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https://mathoverflow.net/questions/419906 | 1 | Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if
$$ \varlimsup\_{\delta \rightarrow 0} \frac{\sup\_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \infty, $$
then, given any $\epsilon>0$, is it true that
$$ \varliminf\_{\delta \rightarrow 0} \frac{\sup\_{0<\lvert y\rvert\leq \d... | https://mathoverflow.net/users/152618 | A problem of the limit of $\frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}}$ | This example is not continuous, but one can replace the jumps with linear pieces of fast growing slopes. Fix $0<\epsilon<a$. Define a sequence $(r\_j)\_{j\in\mathbb N}$ tending to zero inductively as follows:
$r\_1=1$ and $0<r\_{j+1}<r\_j$ so small that
$$
\frac{r\_{j+1}^{a-\epsilon}}{r\_j^{a+\epsilon}}\le 1.
$$
Then s... | 1 | https://mathoverflow.net/users/473423 | 419918 | 170,899 |
https://mathoverflow.net/questions/419929 | 3 | Let us define the following two stopping time $\tau\_B=\inf\{t\geq 0: X\_t\in B\}, \tau'\_B=\inf\{t> 0: X\_t\in B\}$, where $\tau\_B$ is entrance time and $\tau'\_B$ is hitting time. It is clear $\tau\_B=\tau'\_B, \mathbb{P}-a.s.$. By the book [Levy process](http://www.maphysto.dk/oldpages/events/LevyBranch2000/) by Be... | https://mathoverflow.net/users/147009 | Blumenthal 0-1 law | $\newcommand\F{\mathcal F}\newcommand\N{\mathbb N}$First, some preliminary remarks:
1. Your link to Bertoin's book is not very good. Here is a link with a [better reference to the book](https://www.ams.org/bull/1998-35-04/S0273-0979-98-00761-7/S0273-0979-98-00761-7.pdf).
2. It is Blumenthal, not Bulmenthal.
3. It is ... | 2 | https://mathoverflow.net/users/36721 | 419939 | 170,905 |
https://mathoverflow.net/questions/419924 | 4 | Let $A$ and $B$ be self-adjoint operators on some Hilbert space and $B$ is postive. Suppose we have $-B\leq A\leq B$.Is it true then that $\|A\|\_p\leq\|B\|\_p$ where $\|.\|\_p$ is the Schatten-$p$ norm defined as $\|A\|\_p:=(Tr(|A|^p)^{1/p}.$
| https://mathoverflow.net/users/136860 | A trace inequality between self-adjoint operators | Yes, this follows from the fact that $\|B\|\_p^p \geq \sum |\langle Be\_i, e\_i\rangle|^p$ for any orthonormal basis $(e\_i)$ (see [here](https://math.stackexchange.com/questions/2269975/is-this-inequality-on-schatten-p-norm-and-diagonal-elements-true)). If $(e\_i)$ diagonalizes $A$ then we have $\|A\|\_p^p = \sum |\la... | 8 | https://mathoverflow.net/users/23141 | 419945 | 170,909 |
https://mathoverflow.net/questions/419903 | 0 | The diffusion equation with constant diffusion $D$ can be represented as:
\begin{equation}
\frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t)
\end{equation}
where
* $\Delta$ is the Laplace operator and
* $\phi(r,t)$ represents a concentration at a point $r\in\mathbb{R}^n$ at time $t$.
When the diffusion is ... | https://mathoverflow.net/users/142153 | Non-linear diffusion on networks | $\newcommand{\R}{\mathbb R}$You do not need to "discretize $\nabla$". Also, you wrote the diffusion equation incorrectly. The correct version is this:
\begin{equation}
\frac{\partial f(r,t)}{\partial t}=\nabla\cdot[B(r,t)\,\nabla f(r,t)],
\end{equation}
where $f:=\phi$, $B:=D$, and $\cdot$ denotes the dot product. In t... | 2 | https://mathoverflow.net/users/36721 | 419946 | 170,910 |
https://mathoverflow.net/questions/419888 | 8 | Let $\mathcal{A}$ be an abelian category, and let $X$ an object of $\mathcal{A}$. Recall that a *pseudoelement* of $X$ is an equivalence class of arrows $X\_1 \to X$, where $x\_1 \colon X\_1 \to X$ and $x\_2 \colon X\_2 \to X$ are equivalent if there is an object $P$ and *epimorphisms* $p\_1 \colon P \to X\_1$ and $p\_... | https://mathoverflow.net/users/7845 | Pullback and pseudoelements | **The claim stated in the question is false, and the statement of Lemma 1.9.5 in Borceux is unclear, but seems wrong.** To be clear, the claim in this question is that for any Abelian category $\newcommand{\A}{\mathcal{A}}\A$, and maps $f:X \to Z$, $g : Y \to Z$, and pseudo-elements $[x]$, $[y]$ such that $f[x] = g[y]$... | 8 | https://mathoverflow.net/users/2273 | 419951 | 170,911 |
https://mathoverflow.net/questions/419823 | 4 | Let $a\_1,\ldots,a\_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such that
$$
s\_\lambda(a\_1,\ldots,a\_n) \ge 0
$$
for every partition $\lambda = (\lambda\_1 \ge \ldots \ge \lambda\_n)$, whe... | https://mathoverflow.net/users/27013 | Nonnegativity locus of Schur polynomials | The answer to the main question is affirmative. The crucial result is due to M. Aissen, I. J. Schoenberg, and A. Whitney, [*J. Analyse Math.* **2** (1952), 93—103](https://link.springer.com/article/10.1007/BF02786970). For further details see the solution to Exercise 7.91(e) in *Enumerative Combinatorics*, vol. 2. (For... | 9 | https://mathoverflow.net/users/2807 | 419953 | 170,912 |
https://mathoverflow.net/questions/419948 | 3 | Let $\Omega$ be an open set of $\Bbb R^d$: consider the following function spaces
* $H\_0^1(\Omega)$, i.e. the closure of $C\_c^\infty(\Omega)$ in $H^1(\Omega)$
* $H\_\*(\Omega)$, i.e. the closure of $C\_c^\infty(\Omega)$ in $H^1(\Bbb R^d)$.
* $H\_{\Omega}(\Omega)=\{u\in H^1(\Bbb R^d):\ u= 0 \text{ a.e on } \Omega^c\... | https://mathoverflow.net/users/112207 | Possible way to define $H_0^1(\Omega)$ Sobolev spaces | The first two are equivalent, as the $H^1(\Omega)$ norm and $H^1(\mathbb{R}^d)$ norm coincide for $C^\infty\_c(\Omega)$ functions.
The third is in general different:
If you let $d = 1$ and $\Omega = \mathbb{R}\setminus \{0\}$,
you see that $H\_{\Omega}(\Omega) = H^1(\mathbb{R}) \supsetneq H^1\_0(\Omega)$. You can c... | 10 | https://mathoverflow.net/users/3948 | 419956 | 170,913 |
https://mathoverflow.net/questions/419847 | 6 | Fix constant $L,C>0$ and $k\geq 1$ and let $f\in W^{1,k}(\mathbb{R}^d,\mathbb{R}^n)$ with $\|f\|\_{W^{1,k}}\leq C$.
Is there a known estimate on the distance
$$
\|f - \operatorname{Lip}\_L(\mathbb{R}^d,\mathbb{R}^n)\|\_{L^1(\mathbb{R}^d,\mathbb{R}^n)},
$$
*depending on the constants $L,C,$ and $k$*, where $\operatorn... | https://mathoverflow.net/users/36886 | Best approximation of L1 function by Lipschitz function | Let $(\rho\_\epsilon)\_{\epsilon>0}$ be a standard family of mollifiers, with $\rho\_\epsilon$ supported in the ball $B\_\epsilon(0)$. Since $\|\rho\_\epsilon\|\_{L^1}=1$ and $\|\rho\_\epsilon\|\_{L^\infty}=c\_d\epsilon^{-d}$, by interpolation we get $\|\rho\_\epsilon\|\_{L^{k'}}=c\_d^{1/k}\epsilon^{-d/k}$ (for the dua... | 2 | https://mathoverflow.net/users/36952 | 419964 | 170,915 |
https://mathoverflow.net/questions/419718 | 6 | I came across these notes from a talk by Hoschter which talks about superheight of an ideal and it mentions Krull's ideal height theorem on P2-P3 in terms of superheight. Here are the notes: [http://www.math.lsa.umich.edu/~hochster/swb2.pdf](http://www.math.lsa.umich.edu/%7Ehochster/swb2.pdf). Does the Peskine–Szpiro i... | https://mathoverflow.net/users/144294 | Does the Peskine–Szpiro intersection theorem imply Krull's ideal height theorem? | * **Superheight Theorem:** Let $M$ be a non-zero finitely generated $R$-module over a Noetherian ring $R$. Then superheight(ann $M$)$\le$ projdim $M$.
-**The Superheight Theorem implies Krull's Height Theorem:** See page 6 of [Class Notes for Math 918: Homological Conjectures, Instructor
Tom Marley](https://digitalco... | 1 | https://mathoverflow.net/users/127857 | 419971 | 170,918 |
https://mathoverflow.net/questions/419931 | 2 | I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2.22.
Let $\mathcal{L}\_{2g}$ be the linear system of degree $2g+1$ hypersurfaces in $\mathbb{P}^{2g}$ passing through $2g+2$ gener... | https://mathoverflow.net/users/211682 | Question regarding linear system of projective space | The formula for the multiplicity of a linear system along a linear subspace is classical. You can find it for instance in Lemma 2.1 here <https://arxiv.org/pdf/1210.5175.pdf>
This will tell you that a general $D$ contains $H\_J$ with multiplicity one. Now, $D\_{|H\_I}$ has degree $2g+1$ and multiplicity $2g-1$ at $g+... | 2 | https://mathoverflow.net/users/14514 | 419973 | 170,919 |
https://mathoverflow.net/questions/419967 | 6 | It is discussed in this [question](https://mathoverflow.net/questions/316209/deforming-metrics-from-non-negative-to-positive-ricci-curvature) whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there are counter-examp... | https://mathoverflow.net/users/170859 | Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci? | No there are no such examples known. Most known examples of manifolds of nonnegative sectional curvature come from biquotients or cohomogeneity one manifolds. If these are simply connected they are known to admit metrics of positive Ricci curvature by a [result of Schwachhoefer and Tuschmann](https://link.springer.com/... | 8 | https://mathoverflow.net/users/18050 | 419974 | 170,920 |
https://mathoverflow.net/questions/418942 | 6 | Consider the category $\operatorname{Solid}\_{\mathbf{Z}}$ of *solid* abelian groups in the sense of Clausen-Scholze. This category is a full subcategory of *condensed* abelian groups, $\operatorname{Cond}\_{\mathbf{Z}}$. These are, modulo set theoretical technicalities, abelian sheaves on the site of profinite sets, w... | https://mathoverflow.net/users/nan | Solid tensor product of pro-discrete space with Laurent series | This is not true in general, the most important observation being that it fails already when $V$ is discrete. In that case $V\otimes^{\blacksquare} \mathbb Z((T))$ is just the usual algebraic tensor product. This agrees with $V((T))$ only if $V$ is finitely generated.
In a different direction, for those pro-discrete ... | 4 | https://mathoverflow.net/users/6074 | 419975 | 170,921 |
https://mathoverflow.net/questions/419954 | 4 | For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum\_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible to get in closed-form the parity of the arithmetical function $[A,B]$, I mean $[A,B]\text{ mod }2$. I was studying prop... | https://mathoverflow.net/users/142929 | Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures | Please restrict to one question per post (standard policy). Here is a proof of Conjecture 2.
For any $s\in\mathbb{C}$, the function $\sigma\_s(n):=\sum\_{d\mid n} d^s$ satisfies the Hecke multiplicativity relation
$$\sigma\_s(m)\sigma\_s(n)=\sum\_{d\mid(m,n)}d^s\sigma\_s\left(\frac{mn}{d^2}\right).$$
This is straight... | 5 | https://mathoverflow.net/users/11919 | 419982 | 170,923 |
https://mathoverflow.net/questions/419908 | 4 | On Voevodsky's paper 'Cancellation theorem', Lemma 4.8, he stated in the proof that the map
$$\begin{array}{ccc}\mathbb{G}\_m\times\mathbb{G}\_m&\longrightarrow&\mathbb{G}\_m\times\mathbb{G}\_m\\(x,y)&\longmapsto&(y,x^{-1})\end{array}$$
is $\mathbb{A}^1$-homotopic to the identity, i.e., one could find an explicit homot... | https://mathoverflow.net/users/149491 | On the swapping map of $\mathbb{G}_m$ | Recall that $S\_t^1$ is the *reduced* motive $\tilde M(\mathbf G\_m)$ of $\mathbf G\_m$, obtained most explicitly as the kernel of the projector $M(\mathbf G\_m) \to M(\mathbf G\_m)$ given by $\operatorname{id}-1\_\*$ where $1 \colon \mathbf G\_m \to \mathbf G\_m$ is the constant map $1$. In particular, it is a direct ... | 10 | https://mathoverflow.net/users/82179 | 419986 | 170,924 |
https://mathoverflow.net/questions/419981 | 7 | In field theory, the following fact is used in the construction of splitting fields: *Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field extension of $F$ which contains a root of $f$ (namely the congruence class of $\alpha$).*
Let $n$ be a positive integer an... | https://mathoverflow.net/users/15505 | Factorization of an irreducible polynomial in the field extension it defines | Let us show that for the partition $2+1+1=4$, there is no such $f$.
If $f$ were inseparable, then over $K$ it would factor as a constant times $(x-\alpha)^5$. If $f$ were separable, its Galois group $G$ would be a transitive subgroup of $S\_5$ containing a transposition, so $G=S\_5$, but then the stabilizer of a poin... | 15 | https://mathoverflow.net/users/2757 | 419988 | 170,925 |
https://mathoverflow.net/questions/419995 | 11 | In the little galaxy of Category Theory, *Friedrich Ulmer* is known for being one of the authors of **Lokal Präsentierbare Kategorien**, a book that laid the foundations for the theory of locally presentable categories.
Unlike his coauthor, Pierre Gabriel, we do not have much information about him, or at least I can'... | https://mathoverflow.net/users/104432 | Mathematical life of Friedrich Ulmer | I am not sure for how long and where you looked, but it takes less than 30 minutes to figure much more than you mention in your post. Apparently, he continued his academic career as Fritz Ulmer : if you look at the affiliation of the last indexed MathReviews paper
* *Localizations of endomorphism rings and fixpoints*... | 13 | https://mathoverflow.net/users/1306 | 419998 | 170,928 |
https://mathoverflow.net/questions/419960 | 7 | Does there exist a smooth, projective, complex algebraic variety $X$, with two cohomology classes $\alpha,\beta \in H^{\*}(X,\mathbb{Z})$ neither $\alpha$ nor $\beta$ is torsion but the product $\alpha \cup \beta$ is non-trivial and torsion?
| https://mathoverflow.net/users/99732 | Algebraic varieties with certain topological properties | Let $Y$ be a smooth complex projective surface whose cohomology group $H^2(Y;\mathbb{Z})$ has a nonzero torsion element $\gamma$ and whose torsion-free quotient has rank at least $2$, e.g., this holds for an Enriques surface. Let $D$ and $E$ be nontorsion elements whose cup product is zero. By the Hodge Index Theorem, ... | 8 | https://mathoverflow.net/users/13265 | 420004 | 170,931 |
https://mathoverflow.net/questions/420008 | 6 | For a function $f(x,y)$ on $\mathbb{R}^2,$ defined possibly outside the origin, write
$$\int\_\epsilon ' f \,dx\,dy : = \int\_{\mathbb{R}^2\setminus D\_\epsilon}f \, dx\,dy,$$
(the integral on the complement to the $\epsilon$-disk) and
$$\int' f \,dx \,dy : = \lim\_{\epsilon\to 0} \int'\_\epsilon f \,dx\,dy,$$ wh... | https://mathoverflow.net/users/7108 | Elegant proofs of $\bar{\partial}z^{-1} = 2\pi \delta_0$ | The identification of $\partial\_{\bar{z}}z^{-1}$ with a delta function follows directly from the [Cauchy–Pompeiu formula](https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula#Smooth_functions)
$$f(\zeta) = \frac{1}{2\pi i}\int\_{\partial D} \frac{f(z) \,dz}{z-\zeta} - \frac{1}{\pi}\iint\_D \frac{\partial f(z)}{\p... | 9 | https://mathoverflow.net/users/11260 | 420015 | 170,936 |
https://mathoverflow.net/questions/419969 | 10 | My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c\_1>c\_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic function in the right half of the complex plane with
$$ |f(k)|\leq c\_k \quad \forall\, k\in \mathbb N,$$
there holds $ f(z)=0$... | https://mathoverflow.net/users/50438 | On a variant of Carlson’s theorem | Condition
$$\lim\_{n\to\infty}\frac{\log|c\_n|}{n}=-\infty$$
is sufficient for $f=0$.
Since $f(z)=e^{-cz}$ and $c\_n=e^{-cn}$ satisfy all
conditions, we see that this is best possible in certain sense.
This follows for example from a (much more general) theorem of N. Levinson, Gap and density theorems, AMS, 1940, p... | 8 | https://mathoverflow.net/users/25510 | 420018 | 170,937 |
https://mathoverflow.net/questions/419947 | 1 | I have asked a related question on math.SE [here](https://math.stackexchange.com/questions/4417531/finite-free-resolution-of-a-finitely-generated-mathbb-zx-1-ldots-x-n-modul), but the notation is a bit different.
As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x\_1,\dotsc,x... | https://mathoverflow.net/users/149337 | Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module | Let $\mathfrak{a}:=(X\_1-2X\_2,X\_1-2X\_3,X\_1)$ as ideal of $R$. Then the Koszul complex of the mentioned generating set of $\mathfrak{a}$ is not acyclic, because $X\_1-2X\_2,X\_1-2X\_3,X\_1$ is not a regular sequence. However, $X\_1-2X\_2,X\_1-2X\_3,X\_1$ forms a regular sequence in $R'$, thus the Koszul complex $K\_... | 3 | https://mathoverflow.net/users/127857 | 420023 | 170,938 |
https://mathoverflow.net/questions/419936 | 2 | This is more of a reference request in case anyone can direct me to the right literature. I asked originally on MathStack, but I was suggested to better post it here.
If you have an elliptic curve $E/\mathbb Q$, and you consider the $\mathbb Z\_p$ extension, $\mathbb Q\_{\infty}$, then we know that the rank over $\ma... | https://mathoverflow.net/users/124772 | Mordell-Weil rank growth in Iwasawa tower | Thanks to the work of Kato, and his construction of Euler systems for modular forms, this can be extended to all elliptic curves over $\mathbb{Q}$. See [*$p$-adic Hodge theory and values of zeta functions of modular forms*](http://www.numdam.org/item/AST_2004__295__117_0/), Astérisque 295 (2004), Theorem 14.4: for ever... | 2 | https://mathoverflow.net/users/6506 | 420034 | 170,941 |
https://mathoverflow.net/questions/419962 | 7 | Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X\_{y\_0}$ denote a smooth fiber over $y\_0\in Y$.
If $f$ is smooth, then the topological Euler characteristic is multiplicative: $\chi(X) = \chi(Y)\cdot\chi(X\_{y\_0})$. In... | https://mathoverflow.net/users/15242 | Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers | I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $f:X\to Y$. Let me also assume $X$ smooth. Let $h$ be the genus of $Y$, and $g$ the genus of the general ... | 5 | https://mathoverflow.net/users/4144 | 420042 | 170,943 |
https://mathoverflow.net/questions/420046 | 4 | Let $f \in \mathbb{F}\_q[x\_1, \dots, x\_k]$ be a polynomial with $\deg f = n$, and let $\chi$ be a multiplicative character over $\mathbb{F}\_q$.
Is there any known bound, possibly with conditions about $f$ and $\chi$, for
$$\left|\sum\_{c\_1, \dots, c\_k \in \mathbb{F}\_q} \chi(f(c\_1, \dots, c\_k)) \right| ?$$
| https://mathoverflow.net/users/269936 | Bound for sum of multiplicative character calculated over multivariate polynomial | Yes, there are several known bounds. The following statement, due to Katz, has quite strict conditions on $f$, but gives a very strong result. It is perhaps the simplest statement that gives such a strong bound.
Suppose that the equation $f=0$ defines a nonsingular hypersurface in $\mathbb A^k$, and the degree $n$ pa... | 7 | https://mathoverflow.net/users/18060 | 420052 | 170,946 |
https://mathoverflow.net/questions/420050 | 5 | Consider the set
$$\\\{ (A,B) \in \mathbb{P}^{n\times n-1} \times \mathbb{P}^{n\times n -1} : \text{im}(A) \subseteq \text{im}(B)\}.$$
That is, this is the set of pairs of square matrices $(A,B)$ so that the image of $A$ is contained in the image of $B$. Is this Zariski closed? I would be happy if this were at least tr... | https://mathoverflow.net/users/165301 | Is containment of images of linear maps Zariski closed? | I am making my comment an answer. The specified set is not Zariski closed. If it were, then its intersection with every Zariski closed subset $C$ would be relatively closed in $C$. But now let $C$ be the curve, a copy of the affine line, where the first component $A$ is held fixed as the identity $n\times n$ matrix, an... | 12 | https://mathoverflow.net/users/13265 | 420054 | 170,947 |
https://mathoverflow.net/questions/420057 | 5 | I am having some trouble trying to understand the proof of Theorem 7.2.5 in Bhatt and Scholze's paper [The pro-étale topology for schemes](https://arxiv.org/abs/1309.1198). Specifically, I don't quite understand why it was necessary to prove that $F: C \to \mathit{Sets}$ preserves connectedness of objects, and how view... | https://mathoverflow.net/users/143390 | Infinite Galois equivalence | When $X$ is connected, we see that any graph $\Gamma\_f$ for $f \colon X \to Y$ is connected as well, so we get a bijection
\begin{align\*}
\mathscr C(X,Y) &\to \left\{\Gamma \subseteq X \times Y \text{ connected}\ \bigg|\ \Gamma \underset{\pi\_1}{\overset{\sim}\to} X \right\} \\
f &\mapsto \Gamma\_f.
\end{align\*}
The... | 6 | https://mathoverflow.net/users/82179 | 420060 | 170,949 |
https://mathoverflow.net/questions/420000 | 2 | Let $d>1$ be not a square. Then the continued fraction expansion of $\sqrt d$ is $[a\_0; \overline{a\_1,\dots,a\_\ell}]$, where $a\_0=\lfloor \sqrt d\rfloor$ and $a\_\ell=2a\_0$.
Thus, $\ell=\ell(d)$.
About 30 years ago I heard a talk where $\ell(d)$ was somehow related to the class number of $\mathbb Q(d)$ -- in par... | https://mathoverflow.net/users/8131 | Period of continued fraction expansion and class number | Claude Levesque, [On semi-reduced quadratic forms, continued fractions and class number](https://www.kurims.kyoto-u.ac.jp/%7Ekyodo/kokyuroku/contents/pdf/0998-7.pdf), quotes a theorem of Lu, H., On the class number of real quadratic fields, Scientia Sinica II (special number, 1979), 118-130, as follows:
Let $m>1$ be ... | 2 | https://mathoverflow.net/users/3684 | 420061 | 170,950 |
https://mathoverflow.net/questions/368496 | 4 | Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?:
1)For every open set $U\subset H$ and every Frechet differential map
$f:U \to H$ with $Df(x)\in D,\; \forall x\in U$, the mapping $f$ is automatically $C^{\infty}$
... | https://mathoverflow.net/users/36688 | A kind of holomorphicity of maps on Hilbert space | 1. In [this book](https://www.mat.univie.ac.at/%7Emichor/apbookh-ams.pdf), Thm 7.19 (1) $\Longleftrightarrow$ (9) answers this for $D$ the complex linear bounded maps, but you have to assume that $f$ is Gateaux differentiable with the derivative locally Lipschitz.
2. From Thm 7.17 (7) you can deduce an answer, but you ... | 1 | https://mathoverflow.net/users/26935 | 420066 | 170,953 |
https://mathoverflow.net/questions/420089 | 2 | Whether a complete non-compact non-flat Riemannian $n$-manifold $M$ with non-negative sectional curvature has Euclidean volume growth?
That is, whether there is a constant $C>0$ such that $\mathrm{Vol}(B\_x(r))\geq Cr^n$ for all $r>0$ and $x\in M$? Here $\mathrm{Vol}(B\_x(r))$ is the volume of the $r$-ball in $M$.
... | https://mathoverflow.net/users/90512 | Non-negatively curved manifolds and the volume of balls | It is certainly not true that every complete nonflat open manifold of nonnegative curvature has Euclidean volume growth. Counterexamples are trivial to construct. Say, a capped cylinder. More generally any nonnegatively curved manifold $M^n$ with nontrivial soul has slower than Euclidean volume growth. Because its asym... | 6 | https://mathoverflow.net/users/18050 | 420091 | 170,957 |
https://mathoverflow.net/questions/420088 | 14 | $\DeclareMathOperator\Aut{Aut}\newcommand\card[1]{\lvert#1\rvert}$So, after going over the classification of finite abelian groups in a class I was teaching this winter, I got curious about whether it could be used to obtain a 'nice' value for the [groupoid cardinality](https://ncatlab.org/nlab/show/groupoid+cardinalit... | https://mathoverflow.net/users/22131 | Groupoid cardinality of the class of abelian p-groups | This is Corollary 3.8 in Cohen, H.; Lenstra, H. W., Jr. Heuristics on class groups. Number theory (New York, 1982), 26–36, Lecture Notes in Math., 1052, Springer, Berlin, 1984 (MR0750661).
In this paper you will find many formulas for averages of various functions over abelian groups weighted by inverse size of the a... | 19 | https://mathoverflow.net/users/40821 | 420096 | 170,959 |
https://mathoverflow.net/questions/420093 | 2 | My question is:
>
> Is every Polish space image of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$?
>
>
>
Where a Polish space is a separable and completely metrizable space and where $\Bbb{N}^\Bbb{N}$ (aka Baire space) is the space of infinite sequences of natural numbers endowed with the produc... | https://mathoverflow.net/users/141146 | Images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$ | This question is answered positively in [On closed images of the space of irrationals](https://www.semanticscholar.org/paper/On-closed-images-of-the-space-of-irrationals-Engelking/726a02a1075858b78aa322f1dfb504e858d39cca), by Engelking.
| 3 | https://mathoverflow.net/users/172802 | 420099 | 170,960 |
https://mathoverflow.net/questions/420036 | 4 | Let $\pi(x;q,a)$ count the number of primes $\leq x$ congruent to $a$ mod $q$. The Brun-Titchmarsh Theorem states that for all $q< x$, $(a,q)=1$, we have
$$
\tag{1}
\pi(x;q,a) \leq \frac{2x}{\varphi(q)\log(x/q)}.
$$
Let $f(n) = 1$ if $n$ is prime and $0$ otherwise. Then we can rephrase $(1)$ as (almost) saying that
$$
... | https://mathoverflow.net/users/307675 | Generalizations of the Brun-Titchmarsh theorem | Yes, there are several results answering the question you ask about, in great generality.
1. Linnik, in his monograph on the dispersion method, proved an analogue of (2) for the $k$-th divisor function. Concretely,
$$\sum\_{\substack{n \le x \\ n \equiv a \bmod q}} d\_k(n) \ll \frac{x}{q} \left(\frac{\phi^{r-1}(q)}{q... | 3 | https://mathoverflow.net/users/31469 | 420100 | 170,961 |
https://mathoverflow.net/questions/420103 | 9 | I had been pointed to Ramanujan's 1912 article *Note on a set of simultaneous equations* in this [answer](https://mathoverflow.net/a/357941/31310) to my former question about the [Solvability of a system of polynomial equations](https://mathoverflow.net/questions/357138/solvability-of-a-system-of-polynomial-equations).... | https://mathoverflow.net/users/31310 | Impact of Ramanujan's Note on a set of simultaneous equations | *This refers to the third and fourth question in the OP.*
Ramanujan's 1912 paper addresses a problem similar to that considered by Sylvester in 1851 [1]. The method to solve the set of algebraic equations
$$\sum\_{k=1}^{n}x\_kz\_k^j=a\_j,\;\;0\leq j\leq 2n-1$$
in the unknown $x\_k$'s and $z\_k$'s is different in the ... | 11 | https://mathoverflow.net/users/11260 | 420105 | 170,963 |
https://mathoverflow.net/questions/420107 | 7 | Let $A$ be a finitely generated $\mathbb{Z}$-algebra and let
$f: \operatorname{Spec} A \rightarrow \operatorname{Spec} \mathbb{Z}$ be the canonical map.
On pg. 53, Thm. 8.2 of <https://www.math.uni-bonn.de/people/scholze/Condensed.pdf>
one defines a functor $$f\_!: D(A\_\blacksquare) \rightarrow D(\mathbb{Z}\_\black... | https://mathoverflow.net/users/472750 | What is the reason for $f_!$ not preserving discrete objects? | You can see this by a direct calculation, for example in the most basic case $A=\mathbb{Z}[T]$. When you apply $f\_!$ to $A=\mathbb{Z}[T]$ itself, you get the object represented by the two-term complex
$$\mathbb{Z}[T]\to \mathbb{Z}((T^{-1})).$$
with $\mathbb{Z}[T]$ in degree $0$. (This is the "compactly supported cohom... | 6 | https://mathoverflow.net/users/480363 | 420139 | 170,977 |
https://mathoverflow.net/questions/420140 | 1 | Let $K$ be a number field and $\mathcal{O}\_K$ be its ring of integers. For any prime ideal $\mathfrak{p}$ in $\mathcal{O}\_K$ is it true that every residue class in $\mathcal{O}\_K/\mathfrak{p}$ contains an integer? I can prove that it is true if $\mathfrak{p}$ is unramified and has inertial degree 1, but not for gene... | https://mathoverflow.net/users/480366 | Integers in residue classes $\mathcal{O}_K/\mathfrak{p}$ | Let $p$ be the prime number that satisfies $p\mathbb{Z} = \mathbb{Z} \cap \mathfrak{p}$.
Then your claim is equivalent to the inclusion $\mathbb{F}\_p\subset \mathcal{O}\_K/\mathfrak{p}$ being an equality, in other words the prime $\mathfrak{p}$ having inertial degree $1$. So it is not true in general, for example it i... | 3 | https://mathoverflow.net/users/110362 | 420141 | 170,978 |
https://mathoverflow.net/questions/420084 | 5 | Let $X$ be an infinite discrete topological space. Is $$C\_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
| https://mathoverflow.net/users/113200 | Ring of continuous functions is a Jacobson ring | First, let us consider the question of when the ring $C(X)$ of *all* continuous real-valued functions on a topological space $X$ (not necessarily discrete) is Jacobson, keeping in mind that $C\_b(X) = C(\beta X)$ where $\beta X$ is the Stone-Čech compactification.
In fact, every prime ideal of $C(X)$ is contained in ... | 6 | https://mathoverflow.net/users/17064 | 420149 | 170,982 |
https://mathoverflow.net/questions/420156 | 1 | **Question:**
is there an established name for the set $\Big\lbrace\ f {\Large\ \boldsymbol{|}}\ f\in C^\infty\quad {\Large\boldsymbol{\land}}\quad \exists\,{k\in\mathbb{N}^+}:\frac{d^{i+k}}{dx^{i+k}}f(x)=\frac{d^{i}}{dx^{i}}f(x)\ \forall i\in\mathbb{N}^+ \Big\rbrace{\Large\text{?}}$
E.g. $f(x)=e^x\implies k=1;\ ... | https://mathoverflow.net/users/31310 | Functions with periodic sequence of derivative-values | Assuming that $\mathbb N^+$ is defined as $\{1,2,\dots\}$, the set of functions in questions is just the set of all solutions $f$ of all the simple linear ODE's $f^{(k+1)}=f'$ with $k\in\mathbb N^+$, that is, the set of all functions $f$ such that
$$f(x)=a+\sum\_{j=0}^{k-1} c\_{k,j}\exp\{e^{2\pi ij/k}x\}$$
for some $k\... | 5 | https://mathoverflow.net/users/36721 | 420159 | 170,985 |
https://mathoverflow.net/questions/419734 | 4 | Let $X$ be an isolated, Gorenstein singularity of dimension at least $2$ and $\pi: \widetilde{X} \to X$ be a resolution of singularities. Let $E$ be the exceptional divisor and $E\_1,...,E\_r$ be the irreducible components of $E$. Can there exist integers $a\_1,...,a\_r$ not all zero such that $\mathcal{O}\_{\widetilde... | https://mathoverflow.net/users/32151 | Do there exist linear relations between exceptional divisors | I am just posting my comment as an answer. For a quasi-compact, separated, irreducible, smooth $2$-dimensional algebraic space $\widetilde{Y}$ over an algebraically closed field $k$, for a connected, effective Cartier divisor $D$ that is a proper scheme and with irreducible components $(D\_i)\_{i=1,\dots,r}$, there exi... | 2 | https://mathoverflow.net/users/13265 | 420161 | 170,986 |
https://mathoverflow.net/questions/418565 | 7 | This question was also asked [here](https://math.stackexchange.com/questions/4403054/tangent-bundle-of-a-tensor-product-bundle) on math-stackexchange.
Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K\_E: TE \to E\times\_M E$ and... | https://mathoverflow.net/users/98733 | Tangent bundle of a tensor product bundle | Over a point:
$$
T(E\otimes F) = (E\otimes F)\oplus (E\otimes F) = E\otimes (F\oplus F)
$$
which is naturally isomorphic to $(E\oplus E)\otimes F$ using the canonical flip $E\otimes F = F\otimes E$. Likewise
$$
T\operatorname{Hom}(E,F) = \operatorname{Hom}(E,F)\oplus \operatorname{Hom}(E,F)=\operatorname{Hom}(E,TF)
$$
... | 2 | https://mathoverflow.net/users/26935 | 420166 | 170,989 |
https://mathoverflow.net/questions/420162 | 16 | 1. Is it true that if $A\_1 \vee A\_2 \vee .. \vee A\_n = B\_1 \vee B\_2 \vee .. \vee B\_m$, where $A\_i, B\_j$ are homotopy types of complexes not decomposable into a bouquet, then the multisets $A\_i$ and $B\_j$ coincide? That is, is it true that a commutative monoid of homotopy types decomposable into a bouquet of... | https://mathoverflow.net/users/148161 | Is the decomposition of the homotopy type of a complex into a bouquet unique? | In Hilton&Roitberg paper "On principal $S^3$-bundles over spheres" it's proven that if you have a prime order $p \neq 2,3$ class $\alpha$ in $\pi\_k(S^n)$ that is a suspension, then for a prime $q \neq \pm 1 \, mod \, p$ mapping cones $C(\alpha)$ and $C(q \cdot \alpha)$ satisfy $C(\alpha) \vee S^n \cong C(q \cdot \alph... | 17 | https://mathoverflow.net/users/81055 | 420171 | 170,992 |
https://mathoverflow.net/questions/420153 | 2 | I am interested in long-range percolation models with heavy-tailed degree distributions such as [DHH13](http://www.numdam.org/article/AIHPB_2013__49_3_817_0.pdf), [GLM21](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7B85863F4E9E3FB24BA77F538D1A871A/S0001867821000136a.pdf/div-class-title-perco... | https://mathoverflow.net/users/479063 | Which infinite random graphs with percolation threshold $p_c=0$ are transient? | I will first construct a simple deterministic example of a recurrent graph with $p\_c^\mathrm{site}=0$ and then show how it can be modified to be a unimodular random rooted graph (see e.g. Nicolas Curien's lecture notes for the definition <https://drive.google.com/file/d/16qEMGJU2g01g4YWkYtVKTSqetk-SpRmy/view>)
Suppo... | 1 | https://mathoverflow.net/users/41827 | 420173 | 170,994 |
https://mathoverflow.net/questions/420158 | 47 | I would like to ask a question inspired by the title of a book by Sir [Roger Penrose](https://en.wikipedia.org/wiki/Roger_Penrose) ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics.
I'm going to focus the post (and modulate my genuine idea) about an a... | https://mathoverflow.net/users/142929 | Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians | After mathematicians had been been taught for decades that a consistent theory of the calculus based on infinitesimals was impossible, Abraham Robinson was certainly swimming against the tide when he proved otherwise.
Robinson, A. (1961): *Non-standard analysis*, Indagationes Mathematicae 23, pp. 432-440.
Robinson,... | 48 | https://mathoverflow.net/users/18939 | 420175 | 170,996 |
https://mathoverflow.net/questions/420083 | 1 | I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)≅P(E⊗L)$".
The statement can also be seen on the Wikipedia page of the projective bundle. The reference they give is for Hartshorne, Algebraic Geometry. But unfortunately, I am not... | https://mathoverflow.net/users/126899 | Projective bundle is stable under twisting by a line bundle | $\DeclareMathOperator{\Hom}{Hom}$There's a nice explicit map $P\Hom(L,E)\to P(E)$ that takes a homomorphism to its image. Then use that $E\otimes L \cong \Hom(L^\vee,E)$ where $L^\vee$ is the dual.
But if you work through what this does, it's simply the map $P(E\otimes L)\to P(E)$ that takes $[u\otimes v] \mapsto [u]... | 2 | https://mathoverflow.net/users/58888 | 420182 | 171,000 |
https://mathoverflow.net/questions/420183 | 8 | $\newcommand\Legendre{\genfrac(){}{}}$Let $p\equiv 1\pmod 4$ be a prime number, and $x\_{i}\ge 0$ be such that $$x\_{1}+x\_{2}+\dotsb+x\_{p}=1.$$
Show that
$$\sum\_{1\le i<j\le p}\Legendre{i-j}{p}x\_{i}x\_{j}\le\dfrac{p-1}{2p+6}$$
Here $\Legendre\cdot\cdot$ is the Legendre symbol.
This problem was encountered by a ... | https://mathoverflow.net/users/38620 | Prove an inequality related to sums of Legendre symbols | $\newcommand\Legendre{\genfrac(){}{}}$We have $$\sum\_{1\le i<j\le p}\Legendre{i-j}{p}x\_{i}x\_{j} \leq \frac{k-1}{2k }$$ where $k$ is the size of the largest clique in the Paley graph, and this is sharp.
Indeed, if the number of $i$ such that $x\_i>0$ is at most $k$ then
$$\sum\_{1\le i<j\le p}\Legendre{i-j}{p}x\_... | 18 | https://mathoverflow.net/users/18060 | 420188 | 171,002 |
https://mathoverflow.net/questions/416761 | 3 | Assume all algebras are finite dimensional quiver algebras over a field (no restriction of generaltiy if the field is algebraically closed).
>
> Let A be a local Frobenius algebra.
> Is A isomorphic to its opposite algebra?
>
>
>
For non-Frobenius algebras this is false, see [Do you know which is the minimal l... | https://mathoverflow.net/users/61949 | Local Frobenius algebras and their opposite algebras | No. Consider, for example, the quantum complete intersection $A = k\langle X,Y\rangle/(X^2, Y^3, XY-qYX)$, $q\in k\setminus\{0\}$. This is a Frobenius local algebra (see Section 3 of *arXiv:0709.3029*), and the ideal $(X^2, Y^3, XY-qYX)$ in $k\langle X,Y\rangle$ is admissible.
Any isomorphism $f:A\to A^{\rm op}$ must... | 4 | https://mathoverflow.net/users/136180 | 420197 | 171,006 |
https://mathoverflow.net/questions/419157 | 0 | In this paper [N. Ghoussoub](https://www.degruyter.com/document/doi/10.1515/crll.1991.417.27/html),
the author claims the following version of Marino–Prodi perturbation, that is :
Let $H$ a Hilbert space.
Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K\_c$ (that is, the set of critical points with critic... | https://mathoverflow.net/users/166368 | A question about Marino–Prodi perturbation | The proof for $G=Z\_2$ is given in [jde2016](http://dx.doi.org/10.1016/j.jde.2016.09.018) Appendix B Proposition B3.
| 0 | https://mathoverflow.net/users/166368 | 420199 | 171,007 |
https://mathoverflow.net/questions/420048 | 4 | Consider the following [bornologies](https://en.wikipedia.org/wiki/Bornology) $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions from $\mathbb{N}$ to $\mathbb{N}$:
* $\mathbb{D}=\{A: \exists f\in\mathcal{N}\forall g\in A\exists m\in\mathbb{N}\forall n>m(f(n)>g(n))\}$. (**Dominatable** sets)
* $\mathbb{... | https://mathoverflow.net/users/8133 | Comparing bornologies for domination/escaping | Note that $\mathfrak{b}=\mathfrak{d}$ is equivalent to the existence of a $<^\ast$-increasing sequence $(f\_\alpha)\_{\alpha<\mathfrak{d}}$ which is cofinal in $(\mathcal{N},{<^\ast})$, where $f <^\ast g \iff \exists n\,\forall m \geq n\,{f(m) < g(m)}$.
For $\alpha<\mathfrak{d}$, let
$$\begin{aligned}
D\_\alpha &= \... | 6 | https://mathoverflow.net/users/2000 | 420204 | 171,009 |
https://mathoverflow.net/questions/420198 | 3 | Given a matrix group $G$ by its generators i.e. $G =\langle A\_1,A\_2,...,A\_k \rangle \leq GL\_n(q)$, where each $A\_i$'s are matrix in $GL\_n(q)$
>
> **Q.** Does there exist a polynomial time (polynomial in input size) algorithm to find a minimal normal subgroup of $G$ if it exists otherwise it returns $G$ is sim... | https://mathoverflow.net/users/475097 | Algorithm to find a minimal normal subgroup of given group $G$ by matrix group representation | In the paper
```
Holt, D., Leedham-Green, C. R., & O'Brien EA (2020). Constructing composition factors for a linear group in polynomial time. JOURNAL OF ALGEBRA, 561, 215-236. 10.1016/j.jalgebra.2020.02.018
```
the authors consider the question of finding a composition series of a subgroup of ${\rm GL}(n,q)$ - I ... | 1 | https://mathoverflow.net/users/35840 | 420206 | 171,010 |
https://mathoverflow.net/questions/420216 | 2 | A tree $G$ on $n$ vertices $V=\{v\_1,...,v\_n\}$ is a connected undirected graph which is acyclic. For each tree $G$ one can split the set of vertices $V$ into two disjoint subsets $U,W \subset V$ such that $V = U \cup W$ and each edge in $G$ is between a vertex from $U$ and a vertex from $W$. In order to get uniquenes... | https://mathoverflow.net/users/409412 | Is there a formula for the number of trees with this extra condition? | If the trees are labelled, then each tree satisfying the condition corresponds to exactly one spanning tree of the bipartite graph $K\_{n\_1,n\_2}$. Therefore the answer is the number of spanning trees of the bipartite graph $K\_{n\_1,n\_2}$, which is $n\_1^{n\_2-1}n\_2^{n\_1-1}$ according to [this](https://math.stacke... | 5 | https://mathoverflow.net/users/480431 | 420218 | 171,012 |
https://mathoverflow.net/questions/420219 | 2 | Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
>
> We don't assume $X$ and $Y$ are independent!
>
>
>
Let $P\_X$ (resp. $P\_Y$) be the marginal distribution of $X$ (resp. $Y$), defined by $P\_X... | https://mathoverflow.net/users/78539 | Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$? | It is unclear to me what "a principled way" could mean.
However, given $n$ iid pairs $(X\_1,Y\_1),\dots,(X\_n,Y\_n)$, it is easy to get $k:=\lfloor n/2\rfloor$ iid pairs $(X\_1,Z\_1),\dots,(X\_k,Z\_k)$ such that, for each $j\in\{1,\dots,k\}$, (i) the random variables $X\_j$ and $Z\_j$ are independent and (ii) $Z\_j$ ... | 2 | https://mathoverflow.net/users/36721 | 420222 | 171,013 |
https://mathoverflow.net/questions/420177 | 0 | There is an ordered set $M$ with $N$ numbers in it (and $M\_n$ is the $n$-th number in $M$). Let $L\_k$ be the sum of the first $k$ numbers in $M$.
Consider equation $\sum\_{k=0}^{N-1} 3^k 2^{-L\_{k+1}} \equiv 1 \bmod 3^N$
For example, if $N=1 \Rightarrow 3^0 2^{-L\_1} \equiv 1 \bmod 3^1$ or $2^{-L\_1}\equiv 1 \bmo... | https://mathoverflow.net/users/479568 | Is there a general way to solve this modular equation? | There seems to be no simple formula for the solutions to the given congruence. Still, they can be computed iteratively as follows.
First, we notice that $2^m = (3-1)^m \equiv (-1)^m \sum\_{i=0}^{N-1} \binom{m}{i} (-3)^i \pmod{3^N}$. Then the left-hand side of the congruence in question can be rewritten as
$$\sum\_{k=... | 0 | https://mathoverflow.net/users/7076 | 420232 | 171,016 |
https://mathoverflow.net/questions/420238 | -2 | I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is:
Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is invertible and $(A−B)^2=I\_n$, where $I\_n$ is the identity matrix, prove that $tr(A)=tr(B)$ and that $n$ is even. How should I... | https://mathoverflow.net/users/480453 | Proving 2 matrices have the same trace | Let's call $C=A-B$. Then you have $C^2=I\_n$ and $BC-CB$ is invertible. You want to show that $C$ has an equal dimension of $1$ and $(-1)$ eigenspaces, which in turn implies both the equality $tr(A)=tr(B)$ and $n$ being even.
Pick an eigen basis of the space to assume that
$$
C=\left(
\begin{array}{c}I\_m & 0\\ 0&-I\... | 3 | https://mathoverflow.net/users/38468 | 420242 | 171,019 |
https://mathoverflow.net/questions/420233 | 0 | $f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x\_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that:
\begin{equation}
f(x\_{0}+h)=f(x\_{0})+T\cdot h +r(h) \quad\text{donde}\quad \lim\_{h\to 0}\frac{r(h)}{\Vert h\Vert}=0
... | https://mathoverflow.net/users/480450 | Application of the Frechet derivative | If $F:M^{n\times n}\to M^{n\times n}$ is differentiable, then for $A\in M^{n\times n}$, $DF(A)$ is a linear map from $M^{n\times n}$ to $M^{n\times n}$ and we denote by $DF(A)H$ value of this linear map at $H\in M^{n\times n}$.
If $F,G:M^{n\times n}\to M^{n\times n}$ are differentiable maps, then the product of the m... | 2 | https://mathoverflow.net/users/121665 | 420246 | 171,020 |
https://mathoverflow.net/questions/420247 | 6 |
>
> **Question:** For a smooth, bounded domain $\Omega\subset \mathbb R^d$, does there exist a function $u\in L^1(\Omega)$ such that
> $u\not\in L^\Phi(\Omega)$ for *any* Orlicz space $\Phi$?
>
>
>
---
For the definition of Orlicz spaces see [this wikipedia page](https://en.wikipedia.org/wiki/Orlicz_space#Fo... | https://mathoverflow.net/users/33741 | An $L^1$ function but (really) no better? | There is a much more general result of Vallée-Poussin from which a negative answer to your question follows.
Let $(X,\mu)$ be a measure space. We say that a family of function $\mathcal{F}\subset L^1(X)$ is *equi-integrable* if for every $\varepsilon>0$ there is $\delta>0$ such that
$$
\sup\_{f\in\mathcal{F}} \int\_E... | 12 | https://mathoverflow.net/users/121665 | 420248 | 171,021 |
https://mathoverflow.net/questions/420241 | 1 | Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently smooth" function. For simplicity, we may consider $f$ to be an affine function, i.e $f(x) \equiv b-x^\top w$, for some $(w,b) \in \mathbb R^{d}$. Let $\Phi:\mathbb R \to (0,1)$ be the standard Gaussian CDF defind by
$$
\Phi(t):= \frac{1}{\sqrt{2\pi}}\int\_{-\infty}^t... | https://mathoverflow.net/users/78539 | Bound error in approximating $E_x [H(f(x))]$ with random $(1/n) \sum_{i=1}^n \Phi(f(x_i)/h)$ where $H$ is Heaviside function and $\Phi$ is normal CDF | $\newcommand{\si}{\sigma}$Let
\begin{equation\*}
Y:=f(X),\quad Y\_i:=f(X\_i),
\end{equation\*}
so that $Y,Y\_1,Y\_2,\dots$ are iid real-valued random variables (r.v.'s).
Assume that (i) the r.v. $Y$ has a density $p\_Y$ continuous at $0$ and (ii) $E|Y|^k<\infty$ for some real $k>0$. Except possibly for entering ind... | 3 | https://mathoverflow.net/users/36721 | 420251 | 171,022 |
https://mathoverflow.net/questions/420254 | 1 | I am looking for an algorithm with polynomial complexity where, given a strongly connected edge-weighted digraph I can find the minimal subgraph which connects some root vertex v to a known set of other vertices.
As an example, given a strongly connected edge-weighted digraph with vertices labeled a-z, I want to find... | https://mathoverflow.net/users/480464 | Minimum edge-weighted directed subgraph in polynomial time | This is an instance of the Directed Steiner Network Problem and as such it's solvable in time $|V(G)|^{O(|T|)}$ as proved by [Feldman & Ruhl (2006)](https://doi.org/10.1137/S0097539704441241), where $G$ is the given graph and $T\subset V(G)$ is the given subset of vertices (terminals).
| 1 | https://mathoverflow.net/users/7076 | 420259 | 171,025 |
https://mathoverflow.net/questions/420245 | 6 | Q1 : If $X \to Y \to Z$ are maps of schemes, is there a relation such as
$$\omega\_{X/Z} \overset{?}{=} \omega\_{Y/Z}|\_X \overset{L}{\otimes} \omega\_{X/Y}$$
between their dualizing complexes? Or maybe some kind of distinguished triangle?
The reason I think so is that I'm told $\omega\_{X/Z}$ is the determinant $\de... | https://mathoverflow.net/users/86614 | Distinguished triangle of dualizing complexes and/or determinants? | A very good reference for these topics is
Lipman, Joseph: Notes on derived functors and Grothendieck duality. *Foundations of Grothendieck duality for diagrams of schemes*, 1–259, Lecture Notes in Math., **1960**, Springer, Berlin, 2009.
For your first question, the issue is the pseudo functoriality of $(-)^!$ toge... | 4 | https://mathoverflow.net/users/6348 | 420273 | 171,028 |
https://mathoverflow.net/questions/420269 | 1 | I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page 289.
Theorem 10.14.
The Albanese map
\begin{equation\*}
alb\_X:CH\_0(X)\_{hom}\rightarrow Alb(X)
\end{equation\*}
Induces a... | https://mathoverflow.net/users/150116 | Meaning of torsion points in a Roitman's theorem | It occurs to me that your question shouldn't be taken literally, and is simply asking about the meaning of the theorem. To appreciate it, one can ask what $CH\_0(X)$ looks like. As a first attempt, map it to something more concrete like the Albanese (which is just a complex torus over $\mathbb{C}$). It is easy to see t... | 4 | https://mathoverflow.net/users/4144 | 420279 | 171,031 |
https://mathoverflow.net/questions/420280 | 3 | For sets $x, y$ we write $x\leq y$, if there is an injection $\iota: x \to y$, and we write $x \leq^\* y$ if either $x = \emptyset$ or there is a surjection $s: y \to x$. In ${\sf (ZF)}$ we have that $x \leq y$ implies $y \leq^\* y$.
Consider the following statements:
>
> Partition principle (PP): For all sets $x... | https://mathoverflow.net/users/8628 | Does the partition principle imply (DC)? | Yes. This is a combination of facts.
1. $\sf PP$ implies that if a set $X$ can be mapped onto an ordinal $\alpha$, then $\alpha$ injects into $X$. In other words, it implies that $\aleph^\*(X)=\aleph(X)$ for any set $X$.
2. $\sf AC\_{WO}$, that is the axiom of choice from families of sets indexed by an ordinal, is eq... | 8 | https://mathoverflow.net/users/7206 | 420282 | 171,032 |
https://mathoverflow.net/questions/420281 | 2 | Let $R$ be a commutative ring, $M$ be an $R$-module, and $N$ be a submodule of $M$. Assume that both $M$ and $N$ are flat, so we can identify $N\otimes\_RN$, $M\otimes\_RN$, and $N \otimes\_RN$ as submodules of $M\otimes\_RM$.
Is it true that $N\otimes\_R N = (M\otimes\_RN)\cap (N\otimes\_R M)$? If not in general, un... | https://mathoverflow.net/users/nan | Intersection and tensor product of flat modules | This is not true in general: take $M=R$, $N=Rx$ for some $x\in R$. Then $N\otimes \_RN=Rx^2$ while $M\otimes \_RN=N\otimes \_RM=Rx$.
It is true if $M/N$ is flat: this follows from Proposition 7 of §2.6 in Bourbaki's *Commutative algebra*, ch. I.
| 7 | https://mathoverflow.net/users/40297 | 420285 | 171,034 |
https://mathoverflow.net/questions/420283 | 5 | Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with centralizer $Z$.
Does $\rho(Z)$ consist of polynomials in $\rho(x)$?
Examples where I know this is true:
* ~~$x$ is se... | https://mathoverflow.net/users/50929 | Does the centralizer of a regular element in a semisimple Lie algebra act by polynomials? | It is false for the same reason the first example is wrong. Take $\rho$ for example the adjoint representation. As soon as $\mathfrak g$ has rank at least $2$ you can find a regular semisimple $x$ such that some two roots (whose difference is not a root) of $Z$ coincide on $x$, while this is not true for generic $y \in... | 5 | https://mathoverflow.net/users/50929 | 420292 | 171,036 |
https://mathoverflow.net/questions/420208 | 4 | Recall that a basis $(x\_{n})\_{n}$ for a Banach space $X$ is called boundedly complete if for every scalar sequence $(a\_{n})\_{n}$ with $\sup\_{n}\|\sum\_{i=1}^{n}a\_{i}x\_{i}\|<\infty$, the series $\sum\_{n=1}^{\infty}a\_{n}x\_{n}$ converges. It is well-known that $c\_{0}$ has no boundedly complete basis. My questio... | https://mathoverflow.net/users/41619 | $c_{0}$ has no boundedly complete basis | Proposition.
Let $(x\_n)$ be a basis for a Banach space $(X,\|\cdot\|\_X)$ that is not boundedly complete. Then $bc(x\_n) \ge 1$.
Proof: Let $(Z,\|\cdot\|\_Z)$ be the Banach space of all sequences $A=(a\_n)$ of scalars for which
$$
\|A\|\_Z = \sup\_N \|\sum\_{n=1}^N a\_n x\_n\|\_X < \infty.
$$
For $N =1,2,...$, defin... | 2 | https://mathoverflow.net/users/2554 | 420300 | 171,038 |
https://mathoverflow.net/questions/420272 | 7 | The minimal model program aims to find a minimal representative in the birational class of a given variety with reasonable singularities. Assuming this has been done, it seems natural to ask what these minimal models look like.
Is it feasible to ask for a classification of minimal (complex) varieties, say for threefo... | https://mathoverflow.net/users/126543 | Is there a classification of minimal algebraic threefolds? | It depends what you mean by classification.
The key results for surfaces IMO are: 1) Any surface $S$ of general type has a canonical model given by $S\_{can}:={\rm Proj} R(K\_S)$ and a unique minimal model given by the minimal resolution of $S\_{can}$. 2) The canonical volume ${\rm vol}(S)=K\_{S\_{can}}^2$ is an intege... | 15 | https://mathoverflow.net/users/19369 | 420303 | 171,040 |
https://mathoverflow.net/questions/420195 | 3 | I have a linear system
\begin{align\*}
\left[\begin{array}{cccc}
1 & 2 & 1 & -1 \\
3 & 2 & 4 & 4 \\
4 & 4 & 3 & 4 \\
2 & 0 & 1 & 5 \\
\end{array}\right]
\left[\begin{array}{c}
w \\ x \\ y \\z
\end{array}\right] =
\left[\begin{array}{c}
5 \\ 16 \\ 22 \\ 15
\end{array}\right],
\end{align\*}
whose matrix $\bf{... | https://mathoverflow.net/users/99453 | Iterative methods for linear system with non-diagonally dominant matrix | You can see your equation as $f(u)=Au-b$, with $f(u)=0$. You can use some numerical method to solve it. One way is to use the minimization problem
$$\min\_u g(u),\qquad g(u)=\frac{1}{2}\|f(u)\|^2.$$ This gives you the equation $$0=\nabla g(u)=A^T(Au-b)\quad \text{ or } \quad A^TAu=A^Tb$$ which gives you the possible mi... | 1 | https://mathoverflow.net/users/478686 | 420305 | 171,041 |
https://mathoverflow.net/questions/420121 | -1 | In the paper [1] below, among other things, Carlitz introduced weighted Stirling numbers of the second kind $R(n,k,r)$. He also proved that the numbers $R(n,k,r)$ can be generated by
\begin{equation\*}%\label{S(n,k,x)-dfn}
\frac{(\textrm{e}^z-1)^k}{k!}\textrm{e}^{\lambda z}=\sum\_{n=k}^\infty R(n,k,\lambda)\frac{z^n}{n... | https://mathoverflow.net/users/147732 | Could you please confirm or deny two identities involving weighted Stirling numbers of the second kind? | $R(n, k, -\tfrac k2)$ is just the central factorial number $T(n, k)$. (Given the definition of the central factorial numbers, it may be more natural to use them in your context than $R$).
Consider [A136630](https://oeis.org/A136630). We have $\operatorname{A136630}(n, k) = 2^{n-k} T(n,k)$ (this may be stated explicit... | 2 | https://mathoverflow.net/users/46140 | 420309 | 171,044 |
https://mathoverflow.net/questions/420276 | 3 | I am considering the following wave equation (for $\phi=\phi(x,t)$)
$$
\phi\_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}
$$
where $\nabla$ is just spatial gradient, i.e., $\nabla \phi= (\partial\_{x\_1} \phi, \partial\_{x\_2} \phi, \partial\_{x\_3} \phi)$ and $a \in \mathbb{R}... | https://mathoverflow.net/users/468783 | On a nonlinear wave equation | By replacing $\phi$ by $-\phi$ you can set $\alpha$ to be positive (or negative, if you wish). By replacing $\phi$ by $\lambda \phi$ you can rescale away $\alpha$. So you can set $\alpha$ to be either $+1$ or $-1$ as you wish.
So for now let's consider the situation you are solving
$$ \partial^2\_{tt}\phi - \Delta \p... | 5 | https://mathoverflow.net/users/3948 | 420320 | 171,046 |
https://mathoverflow.net/questions/420220 | 1 | Consider pseudo Gaussian densities for $0<s<t$ and $x,y\in\mathbb R$
$$f(s,x,t,y):=\frac{1}{\sqrt{2\pi A(s,t,y)}}\exp\left(-\frac{(y-x)^2}{2A(s,t,y)}\right)\quad\mbox{and} \quad g(s,x,t,y):=\frac{1}{\sqrt{2\pi B(s,t,y)}}\exp\left(-\frac{(y-x)^2}{2B(s,t,y)}\right),$$
where $A(s,t,y):=\int\_s^t k(u,y)/(1+a(u))du$ and... | https://mathoverflow.net/users/nan | Questions on the integral of pseudo Gaussian kernel and its derivative on $(0,\infty)$ | $\newcommand{\De}{\Delta}\newcommand{\vpi}{\varphi}$Let
\begin{equation\*}
A:=A(y):=A(0,t,y),\quad B:=B(y):=B(0,t,y),
\end{equation\*}
so that
\begin{equation\*}
f(0,0,t,y)=\frac1{\sqrt{2\pi}}\,\vpi\_{A(y)}(y), \quad g(0,0,t,y)=\frac1{\sqrt{2\pi}}\,\vpi\_{B(y)}(y),
\end{equation\*}
where
\begin{equation\*}
\vpi\_a(u... | 1 | https://mathoverflow.net/users/36721 | 420321 | 171,047 |
https://mathoverflow.net/questions/420319 | 15 | The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with repeated eigenvalue is of measure zero?
This result feels extremely natural but I do not see an immediate argument for it... | https://mathoverflow.net/users/457901 | Why is the set of Hermitian matrices with repeated eigenvalue of measure zero? | Call $S$ the set of matrices with repeated eigenvalues and fix a hermitian matrix $A\not\in S$. In the vector space of hermitian matrices, any line through $A$ intersects $S$ in at most finitely many points. From this it easily follows that $S$ is negligible (using polar coordinates centered at $A$).
To check the cla... | 10 | https://mathoverflow.net/users/36952 | 420322 | 171,048 |
https://mathoverflow.net/questions/420326 | 3 | It is known that the property of being a worldly cardinal is not absolute (a cardinal $\kappa$ is worldly iff $V\_{\kappa} \vDash \textsf{ZFC}$). See [here](http://jdh.hamkins.org/worldly-cardinals-are-not-always-downwards-absolute/) and [here](https://mathoverflow.net/questions/130019/forcing-mildly-over-a-worldly-car... | https://mathoverflow.net/users/120461 | Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly? | The truth of a first-order sentence $\varphi$ in a structure $\mathfrak{M}$ is absolute between $V$ (= reality) and sufficiently large transitive sets containing $\mathfrak{M}$. In particular, already $V\_{\kappa+2}$ correctly computes the full first-order theory of $V\_\kappa$ for each infinite ordinal $\kappa$, and e... | 7 | https://mathoverflow.net/users/8133 | 420331 | 171,053 |
https://mathoverflow.net/questions/420317 | 2 | I'm trying to show that for an ordinary manifold $X$ and a supermanifold $S$, supermanifold morphisms $\varphi:S\to TX$ are one-to-one to the pairs $(f,F) $ where $f:C^\infty(X)\to C^\infty(S)$ is a super $\mathbb{R}$-algebra homomorphism and $F:C^\infty(X)\to C^\infty(S)$ is an even derivation with respect to $f$, i.e... | https://mathoverflow.net/users/167862 | One-to-one correspondence between super morphisms $\varphi:S\to TX$ and pairs $(f:C^\infty(X)\to C^\infty(S) ,F\in Der_f(C^\infty(X),C^\infty(S))$ | The morphism $$\def\T{{\rm T}} φ:S→\T X$$ can be identified with the homomorphism of algebras $$\def\Ci{{\rm C}^∞} \Ci(\T X)→\Ci(S).$$
The algebra $\Ci(\T X)$ can be identified with the $\Ci$-symmetric algebra $$\def\CiSym{\mathop{\rm\Ci Sym}\nolimits} \CiSym\_{\Ci(X)}(\Gamma(\T^\*X))$$ of the $\Ci(X)$-module $\Gamma(\... | 2 | https://mathoverflow.net/users/402 | 420335 | 171,055 |
https://mathoverflow.net/questions/420337 | 4 | Let $E$ be a Landweber exact ring spectrum. That is, we have a map of homotopy ring spectra $MU\rightarrow E$ and an isomorphism of homology theories $E\_\*X\simeq MU\_\*X\otimes\_{MU\_\*}E\_\*$. Is the homotopy type of $E$ determined by the graded ring $E\_\*$?
My best guess is that the answer is "no" which would id... | https://mathoverflow.net/users/163893 | Are Landweber exact spectra determined by their coefficient ring? | Put $R\_\*=\mathbb{Z}\_{(p)}[x,y,y^{-1}]$ with $|x|=|y|=2$. Define $f,g\colon BP\_\*\to R\_\*$ by
\begin{align\*}
f(v\_i) &= \begin{cases}
y^{p-1} & \text{ if } i = 1 \\
0 & \text{ if } i > 1
\end{cases} \\
g(v\_i) &= \begin{cases}
x^{p-1} & \text{ if } i = 1 \\
y^{p^2-1} & \text{ if } i = 2 \\
0 & \text{ if ... | 9 | https://mathoverflow.net/users/10366 | 420342 | 171,058 |
https://mathoverflow.net/questions/420328 | 9 | In differential geometry, Lie's theorems allow us to integrate any Lie algebra representation to a Lie group representation. The algebraic version of this is more complicated (and I'm not terribly familiar with it), but over a field of characteristic zero we can still get some results from Tannaka duality. This all fai... | https://mathoverflow.net/users/158123 | Can we use formal groups to recover Lie-theoretic representation theory in characteristic p? | The short answer is that the characteristic $p$ picture genuinely has more depth. In particular it's not correct to think of formal groups as "better" than Lie groups. In fact, they can be put on equal footing with each other.
I think it's helpful to put all the objects you're interested in in the same category. In t... | 12 | https://mathoverflow.net/users/7108 | 420346 | 171,061 |
https://mathoverflow.net/questions/420315 | 3 | Let $\mathcal{D}=\mathcal{D}\_X$ be the sheaf of rings of differential operators on a smooth algebraic curve $X$.
Since $\dim X=1$, the D-modules of the form $\mathcal{D}/\mathcal{D}L$ are necessarily holonomic. Conversely, I've verified that a holonomic D-module over $\mathbb{A}^1$ or $\mathbb{G}\_m$ is always cycli... | https://mathoverflow.net/users/131975 | When are simple holonomic D-modules of the form $\mathcal{D}/\mathcal{D}L$? | If $X$ is a smooth proper curve of positive genus, then there exist simple $D$-modules which have no global sections at all. This is because every degree $0$ line bundle admits a flat connection (since the obstruction to a connection is the Atiyah class which vanishes). The line bundle with connection gives a $D$-modul... | 6 | https://mathoverflow.net/users/18060 | 420353 | 171,065 |
https://mathoverflow.net/questions/420213 | 4 | Consider the general semilinear elliptic second-order PDE
$$
u\_t-\mathcal L u=f\left(t,x,u,\nabla u\right)
$$
where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \in [0,T],$ $x$ is in a bounded smooth domain $\Omega,$ and the boundary conditions are
$$
u(T,x)=c, \quad \forall x\in\O... | https://mathoverflow.net/users/480000 | Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds | After a lot of reading, I came across the (enlightening) paper:
**On principally linear elliptic differential equations of the second order, Nagumo 1954**.
Basically, the (classical) results there shows that for any bounded domain $\Omega$ for your space variables, if you have lower and upper solutions for ... | 2 | https://mathoverflow.net/users/480000 | 420368 | 171,068 |
https://mathoverflow.net/questions/420369 | 1 | I hope this is appropriate for the site. I am reading the paper "The analytic rank of a tensor" [S. Lovett, *Discrete Analysis* (2019), #7, 10 pp.] and am a bit confused in one of the applications sections. In the paper, a *tensor* is defined to be a multilinear map from $V^d\to {\bf F}$, where ${\bf F}$ is a field and... | https://mathoverflow.net/users/168142 | The cap set tensor in Lovett (2019) | The cap-set tensor is a tensor $V^3 \to \mathbb F\_3$ where $V$ is $\mathbb F\_3^{ \mathbb F\_3^n}$, not simply $\mathbb F\_3^n$. It is defined by
$$T (e\_a,e\_b,e\_c) = \begin{cases} 1 & \textrm{if }a+b+c = 0 \\ 0 & \textrm{if } a+b+c \neq 0 \end{cases}$$
and extended to arbitrary vectors by linearity.
The indep... | 5 | https://mathoverflow.net/users/18060 | 420370 | 171,069 |
https://mathoverflow.net/questions/420361 | 12 | *This is a crosspost from [this MSE question](https://math.stackexchange.com/q/4395420/39599) from a year ago.*
---
>
> Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form?
>
>
>
If $\omega$ is a symplectic form, then the real cohomolo... | https://mathoverflow.net/users/21564 | Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form? | No, $M$ is not symplectic. Consider a double cover $\tilde{M}$ of $M$ along one of the $S^1$ components. Then it is not hard to prove that $\tilde{M}$ is diffeomorphic with $(S^1\times S^3)\#2(S^1\times S^3)\# 2 (S^2\times S^2)$. Now if $M$ were symplectic then you could pull back the symplectic structure on $\tilde{M}... | 14 | https://mathoverflow.net/users/33064 | 420371 | 171,070 |
https://mathoverflow.net/questions/420087 | 4 | I am trying to prove that the functor
\begin{align\*}
\mathrm{Top} &\longrightarrow \mathrm{Cond}(\mathrm{Set}) \\
X &\longmapsto \underline{X}
\end{align\*}
admits a left adjoint and it is the functor $T \mapsto T(\*)$ where $T(\*)$ has the quotient topology of the map $\bigsqcup\_{\underline{S} \rightarrow T} S \righ... | https://mathoverflow.net/users/130868 | Adjunction between topological spaces and condensed sets | As Wojowu noted in the comments, one should really look at $T\_1$ topological spaces. Consider the functors
\begin{align\*}
G\!: \mathbf{Top}\_{T\_1} &\leftrightarrows \mathbf{Cond}\_\kappa:\!F\\
X &\mapsto \big(\underline X \colon S \mapsto \operatorname{Cont}(S,X)\big)\\
T(\*) &\leftarrow\!\shortmid T,
\end{align\*}
... | 2 | https://mathoverflow.net/users/82179 | 420373 | 171,072 |
https://mathoverflow.net/questions/420378 | 2 | Let $ X $ and $ Y $ be integral curves over some perfect field $ k $ and suppose that $ X $ is smooth. Moreover, let $ \pi\_1 : Y \to X $ and $ \pi\_2 : Y \to X $ be finite morphisms such that the function field $ k( Y ) $ of $ Y $ is the composite field $ \pi\_1^\*( k( X ) ) \cdot \pi\_2^\*( k( X ) ) $ where $ \pi\_i^... | https://mathoverflow.net/users/132492 | Ramification and singular points | Sure. Take $X = \mathbb P^1$, $Y$ a nodal curve in $\mathbb P^1 \times \mathbb P^1$ given by some equation like $x^2 - y^2 + x y^2$. The point $(0,0)$ is singular but splits into two places in the function field.
Each of these two places is unramified since it is a degree $2$ extension in of both $k(x)$ and $k(y)$, s... | 4 | https://mathoverflow.net/users/18060 | 420379 | 171,073 |
https://mathoverflow.net/questions/420382 | 2 | Let $s>1$ be a real number. We look at the zeta probability function / Zipf probability function defined as:
$$P(X = n) = \frac{1}{n^s \zeta(s)}$$
Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is a function such that the following limit exists:
$$\lim\_{s \rightarrow 1} E(f(X\_s)) = \lim\_{s \rightarrow 1} \lim\... | https://mathoverflow.net/users/165920 | Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$? | The answer is no. E.g., if
$$f(k)=\sum\_{j=0}^\infty 1(2^{2j}<k\le2^{2j+1})$$
for natural $k$, then
\begin{equation}
Ef(X\_s)\to\frac13 \tag{1}\label{1}
\end{equation}
as $s\downarrow0$, whereas
$$\frac1n\,\sum\_{k=1}^n f(k)$$
will be forever oscillating between $\frac13$ and $\frac23$ as $n\to\infty$.
Indeed, let $... | 3 | https://mathoverflow.net/users/36721 | 420388 | 171,075 |
https://mathoverflow.net/questions/420297 | 15 | Although I am by no means an applied mathematician, I like to occasionally explain applications of the math I teach to real world problems. Right now I am teaching some students about [longest increasing subsequences](https://en.wikipedia.org/wiki/Longest_increasing_subsequence) of permutations and their connection to ... | https://mathoverflow.net/users/25028 | Longest increasing subsequence as measure of randomness | The Longest Increasing Subsequence has been used as a test statistic for non-parametric tests by García and González-López in
Independence tests for continuous random variables based on the longest increasing subsequence, Journal of Multivariate Analysis, 2014 ([link](https://www.sciencedirect.com/science/article/pii... | 6 | https://mathoverflow.net/users/127599 | 420389 | 171,076 |
https://mathoverflow.net/questions/420304 | 2 | This is a noncommutative version of these three previous questions:
[differential operator power coefficients](https://mathoverflow.net/questions/80828/differential-operator-power-coefficients)
[Сlosed formula for $(g\partial)^n$](https://mathoverflow.net/questions/337330/%D0%A1losed-formula-for-g-partialn)
[A Le... | https://mathoverflow.net/users/1849 | The combinatorics of $(f \partial)^n$ in the noncommutative setting? | In fact, Comtet's formula works almost directly in noncommutative case under an appropriate ordering of the products. Here is just a bit deeper look under the hood.
In umbral form $(R\_f \partial)^n f$ can be written as polynomial
$$f\_n(x\_1,\dots,x\_n) := x\_1(x\_1+x\_2)(x\_1+x\_2+x\_3)\cdots(x\_1+\dots+x\_n),$$
wh... | 1 | https://mathoverflow.net/users/7076 | 420390 | 171,077 |
https://mathoverflow.net/questions/420399 | 0 | I have the following setting:
Let $0 \leq r < 1$ and let $\{z\_i\}\_{i=1}^k$ be $k$ complex numbers such that $|z\_i| \leq r$ for all $i$.
Moreover, $r + \sum\_{i=1}^k 2Re(z\_i) \geq 0$
I am interested to know if the following is true:
$(1-r)\prod\_{i=1}^k |1-z\_i|^2 \leq 1$.
The above stems from trying to boun... | https://mathoverflow.net/users/421678 | Proving maximum value of a determinant of $I - B$, where $B$ is nonnegative matrix | This is not true in general. E.g., take any $r\in(0,1)$ and let
$$z\_i:=x+iy,\quad x:=-\frac r{2k},\quad
y:=r\sqrt{1-\frac{1}{{4k^2}}}$$
for all $i$.
Then all the conditions on $r$ and $z\_i$'s hold, whereas
$$(1-r)\prod\_{i=1}^k|1-z\_i|^2=(1-r)(1+r^2+r/k)^k\to\infty \not\le 1$$
as $k\to\infty$.
Alternatively, let
$... | 2 | https://mathoverflow.net/users/36721 | 420400 | 171,080 |
https://mathoverflow.net/questions/420359 | 1 | For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ be the Discrete Fourier matrix of order $N$, that is $F=\frac{1}{\sqrt{N}}(w^{kl})\_{0\leq k,l\leq N-1}$.
>
> Q. Doe... | https://mathoverflow.net/users/84390 | A particular commutator of the discrete Fourier matrix | If such a pair $(A,\alpha\ne1)$ existed, then the spectrum of $DF$ would be invariant under the operation $\lambda\mapsto\alpha\lambda$, in particular $\alpha$ would be a root of unity.
The answer is therefore **negative** as soon as $N=2$, for then
$$D=\begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix},\qquad
F=\begin{pma... | 3 | https://mathoverflow.net/users/8799 | 420408 | 171,082 |
https://mathoverflow.net/questions/416173 | 5 | I am reading a paper [Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction](http://dx.doi.org/10.1198/jasa.2009.0106) where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.
More specifically, let $B \in \mathbb{R}^{p\times d}$ be a ... | https://mathoverflow.net/users/477109 | Log determinant of quadratic form | Since $\Sigma \succ \Delta$, by operator monotonicity we have $\Sigma^{-1} \prec \Delta^{-1}$ and thus $B^{\top}\Sigma^{-1}B \prec B^{\top}\Delta^{-1}B$. Since log on positive real is increasing, [the trace monotonicity](https://en.wikipedia.org/wiki/Trace_inequality#Convexity_and_monotonicity_of_the_trace_function) ([... | 2 | https://mathoverflow.net/users/18526 | 420410 | 171,083 |
https://mathoverflow.net/questions/419858 | 4 | It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
[Orbits space of real-analytic planar foliations](https://mathoverflow.net/questions/382871/orbits-space-of-real-analytic-planar-foliations)
One can imagine several reason fo... | https://mathoverflow.net/users/36688 | Can a holomorphic vector field have an attractor homoclinic loop? | The answer is 'no' for much the same reason that the OP indicates: the existence of a homoclinic or heteroclinic connection implies that neighboring trajectories are periodic.
First, one needs to have poles in $f$ if one wishes to have heteroclinic connections between (real) saddle singularities. Indeed if $f$ is hol... | 2 | https://mathoverflow.net/users/24309 | 420414 | 171,084 |
https://mathoverflow.net/questions/420423 | 2 | Suppose that $A\subseteq \mathbb{N}$ and suppose that you have an estimate of the form
$$
\sum\_{\substack{a\le x \\ a\in A}}f(a) \sim g(x).
$$
With this information is it possible to get an asymptotic estimate for $\sum\_{\substack{ab\le x \\ a,b\in A}}f(a)f(b)$? I would appreciate any reference in this kind of proble... | https://mathoverflow.net/users/480636 | Asymptotic estimate for $\sum_{\substack{ab\le x \\ a,b\in A}}f(a)f(b)$ | I address your first (general) question, but I am sure it is applicable to your second (specialized) question. (Please restrict to one question per post to avoid confusion and frustration.)
You can try to apply Dirichlet's hyperbola method. First, you can forget about $A$, since you can always re-define $f$ as the re... | 4 | https://mathoverflow.net/users/11919 | 420431 | 171,086 |
https://mathoverflow.net/questions/420415 | 4 | Let $ \Omega\_1 $ and $ \Omega\_2 $ be domains (open and connected) in $ \mathbb{R}^2 $. $ \psi:\Omega\_1\to\mathbb{R} $ and $ \phi:\Omega\_1\to\mathbb{R} $ are $ C^1 $ functions with two variables. Moreover, we assume that map $ (x,y)\to (\phi(x,y),\psi(x,y)) $ is homeomorphism from $ \Omega\_1 $ to $ \Omega\_2 $, i.e... | https://mathoverflow.net/users/241460 | Does the homeomorphism have a non-negative or non-positive determinant? | Let $f=(\phi,\psi):\Omega\_1\to\Omega\_2$.
For every point $p\in\Omega\_1$ consider the curve $\gamma:t\mapsto p+\varepsilon e^{it}$, for $\varepsilon$ so small that the curve is contained in $\Omega\_1$. Let $n(p)$ be the winding number of $f\circ \gamma$ around $f(p)$. As $f$ is a homeomorphism, $n(p)$ is well defi... | 7 | https://mathoverflow.net/users/172802 | 420432 | 171,087 |
https://mathoverflow.net/questions/420434 | 3 | Let $T: {\bf R}^n \rightarrow {\bf R}^n$ be an homeomorphism and $x$ a point in ${\bf R}^n$.
The positive orbit of $x$ is the set $\{T^n(x) \mid n \in {\bf N}\}$ and its
$\omega$-limit set is the set of accumulation points of its orbit.
$$\omega(x) = \{y \mid \exists \, n\_k \rightarrow +\infty \hbox{ such that } T^{n\... | https://mathoverflow.net/users/6129 | Boundedness of orbits and limit sets | We can modify the connectedness argument to show it also holds for homeomorphisms: suppose $\omega(x)$ is contained in an open ball $B(0,R)$, with $R>0$, and let $N$ be so big that $T(B(0,R))\subseteq B(0,N)$.
Then if the positive orbit of $x$ is not bounded, it must contain infinite points inside $B(0,N)\setminus B(... | 6 | https://mathoverflow.net/users/172802 | 420436 | 171,088 |
https://mathoverflow.net/questions/420422 | 1 | Consider the semilinear wave equation in $[0,t\_0] \times \mathbb R^d$ :
$$\square u = \pm |u|^{p-1}u$$
With subcritical/critical power $1 < p \leq \frac{d+2}{d-2}$.
It is easy to show by energy method that any two $C^2$ classical solutions $u(t,x)$ & $v(t,x)$ with same initial conditions on $\{0\} \times B(x\_0,... | https://mathoverflow.net/users/130116 | Finite propagation speed for non-smooth solutions to nonlinear wave equation | You can approach this dually using that for classical solutions the finite propagation speed holds. This argument is similar in spirit to [this answer of mine for low regularity uniqueness for the linear wave equation](https://mathoverflow.net/a/361009/3948).
1
-
First write $w = u-v$. Then you see that $w$ solves ... | 4 | https://mathoverflow.net/users/3948 | 420446 | 171,091 |
https://mathoverflow.net/questions/420441 | 6 | It is well-known that $e/\sqrt{2}$ is irrational.
Indeed, if it was rational, i.e. $p/q$ then $e^2/2 =p^2/q^2.$ Thus, $q^2e^2=2p^2,$ which would imply that $e$ is a root of $q^2x^2=2p^2.$
Now my question is: Does there exist a finite number of complex numbers $a\_1,...,a\_N$ all different from zero such that for ev... | https://mathoverflow.net/users/457901 | Finite set of numbers whose powers sum up to irrational number | Such numbers do not exist. Indeed, let $p\_n$ (resp. $e\_n$) be the $n$-th power sum (resp. elementary symmetric polynomial) of the $a\_i$'s. Since $e\_n=0$ for $n>N$, [Newton's identities](https://en.wikipedia.org/wiki/Newton%27s_identities#Expressing_elementary_symmetric_polynomials_in_terms_of_power_sums) show that
... | 8 | https://mathoverflow.net/users/11919 | 420448 | 171,093 |
https://mathoverflow.net/questions/420449 | 0 | For simplicity, let us set $V\_n:=\{-1+1/2^n, \dotsc, 0 ,\dotsc, 1-1/2^n, 1\}$ with the periodic boundary conditions for each $n \in \mathbb{N}$ and think of the following vector space over this lattice:
\begin{equation}
M(V\_n):=\{f:V\_n \to \mathbb{C} \mid f \text{ is the restriction of some Schwartz function on }
... | https://mathoverflow.net/users/56524 | In what sense does the finite difference operator converge to the continuum differential operator in the hydrodynamic limit? | "In what sense does this $\partial\_n$ converge to the ordinary continuum differential operator?"
It converges in the sense that there exist injective linear maps $\iota\_n:M(V\_n)\to C^\infty\_\text{periodic}([-1,1])$ that take the $2^{n+1}$ eigenspaces of $\partial\_n$ to the first $2^{n+1}$ eigenspaces of $d/dx$, ... | 3 | https://mathoverflow.net/users/5690 | 420451 | 171,094 |
https://mathoverflow.net/questions/420450 | 1 | If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph), and $\kappa \neq \emptyset$ is a cardinal, a map $c: V\to \kappa$ is said to be a *coloring* if the restriction $c|\_e: e\to \kappa$ is non-constant whenever $e\in E$ and $e$ has at least $2$ elements. The least non-empty cardinal such that there... | https://mathoverflow.net/users/8628 | Edge sets on $\omega$ maximal with respect to chromatic number | $\mathcal E\_n$ has maximal elements for every $n\ge1$.
Let $V\_1,\dots,V\_n$ be pairwise disjoint infinite sets such that $V\_1\cup\cdots\cup V\_n=\omega$ and let $E=\{e\in[\omega]^\omega:e\not\subseteq V\_i\text{ for all }i\in[n]\}$. Plainly $\chi(\omega,E)\le n$.
Let $c$ be any proper $n$-coloring of $(\omega,E)... | 3 | https://mathoverflow.net/users/43266 | 420456 | 171,098 |
https://mathoverflow.net/questions/420457 | 1 | I am interested in finding references that develop high probability suboptimality bounds for stochastic gradient descent (SGD) for **general** convex functions in the case where we return the average of the last $\alpha T$ iterates (suffix-averaging) where $\alpha\in(0,1)$ and $T$ denotes the total number of iterations... | https://mathoverflow.net/users/174439 | High probability bounds of SGD for general convex functions with suffix averaging | I didn't look very carefully. I found [this](https://www.cs.ubc.ca/%7Enickhar/papers/GradientDescent/GradientDescent.pdf) paper by Harvey et al (2018) demonstrating the result for general convex, 1-Lipschitz $f(\cdot)$.
| 1 | https://mathoverflow.net/users/174439 | 420458 | 171,099 |
https://mathoverflow.net/questions/420475 | 5 | I think the following statement is not true in the general situations, but consider it:
$R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is isomorphic to $(\dfrac{R}{\mathfrak{p}R})^{\*}\times\dfrac{R}{\mathfrak{p}^{n-1}R}$.
This statement holds if $R=\mathbb{Z}$, ... | https://mathoverflow.net/users/166540 | About the structure of unit groups appearing in number theory | In the case of the ring of integers of a number field, this problem is
studied and solved in complete detail in Section 4.2 of my book "Advanced topics in Computational number theory", Springer GTM 193. In particular,
your statement holds if $e<p-1$, where $p$ is the prime number below
the prime ideal and $e$ the ramif... | 7 | https://mathoverflow.net/users/81776 | 420477 | 171,105 |
https://mathoverflow.net/questions/420473 | 15 | In my answer to [this question](https://mathoverflow.net/q/420359), there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity
$$\sum\_{k=0}^{N-1}z^{2k^2+k}$$
vanish ?
The answer is clearly **No** for $N\le7$. I suspect that it never vani... | https://mathoverflow.net/users/8799 | Vanishing of a sum of roots of unity | For general $N$, we can reason by induction on the $2$-adic valuation of $N$. If $N$ is odd, GH from MO's answer shows that $S\_N :=\sum\_{k=0}^{N-1} \zeta\_{2N}^{2k^2+k} \neq 0$, where $\zeta\_{2N} = z = e^{\pi i/N}$ is a primitive $2N$-th root of unity. The same argument shows that for any odd $N$ and any $b \neq 1$,... | 15 | https://mathoverflow.net/users/6506 | 420496 | 171,113 |
https://mathoverflow.net/questions/418610 | 3 | Hartshorne defines (p. 257 of III.9) an associated point of a scheme $X$ as a point such that $\mathfrak{m}\_x$ is an associated prime of the local ring which he says is equivalent to the maximal ideal consisting of only zero divisors.
I am wondering if these are actually equivalent in the non-noetherian setting? I a... | https://mathoverflow.net/users/154157 | Is Hartshorne's definition of associated points correct in the non-noetherian setting? | They aren't equivalent, unless Hartshorne's usage of "associated prime" is different from e.g. the definition in the Stacks project [[Definition 00LA](https://stacks.math.columbia.edu/tag/00LA)].
Indeed, [[Example 05AI](https://stacks.math.columbia.edu/tag/05AI)] gives the non-Noetherian local ring $A = k[x\_1,x\_2,\... | 4 | https://mathoverflow.net/users/217216 | 420507 | 171,115 |
https://mathoverflow.net/questions/420511 | 0 | Let $f(x)$ be an integer-valued polynomial (when $x\in \mathbb{Z}$, then $f(x)\in \mathbb{Z}$), and $a,b$ be positive integers, and $p$ be a prime number with $(a,p)=1$.
Show that
$$\sum\_{x=0}^{p-1}\left(\dfrac{f(ax+b)}{p}\right)=\sum\_{x=0}^{p-1}\left(\dfrac{f(x)}{p}\right)$$
If $f(x)=x$, it is well known
$$\sum\... | https://mathoverflow.net/users/38620 | Sums of Legendre symbols with integer-valued polynomials | The statment is false. For a counterexample, take $f(x)=\binom{x}{3}$, $p=3$, $a=2$, $b=0$.
The statement is true when $f$ has degree less than $p$, or when $f$ has integral coefficients.
| 8 | https://mathoverflow.net/users/11919 | 420512 | 171,117 |
https://mathoverflow.net/questions/420471 | 3 | EDITED:
A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves.
Is there a similar thing on finite cell complexes? That is, are there “related” notions of equivalence and “similar” theorems “reducing” such equivalence to finite sequences “combin... | https://mathoverflow.net/users/103418 | “Combinatorial” moves between cell complexes | 0. If a pair of finite simplicial complexes are PL manifolds, which are additionally PL homeomorphic, then there is a finite sequence of [bisteller flips](https://en.wikipedia.org/wiki/Pachner_moves) taking one to the other. (These are sometimes also called Pachner moves.)
1. [Kirby calculus](https://en.wikipedia.org/w... | 4 | https://mathoverflow.net/users/1650 | 420515 | 171,118 |
https://mathoverflow.net/questions/415627 | 3 | I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling also didn't help. Since there is some ambiguity in terminology let me state the definition.
Let $\Gamma$ be a directed s... | https://mathoverflow.net/users/53155 | Reference on inductive (direct) limit of ordered vector spaces and vector lattices | There is a paper by Wolfgang Filter form 1980s.
W. Filter, Inductive limits of Riesz spaces, Proceedings of the International Conference held in Dubrovnik, June 23-27, 1987 (Bogoljub Stankovic, Endre Pap, Stevan Pilipovic, and Vasilij S. Vladimirov, eds.), Plenum Press, New York, 1988, pp. 383-392.
Unfortunately I ... | 3 | https://mathoverflow.net/users/480716 | 420517 | 171,120 |
https://mathoverflow.net/questions/420530 | 1 | We know that in characteristic $0$, all algebraic series are differentiably finite.
Is this true in positive characteristic? I look at the [proof](https://www.e-periodica.ch/cntmng?pid=ens-001:1964:10::61), indeed we need to the
characteristic to be $0$ for the proof to work.
If it is not true in positive characteristi... | https://mathoverflow.net/users/122378 | Are algebraic power series in positive characteristics D-finite? | Over a field of characteristic $p$, the $p$-th derivative of any power series is $0$, and so every power series over a field of finite characteristic is $D$-finite.
| 6 | https://mathoverflow.net/users/88679 | 420540 | 171,127 |
https://mathoverflow.net/questions/419320 | 2 | Let $M$ be a large positive integer, $d$ an odd positive integer and $f: \mathbb{Z}\_{>0} \times \mathbb{Z}\_{>0} \to \mathbb{R}$. For a non-principal character $\chi\_d = \chi$ with modulus $d$, I am interested in the following type of sums.
$$S(\chi, f) = \sum\_{\substack{m = M \\ (m, d) = 1}}^{2M}f(m, d) \chi(m). $$... | https://mathoverflow.net/users/167999 | Pólya–Vinogradov like inequality for a character sum with Euler factors | Based on the ideas posted by @OfirGorodetsky and @tomos as comments to this post, I managed to compile this solution.
$$\begin{align\*}
S &= \sum\_{m = M}^{2M} \frac{\phi(m)}{m} \prod\_{p \nmid md}\left(1 - \frac{1}{p^2} - \frac{\chi\_p(m)}{p^2} \right) \; \chi\_d(m) \\
&= \sum\_{m = M}^{2M} \frac{\phi(m)}{m} \left(... | 0 | https://mathoverflow.net/users/167999 | 420546 | 171,129 |
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