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https://mathoverflow.net/questions/194024 | 10 | *Definition of the Grothendieck group and Leftschetz motive.* The Grothendieck group of varieties is a free abelian group generated by classes of algebraic varieties with the following relation:
$$
[X]=[X\smallsetminus Y]+[Y]
$$
for $Y\subset X$ a closed subvariety. Let $\mathbb L$ denote the class of an affine line (... | https://mathoverflow.net/users/19436 | Why do we need localization by Leftschetz motive? | This is more of an opinion than a complete answer but it is too long to leave as a comment. Let $$H : K\_0(\mathcal{M}\_{\mathbb{C}}) \to K\_0(MHM\_{\mathbb{C}})$$ be the Hodge realization from the Grothendieck ring of Chow motives to the Grothendieck ring of mixed Hodge modules, then $H$ kills all $(\mathbb{L}-1)$-tor... | 8 | https://mathoverflow.net/users/5031 | 194046 | 94,738 |
https://mathoverflow.net/questions/194045 | 2 | For $A = \{a\_{ij}\} \in R^{n\times n}$, is finding
$$
\max\_{\sigma \in S\_n}\min\_{1 \le i \le n} a\_{i,\ \sigma\_i}
$$
NP-hard?
| https://mathoverflow.net/users/11263 | NP-hardness of finding maximum of minimum element in diagonal of a matrix | This seems to be polynomial. Here is a proof. It will be convenient to regard $A$ as an edge-weighted complete bipartite graph $G$. Let $m\_1 < \dots < m\_\ell$ be the list of edge weights of $G$, let $E\_i$ be the set of edges of weight $m\_i$, and let $G\_i:=G \setminus (E\_1 \cup \dots \cup E\_i)$. Now test if $G\_1... | 6 | https://mathoverflow.net/users/2233 | 194049 | 94,739 |
https://mathoverflow.net/questions/193988 | 9 | The (infinite) symmetric product of a based topological space $(X,e)$, denoted by $\mathrm{SP}(X,e)$, can be viewed as the topological space of ''multisets'' in $X$ containing the base point $e$ infinitely many times (please see <http://en.wikipedia.org/wiki/Infinite_symmetric_product> for the precise definition). The ... | https://mathoverflow.net/users/66004 | The Metrizability of Symmetric Products of Metric Spaces | The infinite symmetric product of a pointed metric space $(X,e)$ is metrizable iff the basepoint $e$ is isolated. First, if $e$ is isolated and $X=Y\coprod \{e\}$, then $SP(X,e)=\coprod SP^n(Y)$ and each finite symmetric product $SP^n(Y)$ is metrizable (by the metric you define). In fact, in this case it is not hard to... | 7 | https://mathoverflow.net/users/75 | 194063 | 94,742 |
https://mathoverflow.net/questions/186834 | 7 | As discussed in [this MO topic](https://mathoverflow.net/questions/110457/bass-stable-range-condition-for-principal-ideal-domains), every principal ideal domain has stable rank at most 2. The proof in the accepted answer uses the fact that PID is a unique factorization domain, but there can be no irreducibles in case o... | https://mathoverflow.net/users/5018 | Bass' stable range for Bezout rings | In "Rings of continuous functions in which every finitely generated ideal is principal" by L. Gillman and M. Henriksen (Trans. Amer. Math. Soc. 82 (1956), 366-391 [link](http://www.ams.org/journals/tran/1956-082-02/S0002-9947-1956-0078980-4/)), Example 3.4 is of a topological space $X$ such that the ring of continuous ... | 10 | https://mathoverflow.net/users/22989 | 194073 | 94,746 |
https://mathoverflow.net/questions/194056 | 13 | $\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't exactly contrived and yet it seems sort of mysterious.
This is a follow up to an earlier question [What is the etale fu... | https://mathoverflow.net/users/15242 | Some questions about the ring Z((x)) | Question 1: The fraction field is the same as that of $\mathbb{Z}[[x]]$. It can be gotten by inverting all irreducibles. The irreducibles of the UFD $\mathbb{Z}[[x]]$ are described in Theorem 1.4 of <http://arxiv.org/abs/1107.4860>
Question 2: You missed a lot of prime ideals of $\mathbb{Z}[[x]]$. (It has uncountably... | 11 | https://mathoverflow.net/users/17218 | 194074 | 94,747 |
https://mathoverflow.net/questions/194088 | 1 | Let K be a field, let $T = K[X\_1, X\_2,...]$ be a polynomial ring, let $R=K[X\_1^{2}, X\_1X\_2,..,X\_i X\_j,..]$, and let $L = Frac(R)$ = field of fractions of R. How can we prove that $R =T \cap L$ ?
| https://mathoverflow.net/users/66035 | question about a particular Polynomial ring | Consider the involution $\sigma $ of $T$ which maps each $X\_i$ to $-X\_i$; the $\sigma $-invariant subring of $T$ is $R$. Similarly $\sigma $ extends to the field of fractions $E$ of $T$, and the $\sigma $-invariant subfield of $T$ is $L$. Thus $T\cap L=T\cap E^{\sigma }=T^{\sigma }=R$.
| 2 | https://mathoverflow.net/users/40297 | 194092 | 94,750 |
https://mathoverflow.net/questions/194079 | 2 | Today my students asked me the following problem：
Define polynomials $P\_j$ with real coefficients
$$P\_{j}=\sum\_{i\_{1},\cdots,i\_{s}}a^{(j)}\_{i\_{1}\cdots i\_{s}}X^{i\_{1}}\_{1}\cdots X^{i\_{s}}\_{s}, \qquad (1\le j\le s)$$
where $s$ is a given positive integer.
Assmue that $a^{(1)}\_{1,0,0,\ldots,0}, a^{(2)}\... | https://mathoverflow.net/users/38620 | Determine whether a system of polynomials with real coefficients has a real solution? | The system can be put in the form $Y=LX-G(X)$. Here $X$ is the column vector with the unknowns $X\_1,\ldots,X\_s$. $Y$ is the column vector with components $Y\_i=-a\_{0,\ldots,0}^{(i)}$. $L$ is the matrix describing the linear part of your system, namely,
$$
L\_{ij}=a\_{0,\ldots,0,1,0,\ldots,0}^{(i)}
$$
with the $1$ at... | 1 | https://mathoverflow.net/users/7410 | 194095 | 94,751 |
https://mathoverflow.net/questions/194069 | 2 | **Question**:
Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, is there a nice way to work this out?
If it helps, the algebra in question is easily treatable as the $x\_i$ and ... | https://mathoverflow.net/users/65916 | Commutators of Schur polynomials of Lie algebra elements | No, there will be no nice formula for $[V\_{\mu}^{(n)},V\_{\lambda}^{(m)}]$ in general.
Compute first
$$ V\_{\mu}(z\_1) V\_{\lambda}(z\_2)=p(z\_1,z\_2) : V\_{\mu}(z\_1)V\_{\lambda}(z\_2) :$$
$$ V\_{\lambda}(z\_2) V\_{\mu}(z\_1)=q(z\_1,z\_2) : V\_{\mu}(z\_1)V\_{\lambda}(z\_2) :$$
where $: \cdot :$ is the normal ord... | 3 | https://mathoverflow.net/users/66047 | 194098 | 94,753 |
https://mathoverflow.net/questions/193561 | 2 | $R$ is a *local* Noetherian ring. $f\_I(M)$, the *finiteness dimension* of a module $M$ relative to $I$, is defined in `chapter 9 of the book Local Cohomology. An Algebraic Introduction with Geometric Applications` by `M. P. BRODMANN and R. Y. SHARP` as: $f\_I(M) = \inf\ \{i : H\_I^i(M)$ is not finitely generated$\}$. ... | https://mathoverflow.net/users/47763 | finiteness dimension | Ok, I can at least explain one way to do this in the case that $I$ is maximal. Since you are hoping for computers to do this for you, I'm going to assume that $R$ has a (normalized) dualizing complex.
Note that then $H^i\_I(M)$ is Matlis dual to $\text{Ext}^{-i}\_R(M, \omega\_R^{\bullet})$.
Then $H^i\_I(M)$ is fin... | 2 | https://mathoverflow.net/users/3521 | 194100 | 94,754 |
https://mathoverflow.net/questions/193837 | 36 | Let $$\small F\_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and
$ad=bc$, then
$$64 F\_6 F\_{10}=45 F\_8^2$$
This fascinating identity is due to Ramanujan and can be found in ["Ramanujan for Lowbrows", by B.C. Berndt and S. Bhargava](http://www.maa.org/programs/maa-awards/writing-awards/ramanujan-for-low... | https://mathoverflow.net/users/32389 | How did Ramanujan discover this identity? | You have two questions: 1) How Ramanujan discovered it? 2) Is an accidental, isolated result? The second one is easier to answer and may shed light on the first.
>
> **I.** Define $F\_n = x\_1^n+x\_2^n+x\_3^n-(y\_1^n+y\_2^n+y\_3^n),\;$ where $\,\small x\_1+x\_2+x\_3=y\_1+y\_2+y\_3 = 0$.
>
>
>
**Theorem 1:** If... | 34 | https://mathoverflow.net/users/12905 | 194104 | 94,756 |
https://mathoverflow.net/questions/194081 | 3 | Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int\_0^T \langle X(t, B\_t), \mathrm{d} B\_t\rangle$$
and would like for a reference of the following method.
Denote by $M \bowtie M$ the ... | https://mathoverflow.net/users/16702 | Certain construction of the Itô integral on manifolds | A good and general reference on stochastic integrals on manifolds is
[An Invitation to Second-Order Stochastic Differential Geometry](https://hal.archives-ouvertes.fr/hal-00145073/document)
See in particular Theorem 1, page 14. The survey also contains a list of references with some specific pointers to the litera... | 1 | https://mathoverflow.net/users/48356 | 194111 | 94,758 |
https://mathoverflow.net/questions/194107 | 11 | Assume you are given a non-negative, continuous, radial function $f\in L^q(\mathbb{R}^3)$ (for any $q\geq 1$).
Are there any conditions which would guarantee that the Fourier transform of $f$, that is $\hat{f}(p)$, is also non-negative?
| https://mathoverflow.net/users/66052 | Conditions for positivity of Fourier transform | the three-dimensional Fourier transform $F(\vec{p})$, of the radial function $f(r)$ has Fourier transform
$$F(\vec{p})=\int\_0^\infty dr\int\_0^\pi d\theta \int\_0^{2\pi}d\phi\;e^{ ipr\cos\theta}f(r) r^2\sin\theta$$
$$\qquad=\frac{4\pi}{p}\int\_0^\infty rf(r)\sin(pr)\,dr,\;\;{\rm with}\;\;p=|\vec{p}|.$$
so you're a... | 10 | https://mathoverflow.net/users/11260 | 194115 | 94,759 |
https://mathoverflow.net/questions/194060 | 2 | I was wondering if there is any general definition for twistors for spaces of any dimension with a definite (or indefinite) metric; for example, in $\mathbb{R}^3$.
Twistors are spinors of the compactificated minkowski space. Does this work in general? Are twistors the spinors of the compactification of a vectorial s... | https://mathoverflow.net/users/43516 | Twistors for spaces of $n-$dimensions | "Twistors are spinors of the compactified Minkowski space" is not
quite true. Twistors in Minkowski space are spinor fields which
satisfy a particular PDE: the twistor spinor equation. Relative to
flat coordinates and the corresponding global frame for the tangent
bundle, a twistor spinor $\psi$ is given in terms of a ... | 4 | https://mathoverflow.net/users/394 | 194120 | 94,762 |
https://mathoverflow.net/questions/194114 | 6 | For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?
If not, there is a weaker statement that I am interested in:
Is it true that the inequality $\sigma\_n( \sqrt{A} - \sqrt{B}) \... | https://mathoverflow.net/users/24953 | Subadditivity of the square root for matrices | The claim is false. Just try some random psd matrices $A$ and $B$. You can get $\sigma\_n( |A-B|^{1/2}) = \sigma\_n^{1/2}(A-B) = 0$, whereas $\sigma\_n(A^{1/2}-B^{1/2}) > 0$.
Here is an explicit example:
\begin{equation\*}
A = \begin{pmatrix} 19 & 17 & 9\\
17 & 17 & 11\\
9 & 11 & 11\end{pmatrix},\quad
B = \begi... | 4 | https://mathoverflow.net/users/8430 | 194122 | 94,763 |
https://mathoverflow.net/questions/194125 | 2 | We have $\widetilde{KO}(S^4) \cong \mathbb{Z}$ and $\widetilde{K}(S^4) \cong \mathbb{Z}$. There is a map $i:\widetilde{KO}(S^4) \rightarrow \widetilde{K}(S^4)$ that takes a stable vector bundle to its complexification. Question: What is the image of $i$? I've seen it asserted (say, in various sources discussing Rochlin... | https://mathoverflow.net/users/65911 | Complexification of real k-theory gives index $2$ subgroup of complex k-theory | This is the same as the induced map $\pi\_4(KO) \to \pi\_4(KU)$. Via Bott periodicity (see also [this answer](https://mathoverflow.net/questions/192316/the-periodic-values-in-bott-periodicity/192324#192324)), this is the same as the induced map $\pi\_0(KSp) \to \pi\_0(KU)$ coming from the map sending a stable quaternio... | 5 | https://mathoverflow.net/users/290 | 194127 | 94,766 |
https://mathoverflow.net/questions/193951 | 8 | A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of [Kleiner-Leeb](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCAQFjAA&url=http%3A%2F%2Fmath.nyu.edu%2F~bkleiner%2Fsymm.pdf&... | https://mathoverflow.net/users/65993 | Embedding Euclidean buildings into products of trees | The problem of existence/nonexistence of quasi-isometric embeddings $X\to Y$ between symmetric spaces and locally compact Euclidean buildings of rank $\ge 2$ is wide-open in the case when $rank(Y)> rank(X)$ (assuming, of course, that $dim(X)<dim(Y)$ in the setting of symmetric spaces). The most up-to date results and r... | 6 | https://mathoverflow.net/users/21684 | 194129 | 94,767 |
https://mathoverflow.net/questions/194148 | 2 | I have bumped into the following expressions involving $q$-binomial coefficients.
$$
\sum\_{s=0}^a (-1)^s q^{s^2-s} \left(\begin{array}{c}2b+1-2s\\2a-2s\end{array}\right)\_q
\left(\begin{array}{c}b\\s\end{array}\right)\_{q^2}
$$
The expressions depend on $a$ and $b$ and are zero unless $0\leq a\leq b$.
Have these e... | https://mathoverflow.net/users/38468 | Expressions involving $q$-binomial coefficients? | This is a summable ${}\_2\phi\_1$. The standard reference is Gasper and Rahman, Basic Hypergeometric Series. It looks like your series is
$$\frac{(q;q)\_{2b+1}}{(q;q)\_{2a}(q;q)\_{2b+1-2a}}{}\_2\phi\_1\left(\begin{matrix}q^{-2a},q^{1-2a}\\q^{-2b-1}\end{matrix};q^2,q^{4a-2b-2}\right).$$
By the $q$-Gauss summation, this ... | 7 | https://mathoverflow.net/users/10846 | 194151 | 94,771 |
https://mathoverflow.net/questions/194047 | 10 | I am trying to figure out how much one can figure out about an object using category theory. Ideally, any property that is well defined up to isomorphism should be computable using only category theory. Let us say that we are trying to figure out how many elements are in a group? For a set, we could "simply" count the ... | https://mathoverflow.net/users/65915 | A categorical method to, say, determine the cardinality of a group | A bunch of bits and pieces from a bunch of people:
Let us say we are trying to find the cardinality of the Group $G$. First, we select the group, $\bullet$ (unique up to isomorphism) such that for any other group, there is a unique arrow in and out of $\bullet$. Now, we select a group $\mathbb{Z}$, unique up to isomo... | 6 | https://mathoverflow.net/users/65915 | 194152 | 94,772 |
https://mathoverflow.net/questions/194135 | 8 | Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex analytic functions? More information on this object is very appreciated.
| https://mathoverflow.net/users/36688 | Why is this group called "The Holomorph of a group" | I'm not a history expert, but according to Miller, Blichfeldt and Dickson: "Theory and applications of finite groups" (1916), footnote p. 46: "The concept of holomorph was used by many early writers, but the term was introduced by W. Burnside in the first edition of his Theory of Groups, 1897, p. 228."
| 9 | https://mathoverflow.net/users/65801 | 194162 | 94,775 |
https://mathoverflow.net/questions/194165 | 10 | Assume that $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function such that each level set $f^{-1}(c)$ is a convex set.
To what extent such functions are studied?
In particular:
1. Is there a partial or total ordering on $\mathbb{R}^{2}$ such that all functions with this property, must be monotone?
2.Is it t... | https://mathoverflow.net/users/36688 | Continuous functions with convex level sets | Let's call the functions defined by Ali Taghavi to be sliced functions: a continuous function $\ f:\mathbb R^2\rightarrow\mathbb R\ $ is called *sliced* $\ \Leftarrow:\Rightarrow\ \ \forall\_{c\in\mathbb R}\ f^{-1}(c)\ $ is convex.
>
> **NOTATION**:
> $$\ [x;y]\ :=\ \{(1\!-\!t)\cdot x\ +\ t\cdot y\ :\ 0\le t\le 1... | 10 | https://mathoverflow.net/users/8385 | 194171 | 94,778 |
https://mathoverflow.net/questions/194156 | 7 | Consider the following famous theorem by Robert C. James (1964):
>
> Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and only if every continuous linear real-valued functional on $X$ attains its supremum on $C$.
>
>
>
The “only if” p... | https://mathoverflow.net/users/55976 | James' theorem—going from the separable case to the general case | I got it. I just read Pryce (1966), which is more accessible than I had thought. The proofs by Pryce (1966) and by Holmes (1975) use almost identical ideas, it's just that the latter refrains from dealing with the non-separable case for some reason.
Both proofs start out with assuming that $C$ is non-empty, weakly cl... | 6 | https://mathoverflow.net/users/55976 | 194178 | 94,779 |
https://mathoverflow.net/questions/194186 | 0 | Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ 1\}\,\ E\_e\ $ where
$$E\_e\:=\ \{\{(a\ s)\,\ (b\ t)\}\ :\ \{a\ b\}\in V\ \ and\ \ t=e(\{a\ b\})\cdot s\}$$
* Now by... | https://mathoverflow.net/users/36554 | When is a $2$-lift of a graph connected? | Still, I don't understand the question. The matrix determines the graph, hence there is a way to tell whether it's connected. If the question is about **how** to do that, then here is the first thing that comes to my mind:
>
> the double is connected iff
> (1) the original graph $G$ is connected, and (2) the homo... | 2 | https://mathoverflow.net/users/44953 | 194192 | 94,781 |
https://mathoverflow.net/questions/194103 | 3 | Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both *inside* and *outside* mathematics)?
| https://mathoverflow.net/users/nan | Survey on functional equations and inequalities | [Functional Equations and Inequalities with Applications](https://books.google.com.br/books?id=SdZoCM2OeuIC) is a book from 2009 about these topics:
>
> Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations. Nowada... | 2 | https://mathoverflow.net/users/22389 | 194210 | 94,784 |
https://mathoverflow.net/questions/194183 | 9 | Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, and I am interested to know whether there also exists an exponential upper bound. If we restrict to alternating knots, then wo... | https://mathoverflow.net/users/35353 | How many knots are there with hyperbolic volume less than a given constant | Here is an expansion of Ian's answer.
Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of [Thurston's book](http://library.msri.org/books/gt3m/PDF/). Furthermore, the new manifold has smaller hyperbo... | 13 | https://mathoverflow.net/users/1650 | 194211 | 94,785 |
https://mathoverflow.net/questions/194202 | 4 | I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly appreciated as well.
Thanks.
| https://mathoverflow.net/users/3124 | What are Carnot groups? | I learned the theory of Carnot group or more general subRiemannian manifolds from the thesis of Monti, which can be found here:
<http://www.math.unipd.it/~monti/PAPERS/TesiFinale.pdf>
In particular, Section 2 is devoted to the proof of Pansu-differentiability theorem.
Another good notes is by le donne, which [can be ... | 4 | https://mathoverflow.net/users/26608 | 194213 | 94,786 |
https://mathoverflow.net/questions/194215 | 0 | While reading an article about iterative methods for solving nonlinear equations I can't understand what is *exponentially fitted osculating straight line*. Could someone please briefly explain this term to me?
| https://mathoverflow.net/users/64121 | What is exponentially fitted osculating straight line? | it's jargon for a function of the form
$$y(x)=e^{a(x-x\_0)}[b(x-x\_0)+c]$$
where the constants $a,b,c$ are determined such that $y(x)$ [osculates](http://en.wikipedia.org/wiki/Osculating_curve) the function $f(x)$ you are fitting to at $x=x\_0$:
$$y(x\_0)=f(x\_0),\;\;y'(x\_0)=f'(x\_0),\;\;y''(x\_0)=f''(x\_0).$$
Thi... | 4 | https://mathoverflow.net/users/11260 | 194225 | 94,789 |
https://mathoverflow.net/questions/194223 | 3 | Consider the hierarchy given by $\cal S\_0 =$ first-order Peano arithmetic, $\cal S\_{\alpha+1}=\cal S\_{\alpha} + Con(S\_\alpha)$ (a consistency statement for $\cal S\_\alpha$), and if $\alpha$ is a limit ordinal, $\cal S\_\alpha=\bigcup\_{\beta < \alpha} \cal S\_\beta$.
Peano arithmetic proves transfinite inductio... | https://mathoverflow.net/users/66109 | Adding consistency statements to Peano arithmetic allows more instances of transfinite induction? | Over PRA, the transfinite induction schema up to $\epsilon\_0$ for primitive recursive formulas implies not just the consistency of PA, but also its $\Sigma^0\_1$-soundness (i.e., the uniform $\Sigma^0\_1$-reflection principle for PA). As such, it is not provable in any consistent extension of PA by a set of $\Pi^0\_1$... | 4 | https://mathoverflow.net/users/12705 | 194226 | 94,790 |
https://mathoverflow.net/questions/194235 | 6 | Is the following statement true? If so, can anyone provide a reference?
>
> Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
> subcomplex of $X$. Then the following are equivalent:
>
>
> 1. $Y$ is convex in $X$.
> 2. For every cube $C$ in $X$, the intersection $C\cap Y$ is a face of $C$.
>
>
>
... | https://mathoverflow.net/users/6514 | Convex subcomplexes of CAT(0) cubical complexes | Yes, it is true.
You condition (2) implies that $X$ is locally convex;
this can be proved the same way as the flag condition for $\mathrm{CAT}[0]$-ness.
It remains to note that for $\mathrm{CAT}[0]$-spaces local convexity + connectedness implies convexity.
| 9 | https://mathoverflow.net/users/1441 | 194236 | 94,792 |
https://mathoverflow.net/questions/192081 | 7 | Apart from citations all over the internet, the following paper appears to be off-the-grid.
* K. Itô, *A measure-theoretic approach to Malliavin calculus*, in 'New Trends in Stochastic Analysis', Proc. Taniguchi Symposium, Sept. 1994, Charingworth, (eds. K. D. Elworthy, S. Kusuoka and l. Shigekawa),
World Scientific,... | https://mathoverflow.net/users/60641 | Itô's article "A measure-theoretic approach to Malliavin calculus" | I have scanned my copy and sent it to the OP.
| 4 | https://mathoverflow.net/users/4832 | 194237 | 94,793 |
https://mathoverflow.net/questions/194212 | 3 | Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau\_{x, w}$ and $\tau\_{y, w}$ be the parallel transports of the tangent planes at $x$ and $y$ to the tangent plane at $w$, respectively.... | https://mathoverflow.net/users/10747 | Parallel transport on a Hadamard manifold | The following counterexample shows that even for fixed $w,x,y$, the norm in your expression need not be bounded in $z$:
Let $X$ be the hyperbolic plane; I'll use the upper half-space model to identify $X$ with a subset of $\mathbb{C}$. Let $w = x = i - 1$, let $y = i + 1$, and let $z\_n = ni$.
As $n \to \infty$, th... | 2 | https://mathoverflow.net/users/66112 | 194239 | 94,794 |
https://mathoverflow.net/questions/194241 | 1 | Let $\mu\_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have
$$\int fd\mu\_n \to \int fd\mu$$
Then I have seen in some place claiming the following:
$$\int h(x\_n,y)\mu\_n(dy) \to \int h(x,y)\mu(dy)$$
for $h:\Bbb R \times S \to \Bbb R$ and ... | https://mathoverflow.net/users/nan | Problem on convergence in probability measres | **Edit**: Previous answer was bogus.
Let me change your notation a little to let $x\_0$ be the limit of the $x\_n$.
Yes, this is true. It follows from tightness and the general fact that $h(x\_n, \cdot) \to h(x\_0, \cdot)$ uniformly on compact sets.
Without loss of generality, let's suppose that $h(x\_0, \cdot) =... | 1 | https://mathoverflow.net/users/4832 | 194243 | 94,796 |
https://mathoverflow.net/questions/194248 | 2 | If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a *clopen partition cover* if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally finite.
Is there a [paracompact](http://en.wikipedia.org/wiki/Paracompact_space) space $(X,\tau)$ such that
* $(X,\ta... | https://mathoverflow.net/users/8628 | Paracompact zero-dimensional space without clopen partition refinement | I claim that there are such counterexamples. Any Hausdorff space where every open cover can be refined by a partition into clopen sets is called ultraparacompact. In the paper Not every O-dimensional realcompact space is N-compact by Peter Nyikos, he shows that a certain zero-dimensional metrizable space is paracompact... | 4 | https://mathoverflow.net/users/22277 | 194250 | 94,797 |
https://mathoverflow.net/questions/194244 | 10 | Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full subcategory. However, there are serious set-theoretic difficulties: this is not a category, it is too large. One possible solut... | https://mathoverflow.net/users/2841 | Is the functor of points of a scheme cofinally small? | Maybe I'm missing something, but it seems to me like the first question is just a simple definition chase. A morphism $\mathrm{Spec}(A)\to X$ is determined by finitely many affine open subsets $\mathrm{Spec}(B\_n) \subseteq X$ and some ring maps $B\_n\to A\_{f\_n}$ to localizations of $A$ satisfying certain compatibili... | 6 | https://mathoverflow.net/users/75 | 194255 | 94,800 |
https://mathoverflow.net/questions/194173 | 3 | Theorem : Let $A$ be an abelian varieties of dimension $d$ over a field $k$, non-archimedian valued complete, i.e. $\mathbb{Q}\_p$, then $A(k)$ contains a subgroup of finite index analytically isomorphic and homeomorphic to $I \oplus I \oplus \ldots \oplus I$ ($d$ summands) where $I$ is the ring of integers of $k$.
... | https://mathoverflow.net/users/59151 | Abelian varieties over $p$-adic fields |
>
> *Answered in the comments by user74230:*
>
>
>
You omitted the necessary hypothesis that $k$ has characteristic $0$. Over any such field, Serre's book *"Lie groups and Lie algebras"* proves something much more general: every analytic group manifold over such $k$ has a canonically associated Lie algebra (vani... | 6 | https://mathoverflow.net/users/21815 | 194259 | 94,801 |
https://mathoverflow.net/questions/194264 | 5 | Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c\_1})$ inequalities, where $c\_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking for $P$ and $M$ such that the polyhedron $M(P)$ has $\mathcal{O}({c\_2}^n)$ vertices, where $c\_2$ is a constant.
Ar... | https://mathoverflow.net/users/56325 | Examples of Polyhedra with Large Shadows | If I understood your question right, you are looking for a polygon that has a small so-called *extended formulation*.
This problem is studied for example here:
<http://arxiv.org/abs/1107.0371>.
| 5 | https://mathoverflow.net/users/955 | 194268 | 94,803 |
https://mathoverflow.net/questions/194218 | 10 | *All rings are assumed to have unity.*
Let $k$ be a field. Recall the definition of Grothendieck's ring of ($k$-linear) differential operators $D(R;k)$ of a commutative $k$-algebra $R$:
>
> **Definition.** Set $D\_0(R;k)$ to be $R$ viewed as a subring of $\operatorname{End}\_k(R)$ via multiplication, then for al... | https://mathoverflow.net/users/36720 | Ring of differential operators of a quotient ring | I believe the statement is wrong. Here is a counterexample: Let $R=k[x,y]/(xy)$ (the algebra of polynomial functions on a cross) and let $I$ the ideal generated by $y$ in $R$. Then the quotient $S$ is $k[x]$ (affine line) on which we have the standard vector field $\partial\_{x}$. I claim that there is no extension of ... | 7 | https://mathoverflow.net/users/745 | 194269 | 94,804 |
https://mathoverflow.net/questions/194258 | 3 | I've recently encountered the following problem. Given a group $G$, a subgroup $H$ and a sequence $g\_n\in G$, let $$ \liminf\_{j\to\infty}H^{g\_j} :=\bigcup\_{n\ge 1} \bigcap\_{j\ge n} H^{g\_j}.$$ Here $$ H^g=g^{-1}Hg$$ denotes conjugation by the element $g$. The question is whether or not the above subgroup, denote i... | https://mathoverflow.net/users/57021 | Limits of conjugated subgroups | The answer is negative in general, even if you restrict to finitely generated subgroups. Indeed, in a hyperbolic group a f.g. subgroup can be conjugate to a proper subgroup of itself. For example, let $H$ be the free group of rank $2$ and let $\phi:H \to H$ be a non-surjective monomorphism. Define the group $G$ to be t... | 4 | https://mathoverflow.net/users/7644 | 194282 | 94,807 |
https://mathoverflow.net/questions/194279 | 5 | I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$). More concretely, it would be very useful to know:
* The image of the basis elements $e\_k$
* The image of the volume... | https://mathoverflow.net/users/62367 | Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$ | Here's a standard explicit formula: Let $\mathbb{O}\simeq\mathbb{R}^8$ denote the algebra of octonions, and for $x\in\mathbb{O}$, let $L\_x$ (respectively $R\_x)$ denote the linear map from $\mathbb{O}$ to itself generated by left (respectively, right) multiplication by $x$ and let $C:\mathbb{O}\to\mathbb{O}$ be conjug... | 12 | https://mathoverflow.net/users/13972 | 194287 | 94,810 |
https://mathoverflow.net/questions/194276 | 2 | Let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$ or simply $\vert f(w)\vert=1$, show that $\vert \frac{w}2 f(\frac{w}2)\vert\leq1$. Here, $w$ is a complex number.
What happens if we give the problem in its full generality? That is, let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$... | https://mathoverflow.net/users/66131 | polynomial inequality in complex variable (generalized) | If $w = x + i y$, the curve $|f(w)|^2=1$ can be written as $Q(x,y) = 1$ where
$$ Q(x,y) = \dfrac{{x}^{6}}{36}+\dfrac{{x}^{4}{y}^{2}}{12}+\dfrac{{x}^{2}{y}^{4}}{12}+\dfrac{{y}^{6}}{36}+
\dfrac{{x}^{4}}6-\dfrac{{y}^{4}}{6}+\dfrac{{x}^{3}}{9}-\dfrac{x{y}^{2}}{3}+\dfrac{{x}^{2}}{4}+\dfrac{{y}^{2}}4+\dfrac{x}{3}+\dfrac{1}{9... | 5 | https://mathoverflow.net/users/13650 | 194294 | 94,811 |
https://mathoverflow.net/questions/194297 | 10 | Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $P$.
Is it true that there is a dual cell structure, also a regular CW complex, and whose face poset is the opposite (... | https://mathoverflow.net/users/4558 | Dual cell structures on manifolds | [This text](http://www.math.ucla.edu/~cm/tc.pdf) gives some details explaining that even with triangulations this may not work: a construction of a non-combinatorial triangulation (as double suspension of a homology sphere) and further references.
| 9 | https://mathoverflow.net/users/44953 | 194298 | 94,812 |
https://mathoverflow.net/questions/194290 | 6 | I seem to recall that (not necessarily closed) two-sided ideals of $B(H)$ are hereditary. Is that true?
If it is, can anyone post a proof/reference?
| https://mathoverflow.net/users/3698 | Two-sided ideals of $B(H)$ are hereditary | **Yes**, this is true and it's actually due to Calkin himself. See Theorem 1.6 in
>
> J. W. Calkin, [Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space](http://www.jstor.org/stable/1968771?seq=1#page_scan_tab_contents), *Annals of Mathematics* Second Series, **42**, No. 4 (Oct., 1941... | 6 | https://mathoverflow.net/users/15129 | 194300 | 94,814 |
https://mathoverflow.net/questions/194296 | 7 | 1. (Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. The question is, can we detect if $Y$ is reduced as a subscheme on generic points of $Y$? More precisely, assume there ... | https://mathoverflow.net/users/1508 | Is being reduced a generic property of schemes? | I don't think *normal* is quite enough, but something similar should work. So,
* *normal* is equivalent to $S\_2$ and $R\_1$
* *reduced* is equivalent to $S\_1$ and $R\_0$
If $Y$ is a regular hypersurface in something normal and it is not entirely singular which follows from you generically reduced assumption, then... | 8 | https://mathoverflow.net/users/10076 | 194301 | 94,815 |
https://mathoverflow.net/questions/194315 | 8 | Let $u$ be a smooth function on the unit circle $S^1$ such that $\int\_{S^1}ux\_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
| https://mathoverflow.net/users/42326 | Number of critical points of smooth functions on $S^1$ | Yes, in fact the number of critical points of $u$ is at least four.
Expand $u:S^1\to\mathbb R$ as a Fourier series:
$$u(\theta)=a\_0+\sum\_{i=1}^\infty a\_j\cos(j\theta)+b\_j\sin(j\theta)$$
Then your condition may be stated as $a\_1=b\_1=0$ (I assume by $x\_1$ and $x\_2$, you mean to be identifying $S^1$ with $\{x\_1... | 13 | https://mathoverflow.net/users/35353 | 194318 | 94,821 |
https://mathoverflow.net/questions/194323 | 3 | Let $k\subset L$ be an extension of fields of characteristic zero.
Suppose that $X/k$ is an algebraic space such that $X\otimes\_k L$ is representable by a finite type $L$-scheme.
I am sure there are examples where $X/k$ is not representable. I think Mumford's example of the Picard scheme of a degenerating conic ov... | https://mathoverflow.net/users/66144 | What if the base change of an algebraic space is representable | In general $X$ is not a scheme, even if it is proper. But if $X\otimes\_k L$ is a *quasiprojective* scheme, so is $X$.
First, assume $X\otimes\_k L$ quasiprojective. By standard techniques, this still holds for some finite $L/k$. Then the projection $X\otimes\_k L\to X$ identifies $X$ with the quotient (as fppf shea... | 10 | https://mathoverflow.net/users/7666 | 194326 | 94,824 |
https://mathoverflow.net/questions/194338 | 3 | For any graph $G=(V,E)$ we define $\mu(G) = \sup\{|M|: M\subseteq E(G) \text{ is a matching}\}$.
Is there a graph $G=(V,E)$ such that for every matching $M\subseteq E$ we have $|M|<\mu(G)$?
| https://mathoverflow.net/users/nan | Maximum matchings in infinite graphs | Yes. First start off with the disjoint union $G=\bigcup\_{n\in\mathbb{N}}K\_n$ where $K\_n$ is the complete graph on $\{1,\ldots,n\}$ for $n\in\mathbb{N}, n\geq 1$.
Note that this graph contains cliques of size $n$ for every $n\in\mathbb{N}$, but it does not contain a clique of size $\omega$.
Let $G^c$ be the comp... | -1 | https://mathoverflow.net/users/8628 | 194339 | 94,825 |
https://mathoverflow.net/questions/193446 | 12 | This question is inspired by Pace Nielsen's recent question [Does a left basis imply a right basis, without AC?](https://mathoverflow.net/questions/189090/does-a-left-basis-imply-a-right-basis-without-ac).
For any field $k$, the field $k(x)$ of rational functions in one variable has an explicit $k$-basis given by par... | https://mathoverflow.net/users/22989 | Does k(X) have a k-basis for every set X, without AC? | I will show that, if $k$ is countable and has characteristic $0$, then $k(X)$ has a basis. (This whole proof takes place without choice.)
For any $x \in X$, and any $f \in X$, we define an element $r\_x(f)$ of $k(X \setminus \{ x \})$ as follows: We have $k(X) \cong k(X \setminus \{ x \})(x)$, which embeds into the f... | 6 | https://mathoverflow.net/users/297 | 194349 | 94,828 |
https://mathoverflow.net/questions/194333 | 17 | Consider the [object classifier](http://ncatlab.org/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos) of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the [type of types](http://ncatlab.org/nlab/show/type+of+types), let’s denote it as $Type = \coprod\_{[F]... | https://mathoverflow.net/users/447 | What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras? | Let $\mathcal{S}$ denote the $\infty$-category of spaces. For any $\infty$-topos $\mathcal{X}$, there is an essentially unique geometric morphism $\pi^{\ast}: \mathcal{S} \rightarrow \mathcal{X}$. The $\infty$-topos $\mathcal{X}$ has the form $\mathcal{S}\_{/A}$ if and only if the geometric morphism $\pi^{\ast}$ is eta... | 20 | https://mathoverflow.net/users/7721 | 194354 | 94,830 |
https://mathoverflow.net/questions/194337 | 18 | Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology class. (The $\mathrm{CP}^2$ case was proven by Kronheimer and Mrowka '94, the symplectic case by Ozsvath and Szabo '00.)
... | https://mathoverflow.net/users/64043 | Thom conjecture in CP3 | This question was addressed in higher dimensions by Mike Freedman, Surgery on codimension 2 submanifolds. Mem. Amer. Math. Soc. 12 (1977), no. 191. (I think this was his PhD thesis.) Earlier work of Thomas and Wood (On manifolds representing homology classes in codimension 2.
Invent. Math. 25 (1974), 63–89) had given s... | 18 | https://mathoverflow.net/users/3460 | 194356 | 94,831 |
https://mathoverflow.net/questions/194329 | 7 | Let $X = SL\_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi\_j$ on $X$ with $-\Delta$-eigenvalue $\lambda\_j$. Hejhal-Rackner (1991, on the Topography of Maass forms) seems to claim that for any fixed set $A \subset X$ with finite measure,... | https://mathoverflow.net/users/31814 | Mean value of Maass forms | This is an analog of the Riemann-Lebesgue lemma. Let $f$ be a smooth function on $X$. Then by self-adjointness of the Laplacian
$$
\lambda\_j \langle \phi\_j, f \rangle = \langle \Delta \phi\_j ,f \rangle = \langle \phi\_j, \Delta f\rangle ,
$$
and by Cauchy-Schwarz this is bounded in size by $\Vert \phi\_j \Vert \V... | 9 | https://mathoverflow.net/users/38624 | 194357 | 94,832 |
https://mathoverflow.net/questions/194358 | 5 | **EDIT:** Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in X$;
(c) for any holomorphic germ $\gamma\colon (\mathbb{C},0)\to Y$ with image contained in $f(X)$ there exists ... | https://mathoverflow.net/users/16183 | A weak analytic version of the valuative criterion of properness | Here is a counterexample: $Y=\mathbb C^2$, $X=(\mathbb C\setminus\{0\})\cup\mathbb C$, and $f$ is given by $z\mapsto (z,e^{1/z})$ on $\mathbb C\setminus\{0\}$ and is given by $w\mapsto(0,w)$ on the second copy of $\mathbb C$. This map is certainly not proper. It clearly satisfies conditions (a),(b),(d). Let's check con... | 4 | https://mathoverflow.net/users/35353 | 194365 | 94,836 |
https://mathoverflow.net/questions/194322 | 6 | Let $H$ be a separable Hilbert space and $N(0, C)$ and $N(0, D)$ be Gaussian measures on it. Further, for each $v \in H$, define $R\_v = \frac{\left\langle v,Cv \right\rangle}{\left\langle v,Dv \right\rangle}$. Basically, $R\_v$ is the ratio of Covariance of one dimensional Gaussian measures induced by $N(0, C)$ and $N... | https://mathoverflow.net/users/65922 | Equivalence of Gaussian measures | They are not equivalent.
For an explicit counterexample, let $\{e\_1, e\_2, \dots\}$ be an orthonormal basis for $H$, and let $C$ be the diagonal operator $C e\_n = \frac{1}{n^2} e\_n$. Let $D =2C$. Then $R\_v = 1/2$ for every $v$ so (2) is satisfied.
Define random variables $X\_n$ on $H$ by $X\_n(v) = {n} \langle ... | 7 | https://mathoverflow.net/users/4832 | 194375 | 94,838 |
https://mathoverflow.net/questions/194380 | 1 | I need to provide computational complexity for the algorithms in my work. One of the algorithms I have used is Golden Section method for line search. I took a look at "Nonlinear Programming" book by Bertsekas, but it did not mention the computation complexity. I have found from a paper which also has used Golden Sectio... | https://mathoverflow.net/users/64667 | Computation Complexity for Golden Section method | See [E. Bertolazzi's Lecture notes.](http://www.ing.unitn.it/~bertolaz/2-teaching/2011-2012/AA-2011-2012-OPTIM/lezioni/slides-m1D.pdf)
| 0 | https://mathoverflow.net/users/11142 | 194382 | 94,843 |
https://mathoverflow.net/questions/194370 | 4 | I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
* the first one is the adjoint representation on $S$ itself, we write it: $s \in S\mapsto A(s) \in GL(S)$
* the second one is on a given vector space $V$, we ... | https://mathoverflow.net/users/66132 | An algorithm to compare two representations of a simple Lie algebra? | In general, we can write down the polynomial equations for two $S$-modules being isomorphic. By considering characteristic polynomials of similar matrices one can sometimes get necessary conditions which simplify the equations. Also, the non-vanishing of the determinant is useful (the morphism is bijective).
If the dim... | 1 | https://mathoverflow.net/users/32332 | 194384 | 94,844 |
https://mathoverflow.net/questions/194265 | 8 | One recent proof of quadratic reciprocity involves computing various rations of the Gauss sum.
---
In [**Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation**](http://arxiv.org/abs/0808.2447) Gurevich, Hadani and Howe discuss the [Gauss sums](https://en.wikipedia.org/wiki/Gauss... | https://mathoverflow.net/users/1358 | higher reciprocity theorems from ratios of Gauss sums | If you look at bit further in the preprint you will see that the authors also state the generalisation
\begin{equation\*}
\frac{G\_{n\_1} G\_{n\_2}}{G\_{n\_1 n\_2}} = \left(\frac{n\_1}{n\_2}\right) \left(\frac{n\_2}{n\_1}\right)
\end{equation\*}
for any coprime odd numbers $n\_1$, $n\_2$. From this it easy to see that ... | 7 | https://mathoverflow.net/users/65801 | 194386 | 94,846 |
https://mathoverflow.net/questions/194390 | 0 | I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant find any useful literature how this works. Especially, I am confused since the Lie algebra seems to always be defined arou... | https://mathoverflow.net/users/64029 | cartan killing metric | Any inner product on $\mathfrak{g}$ will give rise to a metric on $G$. More specifically, you can use a left trivialization of $TG$ to get a vector bundle isomorphism $TG\cong G\times\mathfrak{g}$. The latter bundle has a metric by virtue of the inner product on $\mathfrak{g}$.
The same construction will work if one ... | 1 | https://mathoverflow.net/users/25358 | 194391 | 94,848 |
https://mathoverflow.net/questions/194360 | 5 | Let $S$ be a compact orientable surface and $p\_1,\dots, p\_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p\_1,\dots, p\_n$.
Is there an algorithm which takes two triangulations and produces a series of flips which transforms one triangulation into another?
WHAT I KNOW:
It is know... | https://mathoverflow.net/users/41487 | How to flip one triangulation on a surface into another | Mosher's proof is, I think, fine. You should re-read the definition of $i(\delta, \{h\})$ given on line -14 of page 38 of Mosher's paper.
The sequence of flips produced by Mosher's algorithm is at most linear in the total intersection number. This worst-case bound is realized when the two triangulations differ by a ... | 5 | https://mathoverflow.net/users/1650 | 194397 | 94,849 |
https://mathoverflow.net/questions/194404 | 2 | Let a formula $\phi$ of the language of first-order Peano arithmetic be total in a theory Th that extends PA iff, for any $k\_1, \dots, k\_n \in \omega$, Th $\vdash \phi(\bar k\_1, \dots, \bar k\_n)$ or Th $\vdash \neg \phi(\bar k\_1, \dots, \bar k\_n)$. Is it true that total formulae in Th are provably equivalent in T... | https://mathoverflow.net/users/66109 | Total formulae in a theory equivalent to $\Delta_0$ formulae in the theory? | No.
Certainly the statement is false for arbitrary theories.
For computably axiomatizable theories, it's still false; what follows is informal. Consider a uniformly computably axiomatizable sequence of theories $T\_i$ which are *independent*: if I have a finite set $\{\varphi\_i: i<n\}$ of formulas with each $\var... | 5 | https://mathoverflow.net/users/8133 | 194406 | 94,852 |
https://mathoverflow.net/questions/162823 | 2 | Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution involution $σ$ in $Aut(G)$. Thus $σ$ is an automorphism of $G$ with $σ^2 = id$ and $H$ is an open subgroup of the set
$G^\... | https://mathoverflow.net/users/nan | Cotangent bundle of symmetric space is symmetric space? | The answer is yes. See this paper <http://arxiv.org/abs/0710.1543> (Introduction, and section 4.2 page 14). The basic ideas are that
* if $M$ is a symmetric space, then $TM$ has a canonical structure of symmetric space (whatever the way you realize your symmetric space, apply the tangent functor to it);
* the dualit... | 0 | https://mathoverflow.net/users/32864 | 194421 | 94,856 |
https://mathoverflow.net/questions/194134 | 4 | I noticed that many results in positive characteristic assumes that the object of the theorem is *excellent*. I have looked up the definition of excellent and have tried to get a feeling for it, but all I could really surmise was that it is nearly harmless to assume that a scheme is excellent as most schemes one natura... | https://mathoverflow.net/users/66066 | Excellent schemes | An excellent reference for questions as yours is Grothendieck's EGA IV.7.8.
| 1 | https://mathoverflow.net/users/11025 | 194424 | 94,858 |
https://mathoverflow.net/questions/194418 | 4 | Let $u\_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u\_0(x)$. How one can apply the maximum principle to prove that the number of critical points of $u(t,x)$ at any given time is not bigger than the number of critical poin... | https://mathoverflow.net/users/42326 | Heat equation and evolution of number of critical points | Since $u$ is nothing but a $2\pi$-periodic solution of $u\_t=u\_{xx}$, looking at critical points amounts to looking at the zeroes of $v:=u\_x$, which is another solution of the same equation. Then your question is solved by H. Matano : Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabo... | 7 | https://mathoverflow.net/users/8799 | 194433 | 94,862 |
https://mathoverflow.net/questions/194437 | 10 | Let $X$ be an algebraic variety over $\mathbb{C}$. If $X$ is smooth, the étale cohomology $H^p\_{\textrm{ét}}(X,\mathbb{Z}/n)$ is isomorphic to the singular cohomology $H^p(X(\mathbb{C}),\mathbb{Z}/n)$. What is the situation if $X$ is not smooth? Are there counter-examples?
| https://mathoverflow.net/users/40297 | Étale cohomology versus classical cohomology | In Katz's review of $\ell$-adic cohomology in the first "Motives" volume, this isomorphism is stated without any smoothness assumptions, with a reference to SGA4, XVI 4.1 (which I don't have easily available).
| 6 | https://mathoverflow.net/users/1310 | 194441 | 94,865 |
https://mathoverflow.net/questions/194442 | 14 | The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when it is possible to avoid using Zorn's lemma, in more simple settings (outside of $\mathbb{R}^n$), because for example, in ... | https://mathoverflow.net/users/56191 | Krein Milman theorem without the axiom of choice | The solution is in the comment of Asaf Karagila, which points to the paper ["A geometric form of the axiom of choice"](http://matwbn.icm.edu.pl/ksiazki/fm/fm77/fm77116.pdf) by Bell and Fremlin.
In fact, this paper asserts that the assertion : "For every normed vector space $X$ over the reals, there exists at least on... | 7 | https://mathoverflow.net/users/56191 | 194445 | 94,867 |
https://mathoverflow.net/questions/162075 | 4 | Given any finite group with multiplication $m(-,-)$ and three permutations $p,q,r$ on the underlying set of the group, we can obtain a quasigroup with binary operation $g\*h:= p(m(q(g),r(h)))$. What is an example of a finite quasigroup that provably does not arise in this way?
| https://mathoverflow.net/users/799 | Finite quasigroup not coming from a scrambled finite group | The construction is called "isotopy" and you ask for quasigroups that are not "isotopic to groups".
It's been proved a long time ago that if a loop (= a quasigroup with a unit element) is isotopic to a group G, it is actually isomorphic to G. Consequently, any non-associative loop is an answer to your question. The ... | 7 | https://mathoverflow.net/users/66148 | 194450 | 94,869 |
https://mathoverflow.net/questions/194454 | 11 | It is referenced that in
Chowla, S., *Proof of a conjecture of Julia Robinson*, Norske Vid. Selsk. Forh. (Trondheim) 34, 100–101 (1961),
it is shown that the equation $\epsilon\_1 + \epsilon\_2 = 1$ has only finitely many solutions, when $\epsilon\_1, \epsilon\_2$ are units of a given number field. Unfortunately, ... | https://mathoverflow.net/users/10591 | An old paper of S.Chowla on unit equations | This is a standard result, and is true for $S$-units. It is proven in many texts, although sometimes the reader is referred elsewhere for the estimate from Diophantine approximation that is needed (such as Thue's or Roth's theorem). The full proof is contained in my book with Hindry, *Diophantine Geometry*, GTM 201, Sp... | 15 | https://mathoverflow.net/users/11926 | 194456 | 94,870 |
https://mathoverflow.net/questions/194434 | 2 | Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
**Question: is it true that if $f$ is onto then $f$ is flat?**
(I am not a specialist, and the answer is not known to me ev... | https://mathoverflow.net/users/16183 | Flatness of a morphism of complex analytic spaces | There are easy counterexamples if you allow "embedded components", e.g. $X\subset D\times\mathbb{C}$ given (in coordinates $z,t$) by the equations $zt=t^2=0$. In the local ring at $x=(0,0)$, the coordinate $z$ from $D$ is a zero divisor, so $\mathscr{O}\_{X,x}$ is not flat over $\mathscr{O}\_{D,0}$.
On the other han... | 4 | https://mathoverflow.net/users/7666 | 194459 | 94,871 |
https://mathoverflow.net/questions/194448 | 2 | Let $X\neq\emptyset$ be a set. A *partition* is a subset $P\subseteq {\cal P}(X)\setminus \{\emptyset\}$ such that $\bigcup P = X$ and any distinct members of $P$ are disjoint. We denote by $\text{Part}(X)$ the set of all partitions of $X$.
We order $\text{Part}(X)$ by the [refinement relation](http://en.wikipedia.or... | https://mathoverflow.net/users/8628 | Infimum of partitions | **Counterexample:** Let $X=[0,1)$; let $\mathcal C=\{C\_n:n\in\mathbb N\}$ where $C\_n=\{[\frac{k-1}{2^n},\frac k{2^n}):1\le k\le2^n\}$;and let $P\_0=\{[\frac{n-1}n,\frac n{n+1}):n\in\mathbb N\}$.
| 4 | https://mathoverflow.net/users/43266 | 194465 | 94,874 |
https://mathoverflow.net/questions/194469 | 3 | One can see that algebraic numbers are dense in the complex plane by just looking at quadratic polynomials. I am interested in a "higher order" density of algebraic numbers.
More specifically: is it known that if $D\_1,\ldots, D\_n \subset \mathbb{C}$ are disjoint disks, then there is an irreducible polynomial in $\... | https://mathoverflow.net/users/11214 | Density of tuples of conjugate algebraic numbers | The idea by Yaakov Baruch works. Take any $q\_i\in D\_i\cap \mathbb Q[i]$ and take the polynomial
$$
g(x)=\prod\_{i=1}^n (x-q\_i)(x-\overline q\_i).
$$
It has rational coefficients.
If a polynomial $f(x)$ of degree $2n$ differs coefficientwise sufficiently small from $g(x)$, then $f(x)$ has a root in each $D\_i$ --... | 4 | https://mathoverflow.net/users/17581 | 194473 | 94,877 |
https://mathoverflow.net/questions/194479 | 3 | Assume we have an Abelian varieties over the p-adic numbers, namely $
k=\mathbb{Q}\_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold.
I am reading a Book by Serre "Lie Algebras and Lie groups". i took the definition of Analytic manifolds from this text, I gu... | https://mathoverflow.net/users/59151 | Abelian varieties as analytic manifolds | Yes, this is true. For any smooth finite type $k$-scheme $X$, $k$ a non-Archimedean field, $X(k)$ has a canonical structure of $k$-analytic manifold in the sense of Serre's book (or Bourbaki). Usually nowadays one calls these *locally $k$-analytic manifolds*. I think this can be proved using "standard smooth affine ope... | 7 | https://mathoverflow.net/users/4351 | 194481 | 94,880 |
https://mathoverflow.net/questions/194453 | 6 | The title is my question.
Alexandrov space here means finite dimensional Alexandrov space with curvature bounded below ,denoted by CBB.
Let $\gamma$ be a simple curve in a $n$ dimensional CBB $M$ with two end points in the manifold part of $M$.
**Analysis**.
$\gamma$ can be covered by a finite collection $\mathc... | https://mathoverflow.net/users/62195 | Whether the manifold part of an Alexandrov space is connected? | Yes.
Assume $A$ is an $m$-dimensional Alexandrov space and $\Omega\subset A$ be the maximal open subset which is a topological $m$-manifold and $A'\subset A$ be the subset of all points with tangent space isometric to Euclidean space.
From Perelmans paper "Beginning of Morse...", we get that $A'\subset \Omega$.
T... | 8 | https://mathoverflow.net/users/1441 | 194493 | 94,884 |
https://mathoverflow.net/questions/194350 | 7 | Let $\gamma$ be a continuous curve in the complex plane without self-intersections and let $\lambda$ be a complex non-real number less than 1 in modulus. Put $\gamma'=\lambda\gamma$.
**Question.** Is it true that the sumset $\gamma+\gamma'$ has a non-empty interior?
| https://mathoverflow.net/users/8131 | The Minkowski sum of two curves | Well, I cannot call it a complete answer, but it seems an idea that can work.
First: take any loop in the (s,t) coordinates, and consider its image on the complex plane under $\gamma(s)+\gamma'(t)$. Any point having nonzero index w.r.t. this loop is then contained in $\gamma+\gamma'$.
Second: take the loop to be th... | 3 | https://mathoverflow.net/users/31371 | 194494 | 94,885 |
https://mathoverflow.net/questions/194490 | 4 | Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H\_2(X,\mathbb Z).$$
I have a very vague question about this situation:
>
> **Q**. If $C$ is *rigid* in $X$, how far is this condition from $C$ being the
> unique curve on $X$ in class $\beta$?
... | https://mathoverflow.net/users/30827 | Rigid curves, and the "richness" of their homology class | Let me spell out examples to keep in mind in both directions. I'll leave it to someone more clueful to give a better general answer.
1) Rigidity does not imply uniqueness in homology. Alex Degtyarev gave an example on ruled surfaces. Here's another. Let $C$ and $D$ be two curves in the same plane in $\mathbb P^3$, me... | 9 | https://mathoverflow.net/users/nan | 194496 | 94,887 |
https://mathoverflow.net/questions/194498 | 2 | Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group structure, number) of rational points $X(\mathbb{Q})$. Can some one explain this relationship ? Thanks
| https://mathoverflow.net/users/66195 | rational point of a curve | Not sure if this is really MathOverflow level, but:
(A) genus X is 0 <==> X(C) is a sphere <==> X(Q) is either empty or looks like P^1(Q)
(B) genus X is 1 <==> X(C) is a 1-holed torus <==> X(Q) is either empty or a finitely generated abelian group (Mordell's theorem)
(C) genus X is g > 1 <==> X(C) is a g-holed to... | 9 | https://mathoverflow.net/users/11926 | 194500 | 94,889 |
https://mathoverflow.net/questions/194491 | 9 | Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D\_T$ as follows:
>
> Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether
> $$
> T \vdash \forall \bar{x}\, p(\bar{x}) \neq 0,
> $$
> i.e., whether $T$ shows that $p$ does not have a root.... | https://mathoverflow.net/users/48332 | Decidability of diophantine equation in a theory | If $T$ contains $I\Delta\_0+\mathit{EXP}$ (or a strong theory like Peano arithmetic for that matter), then by the formalized MRDP theorem [1], $D\_T$ is essentially the $\Pi^0\_1$ fragment of $T$, which is of course undecidable if $T$ is consistent. Curiously, this holds even for some weaker theories that do *not* prov... | 13 | https://mathoverflow.net/users/12705 | 194502 | 94,891 |
https://mathoverflow.net/questions/194509 | 2 | in this paper [on arxiv](http://arxiv.org/pdf/1207.4985v1.pdf) in equation 27, two operators
$$A\_m^\* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A\_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}}$$
belonging to Legendre's ODE are defined. Unfortunately, a spectral a... | https://mathoverflow.net/users/66201 | Proper domain for operators | Minimal and maximal operators will work fine: Define an operator on $C\_0^{\infty}(-1,1)$, using the differential expression of $A$, and then take the operator closure to obtain $A$. Then $A^\*$ will be the corresponding maximal operator, that is,
$$
D(A^\*) = \{ u\in L^2(-1,1)\cap AC(-1,1): (1-x^2)^{1/2}Du + v(x)u \in... | 1 | https://mathoverflow.net/users/48839 | 194512 | 94,894 |
https://mathoverflow.net/questions/194513 | 3 | First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem.
>
> The description of my problem is the following:
>
>
>
Sample a point $x$ uniformly at random from the solutions satisfying linear **equalities** ... | https://mathoverflow.net/users/6146 | Sampling point uniformly at random satisfying equality constraints | The equality constraints determine an affine subspace of $\mathbb R^n$. After suitable change of variables, this becomes $\mathbb R^m$ for suitable dimension $m$, and the nonnegativity of the entries of your probability distribution gives you a convex polytope. So it is exactly the case of uniformly sampling from conve... | 4 | https://mathoverflow.net/users/13650 | 194515 | 94,895 |
https://mathoverflow.net/questions/194267 | 21 | Let $X, Y$ be normed space and $f:X\to Y$ be a mapping. Assume that for all $n\in\mathbf{N}$, $$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$
Under what conditions this map will be an isometry?
Thanks
| https://mathoverflow.net/users/52860 | Under what conditions $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$ implies isometry? | So it turns out my earlier intuition was incorrect, and one can leverage the order properties of ${\bf R}$ to show:
>
> **Theorem**. Let $X, Y$ be real Hilbert spaces, with the dimension of $X$ at least two, and let $f: X \to Y$ be a function such that $\|f(x)-f(y)\|=\|x-y\|$ whenever $\|x-y\|$ is a natural number.... | 24 | https://mathoverflow.net/users/766 | 194516 | 94,896 |
https://mathoverflow.net/questions/194489 | 1 | In the [paper](https://eudml.org/doc/212787) *[Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234*] it is considered the limit
$$
\lim\_{\delta(I)\to0}|I|^{-1}\int\_If(P)\,dP,\qquad\qquad\qquad (1)
$$
where an integrable funct... | https://mathoverflow.net/users/14551 | A question on the Lebesgue differentiation theorem | The form of those collections $\{I\}$ for which the LDTh holds, are studied in several textbooks on Geometric Measure Theory. However, the usual assumption is *bounded eccentricity* (which does not seem to be sufficient here), and I don't have reference handy for Theorem 6. Yet "bounded eccentricity" + "Lebesgue differ... | 3 | https://mathoverflow.net/users/6101 | 194519 | 94,898 |
https://mathoverflow.net/questions/194503 | 4 | The paper [Fusion rules and modular transformations in 2D conformal field theory](http://www.sciencedirect.com/science/article/pii/0550321388906037) by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just $(\mathbb{Z}\_p,+)$:
$$ \phi\_p \times \phi\_{p'} = \phi\_{p... | https://mathoverflow.net/users/1358 | Verlinde Formula and Theta Function Identities | For any integer lattice $L$, you can write a theta function $\theta\_L$ as a generating function for lattice vectors of a given norm. That is,
$$\theta\_L(\tau) = \sum\_{a \in L} q^{(a,a)/2}.$$
The quotient $\theta\_L(\tau)/\eta(\tau)^{\textrm{rank} (L)}$ is the partition function for the lattice model vertex algebra (... | 7 | https://mathoverflow.net/users/121 | 194524 | 94,900 |
https://mathoverflow.net/questions/186682 | 9 | **Hypothesis:** Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$
**Question:** Is it true that $\Gamma$ must either be a complete graph or have prime order?
---
**Terminology and notation:**
* By $N(v)$, I mean the set of neighbours of $v$ in $\Ga... | https://mathoverflow.net/users/22377 | Vertex-primitive graphs with two vertices having almost the same neighbourhood | Pablo Spiga found a proof a few weeks ago.
Together, we then proved a slightly more general result which is now on the arxiv:
<http://arxiv.org/abs/1501.05046>
It is more general in two ways: it deals with digraphs rather than graphs, and it gives some information in general when two vertices have neighbourhoods diff... | 3 | https://mathoverflow.net/users/22377 | 194530 | 94,903 |
https://mathoverflow.net/questions/155347 | 6 | A question about permutation groups: I wonder if someone
who is expert in permutation group theory could answer the
following question.
Let $x \in S\_n$ (the symmetric group) be an involution which
is the product of k disjoint transpositions.
For any permutation $y \in S\_n$, let f(y) be the length of a
shortest **... | https://mathoverflow.net/users/39684 | Permutation Group Question | This is not a complete answer but maybe more an extended comment.
First, it is not hard to find an upper bound that does not depend on $n$, as you claimed.
Note that $x$ has support $2k$ and hence so does $x^y$. This implies that $xx^y$ has support at most $4k$ hence is contained in some copy of $S\_{4k}$. Since the ... | 2 | https://mathoverflow.net/users/22377 | 194535 | 94,907 |
https://mathoverflow.net/questions/193998 | 4 | I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph without diagonalizing it. The graphs I am interested in are not regular (but they have a maximum degree) or bipartite. They may or may not be planar.
1. Are there any bounds for either of the ext... | https://mathoverflow.net/users/66010 | Extremal eigenvalues & eigenvectors of skew-adjacency matrix | 1. Are there any bounds for either of the extremal eigenvalues of the skew-adjacency matrix?
Yes. In particular, the extremal eigenvalue bounds the "asymmetry of arcs" between large subsets of vertices. See for example, "Discrepancy Inequalities for Directed Graphs", Discrete Applied Mathematics, 176 (2014), pp. 30-4... | 3 | https://mathoverflow.net/users/66216 | 194536 | 94,908 |
https://mathoverflow.net/questions/194466 | 1 | Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor which is not necessarily reduced).
Now we assume that $(A/\mathfrak q)\_{\sqrt{\mathfrak q}}$ is regular.
Under these... | https://mathoverflow.net/users/64020 | Reducedness of a ring with prime nilradical | A Noetherian ring is reduced if and only if it is $(R\_0)$ and $(S\_1)$. (See, for example, <http://stacks.math.columbia.edu/tag/031R> ) This condition can be used to provide a simple proof that your ring $A/\mathfrak{p}$ is reduced.
A ring $R$ is $(R\_k)$ iff $A\_P$ is a regular local ring for all primes $P$ of heig... | 2 | https://mathoverflow.net/users/17218 | 194538 | 94,909 |
https://mathoverflow.net/questions/63375 | 23 | Prompted by James Propp's [recent question](https://mathoverflow.net/questions/63320/where-do-surreal-numbers-come-from-and-what-do-they-mean/63364#63364) about surreal numbers, I was wondering whether anyone had investigated the idea of describing surreal numbers (as ordered class) in terms of a universal property, ro... | https://mathoverflow.net/users/2926 | Surreal Numbers as Inductive Type? | Heh, I just ran across this old question again and realized that I now know an answer: it's in section 11.6 of the [HoTT Book](http://homotopytypetheory.org/book/). It's written out there as a higher inductive-inductive type; in other language it says roughly that $\mathbf{No}$ is the initial object among classes $X$ e... | 11 | https://mathoverflow.net/users/49 | 194540 | 94,910 |
https://mathoverflow.net/questions/194175 | 3 | Puzzled by [this still open question](https://mathoverflow.net/questions/193753), I tried comparing the arithmetic mean $A(x,y)=(x+y)/2$ with a mean intermediate between a geometric-type mean $G(X)=(x^a y^{1-a}+x^{1-a} y^a)/2\;$ for $0\le a \le 1$, and an $L^p$-type mean $L(x,y)=(x^p+y^p)^{1/p}\;$ for $p\ge 1$.
Notic... | https://mathoverflow.net/users/2480 | Is this parametric inequality true? | Let me change the notation a bit by calling $m=\frac{1}{2a-1}>1$. Then your inequality reads
$$\left(\frac{x+y}{2}\right)^{m^2}\geq (xy)^{\binom{m}{2}}\cdot\left(\frac{x^m+y^m}{2}\right).$$
This is true for all $x,y\geq 0$, and I would love to see a slick elementary proof. However in the meantime notice that you can pr... | 6 | https://mathoverflow.net/users/2384 | 194560 | 94,915 |
https://mathoverflow.net/questions/194446 | 10 | Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
**Must $V$ have a solvable point?**
The variety $V$ is assumed geometrically irreducible.
A solvable point is a point in $V(L)$ where $L/K$ is a finite Galois extension with $\mathrm{Gal... | https://mathoverflow.net/users/38889 | Is there a solvable point on any variety over the field of complex rational functions? | No, $V$ need not have a solvable point. The proofs I know construct such $V$ via deformation theory. The basic idea is in my paper.
MR2579389 (2011g:14095) Reviewed
Starr, Jason Michael(1-SUNYS)
A pencil of Enriques surfaces of index one with no section. (English summary)
Algebra Number Theory 3 (2009), n... | 10 | https://mathoverflow.net/users/13265 | 194561 | 94,916 |
https://mathoverflow.net/questions/194582 | 4 | Let $K$ be a local field and $D\_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D\_K$? Proposition 1.1 of [this article](http://www.math.ethz.ch/~pink/ftp/SigmaBd.pdf) describes functions on the punctured open disk as certain "Laurent series" (po... | https://mathoverflow.net/users/3544 | The rigid-analytic open disk | About the first part of your question:
What do you mean by "analytic function"?
If I translate "closed unit disc" by "maximal spectrum of the Tate algebra" (but they are not the same), then the open unit disc is the union of the closed discs with radius smaller than one. Let $A\_\varepsilon$ denote the algebra of a... | 9 | https://mathoverflow.net/users/62434 | 194589 | 94,925 |
https://mathoverflow.net/questions/194461 | 0 | Let $U$ be an open subset of $\mathbb{R}^n$ such that
$\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice''
(for simplicity you can assume piecewise smooth). I also want to allow the possibility of $U$ being the whole of $\mathbb{R}^n$, which means there is no boundary. Let $G(x,y,t)$ be the fundamental... | https://mathoverflow.net/users/4463 | Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)? | As Nate already pointed out, the Brownian motion conditioned to stay inside $D$ up to time $1$ (which is what the limit in "Added later" gives you) will not satisfy (2). I suspect (and this what there's a nice theory for) that you really want to consider Brownian motion conditioned to stay inside $U$ forever. (That's o... | 5 | https://mathoverflow.net/users/38566 | 194593 | 94,926 |
https://mathoverflow.net/questions/194541 | 2 | As pointed out by Makoto, on [this question](https://mathoverflow.net/questions/154469/can-we-prove-that-the-ring-of-formal-power-series-over-a-noetherian-ring-is-noet/194416#194416) about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction r... | https://mathoverflow.net/users/3199 | Matching power series to infinity | I believe the answer to your simpler question is "no:" Fix an infinite sequence of sets $A\_i$ ($i\in\omega$) for which no choice function exists. Now let $A$ be the free commutative ring generated by the set $$(\bigcup A\_i)\cup\{d\_i: i\in\omega\}\cup\{c\},$$ modulo the relations $$ca=d\_i\quad\mbox{ for every }i\in\... | 1 | https://mathoverflow.net/users/8133 | 194597 | 94,929 |
https://mathoverflow.net/questions/194596 | 0 | While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p\_{b}-p\_{a}}{b-a}$ where $a<b$ are positive integers:
$$H\_{m}-\log\_{2} m-2\gamma$$
where $m:=\frac{a+b}{2}$, $\gamma$ is Euler-Mascheroni's constant, $\log\_{2}x=\log\log x$ ... | https://mathoverflow.net/users/13625 | Has this formula about prime gaps already been conjectured and/or proven? | Even the weaker statement that $\frac{p\_b-p\_a}{b-a}$ tends to infinity with $a,b\to\infty$ is false, because there are infinitely many bounded prime gaps (Yitang Zhang's theorem).
| 6 | https://mathoverflow.net/users/11919 | 194598 | 94,930 |
https://mathoverflow.net/questions/194584 | 0 | Let $f=\sum\_{n\geq 1}\in S\_2(\Gamma\_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a\_n\colon n\in \mathbb N)$ be the number field generated by the coefficients of $f$. This is either a totally real field or a CM field, namely a totally imaginary quadratic e... | https://mathoverflow.net/users/36370 | Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$ | No, this is not the case. In general these fields will have very little to do with each other.
For instance, let $\varepsilon$ be the unique character of $(\mathbf{Z} / 17 \mathbf{Z})^\times$ mapping 3 to $i$. Then there is a modular form of level $34$, weight $2$ and nebentypus $\varepsilon$ whose $q$-expansion is
... | 4 | https://mathoverflow.net/users/2481 | 194604 | 94,931 |
https://mathoverflow.net/questions/193397 | 0 | **Question:** Two $k$-dimensional subspaces $W\_1,W\_2$ with associated orthogonal projections $P\_1, P\_2$ are isoclinic with parameter $\lambda \ge 0$ if $P\_1P\_2P\_1=\lambda P\_1$ and $P\_2P\_1P\_2=\lambda P\_2$.
I was able to show that if $W\_1\perp W\_2$, then $W\_1,W\_2$ are isoclinic with $\lambda=0$. Now I'... | https://mathoverflow.net/users/64704 | $\epsilon$-nearly isoclinic | Hey I just found a solution on my own.
Let $a$ be an arbitrary Element of a Hilbert space.
We have
$\Vert P\_2 P\_1 a\Vert^2=\langle P\_2 P\_1a,P\_1a\rangle=\Vert P\_2 P\_1 a\Vert\Vert P\_1a\Vert\langle \frac{P\_2P\_1a}{\Vert P\_2 P\_1 a\Vert},\frac{P\_1a}{\Vert P\_1 a\Vert}\rangle<\epsilon\Vert P\_2 P\_1 a\Ve... | 0 | https://mathoverflow.net/users/64704 | 194616 | 94,935 |
https://mathoverflow.net/questions/194614 | 0 | Suppose that the sequence of r.v $\{X\_{n}\}\_{n\geq 1}$ has all the moments, and $X\_{n}\stackrel{D}{\longrightarrow}X\sim N(0,\sigma)$. Assume that $E\left\{(X\_{n})^{K}\right\} \stackrel{n}{\longrightarrow} E(X^{K})$, where $K\geq 1$ is an integer number. Can we say that $E\left\{(X\_{n})^{K+1}\right\} \stackrel{n}{... | https://mathoverflow.net/users/49357 | convergence in distribution and convergence of moments | No, this is not true. Let $X \sim N(0,1)$ and define $Y\_n$ to be independent of $X$, such that $Y\_n = \sqrt{n}$ with probability $1/n$ and 0 otherwise. Set $X\_n = X+ Y\_n$. Since $Y\_n \to 0$ in $L^1$, we have $X\_n \to X$ in $L^1$ and hence also in distribution; in particular, $E X\_n^1 \to E X^1 = 0$. But by indep... | 3 | https://mathoverflow.net/users/4832 | 194617 | 94,936 |
https://mathoverflow.net/questions/194619 | 23 | In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this really true?
Using Poincare duality, it is easy to see that $f$ is a homology equivalence. But has $f$ to induce an is... | https://mathoverflow.net/users/14233 | Is every degree 1 self-map a homotopy equivalence? | I believe that this is an open question in general, and the assertion is an old conjecture of Hopf. Some special cases were considered by Jean-Claude Hausmann, Geometric Hopfian and non-Hopfian situations. Geometry and topology (Athens, Ga., 1985), 157–166, Lecture Notes in Pure and Appl. Math., 105, Dekker, New York, ... | 26 | https://mathoverflow.net/users/3460 | 194627 | 94,941 |
https://mathoverflow.net/questions/194570 | 16 | Mathematical physicists in solid state physics and topological insulators talk a lot about [Walker-Wang models](http://arxiv.org/pdf/1104.2632.pdf), which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original paper is lacking a lot of mathematical details which were promised to appear in a l... | https://mathoverflow.net/users/13767 | How are the Walker-Wang TQFT and the Crane-Yetter TQFT related? | Yes, the Walker-Wang model is related to the Crane-Yetter-Kauffman TQFT in the same way the Levin-Wen model is related to the Turaev-Viro TQFT. See, for example, the table on page 14 of the notes from the talk "Premodular TQFTs" found on [this page](http://canyon23.net/math/talks/)
In general, given an $n$-category w... | 17 | https://mathoverflow.net/users/284 | 194633 | 94,944 |
https://mathoverflow.net/questions/194636 | 0 | I am looking for a method (exact, if possible, but at least asymptotically correct) for generating random variates from a [Cantor Distribution](http://en.wikipedia.org/wiki/Cantor_distribution)? It seems like its abstract definition prevents this. In essence, can one "invert" the [Cantor Function](http://en.wikipedia.o... | https://mathoverflow.net/users/nan | Generating random variables from the Cantor Distribution | If $B\_j$, $j = 1 \ldots \infty$, are independent Bernoulli(1/2) random variables, then $X = 2 \sum\_{j=1}^\infty 3^{-j} B\_j$ has a Cantor distribution.
You could let $U$ be uniform on $[0,1]$ and take $B\_j$ to be the $j$'th base-2 digit of $U$ after the "decimal" point.
| 2 | https://mathoverflow.net/users/13650 | 194638 | 94,945 |
https://mathoverflow.net/questions/194615 | 1 | I am using level method to solve non-smooth convex programming problem (where the objective function is given by an oracle from another program ):
<http://www2.isye.gatech.edu/~nemirovs/Lect_EMCO.pdf>
chapter 8.2.1
Basically, every iteration a projection of a point into a polytope is needed to be done, and I am usin... | https://mathoverflow.net/users/40780 | accelerate convex optimization by proximal projection | There is abundant literature about stuff like this but it can be hard to find the framework that is best suited for your case.
For example there is quite general theory in ["Incremental subgradients for constrained convex optimization: a unified framework and new methods", Elias Salomão Helou Neto, Álvaro Rodolfo De ... | 2 | https://mathoverflow.net/users/9652 | 194645 | 94,947 |
https://mathoverflow.net/questions/194641 | 7 | The only finite connected graphs $G$ that are isomorphic to their [line graph](http://en.wikipedia.org/wiki/Line_graph) $L(G)$ are the [cycle graphs](http://en.wikipedia.org/wiki/Cycle_graph) $C\_n$ (see [this link](http://mathworld.wolfram.com/LineGraph.html) for example).
There are connected countable graphs that a... | https://mathoverflow.net/users/8628 | Infinite graphs isomorphic to their line graph | Note first that $L$ is naturally an endofunctor on the category of graphs and injective graph-homomorphisms that commutes with filtered colimits. Let $G$ be any graph such that there is an embedding $i:G\to L(G)$. This gives rise to an embedding $L(i):L(G)\to L(L(G))$, an embedding $L(L(i)):L(L(G))\to L(L(L(G)))$, and ... | 6 | https://mathoverflow.net/users/75 | 194646 | 94,948 |
https://mathoverflow.net/questions/194640 | 3 | Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F\_q$ the unique field with $q$ elements.
What is known about the structure and properties of $\mathrm{SL}\_2(F\_q[t])$ as opposed to those of $\mathrm{SL}\_2(\mathbb{Z})$? What do they have in common and what not?
Is this group virtually f... | https://mathoverflow.net/users/38889 | Special linear groups over function fields | One important thing that is known about the structure of $SL\_2(\mathbb{F}\_q[t])$ is Nagao's theorem: for any field $k$, in particular $k=\mathbb{F}\_q$, there is an amalgam decomposition
$$
SL\_2(k[t])\cong SL\_2(k)\ast\_{B(k)}B(k[t]),
$$
where $B$ denotes upper triangular matrices. A good reference for statements ab... | 7 | https://mathoverflow.net/users/50846 | 194649 | 94,949 |
https://mathoverflow.net/questions/194648 | 12 | I'm a phd student in number theory, and I'm required (by the funding council) to publish any article I write in open access journals. The problem is that all journals I can find are either out of my league (but open access), or at the right level (but not open access). So I was wondering:
>
> Is there a list of goo... | https://mathoverflow.net/users/66266 | Open access journals in number theory | Here are a few thoughts:
* As several posters have already pointed out in the comments, many funding agencies are happy to accept an Arxiv preprint version as satisfying their "open access" requirement. This is certainly true for the big funders in the UK such as EPSRC, although you should check what your funder's ru... | 13 | https://mathoverflow.net/users/2481 | 194653 | 94,950 |
https://mathoverflow.net/questions/65190 | 5 | Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?
If I am understanding the geometry behind this problem, even if we pick a specific value for $t$ we are left with an elliptic curve, an... | https://mathoverflow.net/users/3199 | Does the following Diophantine equation have nontrivial rational solutions? | (*Edited with more details.*)
I know this is a *really* old question, but this has a nice connection to a problem considered by Euler, what is now known as *Euler bricks*, and I couldn't resist. The OP's equation is equivalent to finding three rationals $a,b,c$ such that,
$$\begin{aligned}
a^2+b^2\; &= u\_1^2\\
a^2... | 7 | https://mathoverflow.net/users/12905 | 194657 | 94,951 |
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