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https://mathoverflow.net/questions/72047 | 19 | Let $LM$ denote "all subsets of $\Bbb{R}$ are Lebesgue measurable", and
$WCH$ (weak continuum hypothesis) denote "every uncountable subset of $\Bbb{R}$ can be be put into 1-1 correspondence with $\Bbb{R}$".
[**Warning:** in other contexts, weak CH means something totally different , i.e., it sometimes means $2^{\... | https://mathoverflow.net/users/9269 | Lebesgue Measurability and Weak CH | This is an expansion of my comments above. In the paper with Zapletal that I reference, we assume a proper class of Woodin cardinals and force over $L(\mathbb{R})$ with a partial order of countable approximations to a certain kind of MAD family (which Jindra named an "improved" MAD family). Although I have yet to write... | 10 | https://mathoverflow.net/users/31807 | 269083 | 120,619 |
https://mathoverflow.net/questions/269064 | 28 | Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true :
$$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - f ( \sqrt{xy} ) \right )=0.$$
Is it true that the only solution to this is the constant function ?
| https://mathoverflow.net/users/109471 | For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant? | Yes. If $f$ were not constant, then (since ${\bf R}^+$ is connected) it could not be locally constant, thus there exists $x\_0 \in {\bf R}^+$ such that $f$ is not constant in any neighbourhood of $x\_0$. By rescaling (replacing $f(x)$ with $f(x\_0 x)$) we may assume without loss of generality that $x\_0=1$.
For any $... | 42 | https://mathoverflow.net/users/766 | 269084 | 120,620 |
https://mathoverflow.net/questions/269048 | 6 | Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think it helps).
>
>
> >
> > Is it true that if $\frac{\partial^2}{\partial z\_i\partial \overline{z\_j}}\log \|f(z)\|=... | https://mathoverflow.net/users/53155 | Derivatives of norm of vector-valued holomorphic functions | Yes, this is true, and this is called the Calabi rigidity, though it was proved
(for finite dimensional Hilbert space)
long before Calabi, see, for example, Polya-Szego, revised edition, part IV, problem 207, where there is a reference on the original paper.
Calabi's paper is: Isometric imbedding of complex manifolds,... | 4 | https://mathoverflow.net/users/25510 | 269087 | 120,622 |
https://mathoverflow.net/questions/268346 | 2 | In Matthias Wulkau's exposition of Scholze's thesis, the term *perfectoid field* is defined as follows:
>
> Let $K$ be a field endowed with a non-archimedian absolute value $\lvert\cdot\rvert$, and let $\mathcal{O}\_K$ and $\mathfrak{m}$ be the closed and open unit balls in $K$, respectively. We say that $K$ is a *... | https://mathoverflow.net/users/94022 | Why is $\mathbb{Q}_p(p^{1/p^\infty})$ a complete topological field? | This was alredy answered in the comments, it is the $p$-adic completion of $\mathbb{Q}\_p(p^{1/p^\infty})$ that is a perfectoid field.
But here is a reference for completeness
* Matthias Wulkau, [Review of perfectoid fields](https://www.mathi.uni-heidelberg.de/~G.QpAsPi1geom/manuscripts/PF.pdf) (2015) [Remark 1]
| 3 | https://mathoverflow.net/users/43108 | 269095 | 120,625 |
https://mathoverflow.net/questions/268847 | 18 | Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let
$$
{\cal D} \subset U(n)
$$
denote the subspace of those matrices having
a non-trivial $(+1)$-eigenspace.
**Background:** It is known that $\cal D$ has vanishing homology
in dimension $n^2$. It is also not difficult to show that $H\_{... | https://mathoverflow.net/users/8032 | Local homology of a space of unitary matrices | Let us first consider the case when $g=e$ is the identity matrix. Let $U$ be an open neighbourhood of the identity in $\mathcal D$. We want to calculate the local homology of $U$ at $e$.
We may assume that $U$ is mapped homeomorphically by the (inverse of) the exponential map onto its image in the tangent space of $... | 10 | https://mathoverflow.net/users/6668 | 269105 | 120,629 |
https://mathoverflow.net/questions/269108 | 6 | Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Let $k \subset L$ be a field extension. Let $b \in \mathrm{Br}(k)$ and denote by $b \otimes L \in \mathrm{Br}(L)$ the base-change of $b$ to $L$.
>
> If $b \otimes L = 0$, then does this exist a subextension $k \subset K \subset L$ such that $K/k$ has ... | https://mathoverflow.net/users/5101 | Brauer groups and field extensions | No: the conic $C:X^2+Y^2+1=0$ splits over the field $L=\mathbb{Q}(x)[y]/(x^2+y^2+1)$, since $(X,Y)=(x,y)$ is an $L$-point of $C$. However $L$ has no subfields algebraic over $\mathbb{Q}$ other than $\mathbb{Q}$ itself, since it is the function field of a geometrically irreducible variety.
| 12 | https://mathoverflow.net/users/17907 | 269109 | 120,630 |
https://mathoverflow.net/questions/263377 | 2 | Suppose that $\boldsymbol{x}\in\mathbb{R}^n$ is subgaussian random vector of variance proxy $\sigma^2$, i.e.,
$$\forall \boldsymbol{\alpha}\in\mathbb{R}^n: \quad \quad \mathbb{E}\left[ \exp\right(\boldsymbol{\alpha}^T\boldsymbol{x} \left) \right] \leq \exp\left(\frac{\sigma^2}{2}\|\boldsymbol{\alpha}\|^2\right).
$$
Not... | https://mathoverflow.net/users/44722 | Bounds on the eigenvalues of the covariance matrix of a sub-Gaussian vector | This serves as a pointer and my thought on the OP's question of bounding the spectrum of covariance matrix of subgaussian (mean zero )random vector. The case of spectrum of covariance matrix of gaussian random vector is discussed in [this post](https://mathoverflow.net/questions/158582/what-kind-of-random-matrices-have... | 3 | https://mathoverflow.net/users/25437 | 269110 | 120,631 |
https://mathoverflow.net/questions/206779 | 14 | Let's call a cardinal $\delta$ an $\text{I1}$-tower cardinal if for each $A\subseteq V\_{\delta}$, there exists a $\kappa<\delta$ such that whenever $\kappa<\alpha<\delta$ there is some $\lambda<\delta$ and a $j:V\_{\lambda+1}\rightarrow V\_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and $j(\kappa)>\alpha$ and $j(V\... | https://mathoverflow.net/users/22277 | Where does this strengthening of I1 stand? | Suppose $\delta$ is an ordinal. We first note that $\delta$ is an $I\_1$-tower cardinal if and only if it has the following superficially weaker property: for all $X\subseteq V\_\delta$ there is some $\kappa < \delta$ such that for arbitrarily large $\lambda < \delta$, there is an elementary embedding $j:V\_{\lambda+1}... | 6 | https://mathoverflow.net/users/102684 | 269112 | 120,632 |
https://mathoverflow.net/questions/269117 | 1 | Let $P$ be a poset with a least element $\bot$ ($\forall x \in P.\ \bot \le x$).
Let $M$ be a set of *monotone* maps $P \to P$.
Call $x \in P$ reachable if $x = f\_1(f\_2(...f\_n(\bot)...))$ for some sequence $f\_i \in M$.
Call $x$ fixed if $x = f(x)$ for all $f \in M$.
If $x$ reachable, $y$ fixed, then $x \le y$. Fo... | https://mathoverflow.net/users/109493 | Random iteration of a set of monotone maps until fixed point | I think yes, to Question #2, which implies Question #1.
Does this work?
Suppose $x$ is reachable and fixed. Pick $F$ with $x = F(\bot)$ to witness $x$'s reachability. Let $s$ be an arbitrary reachable element, with $F' \in M^\*$ witnessing its reachability, i.e. $F'(\bot) = s$.
Observe that $F(s)$ is reachable, s... | 2 | https://mathoverflow.net/users/109497 | 269121 | 120,634 |
https://mathoverflow.net/questions/269052 | 16 | Let $$a(n) = \sum\_{0 \leq k \leq n} {n \choose k}{{n+k} \choose k},$$ and define
$b(n) = \nu\_3 \bigl(a(n)\bigr)$, where $\nu\_3$ is the $3$-adic valuation. About twenty years ago or so, I discovered (empirically) the following conjectured expression for $b(n)$:
$$b(n) = \begin{cases}
b\bigl(\lfloor n/3 \rfloor\big... | https://mathoverflow.net/users/44797 | 3-adic valuation of a sum involving binomial coefficients | The $3$-adic evaluation you seek is compactly given by
$$\nu\_3(a\_{2n})=\nu\_3\left(\binom{2n}n\right) \qquad \text{and} \qquad
\nu\_3(a\_{2n+1})=\nu\_3\left(3(2n+1)\binom{2n}n\right),$$
which can be proved inductively using the well-known recurrence
$$na\_n=3(2n-1)a\_{n-1}-(n-1)a\_{n-2}$$
according to the parity of $... | 11 | https://mathoverflow.net/users/66131 | 269126 | 120,636 |
https://mathoverflow.net/questions/268921 | -2 | One of the most notable features of $ZFC$ is that it builds up sets in recursively defined $V\_i$ stages (where $i$ is an ordinal), however the usual formulation of $ZFC$ has infinitely many stages. The rank of a set $X$ is the ordinal index $i$ of the stage $V\_i$ where $X$ first appears as a subset of. So in $ZFC$ we... | https://mathoverflow.net/users/95347 | Can ZFC be interpreted in a set theory having finitely many ranks? | In relation to the above question, it can be proven that for any theory formulated in first order logic, one can easily have a set theory that interpret it having only three ranks, and even one rank if primitive type level ordered pairs are used. The general workup is to weaken Extensionality as to allow existence of m... | 1 | https://mathoverflow.net/users/95347 | 269138 | 120,643 |
https://mathoverflow.net/questions/266009 | 3 | A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ is a simplicial complex. I would like to know for what values of $n$ the boundary is a flag complex (i.e., a simplicial co... | https://mathoverflow.net/users/7494 | Cyclic polytopes whose boundary is a flag complex | The answer is "never" (except in the obvious case $d=2$, $n\ge 4$).
$C(n,d)$ is neighborly, meaning that every $d/2$ or less vertices define a simplex. In particular, for $d\ge 4$ its graph is complete and the boundary complex cannot be flag. For $d=3$, the boundary complex contains the three edges $1i$, $in$ and $1n... | 4 | https://mathoverflow.net/users/22608 | 269141 | 120,645 |
https://mathoverflow.net/questions/269164 | 3 | Numerous papers/books(citation needed) refer to the operator $$A\_\lambda := \lambda AR\_\lambda (A) = \lambda^2 R\_\lambda(A) - \lambda I$$ where $R\_\lambda(A)=(I+\lambda A)^{-1}$ is the resolvent, as *Yosida approximation* without indicating the original source of it.
Can somebody provide the exact source to a Yo... | https://mathoverflow.net/users/109413 | Reference request: Original source of Yosida approximation |
>
> ...It is well known that the celebrated Hille-Yosida theorem,
> discovered independently by Hille [1] and Yosida [2], gave the first
> characterization of the infinitesimal generator of a strongly
> continuous semigroup of contractions. This was the beginning of a
> systematic development of the theory of sem... | 2 | https://mathoverflow.net/users/25437 | 269167 | 120,652 |
https://mathoverflow.net/questions/269166 | 1 | What is the tightest upper bound one can obtain for the following expression
$$\sum\_{i=1}^kA\_i\log(\frac{A\_i}{e})$$ subject to $\sum\_{i = 1}^k A\_i = C$ in terms of $C$ and $k$?
A very loose upper bound for this expression is $C\log(\frac{C}{e})$. Can we do better than this?
| https://mathoverflow.net/users/109523 | Tighter upper bound for $\sum_{i=1}^kA_i\log(\frac{A_i}{e})$ | The gradient of your function is $(\log A\_1, \dotsc, \log A\_n).$ The function is convex, and so the minimum can be seen, by Lagrange multipliers to occur when all of the $A\_i$ are equal (to $C/k$). So, the minimum value of the function is
$$C \log\left(\frac{C}{ke}\right).$$
On the other hand, the maximum must oc... | 5 | https://mathoverflow.net/users/11142 | 269171 | 120,654 |
https://mathoverflow.net/questions/269168 | 11 | Consider $\mathrm{AD}\_X$, determinacy for games where players pick moves from $X$. We know that it is consistent for $X = \omega$ or $\mathbb{R}$ (under large cardinal assumptions), but inconsistent for $X = \omega\_1$.
Since this implies determinacy is inconsistent for any set with $\omega\_1 \leq X$, this answers ... | https://mathoverflow.net/users/101982 | Is determinacy on an infinite Dedekind finite set consistent? | The answer is no, you cannot have determinacy for all games on an infinite Dedekind-finite set. Indeed, one cannot even have clopen determinacy for games on such a set. So the answer to question 2 is negative.
**Theorem.** If $X$ is an infinite Dedekind-finite set, then there is a non-determined clopen game on $X$.
... | 19 | https://mathoverflow.net/users/1946 | 269178 | 120,656 |
https://mathoverflow.net/questions/269169 | 2 | Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ and $v$ satisfy the CR-equations formally. The necessary condition follows by straightforward formal (algebraic) derivatio... | https://mathoverflow.net/users/1849 | Formal Cauchy-Riemann equations for formal power series without complex analysis | $f\in C[[x+iy]]$ means $f(x,y)=g(x+iy),$ where $g\in C[[z]]$. Then
$$f\_x=g'(x+iy),\; f\_y=ig'(x+iy),$$
therefore $f\_y=if\_x$ and this is Cauchy-Riemann: indeed, if $f=u+iv$ then
$f\_x=u\_x+iv\_x$ and $f\_y=u\_y+iv\_y$, so $f\_y=if\_x$ is equivalent to
$u\_x=v\_y$, $u\_y=-v\_x$.
To prove the converse, make the li... | 9 | https://mathoverflow.net/users/25510 | 269187 | 120,658 |
https://mathoverflow.net/questions/269175 | 2 | Mathworld just says that the lower central series terminates: $\mathfrak{g}\_1= [ \mathfrak{g}, \mathfrak{g}]$,
$\mathfrak{g}\_2= [ \mathfrak{g}, \mathfrak{g}\_1]$ and $\mathfrak{g}\_n= [ \mathfrak{g}, \mathfrak{g}\_{n-1}]$ the best example I could think of is something like:
$$ \left( \begin{array}{cccc}
1 & \mathbb... | https://mathoverflow.net/users/1358 | are all simply connected nilpotent lie groups matrix groups over $\mathbb{R}$? | A connected Lie group admits a faithful finite dimensional representation (i.e. is a matrix group) just when it is a semidirect product $S \rtimes R$ of a connected and simply connected solvable Lie group $S$ and a connected linearly reductive Lie group $R$. See Hilgert and Neeb, **Structure and Geometry of Lie Groups*... | 4 | https://mathoverflow.net/users/13268 | 269190 | 120,660 |
https://mathoverflow.net/questions/269194 | 2 | In principle, associative \*-algebras can be equipped with multiple norms. Can this be done in such a way that (after closure in the respective norm) they are turned into $C^\*$-algebras in multiple, not essentially equivalent ways?
| https://mathoverflow.net/users/56920 | Multiple $C^*$ structures? | There is at most one norm making an associative $\*$-algebra $\mathcal A$ into a C$^\*$-algebra, that is $\mathcal A$ is closed with respect to this C$^\*$-norm. This is because for all $a\in \mathcal A$,
$$ \|a\|^2 = \|a^\*a\| = \textrm{spectral radius of}\: a^\*a = \textrm{sup}\{|\mu| : a^\*a - \mu I \notin \mathcal... | 5 | https://mathoverflow.net/users/76593 | 269201 | 120,662 |
https://mathoverflow.net/questions/269184 | 9 | This problem is a follow up on [my other MO question](https://mathoverflow.net/questions/269174/hooks-in-a-staircase-partition-part-i).
On the basis of experimental data, I'm prompted to ask:
>
> **Question.** Let $R(a,b)$ an $a\times b$ rectangular grid, $h\_{\square}$ the hook-length of a cell $\square$ in the ... | https://mathoverflow.net/users/66131 | Hooks in a rectangle: Part II | As Darij suggests, we may count separately the sum of legs and the sum of arms. Denote the rows sizes $t\_1,t\_1+t\_2,\dots,t\_1+\dots+t\_a$, where $t\_i,1\leqslant i\leqslant a+1$, are non-negative integers and $\sum\_{i=1}^{a+1} t\_i=b$. Then the sum of arms (let's think that the arm includes the half of the source o... | 4 | https://mathoverflow.net/users/4312 | 269206 | 120,665 |
https://mathoverflow.net/questions/269177 | 3 | Let $X$ be an infinite dimensional, reflexive and separable real Banach space. Consider a function $f: X \to \mathbb{R}$, and assume $f$ is sequentially continuous with respect to the weak topology, that is, if $a\_n \rightharpoonup a$ then $f(a\_n) \to f(a)$.
What conditions are needed to show that $f$ is also conti... | https://mathoverflow.net/users/14870 | Sequentially weak-continuous and continuous functions under the weak-topology of a Banach space | Your argument is right. Indeed, a function is weakly sequentially continuous iff its restriction to every ball is weakly continuous. You have proved one direction; the converse is given by the fact that every weakly convergent sequence is bounded (uniform boundedness principle).
As to your last question, [this Math.S... | 4 | https://mathoverflow.net/users/4832 | 269214 | 120,669 |
https://mathoverflow.net/questions/269215 | 8 | Under what all conditions on $(a,b,c)$ where $a,b,c$ are positive integers can we say about the non-existence or existence of any solution to the equation
$$a^a+b^b=c^c$$
(Or in other words :
Solve $a^a+b^b=c^c$ )
$ $
$ $
About this note that : For a solution we must have $gcd(a,b,c) = 1$ (follows from Fermat's ... | https://mathoverflow.net/users/109471 | Solving the equation $a^a+b^b=c^c$ in positive integers | The sequence $(n^n)\_{n \ge 1}$ grows too fast for there to exist a solution:
if $b \ge 2$, then $(b+1)^{b+1} > b^{b+1} \ge 2 b^b$ (and also $2^2 > 2 \cdot 1^1$), so if $1 \le a \le b$, then $b^b < a^a + b^b \le 2 b^b < (b+1)^{b+1}$,
and no $c$ can exist such that $a^a + b^b = c^c$.
| 19 | https://mathoverflow.net/users/21146 | 269217 | 120,671 |
https://mathoverflow.net/questions/269198 | 5 | For a variety over a finite field Deligne define weights by looking at the eigenvalues of the Frobenius. On the other hand, if we take a variety over $\mathbb{Q}$, at least for its constant sheaf we have the weight filtration which gives the mixed Hodge structure. How are these two comparable? Can someone point out som... | https://mathoverflow.net/users/85640 | Comparing Frobenius weights with Mixed Hodge theory | For constant coefficients, the comparison statement, along with a sketch, appears in Deligne's ICM talk, *Poids dans la cohomologie...*.
For things to work the way you seem to want in your second paragraph, you're going to need a lot more structure for $F$ than what you've given. At the very least, when $F$ is a loca... | 4 | https://mathoverflow.net/users/4144 | 269230 | 120,674 |
https://mathoverflow.net/questions/269200 | 3 | Given a densely defined symmetric operator $L$ on a Hilbert space $H$, which is also assumed to be diagonalizable, will there always exist a unique extension of $L$ to a self-adjoint operator?
| https://mathoverflow.net/users/37003 | Symmetric diagonalizable operators and self-adjointness | Yes. If $x\_n\in D(L)$ is an ONB of $H$ and $Lx\_n=\lambda\_n x\_n$, then the operator $T$ acting in the obvious way on $D(T)=\{ \sum a\_n x\_n\in H : \sum \lambda\_n^2|a\_n|^2<\infty\}$ is self-adjoint. It is also an extension of $L$ because if $x\in D(L)$, then
$$
\lambda\_n\langle x, x\_n\rangle = \langle Lx, x\_n \... | 4 | https://mathoverflow.net/users/48839 | 269232 | 120,676 |
https://mathoverflow.net/questions/269226 | -2 | there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix:
$$ \left(
\begin{array}{cccc}
1 & \boxed{\cdot}& \boxed{\cdot}& \boxed{\cdot} \\
0 & 1 & \boxed{\cdot} & \boxed{\cdot} \\
0 & 0 & 1... | https://mathoverflow.net/users/1358 | no classification of nilpotent lie groups | Here is a simple example of an uncountable family of nilpotent Lie groups. Fix real numbers $\lambda\_1, \dots, \lambda\_n$. Fix integers $k\_1, \dots, k\_n$. Let $V$ be the set of all functions of the form $f(x)=\sum\_i e^{\lambda\_i x} p\_j(x)$ for $p\_j(x)$ a polynomial function of degree at most $k\_j$. Let $G=\mat... | 6 | https://mathoverflow.net/users/13268 | 269240 | 120,679 |
https://mathoverflow.net/questions/269227 | 5 | Let $S$ be a set. Is there a monoidal category $TS$ that we can construct from $S$ such that monoidal functors $F: TS \to M$ (up to monoidal natural isomorphism) correspond to $M$-enriched categories with underlying object set $S$ (up to $M$-enriched equivalence)?
The construction I had in mind was to take $S^2$, add... | https://mathoverflow.net/users/84398 | Is there a monoidal category that coclassifies enriched category structures for a given set? | I don't think you can construct $T(S)$ as a monoidal category. It's more common to construct $T(S)$ as a bicategory with object set $S$. (see the end for comments on regarding it as a monoidal category)
Let $I(S)$ denote the indiscrete category on $S$. Then a lax 2-functor $I(S) \to M$ (where $M$ is considered as a 1... | 4 | https://mathoverflow.net/users/2362 | 269243 | 120,680 |
https://mathoverflow.net/questions/269248 | 10 | While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform structures [Johson&Lindenstrauss&Schechtman]. [Ribe] pointed out that two non-isomorphic spaces may have homeomorphic uniform st... | https://mathoverflow.net/users/25437 | Example of Banach spaces with non-unique uniform structures | In Theorem 5.8 of my paper with Lindenstrauss and Schechtman that you reference we show that for every $n=0,1,2,\dots$ there exists a Banach space that is uniformly homeomorphic to exactly $2^n$ mutually non isomorphic spaces. AFAIK, it is still open whether $2^n$ can be replace by any number that is not a power of $2$... | 10 | https://mathoverflow.net/users/2554 | 269249 | 120,681 |
https://mathoverflow.net/questions/269267 | 3 | Assume that $(M, g)$ is a connected Riemannian manifold which is either open or is compact with zero Euler characteristic.
>
> Is there a non vanishing vector field $X$ on $M$ such that all trajectories of $X$ are geodesics, after a possible reparametrization?
>
>
>
The question is somehow a converse question ... | https://mathoverflow.net/users/36688 | A non vanishing vector field compatible to a Riemannian metric | If that would be the case, then $M$ would have a foliation by geodesics. The article
*Zeghib, A.*, [**On continuous geodesic foliations of hyperbolic manifolds**](http://dx.doi.org/10.1007/BF01232666), Invent. Math. 114, No.1, 193-206 (1993). [ZBL0789.57019](https://zbmath.org/?q=an:0789.57019).
shows that for a cl... | 8 | https://mathoverflow.net/users/98590 | 269269 | 120,687 |
https://mathoverflow.net/questions/269268 | 3 | The starting point of this question is that $\sum\_{n=1}^\infty \frac{1}{n^{\alpha}} < \infty$ if and only if $\alpha > 1$.
Let $(a\_n)\_{n\in\mathbb{N}}$ be a non-negative sequence. We say that $(a\_n)$ *slowly converges* to $0$ if $\lim\_{n\to\infty} a\_n = 0$ and $$\sum\_{n=1}^\infty \frac{1}{n^{1+a\_n}} < \infty.... | https://mathoverflow.net/users/8628 | Product of sequences "slowly" converging to $0$ | The answer is no. For example $$a\_n=\frac{2\log\log n}{\log n}$$ "slowly" converges to $0$, because $n^{1+a\_n}=n\log^2n$,
but $$c\_n=\frac{\log\log n}{\log n}$$ converges to $0$ not "slowly".
As $a\_n^2<c\_n$, for large $n$, we conclude that $a\_n^2$ converges to $0$ not "slowly".
| 7 | https://mathoverflow.net/users/25510 | 269271 | 120,689 |
https://mathoverflow.net/questions/269237 | 8 | Recall that an integer $n$ is called *squareful* if for every prime $p$ with $p \mid n$, we also have $p^2 \mid n$.
Any squareful number can be written uniquely as $n= x^2 y^3$ where $y$ is squarefree. From this, it is easy to see that
$$\#\{ n \in \mathbb{Z}: |n| \leq X, \, n \text{ is squareful} \} \ll X^{1/2}.$$
... | https://mathoverflow.net/users/5101 | Squareful values of polynomials | If $f$ is squarefree and of degree at least three, then
the abc conjecture implies $f(x)$ is squareful finite number
of times. For [reference](http://www.math.unicaen.fr/~nitaj/abc.html#Consequences) and several papers of Granville:
>
> Power free-values of polynomials. Langevin noted in [Lan2] the following conjec... | 5 | https://mathoverflow.net/users/12481 | 269281 | 120,691 |
https://mathoverflow.net/questions/269239 | 11 | Let $p$ be an odd prime and $n \geq 2$.
(1) Does there exist an integer-coefficient polynomial $f$ such that $f(x) = x - (x \bmod p)$ for all $x \in \mathbb{Z}/p^n \mathbb{Z}$? The polynomial effectively removes the first base-$p$ digit of $x$. For example, if $n = 2$, then the defining property is $f(a+bp) = bp$.
... | https://mathoverflow.net/users/109563 | lowest degree of polynomial that removes the first digit of an integer in base p | Yes, this can be done, and the bound for the degree given by Neil Strickland (that is, $(n-1)\cdot(p-1)+1$ holds.
Let $F\_A(x)$ be the formal sum in binomial coefficients
$$ F\_A(x):=\sum\_{j=0}^\infty (-1)^j \begin{pmatrix}A-1+j\\j\end{pmatrix}\begin{pmatrix}x\\A+j\end{pmatrix}.$$
Then $F\_A(x)$ converges on ever... | 9 | https://mathoverflow.net/users/61910 | 269282 | 120,692 |
https://mathoverflow.net/questions/269272 | 0 | Given any two positive real numbers $p\_1,p\_2$ and a function $f\_1 : \mathbb{R} \to \mathbb{R}$ such that $f\_1$ has a minimal positive period of $p\_1$ .
Then is it true that whatever be the choice of $f\_1$ there is always a function $f\_2 : \mathbb{R} \to \mathbb{R}$ having smallest positive period of $p\_2$ such... | https://mathoverflow.net/users/109471 | Given reals $p_1,p_2$ and a function $f_1$ with minimal period $p_1$.Existence a function $f_2$ with minimal period $p_2$,$f_1-f_2$ periodic? | If $p\_3$ is a period of $f\_1-f\_2$, we see that $f\_1$ must satisfy an equation $f\_1(x)+f\_1(x+p\_2+p\_3)=f\_1(x+p\_2)+f\_1(x+p\_3)$ for all $x$ (because both $f\_2$ and $f\_1-f\_2$ satisfy this equation). If, say, $f\_1(x)=\sin x$, this rewrites as $\sin(x+\frac{p\_2+p\_3}2)\cos(\frac{p\_2+p\_3}2)=\sin(x+\frac{p\_2... | 3 | https://mathoverflow.net/users/4312 | 269284 | 120,693 |
https://mathoverflow.net/questions/269294 | 33 | Is $L^2(\mathbb R)$ homeomorphic to $L^1(\mathbb R)$?
More generally, are there instances of surprising homeomorphisms between non-isomorphic Banach spaces?
| https://mathoverflow.net/users/5690 | Is $L^2(\mathbb R)$ homeomorphic to $L^1(\mathbb R)$? | According to [Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?](https://mathoverflow.net/questions/196117/are-l-infty-bbb-r-and-l2-bbb-r-homeomorphic?rq=1), the map $f \mapsto \operatorname{sgn}(f) |f|^2$ is a homeomorphism from $L^2$ to $L^1$.
It's clearly a bijection. Suppose $f\_n \to f$ in $L^2$; set $g\_n ... | 40 | https://mathoverflow.net/users/4832 | 269297 | 120,700 |
https://mathoverflow.net/questions/269236 | 11 | I am sorry if my question is stupid (or very hard) or common knowledge, or should be placed at math.stackexchage.com.
As long as a math student read the definition of Lie group, several natural questions appear instantly:
Is it true, that any compact manifold admits Lie group structure? (NO)
Is it true, that there ... | https://mathoverflow.net/users/83980 | Connected compact manifolds with unique Lie group structure | Your question seems to be of a tall order!
There are many,vmany necessary conditions involving the entire
spectrum of algebraic topology such as homotopy and cohomology.
[This 3rd edition book](https://www.degruyter.com/view/product/185231) by Kakl H. Hofmann and Sidney A. Morris *on compact groups* is full of these.... | 4 | https://mathoverflow.net/users/66131 | 269310 | 120,708 |
https://mathoverflow.net/questions/269307 | 24 | Consider the endomorphism ring spectrum $R = \mathrm{End}\_S(H\mathbb{F}\_p)$ of the mod $p$ Eilenberg-MacLane spectrum $H\mathbb{F}\_p$. The homotopy groups of $R$ are the Steenrod algebra $A^\*$ with reversed grading:
$$\pi\_n R = [\Sigma^n H\mathbb{F}\_p, H\mathbb{F}\_p] = A^{-n}.$$
This spectrum $R$ is an associati... | https://mathoverflow.net/users/16109 | Endomorphism ring spectrum of the Eilenberg-MacLane spectrum | No, $A$ is not an $H\Bbb Z$-algebra.
Suppose $R$ is an $H\Bbb Z$-algebra. Then the category of left $R$-modules is $H\Bbb Z$-linear: for any $R$-modules $M$ and $N$, the function spectrum $F\_R(M,N)$ naturally has the structure of an $H\Bbb Z$-module. One reason for this is that $R$ is now an algebra object in the sy... | 31 | https://mathoverflow.net/users/360 | 269314 | 120,710 |
https://mathoverflow.net/questions/269315 | 8 | Let $\mathcal{H}\_2 = \{(x,t) \in \mathbf{R}^2: t > 0\}$ be the upper half-plane, and let $\mathcal{H}\_3$ be the hyperbolic 3-space $\{(x,t) \in \mathbf{C} \times \mathbf{R}: t > 0\}$. Clearly $\mathcal{H}\_2$ embeds in $\mathcal{H}\_3$. There is an action of $PSL(2, \mathbf{C})$ on $\mathcal{H}\_3$, extending the fam... | https://mathoverflow.net/users/2481 | Hyperbolic planes inside hyperbolic 3-space quotients | Let $\Gamma = PSL\_2(\mathcal{O}\_K)$. Note that complex conjugation acts as an orientation-reversing isometry on $\mathcal{H}\_3$, the action being compatible with that on $\Gamma$.
Let $\gamma\in \Gamma$ be such that $a = \mathcal{H\_2}\cap \gamma^{-1}\mathcal{H}\_2$ is a nontrivial geodesic.
For all $x\in a$ we ha... | 10 | https://mathoverflow.net/users/40821 | 269321 | 120,714 |
https://mathoverflow.net/questions/269303 | 9 | Cross posted from MSE at commenter's suggestion: <https://math.stackexchange.com/questions/2269319/definability-of-truth-in-l-without-0>
I'm interested in the relationship between the existence of $0^{\#}$ and the definability of truth in $L.$ My advisor showed me an argument that you don't need $0^{\#}$ for truth in... | https://mathoverflow.net/users/109573 | Definitional complexity of truth in $L$ without $0^{\#}$ | The answer is yes, the theory of $L$ can be definable by a low-complexity definition quantifying over reals, even when $0^\sharp$ does not exist.
Here is one way to achieve this. Let me assume that the theory of $L$ is an element of $L$. This happens, for example, if $L\_\kappa\prec L$ for some ordinal
$\kappa$, bec... | 10 | https://mathoverflow.net/users/1946 | 269323 | 120,715 |
https://mathoverflow.net/questions/269275 | 2 | Maximal green sequences are studied in many papers. For example, [Maximal Green Sequences for Cluster Algebras Associated to the n-Torus](https://arxiv.org/abs/1412.3713) by Eric Bucher, [On Maximal Green Sequences](https://arxiv.org/abs/1205.2050) by Thomas Brüstle, Grégoire Dupont, Matthieu Pérotin, [Minimal length m... | https://mathoverflow.net/users/11877 | Which cluster algebras where the existence of maximal green sequences is still unknown? | First, it may be better to ask which quivers have a maximal green sequence. Or ask which cluster algebras have a initial seed with a maximal green sequence. Greg Muller shows in [The existence of a maximal green sequence is not invariant under quiver mutation](https://arxiv.org/abs/1503.04675) that the existence of a m... | 2 | https://mathoverflow.net/users/51668 | 269325 | 120,716 |
https://mathoverflow.net/questions/269341 | 2 | *(Note: I have asked this question before on math.stackexchangecom, but it wasn't answered, so I am trying again here).*
The question is pretty much in the title; $X$ is an $n\times r$ matrix with $n>r$, and of course I am asking for the limit in $\epsilon \rightarrow 0$.
The interesting case is the case in which... | https://mathoverflow.net/users/109625 | Limit of $(X' (\Delta + \varepsilon I_n)^{-1} X)^{-1}$ with $\Delta$ an $n \times n$ diagonal matrix | By the properties of Schur complements, your limit should be the $(1,1)$ block of
$$
\begin{bmatrix}
0 & X'\\
X & -\Delta
\end{bmatrix}^{-1}.
$$
(when it is invertible). Not sure if this counts as "much better". It also depends on what you are trying to do with it -- numerical computation? You need a closed formula in ... | 2 | https://mathoverflow.net/users/1898 | 269342 | 120,719 |
https://mathoverflow.net/questions/269332 | 8 | I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$,
$$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$
I have tried to approach this using multiple methods and substitutions, but none has gotten me far. I know that Chini equation tend... | https://mathoverflow.net/users/109419 | Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$? | One thing you can try is to select numerical values for $a$ and $b$ (say $a=b=1$ or $a=b^{-1}=2$) and then find a power series solution $y=1+\sum\_{k>0}c\_k(x-1)^k$ and inspect the coefficients $c\_k$. With $a=b=1$ they are
$$ 3, 2, 4, 11, 35, \frac{721}{6}, \frac{18163}{42}, \frac{540391}{336},
\frac{98091}{16}, \fr... | 13 | https://mathoverflow.net/users/10366 | 269345 | 120,720 |
https://mathoverflow.net/questions/269350 | 1 | Is the following statement consistent with ZF (obviously, this is not true in ZFC)?
>
> There are cardinals $\kappa, \lambda$ such that $\kappa \neq \lambda$ and $2^\kappa = 2^\lambda$.
>
>
>
| https://mathoverflow.net/users/60579 | Can the powersets of two different cardinals be of the same cardinality? | Yes, this is possible, even with ZFC. In Cohen's model, where the continuum hypothesis fails, he gets ZFC plus $2^\omega=2^{\omega\_1}$. This statement is known as [Luzin's hypothesis](https://www.encyclopediaofmath.org/index.php/Luzin_hypothesis), an alternative to CH.
More generally, the situation for regular cardi... | 6 | https://mathoverflow.net/users/1946 | 269351 | 120,722 |
https://mathoverflow.net/questions/269356 | -1 | Let $n>1$ be an integer. We consider the line graphs of $K\_{2n}$ and $K\_{2n+1}$.
Since $\chi(L(K\_{2n})) = \omega(L(K\_{2n})) = 2n-1$ we get that removing a point from $L(K\_{2n})$ wouldn't change its chromatic number. (By $\omega(\cdot)$we denote the [clique number](https://en.wikipedia.org/wiki/Clique_(graph_theo... | https://mathoverflow.net/users/8628 | Is $L(K_{2n+1})$ vertex-critical? | It is not vertex-critical for $n>1$, since we may have at most $n$ edges of the same color, thus at most $2n^2<n(2n+1)-1$ edges totally of $2n$ colors.
| 2 | https://mathoverflow.net/users/4312 | 269358 | 120,723 |
https://mathoverflow.net/questions/259305 | 8 | For the characteristic function $\mathbf E e^{i t X}$ of a random variable $X$ with $n+1$ finite moments, there is the well known and easy to prove bound on the remainder of the Taylor series
$$\left\lvert\mathbf E e^{i t X}-\sum\_{k=0}^n \frac{(it)^k}{k!}\mathbf E X^k\right\rvert\le\min\left\{\frac{\lvert t\rvert^{n... | https://mathoverflow.net/users/89934 | Taylor expansion of cumulant generating function |
>
> Can something similar be said for the remainder of the cumulant
> generating function $\log\mathbf E e^{itX}$ with an error bound in
> terms of cumulants?
>
>
>
**Yes and no.**
Your question is whether the union bound for Fourier transform can be somehow generalized to log Fourier transform for the class o... | 3 | https://mathoverflow.net/users/25437 | 269363 | 120,724 |
https://mathoverflow.net/questions/269343 | 1 | Consider $n\times 1$ vector $\alpha = (\alpha\_{1}, ..., \alpha\_{n})$, where $0<\alpha\_{i}<1$, and $\sum\_{i=1}^{n}\alpha\_i = 1$. Construct the $n\times n$ zero-diagonal matrix $A$ with $(i,j)$-th entry
$$A\_{ij} = \begin{cases}\frac{\alpha\_{i}}{1-\alpha\_{j}} & \text{for} \; i\neq j,\\ 0 & \text{otherwise.}\end{... | https://mathoverflow.net/users/18526 | Eigenvector of a nonnegative matrix in closed form | If you write $A\nu=\nu$ as a system of equations it can be written as
$$
\frac{\nu\_i}{1-\alpha\_i}=\sum\_{j=1}^{n}\frac{\alpha\_i\nu\_j}{1-\nu\_j}
\qquad \forall i=1, ..., n.
$$
Then the change of variable $x\_i=\frac{\nu\_i}{1-\alpha\_i}$. This gives the simple system $x\_i=\alpha\_i\sum\_jx\_j$, $i=1,\dots,n$, whic... | 4 | https://mathoverflow.net/users/68572 | 269371 | 120,726 |
https://mathoverflow.net/questions/269380 | 6 | [Perfect number](https://www.britannica.com/topic/perfect-number) is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3.
Is there a sequence of numbers which are equal to the sum of its proper divisors **excluding itself as well as 1**... | https://mathoverflow.net/users/109647 | Kindda-Perfect number: Is there a sequence of numbers which are equal to the sum of its proper divisors excluding itself as well as 1? | Such an abundant number with abundance 1 is called a **quasiperfect** number (which is a more professional way to say "kindda-perfect"). None have been found, according to [Wikipedia](https://en.wikipedia.org/wiki/Quasiperfect_number). This [1982 article](https://www.cambridge.org/core/journals/journal-of-the-australia... | 20 | https://mathoverflow.net/users/11260 | 269382 | 120,730 |
https://mathoverflow.net/questions/242064 | 2 | Let $p$ be prime number and let $A$ be a $k$-elements subset of $\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that $|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where $h$ is an integer with $2 \leq h \leq k$ and $h^{\hat{}}A$ denote the set of all sums of $h$ distinct elements of $A$. What is/are the ... | https://mathoverflow.net/users/93753 | When does the equality hold in Dias da Silva - Hamidoune Theorem? | The lower bound is optimal, and the bound *is* attained on arithmetic progressions: say, for $A=[1,k]$ one has $h^{\hat{}}A=[h(h+1)/2,kh-h(h-1)/2]$ whence $|h^{\hat{}}A|=kh-h^2+1$ (provided, say, $kh<p$).
Characterizing those $A$ for which equality is attained is subtler. In the case where $A$ is a set of integers, ... | 3 | https://mathoverflow.net/users/9924 | 269386 | 120,731 |
https://mathoverflow.net/questions/269378 | 10 | The Cauchy identity states that
$$
\prod\_{i,j} \frac{1}{1-x\_i y\_j} = \sum\_\lambda s\_\lambda(x) s\_\lambda(y),
$$
where $s\_\lambda(x)$ is the Schur function.
Is there a known decomposition of the product
$$
\prod\_{i,j,k} \frac{1}{1-x\_i y\_j z\_k}
$$
as a sum of Schur functions?
Essentially equivalently, let... | https://mathoverflow.net/users/40349 | Cauchy identity in three sets of variables? | Yes, up to the hard problem of determining Kronecker coefficients. Let $\Delta^\lambda$ be the Schur functor for the partition $\lambda$ of $r$ and let $S^\lambda$ be the corresponding irreducible representation of $S\_r$ with character $\chi^\lambda$. Schur–Weyl duality states that
$$ U^{\otimes r} \cong \bigoplus\... | 13 | https://mathoverflow.net/users/7709 | 269388 | 120,732 |
https://mathoverflow.net/questions/269369 | 3 | The derivative of a degree $5$ polynomial $p\in\mathbb{C}[z]$ is a degree four polynomial $p'\in\mathbb{C}[z]$, and as such, the zeros of $p'$ may be found explicitly using the quartic formulae.
One may think of a finite Blaschke product as playing the same role on the unit disk that a polynomial plays on the plane (... | https://mathoverflow.net/users/35158 | Finding the critical points of a degree $5$ Blaschke product | Even for degree $4$ Blaschke products, the critical points need not be contained in a solvable Galois extension of the field of coefficients of $P$. For instance, let $t$ be another variable which is supposed to parametrize the purely imaginary numbers. Take e.g. $w\_k=1/(k+1)+t$. Then the Blaschke product is, up to a ... | 5 | https://mathoverflow.net/users/18739 | 269391 | 120,734 |
https://mathoverflow.net/questions/269395 | 7 | According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the expected distribution of its factors?
By the Turán–Kubilius inquality, we see that this is the expectation:
$$ \left\{\... | https://mathoverflow.net/users/58242 | $\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$) | For precise information on many questions of this type you should consult the Cambridge Tract, *Divisors* by Hall and Tenenbaum. Here is [Hildebrand's review](http://www.ams.org/journals/bull/1990-22-01/S0273-0979-1990-15871-9/S0273-0979-1990-15871-9.pdf) in the Bulletin of this book. A much more precise version of you... | 12 | https://mathoverflow.net/users/38624 | 269398 | 120,736 |
https://mathoverflow.net/questions/269399 | 3 | Let $X$ be a (irreducible) variety, $\tilde{X}$ its normalization, $\pi\colon \tilde{X}\to X$ the natural map. Is there a "nice" characterization of when $\pi$ is a homeomorphism? E.g. is it enough to know that $\pi$ is injective?
If it makes things simpler, I'm most interested in the case that $X$ is an affine toric... | https://mathoverflow.net/users/36720 | When is the morphism to a variety from its normalization a homeomorphism? | It's easy to check that injectivity is enough. Indeed, normalisation is finite, hence closed. Since it's also dominant, it's surjective. A surjective closed injection is a homeomorphism.
**Remark.** We can actually say much more: since $\pi$ is a finite morphism (hence affine), we can reduce to the case of ring maps ... | 9 | https://mathoverflow.net/users/82179 | 269401 | 120,737 |
https://mathoverflow.net/questions/269411 | 2 | Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ a Sub-$\sigma$-Algebra. Moreover, let $X:\Omega\rightarrow\mathbb{R}$ be a random variable and $F:\mathbb{R}^{2}\rightarrow\mathbb{R}$ a measurable function with
$$\mathbb{E}\big[\;F(X,a)\;\big] \;<\; \infty$$
for all ... | https://mathoverflow.net/users/66399 | Measurability of a parametrized conditional expectation | First of all, you should be careful with how you phrase your question: the conditional expectation is defined up to a set of probability $0$, so you can (almost) always destroy the joint measurability by choosing bad slices. The best we can hope for is that we *can make* it measurable and that is, indeed, the case. Sta... | 3 | https://mathoverflow.net/users/1131 | 269417 | 120,738 |
https://mathoverflow.net/questions/269096 | 5 | Consider general Brownian bridge W(0)=0; W(T) = a. (Here ["general"](https://en.wikipedia.org/wiki/Brownian_bridge#General_case) means: $W(T)\ne 0$).
What is the probability W(t) >= b, for all $ t \in [0, T] $ ?
Is there close simple formula in terms of a, b , T ?
**EDIT** Is the answer equal to $1-exp(-2b(a+b)/... | https://mathoverflow.net/users/10446 | Probability of general Brownian (or non) bridge to be higher than given parameter? | For Brownian bridge, this is more or less a standard exercise. It is simpler to do it first for Brownian motion and then move to Brownian bridge. There are two steps. I assume $b<0$, otherwise the answer is $0$. Let $c=a-b>0$.
1) By the reflection principle,
$$P(W\_t\geq b \; \forall t\in [0,T], W\_T\in [a,a+dx])=P^{... | 4 | https://mathoverflow.net/users/35520 | 269436 | 120,743 |
https://mathoverflow.net/questions/239224 | 9 | In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an **impulse response**, a function $h:\mathbb{Z}\to\mathbb{C}$ such that for any input $x:\mathbb{Z}\to\mathbb{C}$ the output is just the convolution $x\ast h$ given by $(x\a... | https://mathoverflow.net/users/5963 | Finite realization of irrational transfer functions | There is at least one (rather trivial) way to formalize "implementability" that has the property that the only implementable functions are rational. Suppose that the implementation is defined as a continuous mapping $f:\mathbb R^{N+1}\to \mathbb R^N$ such that the last coordinate of the image of the current state conca... | 3 | https://mathoverflow.net/users/1131 | 269437 | 120,744 |
https://mathoverflow.net/questions/269444 | 4 | Let $T$ be a torus over a $p$-adic field $F$. Let $q = f(F/\mathbb{Q}\_p)$, and normalize the absolute value $| \cdot |$ on $F$ so that a uniformizer has value $\frac{1}{q}$.
Let $X(T)\_F$ be the group of rational characters of $T$ which are defined over $F$. Define a homomorphism
$$H\_T: T(F) \rightarrow \textrm{H... | https://mathoverflow.net/users/38145 | The unique maximal compact subgroup of a torus | There might be an easier way to see this but here is one way.
Any $F$-torus $T$ can be written as $T = T\_s T\_a$ with finite intersection, where $T\_s$ (resp. $T\_a$) is the unique maximal split (resp. anisotropic) subtorus of $T$ (Springer, LAG, Proposition 13.2.4). Further, the co-characters all come only from th... | 3 | https://mathoverflow.net/users/2720 | 269445 | 120,747 |
https://mathoverflow.net/questions/265882 | 2 | This is, can we find an $m-$dependent ($1\leq m<\infty$) non-stationary, non-independent stochastic process $(X\_{k})\_{k}$ with the property that $\mathbb{P}[X\_{k+1}\in A |X\_{k}]=\nu(X\_{k},A)$ for a fixed (independent on $k$) conditional probability distribution function $\nu:\mathbb{R}\times\mathcal{R}\to [0,1]$? ... | https://mathoverflow.net/users/66825 | Is there a Conditionally Stationary, non-stationary Process which is (strictly) $m-$Dependent? | Let $\{X\_i\}\_{i=0}^{\infty}$ be an i.i.d. Bernoulli sequence with $P[X\_i=1]=P[X\_i=0]=1/2$. Consider:
$$\{X\_1, X\_0, X\_2, X\_1, X\_3, X\_2, X\_4, X\_3, X\_5, X\_4, X\_6, X\_5, ...\}$$
| 0 | https://mathoverflow.net/users/73850 | 269475 | 120,751 |
https://mathoverflow.net/questions/268667 | 5 | Consider a number field $K$ with ring of integers $O\_K$. On the affine scheme $\overline X=\operatorname{Spec}(O\_K)$ we have the well known one dimensional Arakelov geometry.
Let $\overline D=\sum\_{\mathfrak p\neq 0} r\_\mathfrak p\mathfrak p+\sum\_{\sigma} \lambda\_{\sigma}\sigma$ be an Arakelov divisor on $\over... | https://mathoverflow.net/users/109246 | Is there any definition of $H^1$ in one dimensional Arakelov geometry | There is a definition of $h^{1}(\overline{D})$ by many people, and a definition of $H^{1}(\overline{D})$ by [Alexrander Borisov](https://link.springer.com/article/10.1023/A:1023297625434) using the notion of ghost spaces of the second kind. But neither is the same as the "original ones" you are talking about.
As Neu... | 10 | https://mathoverflow.net/users/18850 | 269477 | 120,752 |
https://mathoverflow.net/questions/269476 | 14 | The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
Let's understand the determinant $\det(M\_{ij})\_1^{\infty}$ to mean $\lim\_{n\rightarrow\infty}\det(M\_{ij})\_1^n$. ... | https://mathoverflow.net/users/66131 | "sinc'n determinant" | Let's look at $a\_n=\det\left[\frac{(i-1)!}{(2j-1)!}\binom{i^2-\theta^2}j\right]\_{i,j=1}^{n}$. By taking out common factors from rows we can write
$$a\_n=\left(\prod\_{i=1}^n (i-1)!(i^2-\theta^2)\right)\det\left[\frac{1}{(2j-1)!}\frac{1}{(i^2-\theta^2)}\binom{i^2-\theta^2}{j}\right]\_{i,j=1}^{n}$$
$$=\left(\prod\_{i=1... | 24 | https://mathoverflow.net/users/2384 | 269481 | 120,753 |
https://mathoverflow.net/questions/269480 | 11 | Sorry for this question, but I think it is really important the intuition here.
Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms".
Tannakian categories are very linear objects. Someone could shed some light on the intuition of why (in this case) y... | https://mathoverflow.net/users/83957 | Why linearization leads to arithmetization? | I think:
1. The category of varieties over $\mathbb Q$ is already very arithmetic.
2. One reason that the linearization is considered arithmetic is that so much of the tractable arithmetic information is preserved by linearization, so our arithmetic tools can be used to study it (on the other hand, we have many power... | 13 | https://mathoverflow.net/users/18060 | 269482 | 120,754 |
https://mathoverflow.net/questions/268698 | 6 | If the holomorphic vector bundle of $E$ on a Kähler manifold $M$ admit Hermitian-Yang-Mills metric then we have the following known universal inequality of Yau-Uhlenbeck $$\int\_Mc\_2(End E)\wedge \omega^{n-2}\geq 0$$
Is it known for other chern classes $c\_{2n}$?
$$\int\_Mc\_{2k}(End E)\wedge \omega^{n-2k}\geq 0$$... | https://mathoverflow.net/users/nan | Yau-Uhlenbeck inequality works for higher Chern class? | This is not true, as we can check by calculating the intersection number $\int\_X c\_4(\operatorname{End}(T\_X)) \cup \omega^{n-4}$ for some easy examples of Kahler-Einstein spaces:
This intersection number seems to be always positive for $\operatorname{End}(T\_{\mathbb{P}^n})$ by numerical calculations. There it is ... | 1 | https://mathoverflow.net/users/4054 | 269490 | 120,757 |
https://mathoverflow.net/questions/269484 | 2 | Suppose $\{X\_t; t \in \mathcal{X}\}$ is a centered Gaussian Process with covariance function $k(\cdot,\cdot)$, and let $d(x,y) = \mathbb{E}[(X\_x-X\_y)^2]$.
I am trying to find a tail bound for the suprema of this Gaussian process over a $d$-ball $B(\rho) = \{x \in \mathcal{X}: d(x,0)\leq \rho\}$, i.e., $Pr( \sup\_{... | https://mathoverflow.net/users/85512 | A question about Gaussian Processes suprema | Question 1
>
> My Question is whether we can write an explicit form of these sets
> $B\_t$ for my setting, for which I can get bounds on entropy numbers?
>
>
>
It depends. The thin sets $B\_t$ are thinned from $B$. As you defined, $B$ can be written as $$B(\rho)=\{x\in\mathcal{X}:\mathbb{E}[(X\_{x}-X\_{0})^{2... | 1 | https://mathoverflow.net/users/25437 | 269510 | 120,763 |
https://mathoverflow.net/questions/269538 | 3 |
>
> What is a standard technical term in axiomatic set theory for the operation which sends a given set $A$ to the set $A':=\{\{a\}\colon a\in A\}$?
>
>
>
(Replacement implies that $A'$ is a set.)
Some pointers to relevant places in the literature would also be appreciated, especially if (0) the treatments e... | https://mathoverflow.net/users/108556 | Usual technical term for replacing a set by the set of singletons of its members? | $A'$ is the [discrete partition](https://math.stackexchange.com/questions/153276/a-formal-name-for-smallest-and-largest-partition) of $A$.
That is, we think of it as a partition of $A$ induced by the finest equivalence relation, the identity relation.
| 11 | https://mathoverflow.net/users/4600 | 269541 | 120,770 |
https://mathoverflow.net/questions/219822 | 30 | [It seems](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/) that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.
>
> Is there any way to obtain a copy (online or not) of "La longue Marche à travers la
> théorie de Galois"?
>
... | https://mathoverflow.net/users/43108 | Grothendieck's "La longue Marche à travers la théorie de Galois" | All the manuscripts have been finally made avaible by Montpellier university.
You can find all of them [here](https://grothendieck.umontpellier.fr/archives-grothendieck/), avaible in pdf.
The items related to “La longue Marche" are:
**La "Longue Marche" à travers la théorie de Galois**
* "Longue Marche" [brouil... | 18 | https://mathoverflow.net/users/43108 | 269558 | 120,775 |
https://mathoverflow.net/questions/269026 | 14 | Starting this January, I observe more and more inconsistencies in the MathSciNet citation database. A typical situation is that the number of citations shown in the description of a paper differs from the number of matches that one actually gets by following the link. For example, take papers by Gerd Faltings (sorted b... | https://mathoverflow.net/users/109450 | Inconsistencies in MathSciNet citation database | As @ZachTeitler pointed out by quoting my blog post, one of our new features acted a bit like a bug: *related papers*. We have made adjustments to avoid the "counting with multiplicity" issue, but some unexpected cases still slipped in.
One of the examples involving Faltings given by @AlekVainshtein had related pape... | 12 | https://mathoverflow.net/users/49409 | 269560 | 120,776 |
https://mathoverflow.net/questions/269549 | 1 | Assume $L$ is unbounded, non-self adjoint operator for functions over two space dimensions $(x,y)\in \mathbb{R}^2$, such that upon fourier transforming w.r.t $y$, one can reduce the operator to (for fixed $y$ frequency $\omega$):
$\tilde{L}\_{\omega}:=\partial\_{xx}+f(x)\partial\_x +(i\omega x)$.
Are there any issu... | https://mathoverflow.net/users/106076 | Decomposition of spectrum of a (unbounded, non-self-adjoint) linear operator in two spatial dimensions | What you need are some estimates on the norm of the resolvent, $N(\omega,\lambda) = \| (\tilde{L}\_\omega - \lambda)^{-1} \|$, as a function of $\omega$. In general, if $N(\omega,\lambda)$ grows too quickly for large $\omega$ for a fixed $\lambda$, $\lambda$ may fail to be in the spectrum of $L$ even if $\lambda$ is no... | 2 | https://mathoverflow.net/users/2622 | 269561 | 120,777 |
https://mathoverflow.net/questions/269559 | 2 | I am looking for pointers/references to results of the following kind:
For $M$ a real or integer square matrix drawn at random from some "reasonably" nice set of square matrices (possibly infinite/uncountable), with a "reasonably" nice probability distribution, then with "reasonably" high probability the spectrum of ... | https://mathoverflow.net/users/109743 | Spectrum of random matrices has dominant eigenvalues | The "good" matrices, those whose eigenvalues of largest absolute value are either a single simple real eigenvalue or a conjugate pair of simple eigenvalues, form a dense open set in the $n \times n$ real matrices. In fact, if $B$ is any "good" $n \times n$ matrix and $A$ any $n \times n$ matrix,
$A + t B$ will be "good... | 1 | https://mathoverflow.net/users/13650 | 269567 | 120,780 |
https://mathoverflow.net/questions/269532 | 3 | Let $\Sigma$ be a subshift of finite type (SFT) with the alphabet $\{0,1\}$, which is given by the set of forbidden words $\mathcal F$, all of length $N$.
**Question.** Is there a $\delta>0$ such that for any such $\mathcal F$ of cardinality at most $\delta 2^N$, the topological entropy of $\Sigma$ is positive? If s... | https://mathoverflow.net/users/8131 | Subshifts of finite type of guaranteed positive entropy | Here's a heuristic suggesting that maybe something like $2^N/N$ forbidden words could be enough to give 0 entropy:
Consider $N$-step transitions between blocks of length $N$. That is: if $L\_N$ is the set of all words of length $N$, you're asking: for which $U,V$ in $L\_N$ is the concatenation $UV$ legal.
Clearly ... | 3 | https://mathoverflow.net/users/11054 | 269568 | 120,781 |
https://mathoverflow.net/questions/269274 | 2 | It pertains to the classical literature on the subject (see e.g. [here](https://golem.ph.utexas.edu/category/2014/04/on_two-dimensional_monad_theory.html)) the fact that an accessible 2-monad $T$ on a 2-category $\cal K$ induces an adjunction
$$
\mathbf{Strict}\text{-}T\text{-}\textbf{Alg}\leftrightarrows \mathbf{Pseud... | https://mathoverflow.net/users/7952 | Strictification of the squaring monad | A strict algebra of the squaring monad is precisely a [strict factorization system](https://ncatlab.org/nlab/show/strict+factorization+system). A normal pseudo-algebra (unit holds strictly) is precisely an orthogonal factorization system. See [here](https://doi.org/10.1016/0022-4049(93)90171-O). The pseudo-algebras wil... | 2 | https://mathoverflow.net/users/2362 | 269570 | 120,782 |
https://mathoverflow.net/questions/269552 | 8 | Let $\mathfrak{I}:=\big\{ \, f:=\sum\_{k=0}^\infty f\_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f\_0=0 \;\text{ and }\; f\_1=1\big\}$. A most basic result about linearization states that, for any $f\in\mathfrak{I}$ and for any $\lambda\in\mathbb{C}$ not a root of unity, there exists a unique $h\in\mathfrak{I}$ that l... | https://mathoverflow.net/users/6101 | Linearizing a power series by conjugation | You may find useful information in a recent article by D. Sauzin and al. "Explicit linearization of one-dimensional germs through tree-expansions" [here](https://hal.archives-ouvertes.fr/hal-01053805/), where they use "mould calculus" (introduced by J. Écalle 40 years ago) to write down the coefficients $h\_n$ and expl... | 6 | https://mathoverflow.net/users/24309 | 269579 | 120,786 |
https://mathoverflow.net/questions/243527 | 5 | **Introduction:**
Let $M$ be a Riemann surface, then a Green function on $M$ is an element $g\in C^\infty(V)$ where $V=M\setminus\{x\_1,\ldots,x\_r\}$ and around each point $p\in M$ we have:
$$g=a\log\left|\phi\right|^2+u$$
where $\phi$ is a complex chart centered in $p$, $u$ is a $C^\infty$ function and $a\in \ma... | https://mathoverflow.net/users/65980 | Fiber at infinity of an arithmetic surface $X$ as an element of $\widehat{\operatorname{Div}(X)}$ | Here is another low brow way to look at it by tracing Arakelov's ideas in his paper.
Let $X$ be a curve over $K$, where $K$ is a number field. Let $\infty$ denoting the archimedean valuations of $K$ to $\mathbb{C}$. Let $\sigma\in \infty$, then $X\_{\sigma}=X\_{K}\otimes\_{\sigma}\mathbb{C}$. This is a one dimension... | 1 | https://mathoverflow.net/users/18850 | 269585 | 120,787 |
https://mathoverflow.net/questions/269537 | 4 | I am wondering about a definition of a direct sum in page $31$ of [this paper](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0010437X11005525) by R. Liu.
I am following the notations in page $31$ of the above paper. Let $V$ be a crystalline irreducible representation of the absolute Galois ... | https://mathoverflow.net/users/69289 | A definition of a (amalgamated) direct sum | The key point here is that $\pi(\alpha)$ and $\pi(\beta)$ are isomorphic representations: both of them are the algebraic representation $Sym^{k-2}$ tensored with the smooth representation $Ind\_B^G (\alpha \otimes \beta |\cdot|^{-1})^{sm} = Ind\_B^G (\beta \otimes \alpha |\cdot|^{-1})^{sm}$. (This last isomorphism is a... | 7 | https://mathoverflow.net/users/2481 | 269593 | 120,789 |
https://mathoverflow.net/questions/268530 | 5 | Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes it is not necessarily important to decide which quotients are isomorphic (this is already unclear to me for $\mathbb{Z}/... | https://mathoverflow.net/users/2841 | Classification of the quotients of the ring Z/4 [X] | Let $I$ be an ideal in $(\mathbf{Z}/4\mathbf{Z})[X]$. Let $J\_I$ be its projection to $(\mathbf{Z}/2\mathbf{Z})[X]$, and write its intersection with $2(\mathbf{Z}/4\mathbf{Z})[X]$ as $2K\_I$, for some ideal $K\_I$ of $(\mathbf{Z}/2\mathbf{Z})[X]$ (precisely, first define $K'\_I=\{P \in (\mathbf{Z}/4\mathbf{Z})[X]:2P\in... | 6 | https://mathoverflow.net/users/14094 | 269597 | 120,790 |
https://mathoverflow.net/questions/269581 | -3 | If $G, H$ are finite, simple, undirected graphs, their *categorical product* $G\times H$ is defined by $V(G\times H) = V(G)\times V(H)$ and $$E(G\times H) = \big\{\{(v\_1, w\_1),(v\_2,w\_2)\}: v\_i \in V(G)\land w\_i\in V(H) \land \{v\_1, v\_2\} \in E(G) \land \{w\_1,w\_2\}\in E(H)\big\}.$$
A graph is said to be *cri... | https://mathoverflow.net/users/8628 | Product of critical graphs and Hedetniemi's conjecture | It doesn't imply that the categorical product of critical graphs is critical, and this is not true. For instance, the complete graph on three vertices, $K\_3$, is 3-critical, but $K\_3 \times K\_3$ is not since it is not just an odd cycle (which is the only type of 3-critical graph).
| 3 | https://mathoverflow.net/users/18606 | 269604 | 120,791 |
https://mathoverflow.net/questions/269573 | 2 | Let $\{v\_1,v\_2, \cdots , v\_n, w\_1,w\_2, \cdots ,w\_n\}$ be a basis of $\mathbb C^{2n}$. For a $n$-dimensional subspace $V \in Gr(n,\mathbb C^{2n})$ define another $n$ dimensional subspace $\bar{V} \in Gr(n, \mathbb C^{2n})$ as follows:
If $V$ is generated by $n$ linearly independent vectors $x\_i\in \mathbb C^{2... | https://mathoverflow.net/users/109746 | lines in projective spaces | Since the Grassmannian is an intersection of quadrics, the intersection $Gr \cap L$ is either zero-dimensional of length at most 2, or equals to $L$. So, to prove the claim it is enough to rule out the possibility that $L \subset Gr$.
For this just note that a line on $Gr(n,2n)$ is determined by a flag $U\_{n-1} \sub... | 2 | https://mathoverflow.net/users/4428 | 269606 | 120,792 |
https://mathoverflow.net/questions/269563 | 2 | Let $U \in \mathbb{R}^{n \times n}$ be a unitary matrix, $U$ can be nonsymmetric, its eigenvalues can be complex numbers and all have modulus $1$.
Is there an upper bound for the maximum **singular value** of its skew symmetric part (which is not necessarily unitary) depending on its eigenvalues?
i.e.: Is there an ... | https://mathoverflow.net/users/95387 | Norm of a the skew symmetric part of a unitary matrix | Since I misread the question I will clarify my comments into a formal answer (which is, in the end, quite elementary). Since $U$ is a real orthogonal matrix,
it has a basis of eigenvectors when viewed as a complex matrix say $\{v\_{1},v\_{2}, \ldots,v\_{n} \}.$ Also, the eigenvalues of $U$ all lie on the unit circle, a... | 3 | https://mathoverflow.net/users/14450 | 269609 | 120,793 |
https://mathoverflow.net/questions/263099 | 9 | Is there any reference which studies sets of random variables as independence systems, a type of combinatorial object (see below)?
**Motivation:**
In particular, since independence systems are abstract simplicial complexes, this would allow one to apply homology theory (I think) to study families of random variables.... | https://mathoverflow.net/users/93694 | Is there a combinatorial/topological treatment of statistical independence? | It is aimless to extend "statistical independence" beyond category of $\sigma$-subalgebras and probability-preserving morphisms on a fixed probability space $\prod,\mathcal{A}$ as pointed out by the comment by @Dima Pasechnik.
We restrict ourselves to the category of $\sigma$-subalgebras, whose objects are the colle... | 3 | https://mathoverflow.net/users/25437 | 269615 | 120,795 |
https://mathoverflow.net/questions/269598 | 5 | Is it always possible to put a compatible Kähler metric on an open complex manifold (non-compact, without boundary, say of finite topological type)?
**Edit 1**: Michael Albanese pointed out a nice counterexample (namely the punctured Hopf surface) when this is impossible for topological reasons. **Question**: Would i... | https://mathoverflow.net/users/99732 | Kähler structure on open complex manifold | The punctured Hopf surface is also an example for the revised question. It is easy to see that the Hopf surface $X$ is parallelizable (since it is diffeomorphic to $S^3\times S^1$); hence, $M=X- \{x\}$ admits an immersion in $R^4={\mathbb C}^2$ (Hirsch-Smale theory). Now pull-back the standard (flat) Kahler structure f... | 6 | https://mathoverflow.net/users/21684 | 269628 | 120,798 |
https://mathoverflow.net/questions/269620 | 1 | I am interested in finding the approximate answer to the following infinite sum
\begin{equation}
\sum\_{l=0}^{\infty}( l+a) \exp^{b{(l+c)}^2}
\end{equation}
in the case where $a>0 , b<0 , c>0$ and $a$ and $c$ are both large. Does any one know how to approach this problem and where to look for methods if I need to fin... | https://mathoverflow.net/users/109768 | Approximating an infinite Sum | Approximating the sum by an integral, gives
$$F(a,b,c)=
\int\_0^\infty( l+a) \exp\left[b{(l+c)}^2\right]\,dl=\frac{\sqrt{\pi } (a-c) \text{Erfc}\left(c\sqrt{-b} \right)}{2\sqrt{-b}}-\frac{e^{b c^2}}{2b}.$$
This should be accurate for large $c$. Here is a comparison for $b=-1$, $c=2a$ of the sum (blue) and the integral ... | 4 | https://mathoverflow.net/users/11260 | 269630 | 120,800 |
https://mathoverflow.net/questions/269572 | 1 | This question is cross-posted in MO and MSE <https://math.stackexchange.com/questions/2276064/about-pairwise-distances-of-some-points-in-a-riemannian-manifold-m-of-rm-se>
>
> Assume that there are points $p\_i,\ 1\leq i\leq m$ in a Riemannian
> manifold $(M,d)$ s.t.
>
>
> (1) all sectional curvatures are equal t... | https://mathoverflow.net/users/36572 | About pairwise distances of some points in a Riemannian manifold $M$ of ${\rm sec}\ M\geq 1$ | We can assume that the set $\{p\_1,\dots,p\_n\}$ is maximal.
Note that the set
$$K\_{n,n-1}=\{\,x\in M\mid |x-p\_i|\ge\tfrac\pi2\ \text{for}\ i<n-1\,\}$$
is convex and it contains a point $z$ such that $|p\_i-z|=\tfrac\pi2$ for any $i<n$.
Since $\{p\_1,\dots,p\_n\}$ is maximal, $|p\_n-z|\le \tfrac\pi2$.
By compariso... | 2 | https://mathoverflow.net/users/1441 | 269633 | 120,802 |
https://mathoverflow.net/questions/269643 | 10 | Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively.
**Question:** Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ isomorphic as abstract groups for any reasonable non-discrete space $M$? Say, $M=\mathbb{R}$ or $M=\mathbb{S}^1$.
Obvious... | https://mathoverflow.net/users/89313 | Homeomorphisms vs Borel automorphisms | Let $\bar X = X\cup \{\infty\}$ be the 1-point compactification of a discrete space $X$. Then the autohomeomorphisms of $\bar X$ are all permutations $p$ of $X$ (extended by $p(\infty)=\infty$), whereas the Borel automorphisms are all permutations of $\bar X$ (since every subset of $\bar X$ is Borel, even open$\cup$clo... | 10 | https://mathoverflow.net/users/14915 | 269646 | 120,804 |
https://mathoverflow.net/questions/269626 | 4 | It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}\_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in each dimension $d$, all but a finite number of indecomposable $KQ$-modules occur in a finite number of 1-parameter famil... | https://mathoverflow.net/users/95742 | The explicit indecomposable representations of (any) Euclidean quiver of type E | Crawley-Boevey's notes (<http://www1.maths.leeds.ac.uk/~pmtwc/quivlecs.pdf>) show how to get a family of indecomposable modules indexed by $\mathbb{P}^1$ by a general method. Consider the projective $P$ covering the simple for the extending vertex (this is itself simple in the example above, but in other orientations i... | 5 | https://mathoverflow.net/users/66 | 269650 | 120,805 |
https://mathoverflow.net/questions/269623 | 11 | Let $\{x\_i\}:=\{x\_1=5, x\_2=13, x\_3=29, x\_4=37, x\_5=45, \dots \}$
be the sequence of those positive integers of the form
$$
p^{4\alpha+1}n^2$$
in increasing order where $p\equiv 5\pmod 8$ is prime and $\gcd(n,2p)=1$.
Next, define the sequence $\{y\_i\}$ by letting $y\_i:=\frac{x\_i-5}{8}$.
The first few terms of t... | https://mathoverflow.net/users/66131 | "strange" diophantine and parity of the partition function | Your conjecture is true! Here is one way to get it using some mod 4 generatingfunctionology. The answer got a bit long, so I divided it into two parts, as an attempt to improve readability.
---
**Part1:** Let's denote by $Y(q)$ your generating function $\sum\_{i\geq 1}q^{y\_i}$. We will prove that
$$Y(q)\equiv \... | 13 | https://mathoverflow.net/users/2384 | 269657 | 120,806 |
https://mathoverflow.net/questions/269656 | 4 | I am trying to read Michael Hrusák's ["MAD Families and the rationals"](http://www.matmor.unam.mx/~michael/reprints_files/MAD_families_and_the_rationals.pdf) and I have studied Forcing using Kunen's books. On theorem 1, the author says that no new real is contained in a countable closed set coded in the ground model. I... | https://mathoverflow.net/users/nan | No new real is contained in a countable closed set | While you can code a closed set $A \subseteq 2^\omega$ by any countable dense subset of $A$; I think it is more transparent to code $A$ by the tree $T\_A:=\{ f\upharpoonright n : f\in A, n\in \omega\}$. Then $A=[T\_A] = \{f: \forall n\ f\upharpoonright n \in T\}$.
Any countable closed set $A$ can be decomposed by Ca... | 3 | https://mathoverflow.net/users/14915 | 269658 | 120,807 |
https://mathoverflow.net/questions/188939 | 8 | Recall that a space $X$ is called *locally equiconnected* or *LEC* if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this concept at [this MO question](https://mathoverflow.net/questions/74447/when-is-the-diagonal-inclusion-a-hurewicz-cof... | https://mathoverflow.net/users/8103 | When is the diagonal inclusion a $\Sigma_2$-cofibration? | I ended up needing this again, and eventually proved that $d:X\hookrightarrow X\times X$ is a $\Sigma\_2$-cofibration as long as $X$ is an ENR. Although this doesn't quite answer the original question as asked, it seems general enough to be useful, so I thought I'd post it here.
More generally, if $G$ is a finite gro... | 3 | https://mathoverflow.net/users/8103 | 269659 | 120,808 |
https://mathoverflow.net/questions/269663 | 12 | Two abstract groups $G$ and $H$ are called equivalent, $G\sim H$, if each of them is isomorphic to a subgroup of another.
**Question:** Can a simple group $G$ be equivalent to a non-simple group $H$?
Of course, we are talking about infinite groups here. Thanks.
| https://mathoverflow.net/users/89313 | Can a simple group be equivalent to a non-simple group? | Yes. (This is corrected and expanded since the first version.) There are easy examples of simple groups $G$ such that $G\times G$ is isomorphic to a subgroup of $G$. One example is the group of finitely supported even permutations of a countable infinite set. Another is the quotient of all permutations of a countably i... | 15 | https://mathoverflow.net/users/6666 | 269664 | 120,811 |
https://mathoverflow.net/questions/269649 | 10 | A *Catalan path* of semilength $n$ is a path from $(0,0)$ to $(2n,0)$ that proceeds by taking northeast (1,1) or southeast (1,-1) steps, and never goes below the $x$-axis. The *area* of a path $P$ is the area beneath the path and above the $x$-axis. So, for example, the path of semilength 3 that goes $(0,0)-(1,1)-(2,2)... | https://mathoverflow.net/users/21690 | Distribution of the area statistic for Catalan paths | Notice that the number of Catalan paths of area at least $cn^{\frac{3}{2}+\varepsilon}$ is less than the number of all paths that deviate from the horizontal axis by at least $n^{\frac{1}{2}+\varepsilon}$. Let $(S\_k, k\geq 0)$ be the one-dimensional simple random walk starting from the origin. We need to find an upper... | 5 | https://mathoverflow.net/users/81488 | 269669 | 120,812 |
https://mathoverflow.net/questions/269662 | 0 | I asked this question in MSE about a month ago, and didn't get a reply as of yet.
I'll copy and paste this question here, hopefully to get some response.
I am reading the following paper:
<http://www.kryakin.org/k/butzer_johnen_k-func.pdf>
called: ON THE EQUIVALENCE OF THE K-FUNCTIONAL AND MODULI OF CONTINUITY AND ... | https://mathoverflow.net/users/13904 | Steklov means (A paper by H.Johnen and K. Scherer) | Ok, I think I figured it out.
We can notice that the $|(-1)^r|=1$,so the sum
$$\bigg| \sum\_{k=0}^r (-1)^{r-k}\ldots \bigg| = \bigg| \sum\_{k=0}^r (-1)^{-k}\ldots \bigg| = \bigg| (-1)^r f(x) + \sum\_{k=0}^r (-1)^{r-k} \ldots \bigg|$$
That solves my problem.
Embarrassing... :-D
| 0 | https://mathoverflow.net/users/13904 | 269670 | 120,813 |
https://mathoverflow.net/questions/269668 | 5 | I have just started reading Hain's paper *[On the Universal Elliptic KZB Connection](https://arxiv.org/pdf/1309.0580.pdf)*. I am a bit confused about a comment made there about base points on orbifolds. I am still very new to the idea of orbifolds so I apologise in advance for this likely being very trivial.
Let $\m... | https://mathoverflow.net/users/60535 | Maps to the universal punctured elliptic curve | It is indeed true that a holomorphic map $\mathbb C \to \mathcal E'$ is constant. This is because it must factor through the universal cover of $\mathcal E'$, since $\mathbb C$ is simply connected. But the universal cover is a product of two copies of the complex unit disk, so the result follows by the maximum principl... | 6 | https://mathoverflow.net/users/1310 | 269671 | 120,814 |
https://mathoverflow.net/questions/269241 | 11 | The question is as in the title, but let me explain a bit.
Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm curious if there is a similarly natural inner model for AD$\_\mathbb{R}$.
Now, ZF + $V=L(\mathcal{P}(\mathbb{R}))$ + l... | https://mathoverflow.net/users/8133 | Is there a natural inner model of AD$_\mathbb{R}$? | A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le\_W A$, where $\le\_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathb... | 6 | https://mathoverflow.net/users/6085 | 269690 | 120,821 |
https://mathoverflow.net/questions/269707 | 4 | This "innocent-looking" identity came out of some calculation with determinants, and I like to inquire if one can provide a proof. Actually, different methods of proofs would be of valuable merit and instructional.
>
> **Question.** Can you justify the following identity?
> $$\prod\_{j=1}^n\binom{2j}j=\prod\_{j=1}... | https://mathoverflow.net/users/66131 | An identity for product of central binomials | We have
$$\prod\_{j=1}^n \binom{2j}{j} = \frac{2!4!\cdots (2n)!}{(1!2!\cdots n!)^2}$$
and
$$\prod\_{j=1}^n 2\binom{n+j}{2j} = 2^n\frac{(n+1)!(n+2)!\cdots (2n)!}{(2!4!\cdots (2n)!)\cdot (0!1!\cdots (n-1)!)} = \frac{(n+1)!(n+2)!\cdots (2n)!}{(1!3!\cdots (2n-1)!)\cdot (1!2!\cdots n!)}.$$
Diving the former by the latter, w... | 7 | https://mathoverflow.net/users/7076 | 269711 | 120,828 |
https://mathoverflow.net/questions/269713 | 2 | Consider $\mathbb C^n$ with coordinates $(z\_1,\dots,z\_n)$, $z\_j=x\_j+iy\_j$. Let $\omega=\sum dx\_i\wedge dy\_i$. Let us call by a *radial diffeomorhpism* $\varphi$ of $\mathbb C^n$ a diffemorphism of $\varphi:\mathbb C^n\to \mathbb C^n$ that sends any $z\in \mathbb C^n$ to $f(|z|)\cdot z$, where $f$ is a smooth pos... | https://mathoverflow.net/users/13441 | Pull-backs of $\sum dx_i \wedge dy_i$ under radial diffeomorphisms of $\mathbb C^n$ | Yes, it's true, and it follows immediately from a simple calculation: Since $f$ is assumed to be a function of $|z|$ such that $\phi(z) = f(|z|)z$ is a smooth diffeomorphism, it follows that $\mathrm{d}f$ is a multiple of $\mathrm{d}\bigl(|z|^2\bigr)$, so $\partial f$ is a multiple of $\partial\bigl(|z|^2\bigr)$. Now
$... | 5 | https://mathoverflow.net/users/13972 | 269714 | 120,829 |
https://mathoverflow.net/questions/269174 | 11 | This quest has its impetus in [a paper](https://arxiv.org/abs/1305.6083) by Stanley and Zanello. I became curious about
*What is the sum of all hooks lengths of all partitions that fit
inside the $n$-th staircase partition?*
On the basis of experimental data, I'm prompted to ask:
>
> **Question.** Let $\lambda=... | https://mathoverflow.net/users/66131 | Hooks in a staircase partition: Part I | The following method should give a proof, but I haven't done the
computation. Let $f\_n$ denote the desired sum. I use the result of
Douglas Zare's answer that
$$ f\_n = \sum\_{\mu\subseteq \delta\_n} \sum\_i \mu\_i^2, $$
where $\delta\_n=(n,n-1,\dots,1)$. Let
$$ g\_n = \sum\_{\mu\subseteq \delta\_n} \sum\_i \mu\_i. ... | 4 | https://mathoverflow.net/users/2807 | 269716 | 120,831 |
https://mathoverflow.net/questions/269061 | 8 | Kirszbraun's theorem for $\mathbb{R}^2$ states the following:
>
> Given any set $S\subset \mathbb{R}^2$ and any Lipschitz function $f:S\rightarrow \mathbb{R}^2$ with Lipschitz constant $k$, $0< k< \infty$, for any set $F$ which contains $S$ there exists a function $\tilde f:F\rightarrow C$ such that $Lip(\tilde{f})... | https://mathoverflow.net/users/nan | Bi-Lipschitz version of Kirszbraun's extension theorem | It seems that you are asking the following:
>
> Assume $f\colon\{x\_1,x\_2,x\_3\}\to\mathbb R^2$ is a map with bi-Lipschitz constants $k$ and $1$ ($0<k<1$). Is it possible to extend $f$ to a bi-Lipschitz map $\bar f\colon\mathbb R^2\to\mathbb R^2$ with the same constants?
>
>
>
The answer in "no".
Set $x\_i'... | 6 | https://mathoverflow.net/users/1441 | 269724 | 120,834 |
https://mathoverflow.net/questions/269688 | 5 | Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to ensure that the question isn't answered negatively by exhibiting a not so convincing counter-example.
Let $Coh(A)$ be... | https://mathoverflow.net/users/22810 | Closed symmetric monoidal structure on the derived category of modules whose unit is a dualizing complex? | Let me answer the question in the title and explain the notation $- \otimes^{!}-$, forgive me for ignoring some finiteness conditions.
Let's start with something somewhat simpler - take any field $k$, and for any finitely generated $k$-algebra $A$ with structure map $f:k \to A$, let $R\_A = f^{!}(k)$, the canonical (or... | 5 | https://mathoverflow.net/users/109810 | 269735 | 120,835 |
https://mathoverflow.net/questions/269691 | 4 | I have problems to find out how to do discrete simulation of the Lévy walk. I can sum my doubts in a few questions:
1. According to [Wikipedia](https://en.wikipedia.org/wiki/L%C3%A9vy_flight) it seems to me that Lévy flight can be produced just by integration of samples with Lévy distribution, in other words: $x(k+1)... | https://mathoverflow.net/users/109796 | Simulation of Lévy walk | 1. Indeed, the Lévy flight is a random walk where the step increments $\nu$ are i.i.d. with a Lévy distribution, $p(\nu)\rightarrow 1/\nu^{1+\alpha}$ for $\nu\rightarrow\infty$, with exponent $0<\alpha<2$.
2. [Generating Lévy random variables from Gaussian.](http://andreweckford.blogspot.nl/2011/05/generating-levy-rand... | 3 | https://mathoverflow.net/users/11260 | 269742 | 120,837 |
https://mathoverflow.net/questions/269682 | 7 | Last night I was thinking about some related statements which follow from ZF+DC, but it actually seems they only need DC to hold in some outer model of the universe. In particular, let $M \models ZF.$ Consider the following claims (all relativized to $M$):
1. For any sentence $\sigma$ such that $M \models \sigma,$ th... | https://mathoverflow.net/users/109573 | Absoluteness, reflection to ctms, and choice in outer models | Yes, all three of these statements can be proved in ZF, without any DC assumption.
For statement 1, assume $M\models\newcommand\ZF{\text{ZF}}\ZF+\sigma$. By the reflection theorem, there is some ordinal $\theta$ with $(V\_\theta)^M\models\sigma$. One doesn't need DC to prove the reflection theorem, since the argumen... | 8 | https://mathoverflow.net/users/1946 | 269752 | 120,842 |
https://mathoverflow.net/questions/269491 | 3 | Suppose I have positive semidefinite matrices $A$ and $B$. Then
$$\begin{bmatrix} A & X\\ X^T & B\end{bmatrix} \succeq 0$$
for $X = A^{\frac 12} C B^{\frac 12}$, where $C$ is the contraction matrix with maximum eigenvalue less than $1$.
*Horn, Roger A.; Johnson, Charles R.*, Topics in matrix analysis, Cambridge ... | https://mathoverflow.net/users/109710 | Bounding the norm of a contraction matrix | We have the following linear matrix inequality (LMI)
$$\begin{bmatrix} \mathrm A \,\, & \mathrm X\\ \mathrm X^{\top} & \mathrm B\end{bmatrix} \succeq \mathrm O$$
where $\mathrm X = \mathrm A^{\frac 12} \mathrm C \, \mathrm B^{\frac 12}$ and $\mathrm A, \mathrm B \succeq \mathrm O$. Hence,
$$\begin{bmatrix} \mathr... | 1 | https://mathoverflow.net/users/91764 | 269759 | 120,845 |
https://mathoverflow.net/questions/269526 | 1 | Given a weighted, undirected, bipartite, graph $G(V,E)$. All edge weights are assumed to be non-negative. Let $d(u)$ be the degree of vertex u. Let $c(u,v)$ be the cost of edge $(u,v)$. Goal: compute a maximum weight bipartite matching.
A common 2-approximation algorithm to compute a maximum weight matching $M$ is th... | https://mathoverflow.net/users/106690 | Is greedy matching algorithm with normalized edge weights a 2-approximation | No, this algorithm does not always produce a matching that is within a factor 2 of the optimal matching.
Consider $G = (V,E)$ with
$$\begin{align\*} V &= \{x,y\} \uplus \{z\_1, z\_2, \dots, z\_k\}\\ E &= \{xz\_1, yz\_1, yz\_2, \dots, yz\_k\} \end{align\*}$$
and weights
$$\begin{align}w(xz\_1) &= m \\ w(yz\_1) &= M \\... | 1 | https://mathoverflow.net/users/51668 | 269760 | 120,846 |
https://mathoverflow.net/questions/269757 | 2 | Let $p$ be a prime number and $n$ be a natural number. Does $M\_n({\mathbf{Z}}\_p)$, the ring of all $n\times n$ matrices over the field ${\mathbf{Z}}\_p$ form a Baer\*-ring?
If not, what about for some special $p$ and $n>2$?
| https://mathoverflow.net/users/84390 | Is $M_n(Z_p)$ a Baer*-ring? | If $R$ is a ring such that $M\_n R$ is Baer\*, then all sums of the form $\sum\_{i=1}^n r\_i r\_i^\*$ must be nonzero, unless all $r\_i = 0$. With \*-transpose involution on matrix rings over $R$, this excludes $p \equiv 1 \pmod 4$ and $n \geq 2$, as well as $p\equiv -1 \pmod 4$ and $n \geq 3$, whenever the characteris... | 3 | https://mathoverflow.net/users/42278 | 269762 | 120,847 |
https://mathoverflow.net/questions/269766 | 17 | I've seen plenty of ancient diagrams representing plane figures.\* But I'd like to know how ancient geometers, especially around the time of Euclid, might have represented solids. Did they use diagrams with perspective? Three dimensional models?
\*cf. David Fowler, "The Mathematics of Plato's Academy", see the plate... | https://mathoverflow.net/users/109820 | How did ancient greek geometers represent solids? |
>
> Greek mathematical diagrams are composed of lines, arcs, and letter
> labels, and may be used to represent two- or three-dimensional
> objects. Drawing aids such as ruler and compass were sometimes, but
> not always, used in their construction, while curves in mathematical
> papyri were typically drawn freeha... | 16 | https://mathoverflow.net/users/11260 | 269772 | 120,849 |
https://mathoverflow.net/questions/269756 | 6 | Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ \*}$ a real positive function such that $\displaystyle \lim\_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has compact resolvant, in particular it has a pure discrete spectrum
$(\lambda\_i)\_{i\geq 0}$ such that $\displaystyle... | https://mathoverflow.net/users/109757 | Schrödinger eigenfunctions are bounded | In general, there won't be a uniform bound on all eigenfunctions simultaneously. If $[a,b]$ is a short interval with Dirichlet boundary conditions $y(a)=y(b)=0$ and constant potential $V=c$, then the ground state (normalized) eigenfunction
$$
\phi(x)=\sqrt{\frac{2}{b-a}}\, \sin\pi\frac{x-a}{b-a}
$$
is quite large point... | 4 | https://mathoverflow.net/users/48839 | 269778 | 120,851 |
https://mathoverflow.net/questions/269781 | 7 | Recall $\text{sinc}(x)=\frac{\sin x}x$. It's a familiar exercise that $\int\_0^{\infty}\text{sinc}(x)\,dx=\frac{\pi}2$.
But, at present, I wish to ask about the following claim on a "sinc-ing" product which is supported by extensive numerical computations.
>
> **Question.** Is it true that
> $$\int\_0^{\infty}d... | https://mathoverflow.net/users/66131 | The sinc function strikes again | We can start with a substitution $x=2y$
$$\int\_0^{\infty}dx\prod\_{n=1}^{\infty}\text{sinc}\left(\frac{x}{2n-1}\right)=2\int\_0^{\infty}dy\prod\_{n=1}^{\infty}\text{sinc}\left(\frac{2y}{2n-1}\right).$$
Now the double angle formula
$$\text{sinc}(2a)= \text{sinc}\left(a\right)\cos(a)$$
can be iterated to arrive to the f... | 20 | https://mathoverflow.net/users/2384 | 269783 | 120,853 |
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