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https://mathoverflow.net/questions/285250 | 5 | Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$.
Do there exist two upper triangular matrices $A\in SL(n)$ and $B\in SL(m)$ such that $A\cdot M \cdot B^{T} = \lambda N$ for $\lambda\in K\setminus\{0\}$ ?
| https://mathoverflow.net/users/nan | Action of upper triangular matrices | I think the answer to your question is negative. Consider for instance
$$M = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}, \quad N = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$$
Assume that there exist
$$A = \begin{pmatrix}
a\_{11} & a\_{12} \\
0 & a\_{22}
\end{pmatrix}, \quad B = \begin{pmatrix}
b\_{11} & b\_{12... | 7 | https://mathoverflow.net/users/14514 | 285262 | 126,035 |
https://mathoverflow.net/questions/285266 | 1 | Primes $p$ are integers with no factors (composite allowed) in $[1,p]$. There is a polynomial time test for them.
Given an interval $[a,b]$ what is the best way to test given integer $q$ has no composite factor allowed in $[a,b]$? Can any non-trivial primality test be specialized?
| https://mathoverflow.net/users/10035 | Specializing non-trivial primality tests | Checking whether an integer $n$ has a divisor in a given interval is essentially equivalent to factorization. Factoring is essentially equivalent to finding the smallest prime factor. So suppose that $n$ is not prime. Then check whether $n$ has a prime factor in $[1, n^{1/4}]$. Depending on the outcome check whether $n... | 5 | https://mathoverflow.net/users/37555 | 285267 | 126,037 |
https://mathoverflow.net/questions/285272 | 4 | Vopěnka's principle is commonly used (or at least it was for me) as an intuitionistic approach to large cardinal axioms; that is, there is much intuition to it. This intuition is that for any proper class, there are two distinct elements which are very similar to each other.
At this point, I did not yet understand e... | https://mathoverflow.net/users/115951 | Weaker forms of Vopěnka's principle (using Indiscernables and other forms of Elementarity): How weak are they? | Statement 1 is very nice, it says that in any proper class, there are two distinct objects $X$ and $Y$ that are indiscernible with respect to assertions in the ambiant theory (I assume the language of set theory); they exhibit exactly the same pattern of truths: $\varphi(X)$ if and only if $\varphi(Y)$ for every assert... | 7 | https://mathoverflow.net/users/1946 | 285275 | 126,041 |
https://mathoverflow.net/questions/285115 | 7 | What important results hold for non-metric continua, or where can I find a survey of such results?
There are three definitions of a continuum around: a non-empty topological space that is
(1) connected compact metric, or
(2) connected compact Hausdorff [e.g., *General Topology* by Willard], or
(3) connected com... | https://mathoverflow.net/users/49372 | Non-metric topological continua | A good starting point for studying countinua in the sense of the definition (2), i.e., compact Hausdorff spaces, is the survey paper "Continuum Theory (General)" by Ed Tymchatyn in [Encyclopedia of General Topology](http://www.sciencedirect.com/science/book/9780444503558).
This paper contains further references.
As a... | 3 | https://mathoverflow.net/users/61536 | 285289 | 126,045 |
https://mathoverflow.net/questions/285288 | 5 | Let $G=(V,E)$ be a simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v\in V$ the edges adjacent to $v$ do not all have the same color. (More formally, we want the restriction $c|\_{E(v)}$ to be n... | https://mathoverflow.net/users/8628 | Are all infinite graphs $3$-weak-edge colorable? | If $E$ is countable, then we can define the function $c$ by induction as follows.
Choose an identification $E = \{1,2,\ldots\}$ and choose $c(1) \in \{1,2,3\}$ arbitrarily. Assume that $c(k)$ is defined for every $k < n$. Let $v$ and $v'$ denote the ends of $n$ and let $E(v)$ (resp. $E(v')$) be the edges containing $v... | 7 | https://mathoverflow.net/users/14037 | 285298 | 126,048 |
https://mathoverflow.net/questions/285058 | 4 | I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H\_\*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the integral homology, say $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. How does the Bockstein Spectral Sequenc... | https://mathoverflow.net/users/83274 | Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology? | Each element in the image of $d\_n$ corresponds to an element of order $p^n$ in the homology with integer coefficients. So it suffices to compute the image
of $d\_1=\beta$ on $H\_\*(X;\mathbb{Z}/p)$ to determine the number of summands isomorphic
to $Z/p$ in $H\_\*(X;\mathbb{Z})$.
| 2 | https://mathoverflow.net/users/43326 | 285303 | 126,050 |
https://mathoverflow.net/questions/285304 | 40 | I've always wondered how some "mathematical coincidences" are published, or spread to a wider audience. For instance, the almost integer:
$$
e^\pi -\pi = 19.9990999\ldots
$$
was "*noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe*" ([Weisstein](http://mathworld.wolfram.com/A... | https://mathoverflow.net/users/103722 | Publishing mathematical coincidences | You answered your question yourself: it is published on the web site that you refer to. The author of the web site cites his sources, in most cases these are personal communications. Many results of this sort are spread by correspondence, on Internet, and by oral personal communication. There is also a journal "[Experi... | 28 | https://mathoverflow.net/users/25510 | 285307 | 126,051 |
https://mathoverflow.net/questions/285313 | 7 | Assume we have a set of equations in $x \in \mathbb{R}^n$
$$|a\_i\cdot x|=b\_i$$
where $a\_i \in \mathbb{R}^n$ and $b\_i>0$ are given.
Could such a system be solved efficiently?
1. In a theoretical machine storing reals with perfect accuracy.
2. An approximate solution taking rounding errors into account.
This ... | https://mathoverflow.net/users/6619 | Linear equations with absolute values | The problem is NP-hard over any field not of characteristic 2. We show this by reduction from an NP-complete NAE-3-SAT problem of checking satisfiability of a boolean formula $\land\_{i} \mathrm{NAE}(z\_{i, 1}, z\_{i, 2}, z\_{i, 3})$, where $z\_{i, j}$ are variables $x\_{\cdot}$ or their negations, and $\mathrm{NAE}(\l... | 9 | https://mathoverflow.net/users/106512 | 285344 | 126,066 |
https://mathoverflow.net/questions/285339 | 4 | Let $X$ be a Spherical variety for a reductive group $G$ with a Borel subgroup $B$. A boundary divisor of $X$ is a $G$-invariant divisor and a color of $X$ is a $B$-invariant divisor which is no $G$-invariant. We can also assume that $X$ smooth and the divisor given by the union of the boundary divisors is simple norma... | https://mathoverflow.net/users/nan | Number of boundary divisors and colors of a Spherical variety | There are relations coming from computing Picard groups in different ways. For simplicity let $G$ be semisimple. Let $x\in X$ be in the open $B$-orbit. Then $Bx=B/B\_x$ and $Gx=G/G\_x$ are open in $X$.
We compute $Piс(Gx)$ in $3$ different ways:
a) Since $G$ is semisimple, we have $Pic(Gx)=Pic^G(Gx)=\Xi(G\_x)$. Her... | 7 | https://mathoverflow.net/users/89948 | 285345 | 126,067 |
https://mathoverflow.net/questions/285343 | 6 | Suppose that $\pi: X \to S$ is a proper morphism of schemes, and $G \to X$ is a flat affine algebraic group scheme over $X$. Is the push-forward $\pi\_\* G$ also affine?
I'd be happy to assume more -- say that $\pi$ is smooth, and that $G$ is smooth over $X$, or even a torus...
| https://mathoverflow.net/users/17857 | push-forward of linear algebraic group schemes | Sorry about the mistake in the comment. First of all, for a proper, locally finitely presented morphism of schemes, $\pi:X\to S,$ with $S$ excellent (e.g., a finite type scheme over a field or over $\text{Spec}\ \mathbb{Z}$), for a finitely presented, flat, affine group scheme $\rho:G\to X$, the set-valued functor $\pi... | 9 | https://mathoverflow.net/users/13265 | 285347 | 126,068 |
https://mathoverflow.net/questions/285241 | 9 | Given a smooth manifold $M$, one can consider the Lie algebra $\mathcal{X}(M)$ of vector fields equipped with the standard Lie bracket. This is a standard machinery of differential geometry. Gelfand and Fuchs defined the Lie algebra of formal vector fields at $0 \in \mathbb{R}^n$ as linear combinations
$$\sum\_{j=1}^np... | https://mathoverflow.net/users/24078 | Formal vector fields vs. (standard) vector fields | The Lie algebra of all formal vector fields of a real or complex manifold germ $(M,p)$ (where $p\in M$ is any point), does arise as the Lie algebra of the Lie group of all formal power series automorphisms of $(M,p)$, which is the same as the group of all invertible infinite jets of local diffeomorphisms of $(M,p)$, sm... | 3 | https://mathoverflow.net/users/66983 | 285348 | 126,069 |
https://mathoverflow.net/questions/285323 | 2 | Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an isomorphism on $U$, i.e. $X'\times\_{X}U = U$, and write in this case shorthand $X'\to^UX$. Say a modification $X'\to^UX$ is a... | https://mathoverflow.net/users/7108 | Locally toric resolutions of compactifications | I am just posting my second comment as an answer. I now understand that the equivalence relation of "toric-equivalence" is the smallest equivalence relation generated by the "toric type" relation, rather than the relation where both $X$ and $X'$ are dominated by modifications that are both "toric type."
With the def... | 1 | https://mathoverflow.net/users/13265 | 285351 | 126,071 |
https://mathoverflow.net/questions/284603 | 4 | Let $S$ be a [numerical semigroup](https://en.wikipedia.org/wiki/Numerical_semigroup) and $k[S]$ is the associated semigroup ring. I would like to compute canonical module $\omega$ of $k[S].$
***I want to show that $\omega=k[t^{-n}:n\in\mathbb Z\setminus S]$.***
I have shown that $H^1\_{m}(k[S])=k[t^{n}:n\in\mathbb... | https://mathoverflow.net/users/9485 | Canonical module of a semigroup ring | I'm going to assume you mean $S$ to be graded with the obvious $\Bbb Z$-grading (i.e. $\operatorname{deg}(t)=1$), and that you're asking for the graded canonical module. Otherwise, I don't know what you mean by canonical module.
By graded local duality,
$$ \omega\_S \cong H^1\_\mathfrak{m}(S)^\vee, $$
where $(-)^\vee... | 1 | https://mathoverflow.net/users/36720 | 285352 | 126,072 |
https://mathoverflow.net/questions/285328 | 15 | $$\pi = 3\prod\_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$
| https://mathoverflow.net/users/116870 | Is this equivalent to RH - Riemann hypothesis? | Yes, this is equivalent to RH (but not in any significant way). Recall the completed Riemann $\xi$-function
$$
\xi(s) = s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s),
$$
which, by Hadamard's factorization formula can be written as
$$
e^{A+Bs} \prod\_{\rho}\Big(1-\frac{s}{\rho}\Big) e^{s/\rho},
$$
where the product is o... | 36 | https://mathoverflow.net/users/38624 | 285359 | 126,073 |
https://mathoverflow.net/questions/285330 | 5 | I am reading [this paper](http://www.win.tue.nl/~hochsten/pdf/logcap.pdf) about "Numerical approximation of the logarithmic capacity of domains", and there (on the third page) I found simple formulas for logarithmic capacity of simple figures like squares and equilateral triangles.
For a better understanding, I tried... | https://mathoverflow.net/users/54507 | Calculation of logarithmic capacity? | In two dimensions, you have a powerful tool, the Riemann mapping. If you have a compact set in the plane whose complement is connected, knowing explicitly
the map of the complement onto the exterior of the unit disk gives you the log capacity. Now take a book on complex variables with many examples of conformal maps, a... | 9 | https://mathoverflow.net/users/25510 | 285364 | 126,076 |
https://mathoverflow.net/questions/284515 | 8 | Let $V$ be a normed real vector space and let $K\_1, K\_2\subseteq V$ be closed convex subsets such that the intersection $K\_1\cap K\_2$ is non-empty. Assume that $F\_1$ is a face of $K\_1$ and $F\_2$ is a face of $K\_2$ (face $F$ of a convex set $K$ is a convex subset such that $a=tb+(1-t)c$, for $a\in F$, $b, c\in K... | https://mathoverflow.net/users/64556 | Faces of the intersection of convex sets | For finite-dimensional $V$, your definition of a face is equivalent to the definition of a *poonem*, according to part (i) of Exercise 7 on page 21 of the book [Convex Polytopes by B. Gruenbaum](https://archive.org/download/springer_10.1007-978-1-4613-0019-9/10.1007-978-1-4613-0019-9.pdf); then the positive answer to y... | 3 | https://mathoverflow.net/users/36721 | 285371 | 126,079 |
https://mathoverflow.net/questions/285294 | 3 | **Question:** How to find **the smallest** value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?
Using the Tonelli-Shanks algorithm and the Chinese remainder theorem does not always give me the smallest $x$ satisfying condition.
Is there any solution for calculating the ... | https://mathoverflow.net/users/116244 | Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)? | This is an NP-hard problem. That is, Manders and Adleman [1] proved that given $a$, $b$, and $c$, it is NP-complete to determine if there exists $x\le b$ such that $x^2\equiv a\pmod c$, and that it remains NP-complete even if the prime factorization of $c$ is also given as input.
**Reference:**
[1] Kenneth L. Mande... | 5 | https://mathoverflow.net/users/12705 | 285385 | 126,083 |
https://mathoverflow.net/questions/285226 | 1 | Let $\mathcal{R}$ be the Robba ring and $\mathcal{E}^{\dagger}$ the elements of $\mathcal{R}$ that are bounded at 0 (so the coefficients of the powerseries are bounded. Is it true that $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^\*$ with $x\cdot y= z$ implies $x,y\in (\mathcal{E}^\dagger)^\*$?
If not, is it t... | https://mathoverflow.net/users/104544 | Does $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ imply $x,y\in (\mathcal{E}^\dagger)^*$ | Yes, both statements are true. Note also that $\mathcal{E}^\dagger$ is a field. The simplest way to see this would be to use the fact that if you work over a fixed annulus $A = \{ z, r \leq |z| < 1\}$, then an analytic function on $A$ is bounded if and only if it has finitely many zeroes. You can prove this using the t... | 1 | https://mathoverflow.net/users/5743 | 285390 | 126,085 |
https://mathoverflow.net/questions/285022 | 0 | I am currently trying to prove an elliptic regularity type result for the Laplace Beltrami operator $\Delta\_g$ on a Riemannian manifold $(M^n,g)$. As a matter of convention, I will assume $\Delta\_g$ to be a nonpositive (self-adjoint) unbounded operator on $L^2(M)$, with domain $$
\mathcal{D}(\Delta\_g):=\{f\in L^2(M)... | https://mathoverflow.net/users/94022 | Elliptic regularity of Laplace-Beltrami operator on a manifold | What about M? I will give an answer supposing it is compact without boundary.
You want to prove that for fixed $t\in \mathbb{R}$, $e^{t\Delta}f\in C^\infty(M)$. This comes from an eigenvalue estimate of the Laplacian + elliptic regularity. For $-\Delta$ (positive Laplacian), due to the compactness of the manifold and... | 4 | https://mathoverflow.net/users/89806 | 285406 | 126,091 |
https://mathoverflow.net/questions/285387 | 1 | Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)$ and $\operatorname{Proj}(B)$ ? (even under some additional assumptions you are free to make for exemples)
Many than... | https://mathoverflow.net/users/66686 | If $A\subset B$, what to say about their $\operatorname{Proj}$? | Let $A\_+=\oplus\_{n\ge1}A\_n\subset A$ be the irrelevant ideal. Then the homogeneous ideal $A\_+B\subseteq B$ defines a subscheme $\mathcal N\subseteq{\rm Proj}(B)$ (the ``nullcone'' in case $A$ is a ring of invariants). Then the inclusion $A\subseteq B$ induces a projection $\pi:{\rm Proj}(B)\setminus\mathcal N\to{\r... | 2 | https://mathoverflow.net/users/89948 | 285408 | 126,093 |
https://mathoverflow.net/questions/285273 | 14 | Let $X\subset\mathbb{P}^N$ be a smooth projective variety of dimension $n$, say over $\mathbb{C}$. Assume that $H^i(X,\mathcal{O}\_X)\neq 0$ for some $i$ with $0<i<n$. What is the minimum possible degree of $X$? Some elliptic scrolls have degree $2n+1$, can one do better?
| https://mathoverflow.net/users/40297 | Minimum degree of a variety with $H^i(X,\mathcal{O}_X)\neq 0$ for some $i$ with $0<i<\dim(X)$ | **Proposition:** Let $X$ be a smooth projective $n-$dimensional ($n \geq 2$) nondegenerate subvariety of $\mathbb{P}^{N}$ satisfying the property that $H^{i}(\mathcal{O}\_{X}) \neq 0$ for at least one $i \in \{1, \cdots ,n-1\}.$ Then $${\rm deg}(X) \geq 2n+2.$$
If $X$ is not an elliptic scroll, then ${\rm deg}(X) \geq ... | 12 | https://mathoverflow.net/users/5496 | 285409 | 126,094 |
https://mathoverflow.net/questions/285401 | 0 | What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess?
Ok, I already posted this question, but a couple of notational errors and assumptions were made in the previous version. Hopefully, this new equivalent version will be better understood.
Let $U$ be a nonprinciple $\lambda$-... | https://mathoverflow.net/users/115951 | What strengthenings of measurability do the Mostowski collapses of ultrapowers possess? | Let's adopt Miha's interpretation of your question: which kinds of large cardinal properties can be witnessed by ultrapower embeddings?
Here, there are a variety of things one can say.
If one allows $\kappa$-complete measures on arbitrary sets, then basically all the usual large cardinal properties can be witness... | 4 | https://mathoverflow.net/users/1946 | 285411 | 126,095 |
https://mathoverflow.net/questions/285413 | 3 | I am looking for a suitable set of Dehn-twists generators for the mapping class group of a curve of genus $g$ with $n$ marked points (i.e. the mapping class group of $\mathcal M\_{g,n}$).
For $\mathcal M\_g$ one can use $2g+1$ Dehn twists, and also choose them as non-separating loops (i.e. they do not split the surf... | https://mathoverflow.net/users/116922 | Generators for the mapping class group of pointed curves | The answer to your question can be found on page 114 of Farb-Margalit's Primer on Mapping Class Groups. See Figure 4.10. All the Dehn twists they list are actually nonseparating curves.
| 2 | https://mathoverflow.net/users/317 | 285416 | 126,097 |
https://mathoverflow.net/questions/285396 | 4 | The following conjecture is known as "Table problem on $\Bbb S^2$"
>
> **Conjecture (Table problem on $\Bbb S^2$):** Suppose $x\_1, x\_2,x\_3,x\_4 \in\Bbb S^2 \subseteq \Bbb R^3$ are the vertices of a
> square that is inscribed in the standard $2$-sphere, and let $h : \Bbb S^2\to \Bbb R$ be a smooth function.
> T... | https://mathoverflow.net/users/90655 | Two questions on "Table problem on $\Bbb S^2$" | This is not an answer. but a list of references which wouldn't have fit in a comment (and wouldn't have been as easy to maintain). This is a digest of my colleague (A. Fruchard) answer to my query. Notice that the function $h$ may be assumed to be only continuous in the following first two results.
1. The conjecture ... | 4 | https://mathoverflow.net/users/24309 | 285424 | 126,099 |
https://mathoverflow.net/questions/285434 | 3 | Given $\alpha$ and constant $\mu$,
$$\begin{array}{ll} \text{maximize} & \displaystyle\int\_0^\infty p(x)x^\alpha \,\mathrm d x\\ \text{subject to} & \displaystyle\int\_0^\infty p(x)\,\mathrm d x = 1\\ & \displaystyle\int\_0^\infty p(x)x \, \mathrm d x = \mu\end{array}$$
I had previously posted this problem on math... | https://mathoverflow.net/users/75293 | Maximizing the $\alpha$-moment of a distributution | Let us consider the closely *related* problem: maximize $EX^\alpha$ over all nonnegative random variables (r.v.'s) $X$ with $EX=\mu$. To avoid trivialities, assume that $\mu\in(0,\infty)$. Consider the following cases:
Case 1: $\alpha>1$. Suppose that $P(X=x)=\mu/x=1-P(X=0)$ for some real $x>\mu$. Then $EX=\mu$, whe... | 4 | https://mathoverflow.net/users/36721 | 285438 | 126,103 |
https://mathoverflow.net/questions/285432 | 11 | In a model category, I have tools to show that mapping spaces are *contractible*. But if I want to show a mapping space is *empty or contractible*, is there anything I can do on general grounds?
The idea is this. Suppose I have a type of structure I'm interested in, and a collection of objects. To each object I assig... | https://mathoverflow.net/users/2362 | Property-like structure in a model category | Let us say that a map $f : X \to Y$ in an $\infty$-category is an *embedding* if, for every object $Z$, the map of spaces $\mathrm{Hom}(Z,X) \to \mathrm{Hom}(Z,Y)$ is an embedding (that is, has either empty or contractible fibers). It is easy to define this notion on the level of model categories. A map $f : X \to Y$ i... | 4 | https://mathoverflow.net/users/62782 | 285443 | 126,105 |
https://mathoverflow.net/questions/285441 | 6 | It is known that there are non-amenable groups not containing $F\_2$, the free group on two generators. We can even have that [every 2-generated subgroup is finite](https://mathoverflow.net/questions/278391/does-every-non-amenable-group-contain-a-2-generated-non-amenable-subgroup).
But is there a non-amenable group ... | https://mathoverflow.net/users/4600 | A group, neither amenable, nor having a subgroup that looks like $F_2$ up to level $n$? | Burnside groups of exponent $n$ are length $n$ unfree. If $n\ge 665$, odd, then the free Burnside group of exponent $n$ of rank 2 or more is not amenable (see Adian's book "The Burnside problem" or Olshanskii's book "Geometry of defining relations" or my book "Combinatorial algebra: syntax and semantics", Chapter 5).
... | 12 | https://mathoverflow.net/users/nan | 285444 | 126,106 |
https://mathoverflow.net/questions/285439 | 2 | Suppose $D$ is the Dirac operator on a closed spin manifold $M$, with spinors $S$. One can take the functional calculus of $D$ with respect to the continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ given by $x\mapsto\frac{x}{\sqrt{x^2+1}}$. Let's denote this operator by
$$f(D):L^2(S)\rightarrow L^2(S),$$
which... | https://mathoverflow.net/users/78729 | Why is index unchanged after applying functional calculus? | Perhaps the simplest answer is to use the spectral theorem: $L^2(S)$ decomposes as the orthogonal direct sum of $D$-eigenspaces, and $f(D)$ acts on each $\lambda$-eigenspace as multiplication by $f(\lambda)$. Since $f$ vanishes only at $0$, $f(D)$ has the same kernel as $D$. Moreover $f(D^\*) = f(D)^\*$ since $f$ is an... | 4 | https://mathoverflow.net/users/4362 | 285445 | 126,107 |
https://mathoverflow.net/questions/285449 | 3 | Let $X\neq \emptyset$ be a set and let ${\cal S} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say $C\subseteq X$ is a *choice set* for ${\cal S}$ if $|C\cap s| = 1$ for all $s\in S$. Moreover, we say that
1. $D\subseteq X$ is *shy* if $|D\cap s| \leq 1$ for all $s\in S... | https://mathoverflow.net/users/8628 | Choice sets from above and below | Let $\cal S=\{\{1,2\},\{2,3\},\{3,1\}\}$.
Then $\cal S$ has no choice set, whatsoever.
So there is no asymmetry -- not every shy set is contained in a choice set, and not every gregarious set contains a choice set.
| 9 | https://mathoverflow.net/users/4600 | 285451 | 126,110 |
https://mathoverflow.net/questions/285419 | 3 | Let $F$ be a field and let $G$ be a smooth, commutative and connected $F$-group scheme of finite type. Let $K/F$ be a finite Galois extension of fields. Then there exists a canonical norm (or *trace*, if $G$ is written additively) smooth and surjective morphism of $F$-group schemes
$$
R\_{K/F}(G\_{K})\to G,
$$
where $R... | https://mathoverflow.net/users/5641 | Norm 1 (or trace 0) commutative algebraic groups | You can do the same construction when $K$ is any étale $K$-algebra, not necessarily a field. If $K$ is the trivial $K$-algebra $F^n$, then $R\_{K/F}G\_K\cong G^n$ and the norm map is just the multiplication $G^n\to G$, hence $R^{(1)}\_{K/F}G\_K\cong G^{n-1}$.
Now observe that the construction commutes with ground fie... | 7 | https://mathoverflow.net/users/7666 | 285453 | 126,111 |
https://mathoverflow.net/questions/285423 | 6 | Let $E\_{a,b}$ be an elliptic curve defined by the equation $y^{2}=x^3+ax+b$ where $a,b \in \mathbb{Q}$.
Suppose that for $a=a\_{0}$ and $b=b\_{0}$ the rank of $E\_{a\_{0},b\_{0}}(\mathbb{Q})=1$.
**question:**
is there an $\epsilon> 0$ such that for any $(\alpha,\beta) \in \mathbb{Q}^{2}$ and $|\alpha-a\_{0}|+|\bet... | https://mathoverflow.net/users/110929 | rational points and a local perturbation of an elliptic curve | A counter-example, showing that the answer is "no" for some $(a\_0,b\_0)$ can be constructed as follows. Take an elliptic surface over $\mathbb{Q}$ with rank $2$. By Silverman's specialisation theorem, the rank of each fibre is at least $2$ (but often larger) except for a finite number of fibres. At an exceptional fibr... | 9 | https://mathoverflow.net/users/5015 | 285472 | 126,113 |
https://mathoverflow.net/questions/178535 | 5 | Does there exist any digital copy of the original article (or a transcript) K. Levenberg, *A method for the solution of certain problems in least-squares*, Quart. Appl. Math. 2 (1944): 164-168?
It is (obviously!) cited in numerous works, but the paper itself appears to have completely vanished. There seems to be not ... | https://mathoverflow.net/users/57119 | Levenberg's original article "A method for the solution of certain problems in least squares" | This [link](https://doi.org/10.1090/qam/10666) might be better. The paper is now open access and the pdf is searchable.
| 6 | https://mathoverflow.net/users/113174 | 285473 | 126,114 |
https://mathoverflow.net/questions/285471 | 3 | Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{B}(E)$ the algebra of all bounded linear operators from $E$ to $E$.
I want to show that for $(T\_1,...,T\_d) \in \mathcal{B}(E)^d$ we have:
$$\displaystyle\frac{1}{2}\left\|\displaystyle\... | https://mathoverflow.net/users/113054 | A numerical radius inequality | The inequality you have written cannot be proven.
Gelu Popescu has a big paper on multivariable generalizations of the numerical radius (Memoirs of the AMS, [arXiv](https://arxiv.org/pdf/math/0410492.pdf)).
He calls what you have above the Euclidian operator radius of a tuple:
$$
\omega\_e(T\_1,\dots, T\_d) = \sup\_{... | 10 | https://mathoverflow.net/users/76593 | 285476 | 126,116 |
https://mathoverflow.net/questions/285426 | 1 | I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$. One such example is:
$$
\frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = 1 \, ,
$$
with the (no inflow) boundary condition $u(\cdot,0) = u(0,\cdot) = 0$.
A simple analysis (method of char... | https://mathoverflow.net/users/116929 | Advection equation regularity (2D and time independent) | The problem is that **your boundary conditions are incompatible with your equation**.
Your choice of boundary data implies that, were $u$ to be in $C^1([0,1]\times[0,1])$ (meaning that the derivative extends continuously to the boundary), you must have $\nabla u(0,0) = 0$, since the two partials both vanish. This is... | 0 | https://mathoverflow.net/users/3948 | 285477 | 126,117 |
https://mathoverflow.net/questions/285483 | 5 | Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a non-vanishing holomorphic quadratic differential on $X$. Denote by $\mathbb{H}$ the universal cover of $X$ and by $\widetilde q$ the pullback of $q$ to $\mathbb{H}$. The space $\mathbb{H}$ endowed by the flat singular metric $|\widetilde q|$ is a Cat(0) metric sp... | https://mathoverflow.net/users/nan | Geodesic triangles in particular Cat(0) space | Triangles do not have singular points in their interior ⟺ curvature of any singular point is at least π.
In this case each triangle admits a length-preserving immersion into the plane.
(The interior admits an locally isometric immersion since it is locally flat, then you can extend it to the closure of the interior ... | 1 | https://mathoverflow.net/users/1441 | 285487 | 126,119 |
https://mathoverflow.net/questions/285480 | 2 | Let $N$ be a positive integer and $p$ a prime not dividing $N$.
Let $K$ be a finite Galois extension of $\mathbb{Q}$ which is unramified outside the primes dividing $Np$. Let $G$ be the Galois group of $K$ over $\mathbb{Q}$. Then for any conjugacy class $C \subset G$, there is a prime number $q$ not dividing $Np$ such ... | https://mathoverflow.net/users/116950 | a variant of Cebotarev density theorem | As noted in the comments, this is false in general. For instance, if $K=\mathbb Q(e^{2i \pi/p})$, then $G=(\mathbb Z/p\mathbb Z)^\ast$. If you take $C=\{1\}$, then condition (a) is that $q \equiv 1 \pmod{p}$ and condition (b) is that $q \not \equiv 1 \pmod{p}$ which makes finding such primes $q$ a desperate task.
The... | 8 | https://mathoverflow.net/users/9317 | 285488 | 126,120 |
https://mathoverflow.net/questions/285474 | 0 | I'm having difficulties in proving that the projection of $$(s,y)\in R \times R^{n}$$ onto the second-order cone $$Q^{n+1} = \{(t,x) \in R \times R^n : \|x\|\_2 \leq t \}$$ is $$ \frac{s+\|y\|\_2}{2\|y\|\_2} (\|y\|\_2,y)$$ when $\|y\|\_2 > s $ and $ \|y\|\_2 > -s $.
I tried to first show that to minimize the distance... | https://mathoverflow.net/users/82524 | Projection onto the second-order cone | The proof can be found in [H.H. Bauschke](https://people.ok.ubc.ca/bauschke/)'s 1996 doctoral dissertation: [Projection Algorithms and Monotone Operators (p. 40, Theorem 3.3.6)](http://summit.sfu.ca/system/files/iritems1/7015/b18025766.pdf).
P.S. I wonder what the downvote is for.
| 1 | https://mathoverflow.net/users/18526 | 285490 | 126,121 |
https://mathoverflow.net/questions/284877 | 7 | Every non-zero element in $H\_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A result of Peter Shalen guarantees that a compact, connected, orientable, irreducible 3-manifold $M$ [such a manifold is ... | https://mathoverflow.net/users/33064 | Does there exist a Haken manifold where all its incompressible surfaces are non-separating? | There exist closed orientable hyperbolic 3-manifolds that are surface bundles such that the fiber is the only incompressible surface in the manifold (up to isotopy). Such manifolds can be obtained by Dehn surgery on certain 2-bridge knots, using the classification of incompressible surfaces in 2-bridge knot complements... | 10 | https://mathoverflow.net/users/23571 | 285491 | 126,122 |
https://mathoverflow.net/questions/285469 | 1 | Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:
1. $X'$ is closed, $X/X' \cong \ell\_1 \Rightarrow X'$ is complementary;
2. $X' \cong \ell\_\infty \Rightarrow X'$ is complementary.
For the second one there is an option to try using Hahn-Banach theorem, as ... | https://mathoverflow.net/users/116711 | $\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space | For **(1)**, use **(0)**: for any Banach space $X$, $L(\ell\_1,X)$ is isometrically isomorphic to the space $\ell\_\infty(X)$ of bounded sequences in $X$. You may easily define a concrete isometry and its inverse. As a consequence: any surjective bounded linear operator $S:X\to\ell\_1$ is a left inverse: use **(0)** to... | 3 | https://mathoverflow.net/users/6101 | 285492 | 126,123 |
https://mathoverflow.net/questions/285494 | 11 | Let $M\_1$ and $M\_2$ be two symmetric $d\times d$ matrices. What is the relationship between
$tr(M\_1M\_2M\_1M\_2)$ and $tr(M\_1^2 M\_2^2 )$?
P.S. I tried a few examples and found
$$
tr(M\_1M\_2M\_1M\_2) \le tr(M\_1^2 M\_2^2 )
$$
seems always true. Is there a theorem?
| https://mathoverflow.net/users/88033 | Trace of non-commutable matrices | Your conjecture is a special case of the following result which essentially follows from the Lieb-Thirring inequality.
Let $A$ and $B$ be Hermitian matrices. Then, for every positive integer $p$ we have
\begin{equation\*}
|\text{tr}(AB)^{2p}| \le \text{tr}A^{2p}B^{2p}
\end{equation\*}
| 15 | https://mathoverflow.net/users/8430 | 285499 | 126,126 |
https://mathoverflow.net/questions/240202 | 2 | Matroid $M$ is represented by real vectors, and we know that any base of $M$ generates the same lattice (this is called unimodular representation, I guess.) If we change the sign of any vector, we again get a representation of the same type. Are there examples when two unimodular representations of $M$ are not equivale... | https://mathoverflow.net/users/4312 | Which matroids have not unique unimodular representation? | This is Corollary 10.1.4 in [Oxley](https://books.google.at/books?redir_esc=y&hl=de&id=puKta1Hdz-8C&q=10.1.4#v=snippet&q=10.1.4&f=false). Lenz gives a good account in Section 7 of [*On Powers Of Plücker Coordinates And Representability Of Arithmetic Matroids*](https://arxiv.org/pdf/1703.10520.pdf).
| 3 | https://mathoverflow.net/users/94968 | 285508 | 126,131 |
https://mathoverflow.net/questions/285171 | 10 | It is well known that the absolute value on operators does not satisfy the [triangle inequality](https://mathoverflow.net/questions/173613/how-much-does-the-absolute-value-of-an-operator-behave-like-an-absolute-value).
My question is whether for all positive operators $P,Q \in B(\mathcal H)$ is there a universal con... | https://mathoverflow.net/users/76593 | Comparison of the absolute value of an operator with its positive parts | OK, here goes.
We start with a reformulation of the problem. Assume that everything is invertible and $P+iQ=UBB^\*$ where $U$ is unitary and $BB^\*=|P+iQ|$. We can also re-parameterize $P$ and $Q$ as $BPB^\*$ and $BQB^\*$. Then the question reduces to whether for two positive operators $P,Q$, the relation $P+iQ=B^{-1... | 7 | https://mathoverflow.net/users/1131 | 285512 | 126,134 |
https://mathoverflow.net/questions/285522 | 0 | Let $a, b \in {\cal P}(\omega)/\mathrm{(fin)}$ with $a<b$. Do we have ${\cal P}(\omega)/(fin)\cong [a,b]$?
| https://mathoverflow.net/users/8628 | Is ${\cal P}(\omega)/\mathrm{(fin)}$ order-isomorphic to its intervals? | The answer is yes.
Without loss of generality, we may choose representatives for $a$ and $b$ such that $a \subseteq b$ when construed as elements of $\mathcal{P}(\omega)$ (by taking arbitrary representatives, and changing these as needed at the finitely many indices where the subset relation may not already hold).
... | 1 | https://mathoverflow.net/users/3902 | 285524 | 126,135 |
https://mathoverflow.net/questions/285505 | 4 | Let $A$ be a matrix whose entries are either indeterminates or fixed to zero. For example, consider
$$A = \left( \begin{array}{ccc} 0 & x\_{12} & 0 \\
0 & 0 & 0 \\
0 & x\_{32} & 0 \end{array} \right).$$ It is natural to associate a graph with such matrix, with the directed edge $(i,j)$ present in the graph whenever $A... | https://mathoverflow.net/users/116981 | The generic nullspace of a graph | The answer to question 2 is positive. By König's theorem, you may find $r$ indeterminates belonging to distinct rows and columns, and cover all indeterminates by, say $r\_1$ columns and $r\_2=r-r\_1$ rows. Then the rank of a matrix equals $r$ (it can not be neither more or less), and the kernel has a dimension $n-r$. W... | 1 | https://mathoverflow.net/users/4312 | 285527 | 126,136 |
https://mathoverflow.net/questions/285118 | 3 | I would greatly appreciate any references were they study the stochastic equation in higher dimensions: $u\_{t}=\Delta u+f$ in great detail, especially in dimension 2.
In Hairer's [Spde notes](http://www.hairer.org/notes/SPDEs.pdf) , he mentions that the solutions will not be function-valued, and I am curious what t... | https://mathoverflow.net/users/99863 | Regularity for Stochastic heat equation with additive noise in d=2 | I hope this is an answer to your question. In any case it is too long for a comment.
With which space of distributions you are working depends on where you are solving the equation. Two natural choices are $[0,T]\times \mathbb{R}^d$ or $[0,T]\times \mathbb{T}^d$, the latter being a torus (i.e. periodic boundary condi... | 3 | https://mathoverflow.net/users/33717 | 285535 | 126,138 |
https://mathoverflow.net/questions/283064 | 3 | Given a compact quantum group $(G,\Delta)$, with dense Hopf algebra $H$, is it always true that, up to isomorphism, $H$ will have a countable number of irreducible comodules?
| https://mathoverflow.net/users/81477 | Number of Isomorphism Classes of Corepresentations of A Compact Quantum Group | Take a discrete uncountable group G of cardinality M, then $C^\*\_r(G)$ will be a compact quantum group with M irreducible (1-dimensional) corepresentations.
| 3 | https://mathoverflow.net/users/19471 | 285537 | 126,139 |
https://mathoverflow.net/questions/285513 | 5 | For simplicity, let us consider a function $f$ holomorphic on a domain $D \subseteq \mathbb{C}^2$. We may therefore write $f$ as a sum of power series $$f(z) = \sum\_{\nu\_1 \nu\_2 =0}^{\infty} c\_{\nu\_1 \nu\_2}(z\_1 - w\_1)^{\nu\_1} (z\_2 - w\_2)^{\nu\_2}.$$
We should also note that the domain of convergence of a p... | https://mathoverflow.net/users/105103 | Practically calculating the domain of a power series for function of several complex variables | The usual Cauchy-Hadamard formula has a generalization to several variables.
The numbers $r\_1,\ldots,r\_n$ are called conjugate radii of convergence if the
series converges in the open polydisk $B(r\_1,\ldots,r\_n)$ and does not converge on any open subset of
$\{ z:|z\_j|>r\_j, 1\leq j\leq n\}$. Then we have the formu... | 3 | https://mathoverflow.net/users/25510 | 285543 | 126,143 |
https://mathoverflow.net/questions/285507 | 4 | [Thanks to Gerhard Paseman for helping me reformulate my original question.]
The equation
$$
\frac{a^m-1}{a-1}=b^2
$$
was solved by Ljunggren, building on work of Nagell, who showed that if $a>1$, $b>1$, and $m>2$ are integers, there are exactly two (easy to find) solutions. Can the same methods explicitly give all s... | https://mathoverflow.net/users/3199 | A modification of the Ljunggren-Nagell equation | The equation
$$
\frac{a^m-1}{a-1}=2b^2
$$
does not have solutions in positive integers for $m>2$ as shown below.
First, notice that $a$ must be odd.
Second, one can see that $m$ must be even. Indeed, if $a\equiv 3\pmod4$, then odd $m$ would produce an odd number in the left-hand side. On the other hand, if $a\equi... | 6 | https://mathoverflow.net/users/7076 | 285545 | 126,144 |
https://mathoverflow.net/questions/285452 | 1 | Let $G$ be a finite group and $p\in\pi(G)$. Suppose that
$\quad$(i) for any non-principal $\chi\in\mathrm{Irr}(G)$, $p\nmid\frac{|G|}{|\mathrm{ker}\chi|\chi(1)}$;
$\quad$(ii) $E$ is the unique minimal normal subgroup $G$;
$\quad$(iii) $N$ is a $p'$-normal subgroup of $G$ and $E\leq N$;
$\quad$(iv) $\varphi\in ... | https://mathoverflow.net/users/99750 | Prove some inertia group $T$ is a $p'$-group | Yes. $\DeclareMathOperator{\Irr}{Irr} \renewcommand{\phi}{\varphi}$
Write $\phi^T = \sum\_{\eta} a\_{\eta} \eta$, the sum running over $\eta\in \Irr(T\mid \phi)$. Then $a\_{\eta} = [\phi^T,\eta] = [\phi, \eta\_N]$ and $\eta(1)=a\_{\eta}\phi(1)$. By Clifford theory, $\theta:= \eta^G \in \Irr(G)$ for every $\eta\in \Irr... | 1 | https://mathoverflow.net/users/10266 | 285552 | 126,146 |
https://mathoverflow.net/questions/46986 | 27 | Let me start by reminding two constructions of topological spaces with such exotic combination of properties:
1) The elements are non-zero integers; base of topology are (infinite) arithmetic progressions with coprime first term and difference.
2) Take $\mathbb{R}^{\infty}\setminus \{0\}$ with product-topology and ... | https://mathoverflow.net/users/4312 | Countable connected Hausdorff space | First let us fix the terminology.
The space (1) is known in General Topology as [the Golomb space](https://dml.cz/bitstream/handle/10338.dmlcz/700933/Toposym_01-1961-1_41.pdf). More precisely, the *Golomb space* $\mathbb G$ is the set $\mathbb N$ of positive integers, endowed with the topology generated by the base c... | 28 | https://mathoverflow.net/users/61536 | 285556 | 126,148 |
https://mathoverflow.net/questions/285553 | 2 | Let $M$ denote the space of all complex $n\times n$ matrices. ($M$ is a complex vector space.)
Let $C\subseteq M$ be the set of all conjugate-normal matrices (i.e., matrices $A$ with $AA^\*=\overline{A^\*A}$).
Let $S\subseteq C$ denote all symmetric matrices. (Then $S$ is a subspace of $M$.)
Is there a subspace $... | https://mathoverflow.net/users/101775 | Are symmetric matrices a maximal subspace of conjugate-normal matrices | Yes, the symmetric matrices are a maximal subspace in the conjugate-normal matrices.
Let $B$ be conjugate-normal and assume that $A+B\in C$ for all $A\in S$. Then
\begin{align\*}
0 & = (A+B)(A+B)^\* - \overline{(A+B)^\*(A+B)}
\\ & = (A+B)(\overline A + B^\*) - (A + B^T)(\overline A + \overline B)
\\ & = AB^\* + B\ov... | 4 | https://mathoverflow.net/users/76593 | 285559 | 126,149 |
https://mathoverflow.net/questions/285533 | 2 | Let $G\text{-}PF(X)$ be the set of isomorphism classes of principal topological fibrations over the space $X$ with structural group $G$, and $G\text{-}PF\_{cw} : hCW \to Set$ the contravariant functor $X \mapsto G\text{-}PF(X)$ where $X$ is a CW-complex. Let also $G\text{-}PF\_n(X)$ be the set of isomorphism classes of... | https://mathoverflow.net/users/74372 | Classifying spaces and Brown's representability theorem | **No**, homotopy classes of maps of unpointed spaces are not equivalent to homotopy classes of maps of pointed spaces. You need to be a little bit more clever. What follows is a spelling out of Brown's original argument in section 5.1 of [1]. I'm going to cover the statement for principal bundles of topological groups,... | 10 | https://mathoverflow.net/users/43054 | 285564 | 126,152 |
https://mathoverflow.net/questions/285536 | 5 | As we know, every regular weakly Lindelof space is DCCC. Here DCCC denotes discrete countable chain condition, a space $X$ has discrete countable chain
condition if every discrete family of non-empty open sets of $X$ is countable.
A space $X$ is said to be weakly Lindelof if every open
cover $\mathcal U$ of $X$ cont... | https://mathoverflow.net/users/39873 | Is there a Hausdorff weakly Lindelof space which is not DCCC? | The answer to this problem is negative because of the following
**Theorem.** If a topological space $X$ is weakly Lindelof, then each discrete (more generally, locally countable) family of open sets in $X$ is at most countable.
*Proof.* Let $\mathcal U$ be a locally countable family of open subset of $X$.
Then each... | 4 | https://mathoverflow.net/users/61536 | 285568 | 126,153 |
https://mathoverflow.net/questions/285565 | 5 | Let $G$ be a locally compact abelian group. A *lattice* in $G$ is a discrete subgroup $\Lambda$ such that the quotient $G / \Lambda$ is compact. A *Borel fundamental domain* of a lattice $\Lambda$ in $G$ is a Borel set $F \subseteq G$ that intersects each coset in $G / \Lambda$ in exactly one point, i.e., $F$ is a set ... | https://mathoverflow.net/users/nan | Inclusion of lattices and fundamental domains | The image of $\Lambda\_2$ in $G/\Lambda\_1$ is finite. Take a set of representatives $s\_1, s\_2, \ldots s\_n$ in $\Lambda\_2$. Now consider $\cup s\_i F\_2$. I claim this is a Borel fundamental domain for $F\_1$.
It is Borel as it is a finite union of Borel sets. Let $g\Lambda\_1$ be a coset of $\Lambda\_1$. Then $g... | 8 | https://mathoverflow.net/users/6084 | 285572 | 126,155 |
https://mathoverflow.net/questions/285566 | 7 | The question is maybe a bit technical, but I find the related construction very beautiful.
In the very famous work - "$C^1$-isometric imbeddings" by J.Nash (1954) the
author presented the fundamental theorem (which was especially recognized some years after) about isometric embeddings of Riemannian manifolds.
$\... | https://mathoverflow.net/users/94631 | Short embeddings for open manifolds and dimension reduction of sets | It seems to be a general position argument.
You need to find a $2{\cdot}n+1$-dimensional subspace of $\mathbb{E}^{s\cdot(n+2)}$ such that (1) it contains no vector orthogonal to your submanifold and (2) no two points of your submanifold are projected to the same point.
All you need is to calculate dimensions correc... | 2 | https://mathoverflow.net/users/1441 | 285574 | 126,157 |
https://mathoverflow.net/questions/285550 | 6 | Given a path $P$ in the unit square, and two points $p\_{1},p\_{2}$
located on $P$, let $d\_{P}(p\_{1},p\_{2})$ denote the distance from
$p\_{1}$ to $p\_{2}$ traversed along $P$. Given $a>1$, I am looking
for the shortest $P$ that satisfies the following property:
For any point $x$ in the unit square, there exist $p\... | https://mathoverflow.net/users/70190 | A path in the unit square that "doubles back" on itself in a nice way | This is just about the asymptotic behavior for $a\to+\infty$. I claim that the minimal length $\ell$ is about $\sqrt{2a}$ for large $a$.
***The upper bound***
Consider $2n$ horizontal lines splitting the square into strips of width $\frac 1{2n-1}$ Now travel along the odd-numbered lines (in the natural enumeration ... | 6 | https://mathoverflow.net/users/1131 | 285584 | 126,160 |
https://mathoverflow.net/questions/285570 | 3 | Let $A\_r = \{z\in\mathbb{C}: |z|\leq 1\}\cup\{z\in\mathbb{C}: |z-1|\leq r\}$ be the unit disk with a small "bump" (I'm interested in the regime $r\to 0$). What can be said about the logarithmic capacity of $A\_r$? I can prove that there exists a constant $\gamma>0$ such that $\mathop{\mathrm{cap}}(A\_r)=\gamma r^2 + $... | https://mathoverflow.net/users/81488 | Capacity of a unit disk with a small bump | Both questions have exact explicit answer, which is explained in any good textbook on analytic functions. A region on the Riemann sphere is called a digon if its boundary consists of
two arcs of circles. Let $a$ and $b$ be the common endpoints of these arcs.
Then
$$f(z)=\frac{z-a}{z-b}$$
sends $a,b$ to $0,\infty$, and... | 8 | https://mathoverflow.net/users/25510 | 285594 | 126,162 |
https://mathoverflow.net/questions/285515 | 11 | I am looking at the following function on the domain $x\geq 0$:
$$F(x)=(x+a)e^{x^2}(1-\mathrm{erf}(x))-\frac{b}{\sqrt\pi},$$
where $a>0$, $0<b<1$ are parameters. From plotting this function for different values of $a$ and $b$ it seems that there is at most one root on $x\in[0,\infty)$. But how to prove it?
| https://mathoverflow.net/users/116983 | A problem involving the Error Function | Indeed, there is at most one root on $x\in[0,\infty)$ -- actually, for each real $a$. Consider the equivalent equation
\begin{equation}
r(x):=\frac{f(x)}{g(x)}=\frac{\sqrt\pi}b \tag{1}
\end{equation}
for real $x>z\_a:=0\vee(-a)$,
where $0<b<1$ and
\begin{equation}
f(x):=\frac{e^{-x^2}}{x+a},\quad g(x):=1-\text{erf... | 7 | https://mathoverflow.net/users/36721 | 285598 | 126,165 |
https://mathoverflow.net/questions/285618 | 1 | What is an example of a topological space $(X,\tau)$ with the properties that
1. $X\cong X\setminus \{x\}$ for all $x\in X$, and
2. $(X,\tau)$ is not topologically [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_space)
?
| https://mathoverflow.net/users/8628 | Non-homogeneous space $X$ such that $X\cong X\setminus \{x\}$ for all $x\in X$ | An infinite disjoint union of half-open and open intervals works.
| 8 | https://mathoverflow.net/users/1310 | 285619 | 126,167 |
https://mathoverflow.net/questions/285607 | 3 | Let $(P,\leq)$ be a countably infinite poset with the property that whenever $a<b\in P$ then $P\cong [a,b]$.
**Question.** If $P$ does contain elements $a,b$ with $a<b$, does this imply that $P \cong [0,1]\cap\mathbb{Q}$, or that $P$ is isomorphic to the nonzero countable atomless Boolean algebra?
**Note.** Thanks ... | https://mathoverflow.net/users/8628 | Countably infinite posets isomorphic to its intervals | How about this. Let $P$ have "levels" indexed by $\mathbb{Q} \cap [0,1]$. Levels $0$ and $1$ each contain a single element, all other levels contain a countably infinite set of elements. An element at level $a$ lies below one at level $b$ if and only if $a <b$. It's clearly not a Boolean algebra, but also clearly has t... | 5 | https://mathoverflow.net/users/23141 | 285633 | 126,172 |
https://mathoverflow.net/questions/285578 | 5 | Let $f\_1,\dots,f\_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f\_1^2,\dots,f\_r^2)$ and $J = (f\_1,\dots,f\_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the subschemes of $X$ defined respectively by $I$ and $J$. Assume that $W$ is smooth.
Now, let $X\_Y = Bl\_YX$ and $X\_W = Bl... | https://mathoverflow.net/users/nan | Blowing-up an ideal generated by squares | If $\alpha \in \mathbb{N}^{r}$ satisfies $|\alpha| = r+1$ then $\exists i, \alpha\_i \geq 2$ so that $f^{\alpha} \in f\_i^2 J^{r-1}$.
Thus $J^{r+1} = I J^{r-1}$. In particular $I$ becomes invertible on $X\_W$, so that $X\_W \rightarrow X$ uniquely factors through $X\_{Y}$.
| 2 | https://mathoverflow.net/users/21724 | 285635 | 126,174 |
https://mathoverflow.net/questions/285600 | 6 | I'm mostly interested in the 4d case so I'll state the question in that form. Basically, it comes down to two parts:
1) What simple moves can be performed to a kirby diagram of a 4-manifold that guarantee that the resulting diagram is cobordant (via a 5d manifold) to the original manifold.
2) Can any cobordism be a... | https://mathoverflow.net/users/50693 | Can cobordisms of 3 or 4 manifolds be visualized by moves on kirby diagrams? | I think that a complete set of moves for cobordisms are the following: Kirby moves (that preserve the 4-manifold), handle trading (dotted circles become 0-framed 2-handles), addition/deletion of pairs of trivial circles with framings $\pm 1$ (corresponding to connected sums with $CP^2 \,\#\, {\overline{CP^2}}$). This c... | 7 | https://mathoverflow.net/users/23193 | 285636 | 126,175 |
https://mathoverflow.net/questions/285627 | 10 | In general, when a paper references an object discovered/defined in another paper by author **X**, it goes something along the lines of:
"*Let $\tau$ be the constant defined by **X** in 1999* [1]$\ldots$",
or
"*Let $f\_{\mu}$ denote the function that generalizes the case $\ldots$ (**X**, [1])*".
At what point d... | https://mathoverflow.net/users/103722 | How are constants/functions named after their discoverer? | This is determined by informal consensus of researchers in an area. Anyone can propose a name for a mathematical object, just by using this name in a paper. Then the proposed name either sticks to the object or not. This depends on the opinions of other people working in the area. Eventually the name of the object beco... | 29 | https://mathoverflow.net/users/25510 | 285644 | 126,176 |
https://mathoverflow.net/questions/285135 | 4 | Let $\pi$ be an automorphic representation of $GL\_2$ over a number field. What can I say concerning the order of the pole at $1$ of the $L$-function $L(s, \pi)$? Can we say more about $L(s, \mathrm{Sym}^2 \pi)$ or other constructions depending on $\pi$?
In the case where only holomorphic cusp forms appear, there is ... | https://mathoverflow.net/users/43737 | Poles of $L$-functions associated to Maass forms | Let $\pi$ be a *cuspidal* automorphic representation of $\mathrm{GL}\_n(\mathbb{A}\_F)$, where $F$ is a number field. Then $L(s,\pi)$ is entire and has a zero-free region. The latter is essentially Theorem 5.10 of Iwaniec and Kowalski, while the former is Corollary 13.8 of Godement-Jacquet.
If $\pi$ is not cuspidal, ... | 7 | https://mathoverflow.net/users/3803 | 285653 | 126,179 |
https://mathoverflow.net/questions/285628 | 5 | The Arthur trace formula and its variations provide general results for reductive groups, however to the extent of my knowledge only few specific instances of the formula have been really worked out in details, *i.e.* with precise decomposition of the spectrum, as for $\mathrm{GL}(n)$.
Is there such a formula for $\m... | https://mathoverflow.net/users/43737 | A trace formula for $\mathrm{GSp(4)}$ | There are many kinds of trace formulas on a given group $G$, and different things you could mean by decomposition of the spectrum. As mentioned in the comments, there's Arthur's article in the Shalika volume, though I am not sure whether those results are unconditional as of yet. On the other hand, Arthur's work on the... | 4 | https://mathoverflow.net/users/6518 | 285654 | 126,180 |
https://mathoverflow.net/questions/285648 | 7 | Is there a reasonable notion of an inner automorphism of a Hopf algebra $H$ which in the case of a group ring $H=\mathbb KG$ for a group $G$ reduces to a conjugation by $g\in G$?
| https://mathoverflow.net/users/23935 | Inner automorphisms of Hopf algebras | I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad\_l h)\triangleright k=\sum h\_1 kS(h\_2)$ of a hopf algebra $H$ on itself.
(It is known that the left adjoint action of $H$ on itself is always *inner* (in the sense of the def. 6.1.1, p. 87, in Montgomery... | 8 | https://mathoverflow.net/users/85967 | 285660 | 126,184 |
https://mathoverflow.net/questions/285663 | 2 | The pmf of Ising model is considered as $p(\boldsymbol{x})=\frac{1}{Z(\theta)} exp\left\{ \underset{\left(s,t\right)\in E}{\sum\theta\_{st}}x\_{s}x\_{t}\right\},\quad \boldsymbol{x}\in \{-1,1\}^n$, where $E$ is the set that contains the edges of the graph and $Z(\theta)=\underset{x}\sum{ exp\left\{ \underset{\left(s,t\... | https://mathoverflow.net/users/117055 | Ising model with zero external field - marginalization | Note that the indicator $$ 1\_{X\_i = x} = \frac{1+x X\_i}{2}$$
so that
$$ \mathbb P(X\_i = x\_i,\; X\_j = x\_j) = \mathbb E \left[ \frac{(1+x\_i X\_i)(1+x\_j X\_j)}{4} \right] $$
Expand it out, and note that by symmetry $\mathbb E[X\_i] = \mathbb E[X\_j] = 0$.
| 4 | https://mathoverflow.net/users/13650 | 285664 | 126,187 |
https://mathoverflow.net/questions/285658 | 4 | In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" ([arXiv](http://lanl.arxiv.org/abs/0803.4121), [journal](http://www.ams.org/journals/ert/2009-13-14/S1088-4165-09-00346-X), [MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2525917)), Khovanov and Lauda put a strand with a single d... | https://mathoverflow.net/users/117053 | Why are the divided difference operators of the nil Hecke ring only of degree 1? | It is not true that the crossing is necessarily in degree $1$. More precisely, if $\delta\_{k,\mathbf i}$ is the crossing between strands $k$ and $k+1$ for the sequence $\mathbf i=(i\_1,\ldots, i\_m)$, then $$\deg(\delta\_{k,\mathbf i})=-i\_k\cdot i\_{k+1}=\begin{cases}-2&\text{ if }i\_k=i\_{k+1},\\ 1 &\text{ if } i\_k... | 6 | https://mathoverflow.net/users/805 | 285667 | 126,190 |
https://mathoverflow.net/questions/285668 | 2 | Given real vectors $v$ and $r$ of the same size, what are the following?
1. $\inf\{v'R^{-1}v ~ \colon ~ R>0 \, , \, \text{diag}(R)= r\}$
2. $\sup\{v'Rv ~ \colon ~ R>0\, , \, \text{diag}(R)= r\}$
Note: $R > 0$ denotes positive definiteness, $x'$ denotes transpose, $\text{diag}(R)$ is the vector of the diagonal entr... | https://mathoverflow.net/users/69169 | Extremal Covariance Duality | First, let us answer question 2. Write $R=Q'Q$, where $Q=[q\_1,\dots,q\_n]$, a square matrix with columns $q\_1,\dots,q\_n$. Let $v\_i$ and $r\_i$ denote the coordinates of the vectors $v$ and $r$. Then the problem can be rewritten as follows: maximize $\|Qv\|=\|\sum\_i v\_iq\_i\|$ given $\|q\_i\|^2=r\_i$ for all $i$. ... | 3 | https://mathoverflow.net/users/36721 | 285672 | 126,191 |
https://mathoverflow.net/questions/285638 | 6 | Similar to [Mean minimum distance for N random points on a one-dimensional line](https://mathoverflow.net/questions/1294/mean-minimum-distance-for-n-random-points-on-a-one-dimensional-line), but instead of only N random points, choose N and M random points and find the mean minimum distance between points of N and M, b... | https://mathoverflow.net/users/117039 | Mean minimum distance for M and N uniformly random points on reals between 0 and 1 | Let $D\_1$ be the answer conditioned on the leftmost point being at 0 and the rightmost point at 1 (colors irrelevant). Let $0 = x\_1 \leq \ldots \leq x\_{n + m} = 1$ be the ordered coordinates of points. We can see that $s\_i = x\_{i + 1} - x\_i$ are equidistributed subject to $s\_1 + \ldots + s\_{n + m - 1} = 1$, and... | 4 | https://mathoverflow.net/users/106512 | 285679 | 126,193 |
https://mathoverflow.net/questions/209661 | 8 | Although the box topology is a topology worth studying and is similar to the strong topology in differential topology, the box topology is in many regards very badly behaved since the box product of even nice spaces has many undesirable properties. For instance, unlike ordinary products, non-trivial box products are ne... | https://mathoverflow.net/users/22277 | Which topological properties are preserved under taking box products? | A space $X$ is discretely generated (DG) if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is a discrete set $D \subset A$ such that $x \in \overline{D}$.
For being a convergence-type property (note that Fréchet-Urysohn and even radial spaces are DG) discrete genera... | 3 | https://mathoverflow.net/users/11647 | 285688 | 126,197 |
https://mathoverflow.net/questions/285695 | -1 | Informally speaking, I was wondering whether the relation
>
> $a^k \equiv b \text{ (mod } n)$ for some $k,n$
>
>
>
is computable. More formally: Let $\mathbb{N}$ denote the set of the positive integers and set $$R = \big\{(a,b)\in \mathbb{N}\times\mathbb{N}:(\exists k,n \in\mathbb{N}): (n>\max\{a,b\})\land (a... | https://mathoverflow.net/users/8628 | Computability of a relation connected to the discrete logarithm | Of course since $R = \{ (a,b) \in \mathbb{N} \times \mathbb{N} \ | \ (a=1) \implies (b=1) \} $.
Indeed for $a > 1$ one can take $n = a^k-b$ for $k$ large enough.
| 2 | https://mathoverflow.net/users/21724 | 285697 | 126,198 |
https://mathoverflow.net/questions/285687 | 3 | Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a *cover* if $\bigcup {\cal C} = X$. A subset $D\subseteq X$ is a *choice set* for ${\cal C}$ if $|D\cap c| = 1$ for all $c\in C$. As [Bjørn Kjos-Hanssen](https://mathoverflow.net/users/4600/bj%c3%b8rn-kjos-hanssen) pointed... | https://mathoverflow.net/users/8628 | Subcovers without a choice set | Well order $X$ and at stage $\alpha$ let $d\_\alpha$ be the first element of $X\setminus \bigcup\_{\xi<\alpha}\bigcup\{S \in \mathcal S: d\_\xi \in S \}$ (if this set was empty we stop and let $\gamma=\alpha$). Then $D=\{d\_\alpha: \alpha <\gamma\}$ is a choice set for the subcovering $\mathcal S\_0=\{S \in \mathcal S:... | 5 | https://mathoverflow.net/users/17836 | 285702 | 126,199 |
https://mathoverflow.net/questions/285701 | 5 | While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:
Consider the additive formal group law, $F\_a$, and the multiplicative formal group law, $F\_m$, both defined over a ring $R$.
If $R$ is a $\mathbb{Q}$-algebra, the exponential series defines a morp... | https://mathoverflow.net/users/7519 | Morphisms of formal group laws $\,F_a \rightarrow F_m\,$ and $\,F_m\to F_m$ | Let $D$ be the divided power ring
$$ D = \mathbb{Z}[a\_0,a\_1,a\_2,\dotsc]/(a\_0-1,a\_na\_m-(n,m)a\_{n+m}) $$
(where $(n,m)$ denotes the binomial coefficient $(n+m)!/(n!\,m!)$). Then ring maps from $D$ to $R$ biject with power series $f(t)=\sum\_ka\_kx^k\in R[[x]]$ satisfying $f(s+t)=f(s)f(t)$ and $f(0)=1$, and thus w... | 12 | https://mathoverflow.net/users/10366 | 285704 | 126,200 |
https://mathoverflow.net/questions/285692 | -1 | We know formula of boolean algebra in canonical disjunctive normal form has or may be converted to [Zhegalkin polynomial](https://en.wikipedia.org/wiki/Zhegalkin_polynomial).
Is there any approach to convert first order formula into algebraic function or other functions?
| https://mathoverflow.net/users/14024 | Conversion of logic formula into algebraic formula | Tarski's [cylindric algebras](https://en.wikipedia.org/wiki/Cylindric_algebra) are an algebraization of first-order logic with equality.
| 2 | https://mathoverflow.net/users/1176 | 285708 | 126,201 |
https://mathoverflow.net/questions/285632 | 14 | Let $K^n\_0 := \partial \Delta\_{n+1}$ the simplicial set obtained by removing the $(n+1)$-simplex from the standard simplex. This gives a simplicial decomposition of the sphere $S^n$. More generally, let $K^n\_m$ given by $m$-fold barycentric subdivision of $K^n\_0$.
Given a map of simplicial sets $f: K^{n+k}\_m \r... | https://mathoverflow.net/users/16702 | Realisation of maps between spheres by simplicial maps | First, $\pi\_{4k-1}(S^{2k})$ has an infinite cyclic direct summand for every $k \geq 1$. As a simple example, you can think of $\pi\_3(S^2) \cong \mathbb{Z}$, coming from the Hopf fibration.
We recently wrote a paper where we solved a problem, which is closely related to your question. Namely, we give an algorithm w... | 13 | https://mathoverflow.net/users/82276 | 285712 | 126,203 |
https://mathoverflow.net/questions/285710 | 1 | What’s the relation between Lehmer's Conjecture and Systole?
Lehmer’s conjecture says: There exists $m>1$ such that $M(p)\geq m$ for all noncyclotomic $P$.
Systole is a closed geodesic of the shortest length in Riemannian manifold.
Thanks.
| https://mathoverflow.net/users/111290 | What’s the relation between Lehmer’s Conjecture and Systole | The relationship between Lehmer's Conjecture and systoles is related to the following important conjecture regarding the geometry of arithmetic hyperbolic orbifolds:
*Short Geodesic Conjecture*: There is a universal, positive lower bound for the systole of an arithmetic hyperbolic 2- or 3-orbifold.
To illustrate wh... | 7 | https://mathoverflow.net/users/nan | 285716 | 126,206 |
https://mathoverflow.net/questions/285720 | 3 | It is well known that for a functor $F: \mathcal R \to \mathcal M$, where $\mathcal R$ is a Reedy category and $\mathcal M$ is suitably bicomplete, the following decomposition determines the structure around $y \in \mathcal R$:
$$
Latch(F)(y) \to F(y) \to Match(F)(y);
$$
here we decompose the canonical map between the ... | https://mathoverflow.net/users/42658 | Reedy-indexed diagrams in higher categories | This follows from Proposition A.2.9.14 in Higher Topos Theory. Lurie expresses a Reedy category as inductively obtained by homotopy pushouts in the Joyal model structure, and using this one gets the description you're after. He proves this explicitly for ordinary categories in A.2.9.15 and remarks in A.2.9.16 that the ... | 2 | https://mathoverflow.net/users/6936 | 285733 | 126,208 |
https://mathoverflow.net/questions/285721 | 7 | Let $A\to B$ be a morphism of commutative rings. Let $\mathcal C$ be the category of commutative $A$-algebras augmented over $B$. Let $\mathcal M\_B$ denote the category of $B$-modules. The cotangent complex can be defined like this (I will brush model-categorical considerations under the carpet). Consider the derived ... | https://mathoverflow.net/users/6249 | Is it possible to compute André-Quillen cohomology by resolving the module variable? | The cotangent complex $\Bbb L\_{B/A}$ is a (homologically) bounded-below complex of projective $B$-modules, and the bottom homology group is $H\_0 (\Bbb L\_{B/A}) = \Omega\_{B|A}$.
If we apply $RHom\_{B-mod}(-,M)$ to this, we get a Grothendieck spectral sequence for computing André-Quillen cohomology:
$$
Ext^p\_B(H\_... | 9 | https://mathoverflow.net/users/360 | 285737 | 126,211 |
https://mathoverflow.net/questions/285729 | 3 | Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$, and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the rectangle $R=\{ (x,y) \in \mathbb{R}^2: 0 \leq x \leq a, 0 \leq y \leq b \}$ with velocity of constant magnitude $v$ a according to the ... | https://mathoverflow.net/users/99197 | A Simple Stochastic Dynamic Billiard | To avoid trivialities, assume that the initial position of the particle is in the interior of the rectangle.
Then the limit is actually $0$. This is trivial in the case when the $x$-coordinate of the initial velocity is $0$.
Otherwise, in a nonrandom finite time, depending only on the initial position and velocity,... | 3 | https://mathoverflow.net/users/36721 | 285745 | 126,212 |
https://mathoverflow.net/questions/285739 | 3 | Edited: Given integer $N\geq 0$, let $$I(N):=\Bigl\{(n\_k)\_{k\geq 1}\in {\mathbb N}^\infty \,:\, n\_k\geq 0, \sum\_{k\geq 1}kn\_k = N \Bigr\}$$ be the set of all decompositions of $N$ as a linear combination of $(1, 2, 3,\ldots)$ with nonnegative integer coefficients. Then $$\sum\_{(n\_k)\in I(N)}\prod\_{k\geq 1} \fra... | https://mathoverflow.net/users/91419 | On decompositions of integers as a linear combination of $(1, 2, 3,\ldots)$ | If you multiply both sides by $N!$, then you are just enumerating permutations of $1,2,\dots,N$ according to their cycle type.
| 7 | https://mathoverflow.net/users/2807 | 285747 | 126,213 |
https://mathoverflow.net/questions/285555 | 8 | Let $R$ be a commutative ring with unity. The Bass-Papp theorem states that any countable direct sum of injective $R$-modules is injective iff $R$ is Noetherian . Chase's theorem states that any direct product of projective $R$-modules is projective iff $R$ is Artinian . My question is : Is any characterization for com... | https://mathoverflow.net/users/nan | When is countable direct-product of projective modules again projective ? | Under the assumption that a *countable direct product of modules over $R$* means a direct product of countably many modules over $R$, I answer OP's question when $R$ is **Noetherian**. In addition, I outline a remark for $R$ an arbitrary **countable** ring with identity.
>
>
> >
> > **Claim 1.** Let $R$ be a com... | 7 | https://mathoverflow.net/users/84349 | 285753 | 126,216 |
https://mathoverflow.net/questions/285677 | 3 | Consider the quadratic form $5x^2+6y^2$. This has Conway Sloan $2$-adic symbol $[1^{-1}2^{-1}]\_0$. After a sign walk from $1$ to $2$ the symbol becomes $[1^{+1}2^{+1}]\_4$. However, there doesn't exist a form with this symbol as the oddities of each summand are limited to $1$ and $7$ and there is no solution modulo $8... | https://mathoverflow.net/users/6084 | Normalizing trains without sign walking in compartments | The answer is yes. There is a sign walk entirely within each compartment that removes the minus signs and produces the same normalized symbol for each train. Therefore the compartments of the two different trains are isomorphic as they have the same normalized symbol, even if that normalized symbol isn't the symbol of ... | 0 | https://mathoverflow.net/users/6084 | 285758 | 126,220 |
https://mathoverflow.net/questions/285763 | 4 | Suppose I have an (integer valued, but it probably does not matter) *monotonically increasing* function $f: \mathbb{N} \to \mathbb{R},$ which satisfies
$$\sum\_{i=1}^n f(i) < f(2 n).$$ What can we say about the speed of growth of this function?
| https://mathoverflow.net/users/11142 | lower bounds for growth of a sequence | Here is the trivial part where I'm pretty sure that the way is close to the most elegant requested:
$f(n)=e^{c\log^2 n}$ with $c=\frac 1{2\log 2}$ satisfies the inequality.
Indeed,
$$
f(2n)=e^{c\log^2 n+2c\log 2\log n+c\log^22}\ge f(n)n\ge \sum\_{i=1}^nf(i)
$$
Here is the crude lower bound $f(n)\ge \delta f(1)e^{c\l... | 10 | https://mathoverflow.net/users/1131 | 285764 | 126,222 |
https://mathoverflow.net/questions/285777 | 19 | Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a *fixed point* if $f(x) = x$.
The space $(X,\tau)$ is said to have the *anti fixed point property (AFPP)* if the only continuous maps $f:X\to X$ with fixed points are the identity map $\text{id}\_X:X\to X$, and the constant maps.
... | https://mathoverflow.net/users/8628 | "Anti" fixed point property | Probably, the discrete $\{0,1\}$ is not the counterexample Dominic van der Zypen expected to see :)
A more elaborate CH-example of a AFPP but not strongly rigid space was constructed by van Mill:
**Theorem 4.1 ([van Mill, 1983](http://www.ams.org/journals/tran/1983-280-02/S0002-9947-1983-0716833-2/S0002-9947-1983-... | 20 | https://mathoverflow.net/users/61536 | 285779 | 126,225 |
https://mathoverflow.net/questions/285773 | 9 | Let $A$ be an Artin local ring and let $f:A \to B$ a local ring homomorphism to a Noetherian local one dimensional $A$-algebra $B$.
>
> If $B$ is Cohen--Macaulay and the localization of $B$ at any minimal prime is flat over $A$, then is $B$ flat over $A$?
>
>
>
As far as I can see, this doesn't follow from a... | https://mathoverflow.net/users/519 | Flatness and Cohen-Macaulay rings | That is not true. Let $k$ be a field. A module over the ring of dual numbers, $A=k[\epsilon]/\langle \epsilon^2 \rangle,$ is equivalent to a $k$-vector space $B$ with a square-zero $k$-linear self-map $L\_\epsilon:B\to B.$ For every integer $q\geq 1$, the Tor module equals, $$\text{Tor}^A\_{q\geq 1}(A/\langle \overline... | 5 | https://mathoverflow.net/users/13265 | 285784 | 126,226 |
https://mathoverflow.net/questions/285759 | 6 | Is there any diffeomorphism $x:\mathbb{R}^n\to\text{Im }x=B\_1\left(0\right)\subset\mathbb{R}^n$ such that
1. $x$ is an orthogonal chart, i.e., the coordinate vector fields $X\_i=\partial x/\partial u\_i$ satisfy
$X\_i\left(u\right)\cdot X\_j\left(u\right)=0$ for all $1\leq i\neq j\leq n$ at every point $u=\left(u\... | https://mathoverflow.net/users/117091 | Global orthogonal coordinates on the open unit ball | This problem is just the classical problem of finding global Tchebychev coordinates on hyperbolic $n$-space. By Hilbert's Theorem, this is impossible when $n=2$. The problem remains open in higher dimensions, despite years of work.
Here is how one can see the reformulation: Let $J$ be the Jacobian of $x$ with respect... | 12 | https://mathoverflow.net/users/13972 | 285786 | 126,227 |
https://mathoverflow.net/questions/285795 | 21 | *This is one of those **"what is an $X$?"** questions so let me apologize in advance*.
By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so called "Kdv hierarchy" which is apparently related to enumerative geometry of curves in some ways which are a total ... | https://mathoverflow.net/users/22810 | What is an "integrable hierarchy"? (to a mathematician) | An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively.
[For the definition of a commuting flow, see for example the first part of this [MO question.](https://mathoverflow.net/que... | 16 | https://mathoverflow.net/users/11260 | 285797 | 126,230 |
https://mathoverflow.net/questions/285673 | 6 | Given the Cayley graph of a group $G$ (for some fixed generating set $S$), consider the set $J\_S$ of all the elements which lie on some *infinite* geodesic ray starting at the identity element of $G$, $e\_G$.
Let $d$ denotes the distance in the Cayley graph
A simple example where $J\_S \neq G$ is $G = \mathbb{Z} \ti... | https://mathoverflow.net/users/18974 | Distance to a geodesic ray | Let $G=\mathbb{Z}\times\mathbb{Z}/7\mathbb{Z}$. Let $S$ consist of $(1,1),(1,0),(0,1)$ and their inverses. Let $x=(0,3)$.
It seems clear that the neighbors of $x$ are contained in $\{-1,0,1\}\times\{2,3,4\}$, since each coordinate of a neighbor of $x$ will differ by $0,1$, or $-1$ from the corresponding coordinate of... | 4 | https://mathoverflow.net/users/36155 | 285817 | 126,237 |
https://mathoverflow.net/questions/285755 | 5 | We define $STR(n)$ to be the longest sequence of strings with $n$ symbols such that the $k$th string has at most k symbols, the symbols of the string are taken from an alphabet consisting of $n$ characters, and no string is a subsequence of a later one.
For example, $STR(1)=2$, because the longest sequence is "A", ""... | https://mathoverflow.net/users/65915 | Growth rate of longest sequence of strings where no string is a subsequence of a later one | I suppose it's a good idea to turn my comment into an answer.
The function $STR$ is basically the function $F$ defined by Friedman in [this paper](https://u.osu.edu/friedman.8/files/2014/01/LongFinSeq98-2f0wmq3.pdf) (more precisely, it's easy to show $STR(k)=F(k-1)+1$). Friedman pinpoints the growth rate of this func... | 8 | https://mathoverflow.net/users/30186 | 285821 | 126,239 |
https://mathoverflow.net/questions/285815 | 1 | Do you happen to know a reference for exercise 2.1.18 from page 20 of Zeitouni's et al textbook:
<http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf>
?
| https://mathoverflow.net/users/13904 | Searching for a reference on Wishart matrices | This exercise develops the proof of the Marchenko-Pastur distribution. Worked out proofs along similar lines can be found at various places, some pointers follow. None of these take precisely the same steps as in the exercise, but I presume once you have understood one of these worked out proofs you're done:
1. <http... | 2 | https://mathoverflow.net/users/11260 | 285827 | 126,243 |
https://mathoverflow.net/questions/285783 | 3 | Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$, $c>0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t),z(t))$ be the position at time $t$ of a point which moves in the parallelepiped $R=\{ (x,y,z) \in \mathbb{R}^3: 0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c \}$ with velocity of constan... | https://mathoverflow.net/users/99197 | A Really Simple Stochastic Dynamic Billiard | This answer is a modification of my answer at [2D billiard](https://mathoverflow.net/questions/285783/a-really-simple-stochastic-dynamic-billiard).
To avoid trivialities, assume that the initial position of the particle is in the interior of the box.
Also, at this point, assume that the first time the particle hits... | 2 | https://mathoverflow.net/users/36721 | 285840 | 126,248 |
https://mathoverflow.net/questions/285844 | 0 | I have a bipartite graph, including V1 and V2 vertices, and I would like to convert it to a canonical form. One simple method is converting this graph to a general graph by expanding its adjacency matrix such that the number of rows and columns in the result matrix is equal to |V1| + |V2| and then obtain its canonical ... | https://mathoverflow.net/users/117128 | Canonical form for a bipartite graph | Since you seem to be asking about how to compute it, I'll answer that. With nauty, bliss, Traces, etc, you can specify vertex colours then the canonical form won't mix the colours up. Just use one colour for V1 and one colour for V2.
| 0 | https://mathoverflow.net/users/9025 | 285848 | 126,249 |
https://mathoverflow.net/questions/285855 | -1 | Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int\_{0}^{\infty}f'g dx\geq \int\_{0}^{\infty}g'fdx$.
| https://mathoverflow.net/users/42326 | An elementary question about integration by parts! | Say, if $f/g$ is nondecreasing (plus something to take care of convergence), but that's way too strong.
| 1 | https://mathoverflow.net/users/113161 | 285859 | 126,256 |
https://mathoverflow.net/questions/285774 | 3 | A topological space has *calibre* $\aleph\_1$ if for every uncountable sequence $\langle U\_\alpha\mid\alpha\lt\aleph\_1\rangle$ of nonempty open sets $U\_\alpha\subset X$, there is an uncountable subfamily $\Lambda\subset\aleph\_1$ with $\bigcap\_{\alpha\in\Lambda}U\_\alpha\neq\emptyset$.
Is there a calibre $\aleph... | https://mathoverflow.net/users/39873 | Is there a calibre $\aleph_1$ Moore space which is not separable | Frank Tall proved that under $MA+\neg CH$ there is a regular first-countable space with caliber $\omega\_1$ which is not separable. (see *Tall, Franklin D.*, First countable spaces with caliber $\aleph\_1$ may or may not be separable, Set-theor. Topol., Vol. dedic. to M.K. Moore, 353-358 (1977). [ZBL0382.54003](https:/... | 2 | https://mathoverflow.net/users/11647 | 285865 | 126,257 |
https://mathoverflow.net/questions/285862 | 2 | 1. Is there know set of operations for which uncomputable functions are, let's name it down-unclosed? I mean a set of operations which takes two ( or more) uncomputable functions and return computable function? It is obvious it depends on certain property, like rates of growth, so are this hypothetical internal operati... | https://mathoverflow.net/users/3811 | Internal operations on uncomputable functions | The jump inversion theorem (Friedburg 1957) shows that any Turing degree $d$ above the halting problem is the jump of another degree $d=b'$, which means that $d$ is Turing equivalent to the halting problem relative to $b$.
Friedburg's original construction can be viewed as a specific map from any sufficiently non-co... | 4 | https://mathoverflow.net/users/1946 | 285878 | 126,259 |
https://mathoverflow.net/questions/285874 | 2 | Is it true that there exist $2^{\aleph\_0}$ pairwise non-isomorphic torsion-free countable groups?
| https://mathoverflow.net/users/8628 | Number of countable torsion-free groups | Extended cw answer based on Jason Starr's comment.
>
> The additive group of rationals admits $2^{\aleph\_0}$ non-isomorphic subgroups.
>
>
>
Denote by $P\subset \mathbb{Z}\_{>0}$ the set of positive, integer primes. This is a countably infinite set by Euclid's proof of the infinitude of primes. The set $\math... | 10 | https://mathoverflow.net/users/13265 | 285884 | 126,261 |
https://mathoverflow.net/questions/285885 | 3 | Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically.
$$f(n) = \frac{2}{\pi} \int\_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \cdots \cdot \frac{\sin(x/(2n+1))}{x/(2n+1)} dx$$
Any help would be appreciated.
| https://mathoverflow.net/users/117145 | Evaluation of an interesting Integral | The conjecture $f(n)=1$ is only correct for $n\leq 55$, see H. Schmid, [Two curious integrals and a graphic proof](http://www.schmid-werren.ch/hanspeter/publications/2014elemath.pdf). For $n=56$ an analytical calculation using the Poisson summation formula gives
$$f(56)= 1 − 1.484870809 \cdot 10^{−138},$$
see [More rem... | 10 | https://mathoverflow.net/users/11260 | 285886 | 126,262 |
https://mathoverflow.net/questions/285557 | 28 | The [Golomb space](https://dml.cz/bitstream/handle/10338.dmlcz/700933/Toposym_01-1961-1_41.pdf) $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N\_0$ with relatively prime $a,b$ and $\mathbb N\_0=\{0\}\cup\mathbb N$. It is... | https://mathoverflow.net/users/61536 | Is the Golomb countable connected space topologically rigid? | [Edit, Dec 6, 2019] I have a pleasure to inform that this problem was finally resolved in affirmative by [T.Banakh, D.Spirito and S.Turek](https://arxiv.org/abs/1912.01994) who proved the following
>
> **Theorem.** *The Golomb space is topologically rigid.*
>
>
>
Edit: now published: The Golomb space is topolo... | 21 | https://mathoverflow.net/users/61536 | 285890 | 126,264 |
https://mathoverflow.net/questions/285857 | 6 | Is there an equivalence relation $R$ on $[0,1]\cap \mathbb{Q}$ such that $([0,1]\cap \mathbb{Q})/R$ is connected, Hausdorff, and has more than $1$ point?
| https://mathoverflow.net/users/8628 | Does $[0,1]\cap \mathbb{Q}$ have a connected $T_2$ quotient? | Yes, there exists such a relation on $\mathbb Q$.
Just use the fact that the rational projective space $\mathbb QP^\infty$ from (the answer to) [this question](https://mathoverflow.net/questions/46986/countable-connected-hausdorff-space/285556#285556) is a countable, Hausdorff, connected (and even topologically homog... | 11 | https://mathoverflow.net/users/61536 | 285892 | 126,265 |
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