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https://mathoverflow.net/questions/280397 | 1 | Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm.
Is it possible to evaluate the following integral in closed form?
$$Y = \displaystyle\int\_{0}^{\infty} \left(M + tI\right)^{-1} (\log M) \left(M + tI\right)^{-1}\:\mathrm{d}t.$$
No... | https://mathoverflow.net/users/18526 | Closed form for integral of function of a symmetric positive definite matrix | just go to a basis where $M$ is diagonal, evaluate the integral and find $Y=M^{-1}\log M$.
| 1 | https://mathoverflow.net/users/11260 | 280406 | 124,184 |
https://mathoverflow.net/questions/280363 | 8 | Consider an election with $N$ voters and $k$ candidates, where each voter votes randomly for one of the candidates. What are the odds of a tie?
Here "tie" means that multiple candidates get the highest number of votes, *not* necessarily that all candidates get the same number of votes.
For $k = 2$ the answer is giv... | https://mathoverflow.net/users/114165 | What are the odds of a tie in a random election with k candidates? | For $1\ll k\ll N$, I would roughly estimate the probability of a tie by $O(\sqrt{(k\log k)/N})$. This is a very loose calculation. The number of votes to candidate $i$, $X\_i$, is distributed $Bin(1/k,n)$. The loose part of this answer is assuming the $X\_i$'s were mutually independent. I guess that it shouldn't change... | 7 | https://mathoverflow.net/users/85550 | 280411 | 124,186 |
https://mathoverflow.net/questions/280409 | 9 | Let $f$ be a cuspidal Hecke Eigenform of weight $k \geq 2$ and let $\rho\_{f, \lambda}:G\_{\mathbb{Q}} \rightarrow GL\_2(E\_{\lambda})$ be the corresponding Galois representation with $2 \mid \lambda$ constructed by Deligne. Assume now that $\pi\_f={\otimes}' \pi\_p$ be the corresponding automorphic representation.
... | https://mathoverflow.net/users/25198 | Exceptional primes | No, this does not work: for any modular form of weight $k \ge 2$, the image of the projective representation $\tilde\rho\_{f, \lambda}$ is infinite for every prime $\lambda$.
Proof: if $\tilde{\rho}\_{f, \lambda}$ has finite image, then so does the adjoint representation $\operatorname{Ad}^0 \tilde{\rho}\_{f, \lambd... | 9 | https://mathoverflow.net/users/2481 | 280412 | 124,187 |
https://mathoverflow.net/questions/280407 | 0 | In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties
(N1) $\rho(x)=0\implies x=0$;
(N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$;
(N3) $\rho(x\vee y )\le \rho(x)+\rho(y)$ for all $x\ge 0, y\ge 0$;
(N4) $\rho(\lambda x)\to 0$ if $\lambda \to 0$
Thi... | https://mathoverflow.net/users/113782 | Definition of an Orlicz modular space | How about this?
Let $\alpha, \beta \ge 0$, $\alpha+\beta = 1$.
Then
$$
|\alpha x + \beta y| \le \alpha |x| + \beta |y| \le |x| \vee |y|
$$
so
$$
\rho(\alpha x + \beta y) =
\rho\big(|\alpha x + \beta y|\big) \le \rho\big(|x| \vee |y|\big) \le
\rho\big(|x|\big) +\rho\big(|y|\big) = \rho(x)+\rho(y)
$$
Note: $\rho... | 1 | https://mathoverflow.net/users/454 | 280417 | 124,189 |
https://mathoverflow.net/questions/280414 | 1 | Let $n \geq 1$ be an integer. Let us call $S\subseteq \mathbb{R}^{n+1}$ an *$n$-sphere* if there is $x\in \mathbb{R}^{n+1}$ and $r\in \mathbb{R}$ with $r>0$ such that $$S = \{z\in \mathbb{R}^{n+1}: \|z-x\| = r\}, $$ where $\|\cdot\|$ denotes the Euclidean norm in $\mathbb{R}^{n+1}$.
Suppose that ${\cal S}\neq \emptys... | https://mathoverflow.net/users/8628 | Meeting a set of spheres in $\mathbb{R}^n$ | Let $(S\_i)\_{i \in I}$ be a family of spheres of cardinality $|I| < \mathfrak{c}$ and let $S$ be a sphere distinct from each $S\_i$. Each intersection $S \cap S\_i$ is a (possibly degenerate) circle. Since there is a continuum of circles inside $S$, one can find a circle $C \subseteq S$ distinct from each $S \cap S\_i... | 6 | https://mathoverflow.net/users/21724 | 280423 | 124,191 |
https://mathoverflow.net/questions/280436 | 4 | Let $\{X\_t\}\_{t=-\infty}^{\infty}$ be a sequence of random variables. We are interested in a "random walk" (or more generally, a random field) that can be characterized by
$$
X\_t ~|~ X\_{t-k}, \ldots, X\_{t-1}, X\_{t+1}, \ldots, X\_{t+k} \sim D\_t,
$$
where $k>0$ is a constant and $D\_t$ is some distribution. Here ... | https://mathoverflow.net/users/82358 | Random Walk with "Forward Dependency" | if we assume that the index $t$ in your process is countable then what you are looking for is described in Georgii's book, Gibbs Measures and Phase Transition. In the language of mathematical Statistical Mechanics the processes you are interested in is a one-dimensional spin system on the lattice $\mathbb{Z}$ with fini... | 5 | https://mathoverflow.net/users/2386 | 280438 | 124,196 |
https://mathoverflow.net/questions/280435 | 4 | Let $X$ be a non-compact complete Riemannian manifold and $P$ a first-order elliptic pseudodifferential operator on $X$. Let $Q$ be a parametrix for $P$, so that $PQ - 1 = T$ and $QP - 1 = R$ are lower order operators.
If $X$ were compact, then the operators $Q$, $T$, $PQ$ and $PT$ would all be bounded on $L^2(X)$, ... | https://mathoverflow.net/users/78729 | Choice of parametrix on a non-compact manifold | Yes, at least if $Q(x,y)$ is locally integrable in $y$.
Given two integral kernels $Q(x,y)$ and $Q'(x,y)$ that agree on a neighborhood of the diagonal in $X \times X$, then if one is a parametrix, so is the other one. Hence, you can always restrict a parametrix to have support in an arbitrary neighborhood of the diag... | 2 | https://mathoverflow.net/users/2622 | 280439 | 124,197 |
https://mathoverflow.net/questions/280451 | 4 | **Edit:** According to the interesting comment of Tobias Fritz we revise the question.
Assume that $G$ is a Lie group and $M\subseteq G$ is a closed connected smooth submanifold of $G$ containing the neutral element $e\in G$. Assume that for every $m\in M$ we have $D L\_m (T\_e M)=T\_m M$. This means that the differe... | https://mathoverflow.net/users/36688 | Submanifold of a Lie group whose tangent bundle is invariant under group (left) action | The answer is yes.
Any subspace $V\subset\mathfrak g$ in the Lie algebra of $G$ defines a $G$-invariant distribution of planes in $TG$ via the action you have specified: $V\_m=DL\_m(V)$. This distribution is Frobenius-integrable iff $[V,V]\subset V$, i.e., $V$ is a Lie subalgebra. Note that by $G$-invariance $V$ is i... | 7 | https://mathoverflow.net/users/40950 | 280453 | 124,199 |
https://mathoverflow.net/questions/279264 | 9 | I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the literature of convex analysis could help.
Suppose $P$ is an open bounded convex subset of $\Bbb R^n$ and $f: P \to \B... | https://mathoverflow.net/users/17965 | property of convex functions | *Anyway, If you know a 5 line proof for the first inequality please share it with us*
OK, here goes.
Assume that $\inf\_P f=-1$ and that it is (nearly) attained at the origin. Let $K=\{x\in P: f(x)\le -\frac 12\}$. Then $E=\{x\in P: f(x)<1\}\subset 4K$ by convexity. Also, $|E|\ge |P|/2$ because otherwise $\int\_P ... | 10 | https://mathoverflow.net/users/1131 | 280462 | 124,204 |
https://mathoverflow.net/questions/280275 | 3 | I am reading [this paper](https://doi.org/10.1023/A:1022801220384). I am stuck on something, which I think is something basic but I haven't been able to figure it out yet, and I was hoping someone could explain it to me.
Let
$$
\sigma(u) = \frac{1}{2\pi} \int\_{-T}^T F(t) \frac{u^{1/2 + it} - M^{1/2 + it}}{1/2 + it}... | https://mathoverflow.net/users/84272 | Basic question related to Stieltjes integral | It's two integrations by parts.
$$
\int\_M^N e^{2\pi i\lambda u}\, d\sigma(u) = \sigma e^{2\pi i\lambda u}\bigr|\_M^N - 2\pi i\lambda\int\_M^N e^{2\pi i\lambda u}\sigma(u)\, du
$$
Let me write the asymptotic formula for $\sigma$ symbolically as $\sigma(u)=I(u)+O(\ldots)$ and use it to rewrite this as
$$
I e^{2\pi i\lam... | 1 | https://mathoverflow.net/users/48839 | 280468 | 124,207 |
https://mathoverflow.net/questions/280431 | 1 | Let X be a seminormal variety and S be the singular locus of X. Is Blow$\_S X=$ the normalisation of X?
Is the singular locus given by the conductor ideal?
| https://mathoverflow.net/users/nan | singular locus of semi-normal variety | First, definitely the singular locus is not equal to the conductor. If $X$ is normal, the singular locus of $X$ is definitely not given by the conductor (which is the unit ideal and doesn't vanish anywhere).
For the first question however, Greco-Traverso actually studied blowing up the *conductor* in a seminormal var... | 3 | https://mathoverflow.net/users/3521 | 280470 | 124,208 |
https://mathoverflow.net/questions/280306 | 1 | *Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll give it a certain time loking at reactions on the [discussion in the MO-meta site](https://meta.mathoverflow.net/questions... | https://mathoverflow.net/users/7710 | A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$? | *The answers for b) and c) came out to be trivial and have likely nothing to to with Zsigmondy, so possibly I **should retract** my question.*
---
For the definition of a cycle for some $a\_1=T\_\rho(a\_1;E\_{N,S})$ we have the formula (6.b) in the referred article:
$$ a\_1 = \rho \cdot { Q(E\_{N,S}) \over 2^S -... | 1 | https://mathoverflow.net/users/7710 | 280472 | 124,209 |
https://mathoverflow.net/questions/280478 | 1 | For $A\subseteq\mathbb{N}$ we define the upper density to be $$\text{ud}(A) = \lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$
Is there an infinite set ${\cal S}$ of pairwise disjoint subsets of $\mathbb{N}$ such that $\text{ud}(S) > 0$ for all $S\in {\cal S}$?
| https://mathoverflow.net/users/8628 | Pairwise disjoint subsets of $\mathbb{N}$ with positive upper density | Yes. Take the segments of integers $B\_i=\{2^i,\dots,2^{i+1}-1\}$. Partition naturals onto disjoint infinite sets $I\_1,\dots$. Take the sets $A\_i=\cup\_{k\in I\_i} B\_k$.
| 5 | https://mathoverflow.net/users/4312 | 280480 | 124,211 |
https://mathoverflow.net/questions/280484 | 0 | I have a question about part processes which appear in theory of Markov processes.
Let $\mathbb{M}=(X\_t,P\_x)$ be a Markov prccess on a topological space $E$. Let $A $ be an open subset of $E$. The part process $(X\_{t}^{A},P\_{x})$ of $\mathbb{M}$ on $A$ is defined as follows:
\begin{align\*}
X\_{t}^{A}(\omega)&=X\... | https://mathoverflow.net/users/68463 | Consistency of part processes | What you need is the following fact: the reflecting BM on $\overline{D}$ agrees with the usual BM on $\mathbb{R}^n$ up to the hitting time of $\partial D$. "Agrees" means here that the transition kernels (and resolvents) are the same.
The proof of this fact depends on your favourite definition of the reflecting BM. I... | 1 | https://mathoverflow.net/users/108637 | 280485 | 124,214 |
https://mathoverflow.net/questions/87917 | 5 | I am struggling with following problem. Consider $2A$ class in $Co\_1$ having $819\*759\*75$ elements. Each element $a$ from $2A$ have two representatives in $Co\_0$. Element $a$ corresponds to $E\_8$ sublattice in Leech lattice defined as $\{v: av=-v\}$ where I call by $a$ also proper preimage in $Co\_0$. Now the oppo... | https://mathoverflow.net/users/nan | Geometric interpretation of $2A$ conjugacy class in Conway group $Co_1$ | Here is the answer for $Co\_1$ which I obtained in GAP.
For given $2A$ involution $a$ in $Co\_1$ the number $n$ of such $2A$ involutions $b$ that product $ab$ is in conjugacy class $C$ is presented in format $[C, n]$. I call *hook* the order of product.
$[ [ "1a", 1 ], [ "2a", 12870 ], [ "2c", 60480 ], [ "3b", 573... | 4 | https://mathoverflow.net/users/nan | 280488 | 124,215 |
https://mathoverflow.net/questions/280482 | 10 | It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $E(\Bbb Q)$ are known.
Are there similar results for other... | https://mathoverflow.net/users/114143 | Possible groups of K-rational points for elliptic curves over arbitrary fields | By Mordell-Weil, for any number field $K$ we have
$$C(K)=\mathbb{Z}^r \times E(K)\_{\mathrm{tors}}$$
As you mention, Mazur showed all the possible options for $E(\mathbb{Q})\_{\mathrm{tors}}$ in his famous 1977 paper.
The only other $K$ for which we have a torsion theorem are the quadratic fields. This is the res... | 16 | https://mathoverflow.net/users/43108 | 280490 | 124,217 |
https://mathoverflow.net/questions/280508 | 2 | We work in the NBG set theory (with local choice, but without global choice).
$V=\{x∣x=x\}$ is the universal class and $\emptyset=\{x∣ not x=x\}$ is the empty class. Every class $A$ satisfies $A\subseteq V$; $A$ is a proper class if $A\notin V$ and a set if $A\in V$. For every class the power set of $A$ is $P(A)=\{x∣... | https://mathoverflow.net/users/30395 | More in bijective-equivalent class in NBG set theory (1) | The answer is no, because the existence of an injection of Ord into $W$ implies global choice. From your definitions, $W$ is the proper class of all well-orders of the rank-initial segments of the universe. Your $V(\alpha)$ is what is usually denoted $V\_{\alpha+1}$. Suppose that we had an injection of Ord into $W$. Si... | 3 | https://mathoverflow.net/users/1946 | 280510 | 124,225 |
https://mathoverflow.net/questions/280475 | 3 | Let $(M,g)$ be a $2n$-dimensional almost Hermitian manifold ($n\geq 2$) with a almost complex structure $\cal J$ (*not necessary integrable*). i.e.,
$${\cal J}^2=-I,\quad\qquad g({\cal J} X,{\cal J} Y)=g(X,Y).$$
Suppose that $\{X\_i,{\cal J}X\_i\}$ be any local orthonormal ${\cal J}$-frame and the following relations ... | https://mathoverflow.net/users/90655 | Almost Hermitian manifolds of constant curvature | The answer to the question as asked is 'No', at least when $2n=4$. In that case, the conditions (1) and (2) only involve the Weyl curvature of the underlying metric $g$, so, in particular, when $g$ is conformally flat, then the conditions are satisfied, and there are many conformally flat metrics in dimension $4$ that ... | 4 | https://mathoverflow.net/users/13972 | 280512 | 124,226 |
https://mathoverflow.net/questions/278789 | 16 | The following is a well-know fact:
**Theorem** The real $r$ is Cohen over $V$ iff if it does not belong to any meager Borel
set coded in $V$.
Now suppose that $\kappa$ is an uncountable cardinal and let $(r\_i: i<\kappa)$ be a sequence of reals.
>
> **Question.** Is there a characterization theorem as above for... | https://mathoverflow.net/users/11115 | Characterization of Cohen reals | Chapter 20 of the Handbook of Set Theoretic Topology ("Random and Cohen reals" by Ken Kunen, pp 887-911) deals with such questions.
Quoting from Truss's review:
"Quite a proportion of the paper is devoted to a study of the properties of Cohen extensions of a countable transitive model by $2^I/\mathcal{I}$, where I... | 8 | https://mathoverflow.net/users/18128 | 280514 | 124,227 |
https://mathoverflow.net/questions/280521 | 6 | Let $f(x) = x^m+\sum\_{j=0}^{m-1}f\_{m-j}x^j\in P[x]$ be a **monic** polynomial over a field $P$ and let $f(x) = (x-\alpha\_1)\cdot\ldots\cdot(x-\alpha\_m)$ be a factorization of $f$ over an extension field $Q$ of $P$.
Then it is quite natural to consider a value (called discriminant) $$\prod\_{1\leq i<j\leq m}(\alph... | https://mathoverflow.net/users/85489 | What is the essence of the constant factor in the standard definitions of the discriminant? | As Robert said, if you want everything to work in $\mathbb Z[f\_0,\ldots,f\_m]$, you need that factor. I'll also mention that your polynomial indexing is messed up, you probably meant the sum to go from $j=0$ to $j=m-1$, not $j=1$ to $j=m$.
In any case, things become clearer if you study the theory of *resultants* an... | 11 | https://mathoverflow.net/users/11926 | 280525 | 124,229 |
https://mathoverflow.net/questions/280473 | 3 | Let $P\_n$ be the [Legendre polynomial](https://en.wikipedia.org/wiki/Legendre_polynomials) of degree $n$. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to prove that
$$\sum\_{i=1}^{n}\sqrt i |P\_i(z)|^2 >> n^{a}$$
for... | https://mathoverflow.net/users/114210 | Bounds on Legendre polynomials on the complex plane | Setting $Q\_{n}(z)=(n+1/2)^{1/2}P\_{n}(z)$ for the orthonormalized Legendre polynomial, and
$$\kappa\_{n}(z)=\sum\_{j=0}^{n}|Q\_{j}(z)|^{2},$$
for the inverse of the Christoffel function, it is known that
$$\kappa\_{n}^{\frac{1}{2n}}(z)\to\left|z+\sqrt{z^{2}-1}\right|,$$
uniformly on $\mathbb{C}$ as $n\to\infty$, so ... | 2 | https://mathoverflow.net/users/89429 | 280534 | 124,232 |
https://mathoverflow.net/questions/280471 | 5 | Let $K$ be a number field and let $g$ be an integer. Let $\mathcal{A}(K,g)$ be the set of absolutely simple $g$-dimensional abelian varieties over $K$. Is the set $\{\mathrm{End}^0(A\_{\mathbb{\overline{Q}}}):A\in \mathcal{A}(K,g)\}$ of division algebras a finite set?
| https://mathoverflow.net/users/99726 | Endomorphism Algebras of abelian varieties over a number field | It is a folk conjecture that for each fixed positive integer $g$ and $d$, the set of isomorphism classes of rings arising as $\operatorname{End}(A) := \operatorname{End}(A\_{\overline{F}})$ for a $g$-dimensional abelian variety defined over any degree $d$ number field $F$ is finite.
(For a time when I was in grad sc... | 4 | https://mathoverflow.net/users/1149 | 280539 | 124,234 |
https://mathoverflow.net/questions/257626 | 4 | $$X \longrightarrow \fbox{$\phantom{\int}P\_{Y|X}\phantom{\int}$}\longrightarrow Y$$
The information capacity of this channel is $C=\max\_{P\_X} I(X;Y)$. Any rate $C-\varepsilon$ can be achieved by fixing a large-enough blocklength $n$ and associating each message $m\in \{1,\dots,2^{n(C-\varepsilon)}\}$ with a codewo... | https://mathoverflow.net/users/10668 | Backwards random codebook generation | "Decodable with error probability less than $\varepsilon$" is tautological to "max-a-posteriori decoder $\arg\max\_{\text{message}}P(\text{message}|\text{observation})$ fails with probability less than $\varepsilon$." By the problem's definition no decoder can have error probability less than this one.
It is useful ... | 0 | https://mathoverflow.net/users/10668 | 280541 | 124,236 |
https://mathoverflow.net/questions/280526 | 3 | This is the continuation of [part 1 of this question](https://mathoverflow.net/questions/124494/bijective-equivalent-collections-of-proper-classes-in-set-theory?rq=1), where all useful definitions and notations are given. J. D. Hamkins answered question 1 in the first part, proving that there can be no injection of the... | https://mathoverflow.net/users/30395 | More on bijective-equivalent classes in NBG set theory (2) | The answer to question 2 is yes. To see this, it suffices to produce from any well-ordering of some $V\_\alpha$ a set of ordinals, such that the well-ordering can be reconstructed from the set of ordinals. Given a well-ordering of $V\_\alpha$, this ordering has some length $\kappa$, and so there is a relation $E$ on $\... | 1 | https://mathoverflow.net/users/1946 | 280556 | 124,243 |
https://mathoverflow.net/questions/280554 | 5 | Suppose $T \in V\_1 \otimes \cdots \otimes V\_k$ is a tensor, where each $V\_i$ is a finite dimensional complex vector space. A $1$-flattening (or a flattening) is a realization of $T$ as a matrix in the space of matrices in $k$ essentially different ways as follows:
\begin{equation}
V^\*\_{i} \rightarrow V\_{1} \otime... | https://mathoverflow.net/users/98093 | Is a flattening rank a lower bound for the border rank? | Yes, the flattening rank is a lower bound for border rank.
First note that flattening rank is a lower bound for rank. If $T$ is a decomposable tensor (simple tensor, rank one tensor) then every flattening of $T$ has rank one. If $T$ has rank $r$, then $T$ is a sum of $r$ decomposable tensors, and so every flattening ... | 9 | https://mathoverflow.net/users/88133 | 280559 | 124,245 |
https://mathoverflow.net/questions/280537 | 0 | I'm looking for a reference for the theorem saying that a real-valued lower (upper) semicontinuous function on any metric space can be reached as a pointwise limit by a non-decreasing (non-increasing) sequence of continuous functions.
To provide some context, I'm interested in the special case, where the metric space... | https://mathoverflow.net/users/nan | Approximation of semicontinuous function | As your function has a particularly simple form, it might be the easiest to just explicitly write down a nice approximating sequence.
$$f\_n(x,y) = \begin{cases} g(x,y) & |x| \leq n \\ (n + 1 - |x|)g(x,y) & n \leq |x| \leq n+1 \\ 0 & |x| > |n+1|\end{cases} $$
Homogeniety of $f\_n$ in $y$ is now inherited directly f... | 1 | https://mathoverflow.net/users/15002 | 280575 | 124,250 |
https://mathoverflow.net/questions/280505 | 0 | This question follows up on [Bound on queries to a tree with unusual probabilities](https://mathoverflow.net/q/280202), where @fedja was able to disprove my conjecture under only constraints (1-4) below. I restate the relevant facts here for simplicity.
Consider a tree $\mathcal{T}(r)=(V,E)$ rooted at $r \in V$ and o... | https://mathoverflow.net/users/43307 | Bound on queries to a tree with unusual probabilties -- follow-up | If you pose the problem in the most aesthetically pleasing way, it becomes neat and clean and your conjecture holds. Of course, the reality is not obliged to be aesthetically pleasing, but the chance that it complies with aesthetics is generally higher than the chance that it complies with results of students, so the a... | 1 | https://mathoverflow.net/users/1131 | 280577 | 124,252 |
https://mathoverflow.net/questions/280589 | 2 | Let $p$ be an odd prime, $\zeta $ a primitive $p-$th root of unity and
$${a\_n}(x) = \sum\limits\_{k = 1}^{p - 1} {\prod\limits\_{j = 1}^n {\left( {1 + {\zeta ^{jk}}x} \right)} } .$$ It seems that for $0 \leqslant i < p$
$${a\_{pn + i}}(x) = {b\_i}(x){\left( {1 + {x^p}} \right)^n}$$
for some polynomial $b\_i(x)$ of ... | https://mathoverflow.net/users/5585 | Simplification of a sum with roots of unity | For fixed $k$, the product of $(1+\zeta^{jk}x)$ over $p$ consecutive values of $j$ equals $1+x^p$. Your claim follows.
| 5 | https://mathoverflow.net/users/4312 | 280591 | 124,256 |
https://mathoverflow.net/questions/280598 | 1 | While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is
\begin{equation}
\prod\_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5)
\end{equation}
where $\zeta$ is an arbitrary fifth root of unity and
\begin{equation}
\varphi(q):=\sum\_{n=-\infty}^... | https://mathoverflow.net/users/114251 | How does $\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) $ hold? | First note that $\prod\_{\xi} (1 - \xi^n q^n)$ is equal to $(1- q^n)^5$ if $5 | n$ and to $1 - q^{5n}$ otherwise. Moreover
$$
\varphi(q) = \prod\_{n \geq 1} (1 - q^n)^{e\_n}
$$
where $e\_n = 1,-2,3,$ or $-2$ if $n \equiv 0,1,2$ or $3 \pmod 4$ respectively. In particular $e\_n = e\_{5n}$ $(\*)$
. Thus
$$
\prod\_{\xi} \v... | 4 | https://mathoverflow.net/users/21724 | 280602 | 124,262 |
https://mathoverflow.net/questions/280535 | 2 | Let $\mathcal{C}$ be a simplicial category, such that for any two objects $X, Y\in\mathcal{C}$, $\text{Hom}\_{\mathcal{C}}(X,Y)$ is a simplicial commutative monoid. Is the simplicial nerve $\text{N}(\mathcal{C})$ an $(\infty, 1)$-category?
If for any two objects $X,Y\in \mathcal{C}$, $\text{Hom}\_{\mathcal{C}}(X,Y)$ ... | https://mathoverflow.net/users/nan | Criteria for being an $\infty$-category? | Since this example is kinda fun, let me spell it out. (The intuition should be clear though: the simplicial category I defined is really the result of taking a not-so-exciting (2,2)-category and applying the nerve. This 2-category has non-invertible 2-morphisms, so its nerve shouldn't magically have invertible 2-morphi... | 5 | https://mathoverflow.net/users/6936 | 280613 | 124,265 |
https://mathoverflow.net/questions/280612 | 11 | In Martin-Löf type theory with identity eliminator
$$
J :
\prod\_{B:\prod\_{x,y:A}(x=y)\to\mathcal{U}}\left(
\prod\_{x:A}B(x,x,\mathrm{refl}\_x)\to
\prod\_{x,y:A}\prod\_{p:x=y}B(x,y,p)
\right)
$$
satisfying $J(B,b,x,x,\mathrm{refl}\_x)=b(x)$ we can have terms $p:x=x$ that are not equal to $\mathrm{refl}\_x$.
We... | https://mathoverflow.net/users/114245 | The role of univalence in the homotopy interpretation of type theory | Whenever you’re looking at a logical system, there’s a tension between two main ways of studying it:
* axioms/theorems in the system show what the world it describes *must* look like;
* models show what the world it describes *can* look like.
The interpretation of types as spaces shows that the types of plain Marti... | 18 | https://mathoverflow.net/users/2273 | 280614 | 124,266 |
https://mathoverflow.net/questions/280606 | 1 | Let $\gamma:[a,b]\longrightarrow\mathbb{C}$ be a closed continuous curve in the complex plane satisfies:
$(1)\ \gamma(t)\neq 0,\ \forall t\in [a,b]$;
$(2)\ \{\frac{\gamma(t)}{|\gamma(t)|}:t\in [a,b]\}=\{z\in \mathbb{C}:|z|=1\}$.
For any given positive integer $n$, is that right that we can find $a\leq t\_1<t\_2\l... | https://mathoverflow.net/users/58096 | A question about a closed continuous curve in the complex plane | If $\gamma$ has non-zero winding number, then Nate Eldridge's comment shows that the answer is positive. However, note that for (for example) $n = 2,$ the logarithmic spiral (which is not a closed curve) is a counterexample To make it closed, take the boundary of a thin tubular neighborhood of a long piece of said spir... | 1 | https://mathoverflow.net/users/11142 | 280616 | 124,268 |
https://mathoverflow.net/questions/280562 | 1 | Let $E$ be a Banach space.
It is known that if for any equivalent norm on $E^\*$ the closed unit ball of $E^\*$ is weakly\* closed, then $E$ is reflexive (a very short proof is in the book by Fabian, Habala, Hajek, Montesinos and Zizler).
It is easier to show that if the closed unit ball of any closed subspace of $... | https://mathoverflow.net/users/53155 | Criterion of reflexivity | Question 2 has a negative answer if $E$ contains a reflexive subspace $F$ because the closed unit ball of $F$ under any equivalent norm is weakly compact and hence weak$^\*$ compact.
Question 1 has a positive answer. If $E$ is not reflexive, let $F$ be the kernel of some functional in $E^{\*\*}\sim E$.Then $F$ is nor... | 2 | https://mathoverflow.net/users/2554 | 280621 | 124,270 |
https://mathoverflow.net/questions/280595 | 7 | Let $X$ be the blow-up of $\mathbb{P}^2$ at three general points $p\_1,p\_2,p\_3$, that is a del Pezzo surface of degree six, and let $\pi\_i:X\rightarrow\mathbb{P}^1$ be the morphism induced by the projection from $p\_i$.
Does anyone know a reference for the following classical fact?
Any morphism $f:X\rightarrow\... | https://mathoverflow.net/users/nan | Pencils on del Pezzo surfaces | Let me sketch a proof. Let $D$ be the divisor class giving the map $f$. Then $D^2 = 0$ (because $D$ is the pullback of a point on $\mathbb{P}^1$). By adjunction formula $\deg(K\_D) = K\_X \cdot D$ is negative since $-K\_X$ is ample. Therefore, general fiber of $f$ is a rational curve, hence $K\_X \cdot D = -2$, hence $... | 3 | https://mathoverflow.net/users/4428 | 280622 | 124,271 |
https://mathoverflow.net/questions/280605 | 7 | Assume $f(x)\in\Bbb{R}[x]$ is a polynomial of degree $n$.
**Question.** If $\int\_{-1}^1f^2(x)\,dx=1$, is it true that
$$\vert f(x)\vert\leq \frac1{\sqrt2}(n+1), \qquad \text{for $\vert x\vert\leq1$}\,\,\,?$$
| https://mathoverflow.net/users/66131 | Bounds on polynomial values | This is problem VI.103 in volume 2 of Polya and Szego, where they also characterize the extremal polynomials.
| 10 | https://mathoverflow.net/users/11142 | 280631 | 124,272 |
https://mathoverflow.net/questions/280638 | 2 | A cardinal register machine is like an ordinal register machine but with branching based on cardinal equality rather than ordinal equality. What is the complexity of the halting problem for cardinal register machines (with finite initial values in the registers)?
Is the complexity independent of whether
- at limit... | https://mathoverflow.net/users/113213 | Cardinal Register Machines | *Unbounded Computations*
Independent of the restrictions, the halting problem is $Σ\_1(\mathrm{Card})$-complete, where $\mathrm{Card}$ is the cardinality function (and $Σ\_1$ is $Σ^V\_1$). Also, there are universal machines: For every $Σ\_1(\mathrm{Card})$-predicate $P$ on $ℕ$, there is a cardinal register machine th... | 1 | https://mathoverflow.net/users/113213 | 280639 | 124,274 |
https://mathoverflow.net/questions/280590 | 3 | Let $D$ be a division ring. I have in mind the following result.
> **Theorem**. For every automorphism $f$ of $D$, there is a division ring $E$ extending $D$ such that $f$ extends to an inner automorphism of $E$.
**Q1.** Is the Theorem correct? Any reference?
**Q2.** If $f$ fixes the centre of $D$ pointwise, ca... | https://mathoverflow.net/users/18583 | Extending an automorphism to an inner one | For Q1, you can form the skew polynomial ring $D[t;f]$ (i.e., the ring of polynomials in $t$ with coefficients from $D$, and multiplication satisfying $tr=f(r)t$ for $r\in D$). This is an Ore domain, and has a skew field of fractions $E$ consisting of formal fractions $rs^{-1}$ where $r,s\in D[t;f]$ and $s\neq0$. Then ... | 4 | https://mathoverflow.net/users/22989 | 280651 | 124,276 |
https://mathoverflow.net/questions/242854 | 2 | I am investigating functionals defined over sequences of discrete probability distributions related to dynamical/stochastic system performance. As an initial step, I am searching for references that discuss deriving the time-dependent distribution of queue length in $M\_t/M\_t/1/K$ queues.
Ideally, I'd like to find ... | https://mathoverflow.net/users/nan | Reference Request: $M_t/M_t/1/K$ queue length distributions | Go for William Massey (who had a paper on Mt/Mt/1 queue asymptotics) or Ward Whitt's page (like 400 papers to explore). The latter is working on time-varying single server recently with someone called Ni Ma.
| 1 | https://mathoverflow.net/users/101587 | 280658 | 124,279 |
https://mathoverflow.net/questions/280649 | 0 | Let us call a simple undirected graph $G$ *non-traceable* if there is no Hamiltonian path in $G$.
Are there connected non-traceable graphs $G, H$ such that the [Cartesian produc](https://en.wikipedia.org/wiki/Cartesian_product_of_graphs)t $G{\scriptsize\square} H$ has a Hamiltonian path?
| https://mathoverflow.net/users/8628 | Can the cartesian product of two non-traceable graphs be traceable? | Yes, this is possible and one of the smallest counterexamples is on 6 vertices.
Let $G$ be the graph with edges
```
[(0, 1), (0, 2), (0, 'A'), (1, 2), (1, 'B'), (2, 'C')]
```
This is essentially $K\_3$ with 3 new vertices adjacent to each of the vertices
of the $K\_3$. The cartesian square $G\square G$ has hamil... | 4 | https://mathoverflow.net/users/12481 | 280667 | 124,283 |
https://mathoverflow.net/questions/280496 | 1 | This theorem is obviously true if the set $X$ is finite (so that $\mathop{\rm conv} X$ is a convex polytope). I believe it is true for any set $X\subseteq\mathbb{R}^n$ but I cannot prove it. Can anybody please prove this or give a counter-example? Many thanks!
Notes: $\mathop{\rm conv}X$ denotes the convex hull of th... | https://mathoverflow.net/users/114221 | Let $X\subseteq\mathbb{R}^n$ and let $F$ be a face of $\mathop{\rm conv} X$. Then $F=\mathop{\rm conv}(X\cap F)$ | $\def\conv{\mathop{\mathrm{conv}}}$Recall that $\conv X$ is the set of all convex combinations of points from $X$. In a convex combination
$$
f=\sum\_{i=1}^k\alpha\_ix\_i, \qquad x\_i\in X, \quad \alpha\_i>0, \quad \sum\_{i=1}^k\alpha\_i=1,
$$
for a point $f\in F$, all the points $x\_i$ should lie in $F$ (and hence i... | 1 | https://mathoverflow.net/users/17581 | 280679 | 124,285 |
https://mathoverflow.net/questions/280678 | 17 | By not interpreting arithmetic, I mean it does not interprets enough arithmetic for Godel's argument (coding the syntax, finding the fix point) to work through. In other words, is there any other methods to prove that a theory does not have a computable consistent complete extension, or can we prove the converse that e... | https://mathoverflow.net/users/18879 | Is there a first-order theory who does not interpret arithmetic yet still does not have a computable consistent completion? | Any theory that can represent all recursive functions has no consistent decidable extension, however there are such theories that do not interpret even as weak an arithmetic as Robinson’s theory $R$, see my paper *Recursive functions and existentially closed structures* ([arXiv:1710.09864](https://arxiv.org/abs/1710.09... | 23 | https://mathoverflow.net/users/12705 | 280680 | 124,286 |
https://mathoverflow.net/questions/280624 | 2 |
>
> Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$, with the following properties:
>
>
> 1) The origin is an isolated singularity for $X$ and its linear part is the matrix $J=\begin{pmatrix}0&I\\-I&0 \end{pmatrix}$.
>
>
> 2) There is a Riemannian metric locally defined on... | https://mathoverflow.net/users/36688 | Riemannian metric adapted to singular $1$-dimensional foliation | No, this is not possible. In fact a more general result holds: If a vector field $X$ has an isolated singularity at $x\in M$ for which the linearization $X'(x):T\_xM\to T\_xM$ has no real eigenvalues, then there is no smooth Riemannian metric on an open neighborhood of $x$ for which all the integral curves of $X$ are u... | 10 | https://mathoverflow.net/users/13972 | 280681 | 124,287 |
https://mathoverflow.net/questions/241508 | 0 | Let $k$ be a field of characteristic 0 and let $X$ and $Y$ be smooth, projective and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Does $X\times\_{k}Y$ have a 0-cycle of degree 1?
| https://mathoverflow.net/users/5641 | Products of varieties of index 1 | The referee for one of my papers gave the following argument:
Let $k$ be any field and let $X$ and $Y$ be smooth, proper and geometrically integral k-schemes of finite type. Let $x$ be a $0$-cycle on $X$ and $y$ a $0$-cycle on $Y$. Then $x\times Y$ and $X \times y$ are cycles on $X \times Y$ of complementary dimensio... | 0 | https://mathoverflow.net/users/5641 | 280682 | 124,288 |
https://mathoverflow.net/questions/280668 | 2 | I am currently studying the following inequality involving the square of the modulus of a specific Dirichlet polynomial:
$$\left( \sum\_{1}^{N}\frac{1}{n} \right)^2 \ \ - \ \left| \sum\_{1}^{N}\frac{(-1)^{n-1}}{n^{1/2+it}} \right| ^2 \ > \ \ 0$$
which, for arbitrary combinations of $N$ and $t$, is in general false. ... | https://mathoverflow.net/users/15020 | critical line inequality concerning the square of the modulus of a Dirichlet polynomial | I am assuming $t$ is real. Since
$$ \left|\frac{(-1)^{n-1}}{n^{1/2+it}}+\frac{(-1)^{n+1-1}}{(n+1)^{1/2+it}} \right|
\le|1/2+it|\max\_{0\le s\le1}\left|\frac1{(n+s)^{3/2+it}}\right|
=\sqrt{t^2+1/4}\,\frac1{n^{3/2}},
$$
the series $\sum\_{1}^\infty\frac{(-1)^{n-1}}{n^{1/2+it}}$ converges, whereas the harmonic series di... | 1 | https://mathoverflow.net/users/36721 | 280691 | 124,293 |
https://mathoverflow.net/questions/280670 | -1 | Fixed $0<a<1$, define $f(x):=(1-x)^{a}$ for every $x\in [0,1]$. Recalling that the modulus of continuity of $f$ of order $\varepsilon$ is given by
$\omega(f,\varepsilon):=\sup\{|f(x)-f(y)|:|x-y|\leq \varepsilon\}$, How can I find an upper bound for this function $f(x)$?
Thanks in advance for your comments!
| https://mathoverflow.net/users/76010 | Modulus of continuity an exponential type function | The exact upper bound on the decreasing function $f$ is $\sup\_{x\in[0,1]} f(x)=f(0)=1$. The exact value of $\omega(f,\varepsilon)$ for $\varepsilon>0$ is $f(1-1\wedge\varepsilon)-f(1)=1\wedge\varepsilon^a$, because $f$ is decreasing and concave.
| 1 | https://mathoverflow.net/users/36721 | 280692 | 124,294 |
https://mathoverflow.net/questions/280630 | 8 | There are several scattered statements about fixed points and obstructions which I'd very much like to see unified in some framework.
To state them let $G$ be a group acting on a connected (1-truncated) groupoid $X$. Firstly let's choose a point in $X$ and take $X=BAut(\pi\_1(X))=:BA$. For the purpose of this questio... | https://mathoverflow.net/users/22810 | Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of group $G$ on a groupoid $X$ | I'm not sure exactly what kind of answer you're looking for, but I can try to give some context which may make things sound more reasonable. Let us think of of groupoids as $1$-truncated $\infty$-groupoids, or $1$-truncated spaces. Consider first the case where $A$ is abelian. In this case $BA$ will itself be an $\math... | 4 | https://mathoverflow.net/users/51164 | 280696 | 124,295 |
https://mathoverflow.net/questions/280707 | 3 | What is known about the consistency strength of
ZFC + the continuum is real valued measurable + Martin's maximum?
| https://mathoverflow.net/users/114298 | Joint consistency of continuum real valued measurable and Martin maximum | [Martin's maximum](https://en.wikipedia.org/wiki/Martin%27s_maximum) implies the continuum is $\aleph\_2$, and therefore it is not real-valued measurable, since real-valued measurable cardinals must be limit cardinals and indeed weakly inaccessible and weakly Mahlo and more.
So unfortunately, what is known about you... | 7 | https://mathoverflow.net/users/1946 | 280708 | 124,300 |
https://mathoverflow.net/questions/280629 | 5 | Let $\mathbb{D}\subset\mathbb{C}$ be the unit disk, and remove $n\geq 2$ of its points $P$. The resulting object will be called the punctured disk $\mathbb{D}\_n$ in the following. I am interested in its mapping class group, that is, the isotopy classes of orientation-preserving self-homeomorphisms/self-diffeomorphisms... | https://mathoverflow.net/users/78670 | The action of the mapping class group of a punctured disk on the boundary at infinity of the universal cover | 1. This is an exercise. Your second step is the part that will require some thought. For a related discussion, see Rolfsen's book where he classifies simple closed curves in the annulus.
2. They are not quasi-isometric. The fundamental group of a hyperbolic surface with punctures (and perhaps with boundary) is never is... | 2 | https://mathoverflow.net/users/1650 | 280715 | 124,303 |
https://mathoverflow.net/questions/280720 | 3 | Is there a cardinal $\kappa$ and a connected simple, undirected graph $G^\* = (\kappa, E^\*)$ such that whenever $G = (\kappa, E)$ is a connected graph, there is a [graph homomorphism](https://en.wikipedia.org/wiki/Graph_homomorphism) $f:G^\*\to G$?
**EDIT.** Sorry - I forgot the word "connected" in 2 positions of th... | https://mathoverflow.net/users/8628 | Connected weakly initial graph on $\kappa$ points | **Obsolete answer to a previous edit**: In the present statement the only possible $G^\*$ is the empty graph, for any non-empty $G^\*$ has no homomorphisms to empty $G$.
**After the edit (with additional requirement of connecitivity)**: for every $\kappa$, the initial objects $G^\*$ are exactly the connected bipartit... | 2 | https://mathoverflow.net/users/106512 | 280721 | 124,306 |
https://mathoverflow.net/questions/280727 | 8 | The following is an excerpt from Lawvere's [Some thoughts on the future of category theory](https://link.springer.com/chapter/10.1007/BFb0084208).
>
> To clarify the above considerations, generalize to distributive
> categories and seek philosophical guidance. Even though the determination
> of which maps are epi... | https://mathoverflow.net/users/69037 | In what sense is extensivity a minimal requirement on an opfibration to conform with a notion of "family"? | The only way to know for sure is to ask Lawvere. You could try mailing him.
For (1), I guess what he had in mind is the following.
A family $(X\_i)\_{i \in I}$ of sets indexed by a set $I$ is essentially the same thing as a single set $X$ together with a map $\pi: X \to I$. The idea here is that $X = \coprod\_{i ... | 7 | https://mathoverflow.net/users/586 | 280731 | 124,309 |
https://mathoverflow.net/questions/280730 | 4 | I first asked this question on math.stackexchange, but I think this question is high-level enough that is better suited here.
I'm looking for an efficient algorithm to calculate the Wasserstein-1 distance in the following setting:
Let $G = (V, E)$ be a (locally) finite, undirected graph, with associated weight func... | https://mathoverflow.net/users/114313 | Efficient algorithm for Wasserstein-1 distance in graph setting | See [this PLOS paper](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4195716/) for algorithm and extensive survey. (Gottschlich and Schumacher, 2014)
| 1 | https://mathoverflow.net/users/11142 | 280740 | 124,315 |
https://mathoverflow.net/questions/280729 | 0 | **Edit :** According to the comments of Michael Renardy and Christian Remling I revise the question as follows:
Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such that $X $ has a closed orbit and is in the form $X=f(\bar{z})$ where $f$ is a holomorphic function on $\overline{U}=\{\bar{z}\mid z\... | https://mathoverflow.net/users/36688 | Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function | Yes, $U=\{z:a<|z|<b\}$ ($a>0$) and $f(z)=i/\bar z$. Orbits are circles $\{|z|=c\}$.
| 2 | https://mathoverflow.net/users/75422 | 280744 | 124,318 |
https://mathoverflow.net/questions/280752 | 3 | Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that
\begin{equation}
UV^\top :=\arg\min\_{R \in \mathcal{O}^{r\times r}}\|AR-B\|\_\mathrm{F},
\end{equation}
where $USV^\top$ is the SVD of $A^\top B $ and $\mathcal{O}^{r\times r}$ means the set of $r\times r$ orthonormal matrices.
However, if I... | https://mathoverflow.net/users/90066 | What's the best orthonormal matrix to align two matrices in the operator norm sense? | The operator norm version of this problem is considered in: *[The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms](https://link.springer.com/article/10.1007%2FBF02521606?LI=true),* by G. A. Watson, Advances in Computational Mathematics, **2**(4), pp 393–405, 1994, which actually ... | 7 | https://mathoverflow.net/users/8430 | 280754 | 124,319 |
https://mathoverflow.net/questions/280711 | 2 | A classical theorem of Thierry Aubin states that:
>
> **Theorem ([Aubin, T. 1979](https://books.google.com/books?id=iM7uCAAAQBAJ&lpg=PP1&pg=PA344#v=onepage&q&f=false)):** If the Ricci curvature of a compact Riemannian manifold is
> non-negative and positive at a point, then the manifold carries a metric of positiv... | https://mathoverflow.net/users/90655 | Is Thierry Aubin’s theorem true on Hermitian manifolds? | **EDIT:**
The content below and at the comments rely on Kahler manifolds, and this does not intend to be a complete or even satisfactory answer:
The case when the first Chern class is positive is rather delicate. It was disproved by Yau that even provided the manifold possess positive first Chern class, then there is... | 1 | https://mathoverflow.net/users/94097 | 280758 | 124,321 |
https://mathoverflow.net/questions/280760 | 10 | Recently I'm reading Donaldson's Geometry of four manifolds. It seems to me that the book requires a lot for background. Additionally, the proof in the book is too sketchy without too much detail. I had a really hard time to digest the content in the book. Do we have other textbook demonstrating the same topic with mor... | https://mathoverflow.net/users/110479 | About Simon Donaldson's book on four dimensional manifold | Please do not ignore the other author, Peter Kronheimer. Based on all of the material I've read, I do not agree with your belief about the book. I think it is more detailed than you will find elsewhere which covers *all* of that material. Here are some useful alternatives, though:
Take that book and replace the struc... | 22 | https://mathoverflow.net/users/12310 | 280762 | 124,323 |
https://mathoverflow.net/questions/280764 | 3 | Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X\_{i}, d\_{i})\_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e\_{i}$ for each $i\in I$. We set
$$(X\_{i})\_{\mathcal{U}}:=\{(x\_{i})\_{i}\in \prod X\_{i}:\sup\_{i\in I}d\_{i}(x\_{i},e\_{i})<\infty\}/\sim, $$
wher... | https://mathoverflow.net/users/41619 | Ultraproduct of metric spaces | No, it's not an isometry in general. Let the index set be $\mathbb{N}$ and let each $X\_n = \mathbb{N}$ with its usual metric. Let $\mathcal{U}$ be any free ultrafilter on $\mathbb{N}$. For each $n$ define $f\_n(i) = 0$ for $i < n$ and $1$ for $i \geq n$. Then the ultraproduct of the $f\_n$ is not zero (the distance fr... | 3 | https://mathoverflow.net/users/23141 | 280767 | 124,324 |
https://mathoverflow.net/questions/280775 | 1 | What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of $\mathbb{C}P^2$ arising from a non linear polynomial vector field on $\mathbb{R}^2$ or $\mathbb{C}^2$?
Moreover, is it true to say t... | https://mathoverflow.net/users/36688 | The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$ | For your first question: $dy=y^2$ can be integrated by quadratures. The solutions are homographies.
For your second question: no it is not true. By the uniformization theorem, the universal covering of a leaf $L$ is either the Riemann sphere $\bar {\mathbb{C}}$, the complex line $\mathbb C$ or the unit disc $\mathbb ... | 2 | https://mathoverflow.net/users/24309 | 280780 | 124,329 |
https://mathoverflow.net/questions/280782 | 13 | I'm not sure this question is more suitable for MO or for MSE, so feel free to move it to MSE if necessary.
I work here in ZF theory. Consider the following statements:
$(C)$ Axiom of choice: for any non empty family $(E\_i)\_{i\in I}$ of non empty subsets , there exists a function $f:I\to\displaystyle \bigcup\_{i\... | https://mathoverflow.net/users/36683 | Axiom(s) of choice and bases of vector spaces | There is no fully elementary proof that you are looking for. The reason is that the axiom of regularity is needed in these proofs. Multiple Choice does not imply Choice without it, and the only proofs we know about vector spaces go through Multiple Choice. While regularity is not a "difficult axiom" it does mean the pr... | 13 | https://mathoverflow.net/users/7206 | 280784 | 124,330 |
https://mathoverflow.net/questions/280753 | 8 | I have checked everything "homology of loop spaces"-like, but was not able to find what is $H\_\*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?
| https://mathoverflow.net/users/65937 | Integer homology of double loop space of odd-dimensional sphere | There are homology isomorphisms $K(Br,1)\to \Omega^2\_0 S^2$ and $\Omega^2 S^3\to \Omega^2\_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural inclusions $Br\_n\hookrightarrow Br\_{n+1}$).
As expected there is no neat description with integral coefficients, ... | 6 | https://mathoverflow.net/users/8103 | 280788 | 124,332 |
https://mathoverflow.net/questions/280793 | 6 | Let:
* $K = k(C)$, where $C/k$ is a projective non-singular curve,
* $E/K$ - an elliptic curve,
* $\mathcal{E} \to C$ - the minimal elliptic surface associated to $E$.
Consider the "narrow Mordell-Weil subgroup", as defined in [1.]:
$$E(K)\_0 := \{ P \in E(K) : \tau\_P(\Gamma) = \Gamma \quad \text{ for every
fibr... | https://mathoverflow.net/users/101235 | Two definitions of the narrow Mordell-Weil group | Where you say "trivial action on $\Gamma\_{t0}$" near the end, you mean "carries $\Gamma\_{t0}$ into itself" (i.e., doesn't move it), *not* that the effect on $\Gamma\_{t0}$ is the identity.
The global base is a red herring: the core issue here completely concerns the object over the local ring at $t$ (recall that th... | 8 | https://mathoverflow.net/users/81332 | 280796 | 124,334 |
https://mathoverflow.net/questions/280749 | 2 | Assume that $M$ is a complex manifold.
Let $G$ be the group of all (real) smooth diffeomorphisms $\phi$ of $M$ such that $\phi^\* (X)$ is a holomorphic vector field for all holomorphic vector fields $X$ on $M$. Is $G$ a finite dimensional Lie group?(With respect to a natural smooth structure on $G$).
| https://mathoverflow.net/users/36688 | Real diffeomeorphism preserving the space of Holomorphic vector fields | Take complex vector space $V$, say of complex dimension $n$. Take a complex linear map $A \colon V \to V$ whose eigenvalues $\lambda$ all satisfy $|\lambda|>1$. The group generated by $A$ acts on $V-0$ freely and properly, with quotient a compact complex manifold $M=(V-0)/(z \sim Az)$, called a *primary Hopf manifold*.... | 2 | https://mathoverflow.net/users/13268 | 280805 | 124,338 |
https://mathoverflow.net/questions/189405 | 2 | Working in NBG set theory, with AC but without Global Choice, we ask for two proper classes A and B such that A strictly injects in B and B strictly injects in P(A); so
Question: In NBG set theory, is it possible to have two distinct proper classes A and B such that A injects in B and B injects in P(A), but P(A) does n... | https://mathoverflow.net/users/30395 | An analogue of CH for proper classes | From the answer given by Ali Enayat to my question ["Bijective equivalent collections of proper classes in set theory" on 14/04/2013](https://mathoverflow.net/questions/124494/bijective-equivalent-collections-of-proper-classes-in-set-theory), we see that there exists a model of NBG where the proper class P(On) is such ... | 1 | https://mathoverflow.net/users/30395 | 280813 | 124,342 |
https://mathoverflow.net/questions/188664 | 3 | We work in the set theory NBG (with local choice, but not global choice), because if there is global choice, every proper class is well-ordered, so that every proper class is bijective with the class On of ordinals and the class V universe, and we only have one level of bijective equivalent proper classes.
As it is k... | https://mathoverflow.net/users/30395 | Order in bijective-equivalent collections of proper classes in set-theory | The answer to question 2 is Negative. This is because, answering to my question "Injection of the proper class of ordinals into every proper class", J.D. Hamkins proved on 02/12/2014 the existence in NBG of a proper class W that does not inject into On.He also proved that On does not inject into W.
So that if A were to... | 0 | https://mathoverflow.net/users/30395 | 280814 | 124,343 |
https://mathoverflow.net/questions/280810 | 4 | I don't want to write precisely the formulation of the Calderon-Zygmund theorem for singular integrals. The details are not so important here.
So I consider the operator $T$ given by the following formula:
\begin{equation}
Tf(x) = \int\limits\_{R^n}\dfrac{Y\_{k}^m(\frac{x-y}{|x-y|})}{|x-y|^n}f(y)\,
dy, \, f\in L^2(... | https://mathoverflow.net/users/94631 | Calderon-Zygmund theorem for the kernel of spherical harmonics | Your operator $T$ is a Fourier multiplier with symbol $m = \mathcal{F}[Y^m\_k/r^n]$; that is, $\mathcal{F}[Tu]=m \mathcal{F}[u]$.
It is a relatively simple exercise to show that the norm of $T$ on $L^2$ is *equal* to $\|m\|\_\infty$, the essential supremum of $|m|$. In other words, $C(n)=1$. Therefore, the supremum n... | 4 | https://mathoverflow.net/users/108637 | 280818 | 124,344 |
https://mathoverflow.net/questions/280660 | 1 | I am working on a Markov-modulated Poisson process $\{N\_{t}, t \geq 0\}$, which is itself a Poisson but the rates of which are governed by a CTMC. In my case, the CTMC is a one-class, aperiodic and positive recurrent MC.
My questions are the following
1. Are the interarrival times of $\{N\_t, t \geq 0\}$ i.i.d? My... | https://mathoverflow.net/users/101587 | Are interarrival times of doubly-stochastic Poisson I.I.D.? | If I understand correctly, there is a hidden Markov chain, say $(S\_t)$, whose state $s = S\_t$ describes the rate $\mu\_s$ at which signals of the observed counting process $N\_t$ arrive. If this is correct, then the distribution of the waiting time for the first signal is *much* more complicated. The best way to unde... | 1 | https://mathoverflow.net/users/108637 | 280819 | 124,345 |
https://mathoverflow.net/questions/280820 | 39 | My first language is not English. How can I improve my mathematical writing. I feel like the only things I can write down are numbers and equations. Is there any good suggestion for improving writing, especially for mathematical writing (math-philosophy)?
| https://mathoverflow.net/users/114351 | How to improve writing mathematics? | I want to highlight two tools for learning: imitation and practice.
Read a lot of mathematics.
You will find that some texts are easier to follow than others.
What makes you like a text?
What texts do you like most?
When you write, try to write as your favorite author would.
If you keep on writing mathematics long en... | 33 | https://mathoverflow.net/users/55893 | 280827 | 124,348 |
https://mathoverflow.net/questions/280844 | 4 | An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$
$$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$
On a Hilbert space this is equivalent to saying that $\Re\langle Ax,x\rangle \le 0.$
In particular, if the spectrum of $A$ fulfills $\sigma(A)\subset (-... | https://mathoverflow.net/users/112877 | Dissipative operator on Banach spaces | No, and it's not true on Hilbert space either.
For example, on $\mathbb C^2$ or $\mathbb R^2$ try
$$ A = \pmatrix{0 & 0\cr 1 & 0\cr},\ x = \pmatrix{1\cr -1\cr},\ \lambda = 1$$
The spectrum is $\{0\}$, but $\|(A - \lambda) x\| = 1 < \sqrt{2} = \lambda \|x\|$.
| 6 | https://mathoverflow.net/users/13650 | 280846 | 124,353 |
https://mathoverflow.net/questions/280823 | 4 | Let $\lambda$ be Lebesgue measure on [0,1]. For any $x\_{1},\dots,x\_{k}$ in $[0,1]$, define $$A(x\_1,..,x\_k):=\{(y\_1,\dots,y\_k)\in [0,1]^k: \text{there exist intervals }I\_1,\dots,I\_k \text{ in }[0,1]$$
$$\text{ such that } x\_i,y\_i\in I\_i \text{ and } \lambda(\cup\_iI\_i)\leq\frac12\}$$
My question is: Is i... | https://mathoverflow.net/users/50061 | lower bound volume of a set | Such a constant $c$, not depending on $k$ and $x\_1,\dots,x\_k$, does not exist.
Indeed, suppose $k\ge2$. Let $x\_i:=(i-1)h$ for $i=0,\dots,k+1$, where $h:=\frac1{k-1}$ -- so that $x\_1=0$ and $x\_k=1$.
Take any $(y\_1,\dots,y\_k)\in A(x\_1,\dots,x\_k)$. Then there exist intervals $I\_1,\dots,I\_k$ in $[0,1]$
su... | 2 | https://mathoverflow.net/users/36721 | 280861 | 124,358 |
https://mathoverflow.net/questions/280863 | 2 | Let $E$ be a Banach space. Is it possible that $E$ is super-reflexive and $\ell\_p$ is crudely finitely representable in $E$ for all $p\in (1,2)$?
It seems unlikely but I cannot find an argument off the top of my head.
| https://mathoverflow.net/users/106520 | Finite-representability of $\ell_p$ in super-reflexive spaces | No. If $\ell\_p$ is finitely crudely representable in a Banach space $X$, then $\ell\_p$ is $1+\epsilon$ finitely representable in $X$ for all $\epsilon >0$ by Krivine's theorem. You can find this in the book of Milman and Schechtman.
| 4 | https://mathoverflow.net/users/2554 | 280864 | 124,359 |
https://mathoverflow.net/questions/280857 | 2 | is it true that the Laplacian $\Delta:=\frac{d^2}{dx^2}$ on $(0,1)$ with Neumann boundary conditions is dissipative on $C[0,1]?$
For this we have to show that there is for any $x \in D(\Delta)$a $x' \in C[0,1]'$ such that
$x'(x)=\left\lVert x\right\rVert^2=\left\lVert x'\right\rVert^2$ and
$$\Re \langle \Delta x,x'... | https://mathoverflow.net/users/114378 | Laplacian dissipative? | See the example of Section II.3.30 in
*Engel, Klaus-Jochen; Nagel, Rainer*, [**One-parameter semigroups for linear evolution equations**](http://dx.doi.org/10.1007/b97696), Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). [ZBL0952.47036](https://zbmath.org/?q=an:0952.47036).
| 2 | https://mathoverflow.net/users/12898 | 280869 | 124,362 |
https://mathoverflow.net/questions/280836 | 0 | Let $X$ be a smooth projective variety with an action of $\mathbb{C}^{\*}$. Let us suppose that the set $X^{\mathbb{C}^{\*}}$ is finite. For $x \in X^{\mathbb{C}^{\*}}$, let $A\_{x}$ denote the attractor (under our $\mathbb{C}^{\*}$-action) to $x$ and let $R\_{x}$ denote the repellent. Take $x\_{1}, x\_{2} \in X^{\math... | https://mathoverflow.net/users/113438 | About the connection between repellents and attractors under a $\mathbb{C}^{*}$ action on a projective variety | Counterexample: Take the action of $\mathbb C^\*$ on $\mathbf P^2(\mathbb C)$ given by
$$
t\cdot[a\_0:a\_1:a\_2]=[a\_0:ta\_1:t^2a\_2].
$$
There are three fixed points $p\_0$, $p\_1$, $p\_2$ where $p\_0$ is a sink and $p\_2$ is a source. There are two orbits having $p\_1$ in its closure, namely the lines $L\_0$, $L\_2$ ... | 3 | https://mathoverflow.net/users/89948 | 280871 | 124,364 |
https://mathoverflow.net/questions/280802 | 11 | The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a\_s,b\_s)\_{s \in S}$ one can attach a Hecke algebra $\mathcal{H}(W,S)$ (this is an exercise in Bourbaki's Groupes et algèbres de Lie IV-VI) with gen... | https://mathoverflow.net/users/3824 | Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$? | $PGL\_2(\mathbf{Z})$ is a Coxeter group with presentation $\langle s\_1,s\_2,s\_3 \mid s\_i^2=(s\_1s\_2)^3=(s\_1s\_3)^2=1 \rangle$, i.e. with labels $\{2,3,\infty\}$. To each Coxeter group $W$ with labels in $\{2,3,4,6,\infty\}$ there exists a Kac--Moody $G$ having a (twin) BN-pair with Weyl Group $W$, as constructed b... | 8 | https://mathoverflow.net/users/3380 | 280898 | 124,370 |
https://mathoverflow.net/questions/280858 | 6 | I am interested in understanding (at least roughly, if no such a description exists) the group of automorphisms for the group $UT(n,p)$, of unitriangular matrices over the field $\mathbb{F}\_p$ on $p$-elements. Unfortunately the online searching I've carried out wasn't quite helpful. If no description exists, maybe som... | https://mathoverflow.net/users/94297 | Automorphism group of $UT(n,p)$, the group of unitriangular matrices over the field $\mathbb{F}_p$ | * for size = 3
<https://groupprops.subwiki.org/wiki/Unitriangular_matrix_group:UT(3,p)#Automorphisms>
Automorphisms
The automorphisms essentially permute the subgroups of order
containing the center, while leaving the center itself unmoved.
* over $F\_2$, any size: [Maginnis J. S., (1993/11)."Outer Automorphisms of ... | 3 | https://mathoverflow.net/users/10446 | 280906 | 124,373 |
https://mathoverflow.net/questions/280913 | 6 | I have an optimization problem related, in a certain way, to the expression of a set of points with the least number of points and I don't know if it is NP-hard (or not).
More formally, I have a ground set $U$ of points with coordinates $(x\_i,y\_i)$ for $i=1,...,n$.
Given an integer $k\leq n$, the decision problem c... | https://mathoverflow.net/users/114404 | Is this problem of selecting points NP-hard? | This problem is reducible from VERTEX-COVER. A rough description: let the input graph to VERTEX-COVER be $G = (V, E)$ with $|V| = n$. Choose integer $n << t = O(n^c)$, and create a convex polygon $P$ on $1 + n + 2t$ vertices $C\_0$, $C\_1$, $\ldots$, $C\_n$, $D\_1$, $\ldots$, $D\_t$, $E\_1$, $\ldots$, $E\_t$. Choose ve... | 5 | https://mathoverflow.net/users/106512 | 280916 | 124,375 |
https://mathoverflow.net/questions/280781 | 2 | Given a finite dimensional non-Gorenstein algebra $A$, do we have $Ext^i(D(A),A) \neq 0$ for infinitely many $i$? (We can assume A is local or commutative if that helps).
All I can show is that such algebras would be pretty exotic when this holds only for finitely many i.
| https://mathoverflow.net/users/61949 | Question on Ext for finite dimensional algebras | My modules are right modules.
Let $M$ be a non-projective module for a self-injective algebra $B$ such that $\text{Ext}^i\_B(M,M)=0$ for $i>t$ (i.e., what you've called a "strange" module in some recent questions). Let
$$A=\pmatrix{k&M\\0&B}.$$
Then $A$ is not Gorenstein, as it has a one-dimensional injective module ... | 5 | https://mathoverflow.net/users/22989 | 280918 | 124,376 |
https://mathoverflow.net/questions/280872 | 3 | **Edit:** According to the comment of Prof. Bryant, I revise my question.
Is there a reasonable and non trivial (geometric) interpretation for the following quantity on a compact Riemannian manifold $(M,g)$ of dimension $3$
$$q=\sup\_{\alpha}\int\_M \alpha \wedge d\alpha $$
where $\sup$ is taken over all $1$-fo... | https://mathoverflow.net/users/36688 | Is there a geometric interpretation for this quantity? | There is no reason to believe that there is a supremum of this functional. For example, consider the $3$-torus $M = \mathbb{R}^3/\mathbb{Z}^3$
with the quotient metric and the unit $1$-forms
$$
\alpha\_n = \cos(2\pi n z)\,\mathrm{d} x - \sin(2\pi n z)\,\mathrm{d} y,
$$
where $n$ is an integer, which are well-defined on... | 12 | https://mathoverflow.net/users/13972 | 280923 | 124,377 |
https://mathoverflow.net/questions/280911 | 1 | We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by
$$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$
where $x\in\mathbb{R}^{m}\_{+}$ with $\sum\_{i=1}^{m}x\_{i}=1$, and $\tau\in\mathbb{R}$ are fixed .
**Question:**
Calcu... | https://mathoverflow.net/users/109184 | Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function | The function $\ell$ is concave (it's a minimum of linear functions). Its conjugate $\ell^\*(\theta)$ is the supremum over a linear function minus $\ell$. If I see correctly, $\ell^\*(\theta) = \infty$ for any $\theta$ and thus, $k$ is the supremum of the empty set, usually set to be $-\infty$.
| 0 | https://mathoverflow.net/users/9652 | 280939 | 124,385 |
https://mathoverflow.net/questions/280944 | 4 | Given constants $m\_0,\ldots m\_n$ and a measure $\mu$ on $\mathbb{R}$, how can I "recover" the integrals $\int f x^n d\mu$ of the maximum entropy distribution $f\in L^2(\mathbb{R})$ which satisfies $\int f^2 x^n d\mu=m\_n$ ?
| https://mathoverflow.net/users/404 | minimum information distribution given moments of its square | Let us consider the special case when $\mu$ is the Lebesgue measure on $\mathbb{R}$. The derivation then should be extendable to the general case without much effort.
We need to find the distribution $f$ that solves the following problem: $$\max\_{f\in \mathcal{B}}-\int\_\mathbb{R}f(x)\ln f(x) dx\\\Leftrightarrow\min... | 4 | https://mathoverflow.net/users/64194 | 280949 | 124,387 |
https://mathoverflow.net/questions/280825 | 7 | In my research I have stumbled across the following 1st order complex differential equation for smooth functions $\eta:\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{C}-\lbrace0\rbrace$ defined on the circle,
$$i\frac{\partial\eta}{\partial t}+(re^{it}+\varepsilon i)\bar\eta=0$$
where $\varepsilon\in\mathbb{R}\_+$ is sufficiently... | https://mathoverflow.net/users/12310 | Solution to at least one ODE in a family of ODE's | As I said, let's get rid of some junk like $i$ and conjugation first. Note that in the original equation we can replace $\eta$ by $\zeta\eta$ for any complex number $\zeta$ with $|\zeta|=1$ and can replace $t$ by $t+\tau$. These two degrees of freedom allow us to rotate the coefficients at $\varepsilon$ and $r$ indepen... | 4 | https://mathoverflow.net/users/1131 | 280953 | 124,388 |
https://mathoverflow.net/questions/280956 | 6 | Consider the action of the orthogonal group $\operatorname{O}(d)$ on $k$-way tensors $(\mathbb{R}^d)^{\otimes k}$ defined by
$$Q(x\_1\otimes\cdots\otimes x\_k)=Qx\_1\otimes\cdots \otimes Qx\_k$$
and extending linearly. I would like to locate a proof that the invariant subspace of totally symmetric tensors $\operato... | https://mathoverflow.net/users/29873 | Does the orthogonal group act irreducibly on totally symmetric tensors? | It is classical that, as $O(n)$-representation,
$$
\text{Sym}^k(\mathbf R^n)=H^k\oplus qH^{k-2}\oplus q^2H^{k-4}\oplus\ldots
$$
Here $q=x\_1^2+\ldots+x\_n^2$ is the quadratic form defining $O(n)$ and $H^k\subseteq \text{Sym}^k(\mathbf R^n)$ is the space of harmonic polynomials, i.e., polynomials which are killed by the... | 12 | https://mathoverflow.net/users/89948 | 280962 | 124,393 |
https://mathoverflow.net/questions/280964 | 0 | Let $\omega$ be endowed with the discrete topology, and let $\mathbb{R}$ carry the Euclidean topology. Is there a continuous surjective map $f:\omega^\omega\to \mathbb{R}$?
(I suppose this would entail that there is a contiuous surjective map $f:\omega^\omega \to \mathbb{R}^\omega.$)
| https://mathoverflow.net/users/8628 | Is there a continuous surjection $\omega^\omega\to \mathbb{R}$? | Yes - consider the map sending a sequence of naturals to the corresponding continued fraction representation. (OK fine this hits $[0,\infty)$, but that's easy to fix.)
More is true: any [Polish space](https://en.wikipedia.org/wiki/Polish_space) is the continuous image of $\omega^\omega$. The converse fails, though, e... | 10 | https://mathoverflow.net/users/8133 | 280965 | 124,394 |
https://mathoverflow.net/questions/280833 | 1 | Suppose we have a real-valued smooth function on a complex torus:
$$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$
i.e., this $f$ is a real-valued smooth function on $\mathbb{C}^n$ which is invariant under the tranfsormation of the lattice $(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n$. If the com... | https://mathoverflow.net/users/36974 | a question about complex Hessians on complex tori | Let me copy my remark to close the topic.
If $\partial \bar \partial f$ is a constant matrix on $\mathbb C^n$, then it follows that for any complex line $L\subset \mathbb C^n$, $\Delta (f|\_L)$ is constant, where $\Delta$ is the Laplacian on $L\simeq \mathbb C$ with respect to the restriction of the euclidean metric... | 2 | https://mathoverflow.net/users/5659 | 280966 | 124,395 |
https://mathoverflow.net/questions/280852 | 7 | First I apologize because this is not a research question, but I can't get any answer on MathStackExchange...
Let $\pi \colon E \to B$ and $\pi' \colon E' \to B$ two topological fiber bundles on the same base $B$. A $B$-morphism $f \colon E \to E'$ is an isomorphism of fiber bundles iff for all $b \in B$ the induced ... | https://mathoverflow.net/users/74372 | Are topological fiber bundles on the same base with homeomorphic fibers isomorphic? | If $F$ is locally connected, locally compact and Hausdorff (or alternatively compact Hausdorff), then the inverse function from $Homeo(F, F)$ to itself is continuous. Moreover, such a space $F$ is [exponentiable](https://ncatlab.org/nlab/show/exponential+law+for+spaces). This means, in particular, that a map $B\to Home... | 9 | https://mathoverflow.net/users/6668 | 280968 | 124,397 |
https://mathoverflow.net/questions/280972 | 10 | One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$).
Is there a name for these matrices?
| https://mathoverflow.net/users/1898 | Is there a standard name for (non-square) matrices with orthonormal columns? | ***Orthonormal $\boldsymbol n$-frames*** : <https://en.wikipedia.org/wiki/Stiefel_manifold>.
Added: This terminology of Hirzebruch ([1966](//ams.org/mathscinet-getitem?mr=202713)), Steenrod ([1951](//ams.org/mathscinet-getitem?mr=39258)) translates the ***$\boldsymbol n$-Systeme*** of Stiefel ([1936](//zbmath.org/?q... | 11 | https://mathoverflow.net/users/19276 | 280978 | 124,398 |
https://mathoverflow.net/questions/280447 | 2 | **Disclaimer.** This is a cross-post from [math.SE](https://math.stackexchange.com/questions/2416819/matrix-of-absolute-values-and-largest-eigenvector) where I asked a variant of this question two days ago which has been positively received but not has not received any answers.
---
Let $A$ be a complex Hermitian... | https://mathoverflow.net/users/89934 | Entrywise modulus matrix and the largest eigenvector | It supprised me but your conjecture is true.
First we can suppose that the matrix $B$ is irrecductible. Otherwise we can just keep the $B$-stable subspace which contain the larger eigenvalue.
For $\|A\|\rightarrow \|B\|$, as in your remark we have $\||y|-x\|\rightarrow 0$. With $y\_i=e^{i\phi\_i}|y\_i|$ And we rep... | 1 | https://mathoverflow.net/users/99045 | 280984 | 124,401 |
https://mathoverflow.net/questions/275312 | 2 | I have a somewhat unconventional view of the Prime Number Theorem as a "quantification" of the infinitude of primes. Here I recall the argument of [Furstenberg](https://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes). Define a topology $\mathcal{X}$ on $\mathbb{Z}$.
* $\varnothing \in \mathc... | https://mathoverflow.net/users/1358 | Could Furstenberg's Argument Prove the Infinitude of Primes in Number Fields? | Yes. Just replace arithmetic progressions by translations of a nonzero (integral) ideal. The same argument shows that the union on the RHS is closed, so the set of units $K^\times$ has to be open. Since $1$ is a unit, $K^\times$ must contain $1+\frak a$ for some nonzero (integral) ideal $\frak a$. Take a nonzero elemen... | 4 | https://mathoverflow.net/users/37103 | 280990 | 124,403 |
https://mathoverflow.net/questions/280985 | 14 | The following is a standard combinatorics question:
>
> Any set of $n+1$ numbers from $1, \dotsc, 2n$ contains a pair of
> numbers $a, b$ where $a \left| b \right.$
>
>
>
The argument is by pigeonhole principle: consider $A\_i = \{2^k (2i-1), k\in \mathbb{N}\}.$ The sets $A\_i$ cover $1, \dotsc, 2n,$ and ther... | https://mathoverflow.net/users/11142 | divisibility independence | First, I ran some bruteforce myself (I don't have access to Mathematica at the moment), and I'm fairly sure the mysterious numbers obtained in OP ($1, 2, 2, 2, 4\ldots$) are the *minimal possible numbers that are present in any $S\_m(2n)$*. I'm going to explain the pattern in this assumption.
Long story short, the mi... | 13 | https://mathoverflow.net/users/106512 | 280994 | 124,404 |
https://mathoverflow.net/questions/280849 | 4 | It is a [non-trivial result](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf) that there is a countable connected Hausdorff space.
Let ${\cal T}$ be a set of connected Hausdorff topologies on $\omega$ such that whenever $\tau\_1\neq\tau\_2\in {\cal T}$ we have $(\o... | https://mathoverflow.net/users/8628 | Cardinality of a set of countable connected Hausdorff spaces | It seems that the number of such topologies is $2^{\mathfrak c}$. Such (huge) number of connected Hausdorff topologies can be constructed by a suitable modification of the [Bing's construction](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf) of a connected Hausdorff... | 7 | https://mathoverflow.net/users/61536 | 280997 | 124,406 |
https://mathoverflow.net/questions/280993 | 4 | I have heard the "slogan" that *graded quivers are (derived) equivalent to ordinary quivers* (with this "result" being attributed to Keller) and am looking for a precise statement and a reference.
By a "graded quiver" I would understand the same as an ordinary quiver, except that arrows come with a grading, making th... | https://mathoverflow.net/users/11084 | Graded quivers vs "ordinary" quivers and derived categories | I have not heard the slogan and perhaps do not understand the context, but it seems to me that this has nothing to do with the derived categories. For any graded quiver (with or without relations) there exists an ordinary quiver (respectively, with or without relations) such that the abelian category of representations... | 6 | https://mathoverflow.net/users/2106 | 280999 | 124,407 |
https://mathoverflow.net/questions/280980 | 11 | It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective.
For me, an infinite loop space is a space $Y\_0$ together with a collection of pointed spaces $Y\_1, Y\_2, \dots$ and homotopy equivalences (or home... | https://mathoverflow.net/users/16702 | Connective spectra and infinite loop spaces | Ok, this discussion has grown beyond the level of comments so I'll collect the facts here. A bit of terminology: a $(-1)$-connected space is a space with a choice of basepoint and the category of $(-1)$-connected spaces $Top\_{>-1}$ is the category of pointed homotopy types with basepoint-preserving maps. The category ... | 11 | https://mathoverflow.net/users/10605 | 281001 | 124,408 |
https://mathoverflow.net/questions/281002 | 11 | I have a student who has taken some linear algebra but no abstract algebra, and he wants to learn some interesting math.
I explained how, by using companion matrices, one can represent and work with roots of polynomials via matrices. The resulting expanded number system is a finite dimensional vector space over the b... | https://mathoverflow.net/users/70019 | Number fields without abstract algebra | I guess you are looking for the appendix by Olga Taussky in Harvey Cohn's "A Classical Invitation to Algebraic Numbers and Class Fields". This being said, there is a reason why algebraic number theory is called "algebraic".
| 11 | https://mathoverflow.net/users/3503 | 281003 | 124,409 |
https://mathoverflow.net/questions/270960 | 6 | The Schmidt decomposition theorem says:
>
> If $H\_1,H\_2$ are Hilbert-spaces (for simplicity: of same dimension) and
> $x\in H\_1\otimes H\_2$, then there exist orthonormal bases
> $\alpha\_i,\beta\_i$ of $H\_1,H\_2$, and reals $\lambda\_i\geq0$ such that
> $x=\sum\_i \lambda\_i\alpha\_i\otimes\beta\_i$.
>
>
... | https://mathoverflow.net/users/101775 | Schmidt decomposition on infinite-dimensional Hilbert spaces | Yes. Christian Remling's comment in essence already gave an affirmative answer, and this question is a bit old, but I think it's worthwhile to write a proof that does not require reducing to the separable situation.
I encountered the same question myself and have not found any reference---but with the polar decomposi... | 6 | https://mathoverflow.net/users/114419 | 281012 | 124,413 |
https://mathoverflow.net/questions/281006 | 4 | Let E be a vector bundle of rank 2 over a variety X. Is there a counterexample so that $E$ is not isomorphic to $E^\*\otimes det$ $E$?
| https://mathoverflow.net/users/nan | counterexample related to vector bundles | As Jason Starr wrote in his comment, every rank $2$ vector bundle $E$ is isomorphic to $E^\* \otimes \det E$. Let me give a proof of this fact.
Assume that $E$ is defined by the transition functions $$g \_{\alpha\beta} \colon U\_{\alpha} \cap U\_{\beta} \longrightarrow \mathrm{GL}(2, k), \quad g\_{\alpha \beta} = \b... | 11 | https://mathoverflow.net/users/7460 | 281015 | 124,414 |
https://mathoverflow.net/questions/281026 | 0 | The following fact is known:
>
> If there is a measurable cardinal, then there are only countably many constructible reals.
>
>
>
It is also known that if $ZFC$ + "There is a (two-valued) mesurable cardinal" is consistent, then $ZFC$ + "There exists a (two-valued) measurable cardinal" + $CH$ is also consistent... | https://mathoverflow.net/users/20597 | Real-valued measurability vs. Two-valued measurability in determining whether $CH$ holds or not | First of all, it is **not** the case that if ZFC is consistent, then ZFC + CH + "There is a measurable" is consistent; a measurable cardinal has *much* greater consistency strength than ZFC alone. What **is** true is that if ZFC + "There is a measurable" is consistent, then so is ZFC + CH + "There is a measurable;" I t... | 4 | https://mathoverflow.net/users/8133 | 281028 | 124,418 |
https://mathoverflow.net/questions/281039 | 2 | For any set $X$, let $[X]^2 = \big\{\{a,b\}: a, b\in X \land a\neq b\big\}$. Let $\kappa$ be an infinite cardinal. Is there a set ${\cal E} \subseteq {\cal P}([\kappa]^2)$ such that
1. for all $E \in {\cal E}$ the simple undirected graph $(\kappa,E)$ is connected,
2. if $E\_1\neq E\_2\in {\cal E}$ then the graphs $(\... | https://mathoverflow.net/users/8628 | Mutually non-isomorphic connected graphs on $\kappa$ points | Under the axiom of choice, the answer is yes. Interpret $\kappa$ as its initial ordinal. We take the vertex to consist of distinct vertices $A\_\alpha,B\_\alpha,C\_\alpha,D\_\alpha$ for each $\alpha<\kappa$. In each of the graphs, $B\_\alpha$ is connected to $A\_\alpha$ and $C\_\alpha$ and possibly to $D\_\alpha$, so t... | 4 | https://mathoverflow.net/users/30186 | 281040 | 124,421 |
https://mathoverflow.net/questions/281025 | 1 | Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \in G \} $ is $$\chi(G) := -\sum\_{H \in \mathcal{L}(G)} \mu(H,G)|G:H|.$$ Gaschütz showed that $\chi(G)$ is nonzero for $G$ ... | https://mathoverflow.net/users/34538 | Is an Eulerian subgroup lattice boolean? | **Yes**, and a much more general statement is true.
First, note that if $\mathcal{L}(G)$ is an [Eulerian lattice](https://en.wikipedia.org/wiki/Eulerian_poset) then $\mu(1,G)=\pm 1$.
*Theorem*: $\mu(1,G)=\pm 1$ iff $G$ is cyclic of square-free order iff $\mathcal{L}(G)$ is boolean.
*Proof*: Théorème 3.1. of the ... | 1 | https://mathoverflow.net/users/34538 | 281048 | 124,425 |
https://mathoverflow.net/questions/281049 | 4 | A subgroups $K<G$ is *almost-malnormal* if $\forall g \in G \setminus K$, $K \cap gKg^{-1}$ is finite.
**Question:** Are there infinite groups in which all proper subgroups are almost malnormal?
Any such group $G$ must be fairly closed to being simple (since any normal subgroup is finite). Groups were all normal su... | https://mathoverflow.net/users/18974 | infinite groups in which all proper subgroups are almost-malnormal | Clearly if $G$ is an infinite group such that every proper subgroup of $G$ is finite then every subgroup of $G$ is almost malnormal.
Tarski monster groups are examples of such groups: Tarski monster groups are infinite groups where every proper, non-trivial subgroup is finite cyclic of order $p$ for some fixed prime ... | 4 | https://mathoverflow.net/users/6503 | 281062 | 124,429 |
https://mathoverflow.net/questions/281050 | 4 | Recall that the space $A$ is homotopy dominated by $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ such that $gf\simeq id\_A$.
Suppose that $X$ is a wedge of some spheres and $A$ homotopy dominated by $X$. Let $K$ be a bouquet of circles so that map $\phi :K\longrightarrow A$ induces an isomo... | https://mathoverflow.net/users/114476 | A space homotopy dominated by a wedge of spheres | The case when $A$ is itself equivalent to a wedge of spheres is easy. In the general case, $\pi\_2(\phi)$ is a direct summand of $\pi\_2(\phi')$, where $\phi'\colon K \to X$ is the map $f\phi$. The result follows because projective modules over the group ring of a free group are free modules, by a [theorem of Hyman Bas... | 3 | https://mathoverflow.net/users/6668 | 281063 | 124,430 |
https://mathoverflow.net/questions/281014 | 3 | Let $\lambda$ be Lebesgue measure on $[0,1]$. For $\mathbf{x}=(x\_1,x\_2,..,x\_k)\in[0,1]^k$, define $$A(\mathbf{x}):=\{(y\_1,\dots,y\_k)\in [0,1]^k: \text{there exist intervals }I\_1,\dots,I\_k \text{ in }[0,1]$$
$$\text{ such that } x\_i,y\_i\in I\_i \text{ and } \lambda(\cup\_iI\_i)\leq\frac12\}$$
My question: I... | https://mathoverflow.net/users/50061 | packing with special sets in high dimensional Euclidean space | **[EDIT]** This is an **affirmative** answer to the original question, with $C=2$. The negative answer to the version where $1/2$ is replaced by a smaller constant is kept below.
For every $\mathbf x\in[0,1]^k$, we define $\mathbf a(\mathbf x)\in\{0,1\}^k$ by $a(x)\_i=0\iff x\_i\leq 1/2$. Now, assume that $A(\mathbf ... | 1 | https://mathoverflow.net/users/17581 | 281066 | 124,431 |
https://mathoverflow.net/questions/281024 | 11 | Beyond measure zero we have yet another measure-y notion of smallness: *strong measure zero.* A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}\_{>0}$, there is a sequence $U\_i$ of open sets with
* the diameter of $U\_i$ is $<f(i)$, and
* $S\subseteq\bigcup\_{i\in\mat... | https://mathoverflow.net/users/8133 | The "strong" measure number | Chapter 8 of "Set Theory On the structure of the real line" by Bartoszynski and Judah compiles a lot of combinatorial results about strong measure zero sets. Theorem 8.1.14 gives an old result of Arnie Miller. I'll use your notation $\mathfrak{s}\_-$ when I state it:
Theorem (Miller)
For every cardinal $\kappa$, the ... | 8 | https://mathoverflow.net/users/18128 | 281067 | 124,432 |
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