is yes.
Since $G$ is compact, there is a $G$-invariant Riemannian metric on $M$ (by averaging any metric). The orthogonal distribution to the fiber for this metric is a $G$-invariant Ehresmann connection, and the parallel transport for this connection thus commutes with the $G$-action.
This shows... | 9 | https://mathoverflow.net/users/173096 | 426089 | 172,953 |
https://mathoverflow.net/questions/426096 | 1 | Can two increasing and differentiable convex functions agree exactly on a countable set of cardinality greater than two?
| https://mathoverflow.net/users/473751 | Intersecting points of increasing convex functions | Let
$$f\_\pm(x):=e^x(1\pm c\sin x)$$
for some $c\in(0,1/2]$ and all real $x$.
Then $f'\_\pm(x)=e^x(1\pm c(\cos x+\sin x))>0$ and $f''\_\pm(x)=e^x(1\pm 2c\cos x)\ge0$ for all real $x$.
So, $f\_+$ and $f\_-$ are increasing differentiable convex functions that agree exactly on the countable set $\pi\mathbb Z$.
---... | 1 | https://mathoverflow.net/users/36721 | 426098 | 172,955 |
https://mathoverflow.net/questions/426099 | 5 | The minimal model program attempts to classify algebraic varieties up to birational equivalence. For compact Riemann surfaces, Riemann's uniformization theorem tells us that the geometry of the curve is determined primarily by its genus. In particular, if $M$ is a compact Riemann surface of genus $g$, then $g=0 \implie... | https://mathoverflow.net/users/471309 | Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP | This is a very long comment.
First I should point out that by GAGA, biholomorphic = isomorphic for smooth projective varieties so in this case, classification up to biholomorphism is the same as classification up to isomorphism. Note the non-compact case can be quite wild in higher dimensions (see [this example](http... | 8 | https://mathoverflow.net/users/12402 | 426109 | 172,956 |
https://mathoverflow.net/questions/426103 | 9 | In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple Cauchy complete Archimedean ordered fields, which are not provable to be equivalent to each other: one cannot in general pr... | https://mathoverflow.net/users/483446 | Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field | I think see how this is related to your other questions. These are not resolved last I checked, but the following will answer this question whichever way the other questions end up going.
I believe the following are constructively equivalent for any notion of $\mathbb R$ which is Cauchy complete assuming that unique ... | 9 | https://mathoverflow.net/users/2000 | 426110 | 172,957 |
https://mathoverflow.net/questions/426097 | 10 | What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in the study of modular forms. By summing over lattice points, one obtains a holomorphic modular form
$$E\_{2k}(z) = \su... | https://mathoverflow.net/users/38145 | History of points of view on Eisenstein series | **Q:** *What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?*
**A:** Yes, he did, see [Elliptic Functions According to
Eisenstein and Kronecker](https://ems.press/content/serial-article-files/8714).
The reference is [Beiträge zu Theorie der elliptischen Functionen. VI... | 7 | https://mathoverflow.net/users/11260 | 426117 | 172,958 |
https://mathoverflow.net/questions/426119 | 1 | It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. More precisely, for any $M>0$ there exists a constant $C\_M>0$ such that
\begin{equation}\tag{1}\label{1}
|\widehat{\varphi... | https://mathoverflow.net/users/157356 | Decay of the Fourier transform of a non-differentiable function | I answer question $(1)$, assuming only that $\varphi$ is integrable.
If $(1)$ holds, then $\hat{\varphi}$ is also integrable, so Fourier inversion formula applies. For almost every $x \in \mathbb{R}$,
$$\varphi(x) = \int\_{\mathbb{R}}\hat{\varphi}(\lambda)e^{i2\pi x\lambda}\mathrm{d}\lambda.$$
Since for every $n \in ... | 3 | https://mathoverflow.net/users/169474 | 426122 | 172,960 |
https://mathoverflow.net/questions/426123 | 6 | In *Locally Presentable and Accessible Categories*, page 12 (10),
>
> A topological space is finitely presentable in $\mathbf{Top}$, the category of topological spaces and continuous functions, iff it is finite and discrete.
>
>
>
But the explanation after this sentence makes little sense to me. In particular,... | https://mathoverflow.net/users/239112 | Why finitely presentable objects in Top need to be discrete? | $\newcommand{\colim}{\operatorname{colim}}$I believe there is a mistake in the argument - as written, the obvious maps $A\to D\_n$ are continuous, so the map $A\to\colim D\_n$ certainly factors through the maps $D\_n\to\colim D\_n$. I also don't believe the claim that $\colim D\_n$ is indiscrete is accurate - all open ... | 10 | https://mathoverflow.net/users/30186 | 426127 | 172,961 |
https://mathoverflow.net/questions/426136 | 7 | Question: If $X$ is a simplicial complex that's simply connected and $2$-dimensional, does there always exist a contractible subcomplex $Y$ satisfying $X^{(1)} \subseteq Y$?
The statement is true "down a dimension": If $X$ is connected and $1$-dimensional, then there exists a contractible subcomplex $Y$ satisfying $X... | https://mathoverflow.net/users/164670 | Contractible subcomplex containing 1-skeleton? | I think the answer is no.
In Hatcher's Algebraic Topology there is an example of an acyclic 2-dimensional complex with one 0-cell, two 1-cells and two 2-cells (Example 2.38). You start with a wedge of two circles, labeled $a$ and $b$, and attach two 2-dimensional cells by the words $a^5b^{-3}$ and $b^2(ab)^{-2}$. One... | 8 | https://mathoverflow.net/users/6668 | 426144 | 172,969 |
https://mathoverflow.net/questions/426126 | 19 | Suppose $X$ is an infinite set and $\mathcal{A}$ is a family of subsets of $X$.
(1) We say that $\mathcal{A}$ is **$k$-cover-free** if for every distinct $a\_0, a\_1, \cdots, a\_k \in \mathcal{A}$, $a\_0 \nsubseteq \bigcup\_{1 \leq i \leq k} a\_i$.
(2) We say that $\mathcal{A}$ is **$k$-good** if for every distinct... | https://mathoverflow.net/users/485348 | Family of sets with a covering property | Yes. Let $S$ be a set of cardinality $|S|\ge k+2$. Let $X=\binom Sk$, the set of all $k$-element subsets of $S$. For each $s\in S$ let $a\_s=\{x\in X:s\notin x\}$, and let $A=\{a\_s:s\in S\}$. It is easy to verify that $A$ is $k$-cover-free and $(k+1)$-good.
Given distinct elements $s\_0,s\_1,\dots,s\_k\in S$, we hav... | 13 | https://mathoverflow.net/users/43266 | 426152 | 172,974 |
https://mathoverflow.net/questions/426156 | 6 | For a positive integer $n$, let $\mathcal{P}$ be the power set of $[n]$. Consider the graph $G$ with $\mathcal{P}$ as its vertex set, and, for $S\_1,S\_2 \in \mathcal{P}$, the edge $(S\_1,S\_2)$ exists iff $S\_1 \subset S\_2$.
(That is, $G$ is the transitive closure of the Hasse diagram of the "$\subset$" relation.)
... | https://mathoverflow.net/users/156157 | Coloring of a graph representing the power set | For $k\ge n+1$ there is a proper coloring of $G$ where each set in $\mathcal{P}$ is colored by its cardinality. Then no vertex $v$ has a neighbor with the same color.
| 8 | https://mathoverflow.net/users/24076 | 426161 | 172,977 |
https://mathoverflow.net/questions/423434 | 8 | I recently encountered an interesting space. It is a compactification of the space of $ n$ points in $ \mathbb A^1 $ modulo translation, $ (\mathbb A^1)^n / \mathbb G\_a $.
Let $ n \in \mathbb N $ and let $ p([n]) = \{ (i,j) \in \{1, \dots, n\}^2 : i \ne j \} $. Let $ Y\_n $ be the subscheme of $ (\mathbb P^1)^{p([n]... | https://mathoverflow.net/users/438 | A compactification of the space of points on the affine line | Given a linear space $L \subset \mathbb{A}^n$, the closure of $L$ inside of $(\mathbb{P}^1)^n$ is called the **Schubert variety** of $L$, or of the hyperplane arrangement obtained by intersecting the coordinate hyperplanes with $L$. Your variety is the Schubert variety of the Coxeter arrangement of type $A\_{n-1}$.
T... | 6 | https://mathoverflow.net/users/10273 | 426165 | 172,978 |
https://mathoverflow.net/questions/426162 | 3 | **Background**
Recently I asked [a question](https://mathoverflow.net/questions/425803/simplicial-nerve-of-a-topological-group) on a particular construction ot the classifying space of a topological group. I got an answer, but it relied on nontrivial Quillen equivalences between various models of $(\infty,1)$-categor... | https://mathoverflow.net/users/144250 | Homotopy coherent nerve versus simplicial nerve | Both $\def\W{{\bar W}}\W$ and $\def\N{\mathfrak{N}}\N$ are right Quillen functors from the model category of simplicial groups to the model category of reduced simplicial sets (see the original paper by Dwyer–Kan, or Proposition V.6.3 in Goerss–Jardine).
Thus, to show that the natural transformation $\W→\N$ is a weak e... | 3 | https://mathoverflow.net/users/402 | 426169 | 172,979 |
https://mathoverflow.net/questions/426160 | 3 | Let $T$ be a regular tournament, and $u \in V(T)$. Let $Out(u) \subset V(T)$ denote the set of vertices such that the edges between $u$ and them go out of $u$. Similarly define $In(u)$. Let two distinct vertices $u,v$ be called $\textbf{antipodal}$ if $Out(u) \setminus v = In(v) \setminus u$ (and thus also $In(u) \setm... | https://mathoverflow.net/users/143629 | About regular induced subgraphs of regular tournaments | There are regular tournaments [here](https://users.cecs.anu.edu.au/%7Ebdm/data/digraphs.html) and if you choose one at random except on the smallest sizes you will find a counterexample with high probability.
Enumeration gives a bound (which could be turned into an accurate estimate but I won't). Let $T(n)$ be the nu... | 3 | https://mathoverflow.net/users/9025 | 426171 | 172,980 |
https://mathoverflow.net/questions/148862 | 5 | Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.
We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$
Recall the definition of [ends](http://ncatlab.org/nlab/show/end%E2%80%8E). Taking the end is an operation of signature
$$\mathrm{Dist}(X... | https://mathoverflow.net/users/1261 | Ends and coends – analogues for higher arity – Horn Filling | This is exactly the subject of the paper [Coends of higher arity](https://arxiv.org/abs/2011.13881) by Loregian and de Oliveira Santos.
| 4 | https://mathoverflow.net/users/152679 | 426180 | 172,983 |
https://mathoverflow.net/questions/426066 | 6 | Let $k$ be a nonarchimedean local field and $G$ a reductive $k$-group,
which we assume to be semisimple and simply-connected. Recall that an abstract group $H$ is *perfect* if it is generated by commutators, that is, equals its derived subgroup.
**Question**: Is $G(k)$ perfect?
When $G$ is isotropic, $G(k)$ is know... | https://mathoverflow.net/users/174855 | Is every compact simply-connected reductive p-adic group perfect? | It seems that the answer to the **Question** is "no" in general (in the anisotropic case): for example, if $D$ is a finite-dimensional central division algebra over $k$, then Theorem 1.9 in Platonov-Rapinchuk (based on the article of Riehm quoted above) implies that $$[\textrm{SL}\_1(D), \textrm{SL}\_1(D)] \neq \textrm... | 3 | https://mathoverflow.net/users/120578 | 426211 | 172,994 |
https://mathoverflow.net/questions/426219 | 1 | Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v\_1,v\_2$ be two orthonormal vectors.
I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value
$$\mathbf E\sqrt{4\langle Av\_1,v\_2 \rangle^2 + (\langle Av\_1,v\_1 \rangle-\langle Av\_2,v\_2 \rangle)^2}.$$
We ... | https://mathoverflow.net/users/457901 | Expectation value of random GUE matrix | From the definion of the GUE for $n\times n$ Hermitian matrices $A$, one has the integral expression.
$$\mathbb{E}[f({\rm Re}\,a\_{12},{\rm Im}\,a\_{12},a\_{11},a\_{22})]=\frac{n^2}{2\pi^2}\iint\!\!\!\!\!\iint\_{-\infty}^{\infty} f(x\_1,x\_2,y\_1,y\_2)$$
$$\qquad\qquad\times\exp\left(-n(x\_1^2+x\_2^2+y\_1^2/2+y\_2^2/2)... | 2 | https://mathoverflow.net/users/11260 | 426221 | 172,998 |
https://mathoverflow.net/questions/426199 | 0 | Consider a set of N teams (N even number) that must make a
Round Robin Tournament. To each pair i; j, i ≠ j, of teams there is associated level
of interest si,j ∈ {1;2;3} of the match between them (1 = minimum interest, 2 = medium
interest, 3 = maximum interest). Define a calendar in such a way that:
1. in each day t... | https://mathoverflow.net/users/485367 | Round Robin volleyball Tournament | Assuming binary decision variable $x\_{ijd}$ indicates whether teams $i$ and $j$ play each other on day $d$, introduce a decision variable $z$, and maximize $z$ subject to linear constraints
$$z \le \frac{1}{N/2} \sum\_{i,j} s\_{ij} x\_{ijd} \quad \text{for all $d$}.$$
| 0 | https://mathoverflow.net/users/141766 | 426225 | 173,000 |
https://mathoverflow.net/questions/423925 | 2 | Suppose I have $n$ Boolean variables $x\_1,\dots,x\_n$, and an objective function of the form $f(x\_1,\dots,x\_n) = \sum\_{a\_1,\dots,a\_n}c\_{a\_1,\dots,a\_n} x\_1^{a\_1} \cdots x\_n^{a\_n}$ with $(a\_1,\dots,a\_n) \in \{0,1\}^n$. I know this is a bit vague, but suppose that there aren't that many nonvanishing coeffic... | https://mathoverflow.net/users/3621 | Optimizing a multilinear function over the vertices of the cube | (Making a CW answer to take this off the list of unanswered questions.)
As noted in comments by Jason Gaitonde, MAXCUT for a graph $G=(V,E)$ is equivalent to maximization of the quadratic multilinear polynomial
$$-\sum\_{(i,j)\in E}x\_ix\_j$$
over $\{-1,1\}^V$, hence the problem is NP-hard.
| 2 | https://mathoverflow.net/users/12705 | 426232 | 173,002 |
https://mathoverflow.net/questions/426092 | 4 | I'm looking for the name of what I suspect must be a standard property, and also for a possible statement about that property.
First the property: $W=a\_0\ldots a\_{n-1}$ has *this property* if for all $1\le k<n$,
$a\_0\ldots a\_{k-1}\ne a\_{n-k}\ldots a\_{n-1}$.
In particular, this implies that in any finite or in... | https://mathoverflow.net/users/11054 | Word combinatorics terminology question | Yes you find these in all infinite mixing SFTs. More is true. As mentioned, these words are sometimes called **unbordered**, I'll use that word.
The following is Theorem 8.3.9 in [1].
>
> Theorem. Let $x \in A^\mathbb{N}$ for any alphabet $A$. If $x$ is not periodic, then for any $m$, there exists an unbordered w... | 6 | https://mathoverflow.net/users/123634 | 426233 | 173,003 |
https://mathoverflow.net/questions/426243 | 2 | Let $\mathbb{N}$ denote the set of positive integers, and consider the graph $(\mathbb{N}, E)$ where a set $\{a,b\}$ of two distinct positive integers belongs to $E$ if there is an integer $k>1$ such that $a = k\cdot b$ or $b = k \cdot a$.
**Questions.**
1. Is there a bijection $p:\mathbb{N}\to\mathbb{N}$ such that... | https://mathoverflow.net/users/8628 | Hamiltonian path in divisibility graph | Yes, as for every countable graph on which any two vertices have infinitely many common neighbours. If you constructed a path $v\_1\ldots v\_m$, and $u$ is the first (with respect to a numeration chosen in advance) not visited vertex, then you may proceed with $\dots v\_{m+1}u$, where $v\_{m+1}$ is a not used yet commo... | 5 | https://mathoverflow.net/users/4312 | 426248 | 173,007 |
https://mathoverflow.net/questions/426241 | 1 | Let $\alpha, \beta:\mathbb R\_+\to [0,1]$ be continuous and decreasing functions s.t. $\alpha(0)=1=\beta(0)$ and $\alpha, \beta$ are continuously differentiable on $(0,\infty)$ satisfying for some $c>0$
$$-\frac{c}{\sqrt{t}}\le\alpha'(t), \beta'(t)\le 0,\quad \forall t>0.$$
Let $f,g :\mathbb R\_+\to [0,1]$ be continuou... | https://mathoverflow.net/users/261243 | Do these two pairs coincide at small time? | $\newcommand{\al}{\alpha}\newcommand{\be}{\beta}$The answer is no.
E.g., suppose that for all small enough $t>0$ we have
\begin{equation\*}
f(t)=t,\quad g(t)=0,\quad \al'(t)=-\frac1{\sqrt t}\,\Big(1+\sin\frac1t\Big),\quad \be'(t)=-\frac1{\sqrt t},
\end{equation\*}
with $\al(0)=1=\be(0)$.
Then all your conditions ... | 1 | https://mathoverflow.net/users/36721 | 426253 | 173,008 |
https://mathoverflow.net/questions/426246 | 4 | Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional?
This seems to be true for $\mathfrak{sl}\_n$, since this is the subject of... | https://mathoverflow.net/users/385475 | Are isotypic components of $S(\mathfrak{g})$ finite-dimensional? | I think there must be some different ideas of what your question means, looking at the comments. I understood it to mean as in the decomposition of tensor powers of a $\mathfrak{g}$-representation into irreducible subrepresentations. In which case, the answer is a definite yes. For a start, the action of $\mathfrak{g}$... | 2 | https://mathoverflow.net/users/163024 | 426257 | 173,009 |
https://mathoverflow.net/questions/426268 | 1 | Let $S(\omega)$ denote the collection of "sparse" infinite subsets of $\omega$, that is, $X\subseteq \omega$ is a member of $S(\omega)$ if and only if both $X$ and $\omega\setminus X$ are infinite.
Is there ${\cal S}\subseteq S(\omega)$ such that for all $a\neq b \in \omega$ we have $|\{s\in {\cal S}: \{a,b\} \subset... | https://mathoverflow.net/users/8628 | Steiner-like systems on $\omega$ | Sure, let $K$ be any countably infinite field and let $P$ be the projective plane (or a higher-dimensional projective space) over $K$. Let $S'$ be the set of lines in $P$ (where a line is regarded as a set of points), and transport the family $S'$ of subsets of $P$ to a family $S$ of subsets of $\omega$ via your favori... | 4 | https://mathoverflow.net/users/6794 | 426270 | 173,014 |
https://mathoverflow.net/questions/426261 | 2 | Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty\_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$$
for some constants $A>0,c>0$
| https://mathoverflow.net/users/172078 | The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$ | $\newcommand{\R}{\mathbb R}\renewcommand{\S}{\mathcal S}$This follows from [Theorem IX.14](https://www.google.com/books/edition/II_Fourier_Analysis_Self_Adjointness/Kz7s7bgVe8gC?hl=en), which states the following:
>
> Let $T\in\S'(\R^n)$. Suppose that the Fourier transform $\hat T$ of $T$ can be continued analytica... | 3 | https://mathoverflow.net/users/36721 | 426273 | 173,015 |
https://mathoverflow.net/questions/426236 | 3 | Consider $k\geq 2$ biinfinite arithmetic progressions $\mathcal A\_i=a\_i+b\_i\mathbb Z$ (for $i=1,\ldots,k$) in $\mathbb Z$.
(We suppose that $a\_i$ and $b\_i\geq 2$ are strictly positive integers.
One can assume $a\_i$ and $b\_i$ coprime and $a\_i$ can be reduced modulo $b\_i$ without loss of generality.)
The set $... | https://mathoverflow.net/users/4556 | Density for products of arithmetic progressions | I think Yaakov Baruch is right in general.
Let $D$ be the l.c.m. of all the $b\_i$, and let $S$ be the set of all residue classes in $\mathbb Z/D\mathbb Z$ met by $\mathcal P$. I claim that the density is $|S|/D$.
The upper bound is trivial. In what follows, I assume $(a\_i,b\_i)=1$, otherwise pass to a (1-dim) sub... | 5 | https://mathoverflow.net/users/17581 | 426289 | 173,019 |
https://mathoverflow.net/questions/418976 | 3 | Say that a digraph $(V,E)$ is *reducible* if there exist $x,y\in V$ with $x\ne y$ and such that for all $z\in V$, $(x,z)\in E\leftrightarrow(y,z)\in E$ and $(z,x)\in E\leftrightarrow(z,y)\in E$. It is clear that a reducible digraph has a non-trivial automorphism (swapping $x$ and $y$ and fixing the rest), but the inver... | https://mathoverflow.net/users/30392 | Digraph without "immediately isomorphic" vertices? | Just to provide a conclusion here (summing up the comments): *twin-free* seems to be the best and, dare I say, standard terminology; two points as in my question would then be called *twins*. Otherwise, *point-determining*, *R-thin*, or *mating-type* have been used to denote the same property.
| 0 | https://mathoverflow.net/users/30392 | 426291 | 173,020 |
https://mathoverflow.net/questions/370764 | 3 | It's all in the title: Is there really no name for categories in which all automorphisms are trivial? I've encountered some examples of these, the most prominent being the simplex category.
| https://mathoverflow.net/users/30392 | Name for "Category without nontrivial automorphisms"? | To sum up the comments & discussion: no established terminology seems to exist, but *essentially gaunt* would be a good name for categories without nontrivial automorphisms.
For a related notion, categories without notrivial *isomorphisms* are called *gaunt*, see <https://ncatlab.org/nlab/show/gaunt+category>
On th... | 0 | https://mathoverflow.net/users/30392 | 426299 | 173,021 |
https://mathoverflow.net/questions/426306 | 3 | If $K = \mathbb{Q}(\sqrt{d})$ is a real quadratic field, then any unit $u \in \mathcal{O}\_K^\times$ with $u > 1$ must not be too small: indeed, such a $u = u\_1 + u\_2 \sqrt{d}$ with $u\_1, u\_2 > 0$ must satisfy $u\_1^2 - d u\_2^2 = \pm 1$, so $u\_1 \gg \sqrt{d}$, say. Thus the gap between the smallest unit $u \in \m... | https://mathoverflow.net/users/10898 | Size of a generator of the unit group in a cyclic cubic field | Asking about a smallest unit bigger than $1$ in a unit group of rank greater than $1$ feels like the wrong question, sort of like asking for a smallest algebraic integer of absolute value greater than $1$ in a number field (inside $\mathbf C$) of degree greater than $1$. The ring of integers is discrete when you use al... | 10 | https://mathoverflow.net/users/3272 | 426311 | 173,024 |
https://mathoverflow.net/questions/426307 | 4 | *Note: Here we consider the Lebesgue measure on $[0, 1]$.*
Let $f\_n: [0, 1] \to [0, 1]$ be a sequence of measurable functions.
We say a measurable subset $E$ of $[0, 1]$ is a *condensation set* of the sequence $f\_n$ if there exists a subsequence $f\_{n\_k}$ and a function $f: [0, 1] \to [0, 1]$ (both depending on... | https://mathoverflow.net/users/173490 | A “compactness theorem” for measurable functions | Counterexample. For each $n$ let $k\_n$ be the characteristic function of $[0,\frac{1}{2^n}] \cup [\frac{2}{2^n},\frac{3}{2^n}] \cup \cdots$. Next observe that there are only countably many subsets of $[0,1]$ of the form: a finite union of open intervals with rational endpoints, whose total length is $\frac{1}{10}$. En... | 4 | https://mathoverflow.net/users/23141 | 426312 | 173,025 |
https://mathoverflow.net/questions/426284 | 2 | Let $\psi \in C^{\infty}\_{c}(\Omega)$ where $\Omega$ is a bounded smooth domain, and $\phi$ the solution to
\begin{equation\*}
-\Delta \phi =\psi, ~\phi|\_{\partial \Omega}=0.
\end{equation\*}
My question is how to get the following estimate :
$$\|\phi\|\_{C^1( \overline{\Omega})} \leq C \|\psi\|\_{L^{\infty}(\Omega)... | https://mathoverflow.net/users/166368 | Gradient estimate and $L^1$ theory for the Laplace operator | One way to get both is to use the estimates $
\|\phi\|\_{W^{2,p}(\Omega)} \leq C\|\psi\|\_{L^p(\Omega)}$
which hold when $1<p<\infty$ with a constant $C=C(p,\Omega,n)$. Taking $p>n$ by Sobolev embedding $$\|\phi\|\_{C^1(\Omega)} \leq c\_1 \|\phi\|\_{W^{2,p}(\Omega)} \leq c\_2 \|\psi\|\_{L^p(\Omega)} \leq c\_3 \|\psi\|\... | 2 | https://mathoverflow.net/users/150653 | 426316 | 173,028 |
https://mathoverflow.net/questions/426302 | 44 | A relation $R$ is *implicitly definable* in a structure $M$ if there is a formula $\varphi(\dot R)$ in the first-order language of $M$ expanded to include relation $R$, such that $M\models\varphi(\dot R)$ only when $\dot R$ is interpreted as $R$ and not as any other relation. In other words, the relation $R$ has a firs... | https://mathoverflow.net/users/1946 | Is multiplication implicitly definable from successor? | Contrary to my initial expectation, the answer is Yes.
This answer is based on the idea of Clemens Grabmayer, which makes the observation that addition $+$ is definable from multiplication $\cdot$ and successor.
The idea generalizes to the following:
**Theorem.** Suppose that relation $R$ is implicitly definable ... | 25 | https://mathoverflow.net/users/1946 | 426322 | 173,029 |
https://mathoverflow.net/questions/426272 | 0 | Set $\pi=\frac{1}{4}(\delta\_{(1,0)}++\delta\_{(1,3)}+\delta\_{(1,1)}+\delta\_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.
How to get the disintegration of $\pi$ with respect to $\mu$?
| https://mathoverflow.net/users/168083 | How to compute the unique disintegration w.r.t. the first coordinate? | By disintegration is meant that $\pi$ can by broken down as
$$
\pi(dx,dy) =\mu(dx)\pi\_{x}(dy),
$$
with $\pi\_x$ a Borel measurable family of sub-probability measures on $X$.
In your example:
$\pi\_{x\_1}={1\over 2}(\delta\_{0}+\delta\_{1})$ if $x\_1=1$; $=\delta\_{2}$ if $x\_1=2$; and $=0$ otherwise.
Then you have... | 1 | https://mathoverflow.net/users/42851 | 426324 | 173,030 |
https://mathoverflow.net/questions/426150 | 2 | This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult question, and for that I assume someone smarter than me has already found a better explanation.
To begin, we take a valu... | https://mathoverflow.net/users/133882 | Finding the repelling fixed point of an exponential, knowing only its attracting one | Okay, so I found the answer to this problem! It's a tad different than I thought, so I'll give the run down. Special thanks to @RolandBacher for getting me to think about $e$ as a branching point. The answer was simpler than I thought once I dug through it.
Let
$$
g(y) = y^{1/y}\\
$$
Then, there exists two branch... | 2 | https://mathoverflow.net/users/133882 | 426343 | 173,033 |
https://mathoverflow.net/questions/426326 | 3 | Consider a simple random walk $$\mathcal{X}\_t= \sum\_{n<t} X\_n,$$ where $P(X\_n=1)= P(X\_n=-1)= 1/2.$
If I put an extra condition that excludes cases with more than 5 consecutive +1, or -1 in the sum:
For every $n$, between 1 and t-4:
$$|X\_n+ X\_{n+1}+ X\_{n+2}+ X\_{n+3}+ X\_{n+4}|< 5.$$
Can I still expect t... | https://mathoverflow.net/users/422944 | Simple random walk with an extra condition | Let $Y\_n=(X\_n,X\_{n+1},X\_{n+2},X\_{n+3},X\_{n+4})$. The sequence $\{Y\_n\}$ is an aperiodic irreducible Markov chain on 30 states (vectors of $\pm 1$ that are not all $1$ or all $-1$). Its distribution is known as the Parry measure on a shift of finite type, See [4] or [5]. For the aperiodicity, it is important that... | 4 | https://mathoverflow.net/users/7691 | 426345 | 173,034 |
https://mathoverflow.net/questions/426047 | 13 | Recently Professor Peter Cameron posed a [number theory problem which is related to graphs of groups](https://cameroncounts.wordpress.com/2022/06/17/id-like-to-see-this-solved/). The problem is:
Problem:
>
> Let $n$ be a positive integer. Show that there exist subsets $A\_1, A\_2, …,A\_n$ of $\{1,2,…n\}$ with the... | https://mathoverflow.net/users/19885 | A nice problem by Peter Cameron on subsets of $\{1,\dots,n\}$ | Let $F\_n$ be a set of all irreducible fraction $\frac{p}{q}$ such that $0<\frac{p}{q}\leq 1,1\leq p,q\leq n$ and for $i\in \{1,...,n\}$, $D\_i$ be subset of $F\_n$ which contain all irreducible fraction of the form $\frac{k}{i}$. We have $D\_i$ are pairwise disjoint and $|D\_i|=\varphi(i)$. So we want a function $f:F\... | 33 | https://mathoverflow.net/users/432274 | 426347 | 173,035 |
https://mathoverflow.net/questions/426341 | 17 | Recently, I encountered this problem:
*"Given a sequence of positive number $(x\_n)$ such that for all $n$,
$$x\_{n+1}=x\_n+\frac{n}{x\_1+x\_2+\cdots+x\_n}$$
Find the limit $\lim\_{n \rightarrow \infty} \frac{x\_n}{\sqrt{n}}.$"*
As pointed out in the comment section of the following topic [[1](https://math.stacke... | https://mathoverflow.net/users/168269 | "Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from? | The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the [Emden–Fowler nonlinear differential equation](http://eqworld.ipmnet.ru/en/solutions/ode/ode0302.pdf),
$$f''(t)=t^{p}[f(t)]^q,$$
for $p=1$, $q=-1$. A particular solution is
$$f(t)=\lambda t^{(p+2)/(1-q)},... | 12 | https://mathoverflow.net/users/11260 | 426351 | 173,037 |
https://mathoverflow.net/questions/426269 | 1 | This question is pretty much [this](https://math.stackexchange.com/questions/4477619/a-map-from-the-symmetric-algebra-generated-by-the-first-cohomology-to-the-cohomo) question stated in slightly different way. All projective spaces are complex ones. Let's assume $X$ is a connected CW complex. We have a natural map in t... | https://mathoverflow.net/users/127776 | Dold-Thom and infinite symmetric power of an $H$-space | No, not quite.
Let's set the stage like this:
If $G$ is a connected $H$-space then $Sym^\infty G$ is a ring space, so that $Hom(X,Sym^\infty G)$ is also a ring space and $\pi\_\ast Hom(X,Sym^\infty G)$ is a graded ring.
Also, $Hom^0(X,G)$ is a connected $H$-space, so $Sym^\infty Hom^0(X,G)$ is a ring space and $\... | 2 | https://mathoverflow.net/users/6666 | 426361 | 173,040 |
https://mathoverflow.net/questions/426350 | 1 | They introduce a new correlation. For $\pi\in \Pi(\mu,\nu)$ the set of coupling of two probability measures $\mu$ and $\nu$ on a Polish space $(X,d)$. The author introduces a plugin estimator.
Question: How to understand $\hat{\pi}$?
---
For the estimator of the denominator, I use the same notation as the answe... | https://mathoverflow.net/users/168083 | How to get the estimator? | $\newcommand{\de}{\delta}$We have
\begin{equation\*}
W(\pi):=\frac{N(\pi)}{D(\pi)}, \tag{1}\label{1}
\end{equation\*}
where
\begin{equation\*}
N(\pi):=\int W\_1(\pi\_{x\_1},\nu)\mu(dx\_1)=\int W\_1(\pi\_{x\_1},\pi\_2)\pi\_1(dx\_1),
\end{equation\*}
\begin{equation\*}
D(\pi):=\int d(y,z)\nu(dy)\nu(dz)=\int d(y,z)\pi... | 1 | https://mathoverflow.net/users/36721 | 426367 | 173,043 |
https://mathoverflow.net/questions/425348 | 5 | Let $X$ be a compact metric space, $P\_X$ the set of Borel probability measures on $X$, and $K\_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff convergence" to be the metrisable topology on $P\_X$ whose convergence is given by
\begin{align\*}
& \mu\_n \overset{\text{w+H}}\to... | https://mathoverflow.net/users/15570 | Is the topology of weak+Hausdorff convergence Polish? | The set $MS=\{(\mu,K)\in P\_X\times K\_X:\mathrm{supp}(\mu)=K\}$ is of type $G\_\delta$ in $P\_X\times K\_X$ and hence the weak+Hausdorff topology on $P\_X$ is Polish.
Indeed, fix any countable base $\{U\_n\}\_{n\in\mathbb N}$ of the topology of $X$ and observe that $$MS=\bigcap\_{n\in\mathbb N}\{(\mu, K)\in P\_X\tim... | 2 | https://mathoverflow.net/users/61536 | 426371 | 173,044 |
https://mathoverflow.net/questions/426356 | 2 | Given $C^1([a, b])$ functions $f\_n$ that converge to a continuous real-valued function $f\_n \to f$ on a closed interval $[a, b] \subset \mathbb R$, suppose
$$
\int\_a^b |f\_n'(x)|^{1 + \epsilon} dx < M
$$
for all $n > 0$ and fixed $M < \infty$. For a small $\epsilon$, does this implies
1. $f$ is absolutely continuo... | https://mathoverflow.net/users/92099 | Does that exponent of (absolute value of derivative) is constrained implies Lipschitz continuity? | Q1: Yes. Since $f'\_n$ is bounded in $L^p$, a suitable subsequence will converge $f\_n'\to g$ weakly. Then also
$$
f(x) =\lim f\_n(x) = \lim \left( f\_n(a)+\int\_a^x f'\_n(t)\, dt \right) = f(a) + \int\_a^x g(t)\, dt ,
$$
so $f$ is absolutely continuous (with derivative in $L^p$).
Q2: No. For example $f(x)=|x|^{\alph... | 4 | https://mathoverflow.net/users/48839 | 426372 | 173,045 |
https://mathoverflow.net/questions/426334 | 21 | An [earlier question by Joel David Hamkins](https://mathoverflow.net/questions/426302/is-multiplication-implicitly-definable-from-successor) asked whether multiplication is implicitly definable in the structure $(\mathbb{N},S)$ of the naturals with successor. Here $R$ is implicitly definable if there is a formula $\phi... | https://mathoverflow.net/users/22930 | Is the set of primes implicitly definable from successor? | The set of primes is not implicitly definable in $(\mathbb{N},S)$. This is immediately implied by following:
**Theorem.** A unary predicate $P$ on $\mathbb{N}$ is implicitly definable in $(\mathbb{N},S)$ iff $P$ is a finally periodic set of naturals.
**Proof.** Suppose $P$ is finally periodic, i.e. there are natura... | 20 | https://mathoverflow.net/users/36385 | 426382 | 173,050 |
https://mathoverflow.net/questions/426374 | 1 | Take the $y$-axis and a set of $n$ curves starting from $y$-axis, labelled as $\mathcal{C}:=\{C\_1,C\_2,...,C\_n\}$. These curves fulfill the following conditions:
1. The curves all have a starting point somewhere on the $y$-axis and no two curves share the same starting point.
2. At least one curve is non-$x$-monoto... | https://mathoverflow.net/users/485561 | Representation of $x$-non-monotone curves with one intersection each by $x$-monotone curves | **This answer, as pointed out by Jan in the comment, is incorrect, as the definitions slightly differ. I leave it here as it contains useful pointers.**
If I understood your definitions correctly, your $\mathcal{C}$ would be called an outerstring graph, and your $\mathcal{C}'$ a double outerstring graph. There are ou... | 2 | https://mathoverflow.net/users/955 | 426383 | 173,051 |
https://mathoverflow.net/questions/426332 | 0 | Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}\_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}\_+$ let $$\alpha\_r(n)=\min\{\big|r-\frac{m}{n}\big|: m\in\mathbb{N}\}$$ be the best approximation of $r$ that can be obtained using $n$ as the denominator.... | https://mathoverflow.net/users/8628 | Measuring the quality of real approximation | This certainly can't happen for any $k > 1$. (There is no reason to assume that $k$ is an integer rather than real number). In fact, it can't even happen if you just restrict the $n$ to sufficiently large powers of two.
Suppose you know that for every sufficiently large integer $i$ there is an integer $m$ such that
... | 3 | https://mathoverflow.net/users/485571 | 426386 | 173,054 |
https://mathoverflow.net/questions/424400 | 0 | All,
Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After
performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$?
[Equations (1-18)](https://www.ecmwf.int/sites/default/files/elibrary/2017/Atmospheric_moist_convection.pdf) of dry statics energy... | https://mathoverflow.net/users/483883 | Could variable be still function in x and y after performing Reynolds averaging over area | Based on the comment by @CarloBeenakker, Coarse-grain averaging could be performed on a subset of the domain (spatial or temporal) so that the averaged value still varies with the independent dimension. I just wanted to document some resources (below) that might be helpful for others.
The classical Reynold average in... | 0 | https://mathoverflow.net/users/483883 | 426387 | 173,055 |
https://mathoverflow.net/questions/426331 | 16 | Let $F\_n$ be a free group of rank $n$. Say that $w \in F\_n$ is *non-reversible* if there does not exist any $f \in \text{Aut}(F\_n)$ such that $f(w) = w^{-1}$.
**Original Question.** Intuitively, I expect that most elements of $F\_n$ are non-reversible. However, I have trouble coming up with examples. Does anyone k... | https://mathoverflow.net/users/317 | Elements of a free group that can't be inverted by automorphisms | Proposition 8.7 of <https://arxiv.org/pdf/math/0303386.pdf#page107> says that the nonreversible elements are exponentially generic. This means you should get your limit in 5 with fast convergence. Note that they look only at cyclically reduced words which makes more sense since you can clearly reduce to conjugate eleme... | 9 | https://mathoverflow.net/users/15934 | 426393 | 173,058 |
https://mathoverflow.net/questions/426264 | 5 | Let $\mathsf{RO}(X)$ stand for the collection of regular open subsets of a topological space $X$ and let $d(X)$ be its density. It is well-known (see Theorem~3.3 of Hodel's chapter in the Handbook) that every regular space satisfies the inequality $|\mathsf{RO}(X)|\leq 2^{d(X)}$. What is an example of an infinite $T\_3... | https://mathoverflow.net/users/146942 | $|\mathsf{RO}(X)|$ vs. $2^{d(X)}$ for $T_3$ spaces | In fact, Joseph Van Name's argument gives a ZFC example. For $f \in \{0,1\}^{[0,1]}$ let $supp(f) = \{x \in [0,1] : f(x) = 1\}$. Let $X$ be the $\Sigma$-product given by $X = \{f \in \{0,1\}^{[0,1]}: |supp(f)| \leq \aleph\_0\}$ (with the subspace topology from the product). Then $X$ satisfies the countable chain condit... | 6 | https://mathoverflow.net/users/89233 | 426395 | 173,059 |
https://mathoverflow.net/questions/426407 | 10 | In a [previous question](https://mathoverflow.net/questions/425966/examples-of-locally-hyperbolic-groups), I asked about hyperbolic groups in which every finitely generated subgroup is hyperbolic. I am now curious about the reverse question: what are some examples of hyperbolic groups containing finitely generated non-... | https://mathoverflow.net/users/371866 | Examples of hyperbolic groups with non-hyperbolic subgroups | It is an open problem to find a coherent hyperbolic group with a finitely generated, non-hyperbolic subgroup. See [Wise's survey article](https://www.degruyter.com/document/doi/10.1515/9780691185897-014/html?lang=en) for the state of the art on coherent groups.
Wise, Daniel T. (3-MGL)
**An invitation to coherent grou... | 14 | https://mathoverflow.net/users/1463 | 426413 | 173,064 |
https://mathoverflow.net/questions/426300 | 3 | Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma\_n$ the set of addmisible words with the length $n$. For any $n\in \mathbb{N},$ and $I \in \Sigma\_n$, we define
$$\|A(I)\|:=\sup\_{x\in [I]}\|A^{n}(x)\|.$$
We can... | https://mathoverflow.net/users/127839 | the definition of the topological pressure for matrices | Yes. This is a consequence of the following simple exercise about exponential growth: if $a\_n\geq 0$ is any sequence such that $P = \lim \frac 1n \log a\_n$ exists, then $\frac 1n \log \sum\_{k=1}^n a\_k \to P$ as well. (Apply this with $a\_n = \sum\_{|I|=n} \|A(I)\|$.)
To prove the exercise, observe that $\liminf \... | 1 | https://mathoverflow.net/users/5701 | 426414 | 173,065 |
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