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https://mathoverflow.net/questions/198856 | 4 | Let $G$ be a locally compact group, and let $N$ be a closed, normal subgroup, and let $\pi\colon G\to G/N$ be the quotient homomorphism. It is known that there exists a Borel cross section, i.e., a measurable map $\sigma\colon G/N\to G$ such that $\pi\circ\sigma=\mathrm{id}\_{G/N}$. Moreover, the Borel cross section ca... | https://mathoverflow.net/users/24916 | Coarsely trivial Borel cross section for $G\to G/N$ | No, contrary to what you said, it is not always possible, even if $G$ is a semidirect product. Let us fix some $g \in N$, so $\sigma(gx) = \sigma(x)$ for all $x$. Then $\sigma(x)^{-1} g \sigma(x) = \omega(g,x)$ is a bounded function of $x$. It is easy to construct a counterexample to this, by arranging for the conjugac... | 5 | https://mathoverflow.net/users/68305 | 198875 | 96,152 |
https://mathoverflow.net/questions/198702 | 3 | Let $f:X\longrightarrow B$ be a family of curves, **with $f$ relatively minimal**, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all fibers are smooth and isomorphic, and moreover the number $K^2\_f$ is defined as
$$K^2\_f=K^2\_X-8(g-1)(g(B)-1)$$
where... | https://mathoverflow.net/users/47136 | Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial | What you are saying is not entirely correct.
1. That equation for $K\_f^2$ is not a **definition**. It's a consequence of the definition of $K\_f$ as $K\_X-f^\*K\_B$: If $F$ denotes a fiber of $f$, then since $K\_X\cdot F=\deg K\_F = 2g-2$,
$$
K\_f^2 = K\_X^2 - 2 K\_X\cdot f^\*K\_B = K\_X^2 - 2 K\_X\cdot (2g(b)-2) ... | 5 | https://mathoverflow.net/users/10076 | 198881 | 96,154 |
https://mathoverflow.net/questions/198883 | 5 | There is a basic theorem in the geometry of schemes saying that the Spec of a Noetherian ring is a Noetherian topological space. It can be formulated as the ACC condition implies the ACCR condition (the ascending chain condition for radical ideals). I'm wondering if the converse is true.
In case it is tangentially re... | https://mathoverflow.net/users/37103 | Ascending chain condition on radical ideals | The converse is not true, i.e., there exists a non-noetherian affine scheme whose underlying topological space is noetherian. (Examples: Valuation rings of rank 2.)
More results and examples about such questions can be found for example in the following articles:
>
> J. Ohm, R. L. Pendleton, [*Rings with noetheri... | 8 | https://mathoverflow.net/users/11025 | 198885 | 96,156 |
https://mathoverflow.net/questions/198878 | 7 | Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c \in \{1,\ldots, N\}$ uniformly at random. Then choose further $b\_1, \ldots, b\_r \in \{1,\ldots, N\}$, so that $b\_i$ i... | https://mathoverflow.net/users/7709 | A balls and urns model for a hashing problem | Let $X$ be the number of balls other than the special ball sent to the same urn as the special ball. The distribution of $X$ is close to a Poisson distribution with mean $1$. Condition on $X=x$. There are $x+1$ balls including the special ball sent to the same urn as the special ball. The expected first among $x+1$ obj... | 3 | https://mathoverflow.net/users/2954 | 198891 | 96,159 |
https://mathoverflow.net/questions/198795 | 6 | In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field.
In the case of a local field, Lubin-Tate theory provides an explicit construction of its maximal abelian extension using torsion points... | https://mathoverflow.net/users/39304 | Group laws in class field theory | As mentioned in the comments, this is precisely Hilbert's twelfth problem, for the simple reason that any solution to that problem can be turned into a "group law argument" (or any solution to it is a "group law argument" in disguise in the first place).
For example, you can think of the Kronecker-Weber theorem as sa... | 5 | https://mathoverflow.net/users/43108 | 198901 | 96,163 |
https://mathoverflow.net/questions/198861 | 1 | Let $X$ be a smooth projective variety and let $D$ be a big $\mathbb Q$-divisor on $X$. Assume that for $m$ large $|mD|$ has no fixed components. Is there a $\mathbb Q$-divisor $D'\equiv D$ so that $(X,D)$ is klt?
For $D$ ample, this follows from the usual Bertini theorem, as we can take $D'=\frac1m H$ for $H\in |mD|... | https://mathoverflow.net/users/68722 | Bertini theorem for big divisors and klt pairs | If you want an example where $|mD|$ has no fixed codim-1 components, you can do the following. Let $X \dashrightarrow X^+$ be an Atiyah flop of a curve $C$, and let $H \subset X$ be a general member of a very ample linear system with $H \cdot C \geq 2$. The strict transform $\tilde{H} \subset X^+$ is still big. Let $Y$... | 3 | https://mathoverflow.net/users/nan | 198902 | 96,164 |
https://mathoverflow.net/questions/198870 | 1 | Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u\_n\rVert\_{L^\infty(0,T;H)} + \lVert u\_n \rVert\_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u\_n \rightharpoonup u$ in $L^2(0,T;V)$ and $u\_n \rightharpoonup^\* u$ in $L^\infty(0,T;H)$ for some $u$.
>
> **Does this imp... | https://mathoverflow.net/users/68727 | $u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$? | As it is stated, this property does not hold: indeed consider the sequence of functinos $(u\_n)\_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by
$$
u\_n (t) = \sin (2n\pi t) v,
$$
where $v \in V$ is a fixed vector.
The sequence converges clearly weakly to $0$ in $L^2 (0, T; V)$.
Also since $L^2 (0, T)$ is dense i... | 1 | https://mathoverflow.net/users/42047 | 198922 | 96,172 |
https://mathoverflow.net/questions/198853 | 3 | Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$.
Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal matrix with each block of rank $1$?
Assume that changing even a single $1$ to $0$ in $M$ lowers rank. Does it follow that ... | https://mathoverflow.net/users/10035 | Probe permutationally matrix extreme properties | 1) No. Idea is to take all columns satisfying certain linear relation (i.e. lying in a certain subspace.) Then changing 1 to 0 violates this relation, and there are no immediate reasons that new relation appears. Namely, we may take, say, $n=k^2$, $k\geq 2$, and columns $e\_i+e\_j$ where $1\leq i\leq k$, $k+1\leq j \le... | 3 | https://mathoverflow.net/users/4312 | 198928 | 96,175 |
https://mathoverflow.net/questions/198939 | 1 | For any graph $G=(V,E)$ let $\bar{G}$ be the complement graph. Is $$\text{inf}\big\{\frac{\omega(G)+\omega(\bar{G})}{\chi(G)} : G \text{ is a finite graph}\big\}$$ known? If not, what lower bounds are known?
| https://mathoverflow.net/users/8628 | Relationship of clique, independence, and chromatic numbers | I'll restrict attention to graphs on $n$ vertices and call your parameter $X$. The numerator is at least some multipe of $\log n$ by Ramsey's theorem, and the denominator is at most $n$, so $X \geq \log n /n$. But the random graph $G\_{n,1/2}$ has clique and independence number order $\log n$ and chromatic number order... | 10 | https://mathoverflow.net/users/25485 | 198942 | 96,180 |
https://mathoverflow.net/questions/198810 | 7 | Let $F(M,k)=\{(x\_1,\cdots,x\_k)\mid x\_1\cdots,x\_k\in M,x\_i\neq x\_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$.
My question: is $F(S^\infty,k)$ contractible for each $k$? For what kind of space $X$ can we have $F(X,k)$ is contractible for each $k$?
| https://mathoverflow.net/users/41075 | contractible configuration spaces | The space $S^\infty$ is actually homeomorphic to $\mathbb{R}^\infty$. To see this, put
\begin{align\*}
B\_n &= \{x\in\mathbb{R}^\infty\::\: \|x\|\leq n, x\_k = 0\text{ for } k \geq n\} \\
C\_n &= \{x\in S^\infty\::\: x\_n \leq 1/2,\; , x\_k = 0\text{ for } k > n\}.
\end{align\*}
Then $\mathbb{R}^\infty$ is the colim... | 15 | https://mathoverflow.net/users/10366 | 198954 | 96,182 |
https://mathoverflow.net/questions/198945 | 9 | Why and how publishing a paper in proceedings?
What are the difference with a "classical" journal?
What's the list of the main proceedings in which one can publish?
Do proceedings papers (never, sometimes, often or always) appear on mathscinet?
| https://mathoverflow.net/users/34538 | Publication in proceedings | Proceedings of conferences are often published as special issues of "classical" journals. But even those that are not are usually included in MathSciNet if they include a statement (often a footnote on the first page of each paper) to the effect that the papers are in final form and will not be published elsewhere.
... | 22 | https://mathoverflow.net/users/6794 | 198955 | 96,183 |
https://mathoverflow.net/questions/198933 | 1 | A mathematics professor today asked me if Cramer's conjecture on prime gaps has anything to do with Riemann Zeta function. I did not know but my guess was somehow Cramer's conjecture captures local properties while zeta functions tell global picture (from answer in here <https://math.stackexchange.com/questions/1087381... | https://mathoverflow.net/users/10035 | Zeta functions versus Cramer's conjecture | It was shown [by Heath-Brown](http://www.ams.org/mathscinet-getitem?mr=667711) that a suitable form of the pair correlation conjecture of Montgomery, in conjunction with RH, could improve Cramer's bound $p\_{n+1}-p\_n \ll p\_n^{1/2} \log p\_n$ slightly to $p\_{n+1}-p\_n \ll p\_n^{1/2} \log^{1/2} p\_n$. This appears to ... | 7 | https://mathoverflow.net/users/766 | 198962 | 96,186 |
https://mathoverflow.net/questions/198961 | 13 | Let $E = E\_n$ be the $n$-th Morava $E$-theory with coefficient ring
$$
E\_\* = \mathbb{W}(\mathbb{F}\_{p^n})[\![u\_1,\ldots,u\_{n-1}]\!][u^{\pm 1}].
$$
It is usual to consider the completed co-operations
$$
E^\vee\_\* E := \pi\_\*L\_{K(n)}(E \wedge E)
$$
rather than the 'ordinary' co-operations $E\_\*E$. The latter i... | https://mathoverflow.net/users/16785 | Completed and uncompleted operations for Morava $E$-theory | No, this map is not injective.
To see this, put $W=\mathbb{W}(\mathbb{F}\_{p^n})$, which is a free module of finite rank over $\mathbb{Z}\_p$. It is standard that $\mathbb{Z}\_p\otimes\mathbb{Z}\_p$ contains a rational vector space of uncountable dimension, so the same is true of $W\otimes W$. (Here tensor products ... | 12 | https://mathoverflow.net/users/10366 | 198965 | 96,187 |
https://mathoverflow.net/questions/198927 | 5 | While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably very well-known (even if I couldn't find any reference).
Actually, if we define a regular graph as a planar graph which ... | https://mathoverflow.net/users/17164 | regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations | See "regular tessellations of surfaces and $(p, q, 2)$ triangle groups (Edmonds, Ewing, Kulkarni, annals of math, 1982) and references therein.
| 1 | https://mathoverflow.net/users/11142 | 198975 | 96,190 |
https://mathoverflow.net/questions/198930 | 3 | Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula
$$
\sigma(x)\sigma(y)\alpha(x,y)=\sigma(xy),
$$
for all $x,y\in G/N$.
Then $\alpha$ records how far $\sigma$ is from being... | https://mathoverflow.net/users/24916 | For a cross section $\sigma\colon G/N\to G$, how is $\sigma(y)^{-1}\sigma(x)^{-1}\sigma(xy)$ called? | I believe David Roberts is correct that $\alpha$ is a $2$-cocycle representing an element of $H^2(G/N,N)$, and is the usual $2$-cocycle associated to this extension.
The confusion comes from the fact that $\omega$ is *not* a $2$-cocycle. Rather, it is called a cocycle because it can be thought of as a $1$-cocycle rep... | 6 | https://mathoverflow.net/users/68305 | 198978 | 96,193 |
https://mathoverflow.net/questions/198980 | 3 | Given numbers $0 \leq d\_i \leq 1$ for $i=1,\ldots,m$, it is easy to see that you can always find signs $\varepsilon\_i \in \{-1,1\}$ such that the partial sums $\sum\_{i=1}^k \varepsilon\_i d\_i/2$, for $k=0,\ldots,m$(where $\sum\_{i=1}^0 = 0$) lie in an interval of size at most 1. Is this best possible? i.e. is there... | https://mathoverflow.net/users/39359 | Balancing real numbers in one dimension | Of course, this is best possible. Choose $d\_{2k-1}=1$, $d\_{2k}=1-\delta$ for $k=1,2,\dots$. If two consecutive signs are the same, we already have two partial sums on a distance $(2-\delta)/2$. If signs alternate, partial sums tend to infinity (actually, we need $m=O(1/\delta)$ for getting too large partial sum).
| 5 | https://mathoverflow.net/users/4312 | 198985 | 96,196 |
https://mathoverflow.net/questions/191739 | 4 | Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin\_\ast)$, Lurie says in Remark 2.4.2.7 of Higher Algebra that by identifying $C$ with a functor $N(Fin\_\ast)\to Cat\_\infty$ and composing with the involution $R:... | https://mathoverflow.net/users/11546 | Opposite Symmetric Monoidal Structure on an Infinity Category | First I think it's important to notice one thing: if there is a cocartesian fibration of $\infty$-operads $C^\otimes\to Fin\_\ast$ determining a symmetric monoidal structure on an $\infty$-category $C$, it is *not* the opposite fibration which determines the symmetric monoidal structure on $C^{op}$. In other words, the... | 3 | https://mathoverflow.net/users/11546 | 199007 | 96,206 |
https://mathoverflow.net/questions/198770 | 1 | Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its derivative respect to a connection $\nabla$, evaluating it at a point, taking its Lie derivative, obtaining the curvature of the ... | https://mathoverflow.net/users/66688 | Tensor calculus on the frame bundle | There are some computations in Appendix F of MSM 144 (AMS), following Hamilton's Harnack estimate paper. Under the fair use rule, we've put a link to this at:
www.math.ucsd.edu/~benchow/TensorCalculusFrameBundle.pdf
The material is geared toward an application in Ricci flow.
| 1 | https://mathoverflow.net/users/nan | 199012 | 96,209 |
https://mathoverflow.net/questions/180355 | 25 | I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a subproblem of the classification problem that is already equivalent to an already agreed-upon wild problem.
| https://mathoverflow.net/users/3635 | In what sense is the classification of all finite groups "impossible"? | One can make the argument by wildness much more concrete than in the previous answer: Sergeichuk ["Classification of metabelian p-groups", in: Matrix problems, Inst. Mat. Ukrain. Akad. Nauk, Kiev, 1977, pp. 150-161, in Russian] showed that isomorphism of 2-step nilpotent p-groups is already wild (over $\mathbb{F}\_p$),... | 17 | https://mathoverflow.net/users/38434 | 199014 | 96,210 |
https://mathoverflow.net/questions/199000 | 2 | Consider $(f\_1,\dots,f\_n), (g\_1,\dots,g\_n)\in \mathbb{C}[z\_1,\dots,z\_n]\ $ such that:
i) $\{f\_1=\dots=f\_n=0\}= \{g\_1=\dots=g\_n=0\}=\{0\}\in \mathbb{C}^n\ $ and
ii) $f\_1g\_1+\dots+f\_ng\_n\equiv0$.
What is the relation between
$
\displaystyle\dim \frac{\mathbb{C}[z\_1,\dots,z\_n]}{(f\_1,\dots,f\_n)}
... | https://mathoverflow.net/users/68790 | Relation between intersection multiplicities | Put $S=\mathbb{C}[z\_1,\ldots ,z\_n]$. Since $V(f\_1,\ldots ,f\_n)=\{0\} $, $(f\_1,\ldots ,f\_n)$ is a $S$-regular sequence, so the (partial) Koszul complex
$$\wedge^2S^n\xrightarrow{\ u\ } S^n\xrightarrow{(f\_1,\ldots ,f\_n)} S$$is exact, with $u(e\_i\wedge e\_j)=f\_ie\_j-f\_je\_i$. This implies $(g\_1,\ldots ,g\_n)\s... | 1 | https://mathoverflow.net/users/40297 | 199017 | 96,211 |
https://mathoverflow.net/questions/198991 | 23 | A certain class of books is defined as follows: (1) the book was kept for years in a cafe or mathematics library; (2) the primary contents are research problems and comments, handwritten by resident and visiting mathematicians; (3) the book still exists. Examples include the [Lwów and Wrocław Scottish Books](http://kie... | https://mathoverflow.net/users/61426 | Communal problem books | The math library in Utrecht, the Netherlands, has such a book.
| 3 | https://mathoverflow.net/users/12156 | 199025 | 96,213 |
https://mathoverflow.net/questions/199027 | 5 | I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case.
*Question:*
Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. What is the necessary and sufficient condition $\alpha,\beta$ for which there exists a $u\in C^\infty\_{0}(\mathbb{R}^n)$ su... | https://mathoverflow.net/users/53221 | Function and its Gradient with Prescribed Norms | Regarding $\mathbb{R}^n$, that's a simple matter of scaling. Assume w.l.o.g. that $\alpha = 1$ (the quantity that does matter in your problem is the ratio $\frac{\beta}{\alpha}$). Let $u$ be your favorite smooth cut-off function and assume w.l.o.g. that its $L^p$ norm is equal to $1$. Denote by $N$ the quantity $\|Du\|... | 6 | https://mathoverflow.net/users/62629 | 199029 | 96,216 |
https://mathoverflow.net/questions/199024 | 0 | Let $M$ be the manifold of all matrices in $M\_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.
>
> Does $PM$ satisfy fixed point property?
>
>
>
| https://mathoverflow.net/users/36688 | Fixed point property for the projectivization of manifold of fixed rank matrices | Usually no (check the precise dimensions yourself). In fact, just the map $M\mapsto GM$ for $G\in GL(n)$ typically has no fixed points. For $GM=\lambda M$ implies that the columns of $M$ are eigenvectors of $G$ (with the same eigenvalue $\lambda$). Thus, it suffices to assume that $G$ has no eigenspace of dimension $\g... | 2 | https://mathoverflow.net/users/44953 | 199038 | 96,221 |
https://mathoverflow.net/questions/197765 | 5 | $\newcommand{\cat}[1]{\mathcal{#1}}$
$\newcommand{\cod}{\operatorname{cod}}$
$\DeclareMathOperator{\dom}{dom}$
$\DeclareMathOperator{\colim}{colim}$
The question is about two pushout constructions G.M. Kelly is using in his paper [A unified treatment of transfinite constructions for free algebras, free monoids, colim... | https://mathoverflow.net/users/20356 | Why are pushouts the right tool in these setups | For the second question, it sounds like you have the right idea already.
For the first, one way to view the situation is as follows.
In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(... | 1 | https://mathoverflow.net/users/2503 | 199044 | 96,222 |
https://mathoverflow.net/questions/199042 | 2 | As presented in Oxtoby's book ( <http://link.springer.com/book/10.1007%2F978-1-4615-9964-7> ), there are two notions of largeness for subspace $Y$ of a given space $X$:
1. Topology: $X$ is a topological space, and $Y$ is large if its complement is a countable union of nowhere dense subsets.
2. Measure: $X$ is a proba... | https://mathoverflow.net/users/59239 | Largeness, generic, random points | Note that in the Martin-Löf case, if an ML random point has a property of a suitable kind then the set of points with that property has positive measure, not necessarily measure 1. (There's a whole zoo of related notions coming from algorithmic randomness, which mostly have the same basic structure, but where the kind ... | 2 | https://mathoverflow.net/users/8991 | 199052 | 96,223 |
https://mathoverflow.net/questions/199056 | 2 | Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by
$$T(u\_1,u\_2) := (u\_1' + au\_2',0)$$
on $\textrm{Dom} \,T := \{u=(u\_1,u\_2) \in H \ \vert\ u\_j \in C^1[0,1],j=1,2\}$ is supposed to be non-closable, since the domain of the adjoin... | https://mathoverflow.net/users/33804 | Non-closability of an operator | From the definition of the adjoint, we have that $(v\_1,v\_2)\in D(T^\*)$ precisely if there are $y\_1,y\_2$ such that
$$
\langle v\_1, u'\_1+au'\_2 \rangle = \langle y\_1, u\_1 \rangle + \langle y\_2, u\_2 \rangle
$$
for all $(u\_1,u\_2)\in D(T)$. Since we can vary $u\_1, u\_2$ independently, this means that
$$
\langl... | 2 | https://mathoverflow.net/users/48839 | 199060 | 96,228 |
https://mathoverflow.net/questions/199051 | 4 | Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C\_0(G)$.
>
> Question 1: Let $f\in A(G)$ be a function that is pointwise positive. Does the function $\sqrt{f}$ belong to $A(G)$?
>
>
>
The motivation for this Question is the following:
>
... | https://mathoverflow.net/users/24916 | Root of positive function in Fourier algebra | This won't work. I want to show that we can't take square roots in $A(\mathbb R)$. My function will be of the type
$$
f(x) = \sum h\_n \varphi\left( \frac{x-a\_n}{L\_n}\right) ,\quad\quad\quad\quad (1)
$$
and here the individual summands will have disjoint supports. I will take $h\_n\in\ell^2$, $h\_n\notin\ell^1$. Sinc... | 1 | https://mathoverflow.net/users/48839 | 199061 | 96,229 |
https://mathoverflow.net/questions/188174 | 7 | Geodesics in $\mathbb H^2$ have the following properties:
1. For every two points in the plane there exists a unique geodesic joining them.
2. Every geodesic determines exactly two points on the boundary of $\mathbb H^2$.
3. Conversely, every pair of points on $\partial \mathbb H^2$ determine a unique geodesic
4. An... | https://mathoverflow.net/users/41219 | Teichmuller geodesics vs. geodesics in the hyperbolic plane | I have answers to question 1,2 and 3.
1) True. This is essentially a restatement of existence and uniqueness of Teichmuller maps between two points in Teichmuller space. The Teichmuller map between two marked Riemann surfaces $X$ and $Y$ gives a quadratic differential on $X$. Stretching along the vertical foliation a... | 8 | https://mathoverflow.net/users/24768 | 199062 | 96,230 |
https://mathoverflow.net/questions/185727 | 7 | It is well-known that the naive construction of non-commutative $L^p$-spaces is performed only in tracial case. I would like to know if it is really a necessity.
To wit, let $\varphi$ be a normal state on a von Neumann algebra $M$. Suppose that the triangle inequality for the $L^1$-norm induced by $\varphi$ holds, i.... | https://mathoverflow.net/users/24953 | Triangle inequality for $L^1$-norm with respect to a state | If $\varphi$ is not a trace, M contains a von Neumann subalgebra isomorphic to $M\_2(\mathbf C)$ on which the restriction of $\varphi$ is not a trace. Indeed, if $x \in M$ is such that $\varphi(x^\*x) \neq \varphi(xx^\*)$, by the normality assumption on $\varphi$ we can assume that $|x|$ has a finite spectrum, and by l... | 3 | https://mathoverflow.net/users/10265 | 199070 | 96,234 |
https://mathoverflow.net/questions/185525 | 6 | (**Edit** : see at the bottom of the question for an additional surprising possible hint.)
Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when $n\geqslant 2$ but I haven't managed to prove it:
\begin{equation}
\text{for almost all } t\_1>0,\quad \text{dim}\,\... | https://mathoverflow.net/users/54797 | Proving that the kernel of this matrix is of dimension 2 | Thanks to [@Terry Tao's answer](https://mathoverflow.net/a/198863/54797), I can now give a solution to the question. I am open to comments or precisions.
$Q\_1(t)$ can be conjugated with by $\text{diag}(\omega\_1,\dots,\omega\_n,1,\dots, 1)$ which transforms it into a rotation matrix without affecting $Q\_2$.
It i... | 2 | https://mathoverflow.net/users/54797 | 199078 | 96,236 |
https://mathoverflow.net/questions/199074 | 6 | I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid.
However, I find the [definition](http://en.wikipedia.org/wiki/Rigid_category) of rigid categories somewhat difficult. I don't know how to show that these morphisms are identities.
Is there a way o... | https://mathoverflow.net/users/18060 | Establishing Duality in Tannakian Categories | As Mostafa points in the comments, it suffices to have a canonical isomorphism
$$ \hom(X\otimes Y,Z) \overset?= \hom(X,Z\otimes D(Y)). $$
But if I am not mistaken, you have
$$ \begin{aligned} \hom(X\otimes Y,Z) & = \hom(1,D(X\otimes Y)\otimes Z) \\ & = \hom(1,D(X)\otimes D(Y)\otimes Z) \\ & = \hom(X, D(Y) \otimes Z)\... | 8 | https://mathoverflow.net/users/78 | 199081 | 96,237 |
https://mathoverflow.net/questions/199050 | 9 | Let $K$ be a local field, e.g. $\mathbb{Q}\_p$ or $\mathbb{F}\_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does there exist a global field $F$, a place $v$ in $F$ with $F\_v\simeq K$ and a connected reductive group $\tilde{G}$ ove... | https://mathoverflow.net/users/68805 | Is every connected reductive group over a local field already defined over a global field? | Pick a global field $E$ and finite place $w$ with $E\_w=K$. The fraction field $k$ over $E$ of the henselization of the "algebraic" local ring at $w$ is the direct limit of finite separable sub extensions $F/E$ for which the place $v$ on $F$ from the valuation on $k$ satisfies $F\_v=K$. Thus, it suffices to "algebraize... | 15 | https://mathoverflow.net/users/61939 | 199084 | 96,239 |
https://mathoverflow.net/questions/149149 | 12 | A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set $A^B$ is also finite?
| https://mathoverflow.net/users/42440 | Exponentiation in finite simplicial sets | I think the statement is false for $ X = A = \Delta\_4/\partial \Delta\_4$ and $B = \Delta\_1$:
As Charles Rezk already mentioned, it is enough to consider the growth of $f\_{X^{\Delta\_1}}(n) = |Hom(\Delta\_1\times \Delta\_n, X)|$. Writing the prism $\Delta\_1 \times \Delta\_n$ as a coequalizer over its $(n+1)$- sim... | 7 | https://mathoverflow.net/users/68822 | 199091 | 96,241 |
https://mathoverflow.net/questions/199101 | 1 | Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$
$u\_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $\lim\_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of $u\_0(x,y)=u(0,x,y)$ divides $R^2$ into at least four regions?
| https://mathoverflow.net/users/42326 | Nodal sets under the heat flow | No.
Let $\phi$ be a compactly supported positive smooth function with $\phi(0)>0$.
Now $u\_0(X)=u\_\infty(X)+\phi(X)$ is a possible initial value but the set $\{X;u\_0(X)=0\}$ divides the plane to only three parts.
The function $\phi$ essentially builds a bridge between the first and the third quadrant.
Using a diffe... | 2 | https://mathoverflow.net/users/55893 | 199104 | 96,245 |
https://mathoverflow.net/questions/199121 | 2 | For a better understanding of this question, please see the question and answer [here](https://mathoverflow.net/questions/188455/spin7-as-stabilizer-of-a-4-form).
In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ by the double cover for any $SO(7)<SO(8)$ obta... | https://mathoverflow.net/users/62367 | $Spin(7)$ as stabilizer of a $4$-form revisited | I think that you want to look at the relevant passages in *Spin Geometry* by Lawson and Michelsohn, particularly Chapter IV, Sections 9 and 10, where they explain in general how the square of a spinor can be written as a sum of differential forms. This gives a general way to see how having a parallel spinor (in any dim... | 5 | https://mathoverflow.net/users/13972 | 199125 | 96,250 |
https://mathoverflow.net/questions/199009 | 1 | My singular integral operator is defined by
\begin{align}
Sf(x)=-\int\_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t},
\end{align}
that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast f.$ To prove that this is bounded on $L^p(\mathbb{R})$, $1<p<\infty$, I have tried to make use of the Marcinkiewicz inte... | https://mathoverflow.net/users/68794 | Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$ | We have with $\chi\in C^\infty\_c(\mathbb R)$, equal to $1$ near 0,
$$
\frac{1}{\sinh t}=\frac{\chi(t)}{\sinh t}+\frac{1-\chi(t)}{\sinh t}.
$$
The second function belongs to $L^1$ and Young's inequality implies boundedness on $L^p$ for all $p\in [1,+\infty]$.
We write,
$$
\frac{\chi(t)}{\sinh t}=\frac1t\frac{t\chi(t)}... | 4 | https://mathoverflow.net/users/21907 | 199132 | 96,253 |
https://mathoverflow.net/questions/199023 | 0 | **Edit:** I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M\_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is a canonical $n-k$ bundle $E$ over $M$. $E$ is the kernel bundle:$$\{(A,v)\in M\_{n}(\mathbb{R})\times \mathbb{R}^{n}\mid Av... | https://mathoverflow.net/users/36688 | $\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices | The transposition action does not lift, at least, "typically". The easiest obstruction is in the case $k=1$; in general, I think it's the same, with a bit more characteristic classes (or, "projective space" replaced with other magic words like "Grassmannian"). So, restrict the bundle to matrices with a fixed image $V=\... | 1 | https://mathoverflow.net/users/44953 | 199141 | 96,257 |
https://mathoverflow.net/questions/199157 | 2 | I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q\_{1},\ldots,Q\_{k}$ and $R\_{1},\ldots,R\_{k}$ in $\mathbb{R}^{n^{2}}$ and would like to bound $|| \prod\_{i=1}^{k} Q\_{i} - \prod\_{i=1}^{k} R\_{i} ||\_{\math... | https://mathoverflow.net/users/68845 | Bounds on Hilbert-Schmidt norm of difference of products of matrices | I am not sure that you can get anything better than the following triangle inequality bound:
$$\left\|\prod\_{i=1}^k Q\_i - \prod\_{i=1}^k R\_i\right\|\_{HS} \leq \sum\_{h=1}^k \left\|\prod\_{i=1}^{h-1}Q\_i\right\|\_2 \, \left\|Q\_h-R\_h\right\|\_{HS}\, \left\|\prod\_{i=h+1}^k R\_k\right\|\_2.$$
EDIT: I have conver... | 2 | https://mathoverflow.net/users/1898 | 199160 | 96,261 |
https://mathoverflow.net/questions/199142 | 1 | We know that for any simply connected surface $M$,whose Gaussian curvature $K\leq 0$, for any $p\in M$, $exp\_p: T\_pM\to M$ is diffeomorphism.
We know that for any $v\in T\_pM$ and $w\in T\_v(T\_pM)\simeq T\_pM$,
$$|d(exp\_p)\_vw|\geq |w|$$
That is we have $$\||d(exp\_p)\_v\|\_{op}\geq 1$$
Question: Can we find... | https://mathoverflow.net/users/16031 | Upper bound of derivative of exponential map | To get an upper bound of the kind you seek in general, you need a lower bound on $K$. Thus, for example, if you know that $K\ge -c^2$ on your surface, then you get
$$
\|\mathrm{d}(\exp\_p)\_v\|\_{op} \le \frac{\sinh\bigl(c\,|v|\bigr)}{c\,|v|}
\simeq 1 + \frac{\bigl(c\,|v|\bigr)^2}{3!} + \frac{\bigl(c\,|v|\bigr)^4}{5!}... | 5 | https://mathoverflow.net/users/13972 | 199165 | 96,263 |
https://mathoverflow.net/questions/199143 | 4 | It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive characteristic field? What are the main difficulties in this setting; can one prove certain weaker versions of the main resul... | https://mathoverflow.net/users/2191 | Motivic integration in positive characteristic: how much is known? | There are many theories of motivic integration. The first one, due to Kontsevich (on smooth varieties) and developed by Denef-Loeser (in general, Inventiones Math., 1999) assumed originally that the characteristic was zero. Replacing arc schemes by Greenberg schemes, it has been generalized by Looijenga (Bourbaki semin... | 7 | https://mathoverflow.net/users/10696 | 199166 | 96,264 |
https://mathoverflow.net/questions/199162 | 8 | Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with $2^{a}3^{b} \in \mathbb{N}$. Similarly, if I have any similar unnamed structure I can proceed likewise, as follows:
Firs... | https://mathoverflow.net/users/10110 | Category of Gödel Codings? [Reference Request] | You are about to invent [realizability theory](http://ncatlab.org/nlab/show/realizability). The particular category you are suggesting is going to be some sort of a category of *numbered sets*, see [this answer](https://mathoverflow.net/questions/21947/why-is-kleenes-notion-of-computability-better-than-banach-mazurs/21... | 18 | https://mathoverflow.net/users/1176 | 199170 | 96,266 |
https://mathoverflow.net/questions/195464 | 31 | Let me recall the Standard Conjecture B (see [1,2] below):
>
> The $\Lambda$-operation of Hodge theory is algebraic.
>
>
>
It more or less says that the partial inverse to “cupping with the class of a hyperplane” comes from an algebraic cycle. This is true, for example, for abelian varieties.
**Question**
... | https://mathoverflow.net/users/21815 | On Grothendieck's idea on his Standard Conjecture B | My guess is that he was thinking about **crystalline cohomology**.
It fits rather nicely in Grothendieck research at that time. He had obviously in mind the success of Dwork's p-adic approach to the Weil conjectures, and the limitations of the other cohomologies avaible at the time (see sections 1.5 to 1.8 of "Crysta... | 10 | https://mathoverflow.net/users/43108 | 199175 | 96,268 |
https://mathoverflow.net/questions/199020 | 0 | We are given $X,X\_1,\ldots,X\_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X\_q\in\mathbb{R}^k$ assume without loss of generality that $X\_1,\ldots,X\_m$ are its $m$ nearest neighbors from the $N$ points. Define $$S\_m(X\_q):=\{X\mid D(X,X\_q)\leq D(X\_m,X\_q)\}$$
the sp... | https://mathoverflow.net/users/41187 | Volume of randomly changing sphere follows beta distribution | Fix the query point $X\_q$, and take random variable $\xi=D(X,X\_q)$ for $X$ being distributed w.r.t. your law, and let $F\_{\xi}$ be its distribution function. Then, you are taking a sample of size $N$ of $\xi$, order them by increasing (obtaining $\xi\_{(1)}<\dots<\xi\_{(N)}$) and you evaluate $F(\xi\_{(m)})$.
But... | 1 | https://mathoverflow.net/users/31371 | 199176 | 96,269 |
https://mathoverflow.net/questions/198905 | 8 | Let $A$ be a normal ring (in the sense that its localizations at prime ideals are normal domains), and suppose that a finite group $G$ acts on $A$ by ring automorphisms. Form the subring $A^G \subset A$ of $G$-invariant elements. Is the ring $A^G$ also normal? What if $A$ is Noetherian (so that it is a product of Noeth... | https://mathoverflow.net/users/63877 | Does the ring of invariants inherit normality? | This is the answer of user74230 using localization instead of henselization. Namely, if $\mathfrak p$ is a prime of $A^G$, then we can replace $A^G$ by $(A^G)\_\mathfrak p$ and $A$ by $A\_\mathfrak p$ (same explanation as in the answer of user74230). Thus we may assume $A^G$ is local. Then $A$ has finitely many maximal... | 5 | https://mathoverflow.net/users/60618 | 199177 | 96,270 |
https://mathoverflow.net/questions/199152 | 2 | Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group scheme over $\mathbb P^1$ given by
$x^2 - ty^2 = 1$
I am interested in computing the stalks of the skyscraper $R^1j\_... | https://mathoverflow.net/users/67150 | Stalks of higher direct image under open embedding | $R^1 j\_\*G$ is defined as a sheafification of the presheaf $U \to H^1 ( j^{-1} U, G)$. The stalk is defined as a certain forward limit. The sheafification of a presheaf has the same stalk as the presheaf. Hence the stalk is equal to the forward limit over all neighborhoods $U$ of $0$ of $H^1( j^{-1} U, G)$.
You can ... | 4 | https://mathoverflow.net/users/18060 | 199178 | 96,271 |
https://mathoverflow.net/questions/195690 | 5 | Consider the group scheme $\mathcal T$ over $\mathbf P^1$ given locally (variable $t$) by the equation
$x^2 - f(t)y^2 = 1$
where $f(t)$ is a polynomial of degree $r$ with distinct roots (assume that $r$ is even and the field is algebraically closed). What is the group of torsors for $\mathcal T$, i.e. $\mathrm H^1(... | https://mathoverflow.net/users/58372 | Torsors under the group scheme over projective line | Taking cohomology of the short exact sequence $\mathcal T \to f\_\* \mathbb G\_m \to \mathbb G\_m$ gives a long exact sequence:
$H^0(C, \mathbb G\_m) \to H^0 (\mathbb P^1, \mathbb G\_m) \to H^1(\mathbb P^1, \mathcal T) \to H^1(C, \mathbb G\_m) \to H^1(\mathbb P^1, \mathbb G\_m) $
We can identify most of the terms:
... | 2 | https://mathoverflow.net/users/18060 | 199179 | 96,272 |
https://mathoverflow.net/questions/199189 | 4 | Let $k$ be a field of characteristic zero (I'm only interested in number fields), and let $\mathbb{G}\_{/k}$ be a linear algebraic group defined over $k$ which is almost $k$-simple (all normal subgroups defined over $k$ are finite). This group need not be absolutely almost simple, and one way this situation can arise i... | https://mathoverflow.net/users/327 | Converse to Weil Restriction of Scalars | No. Algebraic groups geometrically isomorphic to $\mathbb G\_m^n$ are classified by homomorphisms from the Galois group to $\operatorname{Aut}( \mathbb G\_m^n) = GL\_n(\mathbb Z)$. Almost simple ones correspond to irreducible representations. You are asking whether the irreducible representations are induced representa... | 7 | https://mathoverflow.net/users/18060 | 199190 | 96,279 |
https://mathoverflow.net/questions/199118 | 5 | We call a topological space $(X,\tau)$ *product-decomposable* if there is an index set $I$ and subsets $X\_i\subseteq X$ for $i\in I$ such that $|X\_i| > 1$ and $X \cong \prod\_{i\in I} X\_i$ where each $X\_i$ is endowed with the subspace topology.
If a space $(X,\tau)$ is not product-decomposable and $|X|>1$, we cal... | https://mathoverflow.net/users/8628 | Product of "prime" topological spaces | This is just a cheap extension of PVAL's comment, but I think it works: Taking for granted that $S^2$ and $TS^2$ are prime, which I believe is true, you can write
$$TS^2 \times (\mathbb R \amalg S^2) = S^2 \times (\mathbb R^3 \amalg TS^2),$$
and now all four spaces appearing are prime and pairwise not homeomorphic.
| 2 | https://mathoverflow.net/users/14233 | 199197 | 96,282 |
https://mathoverflow.net/questions/199212 | 56 | Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces." [She died July 15, 2017](http://www.bbc.com/news/science-en... | https://mathoverflow.net/users/nan | Maryam Mirzakhani's works | A very good expository article (in Farsi) on recent work of Maryam Mirzakhani can be found [here](http://math.ipm.ac.ir/~eftekhary/). ([PDF](http://math.ipm.ac.ir/~eftekhary/files/Mirzakhani-Work.pdf))
| 36 | https://mathoverflow.net/users/38966 | 199216 | 96,288 |
https://mathoverflow.net/questions/199218 | 1 | This question is inspired by a Andrew D. King's comment in [Linear intersection number and coloring (not chromatic) number](https://mathoverflow.net/questions/199128/linear-intersection-number-and-coloring-not-chromatic-number)
---
A *linear hypergraph* is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and ... | https://mathoverflow.net/users/nan | Linear intersection number and maximum degree | Yes there is such a graph. Consider the following hypergraph $\pi = (X, L)$ where
* $X =\{1,2,\ldots 6\}$;
* $L = \big\{\{1,2,3\}\big\} \cup \big\{\{a,z\}: a\in \{1,2,3\}, z\in\{4,5,6\}\big\}$.
Let $v\_0 = \{1,2,3\}$ and we set $G:=G\_\pi$. Then clearly $\ell(G) \leq 6$, but $\Delta(G) \geq \text{deg}(v\_0) = 9$. S... | 0 | https://mathoverflow.net/users/8628 | 199219 | 96,290 |
https://mathoverflow.net/questions/199192 | 3 | Sorry if the question is too long and maybe elementary.
I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension $K(\sqrt{\epsilon},\sqrt{\epsilon^\sigma})$ in part 2-1, he said let $K=\mathbb{Q}\_2(\sqrt{m})$ for $m=2,-2, 10$ or $-10$... | https://mathoverflow.net/users/50859 | Dihedral extension of 2-adic number field | Answer to your first question: The numbers contain a set of representatives of $\mathfrak{o}^\times/(\mathfrak{o}^\times)^2$. (Naito said earlier that "we examine a representative system of $K\_i^\*/(K\_i^\*)^2$", and you copied his equation that shows representatives of this quotient group can be obtained from represe... | 3 | https://mathoverflow.net/users/68305 | 199225 | 96,295 |
https://mathoverflow.net/questions/199230 | 0 | Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R\_+\to\mathbb R\_+$ such that, for every $r>0$ and all paths $\alpha,\beta$ in the ball $B(x,r)$ from $x$ to any $y\in B(x,r)$, there is a homotopy relative to the end points between $\alp... | https://mathoverflow.net/users/19838 | Homotopy bounds in simply connected complete Riemannian manifolds | It will follow if we can find a function $R$ so that the map induced by inclusion $\pi\_1 (B(x,r)) \to \pi\_1 (B(x,R(r)))$ is trivial.
By the density of Morse functions, for any small $\epsilon$ there is a compact smooth manifold with boundary $N$ so that $B(x,r) \subset N \subset B(x, r+ \epsilon)$. The fundamental ... | 2 | https://mathoverflow.net/users/68708 | 199238 | 96,300 |
https://mathoverflow.net/questions/199237 | 11 | Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The [Grothendieck construction](http://ncatlab.org/nlab/show/Grothendieck+construction) is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and $x\in F(c)$. An arrow from $(c,x)\rightarrow (c',x')$ is an arrow ... | https://mathoverflow.net/users/32022 | Topological Grothendieck Construction | This is a standard irritation. The issue is that $Top$ is not a category internal to $Top$, because it doesn't have a space of objects (and I don't mean for set-theoretic reasons), so what do you mean by a functor $F : C^{op} \to Top$?
One solution to this (which I learnt from Section 7 of S. Galatius, I. Madsen, U. ... | 16 | https://mathoverflow.net/users/318 | 199244 | 96,301 |
https://mathoverflow.net/questions/153447 | 3 | The following problem was posted to the [NMBRTHRY mailing list](https://listserv.nodak.edu/cgi-bin/wa.exe?A2=nmbrthry;94f18b90.1312) about a week ago, without eventually getting a satisfactory solution.
Suppose that $p=(n^2+1)/2$ is a prime, with $n\ge 5$ integer. Does the cyclotomic field ${\mathbb Q}(\zeta\_p)$ con... | https://mathoverflow.net/users/9924 | Cyclotomic integers with given modulus | As it turned out by the end of the day, this question admits an exhaustive answer: for ${\mathbb Q}(\zeta\_p)$ to contain an algebraic integer $\alpha$ with $|\alpha|=\sqrt{(n\pm\sqrt p)/2}$ (whatever the sign is), it is necessary and sufficient that for every prime $q$ dividing $p-1=(n^2-1)/2$ to an odd power, the ord... | 2 | https://mathoverflow.net/users/9924 | 199266 | 96,314 |
https://mathoverflow.net/questions/199195 | 1 | I am trying to prove the following Lemma, which seems intuitive, but I still have doubts:
**Lemma**
Given a Brownian motion $\{W\_t,\mathcal F\_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, with $\sigma$ continuous and $\sigma\_0\neq 0$, such that the integral
$$ X\_t=\int\_0^t \sigma\_t dW\_t + \i... | https://mathoverflow.net/users/46907 | $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales? | I think that the lemma is true if you add some additional continuity assumptions on $\sigma$. In the proof you are using the right scaling, however you may want to replace convergence in $L^2$ with convergence in distribution, as $g \_ \infty$ is not defined and does not exist. Note that $\frac{1}{\sqrt{t}} W \_{t} \ov... | 1 | https://mathoverflow.net/users/41071 | 199271 | 96,315 |
https://mathoverflow.net/questions/198507 | 3 | I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my geometry/topology knowledge is too poor to understand the proof. Is there any reference where the proof is explained in d... | https://mathoverflow.net/users/27832 | On Gromov's Theorem on Symplectic Homotopy | I didn't know that the proof of that fact was in McDuff-Salamon (in which one? *Introduction to symplectic topology*?).
You can also find it in Eliashberg-Mishachev, *Introduction to the h-principle*. There it is given as an application of the h-principle for Diff(V)-invariant open differential relations on open mani... | 4 | https://mathoverflow.net/users/67031 | 199277 | 96,316 |
https://mathoverflow.net/questions/196260 | 2 | It seems that
\begin{align}
&\prod\_{\Omega(n)=2}^{}\dfrac{1}{1 - n^{-s}}\approx\zeta (s)\exp \left(P(s)^2/2-P(s)\right)\\
\end{align}
where $P(s)$ is the prime zeta function, $\Omega(n)$ is the number of prime divisors (with mutiplicity) of $n$, and where the RHS is the dominant term in the expansion of the Euler... | https://mathoverflow.net/users/45057 | Euler product approximation for semiprimes | To answer the main question "Is this close enough to be of use in any practical application?" I do not believe so. However, one cannot say "no" with certainty, so it seems unlikely that you will get a concrete answer.
I wanted to add a derivation of a more precise formula of what you gave above. In particular, notice... | 5 | https://mathoverflow.net/users/12176 | 199281 | 96,319 |
https://mathoverflow.net/questions/199249 | 5 | As an application in group theory, I would need an infinite sequence over a finite alphabet, that avoids a sequence of words $w\_i$, where the length of $w\_i$ is such that $l(w\_i) > 10^8 l(w\_{i-1})$.
I have found several results about avoiding patterns, but here I would really just need to avoid the words themsel... | https://mathoverflow.net/users/23232 | Infinite sequence avoiding a countable set of words | I am not sure that it is exactly what you need, but the following is true:
For an alhabet with $q\geq 4$ letters and a sequence of forbidden words with lengths $n\_1<n\_2<\dots$ there exists an infinite word without forbidden subwords (where subword of a word W is a segment of consecutive letters in W, like "hab" is ... | 8 | https://mathoverflow.net/users/4312 | 199285 | 96,322 |
https://mathoverflow.net/questions/193522 | 15 | Suppose $\mathbb P$ is a countably closed forcing, and $\mathbb Q$ is a c.c.c. forcing that adds reals. Is $\mathbb P$ still proper in $V^{\mathbb Q}$?
| https://mathoverflow.net/users/11145 | Preservation of properness | Here is the short version of a negative answer. Let $P$ be the collapse of $\omega\_2$ to $\omega\_1$ with countable conditions. Fix a tree $(N\_\eta:\eta\in 2^{<\omega})$ of quite different models, increasing along each branch. Specifically, make sure that along each branch in $2^\omega$ the union of the models will g... | 12 | https://mathoverflow.net/users/14915 | 199287 | 96,324 |
https://mathoverflow.net/questions/199201 | 10 | A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V\_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ axes, it seems intuitive these integer points should not accumulate in any octant.
The proof involves estimating a... | https://mathoverflow.net/users/1358 | Averages over integer points of the sphere | First of all note that $n$ has to satisfy some trivial local conditions, otherwise $r\_3(n)$ will be zero or too small. Let us assume that $n\equiv 1\pmod{4}$ for simplicity, then the theorem holds. Let me also remark that the best known exponent of $n$ in your second display is $-\frac{1}{12}+\epsilon$, see the work o... | 9 | https://mathoverflow.net/users/11919 | 199294 | 96,327 |
https://mathoverflow.net/questions/199291 | 1 | Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many smooth arcs. Consider a disk $ D $ centered at a point $x \in \Omega $ and denote by $ \Omega\_0 $ the connected componen... | https://mathoverflow.net/users/61506 | Boundary components of a subsurface | (Based solely on the topology:)
Since $\Omega\_0$ is connected and contained within the disc $D$, it must be a planar surface, ie. a disc with holes. It has one 'outer' boundary component that is made up at least in part of $\partial D$ (unless $\partial D$ is disjoint from $\Omega$, in which case $\Omega\_0=\Omega$)... | 6 | https://mathoverflow.net/users/55866 | 199305 | 96,331 |
https://mathoverflow.net/questions/191780 | 3 | **Theorem**
Let $\beta\colon [0,1] \to M$ be a null geodesic. If $\beta(t\_0)$ is conjugate to $\beta(0)$ along $\beta$ for some $t\_0\in (0,1)$, then there is a timelike curve from $\beta(0)$ to $\beta(1)$.
This classical theorem can be found in
* Hawking and Ellis, The large scale structure of spacetime, Prop. 4.... | https://mathoverflow.net/users/40549 | Conjugate or focusing points on null geodesics imply chronality | It seems that your main argument against the correctness of the proof in Beem, Ehrlich and Easley, Global Lorentzian geometry, Theor. 10.72, is that $\frac{d^2}{ds^2}g(T,T)|\_{s=0}=0$ at $t=0$ and $t=1$. Anyway, I don't see why $\frac{d^2}{ds^2}g(T,T)|\_{s=0}=0$ should be zero there. In my opinion the proof by Beem, Eh... | 3 | https://mathoverflow.net/users/68907 | 199306 | 96,332 |
https://mathoverflow.net/questions/199309 | 5 | Do there exist two functions $f$ and $g$ continuous and strictly increasing $[0,1] \to \mathbf{R}$ such that
$$ f^{-1}\left(\frac{1}{3} f(x) + \frac{2}{3} f(y)\right)<g^{-1}\left(\frac{1}{3} g(x) + \frac{2}{3} g(y)\right)$$
and
$$ f^{-1}\left(\frac{2}{3} f(x) + \frac{1}{3} f(y)\right)>g^{-1}\left(\frac{2}{3} g(x) + \fr... | https://mathoverflow.net/users/32898 | Symmetric inequality on generalized means | No. Denote $f(x)=a$, $f(y)=b$, then $g(f^{-1}(t))=h(t)$. Applying $g$ to both parts of first inequality we get $h(\frac13 a+\frac23 b)<\frac13 h(a)+\frac23h(b)$. It means that $h$ is convex (proof below), analogously the second inequality means that $h$ is concave.
Lemma. If $h$ is continuous on $[0,1]$ and $h(tx+(1-... | 6 | https://mathoverflow.net/users/4312 | 199311 | 96,334 |
https://mathoverflow.net/questions/198600 | 3 | Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow \mathbb{P}^n$ be the blow-up of a point $p\in Z$, and let $\widetilde{X}\subset Y$ be the strict transform of $X$ with ex... | https://mathoverflow.net/users/nan | Blowing-up a point in the singular locus | Note that without any assumption on the singularities the answer to your question is yes. For instance, consider the hypersurface $Y = \{x\_0^2+x\_1^3+x\_2^4\}\subset\mathbb{P}^3$. Then $Sing(Y) = [0:0:0:1]$. The projective tangent cone of $Y$ in the singular point is not reduced. Therefore, the singularity is not ordi... | 5 | https://mathoverflow.net/users/14514 | 199323 | 96,341 |
https://mathoverflow.net/questions/199207 | 15 | Is there an abstract structure that characterizes connectedness, analogously to how topological spaces characterize continuity?
Here's one way to make this question more precise: if $(X,T\_X)$ is a topological space with underlying set $X$ and topology $T\_X$, then consider the pair $(X,C\_X)$ where $C\_X$ is the set... | https://mathoverflow.net/users/49 | Abstract connectedness | There seems to be some literature on this already: this [paper](http://www.math.shimane-u.ac.jp/memoir/39/D.Buhagiar.pdf) introduces the notion of a "connective space", i.e., a set equipped with a "connectology", and develops some theory. There was some related Mathematics Stackexchange discussion [here](https://math.s... | 12 | https://mathoverflow.net/users/2926 | 199325 | 96,342 |
https://mathoverflow.net/questions/199327 | 0 | Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$
a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$
is also Cohen-Macaulay. This is known as Hochster-Roberts Theorem.
It is also known that if $X=Spec(R)$, has rational singularities, then
$Y=Spec(R^G)$ has rational s... | https://mathoverflow.net/users/37338 | Hochster-Roberts Theorem reciprocal | This is not true in general.
Indeed, there exists $\mathbb{Q}$-Gorenstein rings $S$ with rational singularities such that their canonical covers do not have rational singularities, see for instance *Cyclic covers of rings with rational singularities.* by Anurag K. Singh. Generally speaking taking Veronese's of grade... | 1 | https://mathoverflow.net/users/3521 | 199331 | 96,344 |
https://mathoverflow.net/questions/199348 | 5 | The well-known Noether's theorem is a vital tool in classical physics. But it assumes some hypothesis, many of which could be removed by a detailed look.
So my question is: In what directions has this theorem has been generalized in the context of Noncommutative geometry, Quantum Groups, Quantum Mechanics, etc? More... | https://mathoverflow.net/users/14555 | What is the current state of generalizations Noether's theorem? | Noether's theorem was successfully generalized to the setting of quantum field theory in the fifties. This is called [Ward-Takahashi identity](http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity).
| 7 | https://mathoverflow.net/users/43108 | 199349 | 96,351 |
https://mathoverflow.net/questions/199342 | 12 | We are going to build $K\_n$ one edge at a time. Begin with the empty graph on $n$ vertices. Take a random permutation of the edges of $K\_n$ and, one at a time, place the edges onto the graph (so, after the $k$th edge in the list is placed, the so-far created graph will have exactly $k$ edges).
Let $H\_k$ be the gra... | https://mathoverflow.net/users/43928 | Probability of a graph procedure | Here's one. You can think of the graph construction process as gradually building a set $S$ of vertices that have been touched so far, beginning with a random two vertices. Let $S\_k$ be the set of the first $k$ vertices in this process.
Now $p(n)$ is exactly the probability that, after the first two vertices, each a... | 11 | https://mathoverflow.net/users/29697 | 199353 | 96,353 |
https://mathoverflow.net/questions/199358 | -4 | Does there exists a good asymptotic formula for
$$A(x) := \prod\_{p\leq x}(1-\frac 1p).$$
By using a heuristic argument one can guess:
$$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$
Here is the argument:
$$\frac{x^2}{2\, \mathrm{ln}(x) } \sim \pi(x^2)-\pi(x) = \sum\_{p\_1<p\_2<\dots<p\_k\leq x}(-1)^k\left\lfloor \f... | https://mathoverflow.net/users/51663 | Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ | Mertens' Theorem says (page 65, *An Introduction to Sieve Methods and Their Applications*, Cojocaru and Murty):
$$
\displaystyle\prod\_{p< x}\left(1-\frac 1p\right)=\frac{e^{-\gamma}}{\log x}\left(1+O\left(\frac{1}{\log x}\right)\right),
$$
where $\gamma$ is the Euler-Mascheroni constant.
| 7 | https://mathoverflow.net/users/62656 | 199359 | 96,355 |
https://mathoverflow.net/questions/199350 | 4 | I suspect the following identity is valid, but I can not prove it. I just calculate it numerically.
$\sum\_{m=0}^\infty\left[\sum\_{n=0}^\infty\frac{(-1)^{n+m}}{(n+1)(n+m+1)}\right]=\sum\_{m=0}^\infty\left[\sum\_{n=0}^\infty\frac{(-1)^{m}}{(n+1)(n+m+2)}\right]$
I would appreciate any idea on how to prove it. Thanks... | https://mathoverflow.net/users/68933 | How to prove this identity on double summation series? | Both are equal to $(\pi^2-6\log^22)/12$.
The inner sum on the right
$$\sum\_{n=0}^\infty\frac{(-1)^m}{(n+1)(n+m+2)}=\frac{(-1)^m}{m+1}\sum\_{n=0}^\infty
\Bigl(\frac{1}{n+1}-\frac{1}{n+m+2}\Bigr)=
\frac{(-1)^m}{m+1}\Bigl(1+\frac12+\frac13+\cdots+\frac{1}{m+1}\Bigr)$$
Therefore the sum on the right is equal to
$$R:=\su... | 14 | https://mathoverflow.net/users/7402 | 199366 | 96,358 |
https://mathoverflow.net/questions/199343 | 7 | I want to ask the question in two parts,
(1)
Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes them?)
(2)
Are there any set of (constant degree) (Abelian) Cayley graphs which are expanders? Do they have any disti... | https://mathoverflow.net/users/36554 | When are (Abelian) Cayley graphs also expanders? | To add to Anthony's comment, one can make an explicit connection between the large number of walks between vertices and the spectra of Abelian Cayley graphs. It turns out that constant-degree Abelian Cayley graph are not only bad expanders,but they tend to be disconnected (they have a positive proportion of their eigen... | 9 | https://mathoverflow.net/users/62709 | 199373 | 96,361 |
https://mathoverflow.net/questions/199144 | 3 | I have a question about the meaning of Yang-Mills Functional.
It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is:
* To have a manifold $M$ together with a smooth vector bundle $E\longrightarrow M$
* To a given connection $A$ over ... | https://mathoverflow.net/users/62367 | Yang-Mills Functional and Energy | The easiest way to see that the norm of the curvature corresponds to the energy is to consider the special case of an abelian U(1)-Yang-Mills theory (i.e. electrodynamics). If you write out the norm squared of the curvature in terms of the $E$ and $B$ fields you get the expression $E^2 + B^2$. This is exactly the famil... | 5 | https://mathoverflow.net/users/17047 | 199379 | 96,365 |
https://mathoverflow.net/questions/163443 | 8 | Let $(X\_t)\_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX\_t = b(X\_t)dt + \sigma(X\_t)dW\_t, ~~ X\_0 = x\_0,
\end{equation}
where $b,\sigma$ are Lipschitz and $(W\_t)\_{t \geq 0}$ is a standard $d$-dimensional Wiener process. Assume $(X\_t)\_{t \... | https://mathoverflow.net/users/49620 | Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion | Here I use the notation given in the question above, and the statements made assume the domain of the SDE problem is unbounded, which seems to be the context of the question.
Assume the following conditions hold.
1. (**Regularity**) Every derivative of $b(x)$ and $\sigma(x)$ exists and is bounded, and $\sigma(x)$ i... | 3 | https://mathoverflow.net/users/64449 | 199380 | 96,366 |
https://mathoverflow.net/questions/199367 | 4 | While it is well known for metric spaces, being separable is equivalent to be second-countable. In [this](http://www.mathcounterexamples.net/a-separable-space-that-is-not-second-countable/) [post](http://bit.ly/1HgwCdl) I give a counterexample for a non metric space.
What are other topological properties that added t... | https://mathoverflow.net/users/41060 | Spaces for which separable is equivalent to second-countable | According to [this paper](http://arxiv.org/abs/1301.5297) under $MA+\neg CH$ a scattered compact space is metrizable if and only if it is separable and hereditarily supercompact. It therefore seems as if to obtain metrizability from separability and some other property, one needs quite strong conditions and one needs t... | 4 | https://mathoverflow.net/users/22277 | 199383 | 96,367 |
https://mathoverflow.net/questions/199297 | 11 | I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum\_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions $(k,Y)$ for $k, Y \in \mathbb{N} $. I have found (using programs) that if this equation has non-trivial solutions, n is ... | https://mathoverflow.net/users/57088 | Sum of consecutive cubes | Found some bigger numbers:
{n,k} as you call them:
{4913 , 11368}
{6591 , 305}
{6859 , 18171}
{8000 , 22534}
{10648 , 33558}
{12167 , 40381}
{13923 , 3010}
{14161 , 1624}
{25201 , 46690}
{33124 , 18551}
{63001 , 11170}
{48841 , 967190}
{277729 , 711785}
Most 'n' are squares, but a strange ones are 6591, 13923 and 25201... | 11 | https://mathoverflow.net/users/68953 | 199396 | 96,376 |
https://mathoverflow.net/questions/194542 | 2 | The paper <http://epubs.siam.org/doi/abs/10.1137/1023098> (Generalizations of Noether’s Theorem in Classical Mechanics, by Willy Sarlet and Frans Cantrijn) mentions "an interesting property of the Noether-invariant, which is
almost never mentioned in the quoted literature, at least not in the context of
Lagrangian mech... | https://mathoverflow.net/users/32389 | Invariance of the Noether charge | Your question is essentially about a special case of a well-known result. True it is more often stated in the Hamiltonian language, but there is no obstacle to restating it in the Lagrangian language. Instead of an ODE, think more generally of a system of a variational PDE system. Let $\rho\_v$ be conservation law corr... | 2 | https://mathoverflow.net/users/2622 | 199401 | 96,378 |
https://mathoverflow.net/questions/199404 | 7 | **Question:**
Let $P\_\pi$ denote the matrix representation of permutation $\pi$. Consider a linear combination of all $n \times n$ permutation matrices
$$U := \sum\_{\pi \in S\_n} c\_\pi P\_\pi$$
where $c\_\pi$ are arbitrary complex coefficients. When is the matrix $U$ unitary? It would be great to have a simple param... | https://mathoverflow.net/users/37211 | When is a linear combination of permutation matrices unitary? | I think this can be done(in principle) in general. The $n \times n$ permutation matrices span a $\mathbb{C}$-vector space of dimension $1 + (n-1)^{2}$ since the natural permutation representation of $S\_{n}$ is the sum of the trivial representation and an irreducible representation of degree $n-1.$ It is necessary for ... | 9 | https://mathoverflow.net/users/14450 | 199409 | 96,382 |
https://mathoverflow.net/questions/199320 | 2 | I am trying to find a reference of a proof of a continuous time version of a result of Dvoretzky and Erdos from their paper "Some problems on random walk in space" that says the probability $\gamma\_2(n)$ that a two dimensional random walk does not return to the origin after first leaving it up until time n is asymptot... | https://mathoverflow.net/users/68920 | The probability that a 2d continuous time random walk avoids the origin | Let $X\_n$ be a discrete-time random walk on the integer lattice, and let $N(t)$ be a rate-1 Poisson process independent of $X\_n$. Then your continuous-time random walk $S\_t$ has the same distribution (as a process) as $X\_{N(t)}$.
Let $T = \min\{n \ge 1 : X\_n = 0\}$ be the return time for $X\_n$, so that $P(T > n... | 4 | https://mathoverflow.net/users/4832 | 199414 | 96,384 |
https://mathoverflow.net/questions/199425 | 9 | Given $n$. Two players in turn mark points on the plane. No three may be collinear, no $n$ may form a convex $n$-gon. The player who does not have legal move loses. Who has a winning strategy?
| https://mathoverflow.net/users/4312 | Erdős-Szekeres game | The problem was raised and discussed in Parikshit Kolipaka and Sathish Govindarajan, Two player game variant of the Erdős-Szekeres problem, Discrete Math. Theor. Comput. Sci. 15 (2013), no. 3, 73–100, MR3141828. It says the second player wins for $n=5$.
| 10 | https://mathoverflow.net/users/3684 | 199428 | 96,387 |
https://mathoverflow.net/questions/199410 | 3 | Let $H$ be a finite group. We write ${{\mathbb{C}}}^{\*n}$ for the $n$-dimensional complex torus $({{\mathbb{C}}}^\*)^n$.
We have a short exact sequence
$$ 0\to {{\mathbb{Z}}}^n\to {{\mathbb{C}}}^n\to{{\mathbb{C}}}^{\*n}\to 1,$$
which gives a connecting isomorphism
$$ \Delta\colon {\rm Hom}(H,{{\mathbb{C}}}^{\*n})\to H... | https://mathoverflow.net/users/4149 | Conjugation of group extensions | Here's an example where they are not isomorphic, with $n=1$.
Let $k$ be the order of the image of $\varphi$, which is also the image of $\sigma\phi$ (since $\mathbf{C}^\*$ has a single subgroup of each given order). Then $\varphi$ can be viewed, after some fixed identification, as the fibre product
$H\times\_{(\math... | 5 | https://mathoverflow.net/users/14094 | 199429 | 96,388 |
https://mathoverflow.net/questions/199415 | 27 | Let $X$ be a compact topological space, $f\_i:Y\_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f\_i^{-1}(U)$ is open for each $i$, for more categorical formulation see <http://en.wikipedia.org/wiki/Final_topology>).
*Does there exist a finite subfa... | https://mathoverflow.net/users/68962 | Possible categorical reformulation for the usual definition of compactness | I think the answer is no. For an example, let $X$ be the unit interval $[0,1]$ (with the usual topology).
For each $i \in \mathbb{N}^+$, let $Y\_i = (1/i,1]$ with the usual topology. The obvious inclusion $f\_i \colon Y\_i \to X$ is continuous.
Let $Y\_0$ be $[0,1]$ with an unusual topology: a set is open in $Y\_0$... | 14 | https://mathoverflow.net/users/68305 | 199430 | 96,389 |
https://mathoverflow.net/questions/199421 | 4 | Let $f : H\_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M\_i (A)}{\det(A)}$ where $M\_i(A)$ is the determinant of the the submatrix formed by the $A\_{kl}$ where $1\leq k, l \leq i$, i.e., it is the $i\times i$ principal minor of $A$.
Is th... | https://mathoverflow.net/users/3709 | Convexity of a (non-symmetric) function of matrices | From [Theorem 9, of this article](http://www-isl.stanford.edu/~cover/papers/paper86.pdf) it follows that $A \mapsto \log\frac{M\_i(A)}{\det(A)}$ is convex on the set of positive definite matrices. The alleged convexity in the OP is a simple consequence of this stronger log-convexity result.
| 2 | https://mathoverflow.net/users/8430 | 199435 | 96,392 |
https://mathoverflow.net/questions/199432 | 15 | Let's say that I have a very large number of the order ($10^{250+}$) which is composite. I have been given one of its factor partially to a significant amount of digits (say 75+). Then, how can I figure out both its factors completely?
That is,
$$X = a b$$
where $X$ is known and $a$ is known to 75+ digits.
| https://mathoverflow.net/users/68971 | Factorization when a factor is partially known | You can use Coppersmith's algorithm [1] (or Howgrave-Graham's [2] simplification) to find the factor, which will be efficient if the number of remaining bits is not too large. The PARI/GP documentation
<http://pari.math.u-bordeaux.fr/dochtml/html-stable/Arithmetic_functions.html#zncoppersmith>
has an explicit examp... | 14 | https://mathoverflow.net/users/6043 | 199437 | 96,393 |
https://mathoverflow.net/questions/199347 | 0 | Consider sums of the form
$S(n)=\sum^{n}\_{m=0}\sum^{m}\_{k=1}2^{2k+m+1}{n-m+k+1 \choose 2k+2}{m \choose k}$
I am interested in the asymptotics of $S(n)$ as $n\to \infty$.
More precisely I would like to know whether it grows purely exponential in $n$ i.e.
whether there are numbers $c>0$ and $R>0$ such that
$R^{... | https://mathoverflow.net/users/12395 | Asymptotic of a certain double sum involving binomial coefficients | The inner summation from $k=1$ looks a bit suspicious, since starting with $k=0$ appears to be more natural. So I assume that the inner summation starts with $k=0$ (the case of $k=1$ easily follows) and to avoid ambiguity I denote this sum by $S'(n)$.
The generating function for $S'(n)$ is
$$\sum\_{n=0}^{\infty} S'(... | 9 | https://mathoverflow.net/users/7076 | 199440 | 96,396 |
https://mathoverflow.net/questions/199453 | 1 | If I let $f(\theta)=((\mathrm{cos} \theta)X+(\mathrm{sin} \theta)Y)^{n-1}$ and view the range of this curve as a subset of the space of homogeneous polynomials of degree $n-1$ in two variables viewed as an $n$-dimensional Euclidean space, then I can show that there are $n$ linearly independent points on the curve, beca... | https://mathoverflow.net/users/15482 | proving that a smooth curve in Euclidean n-space contains n+1 affinely independent points | This isn't true if $n$ is odd. For example, if $n=3$, then your formula is $(a,b,c) = (\cos^2 \theta, 2 \sin \theta \cos \theta, \sin^2 \theta)$ and it always lies in the hyperplane $a+c=1$. More generally, whenever $n$ is odd, the equality $1 = (\sin^2 \theta+ \cos^2 \theta)^{(n-1)/2} = \sum \binom{(n-1)/2}{k} \sin^{2... | 2 | https://mathoverflow.net/users/297 | 199454 | 96,401 |
https://mathoverflow.net/questions/199450 | 1 | Let $G\_n$ be the complete graph whose vertices are the $2^n$ $n$-bit strings. Let $H\_n$ denote the Hamiltonian path through $G\_n$ that uses the maximum number of edges that correspond to a single bit transition $0\mapsto 1$. What is this maximum number? And is there an algorithm that generates this Hamiltonian path?... | https://mathoverflow.net/users/68987 | Hamiltonian Path through $n$-bit strings with maximum number of $0\mapsto 1$ transitions | Then best you can do is $2^n-\binom n {\lfloor n/2 \rfloor}$. The binary strings of length $n$ can be decomposed into $\binom n {\lfloor n/2 \rfloor}$ *chains*, sets that can be ordered so that all transitions are of the form you are interested in. Joining these chains up produces a Hamiltonian path in $G\_n$ achieving... | 2 | https://mathoverflow.net/users/25485 | 199455 | 96,402 |
https://mathoverflow.net/questions/199451 | 2 | I'm thinking about the following question:
If $X\_n$ and $X\_n/Y\_n$ both converge in distribution towards a standard Gaussian random variable and $Y\_n \geq 0$ for all $n$, does then $Y\_n$ necessarily have to converge in distribution towards the constant $1$?
Does it change anything if we assume that all $X\_n$ h... | https://mathoverflow.net/users/57022 | Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$? | Sorry, I don't have the reputation to comment, or I would've written something a bit shorter.
Briefly: I don't think that anything like this can be true without substantial assumptions beyond smoothness (e.g. maybe something like this is true if $X\_{n}$, $Y\_{n}$ are independent). To see this, let's just assume that... | 3 | https://mathoverflow.net/users/68845 | 199459 | 96,403 |
https://mathoverflow.net/questions/199417 | 5 | In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal methods, these oracles are the projection oracle, the (constrained) linear optimization oracle, and the prox. operator,... | https://mathoverflow.net/users/37202 | Analysis of first-order methods for constrained convex optimization with approximate oracles | Building on [Nesterov's work](http://pubsonline.informs.org/doi/pdf/10.1287/moor.1080.0348), in his [Ph.D thesis](http://www.maths.ed.ac.uk/~prichtar/papers/thesis.pdf), Peter Richtarik considers first-order methods with relative error of approximation guarantees. I haven't looked in too closely, but I am sure that a l... | 2 | https://mathoverflow.net/users/8430 | 199460 | 96,404 |
https://mathoverflow.net/questions/199471 | 0 | let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and
$$\omega(\cdot,\cdot)=g(J\cdot, \cdot)\,.$$
I denote with $Ham(M,\omega)$ the group of Hamiltonian symplectomorphisms w.r.t. ... | https://mathoverflow.net/users/37370 | Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact? | By Ascola-Arzelà theorem, the group of isometries of a compact metric space is itself compact and then a compact connected Lie Group. The condition that the isometries preserve the complex structure is a closed condition, so your subgroup is compact.
| 1 | https://mathoverflow.net/users/47274 | 199472 | 96,408 |
https://mathoverflow.net/questions/198246 | 2 | I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of arithmetic groups termed *simplest type*, first defined in Chapter 6 of Vinberg and Shvartzman's *Discrete Groups of Motions of S... | https://mathoverflow.net/users/14835 | Some general properties of arithmetic groups of simplest type | The book *Conformal Geometry of Discrete Groups and Manifolds* by Boris N. Apanasov contains a detailed description of what it means for a group to be arithmetic, and why the definition manifests as it does for topology applications. He includes examples that illustrate the ideas behind statements 1 and 2, and discusse... | 2 | https://mathoverflow.net/users/14835 | 199474 | 96,409 |
https://mathoverflow.net/questions/199467 | 15 | This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (a \pm 1, b)$ or $(a , b \pm 1)$, with a uniform probability (i.e. a uniform random walk on $G$).
Let $P\_n(r)$ be the... | https://mathoverflow.net/users/34538 | In how many steps a random walk visits all the elements of a finite group, with a probability 1/2? | The quantity $R\_n$ is asymptotic to ${4\over \pi}(n\log n)^2$, see "Cover times for Brownian motion and random walks in two dimensions" by Dembo, Peres, Rosen and Zeitouni. This was previously conjectured by Aldous.
| 25 | https://mathoverflow.net/users/38566 | 199481 | 96,412 |
https://mathoverflow.net/questions/199457 | 0 | Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying $k>r + \frac{n}{2}.$ Does this still hold for manifolds or is this no longer true in this context?
I should add that ma... | https://mathoverflow.net/users/68992 | Sobolev's lemma on manifolds | This follows from its counterpart in the Euclidean space by local charts.
If you want to have an estimate on the derivative $D^r f$, then you should impose some bounds on derivatives of the curvature and a lower bound on the injectivity radius.
| 2 | https://mathoverflow.net/users/42047 | 199482 | 96,413 |
https://mathoverflow.net/questions/199487 | 1 | Skewes has proved (without assuming RH) that $\pi(x)<Li(x)$ is violated below $e^{e^{e^{e^{7.705}}}}$ which is clearly a very large number.I was wondering if somewhere else some greater number than Skewes's can be found.
>
> **Question** :Is there any published proof in *number theory* breaking this "large number"... | https://mathoverflow.net/users/38851 | Numbers greater than Skewes's whose existence can be found in number theoretic proofs | The [quantitative version](http://people.maths.ox.ac.uk/greenbj/papers/back-of-an-envelope.pdf) of the [Green--Tao theorem](http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem) says there is an arithemtic progression of length $k$ in the prime numbers below
$$2^{2^{2^{2^{2^{2^{2^{100k}}}}}}}$$
[These are seven 2s,... | 11 | https://mathoverflow.net/users/nan | 199498 | 96,418 |
https://mathoverflow.net/questions/199433 | 7 | What is known about the following decision problem?
Given two finite sets in a finitely generated group G,
decide whether the subgroups generated by them have trivial intersection.
Is this problem decidable for a free non-abelian group G?
| https://mathoverflow.net/users/68935 | Decision problem on triviality of intersection of two subgroups | Let $F$ be a free group of rank $2$. If the intersection triviality problem is decidable for $F\times F$, then using the Mikhailova construction, it would be decidable given a finitely presented group $G=\langle X\mid R\rangle$ and a word $w$ in $X$ whether $w$ has infinite order in $G$. This latter problem was proved ... | 9 | https://mathoverflow.net/users/15934 | 199499 | 96,419 |
https://mathoverflow.net/questions/199489 | 10 | Stirling's approximation is the following well-known asymptotic result:
$$n! \approx \left(\frac{n}{e}\right)^n \sqrt{2 \pi n}$$
This result has several analytical proofs, for example via Laplace's method, the trapezoidal rule (and Euler–Maclaurin formula for an asymptotic expansion) or Hayman's method (essentially... | https://mathoverflow.net/users/31469 | Combinatorial\Probabilistic Proof of Stirling's Approximation | The Wikipedia page [List of probabilistic proofs of non-probabilistic theorems](http://en.wikipedia.org/wiki/List_of_probabilistic_proofs_of_non-probabilistic_theorems) has a reference to the paper:
>
> Blyth, Colin R.; Pathak, Pramod K.
> A Note on Easy Proofs of Stirling's Theorem.
> *Amer. Math. Monthly* **93*... | 6 | https://mathoverflow.net/users/4832 | 199500 | 96,420 |
https://mathoverflow.net/questions/199338 | 6 | This is an awkwardly backwards question, but bear with me here: Suppose I have a graded ring $R$ with unit, which has an invertible element $u$ in degree $2$. The multiplicative formal group law $f(x,y) = x + y + u\,x\,y$ yields a homomorphism $MU\_\* \to R$. Suppose that I know the following things
* The functor $X ... | https://mathoverflow.net/users/3995 | Properties of coefficients of ring spectra | You have given yourself an invertible element $u\in\pi\_2(R)$ and a coordinate $x\in R^2(\mathbb{C}P^\infty)$ with $\psi(x)=x\otimes 1 + 1\otimes x + ux\otimes x$. This means that the class $m=1+ux\in R^0(\mathbb{C}P^\infty)$ satisfies $\psi(m)=m\otimes m$. In other words, $m$ can be regarded as a map of ring spectra f... | 8 | https://mathoverflow.net/users/10366 | 199505 | 96,423 |
https://mathoverflow.net/questions/199476 | 3 | Let $X$ and $Y$ be two complex, irreducible, normal, projective varieties (read: integral, projective, normal $\mathbb C$-schemes of finite type), projective in the sense of Hartshorne.
Let $\phi:X\dashrightarrow Y$ be a birational map. I will think of $\phi$ as a morphism $\phi:U\to Y$ defined on an open set $U\subset... | https://mathoverflow.net/users/9947 | Is every surjective, birational transformation of projective varieties automatically proper? | tl;dr: No. There exists a flop $X \dashrightarrow W$ such that the flopped curve in $W$ is contained in the exceptional locus of a blowndown $W \to Y$, but isn't contracted by it. The complement of the flopping curve in $X$ still surjects onto $Y$ because the map is defined on the rest of the exceptional locus of $W \t... | 4 | https://mathoverflow.net/users/nan | 199519 | 96,430 |
https://mathoverflow.net/questions/199289 | 1 | If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose orbit closure contains a unique polystable orbit?
Edit: It would seem that polystable points are actually those whose orb... | https://mathoverflow.net/users/25590 | Semistability in GIT | If I may answer my own question:
1. Polystable points can be characterized as points that are semistable and with closed orbit in the semistable locus.
2. Yes, If the orbit closure of a point contains a polystable orbit (necessarily unique), then that point is semistable.
However 2. is pretty much saying nothing. I... | 2 | https://mathoverflow.net/users/25590 | 199522 | 96,431 |
https://mathoverflow.net/questions/199520 | 1 | The question is as in the title: is $K\_2(\mathbb{Z}[x,x^{-1}])$ known?
| https://mathoverflow.net/users/69017 | What is $K_2(\mathbb{Z}[x,x^{-1}])$? | Bass/Quillen's Fundamental Theorem of Algebraic K-theory gives:
$$K\_n(R[x,x^{-1}])=K\_n(R)\oplus K\_{n-1}(R)\oplus NK\_n(R)\oplus NK\_n(R)$$
where the two identical nil-terms vanish for $R$ regular.
Because ${\mathbb Z}$ is regular, this gives
$$K\_2({\mathbb Z}[x,x^{-1}])
=K\_2({\mathbb Z})\oplus K\_1({\mathbb Z})... | 8 | https://mathoverflow.net/users/10503 | 199523 | 96,432 |
https://mathoverflow.net/questions/199532 | 3 | Suppose $g\_1$, and $g\_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K\_{g\_1}$ $<$ $K\_{g\_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy classes of closed curves in $S$. The marked length spectrum function is $l\_{g\_i}: \mathcal{C} → \mathbb{R}^{+}$ which... | https://mathoverflow.net/users/51454 | length comparison on negatively curved surfaces | This is false even if $K\_{g\_1} = K\_{g\_2} \equiv -1,$ in case the two surfaces are not isometric.
| 7 | https://mathoverflow.net/users/11142 | 199534 | 96,436 |
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