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https://mathoverflow.net/questions/331765 | 2 | I need a source about the history of algebraic graph theory. I mean for solving which problems or responding to what needs it was created?
Indeed, I want to write a note about the history of the birth of it. Can anybody help me?
Bests.
| https://mathoverflow.net/users/125843 | History of algebraic graph theory | A few comments about the history of algebraic graph theory:
(1) Some early work was due to [William Tutte](https://en.wikipedia.org/wiki/W._T._Tutte) related to the four-color conjecture. The Tutte polynomial (and thus the chromatic polynomial) come out of his work.
(2) A very brief comment about the history appe... | 3 | https://mathoverflow.net/users/11124 | 331798 | 142,038 |
https://mathoverflow.net/questions/331769 | 9 | We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$.
Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^\*:
\mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$ is a coproduct (coalgebraic operation)?
Is there a bialgebra... | https://mathoverflow.net/users/36688 | An inner product approach to Hopf algebras | This doesn't directly answer your question concerning Hopf structures on $\mathbb{C}^n$, but a particularly well-studied class of Hopf algebras for which the product is the adjoint of the coproduct are known as *positive self-adjoint Hopf (PSH) algebras*. These were originally introduced by Zelevinsky to study [represe... | 11 | https://mathoverflow.net/users/76355 | 331801 | 142,041 |
https://mathoverflow.net/questions/331730 | 5 | I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14)
<https://arxiv.org/pdf/1402.4203.pdf>
For completeness, the statement is as follows.
**Theorem:** Let $M$ be a closed manifold and $\rho: \pi\_1(M)\to SL\_n(\mathbb{C})$ be a semisimple representation. Then... | https://mathoverflow.net/users/135446 | Long time existence for heat flow in Corlette-Donaldson Theorem | I won't answer your question "why can he use these results" directly, but let me indicate the structure of the proof and hopefully this will alleviate some of your confusion.
For the long time existence argument, I don't think you need look at Eells-Sampson or Hamilton, or any other paper. The limiting convergence to... | 2 | https://mathoverflow.net/users/49247 | 331804 | 142,043 |
https://mathoverflow.net/questions/331808 | 1 | Let
* $(\Omega,\mathcal A,\operatorname P)$ be a probability space
* $(E,\mathcal E)$ be a measurable space
* $n\in\mathbb N$
* $X\_1,\ldots,X\_n$ be $(E,\mathcal E)$-valued random variables on $(\Omega,\mathcal A,\operatorname P)$
>
> I'm interested in the following question: Given a total$^1$ order $\le$ on $E$... | https://mathoverflow.net/users/91890 | Can we order random variables in a measurable way in a general setup? | $\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\eq}{\,\overset\om\sim\,}
\newcommand{\eqq}{\overset{\,\colon\om}\sim\,}
\renewcommand{\eq}{\,\sim\_\om\,}
\newcommand{\K}{\mathcal K}
$
If $E$ is a [distributive lattice](https://en.wikipedia.org/wiki/Distributive_lattice) with measurable binary operation... | 4 | https://mathoverflow.net/users/36721 | 331811 | 142,046 |
https://mathoverflow.net/questions/331707 | 2 | I have a question about singularities of conformal mappings.
Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to D$ denote a conformal map.
I am concerned with the following quantity:
\begin{align\*}
I(z,r)=\int\_{\mathbb{H} \cap... | https://mathoverflow.net/users/68463 | Conformal mappings and its singularity | **I did not know at all... I am very sorry but can you tell me any references**
Ah-oh. It is hard to find a decent reference now because nobody cares about writing down such trivialities any more: they are talking about domains in $\mathbb R^n$ and quasiconformal mappings, which is a huge overkill for you, so I'll si... | 3 | https://mathoverflow.net/users/1131 | 331827 | 142,050 |
https://mathoverflow.net/questions/331829 | 4 | I have a random variable $x \in [a, b]$ with PDF $f(x)$ and an event $E$ which satisfies the following property for any $x'<b$.
$$\Pr[E\mid x > x'] \geq \Pr[E]$$
My question is whether or not the following inequality holds.
$$\int\_a^b uf(u)\Pr[E\mid x=u] \, du \geq \Pr[E]\int\_a^b uf(u) \, du$$
| https://mathoverflow.net/users/130779 | Expected value of a random variable conditioned on a positively correlated event | The answer is yes.
Indeed, let us write $X$ instead of $x$, according to standard notation, to distinguish between random variables (denoted by upper-case letters) and their values (denoted by lower-case letters). Let us write $P$ instead of $\Pr$, and then let us also write $A$ instead of $E$, to distinguish it from ... | 3 | https://mathoverflow.net/users/36721 | 331833 | 142,051 |
https://mathoverflow.net/questions/331091 | 8 | We all know that $\sum\_{k=0}^n\binom{n}k$ is not divisible by $3$.
>
> **QUESTION.** Is it true that the numerator of $a\_n$ (in reduced form) is never divisible by $3$?
> $$a\_n=\sum\_{k=0}^n\frac1{\binom{n}k}.$$
>
>
>
**POSTSCRIPT.** This the most detailed and pedagogical answer to any question that I aske... | https://mathoverflow.net/users/66131 | Reciprocal sum of binomials and divisibility by $3$ | Here is the answer I hinted at in a comment, in real detail. Took me a while,
but I had no idea how tiresome such arguments are to expose...
Yes, it is true: see Corollary 6 **(b)** below. The proof relies on the
concept of a $p$-adic valuation of a rational number. It is defined as follows:
>
> **Definition.** L... | 12 | https://mathoverflow.net/users/2530 | 331850 | 142,058 |
https://mathoverflow.net/questions/331826 | 3 | I am trying to understand some of the recent research in number theory. There is apparently a certain lemma, called Ihara's lemma, which can be established in some contexts and is unknown in other contexts. Occasionally, one can still prove its consequences unconditionally. This acrobatics is called Ihara avoidance. Wh... | https://mathoverflow.net/users/nan | Papers containing Ihara avoidance arguments | For a gentle(-ish) introduction to the "Ihara avoidance" method, you might want to consult the notes of Toby Gee's course on modularity lifting from the 2013 Arizona Winter School, [www2.imperial.ac.uk/~tsg/Index\_files/ArizonaWinterSchool2013.pdf](http://www2.imperial.ac.uk/~tsg/Index_files/ArizonaWinterSchool2013.pdf... | 1 | https://mathoverflow.net/users/2481 | 331859 | 142,060 |
https://mathoverflow.net/questions/331268 | 2 | I need a reference to the following (known?)
>
> **Fact.** Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of linear metric spaces.
>
>
>
This seems to be a basic fact in the theory of topological vector spaces, but I d... | https://mathoverflow.net/users/61536 | Every linear topological space embeds into the Tychonoff product of linear metric spaces | In Jarchow's *Locally Convex Spaces* Theorems [2.7.3](https://books.google.com/books?id=5tYACAAAQBAJ&pg=PA39) and [2.9.2](https://books.google.com/books?id=5tYACAAAQBAJ&pg=PA43) on pages 39 and 43 together say that a *Hausdorff* topological vector space is linearly homeomorphic to a dense subspace of a projective limit... | 3 | https://mathoverflow.net/users/12643 | 331868 | 142,061 |
https://mathoverflow.net/questions/331874 | 3 | Why the space of all complex structure on a $2n$-dimensional vector space which compatible with a positive definite metric is diffeomorphic to $ O(2n)/U(n) $ ?
| https://mathoverflow.net/users/140797 | The space of complex structure compatible with metric | $O(2n)$ acts transitively on the space $C$ of compatible complex structures, by sending a complex structure $J$ to $OJO^{-1}$, it's easy to check this is a compatible complex structure. To show transitivity use block-diagonalisation for real matrices to show it any complex structure can be conjugated to a standard one ... | 3 | https://mathoverflow.net/users/105734 | 331883 | 142,067 |
https://mathoverflow.net/questions/331897 | 14 | I know that, from compactness theorem, one can prove that there are models of first order arithmetic in which there is some "number" which is not a successor of zero, in the sense that it is strictly bigger than any successor of zero (i.e. any element of the model obtained by applying the successor function to zero fin... | https://mathoverflow.net/users/139854 | Existence of a model of ZFC in which the natural numbers are really the natural numbers | This, in fact, cannot be proven, even in $ZFC+Con(ZFC)$. This is because $ZFC$ proves the following statement:
>
> If we have a model $M$ of $ZFC$ whose natural numbers are standard, then $M$ satisfies $ZFC+Con(ZFC)$.
>
>
>
Indeed, $Con(ZFC)$ is an arithmetic statement. Since we are assuming that $ZFC$ has a m... | 30 | https://mathoverflow.net/users/30186 | 331898 | 142,074 |
https://mathoverflow.net/questions/331889 | 1 |
>
> **Edit:** According to comment of Andre Henriques we revise the question:
> In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.
>
> ... | https://mathoverflow.net/users/36688 | Does fractallity depend on the Riemannian metric? | The answer is no. Any two Riemannian metrics when restricted to a compact set are bi-Lipschitz equivalent and bi-LIpschitz homeomorphism preserves the Hausdorff dimension.
| 5 | https://mathoverflow.net/users/121665 | 331899 | 142,075 |
https://mathoverflow.net/questions/331691 | 3 | In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}\_{2}\*\mathbb{Z}\_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\theta\in\left(-\frac{\pi}{2}\text{, }\frac{\pi}{2}\right)$ the bounded operator \begin{eqnarray} X\_{\theta}:=-8\tan... | https://mathoverflow.net/users/64444 | Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different | The claim is true.
Any difference in norm must be picked up on the span of $(T\_s+T\_t)^ne\_0.$ So we will apply perturbation theory on that subspace. The value of $\langle(T\_s+T\_t)^ne\_0,e\_0\rangle$ should be ${n}\choose{n/2}$ when $n$ is even and zero otherwise. Note that $A=T\_s + T\_t$ is self-adjoint. Moreove... | 3 | https://mathoverflow.net/users/32470 | 331903 | 142,076 |
https://mathoverflow.net/questions/331767 | 9 | In the book *The Geometry of Domains in Spaces* by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M\_\mu f$, where $\mu$ is a Radon measure, using their version of the [Besicovitch covering theorem](https://en.wikipedia.org/wiki/Besicovitch_covering_theorem).
>
> Let $d... | https://mathoverflow.net/users/80191 | A Besicovitch-type Covering Theorem | This trivial counterexample in $\mathbb R^2$ should have taken me five minutes. Instead, I spent almost two days. The moral is the usual one: after 50 you'd better give up on mathematics.
Let $y,z$ be 2 points at distance $1$ from each other. We shall construct by induction a sequence of points $x\_j$ and radii $r\_j... | 11 | https://mathoverflow.net/users/1131 | 331906 | 142,077 |
https://mathoverflow.net/questions/331918 | 1 | Berthelot's comparison theorem connects the algebraic de Rham cohomology of a $\mathbb{Z}\_p$-scheme and the crystalline cohomology of its special fiber. Is there a statement on the level of homotopy types which implies Berthelot comparison?
One would have to define crystalline and de Rham homotopy types first, and ... | https://mathoverflow.net/users/nan | Non-abelian Berthelot comparison? | Yes. A Google search will immediately give you lots of articles in varying generality (see e.g. work of Shiho), but one article I'm particularly fond of is Kim and Hain's *"A de Rham-Witt approach to crystalline rational homotopy theory"* in Compositio 2001. See Theorem 2 for the comparison of crystalline and de Rham h... | 3 | https://mathoverflow.net/users/76409 | 331922 | 142,079 |
https://mathoverflow.net/questions/331924 | 8 | I am desperately trying to understand what is a *conic bundle*. It seems like this is a completely standard term in algebraic geometry, there is even a page on wiki about it, but this doesn't really help. For example, I have the following test question.
**Test question.** Let $S$ be a smooth complex ruled surface $S\... | https://mathoverflow.net/users/13441 | What is a conic bundle and why is it called so? | I think the reasonable definition is to ask for a flat morphism $f$ whose generic fiber is a rational curve. Then you may put more conditions according to your needs (for instance, there is a more strict notion of *standard* conic bundle). But the answer to Question 2 is, I think, quite simple. A rational curve $C$ ove... | 8 | https://mathoverflow.net/users/40297 | 331925 | 142,080 |
https://mathoverflow.net/questions/331790 | 7 | Real-rootedness, log-concavity, and unimodality are intertwined properties. It's in this light that I was prompted to ask the question below.
Suppose the roots of a polynomial $p(x)$ are all real and $p(0)>0$. Fix an integer $k\geq0$ and consider the function $f=\frac1{p^{2k+1}}$. I like to consider the partial sums ... | https://mathoverflow.net/users/66131 | Taylor's polynomials and loss of real roots | It immediately follows from the observations that every even order Taylor polynomial of $e^{ax}$ is strictly positive for any $a\in\mathbb R$ and that $\frac 1{b-x}=\int\_0^\infty e^{ax}e^{-ab}\,da$ and $\frac 1{b+x}=\int\_0^\infty e^{-ax}e^{-ab}\,da$ for $b>0$ and $|x|<b$. The first observation is well-known and almos... | 7 | https://mathoverflow.net/users/1131 | 331932 | 142,082 |
https://mathoverflow.net/questions/331562 | 3 | In the book by [Bensoussan and Lions](https://www.amazon.co.uk/Impulse-control-quasi-variational-inequalities-Bensoussan/dp/2040155775), they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that the... | https://mathoverflow.net/users/91196 | Embedding of weighted sobolev space with exponential weights | Expanding on my comment, let $u\_i\in W^{2,p}\_\mu(\mathbb{R}^d)$ be a bounded sequence, that is, $$
|u\_i|\_{W^{2,p}\_\mu(\mathbb{R}^d)}\leq C
$$
and consider the balls $B\_k(0)$, $k\in\mathbb{N}$. Let $W^{2,p}(B\_k(0))$ be the usual Sobolev space. It is easy to see that
$$
|u\_i|\_{W^{2,p}({B\_k(0)})}\leq C\exp(\mu\... | 3 | https://mathoverflow.net/users/127306 | 331945 | 142,086 |
https://mathoverflow.net/questions/331910 | 1 | When one looks at the way cannon balls and oranges are normally packed by the military and by groceries, it seems intuively clear that there is no way anybody can pack these any tighter. However, it wasn't until recently that Hales proved, using a computer, that this intuition is correct.
Before Hales' proof, it was ... | https://mathoverflow.net/users/7089 | Sphere packing and kissing numbers in 3D | Fedja is right that the gap in the argument is why the kissing number should increase if you increase the density. The 12-sphere kissing configuration is not unique, and one can imagine rearranging the spheres so as to keep the same kissing number but increase the density. Worse yet, one might decrease the kissing numb... | 7 | https://mathoverflow.net/users/4720 | 331946 | 142,087 |
https://mathoverflow.net/questions/331940 | 0 | Suppose $a$ and $b$ are reals such that $a^b=b^a$. If $a$ is algebraic, is $b$ algebraic too?
| https://mathoverflow.net/users/13625 | $a^b=b^a$ and algebraicity | The answer is no. For instance, let $a=3$ and $b\neq 3$ be the real number satisfying $3^b=b^3$. Clearly $b$ is not an integer. It follows that $b$ is irrational -- indeed, if $b$ was a non-integer rational, $3^b$ would be irrational, while $b^3$ would be rational. Finally, $b$ is transcendental, since otherwise $b$ wo... | 8 | https://mathoverflow.net/users/30186 | 331952 | 142,089 |
https://mathoverflow.net/questions/331950 | 13 | I'm a physicist interested in the conformal bootstrap, one version of which was recently [shown to have many similarities](https://arxiv.org/abs/1905.01319) to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been solved in $d=8$ and $24$, and recently those solutions were shown to be [universally op... | https://mathoverflow.net/users/140839 | Illustrating that universal optimality is stronger than sphere packing | In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation ... | 19 | https://mathoverflow.net/users/4720 | 331953 | 142,090 |
https://mathoverflow.net/questions/331921 | 3 | The definition of Markov operator which I am familiar with:
For a graph $G=(V,E)$, Markov's operator upon a function
$\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) is defined in the following manner: $M\varphi(x) = \frac{1}{\deg(x)}\sum\_{y\in N(x)} \varphi(y)$ , when $N(x)$ denotes ... | https://mathoverflow.net/users/140823 | Spectral radius of Markov averaging operator on graphs | The Markov averaging operator $M$ is also known as the transition matrix for simple random walk on the graph (often denoted by $P$). With this terminology, its spectral radius and spectral gap are studied extensively; convenient starting points are the lecture notes [1] and Chapters 12-13 in the book [2] and the refere... | 5 | https://mathoverflow.net/users/7691 | 331954 | 142,091 |
https://mathoverflow.net/questions/331634 | 3 | Suppose $A$ is a non-unital residually finite-dimensional (RFD) $C^\*$-algebra, then the [multiplier algebra](https://en.wikipedia.org/wiki/Multiplier_algebra) $M(A)$ is also RFD. I wonder whether there exists a trace on the corona algebra $M(A)/A$?
| https://mathoverflow.net/users/63864 | Residually finite-dimensional $C^*$-algebra | By "trace" I assume you mean tracial state, and in that case the answer is "not necessarily". A counter example is produced below.
The goal is this: we construct an unital RFD $C^\ast$-algebra $B$ and a $\ast$-epimorphism $\pi \colon B \to \mathcal O\_2$ (Cuntz algebra). Then $A := \ker \pi$ does the trick. In fact, ... | 8 | https://mathoverflow.net/users/126109 | 331955 | 142,092 |
https://mathoverflow.net/questions/331951 | 8 | Let $X$ be a scheme of finite type (say over the complex numbers). The set of points for which the local ring is reduced [is](https://math216.wordpress.com/2011/06/10/favorite-open-or-closed-conditions-2/) then an open subset $U\subseteq X$.
Is it true that there is a closed subscheme $Z\hookrightarrow X$ such that ... | https://mathoverflow.net/users/4721 | Are schemes obtained from reduced schemes by gluing infinitesimal stuff along a reduced closed subscheme? | If this happens then the cotangent bundle $\mathcal I\_{Z\_{red}} / \mathcal I\_{Z\_{red}}^2$ of $Z\_{red}$ in $X$ is equal to the sum of the contangent bundle of $Z\_{red}$ in $Z$ and the cotangent bundle of $Z\_{red}$ in $X$. This is because locally a coproduct of schemes gives a fibered product of rings which theref... | 10 | https://mathoverflow.net/users/18060 | 331964 | 142,095 |
https://mathoverflow.net/questions/331961 | 6 | Say that an inner model $M$ of $V$ is *generically saturated* if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses cardinals.
>
> What is the consistency strength of "$L$ is generically saturated"?
>
>
>
If the answer is $0^\... | https://mathoverflow.net/users/7206 | Generic saturation of inner models | The concept is inconsistent.
Consider $\mathbb{P}=Add(\omega, \kappa)\_L=Add(\omega, \kappa),$ where $\kappa$ is $(2^{\aleph\_0})^+$ of $V$. Forcing with $\mathbb{P}$ over $V$ doesn't collapse cardinals (it is ccc.c. in $V$) and there is no $M$-generic filter for $\mathbb{P}$ in $V$.
| 10 | https://mathoverflow.net/users/11115 | 331967 | 142,097 |
https://mathoverflow.net/questions/331429 | 6 | Weil cohomology theories can be considered as fibre functors from the category of motives. Given two such functors, we have an affine scheme of invertible natural transformations between them, and rational Hodge theory can be considered as providing
$R$-valued points of this scheme (where $R$ is the coefficient ring).... | https://mathoverflow.net/users/nan | Integral $p$-adic Hodge theory and the space of comparisons of cohomology theories | See [the](https://arxiv.org/abs/1802.03261) [works](https://arxiv.org/abs/1602.03148) of Bhatt--Morrow--Scholze. Possibly they provide what you want.
| 2 | https://mathoverflow.net/users/nan | 331972 | 142,099 |
https://mathoverflow.net/questions/331949 | 5 | Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A\_S = \prod\limits\_{v \in S} k\_v \prod\limits\_{v \not\in S} \mathcal O\_v$ for any (large) finite set of places $S$ containing the archimedean ... | https://mathoverflow.net/users/38145 | Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$? | Yes, $G(\mathbb A\_S) G(k) = G(\mathbb A)$ holds when $S$ is sufficiently large and contains the set of archimedean places $\infty$. This is because the double coset space $G(A\_\infty)\backslash G(\mathbb A)/G(k)$ is finite (its cardinality is called the class number), and a set of representatives can be chosen from $... | 7 | https://mathoverflow.net/users/11919 | 331982 | 142,101 |
https://mathoverflow.net/questions/330527 | 3 | Let $X$ be a scheme. Does there exist an open immersion $X\rightarrow Y$ with $Y$ quasi-compact?
| https://mathoverflow.net/users/nan | Quasi-compactifying schemes | Here are two types of counterexamples. They are all locally of finite type over a fixed field $k$. They rely on the following trivial fact: If $X$ is a subscheme of $Y$ and $Y$ is covered by $n$ open affines, then $X$ (and of course every subscheme of $X$) is covered by $n$ open subschemes which are embeddable in affin... | 4 | https://mathoverflow.net/users/7666 | 331985 | 142,102 |
https://mathoverflow.net/questions/331405 | 1 | Let $P=(p\_1,\ldots,p\_k) \in \Delta\_k$ be distribution supported on set of size $k$ and let $\hat{P}\_n$ be an empirical version of $P$ based on an iid sample of size $n$.
Question
========
What's a good **non-asymptotic** tail-bound of the form $\text{Proba}(\|\hat{P}\_n-P\|\_2^2 \le \epsilon) \ge 1 - \delta$ ?
... | https://mathoverflow.net/users/78539 | Finite-sample deviation bound of empirical distribution from true distribution | I will answer the question as stated though I am not sure you wanted to square the norm of $P\_n-P$. Let $e\_1,\ldots,e\_k$ be the standard basis of ${\bf R}^k$. Write $\mu:=\sum\_{j=1}^k p\_j e\_j$. Let $d\_t$ be independent random variables for $t=1,\ldots,n$ with
${\bf Pr}[d\_t=(e\_i-\mu)/\sqrt{n}]=p\_i$ for $i=1,\... | 1 | https://mathoverflow.net/users/7691 | 331990 | 142,105 |
https://mathoverflow.net/questions/331304 | 1 | Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment
($p>0, p \in \mathbb{R}$) of the integral
$$
\int\_0^T |X(t)|^2 dt?
$$
(The integral is the pathwise integral.)
| https://mathoverflow.net/users/47796 | Inequalities for moments of a certain integral | *I will be working with the cumulants rather than the moments.*
To simplify the notation let me choose units so that $T\equiv 1$ and discretize time $t\_n=n/N$, collecting $X(t\_1),X(t\_2),\ldots X(t\_N)$ in a vector $\vec{X}$. Hence
$$\int\_0^1 |X(t)|^2 dt\rightarrow N^{-1}\vec{x}\cdot\vec{x}\equiv Z.$$
The correlat... | 2 | https://mathoverflow.net/users/11260 | 331993 | 142,106 |
https://mathoverflow.net/questions/332001 | 5 | [Wikipedia states](https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(geometry)#Generalizations_and_related_results) that A. D. Alexandrov generalized *Cauchy's rigidity theorem for polyhedra* to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the end of the Wikipe... | https://mathoverflow.net/users/108884 | Alexandrov's generalization of Cauchy's rigidity theorem | The following is Theorem 27.2 of Igor Pak's book [Lectures on Discrete and Polyhedral Geometry](http://www.math.ucla.edu/~pak/book.htm) (which in general is a very nice resource for these sorts of questions):
>
> Let $P,Q\subset\mathbb{R}^d$ (or $P,Q \subset S^d\_+$), $d\geq3$ be two combinatorially equivalent (sph... | 9 | https://mathoverflow.net/users/353 | 332002 | 142,110 |
https://mathoverflow.net/questions/332015 | 7 | I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective.
Note that self-injectivity says that every monomorphism $R \to M$ splits (where $M$ is an arbitrary module), whereas the property I'm looking for looks... | https://mathoverflow.net/users/2362 | Example of a ring where every module of finite projective dimension is free? | The classes of ring you look at are precisly the rings of finitistic dimension zero for the question with projective instead of free. I will give a large class of example for finite dimensional algebras.
Let $A$ be a local finite dimensional (over a field $K$) non-selfinjective algebra. Then every non-projective modu... | 10 | https://mathoverflow.net/users/61949 | 332017 | 142,118 |
https://mathoverflow.net/questions/331994 | 21 | Occasionally I see the claim, that mathematics was constructive before the rise of formal logic and set theory. I'd like to understand the history better.
1. When did proofs by contradiction or by excluded middle become accepted/standard? Can one find them for instance in classical works (Archimedes, Euclid, Euler, G... | https://mathoverflow.net/users/745 | Status of proof by contradiction and excluded middle throughout the history of mathematics? | I want to echo the other answer, that Brouwer presents the first robust challenge to the Law of Excluded Middle (LEM), but I do want to add some history and background, and maybe recommend a related paper.
The main paper by Brouwer is *De onbetrouwbaarheid der logische principes* or in English *The unreliability of ... | 16 | https://mathoverflow.net/users/51771 | 332036 | 142,126 |
https://mathoverflow.net/questions/332022 | 64 | I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory.
Could somebody informed about ... | https://mathoverflow.net/users/nan | Why is the Eisenstein ideal paper so great? | First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the b... | 100 | https://mathoverflow.net/users/140871 | 332050 | 142,130 |
https://mathoverflow.net/questions/331929 | 3 | Let $k$ be a finite extension of $\mathbb{Q}\_p$. In [this](https://arxiv.org/abs/1205.3463) paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says:
>
> ...it should be possible to deduce (log-)crystalline comparison theorems...
>
>
>
1. What s... | https://mathoverflow.net/users/nan | Crystalline comparison for rigid-analytic varieties | The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.
... | 3 | https://mathoverflow.net/users/140875 | 332055 | 142,132 |
https://mathoverflow.net/questions/332068 | 25 | This question was prompted by my answer to [this question](https://math.stackexchange.com/q/3222502/39599).
An exotic $\mathbb{R}^4$ is a smooth manifold homeomorphic to $\mathbb{R}^4$ which is not diffeomorphic to $\mathbb{R}^4$ with its standard smooth structure. An exotic $\mathbb{R}^4$ is said to be *small* if it... | https://mathoverflow.net/users/21564 | Does every large $\mathbb{R}^4$ embed in $\mathbb{R}^5$? | The product of any two smooth open contractible manifolds is diffeomorphic to a Euclidean space, see e.g. remark 5.3 in [my paper](https://arxiv.org/abs/1208.5220); the result is of course not due to me but I don't know any other place where all references are collected.
In particular, this applies to the product o... | 21 | https://mathoverflow.net/users/1573 | 332069 | 142,136 |
https://mathoverflow.net/questions/332006 | 7 | Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the *support of $\mu$*. As is well known, if a sequence of measures $(\mu\_n)\_{n=1}^\infty$ weak$^\*$ converges to some $\mu\in\mathcal M(X)$, then
$$
\o... | https://mathoverflow.net/users/75977 | Comparison of several topologies for probability measures | The topology generated by $d^{PH}$ is, in general, neither coarser nor finer than $d^{TV}$. The Hausdorff distance between the supports of two measures, even when combined with the Prokhorov distance, has very little to do with the total variation distance between them.
To see one direction, let $X$ have at least two... | 6 | https://mathoverflow.net/users/4832 | 332075 | 142,138 |
https://mathoverflow.net/questions/332064 | 6 | Let $L:H \to H$ be a bounded operator on a Hilbert space $H$, with finite dimensional kernel, and whose adjoint also has finite dimensional kernel. Is it true that $L$ is Fredholm if and only if its spectrum is norm bounded below by a non-zero constant?
| https://mathoverflow.net/users/125790 | A spectral description of Fredholm operators | I assume when you say 'the spectrum is bounded below' you mean that there exists $c>0$ so that no $\lambda$ with $0 < |\lambda| \leq c$ is in the spectrum. In fact, for any bounded operator $L$ on a Hilbert space, this condition is equivalent to the demand that $L$ has closed range.
Each of these conditions for $L$ ... | 6 | https://mathoverflow.net/users/40804 | 332076 | 142,139 |
https://mathoverflow.net/questions/332043 | 7 | Under the assumption that any vector space has a basis (so under the assumption of the axiom of choice), we can prove the following algebraic statements :
1) If $\varphi:V\to W$ is an injective linear map between vector spaces, then the exterior powers $\Lambda^k \varphi : \Lambda^k V\to \Lambda^k W$ are injective. ... | https://mathoverflow.net/users/100552 | Exterior powers and choice | As YCor notes in comments, (2) is a special case of (1), so I'll only address (1).
Suppose $\Lambda^k\varphi:\Lambda^kV\to\Lambda^kW$ is not injective, and let $x\neq0$ be in the kernel.
Then $x$ can be written in the form
$$x=\sum\_{i=1}^mv\_{i1}\wedge\dots\wedge v\_{ik}.$$
Let $V'$ be the finite dimensional sub... | 5 | https://mathoverflow.net/users/22989 | 332093 | 142,140 |
https://mathoverflow.net/questions/332077 | 3 | I'm asking for a proof or references of the following claim:
>
> Let $V$ be a rational Hodge structure having CM in the sense that its Mumford-Tate group is abelian. Then there is a filtration $F^{\bullet}\_{\overline{\mathbb{Q}}}$ on $V\_{\overline{\mathbb{Q}}}$ such that the Hodge filtration $F^{\bullet}$ of $V\_... | https://mathoverflow.net/users/43795 | How to show that Hodge filtration of CM type is algebraic? | A CM Hodge structure is a *polarizable* rational Hodge structure whose Mumford-Tate group is commutative, hence a torus $T$. The Hodge filtration is split by a cocharacter of $T$ over $\mathbb{C}$, which is automatically defined over the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$.
Added: it's the cocharacter ... | 1 | https://mathoverflow.net/users/137572 | 332105 | 142,145 |
https://mathoverflow.net/questions/332086 | 3 | Let $G$ be a simple cubic graph (that is, 3-regular). A *dominating circuit* of $G$ is a circuit $C$ such that each edge of $G$ has an endvertex in $C$. The circuit $C$ is *chordless* if no edge which is not in $C$ has both endvertices in $C$ (such edges are called *chords* of $C$). An {\it MO graph} is a simple cubic ... | https://mathoverflow.net/users/130113 | Edge colorability and Hamiltonicity of certain classes of cubic graphs (MO graphs) | This is a NEW EDITION using the condition that two edges incident with a vertex not in the dominating circuit cannot be attached to consecutive vertices in the dominating circuit.
I tried 200 million at random for each size $8, 12, \ldots, 48$ (it must be a multiple of 4). They were all hamiltonian except for 4 graph... | 6 | https://mathoverflow.net/users/9025 | 332106 | 142,146 |
https://mathoverflow.net/questions/332111 | 4 | Given a multivariate Gaussian $\mathbf{X} \sim \mathcal{N}(\mathbf{\mu},\Sigma)$, I believe it is a difficult question to find a closed form for $$ \mathbb{E}[ \max\{X\_1,\ldots,X\_d\}].$$
However, the case I have at hand is perhaps combinatorially nicer: my Gaussian vector is $(X\_1,\ldots,X\_n)$ where $$X\_j = Z\_j... | https://mathoverflow.net/users/69870 | Expectation of maximum of multivariate Gaussian | For $S:=\sum\_1^n Z\_j$, we have
$$X\_j=Z\_j-\frac{S-Z\_j}{n-1}=-\frac{S}{n-1}+\frac{n}{n-1}\,Z\_j,
$$
whence
$$\max\_1^n X\_j=-\frac{S}{n-1}+\frac{n}{n-1}\,\max\_1^n Z\_j
$$
and
$$E\max\_1^n X\_j=\frac{n}{n-1}\,EM\_n,\quad M\_n:=\max\_1^n Z\_j.
$$
In turn,
$$EM\_n=\int\_0^\infty [P(M\_n>x)-P(M\_n<-x)]\,dx
=\int\_0... | 5 | https://mathoverflow.net/users/36721 | 332113 | 142,147 |
https://mathoverflow.net/questions/332096 | 4 | Is anything known about the cohomology past $\mathrm{H}^1$ and $\mathrm{H}^2$ for the trivial module for a finite group of Lie type in cross characteristic?
For the moment I just care about $\dim \mathrm{H}^n(G,k)$ for $n \geq 3$, $G$ a finite group of Lie type defined in characteristic $p$ and $k$ a field with $\ope... | https://mathoverflow.net/users/88690 | Higher cohomology for trivial module for finite groups of Lie type | Lots is known about the cross-characteristic case: it is the same characteristic case that is the difficult one. The method used was introduced by Quillen, who worked out the full answer for $GL\_n$. I think that the best reference for the general results is the book by Fiedorowicz and Priddy `Homology of classical gro... | 4 | https://mathoverflow.net/users/124004 | 332114 | 142,148 |
https://mathoverflow.net/questions/332104 | 9 | The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's *Functional Analysis*) occasionally state and prove more general versions, for instance for $F$-spaces (i.e. metrizable by a com... | https://mathoverflow.net/users/120251 | Open mapping theorem for complete non-metrizable spaces? | Question 1. Such results have been studied in detail—-a good reference is Köthe‘s monograph on topological linear spaces. You could also look up the concept of webbed spaces (de Wilde).
For question 2, you can take the space of bounded, continuous functions on the real line—-it has two distinct complete locally conve... | 7 | https://mathoverflow.net/users/131781 | 332117 | 142,150 |
https://mathoverflow.net/questions/331965 | 5 | I have been thinking of a way to apply the derived algebraic geometry of Toen-Vezzosi to construct virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. This seems to be the natural setting for which to define symplectic topological invariants when the relevant moduli spaces aren't necessarily cut o... | https://mathoverflow.net/users/123002 | Derived algebraic geometry and virtual fundamental cycles: cotangent complexes | I don't know how to define a VFC for the moduli of pseudoholomorphic curves other than using Pardon's paper. Nevertheless here's the general picture (couldn't tell from your question whether you're familiar with this story).
Let $X$ be a Deligne-Mumford stack. Say we want to define the virtual fundamental class $[X]^... | 6 | https://mathoverflow.net/users/85136 | 332126 | 142,153 |
https://mathoverflow.net/questions/332058 | 13 | I am interested in seeing examples of two closed 4-manifolds $X\_1,X\_2$ such that $SW(X\_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which somehow used symplectic geometry and operations on the Seiberg-Witten invariant (knot surgery, rational blow down, st... | https://mathoverflow.net/users/33064 | Example of two exotic closed 4-manifolds s.t. SW(X)=0 | $\mathbb CP^2\#\overline{K3}$ (trivial SW because it is a connected sum with $b^2\_+>0$ for both pieces). This is an example without having to use the Bauer-Furuta invariants (contrast with Kyle's comment), but with the SW invariants of the 4-manifold's *opposite* (reverse $X$'s orientation into $\overline X$). I learn... | 10 | https://mathoverflow.net/users/12310 | 332128 | 142,154 |
https://mathoverflow.net/questions/332125 | 1 | What is the *mathematical* and *physical* meaning of the terms *focusing* and *nonfocusing* when they refer to nonlinear terms in a dispersive equations?
| https://mathoverflow.net/users/139852 | Focusing and nonfocusing nonlinear terms | The terminology focusing versus defocusing comes from electromagnetic wave propagation in a material with a refractive index $n=\alpha|E|^2$ that depends on the energy density of the electric field (proportional to the field strength squared). The corresponding wave equation for propagation along the $z$-axis is the no... | 1 | https://mathoverflow.net/users/11260 | 332140 | 142,159 |
https://mathoverflow.net/questions/332152 | 8 | Let $(M,g)$ be a connected Riemannian manifold.
Let $d\_g$ be the induced distance metric of $g$. Now let $d$ be some other metric on $M$.
Suppose that for each $x \in M$, there is a neighborhood $U$ of $x$ so that $d = d\_g$ on $U \times U$.
**Question:** Does this imply that $d = d\_g$ on $M \times M$?
I am *n... | https://mathoverflow.net/users/41873 | If a (distance) metric on a connected Riemannian manifold locally agrees with the Riemannian metric, is it equal to the induced metric? | The answer is no. Take any non-convex region in the plane, and let the Riemannian metric be the ordinary Euclidean metric $ds^2=dx^2+dy^2$. Then define the new metric as the infimum of Euclidean diameters of curves connecting $a$ to $b$. This new metric coincides with the Riemannian metric locally, but does not coincid... | 14 | https://mathoverflow.net/users/25510 | 332154 | 142,164 |
https://mathoverflow.net/questions/332135 | 5 | My question is motivated by the following little proposition:
**Proposition.** For a vector subspace $V$ of a Banach space $(X, \|\cdot\|\_X)$ the following assertions are equivalent:
(i) There exists a Banach space $Z$ and a bounded linear operator $T: Z \to X$ with range $V$.
(ii) There exists a complete norm $... | https://mathoverflow.net/users/102946 | Characterization of operator ranges | The answer to your question is "No". It can be seen in the following way: If there exists an operator $S$ mentioned in the Question, then, using the standard techniques, one can show that $V$ has to be isomorphic to a quotient space of $X$. So it remains to show that there exists $X$ and an operator range in $X$ for wh... | 6 | https://mathoverflow.net/users/37822 | 332155 | 142,165 |
https://mathoverflow.net/questions/331708 | 1 | Is there a good tail bound for $\operatorname{P}\!\Bigg[\bigg\vert\dfrac{\sum\_{j=1}^n(\sum\_{i=1}^n a\_{i,j})^2}{n^2} -1\bigg\vert > \epsilon\Bigg]\,,$ where all $a\_{i,j}$'s are iid, with $\operatorname{E}[a\_{i,j}] = 0\,, \operatorname{E}[a\_{i,j}^2] = 1\,, \operatorname{E}[|a\_{i,j}|^3] = \rho\,.$
The tail bound... | https://mathoverflow.net/users/140569 | Is there a 1/poly(n) or 1/polylogn upper-bound for this tail bound? | Let $A:=\Bigg\{\bigg\vert\dfrac{\sum\_{j=1}^n(\sum\_{i=1}^n a\_{ij})^2}{n^2} -1\bigg\vert > \epsilon\Bigg\}\,$ denote the event in question. We will show that
$\operatorname{P}(A)\le C\_{\epsilon}/n$
for a suitable constant $C\_{\epsilon}$.
If $a\_{ij}$ had a finite fourth moment, the argument would be easy. Give... | 6 | https://mathoverflow.net/users/7691 | 332158 | 142,167 |
https://mathoverflow.net/questions/332160 | 13 | Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. Is $T$ $\kappa$-categorical?
If $|\mathcal L’| = \kappa = \aleph\_0$, then I can show the answer is yes using the Ryll... | https://mathoverflow.net/users/11145 | Is categoricity retained when reducing the language? | The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:A\to B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
langu... | 17 | https://mathoverflow.net/users/1946 | 332168 | 142,170 |
https://mathoverflow.net/questions/332166 | 1 | Let $G=(V,E)$ be a finite, simple, undirected graph. Let $\Delta(G)$ be the maximal degree of $G$. Is there a finite graph $G'=(V', E')$ with the following properties?
1. $V\subseteq V'$,
2. $E \subseteq E'$, and
3. $G'$ is $\Delta(G)$-regular.
| https://mathoverflow.net/users/8628 | Regularization of arbitrary graphs | Yes. First of all, for any vertex $v\in V$ we draw the edges to $\Delta(G)-\deg(v)$ new vertices so that the degree of $v$ becomes equal to $\Delta(G)$. Now all degrees are equal either to $\Delta(G)$ or to 1. Without loss of generality, the number of vertices with degree 1 is even (else add the disjoint copy of the cu... | 4 | https://mathoverflow.net/users/4312 | 332169 | 142,171 |
https://mathoverflow.net/questions/332171 | 3 | *This question is cross-posted at <https://math.stackexchange.com/questions/3234895/existence-of-transverse-null-vector-bundle>*
Let $(M,g)$ be a Lorentzian manifold. Let $\dim(M)=d$. Given a null hypersurface, i.e., a smooth embedding $(H, \phi )$ of $M$ s.t. $\dim(H)=d-1$, $h:= \phi ^\* g$ over $H$ is degenerate wi... | https://mathoverflow.net/users/74955 | Existence of transverse null vector bundle in a degenerate Lorentzian hypersurface | Note that $CH$ is [complemented](https://ncatlab.org/nlab/show/direct+sum+of+vector+bundles#TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands) in $TH = CH \oplus SH$, with $SH$ some non-unique choice of complementary sub-bundle. Then the restricted metric on $SH$ is non-degenerate (of Riemannian sign... | 1 | https://mathoverflow.net/users/2622 | 332178 | 142,172 |
https://mathoverflow.net/questions/332170 | 0 | We have $a\_1,a\_2,...,a\_n\in (0,1)$ and matrix
M=
\begin{bmatrix}2a\_1&a\_2&a\_3&.&.\\a\_2&2a\_2&a\_3&.&.\\a\_3&a\_3&2a\_3&.&.\\.&.&.&.&.\end{bmatrix}
We need to check if M is positive definite.
I am trying to evaluate it's determinant as a polynomial in $a\_i$ as principal minor are of the same type. And using ... | https://mathoverflow.net/users/41026 | Positive definite matrix | If $D\_n$ is the leading principal minor of order $n$, then it seems to me
you should have
$$D\_n = 2 a\_n D\_{n-1} - a\_n^2 D\_{n-2}$$
| 2 | https://mathoverflow.net/users/13650 | 332194 | 142,175 |
https://mathoverflow.net/questions/332208 | 10 | I have been trying to understand why the term [quantum](https://en.wikipedia.org/wiki/Quantum_calculus) is so easily accepted for calculus based on q-numbers $[n]\_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...).
But the best hints I could find is this question
[Why are quantum ... | https://mathoverflow.net/users/6575 | q-difference equations and quantum mechanics | There exist applications of q-calculus to physics, but there is no direct relation to quantum mechanics. You can find an overview of some of these applications in [q-Calculus and physics](https://link.springer.com/chapter/10.1007%2F978-3-0348-0431-8_12) (paywall).
This should not be a surprise, because the "q" in q-c... | 10 | https://mathoverflow.net/users/11260 | 332213 | 142,179 |
https://mathoverflow.net/questions/332196 | 6 | Let $T=\{1,2,\dots,2n+1\}$. What is the largest $k$ such that we can choose $k$ subsets of size $n$ and $k$ subsets of size $n+1$ of $T$ so that no chosen subset contains another?
$k=\binom{2n}{n-1}$ is possible: Choose all sets of size $n$ containing $1$, and all sets of size $n+1$ not containing $1$. Does it follow... | https://mathoverflow.net/users/140960 | Non-containing subsets of two sizes | Let $\mathcal{A}$ be any collection of subsets of $\{1,\dots,m\}$ such that no subset in $\mathcal{A}$ is contained in another. Let $a\_i$ be the number of $i$-element subsets in $\mathcal{A}$. A complete characterization of the sequences $(a\_0,a\_1,\dots)$ appears as Theorem 2.2 in <https://pdfs.semanticscholar.org/d... | 4 | https://mathoverflow.net/users/2807 | 332218 | 142,182 |
https://mathoverflow.net/questions/332181 | 4 | I need to integrate
$$
\int\limits\_{[1,a]^n} \frac {\chi(\{ b \le x\_1 \cdots x\_n \le c \})} {( x\_1 x\_2 \cdots x\_n)^k} \,dx\_1 \cdots dx\_n,
$$ where $\chi(E)$ is the characteristic function of a set $E$.
We have $a>1$ and $c>b>1$; furthermore $k\in\mathbb R$, but feel free to restrict it to either larger than $0$... | https://mathoverflow.net/users/131653 | Integrate $1/(x_1x_2\cdots x_n)^k$ for $1\le x_i \le a$, where product of coordinates satisfies $ b\le x_1\cdots x_n\le c$ | **UPDATE 2019-05-31.** Formulae are corrected and SageMath code added.
Let us assume that $a$ is fixed, and define:
$$
I\_n^{k,l}(b,c) := \iiiint\_{[1,a]^n} \frac {\chi(\{ b \le x\_1 \cdots x\_n \le c \})\log(x\_1 x\_2 \cdots x\_n)^l} {( x\_1 x\_2 \cdots x\_n)^k} \,dx\_1 \cdots dx\_n.
$$
Hence, the integral in questi... | 6 | https://mathoverflow.net/users/7076 | 332221 | 142,184 |
https://mathoverflow.net/questions/332210 | 15 | I presume this is a GAGA-style result, but I cannot find a reference.
| https://mathoverflow.net/users/123002 | Why are Stein manifolds/spaces the analog of affine varieties/schemes in algebraic geometry? | Also like affine varieties, we have:
>
> Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.
>
>
>
For the "only if" direction, see Hörmander, *An Introduction to Complex Analysis in Several variables*, Theorem 5.3.9. For the converse, an argu... | 24 | https://mathoverflow.net/users/4144 | 332225 | 142,187 |
https://mathoverflow.net/questions/332220 | 3 | Let $N = (N\_t)\_{t\geq 0}$ be a Poisson process and consider random variables $Z\_n$, $n\in N$. Compute the quadratic variations $[X]\_t$ where $X\_t = \sum\_{n=1}^{N\_t}Z\_n$.
What I did was plugging $X\_t$ into formula $V^2\_t(X:τ\_n=∑(X(t\_i)−X(t\_{i−1}))$, but I dont think it is completely right. Could you pleas... | https://mathoverflow.net/users/140968 | Quadratic variation of sum of random variables | You can just use the definition of quadratic variation (see <https://en.wikipedia.org/wiki/Quadratic_variation>) to infer that
$[X]\_t = \sum\_{n=1}^{N\_t} Z\_n^2$.
To see this, it may help to think of the points of the Poisson process $0<S\_1<S\_2<\ldots$ rather than the counting function $(N\_t)$. Recall that $S... | 2 | https://mathoverflow.net/users/7691 | 332233 | 142,190 |
https://mathoverflow.net/questions/332201 | 3 | Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X\_{\alpha} \rightarrow X$, such that for each $\alpha$, there is a scheme $U\_{\alpha}$ and a finite group $\Gamma\_{\alpha}$ acting on $U\_{\alpha}$ with the property that $\mathc... | https://mathoverflow.net/users/123002 | Local quotient covers for derived Deligne-Mumford geometric stacks of Toen-Vezzosi | Yes, these type of local structure theorems also hold for derived stacks, even more general ones such as Theorem 1.2 in <https://arxiv.org/abs/1504.06467>.
If $\mathcal{X}$ is a derived stack (+ adjectives), apply such a result to the classical truncation $t\mathcal{X}$ to get an etale covering $(X\_\alpha^0 \to t\math... | 6 | https://mathoverflow.net/users/85136 | 332237 | 142,192 |
https://mathoverflow.net/questions/332209 | 1 | Let $X$ be a measure space. Let $S\_j$, $j \in \mathbb N$ be an increasing sequence of $\sigma$-algebras on $X$ such that $S := \bigcup\_{j \geq 0} S\_j$ is a $\sigma$-algebra. For every $j$, let $\mu\_j$ be a probability measure on $S\_j$.
Let $f\_{ij}$ $(i, j \in \mathbb N)$ be a double indexed sequence of function... | https://mathoverflow.net/users/132446 | Measure theory problem concerning convergence of integrals | Following Iosif's comment: the family $\sum\_j S\_j$ virtually never is a $\sigma$-algebra.
But this does not matter: even if $S = S\_j$ for all $j$, the claim is clearly false without further restrictions. Indeed, consider $[-1,1]$ with the usual Borel $\sigma$-algebra $S = S\_j$, and
$$
\begin{gathered}
\mu(dx) = ... | 2 | https://mathoverflow.net/users/108637 | 332241 | 142,195 |
https://mathoverflow.net/questions/332235 | 5 | Deligne's theorem on tensor categories states that for any symmetric tensor category $\mathcal{C}$ satisfying the subexponential growth condition, there is a fiber functor to $\mathsf{sVec}$ and that $\mathcal{C}$ is equivalent to the representations of a supergroup. Under what conditions can we draw the stronger concl... | https://mathoverflow.net/users/131358 | Under what conditions is a symmetric tensor category equivalent to $\operatorname{\mathsf{Rep}}G$ for some group $G$? | If you are in characteristic zero and the dimension of every object is a positive integer then it admits a fiber functor to $Vec$ and is equivalent to $Rep(G)$ for some (pro-algebraic) group $G$. This is theorem 7.1 in Deligne's "Categories Tannakiennes".
| 8 | https://mathoverflow.net/users/39120 | 332242 | 142,196 |
https://mathoverflow.net/questions/332188 | 3 | I am a little bit confused by Serre and Hurewicz fibrations in the context of pointed spaces, i.e. in **$Top\_\*$**.
Serre and Hurewicz fibrations are defined in **$Top$**, i.e. for non-pointed spaces, as having the RLP for inclusions $i\_0 \colon I^n \hookrightarrow I^n \times I$ and $i\_0 \colon X \hookrightarrow X... | https://mathoverflow.net/users/74372 | Serre and Hurewicz fibrations definition for pointed spaces? | This all works as long as the basepoint $x\_0$ of $X$ is nondegenerate. The general context for this is due to Arne Strom who showed that the category of topological spaces, with the classic (i.e. Hurewicz) definitions of cofibration and fibration, is a model category. The key theorem goes as follows: Suppose given a c... | 8 | https://mathoverflow.net/users/102519 | 332258 | 142,200 |
https://mathoverflow.net/questions/332038 | 5 | Let $K$ be an algebraically closed field and let $A=K[x\_1,\dots,x\_n]/I$ be a $K$-algebra of finite type which has only an isolated singularity at the origin. Let $\mathfrak{m}=(x\_1,\dots,x\_n)$ and consider the localization $A\_{\mathfrak{m}}$ and its $\mathfrak{m}$-adic completion $B$.
Is $B$ also an isolated sin... | https://mathoverflow.net/users/106706 | Is completion of isolated singularity isolated? | I believe that in your situation, $B$ indeed has an isolated singularity at the maximal ideal $\mathfrak{n} \subseteq B$. Let me first give two possible definitions for “isolated singularity”; please let me know if there are standard definitions for these notions, and I will edit this answer accordingly!
**Definition... | 4 | https://mathoverflow.net/users/33088 | 332260 | 142,201 |
https://mathoverflow.net/questions/242151 | 10 | Given a symmetric monoidal category $C$, we can construct its endomorphism operad (or multicategory) $End(C)$ whose objects are the objects of $C$, and for which the multimorphisms from $\{c\_1,\ldots,c\_n\}$ to $c$ are given by the hom set $Hom\_C(c\_1\otimes\cdots c\_n,c)$. From this symmetric operad we can then prod... | https://mathoverflow.net/users/11546 | Grothendieck Construction, Categories of Operators and Opposites | This question is answered in the affirmative in a recent paper of mine with Liang Ze Wong. In fact, we prove it more generally for a (strictly) monoidal simplicially enriched category. As any simplicially enriched monoidal category can be rigidified up to monoidal equivalence to a strict one, there's nothing lost by ma... | 3 | https://mathoverflow.net/users/11546 | 332264 | 142,202 |
https://mathoverflow.net/questions/332112 | 6 | Let $M$, $N$ be compact, connected, oriented manifolds without boundary embedded in $\mathbb{R}^m$ of dimensions $p$ and $q$ respectively. I know that when $m=p+q+1$ we can define the linking between $M$ and $N$ in the sense given [here](https://math.stackexchange.com/questions/1653975/relation-between-alexander-dualit... | https://mathoverflow.net/users/103418 | Notion of linking between two general $p$ and $q$ manifolds embedded in a higher dimensional manifold | Sure, there are ways to make sense of linking beyond the constraints you mention.
I suppose the most basic would be: two disjoint submanifolds $A$ and $B$ of a manifold $M$ are **unlinked** if you can find disjoint embedded $m$-dimensional discs $D\_1, D\_2 \subset M$ such that $A \subset D\_1$ and $B \subset D\_2$.... | 7 | https://mathoverflow.net/users/1465 | 332289 | 142,216 |
https://mathoverflow.net/questions/332277 | 5 | I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers which refer to complicated differential geometry or number theory.
The Poisson summation formula (also called Jacobi In... | https://mathoverflow.net/users/140999 | Poisson summation formula and its implication for the spectrum of the flat torus | The theta function (the left-hand side of the Jacobi identity) uniquely determines the values of ${||\gamma^\*||:\gamma^\*\in\Gamma^\*}$ counted with multiplicities, just by looking inductively at its asymptotic expansion as $t\to +\infty$: the leading term will be $1$ since the multiplicity of $0$ is 1, the next term ... | 5 | https://mathoverflow.net/users/56624 | 332295 | 142,219 |
https://mathoverflow.net/questions/332269 | 8 | Let $G$ be a finite group $S\subset G$ a generating set $|g|=$ word length with respect to $S$. Set
$$ \sigma(G) = \sum\_{H \le G} [G:H]$$
Let $\rho$ be the regular representation and set $A\_G := \sum\_{g \in G} \frac{1}{1+|g|} \rho(g)$. Then $|A\_G| = H\_G = \sum\_{g \in G} \frac{1}{1+|g|}$, $A\_G$ is a normal ma... | https://mathoverflow.net/users/nan | Riemann Hypothesis, Primes and Groups | Unless I'm mistaken, the group that you're constructing here is an extension of $\mathbb{Z}$ by $G$. It would cause less confusion if you reserved the notation $\mathbb{Z} \times G$ for the direct product. A fairly common notation for this extension is $\mathbb{Z} : G$.
As far as your question on the relationship bet... | 8 | https://mathoverflow.net/users/19729 | 332297 | 142,220 |
https://mathoverflow.net/questions/332021 | 5 | Let us say a sequence $(x\_n)\_{n=1}^\infty$ in some Banach space $X$ has $S\_C$ if there exist $k\_1<k\_2<\ldots$ such that for any $t\in \mathbb{N}$ and scalars $(a\_n)\_{n=1}^t$, $$\|\sum\_{n=1}^t a\_n x\_{k\_n}\|\leqslant C\Bigl(\sum\_{n=1}^t |a\_n|^2\Bigr)^{1/2}.$$
Let us say the Banach space $X$ has $HSP$ if e... | https://mathoverflow.net/users/nan | Non-uniform property of sequences | No, and the reflexivity plays no role. This is actually a theorem of [Knaust and Odell](http://helmut.knaust.info/paper/KOulp.pdf).
| 0 | https://mathoverflow.net/users/3675 | 332298 | 142,221 |
https://mathoverflow.net/questions/332305 | 11 | I have wondered for a while if there are any interesting rational solutions to $a^b = b^a$. I have tried but cannot find any solutions other than $a=2$ and $b=4$, or vice versa.
Thank you in advance.
| https://mathoverflow.net/users/141013 | Are there more than two rational solutions to $a^b= b^a$? | There is an infinite number of rational solutions
$$a=\left(\frac{n+1}{n}\right)^n,\;\;b=\left(\frac{n+1}{n}\right)^{n+1},\;\;n\in\mathbb{Z},\;\;0\neq n\neq -1.$$
For a proof that these are *all* the rational solutions of $a^b=b^a$ with $a\neq b$, see [Marta Sved's article](https://www.maa.org/sites/default/files/Sve... | 27 | https://mathoverflow.net/users/11260 | 332307 | 142,226 |
https://mathoverflow.net/questions/332315 | 0 | Let $p\equiv 3\pmod 4$ and $G$ be the set of nonzero quadratic residues modulo $p$ (so $G=(p-1)/2$). For $1\leq a\leq p-1$, let $G\_a=\{(a+g)\pmod p\mid g\in G\}$. What is the size of $G\_0\cap G\_a$?
I tested for $p=3,7$ and $11$, and the size of the intersection is always $(p-3)/4$. Is this always true? If so, it s... | https://mathoverflow.net/users/140960 | Shifting quadratic residues | Yes, the intersection has $(p-3)/4$ elements and this is standard, probably this specific claim also is written somewhere. We have $$\left|G\_0\cap G\_a\right|=\sum\_{x} \chi\_{G\_0}(x)\cdot \chi\_{G\_0}(x-a)=\sum\_x \frac{1-(\frac{x}p)-\delta(x)}2\cdot \frac{1-(\frac{x-a}p)-\delta(x-a)}2.$$
Expand the brackets and cal... | 1 | https://mathoverflow.net/users/4312 | 332319 | 142,231 |
https://mathoverflow.net/questions/332176 | 0 | I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . Similar results related to continuous time Markov chains, that allow discretization, are welcome. Thank you.
| https://mathoverflow.net/users/40906 | Lower bounds on discrete time finite Markov chains hitting probabilities | Chapter 10 in the book "Markov chains and mixing times (see <http://www.ams.org/publications/authors/books/postpub/mbk-107>
and <https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf> ) is all about bounding hitting times for discrete chains.
| 2 | https://mathoverflow.net/users/7691 | 332327 | 142,234 |
https://mathoverflow.net/questions/330990 | 1 | This question is followup to the [previous similar question](https://mathoverflow.net/questions/330088/proving-that-preorder-on-the-set-of-measurable-functions-is-symmetric/). There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of my sp... | https://mathoverflow.net/users/48726 | Again, proving that specific preorder on the set of measurable functions is symmetric | I did it, finally. Without going into much detail, it is possible to prove following fact - if there exists at least one pair of measurable $f\_1, f\_2 : \Omega \rightarrow X, f\_1 \prec f\_2 \equiv f\_1 \precsim f\_2 \cap \neg (f\_1 \succsim f\_2)$, then for any measurable $g\_1 : \Omega \rightarrow Y$ it is possible ... | 0 | https://mathoverflow.net/users/48726 | 332330 | 142,235 |
https://mathoverflow.net/questions/332317 | 5 | There are several expository articles with the title "You could have invented [insert something mysterious here]" (a notable one being about spectral sequences, possibly it even started this genre). This question is somewhat similar in spirit to them.
[Here](https://ncatlab.org/nlab/show/motive) it is stated that "D... | https://mathoverflow.net/users/nan | Why is the triangulated category of motives easier than the abelian one? | I believe that there is more than one answer to this question.
1. Even if you don't know anything about the success of Voevodsky and others, you may start with a certain "category of complexes of varieties" and then "impose relations" by localizing by certain elements. Then you will obtain a certain triangulated cate... | 1 | https://mathoverflow.net/users/2191 | 332331 | 142,236 |
https://mathoverflow.net/questions/233476 | 10 | According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known that SMITH is in PPA, but it is unknown whether is it PPA-complete.
More details of this problem can be found here: <h... | https://mathoverflow.net/users/nan | What is the complexity of finding a third Hamilton Cycle in cubic graph? | Thomason's algorithm surely is superpolynomial, and shows that the problem is in PPA. In [3] I described another algorithm, also exponential and shows PPA, which is just as simple and has the added feature that it is easier to show that it is superpolynomial. Towards your question, given the way in which each of those ... | 6 | https://mathoverflow.net/users/141026 | 332333 | 142,237 |
https://mathoverflow.net/questions/332346 | 3 | The general position theorem asserts that any $M$ $m$-manifold unknots in $R^n$ provided $n\geq 2m+2$. The general position theorem assumes a smooth setting. Is unknotting still hold in the PL setting? what is the lower bound on n in that case? what is the argument for unknotting a general $m$-manifold (compact, closed... | https://mathoverflow.net/users/103418 | Unknotting a compact manifold in the PL setting | Yes, if two PL-embeddings $f,g:M^k\hookrightarrow N^n$ of a compact $PL$ manifold of dimension $k$ are homotopic and $n\ge 2k+2$, then they are PL-isotopic.
This is Corollary 5.9 in
*Rourke, C. P.; Sanderson, B. J.*, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete. Band ... | 3 | https://mathoverflow.net/users/8103 | 332355 | 142,245 |
https://mathoverflow.net/questions/332354 | 4 | Given a manifold $M$, the collection of all automorphisms of $M$, denoted by $\text{Aut}(M)$ forms a Lie group.
Do we have similar setting in case of Lie groupoid?
Is there "a structure" whose "automorphisms" forms a Lie groupoid?
| https://mathoverflow.net/users/118688 | Automorphisms of which structure form a Lie groupoid | Here is a construction due to Ehresmann, and covered in detail by Mackenzie in either of his Lie groupoids books.
Take a principal $G$-bundle, $\pi\colon P\to M$, everything here in smooth manifolds. Then there is a Lie groupoid with object manifold $M$ and and a morphism from $m\_1$ to $m\_2$ is a triple $(m\_1,m\_... | 5 | https://mathoverflow.net/users/4177 | 332360 | 142,246 |
https://mathoverflow.net/questions/331299 | 5 | The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of the space of all continuous maps between the underlying topological spaces.
My question is that, regarding homotopy theo... | https://mathoverflow.net/users/140522 | Topological category of topological monoids / operads | I don't think that there's necessarily a right answer to this question. Any category with spaces of maps like you describe has a homotopy category, as well as a lot of other attendant structure. You have to decide what you are interested in.
Let's step back from topological monoids and just talk about topological spa... | 4 | https://mathoverflow.net/users/360 | 332365 | 142,249 |
https://mathoverflow.net/questions/328609 | 8 | Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread [Research topics in distribution theory](https://mathoverflow.net/questions/297186/research-topics-in-distribution-theory) since a good connection exis... | https://mathoverflow.net/users/86432 | Research topics in Microlocal Analysis | I can give a brief description of a (perhaps, currently the main) topic pertaining the application of microlocal analysis to the study of nonlinear PDEs, which seem the argument you are most interested to, since it is also the one to which I am more accustomed, having been interested in it many years ago: to my knowled... | 6 | https://mathoverflow.net/users/113756 | 332373 | 142,251 |
https://mathoverflow.net/questions/332368 | 3 | Let $n>0$ be an integer, and let $[n] = \{1,\ldots,n\}$. A function $f:[n]\to \mathbb{Z}$ is said to be *in- and out-degree-realizable* (or io-realizable for short) if there is a directed graph $G = ([n], E)$ where $E \subseteq [n]\times[n]$ such that for all $k\in[n]$ we have $$f(k) = \text{deg}\_+(k) - \text{deg}\_-{... | https://mathoverflow.net/users/8628 | Finite sequences realizable by degree difference in digraphs | Yes. Case $n\leqslant 2$ is clear, so assume that $n\geqslant 3$. We may use the following well-known
**Lemma**. Let $G=(V,E)$ be an undirected graph (multiple edges allowed, loops not allowed) and $g:V\to \{0,1,2,\dots\}$ be a function such that $\sum\_{v\in V} g(v)=|E|$. Then there exists an orientation of $G$ with... | 2 | https://mathoverflow.net/users/4312 | 332378 | 142,253 |
https://mathoverflow.net/questions/332359 | 3 |
>
> Suppose we have two quadratic equations in $\mathbb R[x\_1,\dots, x\_n]$. What is the expected dimension of their intersection?
>
>
>
In general what can we say about intersection of $k$ quadratics? How many intersections we expect no real solutions in the best case (which I think should be treated similar t... | https://mathoverflow.net/users/10035 | Intersection of quadratic equations with planted solutions? | First (highlighted) question: the dimension is at most $n-2$. Each equation is expected to
decrease the dimension by $1$.
Second question: the dimension of intersection of generic (random) $k$ hypersurfaces is at most $n-k$. (For complex solutions it is equal). It does not matter that they are quadratics.
Third qu... | 2 | https://mathoverflow.net/users/25510 | 332384 | 142,255 |
https://mathoverflow.net/questions/331864 | 4 | Let $\mathcal P$ be a family of nonempty subsets of a topological space $X$. A subset $D\subset X$ is called *$\mathcal P$-generic* if for any $P\in\mathcal P$ the intersection $P\cap D$ is not empty.
A topological space $X$ is called *air* if $X$ has a countable family $\mathcal P$ of infinite subsets of $X$ such th... | https://mathoverflow.net/users/61536 | Is there a universally meager air space? | Lyubomyr Zdomskyy (in private communication) sent me the proof of the following result giving a consistent answer to Problems 1 and 2.
>
> **Theorem (Zdomskyy).** Under $\mathfrak b=\mathfrak c$ there exists a $\mathfrak b$-scale space $X$ which is air and universally meager.
>
>
>
| 1 | https://mathoverflow.net/users/61536 | 332395 | 142,258 |
https://mathoverflow.net/questions/332349 | 5 | A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference that essentially states this as a result? It has been difficult to locate precise definitions. In addition, a sketch of how... | https://mathoverflow.net/users/123002 | Casson invariant and Euler characteristic | Just to finalize comments since people are upvoting the question: The canonical reference is Taubes' *"Casson's invariant and gauge theory"* which makes the statement rigorous and has all the definitions (based on the relevant Chern-Simons functional). Floer's paper *"An instanton-invariant for 3-manifolds"* does build... | 6 | https://mathoverflow.net/users/12310 | 332396 | 142,259 |
https://mathoverflow.net/questions/332329 | 4 | The following result is well-known:
>
> Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have
> $$H \left( (1 - \lambda)(x\_1,y\_1) + \lambda (x\_2,y\_2) \right) \geq H(x\_1,y\_1)^{1 - \lambda} H(x\_2,y\_2)^{\lambda},$$
> and let $M(y)$ denote the... | https://mathoverflow.net/users/119875 | Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization | By the [Brascamp–Lieb concentration inequality](https://en.wikipedia.org/wiki/Brascamp%E2%80%93Lieb_inequality#The_concentration_inequality), we have $$
\operatorname{ Var}\_z (D\_{y}f) \le \langle (D\_{zz}f)^{-1} (D\_{zy} f)^2 \rangle\_z \;,
$$ and hence, $$
g''(y) \ge \langle (D\_{zz}f^{-1}) (D\_{zz} f D\_{yy} f - (... | 6 | https://mathoverflow.net/users/64449 | 332398 | 142,260 |
https://mathoverflow.net/questions/332404 | 6 | I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was told that's for engineers and physicist mostly, not for mathematicians. I also looked for Fisher and Flanigan, but they d... | https://mathoverflow.net/users/129008 | What is a really good book for complex variables? | There are many good books, but the choice depends on your background and on your needs and on your taste. For what purpose do you study complex variables? Do you like geometry or formulas?
If your aim is to use complex variables (for example in engineering and physics problems) Whittaker and Watson is an excellent ch... | 22 | https://mathoverflow.net/users/25510 | 332408 | 142,261 |
https://mathoverflow.net/questions/332342 | 6 | Suppose the usual modular curve $E=X\_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer:
$$M=\prod\_{p}p^{f(E/\mathbb{Q}\_p)},$$
where
$$f(E/\mathbb{Q}\_p)=\begin{cases}0 & E\text{ has good reduction mod }p\\1 & E\text{ has multiplicative reduction mod ... | https://mathoverflow.net/users/140562 | When is the conductor of an elliptic modular curve equal to its level? | Already in the beginning of the 20th century, Fricke had determined explicit equations of the modular curves $X\_0(N)$ which are of genus 1. To do this he constructed, in each case, two explicit functions $\sigma, \tau$ on $X\_0(N)$ such that $j$ and $j\_N$ are rational functions of $\sigma,\tau$ with coefficients in $... | 5 | https://mathoverflow.net/users/6506 | 332409 | 142,262 |
https://mathoverflow.net/questions/332376 | 12 | In ["Quelques propriétés globales des variétés différentiables"](https://eudml.org/doc/139072), Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. Theoreme II.27).
I have the following variation of this question:
>
> Let $M$ be a ... | https://mathoverflow.net/users/17047 | Realizing cohomology classes by submanifolds | Your question is just a reformulation of what Thom did, so the answer is always yes.
Since the Stokes map from de~Rham cohomology to singular cohomology (with real coefficients) is an isomorphism, your problem is equivalent to that of understanding the degree to which the map
$$
\Omega\_\bullet(M)\otimes \Bbb R \to H... | 18 | https://mathoverflow.net/users/8032 | 332412 | 142,264 |
https://mathoverflow.net/questions/332318 | 1 | Consider a linear system $\lvert D \rvert \subset H^0(\mathbf{P}^2 \times \mathbf{P}^2, \mathcal{O}(a,b))$ with nonempty base locus $B$. This linear system defines a rational map $\mathbf{P}^2 \times \mathbf{P}^2 \dashrightarrow \mathbf{P}^n$, defined on the complement of $B$. (Here $n$ is of course one less than the d... | https://mathoverflow.net/users/nan | Resolving the base locus of a linear system on product projective space | The main problem in your approach is that you are only looking at the **set** of points in the base locus. You really have to look at it with its natural scheme structure.
By the way, a linear system $|D|$ forms a projective linear subspace of $\mathbb P(H^0(X, \mathscr O\_X(D)))$. (I.e., it is not a subspace of $H^... | 2 | https://mathoverflow.net/users/10076 | 332417 | 142,268 |
https://mathoverflow.net/questions/332410 | 12 | It is easy to see that $f(x)=x^n-x^{n-1}-\cdots-x-1$ has only one positive root $\alpha$ which lies in the interval $(1,2)$. But it is claimed that this root is a Pisot number (a.k.a. [PV number](https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number)), i.e., the other roots are in the open disk $\{z\in \mat... | https://mathoverflow.net/users/91357 | Roots of $x^n-x^{n-1}-\cdots-x-1$ | Let $r\in\mathbb{R}$ be slightly larger than $1$. Then, on the circle $|z|=r$, we have
$$|z^{n+1}+1|\leq r^{n+1}+1<2r^n=|2z^n|,\qquad |z|=r.$$
By Rouché's theorem, it follows that $P(z):=z^{n+1}-2z^n+1$ has the same number of zeros in the open disk $|z|<r$ as $2z^n$ does (zeros are counted with multiplicity). Therefore... | 21 | https://mathoverflow.net/users/11919 | 332421 | 142,269 |
https://mathoverflow.net/questions/331884 | 8 | In Mori program in dimension $3$ there is a class of Mori contractions $\phi: X\to C$ called *quadric bundles*, where $X$ is a three-dimensional manifold and $C$ is a curve. As far as I understand, such contractions have the property that the morphism $\phi$ is flat and $\phi^{-1}(C)$ is a smooth quadric surface for a ... | https://mathoverflow.net/users/13441 | What are quadric bundles? | have you consulted chapter one of:
Variétés de Prym et jacobiennes intermédiaires
Beauville, Arnaud
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 10 (1977) no. 3, p. 309-391
| 3 | https://mathoverflow.net/users/9449 | 332425 | 142,270 |
https://mathoverflow.net/questions/332428 | 6 | It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s..
From a physics perspective it seems reasonable that when the disorder of the path of a particle decreases and the motion becomes more deterministic, then the number of maxima should decrease.
But I could not fin... | https://mathoverflow.net/users/119875 | Maxima of Brownian motion | A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B\_\cdot$, and if the drift is nice enough to have absolute continuity, the same holds for your drifted diffusion.
This however ch... | 9 | https://mathoverflow.net/users/35520 | 332430 | 142,271 |
https://mathoverflow.net/questions/332338 | 0 | Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,Y)\\
&f\mapsto f|\_Z.
\end{align}
Is the map $\rho$ continuous?
I see this "type of" operation used all the time in Sheaf... | https://mathoverflow.net/users/36886 | Continuity of the Restriction Map Between Function Spaces | If $K$ is compact in $Z$ and $U$ open in $Y$, then $K$ is still compact as
a subset of $X$ as well (compactness is absolute, or maybe use that $i[K]$ is compact where $i: Z \to X$ is the continuous canonical embedding..)
$\rho^{-1}[[K,U]]=\{f \in C(X,Y)\mid (f\restriction\_Z)[K] \subseteq U\} = \{f \in C(X,Y)\mid f[... | 5 | https://mathoverflow.net/users/2060 | 332443 | 142,276 |
https://mathoverflow.net/questions/332304 | 2 | Assume for $\xi\in S^{n-1}$ the parametrization of a closed hypersurface is given by $x(\xi)=R(\xi)\xi\in \mathbb R^n$. Here $R: S^{n-1}\to \mathbb R$ is a positive function. Is there a reference for a proof of the formula
\begin{eqnarray\*}
dS\_{R}=R^{n-2}\sqrt{R^2+\vert\nabla R(\xi)\vert^2}\:dS\_{\xi}\:?
\end{eqnarra... | https://mathoverflow.net/users/79956 | Area formula for parametric surfaces | Other than the chain rule, I think the only ingredient needed for this is the following formula for the determinant of a rank 1 perturbation of an invertible matrix; for $d\in \mathbb{N}$, $A\in \mathrm{GL}(d)$ and $v\in \mathbb{R}^d$,
$$
\mathrm{det}(A+vv^T) = \mathrm{det}(A)(1 + v^TA^{-1}v).
$$
Writing $\hat x$... | 2 | https://mathoverflow.net/users/61771 | 332451 | 142,280 |
https://mathoverflow.net/questions/332356 | 4 | There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only vector killed by generators corresponding to positive roots) and atypical (not typical) but there is no such distinctio... | https://mathoverflow.net/users/33047 | Typical and atypical modules for Lie superalgebras | Regarding the "*what is happening in the super case*"; yes i agree that in some sense, it has to do with the odd simple roots but i think it is deeper than that:
In the case of semisimple, complex, Lie algebras, every reducible representation is completely reducible. However, this result is not true for the basic, c... | 4 | https://mathoverflow.net/users/85967 | 332466 | 142,283 |
https://mathoverflow.net/questions/332467 | 1 | I have a set $P$ of points in a Banach space. Consider the following two cones:
* The closure of the set of all (finite) nonnegative linear combinations of $P$. (I.e., the topological closure of $\{\sum\_{i=1}^n a\_ip\_i: a\_i\geq0, p\_i\in P\}$.)
* The set of all infinite nonnegative linear combinations of $P$. (I.e... | https://mathoverflow.net/users/101775 | Closed convex cone - equivalence of definition via closure and via infinite sums | The answer to your question is "No" (but the first set, obviously, always contains the second).
Example showing that the second set can be strictly smaller: Denote by $\{e\_n\}$ the unit vector basis in $\ell\_1$ and consider the following set $P:=\{e\_1+\frac1ne\_n\}\_{n=2}^\infty$ in $\ell\_1$. It is clear that $e... | 4 | https://mathoverflow.net/users/85406 | 332477 | 142,289 |
https://mathoverflow.net/questions/332461 | 5 | Let $n>1$ be an integer and set $[n]=\{1,\ldots,n\}$. We say that $n$ has a "Hamiltonian Square Path" if there is a bijection $\varphi:[n]\to[n]$ such that for all $k\in [n-1]$ we have that $\varphi(k)+\varphi(k+1)$ is a square number.
For instance $15$ and $16$ have this property.
**Question.** Is there an integer... | https://mathoverflow.net/users/8628 | Integers with a Hamiltonian Square Path | Post #22 at <https://mersenneforum.org/showthread.php?p=477787> by R. Gerbicz claims a proof that $N=25$ is the answer, and that for $N\ge32$ there is a Hamiltonian cycle. See also the tabulation and discussion at the [Online Encyclopedia of Integer Sequences](http://oeis.org/A090461).
| 7 | https://mathoverflow.net/users/3684 | 332479 | 142,290 |
https://mathoverflow.net/questions/82087 | 53 | Edit: Infos on the current state by Lieven Le Bruyn: <http://www.neverendingbooks.org/grothendiecks-gribouillis>
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible (+ Tha... | https://mathoverflow.net/users/451 | Grothendieck's manuscript on topology | In the light of past events ("[Les Archives Grothendieck](https://grothendieck.umontpellier.fr/archives-grothendieck/)"), we now have:
**Vers une Géométrie des Formes** (1986)
* I. *Vers une géométrie des formes (topologiques)* : notes manuscrites (05/06/1986).
Cote n° 156-1 (26 p.)
* II. *Réalisations topologi... | 11 | https://mathoverflow.net/users/nan | 332481 | 142,292 |
https://mathoverflow.net/questions/332493 | 0 | Let $a$, $c$, $d$, $v \in \mathbb{R}^n$ are vectors, and $A, B \in \mathbb{R}^{n \times n}$ are matrices. Suppose that $ v = Ac-d $, and $a = ABc- \|B\| d$ where $\| B \|$ is the maximum value of the norms of the eigenvalues of $B$.
Is it true that $\| a \| \leq \|B\| \| v\|$?
| https://mathoverflow.net/users/140940 | Upper-bounds for a vector equation | No, it is not true in general.
Example: $A:=I$ (identity). $B:=-I$. $c\neq 0$ arbitrary. $d:=c$.
Then $v=Ac-d=0$ and $a=ABc-\lVert B\rVert d=-c-1d=-2c\neq 0$. Then $\lVert a\rVert > 0$ and $\lVert B\rVert\lVert v\rVert=0$.
| 0 | https://mathoverflow.net/users/101775 | 332498 | 142,295 |
https://mathoverflow.net/questions/332515 | 5 | Let $F(x,y,z)$ be the degree 12 homogeneous polynomial:
$$x^{12} - x^9 y^3 + x^6 y^6 - x^3 y^9 + y^{12} - 4 x^9 z^3 + 3 x^6 y^3 z^3 - 2 x^3 y^6 z^3 + y^9 z^3 + 6 x^6 z^6 - 3 x^3 y^3 z^6 + y^6 z^6 - 4 x^3 z^9 + y^3 z^9 + z^{12}$$
Over the rationals it is irreducible and $F=0$ is genus 1 curve.
Numerical evidence i... | https://mathoverflow.net/users/12481 | Does this degree 12 genus 1 curve have only one point over infinitely many finite fields? | The polynomial you wrote is the product of the four polynomials $x^3 - r y^3 - z^3$, where $r$ is a root of the polynomial $t^4 - t^3 + t^2 - t + 1$. I did not read your reference, but likely they assume that the curves are geometrically irreducible.
| 14 | https://mathoverflow.net/users/141120 | 332519 | 142,298 |
https://mathoverflow.net/questions/332374 | 5 | Consider $P$ the complex projective plane, and fix a line $L$ in $P$
I had a conjecture, that prof. I. Dolgachev showed me how to prove, that $3$ quadratic polynomials depending on a variable $z \in L$, say $p\_1$, $p\_2$ and $p\_3$, are linearly dependent over $\mathbb{C}$ if and only if:
there exist $4$ points $A... | https://mathoverflow.net/users/81645 | Is there a general geometric characterization for polynomials to be linearly dependent? | For an arbitrary $n \geq 3$, consider a collection $X$ of $n(n-1)/2 + 1$ points in general position in $P^2(\mathbb{C})$. Subdivide this collection of points into a set $S$ with $n(n-1)/2 - 1$ points and a set $T$ with the remaining $2$ points.
There is a unique planar curve of degree $n-2$ passing through all the po... | 1 | https://mathoverflow.net/users/81645 | 332522 | 142,299 |
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