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https://mathoverflow.net/questions/402178 | 5 | Euclid proof of the infinitude of primes can be extended into this.
Assuming there is a finite number of primes, $k$, sort them in increasing order and split the series after any prime at $t$. Create the difference between the products of each group.
$$ \left | \prod\_{t< m \leq k} p\_m - \prod\_{1 \leq n \leq t} p... | https://mathoverflow.net/users/nan | Extended Euclid proof and primes in form $|\prod\limits_{n \neq m} p_n -\prod\limits_{m \neq n} p_m|$ | See Guy, Lacampagne, and Selfridge, [Primes at a Glance](https://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866108-3/S0025-5718-1987-0866108-3.pdf), Mathematics of Computation, volume 48, number 177, January 1987, pages 183-202.
Abstract. Let $N = B - L$, $B \ge L$, $\gcd(B,L) = 1$, $p \mid BL$ for all pr... | 1 | https://mathoverflow.net/users/3684 | 402201 | 165,061 |
https://mathoverflow.net/questions/402207 | 3 | Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may expect the elements of a (normalized) Schauder basis $\{u\_n\}\_{n=1}^\infty$ of $W^{1,p}\_0(\Omega)$ (for $p>1$) to be... | https://mathoverflow.net/users/80191 | Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory? | Even the modified question does not hold.
Let $u\_n$ be a basis such that $\mathcal{L}^d(\operatorname{spt} u\_n) \to 0$, e.g. a wavelet basis and let $\phi \in C\_0^\infty(\Omega)$ a function such that $$\int\_{\{|\nabla \phi| \leq \varepsilon\}} |\nabla \phi|^p dx > 0$$ for all $\varepsilon > 0$, e.g. a smooth bump... | 3 | https://mathoverflow.net/users/51695 | 402213 | 165,064 |
https://mathoverflow.net/questions/402140 | 2 | I asked the following [question](https://math.stackexchange.com/questions/4223194/a-uniform-continuity-type-condition-on-an-integral) on MathStackExchange, but I have not received the answer that I'm looking for. Although it may not be a research-level question, I thought I could ask it here.
---
I'm currently re... | https://mathoverflow.net/users/102228 | A "uniform continuity" type condition on a Hammerstein integral equation | This does certainly not follow from your other hypotheses, as what you want to conclude is not much weaker than the equi-integrability of $\{K(t,\cdot):t\in I\}$ (sometimes also called absolute continuity in the $L\_1$-norm), but what you assume is only a continuity in one variable which implies nothing about the $L\_1... | 1 | https://mathoverflow.net/users/165275 | 402214 | 165,065 |
https://mathoverflow.net/questions/402218 | 6 | It is known that the logarithm of the modulus of an *analytic* function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions:
(1) Are there some weaker conditions than analyticity that ensure the same result?
(2) Is there any characterization of functions $f$ such th... | https://mathoverflow.net/users/151918 | $\log |f|$ is subharmonic | Conditions for a log-subharmonic function $f$ on $D\in\mathbb{R}^n$ are described by Mochizuki in [A Class of Subharmonic Functions and Integral Inequalities](https://www.jstage.jst.go.jp/article/iis/10/2/10_2_153/_pdf) (2004).
The conditions are phrased in terms of an inequality for the volume average $A\_p$ of $|f|... | 9 | https://mathoverflow.net/users/11260 | 402219 | 165,066 |
https://mathoverflow.net/questions/402211 | 4 | Let $\mathbb{R}\_\*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of [this question](https://mathoverflow.net/q/72612/118366) and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}\_\*$: $(\omega,\omega\_1),~(\omega\_1,\omega),~(\omega\_1,\omega\_1)$. And only $(\omega... | https://mathoverflow.net/users/118366 | On a completeness property of hyperreals | This is also called Cauchy-completeness, and it coincides for non-Archimedean ordered fields with the natural valuation to the valuation-theoretic notion of completeness. Also, this is the same as having no proper dense ordered field extension.
I will say that an ordered pair $(A,B)$ of subsets of an ordered field $F... | 7 | https://mathoverflow.net/users/45005 | 402223 | 165,068 |
https://mathoverflow.net/questions/402227 | 25 | A famous result of Galois, in his letter to Auguste Chevalier, is that for $p$ prime $>11$ the group $\operatorname{PSL}(2,\mathbb{F}\_p) $ does not embed in the symmetric group $\mathfrak{S}\_p$. The standard proof nowadays goes through the classification of subgroups of $\operatorname{PSL}(2,\mathbb{F}\_p) $ (Dickson... | https://mathoverflow.net/users/40297 | $\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in $\mathfrak{S}_p$ for $p>11$ | A few months ago Péter Pál Pálfy has given a talk about this exact topic. The abstract of the talk was the following:
>
> In his "testamentary letter" Galois claims (without proof) that PSL(2,p) does not have a subgroup of index p whenever p>11, and gives examples that for p = 5, 7, 11 such subgroups exist. The att... | 31 | https://mathoverflow.net/users/30186 | 402234 | 165,072 |
https://mathoverflow.net/questions/401083 | 7 | This is related to [my earlier (unanswered) MO post](https://mathoverflow.net/questions/400879/fibonacci-embedded-in-catalan). Preserve notations from there.
We take advantage of the one-to-one correspondence between the $(s,s+1)$-core partitions and $(s,s+1)$-Dyck paths. Let $\mathcal{F}\_s$ denote the set of $(s,s+... | https://mathoverflow.net/users/66131 | $(q,t)$-Fibonacci polynomials: area & bounce statistics | I mentioned in [my previous answer](https://mathoverflow.net/questions/400879/fibonacci-embedded-in-catalan/402188#402188) that the order ideals corresponding to $(s,s+1)$-cores with distinct parts are precisely the subsets of $\{1,2,\dots,s-1\}$ that contain no consecutive elements. This means that the Dyck paths unde... | 3 | https://mathoverflow.net/users/2384 | 402250 | 165,076 |
https://mathoverflow.net/questions/402255 | 3 | I'm interested in the following operator $T$, close relative of the standard logarithmic derivative:
$$f(x)\to Tf(x)=\frac{\text{d}(\log {f})}{\text{d}(\log {x})}=\frac{xf'}{f},$$
where $f$ is an increasing, positive $C^{\infty}(\mathbb R^+)$ function.
Does anyone know whether this pops up, say, in functional analy... | https://mathoverflow.net/users/167834 | On the operator $f\to xf'/f$ | You see this in the discussion of modular forms and related topics. When complex variable $\tau$ is in the upper half-plane $\operatorname{Im} \tau > 0$, the related complex variable $q = e^{2\pi i \tau}$ is in the (punctured) unit disk $0 < |q| < 1$.
An important derivation in this setting [call it say $\vartheta$]... | 6 | https://mathoverflow.net/users/454 | 402259 | 165,081 |
https://mathoverflow.net/questions/401905 | 0 | Consider a $6\times 1$ continuous random vector
$$
\eta\equiv (\eta\_1,\eta\_2,..., \eta\_6)
$$
satisfying the following property:
$$
\underbrace{\begin{pmatrix}
\eta\_1\\
\eta\_2\\
\eta\_3
\end{pmatrix}}\_{\equiv x\_1} \sim \underbrace{\begin{pmatrix}
\eta\_1\\
\eta\_4\\
\eta\_5
\end{pmatrix}}\_{\equiv x\_2} \sim \und... | https://mathoverflow.net/users/42412 | Conditions for existence of a distribution with full support | $\newcommand{\ep}{\epsilon}\newcommand{\R}{\mathbb R}$There is no necessary and sufficient condition **in terms of the support of $G$** for the following: there exists a $4\times 1$ random vector $\ep:=(\epsilon\_0, \ep\_1,\ep\_2,\ep\_3)$ having an absolutely continuous distribution with full support on $\R^4$ and such... | 1 | https://mathoverflow.net/users/36721 | 402261 | 165,082 |
https://mathoverflow.net/questions/402049 | 2 | Let $C$ be a smooth projective curve of genus $g$ with an involution $\iota: C \to C$. We have the quotient map $\pi: C \to C/\iota$, with $C/\iota$ a smooth curve of genus $h$.
The pullback map $\pi^{\*}: \text{Jac}(C/\iota) \to \text{Jac}(C)$ in this case is injective, and we define the Prym variety as the cokernel:
... | https://mathoverflow.net/users/105661 | Induced action on Prym variety | Kapil's suggestion to express the tangent space as $H^1(C,\mathcal O\_C)$ is great in characteristic $0$. In characteristic $p$, and in particular characteristic $2$, the statement is still true, but you can't detecet it from the tangent space.
Instead, note that the statement is equivalent to the claim that for $L$ ... | 2 | https://mathoverflow.net/users/18060 | 402280 | 165,086 |
https://mathoverflow.net/questions/402268 | 1 | I am reading Louigi's lecture note on random trees and graphs [here](http://problab.ca/louigi/notes/ssprob2021.pdf). I get stuck on part (b), Exercise 1.2.3 on page 19, which says the following:
>
> Let $T\_n$ be uniformly drawn from $\mathcal{T}\_n$, the set of $n$-vertices labeled rooted trees, and $c(v;t)$ be th... | https://mathoverflow.net/users/174600 | Empirical degree distribution of random $n$ vertices labeled rooted tree converges to Poisson distribution | Essentially, you want to show that
$$p\_{12}-p\_1^2\to0,\tag{1}$$
where
$$p\_{12}:=P(c(1;T\_n)=c(2;T\_n)=c),\quad p\_1:=P(c(1;T\_n)=c).$$
We have
$$p\_1=\binom{n-1}c\frac1{n^c}\Big(1-\frac1n\Big)^{n-1-c}
=\Big(1-\frac1n\Big)^{n-1-c}\frac1{c!}\prod\_{j=1}^c\Big(1-\frac jn\Big)\to\frac{e^{-1}}{c!}.$$
If your calculation ... | 1 | https://mathoverflow.net/users/36721 | 402283 | 165,087 |
https://mathoverflow.net/questions/402066 | 3 | I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on Semigroups of Linear operators I found on many places properties of the Neumann Laplacean.
In W. Arendt, *Semigroups and evolution equations: functional calculus, regularity and kernel estimates*, Handbook of Diffe... | https://mathoverflow.net/users/61629 | Contractivity of Neumann Laplacean | It might be helpful to point out the following conceptual reasons why ultracontractivity estimates are mainly interesting for times close to $0$.
Let us consider the following general setting: We have a finite measure space $(\Omega,\mu)$, two integrability indices $1 \le p < q \le \infty$ and a $C\_0$-semigroup $(S(... | 5 | https://mathoverflow.net/users/102946 | 402285 | 165,089 |
https://mathoverflow.net/questions/402274 | 5 | Let $h(n,t) = \sum\limits\_{j = 0}^n {\binom
{\lfloor {\frac{n}{2}} \rfloor }{j}\binom
{\lfloor {\frac{n+1}{2}}\rfloor }{j}t^j \\ }.$
I am interested in the Hankel determinants $${D\_k}(n,t) = \det \left( {h(k + i + j,t)} \right)\_{i,j = 0}^{n - 1}.$$
These can easily be computed for $0 \leq k \leq 3.$
It seems t... | https://mathoverflow.net/users/5585 | An interesting Hankel determinant | Denote $a\_n=a(n,t)$ and $b\_n=b(n,t)$. To help avoiding the **min** function, write
$$b\_n=\binom{n+3}3t^n+\sum\_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$
Notice that $a\_n=\frac{nt^{n+1}-(n+1)t^n+1}{(1-t)^2}$ and $\sum\_{j=0}^nt^j=\frac{1-t^{n+1}}{1-t}$. Your identity takes the form
$$(nt^{n+1}-(n+1)t^n+1)^... | 5 | https://mathoverflow.net/users/66131 | 402289 | 165,090 |
https://mathoverflow.net/questions/402277 | 6 | Does simple theory of types + ambiguity prove axiom of infinity?
The simple theory of types known as $\sf TST$ is a multi-sorted first order theory, syntactical restrictions include $\in$ being a dyadic symbol where the symbol on the right of it is one sort (type) higher than the one on the left, while the two symbol... | https://mathoverflow.net/users/95347 | Does simple theory of types + ambiguity prove axiom of infinity? | The development of TST + Amb is arguably motivated by the logicist program, ultimately.
But Amb is not a purely logical principle, it is a conjecture past the logically provable facts, with unexpected consequences.
I dont think that TST + Amb can be taken to motivate logicism. It is a by-product of this program.
... | 3 | https://mathoverflow.net/users/345616 | 402292 | 165,092 |
https://mathoverflow.net/questions/402262 | 12 | This is the second in a pair of questions. For the other see [Are representations in computable analysis the equivalent to countably-generated condensed sets?](https://mathoverflow.net/questions/402260/are-representations-in-computable-analysis-the-equivalent-to-countably-generated).
Dustin Clausen and Peter Scholze ... | https://mathoverflow.net/users/12978 | Are the “topologies” arising from constructive type theories with quotients actually condensed sets? | Not quite a complete answer, but:
There are models of constructive mathematics where sets are $T\_0$ second-countable (optionally zero-dimensional) topological spaces equipped with equivalence relations, morphisms are continuous maps which respect the equivalence relation, and pointwise-equivalent morphisms are equal... | 4 | https://mathoverflow.net/users/100508 | 402293 | 165,093 |
https://mathoverflow.net/questions/402295 | 3 | Let $d\_{KS}(F,G)= \sup\_{x} |F(x) -G(x)|$ be the Kolmogorov-Smirnov distance between two cdfs $F$ and $G$.
Question: Let $F\_m$ be a cdf of distribution with $m$ atoms and let $\Phi$ bet the distribution of standard normal. What can we say about
\begin{align}
g(m)=\inf\_{F\_m} d\_{KS}(F\_m,\phi)
\end{align}
In other... | https://mathoverflow.net/users/69661 | Best approximation of normal with $m$ atoms in Kolmogorov-Smirnov distance | The best difference is $\frac{1}{2m}$, attained by the distribution with $m$ atoms, each with mass $\frac{1}{m}$, at the points where the cdf of the normal distribution takes the values $\frac{2i-1}{m}$ for $i$ from $1$ to $m$.
Proof that this is optimal: One atom must have mass at least $\frac{1}{m}$. Call this $x$.... | 4 | https://mathoverflow.net/users/18060 | 402298 | 165,095 |
https://mathoverflow.net/questions/402245 | 3 | Here's a fairly easy fact from point-set topology that I'm having trouble finding a reference for. Say $X$ is a total order satisfying the least-upper bound property, and $S$ is a closed subset of it. Then the subspace topology on $S$ and the order topology on $S$ coincide.
Does anyone know what would be a reference ... | https://mathoverflow.net/users/5583 | A closed subset of a Dedekind-complete order has subspace topology equal to order topology | While I agree that it's pretty direct to show, I was unable to find a reference for a proof of this fact myself (I thought it was in Willard, but I thumbed through my copy and failed to find it). So here's a proof in any case.
Let $S$ be a closed subset of a linear order $X$ with the least upper bound property, that ... | 3 | https://mathoverflow.net/users/73785 | 402304 | 165,098 |
https://mathoverflow.net/questions/402287 | 6 | **Question 1.** Let $\epsilon > 0$ and $V > 0$. Is there always a complete connected Riemannian manifold $M$ with
$$
\operatorname{diam} M < \epsilon\quad\text{ (small diameter)} \quad \text{and} \quad \operatorname{vol}M > V\quad\text{(large volume)}?
$$
In other words, can we construct worlds with room for arbitraril... | https://mathoverflow.net/users/123207 | Superconnected spaces | The answer to both questions is positive, even in dimension 2.
Take a round sphere of diameter $\epsilon$, and make many, say $N$ little holes in it.
Then take $N$ spheres of diameter $\epsilon$ and make one little hole in each.
Then glue these $N$ spheres to the first sphere along the boundaries of the holes. The diam... | 6 | https://mathoverflow.net/users/25510 | 402307 | 165,100 |
https://mathoverflow.net/questions/402315 | 4 | Does there exist an analog of Lagrange inversion formula in positive characteristic? Obviously, the formula is still valid for coefficient with index not divisible by the characteristic, but for the other ones I did not manage to find one.
| https://mathoverflow.net/users/33128 | Lagrange inversion formula in positive characterisic | There are many forms of Lagrange inversion. The ones that don't involve division by integers are valid in positive characteristic. For example:
Given a power series $R(t)$, there is a unique power series $f=f(x)$ such that
$f(x) = x R(f(x))$, and for any Laurent series $\phi(t)$ and $\psi(t)$ and any integer $n$ we hav... | 7 | https://mathoverflow.net/users/10744 | 402318 | 165,103 |
https://mathoverflow.net/questions/402333 | 4 | This is a repost of the [same question on math.SE](https://math.stackexchange.com/questions/4227470/unbounded-set-in-vg-has-an-unbounded-subset-in-v), which received several comments but no answers/comments on the first question.
---
Suppose $\kappa$ is a cardinal preserved in the generic extension $V[G]$. Let $Y... | https://mathoverflow.net/users/146831 | Unbounded set in $V[G]$ has an unbounded subset in $V$? | Let $\mathbb{P} = \textsf{Col}(\delta^{+ \omega + 2}, < \kappa)$ and $\mathbb{Q} = \textsf{Col}(\omega, \delta^{+ \omega})$. Then $\mathbb{P}$ is $< \delta^{+ \omega + 2}$-closed so it doesn't add any new set of ordinals of order type $\leq \delta^{+ \omega + 1}$. Put $V\_1 = V^{\mathbb{P}}$ and note that all cardinals... | 3 | https://mathoverflow.net/users/345950 | 402337 | 165,108 |
https://mathoverflow.net/questions/402339 | 8 | It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M\_{q}:=2^{q}-1$ a Mersenne prime.
As the very defining property of such a perfect number is to fulfill the equality $\sigma(N)=2N$, one can see that this value is almost the sum of a geometric series. But another conceptual fr... | https://mathoverflow.net/users/13625 | Can perfect numbers be seen $p$-adically? | (Not a complete answer but a bit too long for a comment.)
There's a fundamental difficulty here in proving the sort of result you envision. If there were some prime $p$ which had to divide every odd perfect number and larger odd perfect numbers had to be divisible by higher powers, that would work. But we can't right... | 8 | https://mathoverflow.net/users/127690 | 402351 | 165,114 |
https://mathoverflow.net/questions/402341 | 1 | This is related to discrete dynamical systems, with the initial condition $X\_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X\_{n+1} = g(X\_n)$ for some rather smooth mapping $g$. The purpose here is to find a mapping $g$ so that the invariant distribution (also cal... | https://mathoverflow.net/users/140356 | Invariant distributions for iterated random variables (stochastic dynamical systems) | To have here the invariant distribution with cdf $F$ given by $F(x)=x^2$ for $x\in[0,1]$, all that is needed is a change of variables.
More generally, let $F$ be the cdf of any non-atomic distribution supported on an interval $I$ in $\mathbb R$. Let $F^{-1}$ denote the inverse of the restriction of $F$ to $I$. Let $U... | 3 | https://mathoverflow.net/users/36721 | 402354 | 165,116 |
https://mathoverflow.net/questions/402353 | 1 | I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ of finite dimension (in the following by "representation" I mean a representation of finite dimension). There is a complete classification of complex representations of $\mathrm{SL}(2,\mathbb{C}).$ More precisely, if $V... | https://mathoverflow.net/users/346215 | Irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ | First, the classification of *complex* representations of $\mathrm{SL}\_2(\mathbb{R})$ is the same as that of $\mathrm{SL}\_2(\mathbb{C})$. This is because any representation of the Lie group $\mathrm{SL}\_2(\mathbb{R})$ gives a representation of the Lie algebra $\mathfrak{sl}\_2(\mathbb{R})$. But by the universal prop... | 3 | https://mathoverflow.net/users/22 | 402362 | 165,119 |
https://mathoverflow.net/questions/402336 | 3 | From what I understand, the [Borel construction](https://ncatlab.org/nlab/show/Borel+construction) takes a $G$-space $X$ and produces a topological space $X\times\_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category of spaces $\mathcal{S}$―satisfying
$$\mathrm{H}^\*(X\times\_G\mathbf... | https://mathoverflow.net/users/130058 | What is the pointed Borel construction of the $0$-sphere? | Let's apply your definition (which I think has typos on the RHS - the two "+" subscripts on the EG should not be there I think). Let's model everything as topological spaces and do the calculation there. Let's model $S^0$ as the discrete set $\{p,q\}$ with basepoint $p$.
$$EG\_+\wedge\_GS^0:=\frac{EG\times\_G\{p,q\}}... | 2 | https://mathoverflow.net/users/163893 | 402378 | 165,125 |
https://mathoverflow.net/questions/402375 | 4 | Let $A$ be a ring. Is the sequence \begin{align} \textstyle A \to \prod\_{\mathfrak{p}} A\_{\mathfrak{p}} \rightrightarrows \prod\_{\mathfrak{p}\_{1},\mathfrak{p}\_{2}} A\_{\mathfrak{p}\_{1}} \otimes\_{A} A\_{\mathfrak{p}\_{2}} \end{align} exact? Here the products are over all prime ideals of $A$.
*Thoughts:*
1. Si... | https://mathoverflow.net/users/15505 | Descent for the "localizations at all primes" ring map | Let $S$ be a compact, totally disconnected topological space whose topology is not discrete.
Let $k$ be a field and let $A$ be the ring of locally constant $k$-valued functions on $S$.
Then I claim the kernel of $\prod\_{\mathfrak p} A\_{\mathfrak p} \to \prod\_{\mathfrak p\_1,\mathfrak p\_2} A\_{\mathfrak p\_1} \o... | 7 | https://mathoverflow.net/users/18060 | 402379 | 165,126 |
https://mathoverflow.net/questions/402372 | 6 | Consider this ODE on $[1, \infty)$
$(r^2 - 2ar)f''(r) + 2(r-a) f'(r) - ({4a} + m(m+1))f(r) = -4af(1) $
with initial conditions
$\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim\_{r\to \infty} f(r) = 0$
where $0\leq a < \frac{1}{2}$, $m$ is a positive integer, and $C \in \mathbb{R}$.
I want to ask if there exists ... | https://mathoverflow.net/users/138705 | Existence and uniqueness of an Euler-type ODE with varying parameters | As Iosif said, in general the system you specified does not admit a solution. Here we will give a more pedestrian argument using only comparisons.
Monotonicity
------------
**Claim**: if a solution exists, and $f(1) > 0$, then the function is monotonically decreasing; if $f(1) < 0$, then the function is monotonical... | 6 | https://mathoverflow.net/users/3948 | 402384 | 165,129 |
https://mathoverflow.net/questions/402385 | 1 | Let us define $M\_0=2^n$ for $n\in\mathbb{N}$. Let $\ell\in\mathbb{N}$ be the number of random variables we are working with. For $1\leqslant i\leqslant\ell$, we define $M\_i$ to be a random variable following a binomial distribution with parameters $M\_{i-1}$ and $2^{-n}$.
I'm interested in computing $\mathbb{P}\lef... | https://mathoverflow.net/users/178595 | Distribution of a random variable obtained by chaining distribution | Abbreviate $N=M\_0=2^n$ and $b=1/N$. As described, the conditional distribution of $M\_i$, given that $M\_{i-1}=n$, is binomial with parameters $n$ and $b$. The conditional generating function of $M\_i$ is therefore
$$
E[s^{M\_i}\mid M\_{i-1}]=(1-b+bs)^{M\_{i-1}}.
$$
Using $g\_i$ to denote the generating function of $M... | 1 | https://mathoverflow.net/users/42851 | 402386 | 165,130 |
https://mathoverflow.net/questions/402374 | 2 | Assume $\varepsilon \in [0,1/2]$. Consider the discrete-time random walk $X\_0 = 0$, $X\_{t+1} - X\_t \sim f(X\_t) \delta\_0 + (1-f(X\_t))\operatorname{Rademacher}$, where $\delta\_0$ is the Dirac delta on zero and $\operatorname{Rademacher}$ is the Rademacher distribution, and
$$
f(x) = \begin{cases} 1-\varepsilon & x... | https://mathoverflow.net/users/346420 | Occupation time of non-stationary random walk | Here is a back of envelope computation that you should be able to make rigorous without too much trouble.
Let's condition upon the last hit of $0$. The passing time from it to $T$ is large with probability close to $1$ (all you need to show for this is that the probability to be in any particular position tends to $0... | 3 | https://mathoverflow.net/users/1131 | 402393 | 165,132 |
https://mathoverflow.net/questions/402397 | 9 | I am looking for a proof of the following claim:
>
> Let $H\_n$ be the nth [harmonic number](https://en.wikipedia.org/wiki/Harmonic_number). Then,
> $$\frac{\pi^2}{12}=\ln^22+\displaystyle\sum\_{n=1}^{\infty}\frac{H\_n}{n(n+1) \cdot 2^n}$$
>
>
>
The SageMath cell that demonstrates this claim can be found [here... | https://mathoverflow.net/users/88804 | An infinite series involving harmonic numbers | Denoting $H\_0=0$, we have $$\sum\_{n=1}^\infty \frac{H\_n}{n(n+1)2^n}=\sum\_{n=1}^\infty \left(\frac1n-\frac1{n+1}\right)\frac{H\_n}{2^n}=\sum\_{n=1}^\infty \frac{1}n\left(\frac{H\_n}{2^n}-\frac{H\_{n-1}}{2^{n-1}}\right)\\=\sum\_{n=1}^\infty\frac1{n^22^n}-\sum\_{n=1}^\infty\frac{H\_n}{(n+1)2^{n+1}}.$$
It is well known... | 17 | https://mathoverflow.net/users/4312 | 402405 | 165,134 |
https://mathoverflow.net/questions/401735 | 5 | Let $W^{2,2}(\mathbb R)\newcommand{\R}{\mathbb R}\newcommand{\C}{\mathbb C}\newcommand{\N}{\mathbb N}$ denote the Sobolev space as defined in chapter 5 of [Evans' PDE book](https://bookstore.ams.org/gsm-19-r) and consider the linear operator
\begin{equation\*}\begin{split}T: D(T)&\to L^2(\mathbb R), \\ \phi&\mapsto \... | https://mathoverflow.net/users/129831 | Is the second weak derivative a self-adjoint operator? | In Theorem 4.7b) of [these lecture notes](https://www.math.kit.edu/iana3/%7Eschnaubelt/media/st-skript15.pdf), the following general result is proved:
>
> Let $X$ be a Hilbert space and let $A$ be densely defined, closed and symmetric. Then $A$ is self-adjoint if and only if the spectrum of $A$ is contained in the ... | 1 | https://mathoverflow.net/users/136913 | 402406 | 165,135 |
https://mathoverflow.net/questions/402399 | 4 | I have seen the claim that Beilinson Lichtenbaum implies that higher algebraic $K$ groups coincides with etale ones integrally in high enough degrees. Is this statement accurate? What conditions are required and how to derive it?
| https://mathoverflow.net/users/127776 | Etale $K$ theory coincides with algebraic one in high enough degrees | To my knowledge the most general known statement has been proven by Clausen and Mathew in their paper [Hyperdescent and étale K-theory](https://arxiv.org/abs/1905.06611) as Theorem 1.2. The precise conditions on your commutative ring (or more generally algebraic space) are a bit technical to summarize, but are very gen... | 5 | https://mathoverflow.net/users/2039 | 402408 | 165,136 |
https://mathoverflow.net/questions/402308 | 11 | In ZF(C), one can easily get a class partition of $V$, we can even get an $\mathrm{Ord}$-partition using the Cumulative hierarchy: $P=\{V\_{α+1}\setminus V\_α\mid α∈\mathrm{Ord}\}$, such a partition let us do stuff like Scott's trick: given a class $A$, we can look at $A∩V\_β$ where $β$ is the minimal $β$ so that inter... | https://mathoverflow.net/users/113405 | Scott's trick without regularity | Yes! Extend $\sf ZF - Reg.$ with the existence of a unique Quine atom $\sf Q=\{Q\}$, take $V$ to be the hierarchy over $\sf Q$, that is: $$\begin{align} & V\_\emptyset = \sf Q \\ & V\_{\alpha+1}= \mathcal P (V\_{\alpha}) \\ & V\_\lambda= \bigcup\_{\alpha < \lambda} V\_\alpha, \text {for limit } \lambda \\ & V= \bigcup ... | 5 | https://mathoverflow.net/users/95347 | 402412 | 165,138 |
https://mathoverflow.net/questions/402366 | 0 | Given two freely independent random hermitian matrices $A$ and $B$ following laws $\mu, \nu$, one can compute the empirical spectral distribution of $AB$ by their free multiplicative convolution $\mu\boxtimes\nu$ using the $S$-transform. Is there a way to compute the empirical spectral distribution of other products of... | https://mathoverflow.net/users/346351 | Free multiplicative convolution of two random matrices | There is the possibility of dealing with any non-commutative rational function in A and B, by using more general (operator-valued) versions of free probability. Usually the corresponding equations cannot be solved analytically, but they are nice fixed-point equations which can be addressed numerically. See for example,... | 2 | https://mathoverflow.net/users/112626 | 402413 | 165,139 |
https://mathoverflow.net/questions/402401 | 0 | In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu\_f=f$ where $u\_f=G\*f$.
Is the same thing hold for non-linear PDE? Even if not, I wanted to know if we have quasilinear PDE is that holds? If this is not true at all... | https://mathoverflow.net/users/119011 | Application of Green function for non linear PDE | The equality $u\_f= G\ast f$ uses linearity in an essential way since
$$
L\_y G(x,y)= \delta(y-x).
$$
The function $f$ is a superposition of $\delta$'s
$$
f(\bullet)=\int f(x)\delta(\bullet-x) dx.
$$
On the other hand the Green function has indirect uses in nonlinear equations. The solution on the [Yamabe problem](http... | 1 | https://mathoverflow.net/users/20302 | 402421 | 165,143 |
https://mathoverflow.net/questions/402388 | 1 | I asked this question on MSE a while ago but didn't receive any useful answers.
Suppose I have a $1$-parameter family continuous maps $f\_t: \mathbb{S}^2\rightarrow \mathbb{C}P^1$ from a topological $2$-sphere to the Riemann sphere which is a local homeomorphism away from isolated points (for example, imagine a conti... | https://mathoverflow.net/users/125534 | Existence of continuous family of uniformising parameters | That your $f\_t$ are local homeomorphisms away from isolated points is not sufficient for the conclusion you want. Your $f\_t$ must be at least topologically holomorphic. (A continuous map is called topologically holomorphic if it is open and discrete).
Now one needs some stronger restrictions on $z\mapsto f(z)$ (how... | 1 | https://mathoverflow.net/users/25510 | 402426 | 165,144 |
https://mathoverflow.net/questions/402294 | 2 | Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E\_1,E\_2)$ be the orthonormal frame on $D$, such that at each point $p$ the image vectors $\mathrm{d}f\bigl(E\_i(p)\bigr)$ are orthogonal and... | https://mathoverflow.net/users/171439 | Signs of curvatures of integrals lines of frames with constant principal values | Here's how one can construct a specific example to illustrate what can happen:
First, recall from my answer to [this question](https://mathoverflow.net/questions/351546/are-all-maps-mathbbr2-to-mathbbr2-with-fixed-singular-values-affine) that, if you have a smooth map $f:D\to\mathbb{R}^2$ with constant positive singu... | 2 | https://mathoverflow.net/users/13972 | 402432 | 165,146 |
https://mathoverflow.net/questions/402442 | 2 | Given a finite number of algebraic curves over $\mathbb{Q}$ is there a curve that covers all of them?
| https://mathoverflow.net/users/343850 | Any finite number of curves over $\mathbb{Q}$ have a common cover | **If we are talking about branched covers, yes**: Let the curves be $C\_1$, $C\_2$, ..., $C\_r$. Take the product $\prod C\_i$, embed it into projective space, and intersect with a general codimension $r-1$ plane $H$. Then (by Bertini) $H \cap \prod C\_i$ will be a smooth curve and, for $G$ chosen generically, the proj... | 3 | https://mathoverflow.net/users/297 | 402445 | 165,148 |
https://mathoverflow.net/questions/402435 | 10 | Why are [W-types](https://ncatlab.org/nlab/show/W-type) called "W"?
Probably "W" means either "wellordered" or "wellfounded". ([Martin-Löf](https://www.csie.ntu.edu.tw/~b94087/ITT.pdf) uses the term "wellorder".) But these are notions associated to order theory, whereas W-types don't directly have to do with order re... | https://mathoverflow.net/users/347155 | Why are W-types called "W"? | You write:
>
> Probably "W" means either "wellordered" or "wellfounded". […] But these are notions associated to order theory, whereas W-types don't directly have to do with order relations (if at all).
>
>
>
I don’t know an official source for this, but I’ve always assumed W stands for “well-founded” as you s... | 17 | https://mathoverflow.net/users/2273 | 402453 | 165,151 |
https://mathoverflow.net/questions/402443 | 3 | I am working on some non-local differential equations that appear in geometric analysis.
One of which I posted [here](https://mathoverflow.net/questions/402372/existence-and-uniqueness-of-an-euler-type-ode-with-varying-parameters) and was answered by @WillieWong and @losifPinelis.
Consider this non-local differential... | https://mathoverflow.net/users/138705 | Existence and uniqueness of an Euler-type ODE with varying parameters part 2 | Reformulate
-----------
Supposing $f$ is a solution to your first formulation, with $f(1) = \lambda$. Let $\tilde{f}(r) = \lambda^{-1} f(r)$. Then $\tilde{f}$ solves the differential equation
$$ \tag{1} r(r-2a) \tilde{f}'' + 2(r-a) \tilde{f}' - m(m+1) \tilde{f} = \frac{4a^2}{r(r-2a)} (\tilde{f} - 1) + \frac{4a(1-2a... | 2 | https://mathoverflow.net/users/3948 | 402461 | 165,155 |
https://mathoverflow.net/questions/402383 | 4 | Consider a collection of $C^\*$-algebras $\{A\_i\}\_{i \in I}$. We can form the direct sum $$\bigoplus\_{i \in I}^{c\_0} A\_i:= \left\{(a\_i)\_{i \in I} \in \prod\_{i\in I} A\_i: \lim\_{i \in I} \|a\_i\| = 0\right\}$$
which is an ideal in the $C^\*$-algebra
$$\bigoplus\_{i \in I}^{\ell^\infty} A\_i:= \left\{(a\_i)\_{i ... | https://mathoverflow.net/users/216007 | Direct sum of multiplier algebras | Let's try to flesh out your "sketch". Set $A=c\_0-\oplus\_i A\_i$ and consider this as a Hilbert $C^\*$-module over itself. Let $\iota\_i:A\_i\rightarrow A$ be the inclusion, and $\jmath\_i:A\rightarrow A\_i$ the left inverse to $\iota\_i$. These are both non-degenerate $\*$-homomorphisms, and so extend to unital $\*$-... | 4 | https://mathoverflow.net/users/406 | 402462 | 165,156 |
https://mathoverflow.net/questions/402346 | 4 | Let $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^{\infty}$ vector field. Fix a *(single)* real number $d$ such that
$$
1\leq d\leq n
.
$$
Under what conditions is the flow map $\Phi\_V$ defined as sending any $x\_0\in \mathbb{R}^n$ to the time $1$-solution of autonomous system of ODEs:
$$
\dot{x(t)} = V(x(t)) \qqua... | https://mathoverflow.net/users/346124 | When is a smooth field's flow map volume preserving diffeomorphism | I would like to know why you are interested in the specific $d$-dimensional measures given by the Hausdorff ones. For $d=n$ such a choice is understandable since the Lebesgue measure in $\mathbb{R}^{n}$ is proportional to $\mathcal{H}^{n}$. But (to the best of my knowledge) there is no nice description in the case of $... | 3 | https://mathoverflow.net/users/85336 | 402488 | 165,164 |
https://mathoverflow.net/questions/402471 | 3 | This [paper](https://homepages.inf.ed.ac.uk/gdp/publications/trace.pdf) says that $FinVect\_k$ is collectively complete for traced symmetric monoidal categories, in the sense that given distinct arrows in the free traced SMC (over some generating monoidal signature) there exists a strong functor (from the free traced S... | https://mathoverflow.net/users/156811 | Is Set (collectively) complete for Cartesian categories? | If I understand what you are asking, the answer is yes.
Indeed:
**Proposition.**
Let $\mathcal{C}$ be a locally small cartesian monoidal category and let $f\_0, f\_1 : X \to Y$ be a parallel pair of morphisms in $\mathcal{C}$.
Then there is a cartesian monoidal functor $F : \mathcal{C} \to \textbf{Set}$ such that $F ... | 7 | https://mathoverflow.net/users/11640 | 402492 | 165,166 |
https://mathoverflow.net/questions/402501 | 1 | Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$.
I would like to ask if there exists a compact moduli space $\overline{M}\_{g,d}(X)$ such that all points of $\overline{M}\_{g,d}(X)$ repr... | https://mathoverflow.net/users/14514 | Moduli spaces of horizontal curves | It is not possible to have such a moduli space that contains all the smooth curves of genus $g$ and degree $d$ and over which the universal family of curves is proper.
Let $X = \mathbb P^1 \times \mathbb P^1$ with coordinates $x,y$, $Y= \mathbb P^1$, $f$ the projection onto the $x$ coordinate. Let $C\_t$ be given by ... | 3 | https://mathoverflow.net/users/18060 | 402503 | 165,169 |
https://mathoverflow.net/questions/402446 | 6 | Let $\mathcal{C}$ and $\mathcal{D}$ be two tensor categories (if necessary, assume they are fusion categories). Is the canonical braided monoidal functor $$\mathcal{Z}(\mathcal{C})\boxtimes\mathcal{Z}(\mathcal{D})\rightarrow\mathcal{Z}(\mathcal{C}\boxtimes\mathcal{D})$$ an equivalence?
NB: The two monoidal categories... | https://mathoverflow.net/users/105094 | Drinfeld center of a Deligne tensor product | By Cororllary 3.26 of arxiv:1009.2117, any braided tensor functor out of a non-degenerately braided fusion category is automatically fully faithful. Since $Z(\mathcal{C})\boxtimes Z(\mathcal{D})$ is non-degenerate when $\mathcal{C}, \mathcal{D}$ are fusion, the result follows immediately from your FP dimension observat... | 7 | https://mathoverflow.net/users/351 | 402509 | 165,170 |
https://mathoverflow.net/questions/402495 | -1 | From *Surface subgroups of Coxeter and Artin groups* (Gordon, Long and Reid, 2003) [DOI link](https://doi.org/10.1016/j.jpaa.2003.10.011), we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi\_1$ of a closed orientable surface of genus $\ge 1$).
My question is:
Are... | https://mathoverflow.net/users/197544 | Are all Coxeter groups virtually free or virtually surface groups? | When asking a question about Coxeter groups, it might be useful to focus on right-angled Coxeter groups first: usually, they are easier to handle. Below are a few criteria you can play with in order to create various examples of (right-angled) Coxeter groups.
Let $\Gamma$ be a finite simplicial graph. The right-angle... | 4 | https://mathoverflow.net/users/122026 | 402510 | 165,171 |
https://mathoverflow.net/questions/402499 | 2 | Given a field $k$ with characteristic $p$ and a finite cyclic $p$-group $G$ of order $p^a$, it is well-known that all the indecomposable representations of $kG$ are given by mapping a generator $x$ of $G$ to the Jordan matrix $J\_s\in M\_s(k)$ with all eigenvalues one for $1\leq s\leq p^a$. If we replace $k$ by a commu... | https://mathoverflow.net/users/134942 | Indecomposable representations for group ring $RG$ over commutative ring $R$ with characteristic $p$ | One might ask whether one can classify all indecomposable $RG$-modules when one knows all indecomposable $R$-modules but the example $R=K[x,y]/(x^2,y^2)$ shows that this is not possible.
The answer will in general be that one can not classify the indecomposable representations as those algebras are most often of "wil... | 2 | https://mathoverflow.net/users/61949 | 402511 | 165,172 |
https://mathoverflow.net/questions/402497 | 27 | In so-called 'natural unit', it is said that physical quantities are measured in the dimension of 'mass'. For example, $\text{[length]=[mass]}^{-1}$ and so on.
In quantum field theory, the dimension of coupling constant is very important because it determines renormalizability of the theory.
However, I do not see w... | https://mathoverflow.net/users/56524 | How do we give mathematical meaning to 'physical dimensions'? | Mathematically, the concept of a physical dimension is expressed using one-dimensional vector spaces and their tensor products.
For example, consider mass.
You can add masses together and you know how to multiply a mass by a real number.
Thus, masses should form a one-dimensional real vector space $M$.
The same rea... | 55 | https://mathoverflow.net/users/402 | 402515 | 165,174 |
https://mathoverflow.net/questions/400747 | 9 | I was reading a [physics paper](https://www.sciencedirect.com/science/article/abs/pii/0550321386901550) where it was mentioned that the basic framework of Connes' differential non-commutative geometry (or actually, a slight modification of Connes in that paper) would need some extensions in order to arrive at a theory ... | https://mathoverflow.net/users/119114 | Non-commutative complex geometry | I don't think there are any obvious ``technical obstacles'' to extending noncommutative geometry to the a theory of noncommutative complex geometry. Instead I would say that there are a number of differing points of view on what form noncommutative complex geometry should take, and it's not completely clear how the dif... | 7 | https://mathoverflow.net/users/3072 | 402517 | 165,175 |
https://mathoverflow.net/questions/402523 | 6 | For any fixed $\frac{1}{2} < \sigma < 1$, let
$$\int\_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$
It is clear that $\theta > 0$, since we have the classical asymtotic
$$\int\_0^T \frac{|\zeta(\sigma+it)|^2}{T} \ dt \sim \zeta(2\sigma), \qquad T \uparrow \infty. $... | https://mathoverflow.net/users/345624 | Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function | Let us introduce the notation
$$M(T):=\int\_0^T|\zeta(\sigma+it)|^2\,dt.$$
Then
$$\int\_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \,dt=\int\_0^T\frac{dM(t)}{\sqrt{1+t^2}}=\frac{M(T)}{\sqrt{1+T^2}}+\int\_0^T\frac{tM(t)}{(1+t^2)^{3/2}}\,dt$$
by writing this as a Riemann-Stieltjes integral and then integrating by part... | 9 | https://mathoverflow.net/users/11919 | 402526 | 165,179 |
https://mathoverflow.net/questions/402534 | 9 | Suppose that $\mathcal{V}$ is a symmetric monoidal model category, and that $\mathcal{C}$ is a $\mathcal{V}$-enriched model category. Write $\Bbb{R}\!\operatorname{Hom}(-,-)$ for the derived Hom functor
$$
\Bbb{R}\!\operatorname{Hom}(-,-) : \operatorname{Ho}(\mathcal{C})^{\textrm{op}}\times\operatorname{Ho}(\mathcal{C}... | https://mathoverflow.net/users/29322 | Does derived hom commute with homotopy limits? | Yes, this is always true.
Replacing $X$ by its cofibrant replacement if necessary, we can assume $X$ to be cofibrant.
In this case, $\def\Hom{\mathop{\rm Hom}} \Hom(X,-)\colon C→V$ is a right Quillen functor.
The right derived functor of this right Quillen functor computes the derived hom $\def\RHom{\mathop{\rm RHom}... | 10 | https://mathoverflow.net/users/402 | 402539 | 165,182 |
https://mathoverflow.net/questions/402540 | 0 | I have a set of equations with some inequality constraints that I expect generally does not have a unique solution.
The equations take the form below:
$$\alpha/N+(1-\alpha)x\_1=a\_1$$
$$\alpha/N+(1-\alpha)x\_2=a\_2$$
$$\vdots$$
$$\alpha/N+(1-\alpha)x\_N=a\_N$$
$$x\_1+x\_2+\dots+x\_N=1$$
$$0<x\_i<1$$
$$1>\alpha>0$$
... | https://mathoverflow.net/users/69486 | Bounding parameter satisfying a collection of inequalities | Let $n:=N$ and $t:=\alpha$. We have
$$0<x\_i=\frac{a\_i-t/n}{1-t}<1$$
for all $i\in[n]:=\{1,\dots.n\}$ --
or, equivalently,
$$t<t\_{n,a}:=\min\_{i\in[n]}\min\Big(\frac{1-a\_i}{1-1/n},na\_i\Big)
=\min\Big(\frac{1-a\_{\max}}{1-1/n},na\_{\min}\Big),$$
where $a\_{\max}:=\max\_{i\in[n]}a\_i$ and $a\_{\min}:=\min\_{i\in[n]}a... | 3 | https://mathoverflow.net/users/36721 | 402541 | 165,183 |
https://mathoverflow.net/questions/402536 | 8 | Given matrices $A, B \in \mathbb{R}^{n\times n}$, I would like to solve the following optimization problem,
$$\begin{array}{ll} \underset{v \in \mathbb{R}^n}{\text{maximize}} & \|Av\|\_2+\|Bv\|\_2\\ \text{subject to} & \|v\|\_2 = 1\end{array}$$
I'm hoping to solve this with some sort of convex optimization approach... | https://mathoverflow.net/users/97603 | Maximizing sum of vector norms | I have no doubt that someone will come with some brighter idea but here are my 2 cents anyway.
If you don't aim at something very fast, I would just use the inequality $(a+b)^2\le (ta^2+(1-t)b^2)(t^{-1}+(1-t)^{-1})=F\_t(a,b)$ and try to find $\min\_{t\in[0,1]}\max\_v F\_t(\|Av\|,\|Bv\|)$. The maximum inside is just $... | 5 | https://mathoverflow.net/users/1131 | 402544 | 165,185 |
https://mathoverflow.net/questions/402508 | 6 | I am interested in the complexity of multiplying two matrices $A$ and $B$, i.e. to compute $AB$.
From [Le Gall and Urrotia], I know that:
* if $A$ and $B$ are square-matrices of size $n$, then this can be done in $O(n^{\omega})$ where $\omega\approx 2.372$.
* if $A$ has size $n\times n^{k}$ and $B$ has size $n^k \t... | https://mathoverflow.net/users/90045 | Complexity of rectangular matrix multiplication | Assuming that *efficient* means better than the naive $O(n^{2+k})$ multiplication, let us review some possibilities.
**Padding.** For $k > \omega-2$, just pad $A$ with $n-n^k$ zero or garbage rows, perform square matrix multiplication in $O(n^\omega)$ time, and discard the extra rows from the output. This is already ... | 8 | https://mathoverflow.net/users/171662 | 402552 | 165,187 |
https://mathoverflow.net/questions/100650 | 7 | Quantum motivation
------------------
[Noncontextuality inequalities](http://arxiv.org/abs/1102.0264) (and in particular Bell inequalities) can be [mapped into graphs](http://arxiv.org/abs/1010.2163), in such a way that its relevant properties can be calculated via some simple graph-theoretical functions. In particul... | https://mathoverflow.net/users/9211 | Lovász function of the Möbius ladder | Now, 9 years later, this conjecture has been proven by [Bharti et al.](https://arxiv.org/abs/2104.13035) (one of the authors is Adán Cabello, who was in the paper that originally posed the conjecture).
The technique used was semidefinite programming (SDP) duality: computing the Lovász function is an SDP, and as all S... | 1 | https://mathoverflow.net/users/9211 | 402555 | 165,188 |
https://mathoverflow.net/questions/402554 | 1 | Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not necessarily finite rank) locally free $\mathcal{O}\_X$-modules
| https://mathoverflow.net/users/99988 | Are there nonaffine schemes over which every exact sequence of vector bundles is split? | Affine plane with double origin works.
| 5 | https://mathoverflow.net/users/348986 | 402559 | 165,189 |
https://mathoverflow.net/questions/298248 | 1 | Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$, $g \in C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times\partial\Omega)$, $u\_0 \in ... | https://mathoverflow.net/users/58541 | Schauder regularity heat equation | For a partial answer, see Theorem 6.48 in *Second order parabolic differential equations*, 1996, by Gary Lieberman.
There is an additional assumption that $f$ belongs to the Morrey space $M^{1,n+1+\alpha}$ defined page 130 of the book. In particular, $L^{\infty}\subset M^{1,n+1+\alpha}\subset L^1$.
I do not know if... | 1 | https://mathoverflow.net/users/349090 | 402570 | 165,193 |
https://mathoverflow.net/questions/402549 | 6 | Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). Moreover let $G$ act on $\mathbb{R}^{n}$ by standard left multiplication. We assume that $f$ is equivariant with respect to $G... | https://mathoverflow.net/users/348912 | Equivariant implicit function theorem | The equivariant version of the implicit function theorem is the following.
>
> Let $f: \mathbb{R}^p \times \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function (possibly only defined on open neighborhoods) which is equivariant with respect to the action of a compact Lie group $G$ on $\mathbb{R}^n$ and $\mathbb{R}^m$... | 7 | https://mathoverflow.net/users/17047 | 402581 | 165,197 |
https://mathoverflow.net/questions/402422 | 15 | There is a general result which holds for the rational numbers $ \mathbb Q $ (as well as number fields in general):
>
> For any completion $ K $ of $ \mathbb Q $ and any finite extension $ L/K $ of degree $ n $, the function $ L \to \mathbb R $ defined by $ x \to \sqrt[n]{|N\_{L/K}(x)|} $ gives a norm on $ L $.
>
... | https://mathoverflow.net/users/131052 | Why does the field norm on the field extension $ \mathbb C/\mathbb R $ induce a vector space norm? | The map $|N(\cdot)|^{1/n}$ is a continuous multiplicative extension of $|\cdot|$.
By a multiplicative function I mean a function $\chi:L\to [0,\infty)$
such that $\chi(0)=0$, $\chi(1)=1$ and for every $x,y\in L$, $\chi(xy)=\chi(x)\chi(y)$.
A multiplicative function which satisfies for every $x,y\in L$, $\chi(x+y)\leq \... | 14 | https://mathoverflow.net/users/89334 | 402589 | 165,198 |
https://mathoverflow.net/questions/402598 | 3 | Is there an open connected orientable 3-manifold $M$ with the following properties:
1. $M$ admits a complete hyperbolic metric with finite hyperbolic volume.
2. $H\_{i}(M,\mathbb{Z})=0$ for any $i>0$.
| https://mathoverflow.net/users/17895 | Special kind of 3-manifolds | No. Suppose that $M$ is a finite volume oriented hyperbolic three-manifold. In the closed case, as $M$ is oriented, we have $H\_3(M, \mathbb{Z}) \cong \mathbb{Z}$ generated by the fundamental class. In the open case, $M$ has torus cusps. Appealing to "one-half lives, one-half dies" we find that $M$ has non-trivial (in ... | 5 | https://mathoverflow.net/users/1650 | 402601 | 165,202 |
https://mathoverflow.net/questions/402623 | 11 | By Serre's theorem, we know the only nontorsion parts of the homotopy groups of spheres occur as $\pi\_n(S^n)$ and $\pi\_{4n-1}(S^{2n})$. The first of these are trivial to describe, but the second have very interesting, symmetric incarnations, they are the generalised hopf fibrations, at least for $n=1,2,4$, associated... | https://mathoverflow.net/users/128502 | Is there a concrete description of the nontorsion elements in the homotopy groups of spheres? | For $n \neq 1,2,4$, the minimal positive Hopf invariant of an element of $\pi\_{4n-1}(S^{2n})$ is $2$.
An explicit element of Hopf invariant $2$ can be constructed as follows: consider the attaching map $S^{4n-1} \to S^{2n} \vee S^{2n}$ of the $4n$-cell of the CW-complex $S^{2n} \times S^{2n}$ and compose it with the... | 21 | https://mathoverflow.net/users/14233 | 402624 | 165,205 |
https://mathoverflow.net/questions/402618 | 4 | [**Semiring categories**](https://ncatlab.org/nlab/show/rig+category), also called **rig categories** or **bimonoidal categories**, are pseudomonoids in the symmetric monoidal bicategory $(\mathsf{SymMonCats},\otimes\_{\mathbb{F}},\mathbb{F})$¹. These are a categorification of semirings, the monoids in $(\mathsf{CMon},... | https://mathoverflow.net/users/130058 | Day convolution for bimonoidal categories | Regarding Q2: probably there is a way to avoid going deep into coherence conditions: instead of proving by hand the equivalence between promonoidal structures on $C$ and biclosed monoidal structures on $\hat C$, one can resort to a more conceptual pov.
What happens for pro/monoidal categories is that there is a pseud... | 4 | https://mathoverflow.net/users/7952 | 402628 | 165,207 |
https://mathoverflow.net/questions/402191 | 5 | Perhaps the characteristic feature of the theory of ends is that they are extremely useful for computing sets of transformations between two functors. For example, one has the formulas
\begin{align\*}
\mathrm{Nat}(F,G) &\cong \int\_{A\in\mathcal{C}}\mathrm{Hom}\_{\mathcal{C}}\left(F\_{A},G\_{A}\right),\\
\mathrm{DiNat}... | https://mathoverflow.net/users/130058 | End formulas for sets of monoidal natural transformations | ### Monoidal ends
Let $(\mathcal{C},\otimes)$ be a monoidal category and let $(\mathcal{D},\times)$ be a *cartesian* monoidal category. Let $(X,\eta,\mu) : (\mathcal{C}^{\mathrm{op}},\otimes) \times (\mathcal{C},\otimes) \to (\mathcal{D},\times)$ be a lax monoidal functor. A wedge to $(X,\eta,\mu)$ is a wedge to the ... | 2 | https://mathoverflow.net/users/2841 | 402630 | 165,208 |
https://mathoverflow.net/questions/402637 | 19 | The one-point compactification $\mathbb{N}\_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of Cantor space
$$
\mathbb{N}\_\infty = \{ \alpha \in \{0,1\}^\mathbb{N} \mid
\forall n \,.\, \alpha\_n \geq \alpha\_{n+1} \}.
... | https://mathoverflow.net/users/1176 | Is the one-point compactification of $\mathbb{N}$ computably countable? | The answer is no. Suppose that there are computable functions $q$ and $s$ as you describe.
Let $k$ be a program that performs the following task. It starts enumerating $1$s at the start of the sequence until it discovers that $s(k)$ is defined. (We use the Kleene recursion theorem to know that there is such a self-re... | 17 | https://mathoverflow.net/users/1946 | 402641 | 165,210 |
https://mathoverflow.net/questions/402577 | 4 | Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms by Clifford multiplication (say, denoted by $\gamma$). Now for a real $3$-form $\beta$ on $M$, one can see that $\gamma(\... | https://mathoverflow.net/users/131004 | Identifying a $4$-form on a $6$-dimensional manifold | $\newcommand{\R}{\mathbb{R}}$As you state, there is a $SO(6)$-equivariant map $\delta:\operatorname{Sym}^2(\Lambda^3\R^6)\to \Lambda^4 \R^6$ such that $\gamma(\delta(\beta^{\otimes 2})) = \gamma(\beta)^2 - |\beta|^2\operatorname{id}$ on the positive spinors (the central $U(1)\subset\operatorname{Spin}^c(6)$ cancels, so... | 2 | https://mathoverflow.net/users/35687 | 402642 | 165,211 |
https://mathoverflow.net/questions/402631 | 0 | Let $\mathbf{v}\_1, \mathbf{v}\_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral:
$$\int\_{-\infty}^{ \infty} \int\_{-\infty}^{\infty}
\int\_{[-1,1]^n}
e(\theta\_1 \mathbf{v}\_1.\mathbf{x} +\theta\_2 \mathbf{v}\_2.\mathbf{x} ) d \mathbf{x} d \theta\_1 d\theta\_2$$
I vaguel... | https://mathoverflow.net/users/84272 | How to compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{[-1,1]^n}\exp[2\pi i(\theta_1 v_1.x+\theta_2v_2.x)]d^nx d\theta_1d\theta_2$ | First integrate over $\theta\_1,\theta\_2$. Use the delta function representation (for $k\in\mathbb{R}$)
$$\int\_{-\infty}^\infty e^{2\pi i k\theta}\,d\theta=\delta(k),$$
to evaluate
$$\int\_{-\infty}^\infty \int\_{-\infty}^\infty e^{2\pi i (v\_1\cdot x)\theta\_1+2\pi i(v\_2\cdot x)\theta\_2}\,d\theta\_1 d\theta\_2=\de... | 2 | https://mathoverflow.net/users/11260 | 402646 | 165,212 |
https://mathoverflow.net/questions/398496 | 8 | Corollary 12.14 of Digne-Michel's book *Representations of finite groups of Lie type* gives various decompositions of the regular representation $\operatorname{reg}\_G$ in terms of the Deligne-Lusztig characters. The two I am primarily interested in are $$\operatorname{reg}\_G = \frac{1}{|G^F|\_p} \sum\_{T \in \mathcal... | https://mathoverflow.net/users/175051 | Intuitive reason that the regular representation is a uniform function | This is an interesting question, but the kind of geometric or structural intuition your are looking for may not exist. To put it another way, the reason behind the fact in the OP is a non-trivial combination of several other central facts, but can't conceptually be reduced to either of them.
If we want to guess that ... | 5 | https://mathoverflow.net/users/2381 | 402653 | 165,213 |
https://mathoverflow.net/questions/402632 | 1 | I want to formulate something using the language of (possibly higher) category theory, but my knowledge in category theory is what most graduate students have learned in a first course in algebraic topology. So hopefully someone can help me out.
Let me provide some background to my question. A paper I am reading says... | https://mathoverflow.net/users/41686 | A question about possibly $\infty$-category or functors | $T$ can be formalized as a natural transformation $\def\Vect{{\rm Vect}} \def\Vectc{\Vect\_\nabla} \Vectc→Ω^n$ of functors $\def\Man{{\sf Man}} \def\op{{\sf op}} \def\Grpd{{\sf Grpd}} \Man^\op → \Grpd$.
The functor $\Vectc$ sends a smooth manifold $M$ to the groupoid $\Vectc(M)$
of vector bundles with connection over... | 2 | https://mathoverflow.net/users/402 | 402658 | 165,214 |
https://mathoverflow.net/questions/402661 | 0 | Is there a definition Df(g) of uniform continuity of g, without using the notion of metric?
Let $(E,d\_E)$ and $(F, d\_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$
We must have :
Df$(f)$ iff $f$ is uniform continue.
| https://mathoverflow.net/users/110301 | About uniform continuity | The notion of uniform continuity does not require all of the structure behind metric spaces. Every metric space is automatically a uniform space, and the uniformly continuous functions are the morphisms in the category of uniform spaces.
The two most common ways of defining uniform spaces are in terms of entourages o... | 4 | https://mathoverflow.net/users/22277 | 402664 | 165,217 |
https://mathoverflow.net/questions/402643 | 2 | Let $X$ be a closed complex manifold. Let $L$ be the trivial holomorphic line bundle. Can there be a short exact sequence of holomorphic line bundles $0\to L\to L\oplus L\to L\to 0$ that does not split?
| https://mathoverflow.net/users/349872 | Short exact sequence of trivial holomorphic line bundles not splitting | For any line bundle $L$, such exact sequence splits. Indeed the map $L\rightarrow L$ induced by the surjection $p:L\oplus L\rightarrow L$ must be nonzero on one of the summands, say the first one; hence it is the multiplication by a nonzero scalar $\alpha $. Then the map $L\xrightarrow{\ (\alpha ^{-1},0)\ } L\oplus L$ ... | 4 | https://mathoverflow.net/users/40297 | 402667 | 165,218 |
https://mathoverflow.net/questions/402663 | 3 | In a 2017 article [More on supersymmetric and 2d analogs of the SYK model](https://link.springer.com/article/10.1007%2FJHEP08%282017%29146) by Murugan, Stanford and Witten, the authors take a model called the SYK model (named after Sachdev, Ye and Kitaev) and study supersymmetric versions in dimensions one and two.
T... | https://mathoverflow.net/users/119114 | Supersymmetric SYK Model in 3D? | The disordered SYK model in three dimensions with supersymmetry was studied by Fedor Popov in [Supersymmetric tensor model at large $N$ and small $\varepsilon$.](https://arxiv.org/abs/1907.02440)
The complications are discussed in [A 3d disordered superconformal fixed point](https://arxiv.org/abs/2108.00027):
>
>... | 2 | https://mathoverflow.net/users/11260 | 402668 | 165,219 |
https://mathoverflow.net/questions/402607 | 0 | Let $X$ be a compact and convex space and let $T=[0,1]$ be some parameter space. Let $F:X\times T\rightrightarrows X$ be a correspondence that is compact-valued, convex, and upper-hemicontinous. By Kakutani's fixed point theorem, there is a fixed point $x(t)=F(x(t),t)$ for each parameter $t\in T$.
Suppose we also kno... | https://mathoverflow.net/users/121674 | Continuity of Kakutani fixed points | I assume that you mean that $F$ is upper semicontinuous on the product space. Then in particular (since $X$ is compact, Hausdorff and $F$ has closed values), $F$ has a closed graph. This implies that also the set $\{(x,t):x\in F(x,t)\}$ is closed. This means that the multimap $$t\mapsto\{x:x\in F(x,t)\}$$ has a closed ... | 1 | https://mathoverflow.net/users/165275 | 402673 | 165,222 |
https://mathoverflow.net/questions/402596 | 2 | This question is related to the matrices described in [Deyi Chen's recent MO post](https://mathoverflow.net/questions/402572/euler-numbers-and-permanent-of-matrices) (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, plus we have added variables. Defin... | https://mathoverflow.net/users/66131 | Determinants of striped Hankel matrices | **to Question 1:** Yes.
To prove this, let me fix a positive integer $n$ and denote your matrix (whose
determinant $f\_{n}$ is) by $A$. The notation $\left[ k\right] $ shall be
used for the set $\left\{ 1,2,\ldots,k\right\} $ whenever $k$ is an integer.
The notation $M\_{i,j}$ will be used for the $\left( i,j\right) ... | 4 | https://mathoverflow.net/users/2530 | 402683 | 165,225 |
https://mathoverflow.net/questions/402682 | 2 | In Silverman's book AEC, question 7.6 asks to prove $E\_0(K)$ has finite index in $E(K)$ for $K$ a local field. For part (a), I know the topology on $P^{n}(K)$ is the quotient topology on $K^{n+1}$, and the topology on $K^{n+1}$ is induced by the absolute value. However, I do not know how to prove the compactness of $P... | https://mathoverflow.net/users/350297 | Why is $P^n(K)$ compact, when $K$ is a local field? | Because the quotient mapping $K^{n+1} - \{\mathbf 0\} \to {\mathbf P}^n(K)$ (note $\mathbf 0$ is not in the domain) is continuous by the definition of the quotient topology, it suffices to show there is a compact subset $C$ of $K^{n+1} - \{\mathbf 0\}$ such that every element of ${\mathbf P}^n(K)$ is hit by some elemen... | 7 | https://mathoverflow.net/users/3272 | 402687 | 165,226 |
https://mathoverflow.net/questions/402689 | 6 | Can you prove or disprove the following claim:
First, define the function $\xi(n)$ as follows: $$\xi(n)=\begin{cases}-1, & \text{if }\varphi(n) \equiv 0 \pmod{4} \\
1, & \text{if }\varphi(n) \equiv 2 \pmod{4} \\ 0, & \text{if otherwise }
\end{cases}$$
where $\varphi(n)$ denotes [Euler's totient function](https://en.w... | https://mathoverflow.net/users/88804 | An infinite series involving the mod-parity of Euler's totient function | The only odd values of $\phi(n)$ are $\phi(1)=\phi(2)=1$.
$\phi(n)$ is even but not divisible by $4$ when:
1. $n=4$
2. $n=2^{\left\{0,1\right\}}p^m$, where $p=4k+3$ is prime, $m=1,2,3,...$
We have
$$
\frac{\pi^2}{6}=1+\frac14+\sum\_{\substack{n=1\\\phi(n)\equiv 0}}^\infty\frac{1}{n^2}+\sum\_{\substack{n=1\\\phi(n... | 8 | https://mathoverflow.net/users/82588 | 402696 | 165,229 |
https://mathoverflow.net/questions/402303 | 7 | Let $\mathcal C$ be a category whose skeleton has $\lambda$-many objects and $\kappa$-many morphisms. Then the skeleton of the endofunctor category $\mathcal C^{\mathcal C}$ has at most $\kappa^{3 \times \kappa}$ many morphisms.
My guess is that in most cases, this upper bound is achieved.
**Question:** What is an ... | https://mathoverflow.net/users/2362 | Category with few endofunctors? | An example of (1),(2),(3),(4) with $\kappa=\lambda=|\mathbb R|$ is to take $\mathcal C$ to be the posetal category $\mathbb R.$ This is already skeletal. Its category of endofunctors is the poset of non-decreasing functions $f:\mathbb R\to\mathbb R,$ with the pointwise order. There are at most $|\mathbb R^{\mathbb Q}|=... | 5 | https://mathoverflow.net/users/164965 | 402698 | 165,231 |
https://mathoverflow.net/questions/402621 | 4 | I am looking to find real quadratic fields whose Hilbert class field is abelian over $\Bbb Q$. Then I learned about genus numbers and genus field of the number field. It is enough to find a number field whose class number is equal to the genus number. In [YOSHIOMI FURUTA article](https://projecteuclid.org/download/pdf_... | https://mathoverflow.net/users/131448 | How to calculate genus number of number field using sage? | I may as well promote my comment to an answer, so this is closed. In a short paper, H. Hasse [1] computed the genus field and genus number of every real quadratic field. In particular, he shows that, given a real quadratic field $\Omega = \mathbb{Q}(\sqrt{d})$, the genus number of $\Omega$ is $2^{r-1}$, where $r$ is th... | 2 | https://mathoverflow.net/users/120914 | 402704 | 165,234 |
https://mathoverflow.net/questions/402505 | 7 | Let $\Gamma$ be a discrete group and $A$ be a $C^\*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and Ozawa's book "C\*-algebras and finite-dimensional approximations", section 4.3), but how should one think intuitively about... | https://mathoverflow.net/users/216007 | Amenable action intuition | It is impossible to understand the motivation behind the definition of an amenable action without first understanding the definition of amenable groups, so let me first talk about groups (for simplicity, just countable ones).
Finite groups are precisely the ones for which there is an invariant probability measure (th... | 14 | https://mathoverflow.net/users/8588 | 402712 | 165,237 |
https://mathoverflow.net/questions/402639 | 4 | Let $M$ be a compact $3$-manifold such that no component of $\partial M$ is $S^2$ and one component $F$ of $\partial M$ is the projective plane.
If $i\_\*:\pi\_1(F) \to \pi\_1(M)$ is an isomorphism, can we prove that $M$ is homeomorphic to $F \times [0,1]$?
| https://mathoverflow.net/users/280895 | 3-manifold with boundary containing a projective plane | Yes, the result follows from a theorem of [Livesay](https://doi.org/10.2307/1970543) and the Poincaré Conjecture that if $M$ is a compact connected non-orientable 3-manifold with $\pi\_1(M)$ finite, then $M$ is homeomorphic to $P^2\times I$ minus a collection of disjoint open 3-balls.
---
The first answer to the ... | 3 | https://mathoverflow.net/users/126206 | 402723 | 165,240 |
https://mathoverflow.net/questions/402717 | 13 | Let $S(\alpha) = \sum\_{n\leq N}f(n) e^{2\pi i \alpha n}$ for some arithmetic function $f$. Suppose $\alpha\_1, \ldots, \alpha\_R$ are real numbers that are $\delta$-spaced modulo $1$, for some $0 < \delta < 1/2$. The large sieve inequality then gives
$$
\sum\_{r=1}^R \left| S\left(\alpha\_r \right)\right|^2 \ll (N + \... | https://mathoverflow.net/users/75932 | Large sieve inequality for sparse trigonometric polynomials | If $f$ is $M$-sparse, then from Cauchy-Schwarz one has $|S(\alpha)|^2 \leq M \sum\_{n \leq N} |f(n)|^2$ which gives the bound
$$ \sum\_{r=1}^R |S(\alpha\_r)|^2 \leq R M \sum\_{n \leq N} |f(n)|^2$$
which is superior in the regime $RM \leq N$ (i.e., below the range of the Heisenberg uncertainty principle). For $RM \geq N... | 16 | https://mathoverflow.net/users/766 | 402734 | 165,248 |
https://mathoverflow.net/questions/402722 | 7 | Let $S$ be a subset of ${1,2,...,n}$ such that for every $a,b$ in $S$ the numbers of form $a^k+b^k$ are distinct ($k$ is positive integer)
What is the maximum cardinality of $S$
| https://mathoverflow.net/users/174530 | Sidon sets with k >1 | The $k=2$ case was considered by Alon and Erdős (*European J. Combin.* 6 (1985) 201-203, MR0818591) and improved by Lefmann and Thiele (*Combinatorica* 15 (1995) 379-408, MR1357284).
They expressed the problem as looking for the largest Sidon set of integers squared. The 1995 result: There exists a Sidon set $S \subs... | 10 | https://mathoverflow.net/users/14807 | 402736 | 165,250 |
https://mathoverflow.net/questions/402741 | 6 | Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
* Is there a natural way to introduce topology on $H^i(X, \mathcal{F})$?
My guess is that for each open covering $\mathcal{U}$ the space of Čech cochains $\check{C}(\mathc... | https://mathoverflow.net/users/82309 | Topology on cohomology of a sheaf of topological groups | Both cases ($F$ is a sheaf of abelian topological groups or abelian Lie groups) can be treated using the same machinery.
The Yoneda embedding embeds abelian Lie groups as a fully faithful subcategory of the category of sheaves of abelian groups on the site of smooth manifolds, and the embedding functor preserves smal... | 5 | https://mathoverflow.net/users/402 | 402744 | 165,253 |
https://mathoverflow.net/questions/402714 | 7 | Let $K\_{n,n}$ be a complete bipartite graph with two parts $\{u\_1,u\_2,\ldots,u\_n\}$ and $\{v\_1,v\_2,\ldots,v\_n\}$, and let $K^-\_{n,n}$ be the graph derived from $K\_{n,n}$ by delete a perfect matching $\{u\_1v\_1,u\_2v\_2,\ldots,u\_nv\_n\}$.
Since $K^-\_{n,n}$ is now $(n-1)$-regular, it has $n-1$ disjoint perf... | https://mathoverflow.net/users/148974 | Disjoint perfect matchings in complete bipartite graph | The answer is that this is possible for all $n>4$.
Your question is equivalent to asking whether there exists a unipotent Latin square $L$ of order $n$ with $L\_{ij}\ne L\_{ji}$ for $i\ne j$. The equivalence is obtained by using $L\_{ij}$ to record the index of the matching that contains the edge $u\_i v\_j$ (and put... | 8 | https://mathoverflow.net/users/351290 | 402749 | 165,254 |
https://mathoverflow.net/questions/402688 | 16 | The Cayley-Menger determinant gives the squared volume of a simplex in $\mathbb{R}^n$ as a function of its $n(n+1)/2$ edge lengths:
$$v\_n^2 = \frac{(-1)^{n+1}}{(n!)^2 2^n}
\begin{vmatrix}
0&d\_{01}^2&d\_{02}^2&\dots&d\_{0n}^2&1\\
d\_{01}^2&0&d\_{12}^2&\dots&d\_{1n}^2&1\\
d\_{02}^2&d\_{12}^2&0&\dots&d\_{2n}^2&1\\
\vd... | https://mathoverflow.net/users/23829 | Is there a degenerate simplex in $\mathbb{R}^{8 k-1}$ with odd integer edge lengths? | This is answered in a paper by R. L. Graham, B. L. Rothschild & E. G. Straus ["Are there $n+2$ Points in $E\_n$ with Odd Integral Distances?"](https://sci-hub.ru/https://www.tandfonline.com/doi/abs/10.1080/00029890.1974.11993491). Such simplexes exist iff $n+2 \equiv 0 \pmod {16}$. They also consider the related proble... | 12 | https://mathoverflow.net/users/795 | 402750 | 165,255 |
https://mathoverflow.net/questions/402691 | 17 | Recall that a compact Riemann surface/algebraic curve $C$ is **hyperelliptic** if it admits a branched double cover $C \to \mathbb P^1$, where $\mathbb P^1$ is the complex projective line/Riemann sphere. Among those curves of hyperbolic type ($g \ge 2$), the only genus that admits such a double cover in general is $g =... | https://mathoverflow.net/users/27219 | How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and only if $g=2$"? | Here is a small variant on Eremenko's answer.
The "Fenchel-Nielsen" coordinates on the space of hyperbolic metrics on a surface $\Sigma\_g$ can be described via a pants decomposition. This is a decomposition of the surface along a collection of curves, that split the surface into a union of disjoint $3$-punctured sph... | 8 | https://mathoverflow.net/users/1465 | 402752 | 165,256 |
https://mathoverflow.net/questions/402746 | 3 | We know for morphisms of schemes $f:X \rightarrow S, u:S’ \rightarrow S, X’=X \times\_S S’$, (coherent) sheaf $F$ over X,we have natural morphism
$$u^\* R^q f\_\*F\rightarrow R^q u’\_\* f’^\*F$$
where $f’, u’$ are basechanges. (Or using derived category if you like.)
The base change theorems says under certain restrict... | https://mathoverflow.net/users/170335 | What‘s the obstruction to base change | The composition of functors $Rf\_\* \colon D(X) \to D(S)$ and $Lu^\* \colon D(S) \to D(S')$ is a Fourier--Mukai functor given by an explicit object $K \in D(X \times S')$. It is easy to check that it has no cohomology sheaves in positive degrees, its zero cohomology sheaf is isomorphic to the structure sheaf of the fib... | 11 | https://mathoverflow.net/users/4428 | 402755 | 165,257 |
https://mathoverflow.net/questions/402761 | 6 | Let $A$ a square real matrix such that the largest singular value $\sigma\_\text{max}(A) = \sigma < 1$. I want to find a lower bound on $\langle (I + A)^{-1}x, x\rangle$ where $x$ is a vector of euclidean norm $1$: $\langle x, x\rangle=1$.
I empirically find that a seemingly tight lower bound is
$$
\langle (I + A)^{-... | https://mathoverflow.net/users/173967 | Lower bound $\langle (I + A)^{-1}x, x \rangle$ given that $\sigma_\text{max}(A) < 1$ | The map $f(z)=(1+z)^{-1} - (1+\sigma)^{-1}$ maps the disk of radius $\sigma$ into the right half plane as a function of one complex variable.
Therefore, essentially by von Neumann's inequality, we get that $$\frac{f(A)+f(A)^\*}{2}=\mathrm{Re }f(A)\geq 0$$ since $\|A\|\leq \sigma.$ Assuming $A$ has real entries, this ... | 6 | https://mathoverflow.net/users/32470 | 402770 | 165,261 |
https://mathoverflow.net/questions/402713 | 8 | In [Zhu's seminal paper](https://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00182-8/), he proves (5.3.2) that if $V$ is a vertex algebra the character of all of its modules are ***modular forms***! (This is not literally true- there are conditions).
I have always found this statement very mysterious. How on ea... | https://mathoverflow.net/users/119012 | Why are VOA characters modular forms (geometrically)? | I'm sure someone here can handwave the intuition behind these statements much better than me. This handwaving version goes like this, one is interested in computing vacuum 1-point functions on a torus, but you can cut a torus along an $S^1$ making it into a cylinder, put a module `M` on the boundaries, now you start wi... | 11 | https://mathoverflow.net/users/17980 | 402786 | 165,264 |
https://mathoverflow.net/questions/402718 | 0 | Let $M$ be a complete non-compact manifold (possibly with boundary). Let $E$ be an open proper connected non-precompact subset of $M$ with smooth topological boundary, so that $\overline{E}$ is a non-compact complete manifold with boundary. Suppose that $E=M\setminus K$, where $K$ is a compact set that is the closure o... | https://mathoverflow.net/users/163368 | Is this a manifold of bounded geometry? | Unless I am terribly mistaken, the answer is yes and the proof strategy is rather simple. The crucial observation is that the definition of bounded geometry depends on quantities that are continuous.
Consider first the injectivity radius function of the boundary, $r\_{b}\colon\delta X\to \mathbb{R}$,
$$r\_b(x)=\sup\{... | 2 | https://mathoverflow.net/users/44172 | 402791 | 165,267 |
https://mathoverflow.net/questions/402767 | 4 | The Hankel determinants of the Catalan numbers are well known and can be written as
$d(k,n)= \det \left( C\_{k + i + j} \right)\_{i,j = 0}^{n - 1}=\prod\_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$ with $d(k,0)=1.$
Computations suggest that
$$D\_k(x)=\sum\_{n\geq 0}d(k,n)x^n=\frac{A\_{k}(x)}{(1-x)^{\binom{k}{2... | https://mathoverflow.net/users/5585 | Generating functions for Hankel determinants of Catalan numbers | I'm upgrading my comments to an answer.
As I've mentioned in comments/answers to some of your previous MO questions (e.g. [Number of bounded Dyck paths with negative length as Hankel determinants](https://mathoverflow.net/questions/372811/number-of-bounded-dyck-paths-with-negative-length-as-hankel-determinants) and [... | 6 | https://mathoverflow.net/users/25028 | 402795 | 165,269 |
https://mathoverflow.net/questions/402797 | 1 | What is the name of the following combinatorial game:
Two players, moving in turn.
Positions: $0,1,2,\ldots$.
Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$
if $n>0$.
No move for $0$ which loses.
The determination of the winning strategy is easy:
$n=2^k(2m+1)>0$ wins if and only if $k$ is even... | https://mathoverflow.net/users/4556 | Name for an easy combinatorial game | This is "Mark", supposedly due to Mark Krusemeyer; see the first sentence of the introduction to <https://arxiv.org/abs/1509.04199> and section 2 of <https://doi.org/10.37236/2015>.
| 3 | https://mathoverflow.net/users/3075 | 402805 | 165,272 |
https://mathoverflow.net/questions/402789 | 9 | $\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
* A **monoid** is a monoid in $(\mathsf{Sets},\times,\pt)$;
* A **semiring** is a monoid in $(\mathsf{CMon},\otimes\_{\N},\N)$;
* A **r... | https://mathoverflow.net/users/130058 | Are differential rings monoids in a monoidal category? | $\newcommand{\defeq}{\overset{\mathrm{def}}{=}}\newcommand{\id}{\mathrm{id}}\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\d}{\mathrm{d}}\newcommand{\dAb}{\mathsf{End}(\mathsf{Ab})}$DGAs are monoids in chain complexes. To get differential r... | 7 | https://mathoverflow.net/users/130058 | 402816 | 165,274 |
https://mathoverflow.net/questions/402801 | 2 | Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha\_1, \cdots, \alpha\_r$ be the corresponding choice of simple roots, $X\_{\alpha\_i}, H\_{\alpha\_i}, X\_{-\alpha\_i}$ the Cartan--Weyl basis, and $\pi\_1, \cdots, \pi\_l$ the fundamental weights. For th... | https://mathoverflow.net/users/352001 | Action of the negative Cartan-Weyl generators on a highest weight element | Let α be a simple root that is not αk. Associated to this simple root is a subalgebra isomorphic to $\mathfrak{sl}\_2$. For this subalgebra, an element of weight ωk has weight zero. In the fundamental representation you are considering, such an element is highest weight by assumption and therefore by the classificaitio... | 4 | https://mathoverflow.net/users/425 | 402820 | 165,277 |
https://mathoverflow.net/questions/402847 | 4 | I'm trying to compute the following Haar integral over the unitary group:
$$
\int\limits\_{\mathbb{U}(d)}\dfrac{1}{\sum\_{k,l=1}^d u\_{ik}\overline{u\_{il}}c\_{kl}}dU.
$$ Is there anything known about the value of such integrals? I haven't been able to find anything that studies the integrals of either non polynomial f... | https://mathoverflow.net/users/71033 | Haar integral of rational function of unitaries | There exist no closed-form expressions for arbitrary $d$ for the integral over the unitary group $\mathbb{U}(d)$ of a rational function of the matrix elements.
There are asymptotic results for large $d$, see for example [J. Math. Phys. 37, 4904 (1996)](https://arxiv.org/abs/cond-mat/9604059). The leading order term ... | 7 | https://mathoverflow.net/users/11260 | 402848 | 165,284 |
https://mathoverflow.net/questions/402802 | 3 | Informally, an $\infty$-category should be the following data:
* A collection of objects
* A space of morphisms between any two objects
* Weak associativity rules: Coherent homotopies between all of the different ways to compose morphisms
As far as I understand, simplicially enriched categories can be used as a som... | https://mathoverflow.net/users/125868 | Weak composition rule for simplicial categories | The most obvious approach is to consider simplicial $\def\Ai{{\sf A}\_∞}\Ai$-categories, where $\Ai$ denotes a nonsymmetric operad in simplicial sets that is weakly equivalent to the terminal operad, i.e., the associative operad.
In such a structure, instead of the usual compositions we have maps
$$\Ai(n)⨯C(X\_{n-1},... | 6 | https://mathoverflow.net/users/402 | 402849 | 165,285 |
https://mathoverflow.net/questions/402851 | 4 | $\newcommand{\R}{\mathbb R}$Consider the following [construction](https://www.mathcounterexamples.net/pointwise-convergence-not-uniform-on-any-interval/). For real $u$, let
\begin{equation}
f(u):=\frac{2u^2}{1+u^4},
\end{equation}
so that the function $f\colon\R\to\R$ is continuous, $0\le f\le1=f(\pm1)$, and $f(u)\to0... | https://mathoverflow.net/users/36721 | How bad can pointwise convergence in $C$ be? | This is just a standard application of the Baire category theorem.
Proposition: Suppose that $X$ is a topological space where the Baire category theorem holds and $g\_{n}:X\rightarrow[0,\infty]$ for each natural number $n$. Suppose that $\overline{\lim}\_{n}\sup\_{I}g\_{n}\geq 1$ for each non-empty open set $I$. Then... | 12 | https://mathoverflow.net/users/22277 | 402855 | 165,288 |
https://mathoverflow.net/questions/402742 | 3 | Consider the split monic $f=\prod\_{i=1}^n(x-x\_i)\in \mathbb Z[x\_1 ,\dots ,x\_n,x]$. Its discriminant is usually defined as $$(-1)^{n(n-1)/2}\prod\_{i=1}^nf^\prime(x\_i)=\prod\_{1\leq i<j\leq n}(x\_i-x\_j)^2.$$
What is the reason for taking this definition as opposed to $\prod\_{i=1}^nf^\prime(x\_i)$? The product o... | https://mathoverflow.net/users/69037 | Why the sign in the definition of the discriminant? | The reason is that the formula on the *right side* should be considered more fundamental, not the formula on the left, when seeking a symmetric expression in the roots. Don't use a product of anything "at" the roots, but a symmetric expression in the roots that vanishes if any pair of roots are equal. That explains the... | 7 | https://mathoverflow.net/users/3272 | 402862 | 165,292 |
https://mathoverflow.net/questions/402824 | 2 |
>
> **Question 1.** Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is generated in degree $\leq a$ (that is, there is a set of generators of $I$ that have degrees $\leq a$) and $J$ is ge... | https://mathoverflow.net/users/2530 | Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring | Take $I=(a^3,b^3)$ and $J=(ac^2-bd^2)$. Then according to Macaulay2, $I\cap J$ has generators in degrees $7,8,9$, for instance $a^3c^6-b^3d^6$. So the answers to Question 3 and 1 are no.
| 2 | https://mathoverflow.net/users/2083 | 402871 | 165,297 |
https://mathoverflow.net/questions/402872 | 1 | Let $X,Y$ be Banach spaces and let $A:X\rightarrow Y$ be a linear operator. Does it suffice to show that there exists a sequence $x\_n\in X$ such that $\lim\_{n\rightarrow\infty}Ax\_n = 0$ with $||x\_n||=1\quad\forall n\in\mathbb{N}$ to proof non-injectivity of $A$?
| https://mathoverflow.net/users/153356 | Showing non-injectivity | There are examples of Banach spaces $X,Y$ along with bounded linear mappings $L:X\rightarrow Y$ and sequences $(x\_{n})\_{n}$ of elements in $X$ such that
$^{\lim}\_{n\rightarrow\infty}L(x\_{n})=0$ in the metric space induced by the norm on $Y$ but where $\|x\_{n}\|=1$ for each $n$. For instance, if $X=Y$ and $X$ is a ... | 4 | https://mathoverflow.net/users/22277 | 402875 | 165,299 |
https://mathoverflow.net/questions/397542 | 12 | There is a [construction](https://iopscience.iop.org/article/10.1070/IM1971v005n04ABEH001121/meta) of the algebraic K-theory groups $K\_i(R)$ of a ring $R$ by Volodin. He gave an explicit construction of the plus-construction $BGL(R)^+$ as the quotient of the bar construction $BGL(R)$ by the union $\bigcup\_{n,\sigma} ... | https://mathoverflow.net/users/157284 | Uses of Volodin's construction of algebraic K-theory | It has several uses:
1. Volodin $K$-theory was used by Igusa in the late 1970s/Early 1980s to define $K$-theoretic invariants of families of pseudoisotopies.
2. In the 1980s, the Volodin construction (actually a variant of it) was used by Goodwillie to relate rational relative K-theory to rational relative THH. This ... | 12 | https://mathoverflow.net/users/8032 | 402876 | 165,300 |
https://mathoverflow.net/questions/402821 | 7 | Let $M$ be a submanifold of a symmetric space $Q$. The normal bundle $NM$ is called *abelian* if $\exp(N\_{p}M)$ is contained in some totally geodesic and flat submanifold of $Q$ for all $p \in M$; see Terng & Thorbergsson, "Submanifold geometry in symmetric spaces", J. Differential Geom. 42 (1995), 665–718.
It is cl... | https://mathoverflow.net/users/74033 | Submanifolds of Lie groups with abelian normal bundle | One class of examples in a compact, connected Lie group $G$ of rank $r$ are the conjugacy classes of codimension $r$. Taking an element $a\in G$ whose centralizer is a maximal torus (a generic condition), the conjugacy class of $a$ is a submanifold $C\_a\subset G$ of codimension $r$ whose normal plane at $a$ is the tan... | 6 | https://mathoverflow.net/users/13972 | 402877 | 165,301 |
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