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https://mathoverflow.net/questions/360492 | 3 | I was studying Sheaves on Manifolds by Kashiwara and Schapira, and while the singular support seems like a complicated invariant I cannot seem to find a counterexample to the following:
Let $X$ be a smooth complex variety and $\mathcal{F}=IC(U,\mathcal{L})$ be an irreducible perverse sheaf, where $\mathcal{L}$ is a l... | https://mathoverflow.net/users/148583 | Singular support of an irreducible perverse sheaf | Take $X=\mathbb C$, $U=\mathbb C^\times$, and $\mathcal L$ a nontrivial rank 1 local system (with monodromy $\mu \neq 1$, say).
Then the singular support of $IC(U,\mathcal L)$ is the union $T^\ast \_X X \cup T^\ast\_0 X$ of the zero section (which is what your conjecture would give in this case) with the cotangent fi... | 5 | https://mathoverflow.net/users/7762 | 360494 | 151,748 |
https://mathoverflow.net/questions/359514 | 6 | In Macdonald's book, the Jack symmetric function $J\_{\lambda}(x\_1,\ldots, x\_n)$ for a partition $\lambda$
is defined by three properties (orthogonality, triangularity, and normalization). In the following paper (<http://www-math.mit.edu/~rstan/pubs/pubfiles/73.pdf>) its existence and uniqueness appear as Theorem 1.... | https://mathoverflow.net/users/45170 | Jack function in power symmetric basis | This explanation can be found in Macdonald's book "Symmetric Functions and Hall Polynomials" by looking at Ex.VI.4.3.
Note that Stanley's Laplace-Beltrami operator $D(\alpha)$ depends on $n$ and acts on the algebra $\mathbb{Q}(\alpha)\otimes \Lambda^n$, where $\Lambda^n$ denotes the algebra of symmetric polynomials i... | 3 | https://mathoverflow.net/users/51620 | 360508 | 151,751 |
https://mathoverflow.net/questions/357515 | 3 | Let $\Omega \subset \mathbb C^n$ be a bounded domain which is biholomorphic to the unit ball $B^n=\{|z|<1 \mid z\in \mathbb C^n\}$. Can we show $\Omega$ must be a Runge domain? By definition, $\Omega$ is a Runge domain if any analytic function in $\Omega$ can be approximated by polynomials.
Notice that J. Wermer gave... | https://mathoverflow.net/users/105900 | Is a domain biholomorphic to the unit ball a Runge domain? | The answer to your question is NO. Every domain in $\mathbb{C}^n$ can be embedded into $\mathbb{C}^n$ such that its image is non-Runge.
See:
Wold, E. F.: A Fatou–Bieberbach domain in $\mathbb{C}^2$ which is not Runge.\* Math. Ann. 340 (2008) 775–780
| 2 | https://mathoverflow.net/users/47862 | 360512 | 151,752 |
https://mathoverflow.net/questions/360384 | 9 | Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim\_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$.
A well known result of
Szpilrajn (He changed his name to Marczewski while hiding from Nazi persecution) proved in [S] asserts that
if $\mathcal{H}^{n+1}(X)=0$, then the topological di... | https://mathoverflow.net/users/121665 | Unknown work of Nöbeling on topological/Hausdorff dimension | So, the sought for paper is:
Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931).
And here is a ``translation" (to English and to modern math exposition standards, if such a thing exists.) What is called the set-theoretical dimension is defined inductively:... | 12 | https://mathoverflow.net/users/91442 | 360522 | 151,756 |
https://mathoverflow.net/questions/360528 | 6 | It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it.
I'm sure this can be done not only for $\text{Set}$-enriched categories but for general $\mathcal{V}$-enriched categories, with the appropriate notions of $\mathcal{V}... | https://mathoverflow.net/users/139854 | Universal model category as a $\text{sSet}$-enriched co-completion | Given $C$ a small category (eventually, a small simplicial category) I denote by $UC$ the projective model structure on the category of simplicial presheaves on $C$ as in the paper. Using the kind of argument you have in mind we obtain the following theorem:
**Theorem:** If $M$ is a simplicial model category, then th... | 3 | https://mathoverflow.net/users/22131 | 360532 | 151,759 |
https://mathoverflow.net/questions/360520 | 1 | Fix $u\_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\mathbb{R}$ satisfying the following variational equation
$$
\begin{cases}
\langle \nabla u, \nabla v\rangle\_{L^2(\Omega... | https://mathoverflow.net/users/105925 | A time dependent variational problem coming from a second order linear PDE | As you correctly pointed out, since the operators appearing here are only acting in space your problem amounts to solving an elliptic equation for each time. In particular assigning $u(0)=u\_0$ is meaningless: Even if the right-hand side $f$ is continuous in time, and without further compatibility assumptions between $... | 1 | https://mathoverflow.net/users/33741 | 360539 | 151,761 |
https://mathoverflow.net/questions/360363 | 0 | (The following question arises from my Math.SE question <https://math.stackexchange.com/questions/3643865>.)
---
Let $\rho$ be a probability measure on $\mathbb{R} \times (0,\infty)$, and writing $\ \pi\_1 \colon \mathbb{R} \times (0,\infty) \to \mathbb{R}\ $ and $\ \pi\_2 \colon \mathbb{R} \times (0,\infty) \to ... | https://mathoverflow.net/users/15570 | Is there a generalised version of the Donsker invariance principle for a "sort-of continuous-time-random-walk"? | I think Iosif Pinelis is correct, but his comment should be expanded as follows.
---
Notation: let
$$ p(t) = \lfloor t + 1\rfloor - t , \qquad q(t) = t - \lfloor t\rfloor . $$
Whenever we have a discrete-time process $Z\_n$, we extend it into a continuous one, piecewise linear, defined by:
$$ Z\_t = p(t) Z\_{\lfl... | 2 | https://mathoverflow.net/users/108637 | 360542 | 151,762 |
https://mathoverflow.net/questions/360536 | 11 | Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[t^{\pm 1},(t-1)^{-1}]$ obtained by inverting $t-1$. More specifically, I know that finitely generated projective $R$-mod... | https://mathoverflow.net/users/36098 | Are projective modules over a certain localised Laurent polynomial ring free? | The answer is yes. Given any projective module $P$ over $S^{-1}A$, where $A=\mathbb{Z}[t]$ (and works for many other rings too), it is the localization $S^{-1}M$ of a projective module over $A$. The reason is, you can always find such a finitely generated module $M$ with $S^{-1}M=P$, but you may replace $M$ with its do... | 10 | https://mathoverflow.net/users/9502 | 360543 | 151,763 |
https://mathoverflow.net/questions/360537 | 3 | I'm having a hard time proving the following:
>
> If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric on $TM$ whose index is $2s$. Let $J'$ be the natural almost complex structure on TM, then $TM(J',g^{D})$ is an indef... | https://mathoverflow.net/users/156541 | Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat | For a proof of this result (and a more general version), there's a paper by Satoh [1] which has a lot of detail. The main idea is that for $TM$ to be Hermitian, the Nijenhuis tensor of $J^\prime$ needs to vanish. However, when you calculate the Nijenhuis tensor, you find that it vaishes if and only if the torsion and c... | 4 | https://mathoverflow.net/users/125275 | 360544 | 151,764 |
https://mathoverflow.net/questions/360538 | 2 | I am studying in PDE and I have next definition :
>
> ***Definition***. Let $\Omega\subset\mathbb{R}^n$ open, connected. Then $\xi\in\partial\Omega$ is **regular** if there exists a superharmonic function $p$ in $\Omega$ such that $p>0$ in $\overline{\Omega}\backslash\{\xi\}$ and $p(\xi)=0$.
>
>
>
And with thi... | https://mathoverflow.net/users/151368 | What is the example of non-regular boundary point? | Example 1. In dimension 2, all isolated boundary points (punctures) are irregular.
Example 2. (Generalization) In dimension $n$ if you remove from a region $D$ a smooth
$n-2$ dimensional surface $S$, which does not separate $D$ then all points of this surface $S$ are irregular for
$D\backslash S$.
Example 3. (Furth... | 6 | https://mathoverflow.net/users/25510 | 360545 | 151,765 |
https://mathoverflow.net/questions/360515 | 3 | Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If we assume that for any $\omega \in \Gamma$, $f^{-1}(\{\omega\}) \ne \emptyset$, then my question is that, can we find a ... | https://mathoverflow.net/users/51546 | A question on preimage of a locally injective meromorphic function | No, this is not true. The simplest example is $f(z)=\int\_0^ze^{-\zeta^2}d\zeta$. Preimage of the real line consists of infinitely many curves, each of them is mapped homeomorphically onto one or two intervals of
the three intervals $(-\infty,-\pi/2),\; (-\pi/2,\pi/2),\; (\pi/2,+\infty)$. But none of the curves is mapp... | 4 | https://mathoverflow.net/users/25510 | 360547 | 151,766 |
https://mathoverflow.net/questions/360566 | 3 | **Motivation for my question:**
It is a well-known fact that there exists a bijection between the set of isomorphism class of
principal $G$ bundles over a nice topological space $X$ and the set $[X,B'G]$ of homotopy class of continuous maps from $X$ to the classifying space $B'G$ (using the different notation than co... | https://mathoverflow.net/users/86313 | What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)? | It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E\_1$-space). Pulling back the overcategory projection $EG=BG\_{/\ast}\to BG,$ where $\ast$ is the unique object of $... | 3 | https://mathoverflow.net/users/1353 | 360567 | 151,769 |
https://mathoverflow.net/questions/360560 | 12 | Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^0(X)\to\mathbb{R},$ such that $\phi(f)\geq 0$ if $f\geq 0$ ($\phi$ is called a positive linear functional), then there ex... | https://mathoverflow.net/users/125982 | Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces | The answer is yes.
First, it follows from the following result and the Riesz–Markov–Kakutani representation theorem that we can always find a suitable Baire measure representing a positive linear functional.
>
> **Theorem:** Let $X$ be any topological space. Then there exists a completely regular Hausdorff space... | 9 | https://mathoverflow.net/users/35357 | 360576 | 151,772 |
https://mathoverflow.net/questions/360578 | 71 | This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as precise as possible, I am outlining the background and nature of my questions here:
I did my Ph.D. in probability & stat... | https://mathoverflow.net/users/156936 | Category theory and set theory: just a different language, or different foundation of mathematics? | Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, exis... | 44 | https://mathoverflow.net/users/1353 | 360585 | 151,776 |
https://mathoverflow.net/questions/360597 | 6 | Is there anything special about the classes of affine varieties in the Grothendieck ring of varieties (over $\mathbb{C})$?. Is there some specialisation that allows us to distinguish classes of affine varieties from general classes?
After R. van Dobben de Bruyn's answer, it might be more interesting to consider the m... | https://mathoverflow.net/users/64302 | Detecting affine varieties in the Grothendieck ring | Being affine is not invariant under scissors relations. In other words, it is possible that $[X] = [Y]$ in the Grothendieck ring, where $X$ is affine and $Y$ is not.
For example, the diagonal $\Delta \subseteq \mathbf P^1 \times \mathbf P^1$ is ample, so $X = \mathbf P^1 \times \mathbf P^1 \setminus \Delta$ is affine... | 16 | https://mathoverflow.net/users/82179 | 360601 | 151,782 |
https://mathoverflow.net/questions/357199 | 2 | We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\tag1
$$
with $$(s+i+r)\big|\_{t=0}=1,\ s(0)\ge0,\ i(0)\ge0,\ r(0)\ge0,$$
where prime denotes derivative w.r.t. time, $s,i,r$ represent ... | https://mathoverflow.net/users/32660 | Seeking a Lyapunov function for a SIR model with immunity loss | We examine the local stability of this system. Since the first two equations of System $(1)$ form the largest set of independent equations, the Jacobian of this system is
$$J(s,i) := \begin{bmatrix}
-i-\alpha & -s-\alpha \\
i & s-\gamma
\end{bmatrix}.
$$
At Fixed Point $(3)$, the eigenvalues are
$$x=-\frac{1+\alpha}... | 2 | https://mathoverflow.net/users/32660 | 360604 | 151,783 |
https://mathoverflow.net/questions/360172 | 4 | This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either [there](https://math.stackexchange.com/questions/958531/why-do-adk-orbits-in-the-1-eigenspace-of-a-cartan-decomposition-intersect-t) or here.
$G$ is a semisimple Lie group with a choice of Cartan decomposition on ... | https://mathoverflow.net/users/105628 | A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$ | This has been answered [here](https://math.stackexchange.com/questions/958531/why-do-adk-orbits-in-the-1-eigenspace-of-a-cartan-decomposition-intersect-t/3678116#3678116) on MSE.
FYI, this is Theorem 4.21 (without proof) on page 74 in [Bekka and Mayer - Ergodic theory and topological dynamics of group actions on homo... | 1 | https://mathoverflow.net/users/105628 | 360608 | 151,784 |
https://mathoverflow.net/questions/360581 | 2 | Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}\_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves
$$\nabla: E \to E \text{ }\otimes\_{\mathcal{O}\_X} \Omega\_{X}^{1} $$ satisfying the product rule $\nabla(fs) = s \otimes df... | https://mathoverflow.net/users/149325 | Simple example of non-integrable holomorphic connection | Probably this answer intersects with the previous ones. Consider the complex Heisenberg group $H$ of the $3\times 3$ complex matrices with 1 on the diagonal and 0 under the diagonal. Let $x, y, z$ be the other entries, $z$ being in the corner. The center $Z$ is $\{x=y=0\}$. One has the (trivial) line bundle
$$Z\to H\to... | 3 | https://mathoverflow.net/users/105095 | 360609 | 151,785 |
https://mathoverflow.net/questions/360614 | 3 | What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}\_2(\mathbb{Q}\_p)$ for some prime $p$?
Proposition 3.4 in [Loeffler and Weinstein - On the computation of local components of a newform](https://arxiv.org/abs/1008.2796) refers to [Breuil and Mézard - Multiplicité... | https://mathoverflow.net/users/140336 | Level vs. conductor of a supercuspidal representation | There's probably a more elementary reference, but, according to [Bushnell, Henniart, and Kutzko - Local Rankin–Selberg convolutions for $\operatorname{GL}\_n$](https://www.ams.org/journals/jams/1998-11-03/S0894-0347-98-00270-7), (6.1.2), if $m$ is the level of $\pi$, then the conductor of $\pi$ depends on a choice of a... | 2 | https://mathoverflow.net/users/2383 | 360617 | 151,789 |
https://mathoverflow.net/questions/360611 | 2 | Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid\_M$ is invertible. We will say that $M'$ is ind-torsion if it belongs to the smallest Serre subcategory of $A$ that contains all torsion objects (that is, those $N\in A$ for... | https://mathoverflow.net/users/2191 | When uniquely divisible objects can be embedded into ind-torsion ones? | If your category $A$ is AB5, the answer is positive, because the ind-torsion objects are only the objects $X$ of $A$ such that the canonical map from the colimit $\_\infty X$ of $\_n X$ to $X$ is an isomorphism, where $\_n X$ is the kernel of multiplication by $n$ on $X$ and the (filtered) colimit is taken over positiv... | 2 | https://mathoverflow.net/users/76506 | 360646 | 151,799 |
https://mathoverflow.net/questions/360582 | 5 | I am interested in the finite unitary reflection group $G= G\_{32}$, the group No. 32 in Table VII on page 301 of the paper:
[Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. Math. 6 (1954), 274–304](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/finite-unitary-... | https://mathoverflow.net/users/4149 | Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation | We can take $H={\rm Sp}(4,3)$ to be the group $\{ A \in {\rm GL}(4,3) \mid AFA^{\mathsf T} = F\}$, where $$F=\left(\begin{array}{rrrr}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{array}\right)$$ is the matrix of the preserved symplectic form.
The the matrix $$C =\left(\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&... | 6 | https://mathoverflow.net/users/35840 | 360650 | 151,800 |
https://mathoverflow.net/questions/360631 | 0 | We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a nontrivial character and if $\chi$ is trivial then $\log L(s, \chi) = \log \frac{1}{s-1}+g(s)$ for some holomorphic function $g... | https://mathoverflow.net/users/157984 | Logarithms of $L$-functions of irreducible characters of Galois group | Yes this is the fact that for a non-trivial irreducible Artin character, the associated Artin L-function is holomorphic and non-zero on $\rm{re}\, s \geq 1$.
For the trivial character, one just obtains the Riemann zeta function, where there is a pole of order $1$.
| 1 | https://mathoverflow.net/users/5101 | 360653 | 151,802 |
https://mathoverflow.net/questions/360664 | 2 | On p. 20 of an [article](https://www.sciencedirect.com/science/article/pii/S0377042700003368?via%3Dihub) by Borwein et al., it is stated that Ramanujan could generalize the following formula due to Glaisher $$\gamma = 2 - 2\log2 -2\sum\_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$ to infinitely many formulae for $\g... | https://mathoverflow.net/users/93724 | What is the collection of series that amount to $\gamma$ deduced by Ramanujan? | Here is a scan from Ramanujan's 1917 paper on the generalized $\gamma$ formulas (equations 5 and 7). The reference in [Messenger of Mathematics](https://en.wikipedia.org/wiki/Messenger_of_Mathematics) is a journal that no longer exists. Only the volumes through 1901 are freely accessible [online.](https://gdz.sub.uni-g... | 3 | https://mathoverflow.net/users/11260 | 360676 | 151,810 |
https://mathoverflow.net/questions/360683 | 4 | *This is a copy from [MSE](https://math.stackexchange.com/questions/3677192/confusion-over-spin-representation-and-coordinate-ring-of-maximal-orthogonal-gra) where the question did not attract much attention.*
I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the ma... | https://mathoverflow.net/users/25028 | Confusion over spin representation and coordinate ring of orthogonal Grassmannian | Let me first treat the case $n = 1$. Then $G\cong PSL(2,\mathbb C),P\cong (GL(1,\mathbb C)/\mathbb Z/2)\ltimes \mathbb C$ embedded as upper triangular matrices, $G/P\cong \mathbb{CP^1}$, and $\omega\_1$ defines the line bundle $O(1)$ over $\mathbb{CP}^1$. The root of your confusion is that this line bundle is not $G$-e... | 3 | https://mathoverflow.net/users/35687 | 360684 | 151,812 |
https://mathoverflow.net/questions/360688 | 1 | Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}\_X$-modules. Then $Hom\_{\mathcal{O}\_X}(\mathcal{L}, \mathcal{M}) \cong Hom\_{\mathcal{O}\_X}(\mathcal{O}\_X, \mathcal{M}\otimes \mathcal{L}^{\vee}) \cong H^0(X, \mathcal{M}\otimes \mathcal{L}^{\vee})$.
Therefore der... | https://mathoverflow.net/users/108274 | Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication | Remark. Exact sequence $0 \to L \to E \to M \to 0$ corresponds to $Ext^1(M,L)$, not to $Ext^1(L,M)$.
Q1. $a \in k^\times$ acts on $Ext^1(L,M)$ via pullback along $a:L \to L$ or via pushout along $a: M \to M$.
Q2. There are two options: either one can check that the split sequence is the neutral element for the addi... | 5 | https://mathoverflow.net/users/4428 | 360690 | 151,813 |
https://mathoverflow.net/questions/360670 | 2 | This is an exercise in §3.13 Beilinson's [notes](https://drive.google.com/file/d/1p_msX4oxY6WFv1ITpbzjOjQIXh--yr3c/view) on homological algebra. He doesn't specify but I'm pretty sure $K\_0(\mathcal{A})$ is defined as the free group on the isomorphism classes of $\mathcal{A}$ modulo the relations generated by finite (c... | https://mathoverflow.net/users/157320 | For an additive category $\mathcal{A}$, how does one show $K_0(\mathcal{A})\cong K_0(\mathcal{K}^b(\mathcal{A}))$? | Consider the maps
\begin{align\*}
i \colon K\_0(\mathscr A) &\to K\_0\big(K^{\text{b}}(\mathscr A)\big) & & & \chi \colon K\_0\big(K^{\text{b}}(\mathscr A)\big) &\to K\_0(\mathscr A)\\
[A] &\mapsto\big [A[0]\big] & & & \big[K^\*\big] &\mapsto \sum\_i (-1)^i \big[K^i\big].
\end{align\*}
It is clear that $i$ is well-defi... | 2 | https://mathoverflow.net/users/82179 | 360698 | 151,815 |
https://mathoverflow.net/questions/360694 | 1 | I have seen several references to the so-called ***Extension Theorem*** in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
>
> The Extension Theorem [...] gives a criterion for a finite
> monohedral complex of polytopes to be extendable to a global isohedral t... | https://mathoverflow.net/users/108884 | What does the extension theorem for tilings state? | The first source is in English in Discrete Mathematics. You can find it here
* Nikolai P. Dolbilin, *The Extension Theorem*, Discrete Mathematics **221** Issues 1–3 (2000) pp 43–59, <https://doi.org/10.1016/S0012-365X(99)00385-4>.
| 2 | https://mathoverflow.net/users/137713 | 360699 | 151,816 |
https://mathoverflow.net/questions/360657 | 5 | Let $Sp(2n)$ be the group of complex symplectic $2n\times 2n$ matrices, and $O(2n)$ the group of complex orthogonal $2n\times 2n$ matrices.
Consider $Sp(2n)\cap O(2n)\subset Sp(2n)$ and the quotient $X=Sp(2n)/(Sp(2n)\cap O(2n))$. How could one compute the Picard group of $X$?
EDIT. Consider the action of $Sp(2n)$ o... | https://mathoverflow.net/users/14514 | Picard group of symplectic group modulo orthogonal group |
>
> **Answer:** ${\rm Pic\,} X={\Bbb Z}/2{\Bbb Z}$; see Corollary 4 below.
>
>
> **Theorem 1.** *Let $G$ be a simply connected semisimple group over a field $k$ of characteristic 0.
> Let $H\subset G$ be an algebraic subgroup defined over $k$, not necessarily connected. Set $X=G/H$.
> Then there is a canonical is... | 5 | https://mathoverflow.net/users/4149 | 360711 | 151,820 |
https://mathoverflow.net/questions/360701 | 4 | $$
\min\_{f} \sum\_{i=1}^n \max \left( 0, 2f(i) - f(i-1) -f(i+1)\right),
$$
where the minimum is taken over all the functions $f$ from $\{0,1,2,\ldots,n+1\}$ to $[0,x]$, $x <1$, such that $f$ is non-decreasing over $\{1,2,\ldots,n\}$, $f(0)=f(n+1)=0$, and $f(n)=x$.
| https://mathoverflow.net/users/nan | Minimization of a discrete valued function | $\newcommand{\De}{\Delta}$Letting
\begin{equation}
\De^2f\_i:=2f(i)-f(i-1)-f(i+1),
\end{equation}
we rewrite the target of the minimization as
\begin{equation}
s(f):=\sum\_{i\in[n]}(\De^2f\_i)\_+,
\end{equation}
where
$[n]:=\{1,\dots,n\}$ and $u\_+:=\max(0,u)$ for real $u$.
Let now $f$ be a minimizer of $s(\cdot)... | 0 | https://mathoverflow.net/users/36721 | 360717 | 151,821 |
https://mathoverflow.net/questions/360714 | 0 | I came across the statement in a book:
Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G\_k:=\operatorname{Gal}(K/k)$. Let $\chi$ denote the character of the permutation representation of $G$ in $G/G\_k$. Then the Ar... | https://mathoverflow.net/users/157984 | On $L$-function of permutation representation | It is not easy to give all the details so I'll give a sketch in the case of unramified prime
* For $p$ an unramified prime number, $Q\subset O\_K$ a prime ideal above $p$, those of $O\_k$ are of the form $P\_g =g(Q)\cap O\_k$ for $g\in G$ with norm $N(P\_g)=p^{f\_g}$ for some integers $f\_g$
$\sigma$ a Frobenius su... | 2 | https://mathoverflow.net/users/84768 | 360723 | 151,825 |
https://mathoverflow.net/questions/360731 | 3 | For any set $X$, let $[X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$.
Is there a finite, simple, undirected, connected graph $G=(V,E)$ with the following properties?
1. There is $\{v, w\}\in [V]^2\setminus E$ such that collapsing $v,w$ increases the chromatic number, but
2. for all $\{a, b\}\in [V]^2\setminus E$ we h... | https://mathoverflow.net/users/8628 | The effects of collapsing vs joining non-adjacent vertices on the chromatic number | **Yes,** such a graph does exist. Let $G$ be obtained from the complete graph $K\_{100}$ by adding two non-adjacent vertices $v$ and $w$ such that $|N\_G(v)|=|N\_G(w)|=50$ and $N\_G(v) \cup N\_G(w)=V(K\_{100})$. Here, $N\_G(v)$ denotes the set of vertices of $G$ which are adjacent to $v$. Then collapsing $v$ and $w$ in... | 4 | https://mathoverflow.net/users/2233 | 360734 | 151,829 |
https://mathoverflow.net/questions/360663 | 11 | Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap \mathcal{D}^{\geq 0}$, is an abelian category.
I know that in general there is not a natural functor from the derived... | https://mathoverflow.net/users/155635 | The relation between t-structures and derived category | Assume $\mathcal D$ is a presentable stable $\infty$-category with a $\mathrm t$-structure (which is accessible and compatible with filtered colimits), and let $\mathcal A$ be its heart, $\mathcal{D(A)}$ its derived $\infty$-category.
Note that under those hypotheses, $\mathcal A$ is Grothendieck abelian (*Higher Alg... | 9 | https://mathoverflow.net/users/102343 | 360737 | 151,831 |
https://mathoverflow.net/questions/186253 | 1 | Say I have a B-spline function (or curve) of order $k\_1$, defined over some knot vector
$\mathbf{t} = \{ t\_i\}\_1^{n\_1}$, i.e. $$f(x) = \sum\_i a^i B\_{i,k\_1}(x).$$
Do you know of a process of finding another B-spline function, say $g(u) = \sum\_j b^j B\_{j,k\_2}(u)$, of order $k\_2$ defined on some other knot ve... | https://mathoverflow.net/users/61077 | General reparameterization of a B-spline | B-splines are a basis-function representation for piecwise polynomial functions. Therefore, if the reparameterization you seek cannot be represented as a piecewise polynomial it cannot, in general, be represented as a B-spline.
This can be shown with the example you gave. For the reparameterization $x(u) = \sqrt{u^2+... | 1 | https://mathoverflow.net/users/123142 | 360739 | 151,832 |
https://mathoverflow.net/questions/360743 | 0 | This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to the Lebesgue measure). For any subset $A \subset \mathbb{R}^d$ and vector $t \in \mathbb{R}^d$, define
$$
A+t=\{x+t=(x... | https://mathoverflow.net/users/148849 | Is this probability inequality true? | This is false:
Let the probability distribution be $P(X = 0, Y = 0) = P(X = 1, Y = 1) = \frac{1}{2}$ and let $A = \{ 0\}$, $B = \{ 0, 1\}$. Then the left-hand side of your inequality is $1$ while the right-hand side is $\frac{1}{2}$.
If you want your densities to be continuous just convolve with some smooth highly ... | 3 | https://mathoverflow.net/users/104330 | 360750 | 151,834 |
https://mathoverflow.net/questions/360082 | 4 | A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following examples:
(i) A non-compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \... | https://mathoverflow.net/users/105103 | Fun examples relating to Hopf surfaces | **(ii)** I would like to prove that there are no complex surfaces that satisfy (ii).
Indeed, suppose that the universal cover $\widetilde X$ of a complex surface $X$ is diffemorphic to $\mathbb C^2\setminus 0$. Let's prove that $\widetilde X$ is biholomorphic to $\mathbb C^2\setminus 0$.
First, we note that $X$ has... | 6 | https://mathoverflow.net/users/943 | 360784 | 151,844 |
https://mathoverflow.net/questions/357577 | 5 | Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or Dirichlet boundary conditions. Here $f$ is globally bounded and Lipschitz in the second argument. Is it true that if $u$,$v$ ar... | https://mathoverflow.net/users/7646 | Backward uniqueness for a heat equation with a drift | The result follows by an extension of the method of **logarithmic convexity** which is well-known for the heat backward problem.
Let $H$ be a Hilbert space. Consider the following inequality
\begin{equation}
\|\partial\_t u + Au\| \leq \alpha\|u\|, \qquad \text{ on } (0,T), \qquad (1)
\end{equation}
with $\alpha=\mat... | 2 | https://mathoverflow.net/users/124904 | 360786 | 151,845 |
https://mathoverflow.net/questions/360779 | 2 | Problem Setup
-------------
Suppose we have the following scalar, linear time-varying (LTV) system with parameter $\mu \in [0,\pi[$:
\begin{cases}
\dot{x\_1}(t,\mu) = a(t,\mu)x\_1(t,\mu) + b(t,\mu) \\
x\_1(0,\mu) = 0
\end{cases}
where
* $a(t + \pi,\mu) = a(t,\mu)$ and $a(-t,\mu) = -a(t,\mu)$ $\forall (t,\mu)$
*... | https://mathoverflow.net/users/158302 | Does a scalar LTV system with odd-periodic coefficients and even-periodic inputs have no periodic solutions? | In fact, $x\_1$ can be $\pi$-periodic. Indeed, let $m:=\mu$,
$$a(t,m):=m\sin2t,\quad b(t,m):=m\cos2t-c\_m,$$
where $c\_m$ is the unique solution to the equation
$$\int\_0^\pi e^{-A(s,m)}(m\cos2s-c\_m)\,ds=0,$$
with
$$A(t,m)=\int\_0^t a(s,m)\,ds=\frac m2\,\sin^2t. $$
Then all your conditions on $a$ and $b$ hold, and
... | 1 | https://mathoverflow.net/users/36721 | 360787 | 151,846 |
https://mathoverflow.net/questions/360804 | 4 | Let $M\_p(i)$ be the mod $p^i$ Moore spectrum, i.e. the cofiber of $p^i: \mathbb S \to \mathbb S$. Upper and lower bounds on the $n$ for which $M\_p(i)$ admits an $A\_n$ structure are known, cf. [Bhattacharya](https://arxiv.org/abs/1607.02702). I gather from this that $M\_p(i)$ admits at least an $A\_2$ structure for a... | https://mathoverflow.net/users/2362 | Is the mod-2 Moore spectrum a retract of a shift of its tensor square? | The mod $2$ cohomology of $S^0/2 \wedge S^0/2$ is a $\mathbf{F}\_2$-vector space on generators in degrees 0, 1, 1, and 2. The classes in degrees 0 and 2 are connected by a nontrivial $\mathrm{Sq}^2$, so you cannot split $S^0/2$ off (any shift of) $S^0/2 \wedge S^0/2$. The topological version of this statement is the fa... | 6 | https://mathoverflow.net/users/102390 | 360818 | 151,855 |
https://mathoverflow.net/questions/360832 | 1 | This post is a continuation of [Weirdos but algebraic](https://mathoverflow.net/questions/360746/weirdos-but-algebraic).
Logically, the quoted post could follow the present one rather than precede it.
**Question** Does there exist an indecomposable weirdo which is neither an Abelian group nor is based on a module o... | https://mathoverflow.net/users/110389 | Indecomposable weirdos (cnt.) | Not sure if the following example counts as being different
from those listed in the problem, but consider the *weirdo*
with universe $\mathbb N = \{0,1,2,\ldots\}$ and
$\sigma(x,y)=x+y$
$\lambda(x,y)=\rho(y,x) = x\ominus y$,
where $x\ominus y = x-y$ if $x\geq y$
while $x\ominus y = 0$ if $x<y$.
**Comments.**
... | 2 | https://mathoverflow.net/users/75735 | 360837 | 151,862 |
https://mathoverflow.net/questions/360843 | 6 | Let $K\_1=\Bbb Q(\sqrt{d\_1})$ , $K\_2=\Bbb Q(\sqrt{d\_2})$ and $K=\Bbb Q(\sqrt{d\_1},\sqrt{d\_2})$.Suppose $h\_1,h\_2,h$ be class number of $K\_1,K\_2,K$ respectively.
(i) Can we express $h$ in terms of $h\_1,h\_2$?
(ii) Knowing the divisibility properties of $h\_1,h\_2$, I want help with concluding about the divi... | https://mathoverflow.net/users/131448 | How is class of composition of two quadratic fields is related class numbers of quadratic field? | I assume you mean $K\_1=\mathbb Q(\sqrt{d\_1})$.
1) If by $K$ you mean $\mathbb Q(\sqrt{d\_1d\_2})$, then there is no simple relation
between $h$ and $h\_1$ and $h\_2$.
2) If by $K$ you mean the quartic biquadratic field $\mathbb Q(\sqrt{d\_1},\sqrt{d\_2})$, a theorem of Herglotz says that $h=h\_1h\_2h\_3/2^j$, whe... | 11 | https://mathoverflow.net/users/81776 | 360849 | 151,868 |
https://mathoverflow.net/questions/360851 | 1 | If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance
| https://mathoverflow.net/users/158331 | Do there exist graphs whose adjacency matrix is positive semi-definite? | If you don't allow self-loops in the graph, then the trace is $0$. If the adjacency matrix is PSD then $0$ is the only eigenvalue. Since your adjacency matrix is symmetric, it must equal to $0$. So you essentially have the empty graph.
If you allow self-loops then you can also get a PSD adjacency matrix by adding so... | 6 | https://mathoverflow.net/users/158327 | 360853 | 151,869 |
https://mathoverflow.net/questions/360858 | 4 | Let $f$ be a newform of weight $k$ and level $N$ with integer coefficients. Deligne-Serre theorem theorem says there exist a nice associated representation $\rho\_{f}^{(\ell)}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}\_2(\mathbb{Q}\_{\ell}),$ for any prime $\ell$ not dividing $N$. In particular, this in... | https://mathoverflow.net/users/100578 | Surjectivity in Deligne-Serre | In the setting that $f$ has integer Hecke eigenvalues, then the result is true, so long as $f$ has weight $k\ge 2$ and does not have complex multiplication. See Theorem 3.1 of [this paper by Ribet](https://math.berkeley.edu/~ribet/Articles/rankin.pdf): in this setting, $R=\mathbb Z$.
In general, the Hecke eigenvalues... | 5 | https://mathoverflow.net/users/54339 | 360864 | 151,874 |
https://mathoverflow.net/questions/360823 | 1 | I am trying to find a closed form formula for the following recursive function:
$$f\_n(h)= \sum\_{i=1}^{n-1} \binom{n-2}{i-1} \cdot (0.5)^{n-2} \cdot [ (f\_{n-i}(h-1)\cdot \sum\_{j=0}^{h-1}f\_i(j)) + (f\_{i}(h-1)\cdot \sum\_{j=0}^{h-2} f\_{n-i}(j))] $$
The base cases are the following:
$$ f\_1(h)= \begin{cases}
1 & h... | https://mathoverflow.net/users/158321 | Solving recursion of a complex function | Define $g\_k(m) := \sum\_{j=0}^m f\_k(j)$. Then the given recurrence becomes
\begin{split}
g\_n(h)-g\_n(h-1) &= 0.5^{n-2} \sum\_{i=1}^{n-1}\binom{n-2}{i-1} [(g\_{n-i}(h-1)-g\_{n-i}(h-2))g\_i(h-1)+(g\_{i}(h-1)-g\_{i}(h-2))g\_{n-i}(h-2)] \\
&=0.5^{n-2} \sum\_{i=1}^{n-1}\binom{n-2}{i-1} [(g\_{n-i}(h-1)g\_i(h-1)-g\_{i}(h-2... | 1 | https://mathoverflow.net/users/7076 | 360875 | 151,879 |
https://mathoverflow.net/questions/360860 | 1 | Let $G$ be a finite perfect group, and let $N$ be the solvable radical
of $G$. If $G/N$ is a non-abelian simple group, then is it true that $N$
is contained in the Schur multiplier of $G/N$?
If this is not true in general, then does it hold at least in case
$G/N$ is either of type ${\rm PSL}(2,2^p)$ ($p$ prime) or is... | https://mathoverflow.net/users/128342 | Is the solvable radical of a finite perfect group contained in the Schur multiplier of the quotient of the group modulo the solvable radical? | The answer to the question as asked is definitely "no" in general. For any value of $n>1 $ and any odd prime $p$, we may take a perfect group $G$ which is a semidirect product of the form $E.{\rm Sp}(2n,p),$ where $E$ is extra special of order $p^{2n+1}$ and the action of the given symplectic group on $E$ is the natura... | 2 | https://mathoverflow.net/users/14450 | 360883 | 151,882 |
https://mathoverflow.net/questions/360880 | 19 | It is historically known that Jean Leray gave a course on algebraic topology while captive in the Officer's detention camp XVI in Edelbach, Austria during WW2. (References to this topic include an article by Sigmund, Michor and Sigmund in the mathematical Intelligencer, 27/2 (2005), 41-50.)
Is there a digitalized cop... | https://mathoverflow.net/users/21985 | Digitalized version of "Cours de topologie algébrique professé en captivité" | The course has been published in the [Journal de Mathématiques Pures et Appliquées](https://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es), volume 24 (1945), and can be found here: [first part](https://gallica.bnf.fr/ark:/12148/bpt6k97042399/f101.image) and [second part](https://gallica.b... | 25 | https://mathoverflow.net/users/11260 | 360885 | 151,883 |
https://mathoverflow.net/questions/360882 | 3 | In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ has a finite volume it seems natural to ask whether there also exists a maximally symmetric hyperbolic counterpart which a... | https://mathoverflow.net/users/158357 | Maximally symmetric hyperbolic 3-manifolds with finite volume | [Kojima](https://pdf.sciencedirectassets.com/271523/1-s2.0-S0166864100X02064/1-s2.0-0166864188900272/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEMP%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJGMEQCIESPHm0uX8EZMiqb6338fUic%2BRqVJVS9xhy9KnZHrUV9AiAPNfg6RNp6n%2FEZwlr0noHWvQPjoKhiQWuXso9vK6YKYyq0AwgcEAMaDDA1OTAwMzU0N... | 2 | https://mathoverflow.net/users/39082 | 360895 | 151,886 |
https://mathoverflow.net/questions/360884 | 3 | When $f\in L\_\text{loc}^1$, it's distributional derivative can be defined as $D\_{f'}\in\mathfrak{D}'$, such that $D\_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$ is the space of test functions. Then from what I understood, $f$ is said **weakly differentiable**, if there exists... | https://mathoverflow.net/users/151368 | Is there any nontrivial characterization of weakly differentiable functions? | **Definition.**
If $U\subset\mathbb{R}$ is open, we say that $u\in {AC}(U)$
if $u$ is absolutely continuous on every compact interval in
$U$. Let $\Omega\subset\mathbb{R}^n$. We say that
$u$ is *absolutely continuous on lines*, $u\in {ACL}(\Omega)$,
if the function $u$ is Borel measurable and for almost every line
$\el... | 8 | https://mathoverflow.net/users/121665 | 360899 | 151,889 |
https://mathoverflow.net/questions/360744 | 0 | Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I\_B$ the identity on $B$, and define $L\_k: =k (I\_B - L)$. When is $L\_k$ hypercyclic on $B$? Can anything else be said about $L\_k$?
| https://mathoverflow.net/users/36886 | Difference of hypercyclic operator and identity | I don't know if anything can be said in general. If $k>0$ is small enough, then the norm of $L\_k$ will be less than $1$ and hence $L\_k$ cannot be hypercyclic. For other values of $k$, the spectrum of $L\_K$ will be
$$
k(1-\sigma(L))
$$
and thus it may not be true that every component of $\sigma(L\_k)$ intersects the ... | 1 | https://mathoverflow.net/users/20484 | 360907 | 151,891 |
https://mathoverflow.net/questions/360903 | 1 | I asked this question a couple of days ago on [Math.SE](https://math.stackexchange.com/questions/3680748/are-conditional-expectation-values-the-same-if-expectation-values-are-part-ii) but without any echo (no upvotes, although I offered a bounty). But because I did it for oversight from a reputable/professional source ... | https://mathoverflow.net/users/158368 | Conditional expectation values defined by expectation values | Your desired conclusion does hold without the assumption of a probability density of $\bf x$ and $\bf z$, provided that the functions $f$ and $g$ are assumed to be Borel measurable.
Indeed, let $X:=f(\bf x)-g(\bf x)$ and $Z:=\bf z$, so that $X$ and $Z$ are bounded random variables with values in $\mathbb R$ and $\ma... | 1 | https://mathoverflow.net/users/36721 | 360909 | 151,893 |
https://mathoverflow.net/questions/360888 | 6 | The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below:
1. Ginot and Stiénon's paper [$G$-gerbes, principal $2$-group bundles and characteristic classes](https://webusers.imj-prg.fr/~gregory.ginot/papers/2-bundles.pdf) defines a Lie $2$-groupoid to be a double Lie ... | https://mathoverflow.net/users/118688 | Notions of Lie 2-groupoids | For your first question:
They are essentially all the same thing: some globular, some simplicial (taking the nerve goes from the former to the latter). The only subtlety is perhaps in the requirement on the maps $d\_{2,0}, d\_{2,2}$ to be surjective submersions in the del Hoyo–Stefani paper. This is not unusual for t... | 5 | https://mathoverflow.net/users/4177 | 360912 | 151,894 |
https://mathoverflow.net/questions/360894 | 0 | Let $d\_i\in\mathbb N$, $I\_i:=\{1,\ldots,d\_i\}$ and $u\in\mathbb R^{d\_1}\otimes\mathbb R^{d\_2}\otimes\mathbb R^{d\_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see Corollary 3 below) and hence consider its entry $u\_{i\_1i\_2i\_3}$ at the index $(i\_1,i\_2,i\_3)\in I\_1\times I... | https://mathoverflow.net/users/91890 | Cores in the tensor-train decomposition | Yes, mapping $A \otimes B \otimes C$ to $A \otimes C$ by choosing a coordinate of $B$ (equivalently, basis element of dual space $B^\*$) is indeed a special case of choosing any element of $B^\*$, considering it as a map $B \to k$ (where $k$ is the field), and then extending that map to $A \otimes B \otimes C \to A \ot... | 2 | https://mathoverflow.net/users/88133 | 360916 | 151,896 |
https://mathoverflow.net/questions/360889 | 21 | What can be said about publishing mathematical papers on e.g. viXra if the motivation is its low barriers and lack of experience with publishing papers and the idea behind it is to build up a reputation, provided the content of the publication suffices that purpose.
Can that way of getting a foot into the door of pu... | https://mathoverflow.net/users/31310 | Can the place of publication be harmful to one's reputation? | Yes, the place of publication can absolutely hurt your reputation. Specifically, I can tell you from having served on many hiring committees (and from conversations with professors at other universities about their hiring committees and tenure processes), that publications in predatory journals can hurt you. I'm talkin... | 29 | https://mathoverflow.net/users/11540 | 360917 | 151,897 |
https://mathoverflow.net/questions/360920 | 1 | For the purposes of a project, I've been looking for the following two papers referred to in Serre's "Divisibilité de certaines fonctions arithmétiques":
Landau (E.), - Über die Eitenlung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, A... | https://mathoverflow.net/users/157984 | Looking for a paper by Landau and one by Watson | Here's the Landau paper at the Internet Archive: <https://archive.org/details/archivdermathem37unkngoog/page/n324/mode/1up>
Here's the Watson paper at the EuDML: <https://eudml.org/doc/168581>, which links to the GDZ for a scan.
| 4 | https://mathoverflow.net/users/4177 | 360922 | 151,899 |
https://mathoverflow.net/questions/360859 | 1 | Suppose $(-\Delta)^s u=f \geq 0$ in a ball $B\_2$ and $u=0$ in $ R^N \setminus B\_2.$ Also suppose $u$ is $C^{s}$ non-negative and $(-\Delta)^s u=0$ in $B\_2 \setminus B\_1$ and $u\leq a$ on $\partial B\_1$ where $B\_1, B\_2$ is a ball of radius $1$ and $2$ and $a$ is a positive constant. Can one claim that $u\leq a$ i... | https://mathoverflow.net/users/139853 | fractional Laplacian estimates | **Yes**.
We have
$$ u(x) = \int\_{B\_1} G\_{B\_2}(x,y) f(y) dy , $$
where
$$ G\_{B\_2}(x,y) = C\_{N,s} \frac{1}{|x - y|^{N - 2s}} \int\_0^{T(x,y)} \frac{t^{s-1}}{(t+1)^{N/2}} dt , \\ T(x, y) = \frac{(4-|x|^2)(4-|y|^2)}{4|x-y|^2} $$
is the corresponding Green function. Fortunately, $1/|x - y|^{N-2s}$ and $T(x,y)$ are ... | 1 | https://mathoverflow.net/users/108637 | 360925 | 151,900 |
https://mathoverflow.net/questions/360341 | 4 | Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$.
A function $\lambda\colon G\to \mathbb C$ is an *eigenvalue* of the action of $G$ if there exists a function $f\in L^2(X,\mu)$ such that for every $g\in G$ one has $\lambda(g... | https://mathoverflow.net/users/115744 | Properties of the spectrum of the Koopman representation | The answer to both question is negative.
Take $G=S\_3$, the symmetric group of the set $X=\{1,2,3\}$.
Then $L^2(X)$ is decomposed to the trivial representation and a another two dimensional irreducible representation.
As a $G$-space, $X\times X\simeq X \cup G$ where $X\subset X\times X$ is the diagonal and $G$ corres... | 2 | https://mathoverflow.net/users/89334 | 360929 | 151,902 |
https://mathoverflow.net/questions/360930 | 0 | The Wikipedia article about p-boxes only talks about cumulative probability density functions, which are meaningful for continuous sample spaces. <https://en.wikipedia.org/wiki/Probability_box>
Just out of curiosity, is it possible, and meaningful, to define p-boxes for discrete sample spaces? What axioms should they... | https://mathoverflow.net/users/158381 | Are p-boxes for discrete sample spaces meaningful? | There is no such thing as a "cumulative probability density function". You seem to get confused between the notions of (i) the (cumulative) probability distribution function, (ii) the probability density function, and (iii) the probability mass function. I suggest you read an introductory textbook on probability or sta... | 0 | https://mathoverflow.net/users/36721 | 360950 | 151,910 |
https://mathoverflow.net/questions/360964 | 36 | Sorry if this question is not well-suited here, but I thought research in mathematics can be identified from other science field, so I wanted to ask to mathematicians.
I am just starting graduate study in mathematics (and my bachelor was in other field) so I have no research experience in mathematics. Recently I cam... | https://mathoverflow.net/users/151368 | How do you check that your mathematical research topic is original? | (1) It depends a lot on the field. In fields that rely on specialized techniques discovered relatively recently or known only to a few, or fields where the questions involve recently-introduced objects, it's much easier to keep abreast of current research.
On the other hand, in fields with elementary questions that c... | 28 | https://mathoverflow.net/users/18060 | 360968 | 151,917 |
https://mathoverflow.net/questions/360972 | 1 | I'm searching for a recommendable reference dealing with theory of
non-Archimedean local fields where I can find proofs of the following claims about
finite extensions $L/K$ of non-Archimedean local fields with finite
residue fields $l / k$. I'm pretty sure that this request might not have a research level but up to ... | https://mathoverflow.net/users/108274 | Theory of extensions of non-archimedian local fields | See [Fesenko and Vostokov - Local fields and their extensions](http://www.ams.org/books/mmono/121). (i) is Proposition 3.3(2). (ii) is Proposition 3.2(1). (iii)(1) is Proposition 3.5(1) (and, yes, $b$ may be chosen as a uniformiser). For (iii)(2), I think you meant $e \mid \lvert k\rvert - 1$, not $p \mid \lvert k\rver... | 3 | https://mathoverflow.net/users/2383 | 360974 | 151,919 |
https://mathoverflow.net/questions/360994 | 3 | Let $\mathcal{A}$ be an abelian category and let $X$ and $Y$ be objects in $\mathcal{A}$. The Yoneda $\text{Ext}^{n}(Y,X)$ is defined by the following:
First we consider the class $\text{E}^{n}(Y,X)$ of all exact sequences in $\mathcal{A}$ of the form $E : 0 \rightarrow X \rightarrow Z\_{n} \rightarrow \cdots \righta... | https://mathoverflow.net/users/nan | Definition of the Yoneda Ext | You can always reduce an arbitrary "zigzag" as in the first definition, to one of length two, as in the second definition, by applying the following trick:
1. Whenever you encounter morphisms $E\_{j-1}\to E\_j\to E\_{j+1}$ or $E\_{j-1}\leftarrow E\_j\leftarrow E\_{j+1}$ that go in the same direction, take their compo... | 4 | https://mathoverflow.net/users/39747 | 360998 | 151,928 |
https://mathoverflow.net/questions/360997 | 7 | Let $X$ be a CW-spectrum. It is well-known that $[\\_ ,X]$ is a generalized cohomology theory and, by Brown's representability theorem, every generalized theory is $H$ represented by a spectrum (namely, $H$ has this form).
What about $[X, \\_]$? Is it a homology theory? (I do not claim every homology is corepresented... | https://mathoverflow.net/users/112348 | Is $[X, \_]$ a homology theory? | This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a homology theory from a spectrum $X$ is to consider $\pi\_\*(X\otimes -)$, note that for compact $X$ your $[X,-]$ is of th... | 22 | https://mathoverflow.net/users/39747 | 361000 | 151,929 |
https://mathoverflow.net/questions/360886 | 3 | Consider the wave equation
$$\frac{\partial^2 u}{\partial t^2}-\sum\_{i=1}^n\frac{\partial^2 u}{\partial x\_i^2}=0$$
with initial conditions
$$u|\_{t=0}=\frac{\partial u}{\partial t}|\_{t=0}=0$$
>
> Does it follow that $u\equiv 0$? If not, are there sufficient extra conditions which guarantee that?
>
>
>
**Rem... | https://mathoverflow.net/users/16183 | Uniqueness of solution of the wave equation | Since I haven't been able to [track down Selberg's lecture notes](https://mathoverflow.net/questions/360886/uniqueness-of-solution-of-the-wave-equation/360893#comment909463_360901) since he moved to Bergen, and since the proof of the result I mentioned in [this comment](https://mathoverflow.net/questions/360886/uniquen... | 7 | https://mathoverflow.net/users/3948 | 361009 | 151,933 |
https://mathoverflow.net/questions/361005 | 2 | Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure?
I've seen examples of higher-dimensional foliations not admitting transverse invariant measures, but I'd imagine the same question is much easier t... | https://mathoverflow.net/users/43158 | Transverse invariant measures to vector fields | I think that the best reference for this question is still the (relatively) old paper by Plante *Foliations with measure preserving holonomy* Ann. of Math. (2) 102 (1975), no. 2, 327–361, although it is a bit of an overkill for one-dimensional foliations. For instance, by Theorem 4.1 holonomy invariant measures exist f... | 4 | https://mathoverflow.net/users/8588 | 361010 | 151,934 |
https://mathoverflow.net/questions/361001 | 0 | Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C\_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\ast}$ (for $t\in X$). It is well known that $C\_0(X,A)$ is $C^{\ast}-$ Algebra.
>
> What’s known about ideals and r... | https://mathoverflow.net/users/129638 | Need reference for ideals and representations of $C_0(X,A)$ | For each $x \in X$ let $I\_x$ be a closed ideal of $A$. Then the set of $f \in C\_0(X,A)$ satisfying $f(x) \in I\_x$ for all $x$ is clearly an ideal of $C\_0(X,A)$, and it shouldn't be too hard to show that every closed ideal has this form.
I assume this is "well known" but I don't have a reference.
As for represen... | 1 | https://mathoverflow.net/users/23141 | 361016 | 151,937 |
https://mathoverflow.net/questions/361012 | 5 | Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.
| https://mathoverflow.net/users/1573 | Whitney sum formula for topological Pontryagin classes | Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles.
Edit: The argument for this is as follows. Let $\mu : BTOP \times BTOP \to BTOP$ be the map corresponding to Whitney sum of (sta... | 8 | https://mathoverflow.net/users/318 | 361018 | 151,938 |
https://mathoverflow.net/questions/360681 | 2 | EDIT: I thought on rephrasing the question in another way:
I have been working recently with a tensor that satisfies
$A\_{ijkl}=A\_{i+b,j+b,k+b,l+b}$ $\forall$ $i,j,k,l$ $\in$ Z
$$dist(i,j,k,l)\leq M$$
where all indices are meant to be integers (also b with $b\geq 0$), and dis(i,j,k,l) is the distance between... | https://mathoverflow.net/users/157629 | Symmetric tensor components |
>
> Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$ and generate the other elements from the symmetry relation above.
>
>
>
The issue is very simple. For each integer $i’$ there exists $i $ defined above such that $i’=i+tb$ for some integer $t$. The... | 0 | https://mathoverflow.net/users/43954 | 361036 | 151,943 |
https://mathoverflow.net/questions/361035 | 5 | Let $(\Omega,\Sigma)$ be a measurable space (no reference measure is chosen!), and $V$ a finite-dimensional normed vector space.
Note carefully that I am not choosing any topology on $\Omega$, so the $\sigma$-algebra $\Sigma$ is a priori not induced by any Borel structure whatsoever.
The total variation $|\mu|$ of a ... | https://mathoverflow.net/users/33741 | completeness of $\mathcal M(\Omega)$ without any topological assumptions? | Indeed $(\cal{M}(\Omega),\|\cdot\|)$ is a Banach space. For $V = \mathbb{R}$ or $V = \mathbb{C}$ you can find this result in Dunford/Schwartz (1957), Linear Operators I, ch. III.7.4, in particular p. 161. For arbitrary Banach space $V$ this also holds true, but with a sligthly different norm (see p. 160). For finite di... | 6 | https://mathoverflow.net/users/100904 | 361039 | 151,944 |
https://mathoverflow.net/questions/361043 | 1 | Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?
| https://mathoverflow.net/users/139843 | Computing the fractional Laplacian of power function | Here it is:
>
> **Proposition:** Let $\alpha \in (0, \infty)$, $p \in (-n, \alpha)$, and
> $$
> f(x) = |x|^p .
> $$
> Then $(-\Delta)^{\alpha/2} f(x)$ is well-defined for $x \ne 0$, and
> $$
> (-\Delta)^{\alpha/2} f(x) = 2^\alpha \frac{\Gamma(\frac{p+n}{2}) \Gamma(\frac{\alpha-p}{2})}{\Gamma(\frac{p+n-\alpha}{... | 5 | https://mathoverflow.net/users/108637 | 361046 | 151,946 |
https://mathoverflow.net/questions/360866 | 4 | Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric
>
> $$ d\_Q([x],[y]) = \inf\{d(p\_1,q\_1)+d(p\_2,q\_2)+\dotsb+d(p\_{n},q\_{n})\}
> > , $$ where the $\inf$ is taken ove... | https://mathoverflow.net/users/36886 | Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric | This is a partial answer: we show that Fubini-Study metric does not exceed the quotient metric(and some ideas for other direction).
Let $(X,d)$ and $(Y,h)$ be metric spaces and let $q:X\to Y$ be a bijection. This map generates an equivalence relation on $X$: $x\sim z\Leftrightarrow q(x)=q(z)$. Moreover, we can view $... | 2 | https://mathoverflow.net/users/53155 | 361049 | 151,947 |
https://mathoverflow.net/questions/317127 | 1 | In the study of nonlinear conservation laws a lot of time I work on the two problems given bellow:
$$(1) \hspace{1cm} \begin{cases}
u\_t+(f\_{1}(u))\_x=\lambda \cdot g(u) \\[2ex]
u(x,0)=h\_{1}(x)
\end{cases}
$$
$$(2) \hspace{1cm} \begin{cases}
u\_t+(f\_{2}(u))\_x=0 \\[2ex]
u(x,0)=h\_{2}(x)
\end{cases}
$$
Here u... | https://mathoverflow.net/users/117762 | Transformation from the PDE problem with a source to the PDE problem without it and viceversa | When $n=1$, you can always do this, at least near $t=0$, by solving a single inhomogeneous, linear first-order PDE; you can even arrange that $h\_2 = h\_1$. When $n>1$, there is a geometrical obstruction that can be computed in terms of $f\_1$ and $\lambda g$. This is a classical fact in the geometry of PDE and charact... | 4 | https://mathoverflow.net/users/13972 | 361050 | 151,948 |
https://mathoverflow.net/questions/361015 | 2 | Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying some conditions? To be more specific, for example, can we construct a vector field $ X$ on $\Omega$ satisfying
(1) $X|\_{... | https://mathoverflow.net/users/120509 | Extension of outer unit normal vector to interior | If $\partial\Omega\in C^2$, then the answer is yes. We need $C^2$ as this condition implies that the vector normal to the boundary is $C^1$ (since the normal vector id defined through derivatives).
According to the collar neighborhood theorem, there is $d>0$ and a diffeomorphism of class $C^1$:
$$
\Phi:\partial\Omega... | 2 | https://mathoverflow.net/users/121665 | 361052 | 151,949 |
https://mathoverflow.net/questions/361021 | 1 | Let $\alpha \in (0,1)$ and $\delta \in (0,1/2)$ be fixed, and consider the following integrals for each integer $j \geq 0$:
$$I\_j(u):= \frac{e^u}{u^{j+\alpha}} \int\_{-u\delta}^0 e^t t^{j-1+\alpha}\left(1+\frac{t}{u}\right)^{-1}dt, \hspace{2mm} u>0$$
Show that for any integer $k \geq 0$ and for any integer $0 \leq j \... | https://mathoverflow.net/users/157984 | Asymptotic development of Integral of $e^xx^r$ | You want to show that for some complex $b$ we have
$$\int\_{-v}^0 e^t t^a\,dt=b+O(v^{-c})$$
where $v:=u\delta\to\infty$, $a:=j-1+\alpha+n$, $c:=k+1$.
This is true. Indeed, let $b:=\int\_{-\infty}^0 e^t t^a\,dt$. Then, by l'Hospital's rule, for any real $c$
$$\Big|b-\int\_{-v}^0 e^t t^a\,dt\Big|\le\int\_{-\infty}^{-v... | 1 | https://mathoverflow.net/users/36721 | 361063 | 151,956 |
https://mathoverflow.net/questions/361070 | 3 | Let $Z$, $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous surjection then is the induced map $g \to f\circ g$ from $C(Z,X)$ to $C(Z,Y)$ is continuous. But is it still a surjection?
My issue is that it's not clear if it has a right-inverse...
| https://mathoverflow.net/users/36886 | Surjection in compact-open topology | In many cases, $f\_\ast: C(Z,X)\to C(Z,Y)$, $g\mapsto f\circ g$ is not surjective: Put $Z=Y$ and $h=id\_Y \in C(Z,Y)$, then $h$ is in the range of $f\_\ast$ only if $f$ has a right inverse.
| 5 | https://mathoverflow.net/users/21051 | 361075 | 151,961 |
https://mathoverflow.net/questions/360541 | 42 | Consider the category of abstract $\sigma$-algebras ${\mathcal B} = (0, 1, \vee, \wedge, \bigvee\_{n=1}^\infty, \bigwedge\_{n=1}^\infty, \overline{\cdot})$ (Boolean algebras in which all countable joins and meets exist), with the morphisms being the $\sigma$-complete Boolean homomorphisms (homomorphisms of Boolean alge... | https://mathoverflow.net/users/766 | In the category of sigma algebras, are all epimorphisms surjective? | $\require{AMScd}$
A 1974 paper of R. Lagrange, [*Amalgamation and epimorphisms in $\mathfrak{m}$-complete Boolean algebras*](https://scihub.wikicn.top/10.1007/BF02485738) (Algebra Universalis 4 (1974), 277–279, [DOI link](https://doi.org/10.1007/BF02485738)), settled this affirmatively. In the cited paper, Lagrange s... | 26 | https://mathoverflow.net/users/97635 | 361099 | 151,971 |
https://mathoverflow.net/questions/361101 | 0 | Is there an easy way to "launder" a PDF file so that it won't appear to have been generated from LaTeX?
(I have a good reason for wanting to do this: I just tried to post an article to the arXiv, but the arXiv software isn't processing my latex source correctly, so I have to circumvent the usual way the arXiv creates... | https://mathoverflow.net/users/3621 | Laundering PDF files | One solution that will always work is to physically print it and scan it back as a pdf. Alternatively, you can probably open it in Adobe or Preview (on a mac) and then use the print feature to "save as pdf." I did this latter solution once successfully to solve the exact sort of problem you are asking. I can't remember... | 0 | https://mathoverflow.net/users/11540 | 361106 | 151,974 |
https://mathoverflow.net/questions/361064 | 7 | I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch\_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough projectives, although I'm willing to add more hypotheses, because the $\mathcal A$ I want to use it for is the categor... | https://mathoverflow.net/users/102343 | Reference for homotopy colimit = total complex | See Problem 4.23 and Problem 4.24 (with proofs) of Ulrich Bunke's [Differential cohomology](http://arxiv.org/abs/1208.3961).
The underlying abstract machinery for computing homotopy (co)limits
via homotopy (co)ends is presented by
Sergey Arkhipov and Sebastian Ørsted
in [Homotopy (co)limits via homotopy (co)ends in g... | 3 | https://mathoverflow.net/users/402 | 361114 | 151,977 |
https://mathoverflow.net/questions/361090 | 2 | Let $(X,\omega )$ be a compact Kahler manifold. For any $d>0$ are there only finitely many families of curves $C\_i$ such that $C\_i\cdot \omega <d$? (More precisely, if $C$ is any curve such that $C\cdot \omega <d$, then $C$ belongs to one of the families $C\_i$.) I believe the analogous statement for projective varie... | https://mathoverflow.net/users/44610 | Curves on a Kahler manifold | From "Bounded sets of sheaves on Kähler manifolds"
By Matei Toma, J. reine angew. Math. 710 (2016), 77–93
Lemma 4.4. Let X be a Kähler manifold, r be an integer and F be a set of compact reduced subspaces of X of bounded degree and all of whose components are of dimension r and contained in a fixed compact subset of X... | 4 | https://mathoverflow.net/users/19369 | 361125 | 151,982 |
https://mathoverflow.net/questions/352164 | 3 | I wonder if the following graph-theoretical concepts have been considered before, and if so, under which name.
---
Consider a directed graph $G$ with $n$ nodes.
Let the **cycle number** $\gamma(\nu)$ be the length of the shortest directed cycle from node $\nu$ to itself. $\gamma(\nu) = 1$ when $\nu$ is connect... | https://mathoverflow.net/users/2672 | Yet another graph characteristic | The "cycle number of a graph" is (roughly) equivalent to (the output of) [Dijkstra's algorithm](https://en.wikipedia.org/wiki/Dijkstra's_algorithm) in the following sense
>
> If you fix $i\in V$ and run Dijkstra's algorithm for the pairs $(i,j)$ and $(j,i)$ for $j \in V$ and let $D\_{i,j}= $"the output on $i,j$" t... | 4 | https://mathoverflow.net/users/157298 | 361128 | 151,983 |
https://mathoverflow.net/questions/361024 | 4 | I am reading [Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems](https://onlinelibrary.wiley.com/doi/abs/10.1002/oca.2190), p.753.
Consider the following PDE: $$q\_1x\_2^2+q\_2x\_2^2+V\_{x\_1}x\_2-\frac{V^2\_{x\_2}b^2... | https://mathoverflow.net/users/93600 | Reduce PDE to ODE by dilation symmetry | Here's an answer, at least as I understand your question.
Geometrically, you can understand the solutions to your PDE as graphs of surfaces in $\mathbb{R}^3$ given by $(x\_1,x\_2,V(x\_1,x\_2))$ (at least locally). From this viewpoint, to say that a PDE has a symmetry, is to say that a solution surface "moved" in the... | 4 | https://mathoverflow.net/users/103158 | 361129 | 151,984 |
https://mathoverflow.net/questions/361103 | 7 | I am looking for resources (books, notes, lecture video, etc. anything will do although printed material in English is preferable) on foliations which satisfy some or all of the following constraints.
* Prerequisites: I am familiar with algebraic topology (in the geometric style, as in Hatcher), differential topolog... | https://mathoverflow.net/users/152049 | Books on foliations | Geometric Theory of Foliations by César Camacho and Alcides Lins Neto in Portuguese, or in English thanks to Sue Goodman's fantastic translation.
I think it does almost everything you're asking for in terms of pictures, examples, and lovely exercises beyond just letting readers fill in details, though some theorems ... | 13 | https://mathoverflow.net/users/157042 | 361141 | 151,987 |
https://mathoverflow.net/questions/361145 | 6 | For a Serre fibration of pointed topological spaces $f:X \to B$, there is an action of $\pi\_1\left(B,b\_0\right)$ on the fiber $F$. The construction of this action I'm familiar with uses a lift $F\times I \to X$ of the map $F \times I \xrightarrow{\pi} I \xrightarrow{\gamma} B$ for any $\left[\gamma \right] \in \pi\_1... | https://mathoverflow.net/users/125868 | Action of fundamental group on homotopy fiber | (This answer is written in a model-independent fashion -- translate to your favourite formalism).
For every path $\gamma:[0,1]\to B$ you get an isomorphism in the homotopy category $X\_{\gamma0}\xrightarrow{\sim} X\_{\gamma1}$ (where with $X\_b$ I denote the homotopy fiber over $b\in B$). Probably the easiest and mos... | 6 | https://mathoverflow.net/users/43054 | 361152 | 151,992 |
https://mathoverflow.net/questions/361159 | 2 | Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator
\begin{equation\*}
\begin{array}{rccc}
R : & L\_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) & \to & L\_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) \\
& f & \mapsto & \int f(x)\, N(\cdot,\sigma^2)(\mathrm{d}x).
\end{ar... | https://mathoverflow.net/users/134012 | Eigenspace of Gaussian Markov operator | If I understand correctly, your operator $R$ is the convolution operator with the Gauss–Weierstrass kernel. This is a Fourier multiplier with symbol $\lambda(\xi) = \exp(-\tfrac{1}{2} \sigma^2 |\xi|^2)$:
$$ \widehat{R f}(\xi) = \lambda(\xi) \hat f(\xi). $$
If $f$ is a tempered distribution, then $R f = f$ if and only i... | 5 | https://mathoverflow.net/users/108637 | 361161 | 151,994 |
https://mathoverflow.net/questions/361135 | 1 | Let a Cayley graph $G$ of a group $H$ with respect to the generating set $\{s\_i\}$ have a clique of order $> 2$. In addition assume the graph $G$ is non-complete. If the clique size is less than half the order of $G$, then is it possible for some group $H$ that $G$ has a unique "disjoint maximal clique". By "disjoint ... | https://mathoverflow.net/users/100231 | Cayley graphs do not have isolated maximal cliques | Let $G$ be the linegraph of the complete graph $K\_n$ for $n\geq 5$. For some but not all $n$, $G$ is a Cayley graph, see Chris Godsil's answer to [another question](https://mathoverflow.net/questions/150744/the-line-graphs-of-complete-graphs-and-cayley-graphs).
$G$ has $\binom n2$ vertices and degree $2n-4$. The max... | 1 | https://mathoverflow.net/users/9025 | 361162 | 151,995 |
https://mathoverflow.net/questions/361165 | 5 | The question is in the title:
>
> **Question:** Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the [120-cell](https://en.wikipedia.org/wiki/120-cell)?
>
>
>
I consider only *convex* polytopes (convex hull of finitely many points) that are full-dimensional (... | https://mathoverflow.net/users/108884 | Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell? | There are other polytopes. To construct one let's do the following. Remember first that in the hyperbolic $4$-space there exists a regular compact *right-angled* 120-cell. Here, right-angled means that any two adjacent faces intersect under angle $\frac{\pi}{2}$. Regular means, that all the faces are isomeric, and the ... | 8 | https://mathoverflow.net/users/943 | 361166 | 151,996 |
https://mathoverflow.net/questions/355464 | 3 | Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/unstable disks corresponding to th flow of $-\nabla f$ intersect transversely) are dense in the appropriate topology. Howe... | https://mathoverflow.net/users/147463 | Finite-dimensional argument for Morse-Smale pairs? |
>
> The Sard-Smale result certainly guarantees that this will be true for a generic , but will it hold for any ?
>
>
>
If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $f$ so that the pair $(f,g)$ is Morse-Smale.
And yes, there's also a proof with less he... | 2 | https://mathoverflow.net/users/119609 | 361168 | 151,997 |
https://mathoverflow.net/questions/361023 | 6 | Let $\text{M}\_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in
>
> C.R. Yohe, *Triangular and Diagonal Forms for Matrices over Commutative Noetherian Rings*, J. Algebra **6** (1967), 335-368
>
>
>
provides a characterization of the *Noetherian* rings $R$ w... | https://mathoverflow.net/users/16537 | Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix | As Luc Guyot mentioned, check out Kaplansky's paper *[Elementary Divisors and Modules](https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-0031470-3.pdf)* from 1949.
Kaplansky calls a ring Hermite when every $1 \times 2$ matrix is equivalent to a diagonal matrix, and shows that equ... | 3 | https://mathoverflow.net/users/97635 | 361173 | 152,000 |
https://mathoverflow.net/questions/361081 | 11 | Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})\_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d\_1,...,d\_n\}$ be the degree matrix where $d\_i=\sum\_{j\neq i}e^{-|i-j|}$. Then $L=D-A$ is the Laplacian. Let $L^\dagger$ be the Moore-Penrose inverse of the Laplacian. I'm inte... | https://mathoverflow.net/users/123075 | Exponential decay of voltage potential difference | **Edit:** This turns out to be quite simple. Observe that $a\_{1i} / a\_{2i} = q$ does not depend on $i \in \{3, 4, \ldots, n\}$. Thus, if $x\_1 = 1$, $x\_2 = -q$ and $x\_i = 0$ for $i \in \{3, 4, \ldots, n\}$, then we clearly have $L x = c e\_1 - c e\_2$, where $c = \sum\_{i=3}^n a\_{1i}$. It follows that $$L^\dagger ... | 3 | https://mathoverflow.net/users/108637 | 361181 | 152,003 |
https://mathoverflow.net/questions/357612 | 4 | The intrinsic volume functions on $\mathbb{R}^d$ known from the Steiner formula and Hadwiger's Theorem can be extended to the domain of definable sets of an o-minimal structure $\text{Def}(\mathbb{R}^d)$ as:
$$\mu\_k(A):=\int\_{G\_{d,d-k}}\int\_{\mathbb{R}^k(L)} \chi(A\cap (L+x)) \text{ } dx \text{ }d\gamma(L) $$
w... | https://mathoverflow.net/users/140709 | Is the intrinsic volume always positive for maximum dimension? | $\DeclareMathOperator\dim{dim}$To answer my own question, I have put below a proof of both my conjectures (positivity and agreement with Hausdorff measure).
Let $A$ be a definable subset of $\mathbb{R}^{m+n}$ where $A$ is of (o-minimal) dimension $m$. We show $\mu\_m(A)>0$.
Define the set $A\_x=\{y\in \mathbb{R}^n ... | 1 | https://mathoverflow.net/users/140709 | 361208 | 152,012 |
https://mathoverflow.net/questions/361182 | 3 | Are there any classes of (Arens regular) Banach algebras that are not operator algebras whose bidual comes from a “universal representation”, as in the case of C\*-algebras?
| https://mathoverflow.net/users/99234 | Classes of Banach algebras (that aren't operator algebras) whose bidual comes from a "universal representation" | This sort of depends on what you mean by "universal representation". For $C^\*$-algebras, I think the statement is usually the following: Given a $C^\*$-algebra $A$ and a representation $\pi:A\rightarrow B(H)$, let $M(\pi) = \pi(A)''$ denote the von Neumann algebra generated by $\pi(A)$. There is a unique surjective no... | 3 | https://mathoverflow.net/users/406 | 361209 | 152,013 |
https://mathoverflow.net/questions/361191 | 5 | I was wondering if someone could explain some of the concrete applications of model categories. My possibly naive understanding of the motivation is that one wants to mimic the category of topological spaces in some sense or to define a homotopy theory for a category.
For example, more concretely on the [Wikipedia pa... | https://mathoverflow.net/users/119114 | Applications of model categories | There are many references where model categories, and their connection to homology, are described more. See [this MO question](https://mathoverflow.net/questions/132139/what-is-a-good-basic-reference-on-model-categories) for a list. For the example of $Ch(R)$, there are several model structures. Those that have quasi-i... | 1 | https://mathoverflow.net/users/11540 | 361219 | 152,014 |
https://mathoverflow.net/questions/357733 | 1 | Let $\{0,1\}$ be equipped with the Sierpiński topology $\{\emptyset, \{0,1\},\{1\}\}$, and $\mathbb{R}^d$ with the usual Euclidean topology. Then is the pointwise-convergence (point-open) topology on $C(\mathbb{R}^d,\{0,1\})$ indeed weaker than the compact-open topology?
I have in mind the case where $n>1$.
| https://mathoverflow.net/users/36886 | Comparison of topology of pointwise convergence and compact-open topologies for Sierpiński space | Functions $\mathbb R^d\to \{0,1\}$ are indicator functions $f=I\_{A}$ with $A=f^{-1}(\{1\})$ and continuity with respect to the Sierpiński topology precisely means that $A$ is open in $\mathbb R^d$. The topology of pointwise convergence on $C(\mathbb R^d,\{0,1\})$ is strictly coarser than the compact-open topology: Sin... | 3 | https://mathoverflow.net/users/21051 | 361228 | 152,019 |
https://mathoverflow.net/questions/361041 | 6 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx\_i\wedge dy\_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$.
We consider the following two
extensions of $\Sp(2n,\mathbb{R})$, the group of linear isomorphisms of $\mathbb{R}^{2n}$ preserving... | https://mathoverflow.net/users/36688 | An extension of symplectomorphism group | **Short answer:** $G = H$ is the group of conformal symplectic linear maps. What follows is a proof of this (which I've simplified slightly from what I originally wrote):
---
**1) $G$ is the group of conformal symplectic linear maps**
First, some notation. We write $\{e\_i\} \cup \{f\_j\}$ for the standard basi... | 5 | https://mathoverflow.net/users/66405 | 361236 | 152,023 |
https://mathoverflow.net/questions/361225 | 6 | Given $X\_i, Y\_i$ Banach spaces, $f\_j, g\_j, T\_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram
$\require{AMScd}$
\begin{CD}
0 @>>> X\_1 @>f\_1>> X\_2 @>f\_2>> X\_3 @>>> 0\\
@V VV @V T\_1 VV @V T\_2 VV @V T\_3 VV @V VV \\
0 @>>> Y\_1 @>>g\_1> Y\_2 @>>g\_2> Y\_3 @>>> 0
\end{CD}
... | https://mathoverflow.net/users/2258 | Are nuclear operators closed under extensions? | **The answer is no:** you can even have $T\_1=T\_3=0$ and $T\_2$ equal to the identity $id$ on an infinite dimensional Banach space.
Indeed, consider the following commutative diagram with exact rows:
$$\begin{CD}
0@>>> 0 @>0>> X @>id>> X @>>> 0\\
&&@V0VV @VV{id}V @VV0V\\
0@>>>X @>>id> X @>>0> 0 @>>> 0
\end{CD}
$$... | 7 | https://mathoverflow.net/users/39421 | 361238 | 152,025 |
https://mathoverflow.net/questions/361187 | 4 | From [nlab](https://ncatlab.org/nlab/show/K%C3%A4hler+differential#AbstractDef), the module of Kähler differentials over some category $\mathcal{C}$ is the free functor:
$$\Omega: \mathcal{C} \to \mathsf{Mod\_{\mathcal{C}}}$$
left-adjoint to the (forgetful) embedding:
$$u: \mathsf{Mod}\_{\mathcal{C}} \cong \mathsf{Ab}(... | https://mathoverflow.net/users/143390 | Categorical Kähler differentials and the Leibniz rule | 1. The Leibniz rule follows immediately from the last description
of derivations as morphisms of commutative rings X:R→u(M).
Indeed, u(M) is the square-zero extension of some R-module M'
(in the traditional sense), i.e., u(M)=R⊕M'.
Now a morphism of commutative rings f:R→R⊕M' in the slice category C/R
(not in C, as... | 2 | https://mathoverflow.net/users/402 | 361243 | 152,028 |
https://mathoverflow.net/questions/361239 | -2 | I have a random graph/network described by the adjacency matrix $(a\_{ij})\_{N\times N}$ where $a\_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta\_i=\sum\_j a\_{ij}\eta\_j$ in $\mathbb{R}$. In this framework the *ensemble average* of $\eta\_i$, denoted by $\bar{\eta}... | https://mathoverflow.net/users/158444 | Ensemble averaging in a random graph (or network) in the large $N$ limit | Ok, it seems that this section of StackExchange is really made for you to figure out your own answer... After some lengthy frustrating trial and error, there is no typo in the given solution. The mistake I was doing was in the calculation of $\mathrm{E}[(\sum\_ja\_{ij}\eta\_j)^2]$. In the development of this multinomia... | 1 | https://mathoverflow.net/users/158444 | 361252 | 152,032 |
https://mathoverflow.net/questions/361234 | 1 | I am fairly new to generating functions and have been trying to solve the following recurrence for a computer science problem.
$$ f(k,d,n) = \sum\_{i=1}^{n-1} \binom{n-2}{i-1} \left(\frac{1}{2}\right)^{n-2} \left(\sum\_{j=0}^{k} f(k-j,d-1,n-i)\cdot f(j,d-1,i)\right)$$
Note that
$$
f:\mathbb{N}^3 \rightarrow [0,1]
$$
w... | https://mathoverflow.net/users/158321 | Solving recurrence of a three variable function | Define
$$F\_d(x,y) := \sum\_{k\geq 0}\sum\_{n\geq 1} f(k,d,n) x^k \frac{y^{n-1}}{(n-1)!}.$$
Then the recurrence is equivalent to
$$\frac{\partial F\_d}{\partial y}(x,y) = F\_{d-1}(x,y/2)^2,$$
while the initial conditions imply
$$F\_0(x,0)=x,\quad F\_d(x,0)=1\ \text{for}\ d\ne 0$$
and
$$\frac{\partial F\_1}{\partial y}(... | 1 | https://mathoverflow.net/users/7076 | 361257 | 152,034 |
https://mathoverflow.net/questions/360923 | 2 | The followings are from Mnev's [paper](https://arxiv.org/abs/1707.08096) about BV formalism.
>
> **Example 4.15** (Definition of split supermanifold)
>
>
> Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a split $(n|m)$-supermanifold $\Pi E$ with body $M$ and structure sheaf $\... | https://mathoverflow.net/users/133793 | Question about Berezin line bundle of odd cotangent bundle of supermanifold $\text{Ber}(\Pi T^*\mathcal N)$ | The Berezinian of a vector bundle $E$ arises via an associated bundle construction: it is reasonably easy to check that
$$
\begin{pmatrix}A&B\\C&D\end{pmatrix}\mapsto |\operatorname{det}(A)|\operatorname{det} ^{-1}(D- CA^{-1}B)
$$
defines a super Lie group homomorphism $\operatorname{Ber}:GL(\mathbb R^{m|n})\to\mathbb ... | 3 | https://mathoverflow.net/users/35687 | 361261 | 152,036 |
https://mathoverflow.net/questions/361017 | 3 | Let $T=\operatorname{PSL}\_n(q)$ with $n$ a prime number. Then the $\mathscr{C}\_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}\_{\frac{q^n-1}{(q-1)(n,q-1)}}{:}\mathbb{Z}\_n$, where $x$ comes from the Singer cycle.
Note that $\sigma$ has a matrix which is a permutation mat... | https://mathoverflow.net/users/131819 | Is the Singer cycle preserved by field automorphisms and graph automorphisms? | This is true by Proposition 4.3.6.(I) of Kleidman and Liebeck's book "The Subgroup Structure of the Finite Classical Groups", which says that, in all cases for the linear and unitary groups, there is a unique conjugacy colass of maximal ${\mathscr C}\_3$-subgroups.
In fact in your situation it is easy to prove it dir... | 1 | https://mathoverflow.net/users/35840 | 361295 | 152,046 |
https://mathoverflow.net/questions/361296 | 16 | This question is purely out of curiosity, and well outside my field — apologies if there is a trivial answer. Recall that a *Vitali set* is a subset $V$ of $[0,1]$ such that the restriction to $V$ of the quotient map $\mathbb{R}\rightarrow \mathbb{R}/\mathbb{Q}$ is bijective. It follows easily from the definition that ... | https://mathoverflow.net/users/40297 | Topological proof that a Vitali set is not Borel | Sometimes a convenient substitute for Lebesgue measurability is the [property of Baire](https://en.wikipedia.org/wiki/Property_of_Baire). Just like Lebesgue measurability, the class of sets with this property is a $\sigma$-algebra containing the open subsets - indeed, open sets clearly have property of Baire, this clas... | 18 | https://mathoverflow.net/users/30186 | 361301 | 152,050 |
https://mathoverflow.net/questions/361294 | 4 | It most books dealing with Cholesky decomposition, or it is variants, one finds a statement of the form if $A$ is symmetric $k\times k$ positive **semi-definite** (non-negative definite) then the $k\times k$ matrix $L$ solving
$$
A=RR^{\top}.
$$
**Note:** I do not require that $A$ is positive *definite*, so $A^{-1}$ m... | https://mathoverflow.net/users/36886 | Reference request: continuity of Cholesky factor | A subtle issue is that $\Pi$ is not unique here. For instance, if
$$
A = \begin{bmatrix}
1 & 0 & 0\\\\
0 & 0 & 0\\\\
0 & 0 & 0
\end{bmatrix}
$$
then you can take both the identity and $(23)$ as the permutation. Similarly, if $A=I$, then any $\Pi$ will work (and $R=I$).
I don't think you can speak about continuity... | 2 | https://mathoverflow.net/users/1898 | 361303 | 152,051 |
https://mathoverflow.net/questions/361318 | 10 | I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^n)^G \rangle$ is hyperbolic for all sufficiently large $n > 0$.
I have been searching for references, but without succ... | https://mathoverflow.net/users/35840 | hyperbolic quotient of hyperbolic group | This is contained in at least Delzant's paper Sous-groupes distingués et quotients des groupes hyperboliques. [Distinguished subgroups and quotients of hyperbolic groups] Duke Mathematical Journal, vol. 83 (1996), no. 3, pp. 661–682, and also in Ol'shanskii's paper SQ-universality of hyperbolic groups, Mat. Sb. 186 (19... | 10 | https://mathoverflow.net/users/10265 | 361321 | 152,056 |
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