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https://mathoverflow.net/questions/224519 | 9 | Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
**Question:** Is it true that for the 26 sporadic finite simple groups, all real elements of odd order are strongly real, apart from elements i... | https://mathoverflow.net/users/20764 | Strongly real elements of odd order in sporadic finite simple groups | If my coding is correct, then the answer to your question is **Yes**: All real elements of odd order in the sporadic simple groups are strongly real, with the exception of 3a, 5a in McL. With GAP, it takes only about a second to check the tables.
The following GAP function returns the class position of all strongly ... | 11 | https://mathoverflow.net/users/10266 | 224604 | 105,055 |
https://mathoverflow.net/questions/36174 | 2 | Suppose we have a vector of probabilities $\mathbf{p}=(p\_1,...,p\_n)$, where $p\_i>0$ for $i=1,...n$ and $\sum p\_i=1$. Define new vector $\mathbf{r}=(r\_1,...,r\_{n-1})$ in a following way:
$r\_i=\log(p\_i/p\_n)$
This defines the transformation $T:(0,1)^n\to\mathbb{R}^{n-1}$, $\mathbf{r}=T\mathbf{p}$. This transf... | https://mathoverflow.net/users/8652 | Multinomial transformation for matrices | At the suggestion of the original poster, I am summarizing an alternate answer that has a few strengths relative to my original answer. It is related to the question and answer at [this MathOverflow Question](https://mathoverflow.net/questions/156983/injectivity-of-matrix-fingerprint/157174).
For any matrix $A$, def... | 2 | https://mathoverflow.net/users/8938 | 224611 | 105,056 |
https://mathoverflow.net/questions/224598 | 3 | Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega\_1$ for practical purposes.
Let $\cal U$ be an ultrafilter on $\omega\_1$ in $V$ which is non-principal and even uniform. Let $\Bbb P$ be a forcing such that:
1. $\Bbb P$ does not add bounded subsets to $\kappa$, but its generic de... | https://mathoverflow.net/users/7206 | When can you canonically extend an ultrafilter after forcing? | Here is a necessary and sufficient condition:
**Theorem.** If $U$ is an ultrafilter in $V$, then the following
are equivalent:
1. $U\cup\{G\}$ is an ultrafilter base.
2. $G$ is not disjoint from any element of $U$ and $U$ decides every subset of $G$ in $V[G]$, meaning that for every $A\subset G$ in $V[G]$
there is ... | 7 | https://mathoverflow.net/users/1946 | 224614 | 105,057 |
https://mathoverflow.net/questions/219794 | 3 | In combinatorial game theory: The birthday of a game is defined recursively as 1 plus the maximal birthday of its options, with the zero game having birthday 0.
Suppose we define the *quasi-birthday* of a game to be 1 + the minimal birthday of its options, with the zero game having birthday 0. Is anything known about... | https://mathoverflow.net/users/75293 | Minimal Birthdays | Since your definition of birthday just said "1+…", I assume you're not talking about transfinite games. In this context, the birthday is the longest a game could go on (in number of moves), and the quasi-birthday is the shortest a game could go on.
It's notationally convenient to treat a short impartial game as the ... | 3 | https://mathoverflow.net/users/28209 | 224620 | 105,058 |
https://mathoverflow.net/questions/224612 | 0 | Is following true?
For every given $c>0$ there is an $n\_c>0$ such that for every $n>n\_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log n)^c}\approx\alpha<\beta\approx2\alpha$ such that
for every $x\in[\alpha,\beta]$ and $y\in[\alpha,\beta]$ we have
$$ax+by$$ always represe... | https://mathoverflow.net/users/nan | Linear forms that avoid numbers with lot of factors | There is no such $c>0$. Indeed, assume that $n$ is large and $a,b,\alpha,\beta$ have the given property. Put $k:=\log n/\log\log n$. The number of prime factors of $a$ is at most $k+o(k)$, hence up to $(3/2)\log n$ there are at least $k/2-o(k)$ primes that don't divide $a$. So pick any $\lfloor k/3\rfloor$ such primes,... | 2 | https://mathoverflow.net/users/11919 | 224624 | 105,060 |
https://mathoverflow.net/questions/224576 | 0 | Suppose we want to minimize a linear objective inside an ellipsoid that is,
$\min \_x l^Tx$
such that $(x - \mu)^TA(x - \mu) \leq \beta ^2$.
Here, A is PSD and $\mu$ is a fixed vector. Can this be written as a SDP ?
| https://mathoverflow.net/users/51716 | Convex Optimization in an Ellipsoid | There is a closed-form expression for that value.
Indeed, a straight-forward computation yields
\begin{equation}
\begin{split}
\min\_{\langle A(x-\mu),x-\mu\rangle \le \beta^2} \langle l, x\rangle &= \min\_{\|v\|^2 \le \beta^2}\langle l, \mu + A^{-1/2}v\rangle = \langle l, \mu \rangle + \min\_{\|v\| \le \beta}\langle A... | 2 | https://mathoverflow.net/users/78539 | 224633 | 105,063 |
https://mathoverflow.net/questions/224619 | 8 | Suppose that a linear system of inequalities $Ax \le b$, where $A\in Z^{m\times n}$ and $b\in Z^m$ have integral coefficients, has an infinite number of integral solutions $x$.
Can one conclude that there is a ray $\{x+tp\mid t\ge 0\}$ containing infinitely many integral solutions? (If one drops the integrality cond... | https://mathoverflow.net/users/56920 | On linear integer inequalities with infinitely many solutions | Let the constraints be numbered 1 to $m$. Let the $i$th constraint be $a^{(i)}\cdot x\le b$. Let $S$ be the set of solutions. Inductively refine the constraints as follows: either there are $\limsup\_{x\in S, x\to\infty} a^{(1)}\cdot x=-\infty$ or not. If $\limsup\_{x\in S,x\to\infty} a^{(1)}\cdot x=c^{(1)}>-\infty$, t... | 2 | https://mathoverflow.net/users/11054 | 224646 | 105,065 |
https://mathoverflow.net/questions/224475 | 1 | Is there anything known about uniqueness of classical solutions to
$$
\partial\_t u -u\Delta u=0\quad u(0,\cdot)=1
$$
on smooth domains $[0,T]\times D$ without boundary conditions? I know that $u(0,\cdot)=0$ implies that $u=0$ even without boundary conditions.
However, does $u(0,\cdot)=1$ imply that $u=1$? It is not so... | https://mathoverflow.net/users/75786 | Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$ | I found that if one has $\partial\_n u=0$ or $u=1$ on $[0,T]\times \partial D$ it is relatively easy to prove that $u=1$ is the only regular solution.
This can be seen by observing that $e:=u-1$ satisfies
$$
\partial\_t e -e\Delta e -\Delta e =0.
$$
Multiplication by $e$, the identity $\Delta e^2 = 2e\Delta e + |\nabla... | 0 | https://mathoverflow.net/users/75786 | 224649 | 105,066 |
https://mathoverflow.net/questions/224634 | 11 | Let $K$ be a (commutative) field.
We can define the free $K$-algebra of polynomials in non commutative variables $x\_1, \cdots, x\_n$. It is usually denoted by $K\langle x\_1, \cdots, x\_n \rangle$.
Fix a non commutative polynomial $P \in K\langle x\_1, \cdots, x\_n \rangle$.
For every natural number $m$ and every ch... | https://mathoverflow.net/users/83320 | non commutative polynomial which is zero for all matrix evaluation | This is an algebraic elaboration on Emil's answer.
Let $A = K \langle x\_1,\ldots,x\_n \rangle$ and let $\hat{A} = K \langle\langle x\_1,\ldots, x\_n \rangle \rangle$. Since $A$ is a subring of $\hat{A}$, a positive answer for $\hat{A}$ implies one for $A$.
Now for each $d \in \mathbb{N}$, let $\hat{A}\_{d}$ be the... | 11 | https://mathoverflow.net/users/6827 | 224659 | 105,070 |
https://mathoverflow.net/questions/102874 | 8 | Let $n\in\mathbb N$. Let $k$ be a commutative ring in which $1,2,3,\ldots$ are invertible. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x\_1$, $x\_2$, ..., $x\_n$ over $k$. (The multiplication in this algebra is the composition of differential operators.)
Let $M:k\left[x\... | https://mathoverflow.net/users/2530 | Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$? | Yes, the multiplication $\boxdot$ on $U\left(\mathfrak{gl}\_n\right)$ exists. Moreover, Alexander Chervov's conjecture is true: There is a commutative multiplication $\boxdot$ on $U\left(\mathfrak{gl}\_n\right)$ such that, for every $m \in \mathbb{N}$, the $k$-algebra homomorphism
$$
U\left(\mathfrak{gl}\_n\right) \to ... | 3 | https://mathoverflow.net/users/2530 | 224663 | 105,071 |
https://mathoverflow.net/questions/224662 | 1 | I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here.
Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial\_tu-\Delta u$. Let $f\in C^{0,0,\alpha}((0,T)\times M,\mathbb{R})$, $u\_0\in C^{2,\alpha}(M,\mathbb{R})$ and let $u\in C^{1,2,\alpha}((0,T)\time... | https://mathoverflow.net/users/83341 | Schauder estimate for the heat equation on compact manifolds | The constant $C(T)$ depends on $T$, consider $f=1$, $u\_0=0$ and the solution $u=t$. But it is bounded for $T\to0$. The rhs $f\in C^{0,\alpha}([0,T]\times M)$ can be continued to $\tilde f\in C^{0,0,\alpha}([0,1]\times M)$, $||\tilde f||\_{C^{0,0,\alpha}((0,1)\times M)}= |f||\_{C^{0,0,\alpha}((0,T)\times M)}$. Denote $... | 1 | https://mathoverflow.net/users/14551 | 224667 | 105,074 |
https://mathoverflow.net/questions/224605 | 7 | Let $X=(X\_1,\dots,X\_n)$ and $Y=(Y\_1,\dots,Y\_n)$ be centered Gaussian vectors
with variance matrix $\Gamma\_X$ and $\Gamma\_Y$. We assume that the matrix
$\Gamma\_Y-\Gamma\_X$ is positive definite. Is it possible to prove that
$$ \forall x \ge 0, \mathbb{P}(\max \lbrace |Y\_1|,\dots,|Y\_n| \rbrace \ge x) \ge \mat... | https://mathoverflow.net/users/83300 | Inequality for the maximum of Gaussian variables | I think that what you are writing is precisely Anderson's correlation inequality, see "T. W. Anderson. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc., 6(2):170–176, 1955."
| 6 | https://mathoverflow.net/users/54263 | 224680 | 105,076 |
https://mathoverflow.net/questions/214354 | 18 | What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?
I can think of more than one natural way to generalize the notion of a rational number, but "ratio of omnific integers" is not one of them, since every real number is a r... | https://mathoverflow.net/users/3621 | Nice sign-expansions of special surreal numbers | (This answer has been substantially edited since its original version.)
### 1.
The surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) can be rewritten in a tidy form, which shows that it *does* lie in the field generated by $\omega$.
As you read o... | 11 | https://mathoverflow.net/users/28209 | 224694 | 105,083 |
https://mathoverflow.net/questions/224638 | 6 | This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an elliptic curve $E / \mathbb{Q}$ given by an equation of the form $y^{2} = x(x - 1)(x - \lambda)$ with $\lambda \in \mathbb{Q}$... | https://mathoverflow.net/users/24757 | realizing uniform boundedness of Galois representations associated to elliptic curves | $\def\Q{\mathbf{Q}}$
$\def\Qbar{\overline{\Q}}$
$\def\Z{\mathbf{Z}}$
$\def\GL{\mathrm{GL}}$
There are many confusions in your post, let me point out just a few
It is dangerous to fix embeddings of $\Qbar$ into $\Qbar\_p$ (equivalently, to fix decomposition groups $D\_p$ inside the Galois group $G\_{\Q}$ for all $p$... | 9 | https://mathoverflow.net/users/83361 | 224696 | 105,084 |
https://mathoverflow.net/questions/224682 | 4 | Suppose $(a\_n)\_{n=1}^\infty$ and $(b\_n)\_{n=1}^\infty$ are two decreasing sequences of positive numbers such that
$a\_1<b\_1$ and
$$\sum\_{k=1}^\infty a^p\_k\leq \sum\_{k=1}^\infty b^p\_k<\infty$$ for $p=2,3,\cdots.$
Does it follow that
$$\sum\_{k=1}^\infty a^p\_k\leq \sum\_{k=1}^\infty b^p\_k$$
for all real $p... | https://mathoverflow.net/users/48438 | Extending inequality for $\ell^p$ from integer $p$ to real $p$ | All we have to do is solve some equations. If I did it right, then:
Let $q \approx 2.45$ satisfy $204q^4-580q^3+867q^2-4028=0$. Let
$$
r = \frac{578}{4323}q^3+\frac{2465}{12969}q^2-\frac{8056}{12969}q+\frac{41905}{2593}\approx 3.20 .
$$
Let
$$
a\_1 :=3 ,\qquad a\_2 := q,\qquad a\_3 := \frac{1}{2},
$$
$$
b\_1 := r,... | 4 | https://mathoverflow.net/users/454 | 224701 | 105,086 |
https://mathoverflow.net/questions/224695 | 4 | I have seen in the literature that Irreducible outer automorphisms of a free group $F\_n$ are "generic".
I would like to ask that if for example : it is true that for any generating set $X$ of $Out(F\_n)$ if we take the $k$- closed ball in the Cayley graph (with respect to $X$) $B(k)$ and we denote by $R$ the set of ... | https://mathoverflow.net/users/42001 | Genericity of irreducible automorphisms of free groups | No, this is not known. The results for random elements (which say that a random autormorphism is irreducible with irreducible powers (iwip)) sadly do not imply this.
| 4 | https://mathoverflow.net/users/11142 | 224703 | 105,087 |
https://mathoverflow.net/questions/220887 | 10 | The simplest form of Finsler metric is:
$ds (X, dx)=\sqrt[4]{g\_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g\_{\alpha, \beta, \gamma, \delta}$ is a fourth degree polynomial that smoothly depends on $X$.
My question is: is it always true that $s^4(X)$, the fourth power of ... | https://mathoverflow.net/users/56819 | Smoothness of the fourth power of the geodesic distance in a Finsler geometry | It turns out that the answer is 'No, the fourth power of the geodesic distance from a point $p$ in a Finsler space $M$ whose norm, raised to the fourth power, is a smooth convex quartic form is *not* generally a smooth function on a neighborhood of $p$'.
While the desired smoothness does hold in the Minkowski case (... | 9 | https://mathoverflow.net/users/13972 | 224704 | 105,088 |
https://mathoverflow.net/questions/224716 | 2 | Okay the question is really soft. But I am wondering about the relationship between one's productivity (namely quality of papers, number of papers published) and prizes received.
So here is my question:
>
> What is your opinion on the following statement: Mathematician tends to have their productivity declined af... | https://mathoverflow.net/users/76219 | How does your productivity change after receiving prizes? | [Prizes and Productivity: How Winning the Fields Medal Affects Scientific Output](http://www3.nd.edu/~tjohns20/RePEc/deendus/wpaper/022_Fields.pdf) (2013)
>
> Knowledge generation is key to economic growth, and scientific prizes
> are designed to encourage it. But how does winning a prestigious prize
> affect fut... | 11 | https://mathoverflow.net/users/11260 | 224717 | 105,090 |
https://mathoverflow.net/questions/224725 | 28 | I would like to see a simple example which shows how mathematical notation were evolve in time and space.
Say, consider the formula
$$(x+2)^2=x^2+4{\cdot}x+4.$$
If I understand correctly, Franciscus Vieta would write something like this. (Feel free to correct me.)
$\overline{N+2}$ quadr. æqualia $Q+N\,4+4$.
-------... | https://mathoverflow.net/users/1441 | History of Mathematical Notation | For a quite extensive overview with many examples, you might want to check out [The origins and development of mathematical notation](http://math.unipa.it/~grim/mathnotation.pdf).
I also enjoyed reading Stephen Wolfram's take on [Mathematical Notation – past and future,](http://www.stephenwolfram.com/publications/mathe... | 29 | https://mathoverflow.net/users/11260 | 224730 | 105,092 |
https://mathoverflow.net/questions/224729 | 1 | This was also posted in [stackexchange](https://math.stackexchange.com/q/1549745/294077). However, I have no idea how difficult it is. All hints or references are appreciated!
Consider a set $S$ of $n$ red balls and $m$ blue balls. It is well known that the number of partitions of this set is the Bell number $B\_{n+m... | https://mathoverflow.net/users/83377 | The number of good partitions | There is no simple closed form expression for $B\_{m+n}$, so why should
we expect one in this situation? However, an explicit generating
function can be given as a routine application of the exponential
formula. Let $X$ be the set of all pairs $(n,m)$ where we are allowed
to have a block with $n$ red balls and $m$ blue... | 4 | https://mathoverflow.net/users/2807 | 224746 | 105,100 |
https://mathoverflow.net/questions/224741 | 4 | Say I have a set of oriented points at locations $\vec{v\_i}$ with each some direction $\vec{n\_i}$, in practice they represent a surface with its normals. How would I calculate the divergence of this vector field? Standard finite-difference methods would obviously not work because there is no grid.
Can I perhaps as... | https://mathoverflow.net/users/53035 | Numerically calculating the divergence of a set of oriented points | Interpolation of vector fields (a notoriously tricky process) to determine the divergence can be avoided by using a line integral definition of the divergence via Stokes' theorem. You can find a discussion of this method in [On the interpolation of a vector field](http://www.spc.noaa.gov/publications/schaefer/interpol.... | 2 | https://mathoverflow.net/users/11260 | 224750 | 105,103 |
https://mathoverflow.net/questions/224768 | 11 | For $i=1,2$, let $j\_i$ denote the inclusion of $S^n$ into the product $S^n \times S^n$ as the $i^{\text{th}}$ factor. I would very much like to know the answer to the following question, which seems to be very basic:
**For which $(a\_1,a\_2) \in \mathbb Z^2$ is there a map $$m \colon S^n \times S^n \to S^n,$$ such ... | https://mathoverflow.net/users/14233 | Product-like structures on spheres | Your condition determines the map $a\_1 \vee a\_2 : S^n \vee S^n \to S^n$ on the $n$-skeleton, so the question is when does this extend over the $2n$-cell of $S^n \times S^n$. The $2n$-cell is by definition attached along the universal Whitehead product $[\iota\_1, \iota\_2 ] \in \pi\_{2n-1}(S^n \vee S^n)$, so the ques... | 18 | https://mathoverflow.net/users/318 | 224770 | 105,112 |
https://mathoverflow.net/questions/224775 | 5 | Consider the effect of $f(x)=\frac12(x-x^{-1})$ on the residues mod $p$ (plus $\infty$) of a Mersenne prime $p$. You get the following tree (example $p=7$):
$$
\begin{array}{ccccccc}
4\\
&\searrow\\
&&1\\
&\nearrow&&\searrow\\
5\\
&&&&0&\rightarrow&\infty\\
2\\
&\searrow&&\nearrow\\
&&6\\
&\nearrow\\
3\\
\end{array}... | https://mathoverflow.net/users/11504 | A property of Mersenne primes | The critical points of $f$ are at $x=\pm i$, so we try a projective
(a.k.a. fractional linear) change of variable that puts these
critical points at $0$ and $\infty$, namely $x = \alpha(y)$ where
$\alpha(Y) = i(Y+1) \, / \, (Y-1)$, $\alpha^{-1}(X) = (X+i) \, / \, (X-i)$,
and find that $f(\alpha(y)) = \alpha(y^2)$.
Ther... | 10 | https://mathoverflow.net/users/14830 | 224780 | 105,117 |
https://mathoverflow.net/questions/224789 | 8 | Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values in $B$,
$\qquad \qquad \| x=y \|, \qquad \| x \in y \|$
that safisfy the following:
$ (i)\ \ \ \| x=x \| = 1 \\
(ii)... | https://mathoverflow.net/users/nan | Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$? | One way to show this is by first noting that (i)-(iv) ensure the atomic instances of the identity axioms:
$x = y \to [\phi(x) \leftrightarrow \phi(y)]$
get value 1. Then observe, by a simple induction on the complexity of $\phi$, that the non-atomic instances follow from the atomic instances in first-order logic. S... | 3 | https://mathoverflow.net/users/17968 | 224791 | 105,123 |
https://mathoverflow.net/questions/224820 | -2 | Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative $\lim\_{\epsilon->0}\frac{h(x+\epsilon)-2h(x)+h(x-\epsilon)}{\epsilon^2}$ should exist?
| https://mathoverflow.net/users/82978 | Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? | Suppose $h(x)=x^2\sin(1/x)$ for $x\neq 0$, $h(0)=0$. Then it's derivative exists everywhere and is finite, but $$\frac{h(0+\varepsilon)-2h(0)+h(0-\varepsilon)}{\varepsilon^2}=\frac{2\varepsilon^2\sin(\frac{1}{\varepsilon})}{\varepsilon^2}=2\sin(\frac{1}{\varepsilon})$$
has no limit as $\varepsilon\rightarrow 0$.
| 1 | https://mathoverflow.net/users/30186 | 224821 | 105,130 |
https://mathoverflow.net/questions/224809 | 7 | Let $X$ be a complex manifold (not necessarily Kahler or even compact).
For the first Chern class $c\_1(E)\in H^2(X,Z)$ of a holomorphic vector
bundle $E\to X$ there is an obvious lift to $H^1(X, O^\*)$, namely the determinant.
Question: Is there a natural (in some sense) lift of the second Chern class to $H^3(X, ... | https://mathoverflow.net/users/9833 | A lift of the second Chern class | There are various instances of the first Chern class:
* In Betti cohomology $c\_1\in H^2(X, \mathbf{Z})$,
* In Dolbeault cohomology $c\_1\in H^1(X, \Omega^1)$,
* In de Rham cohomology $c\_1\in H^2\_{dR}(X)$.
These can be combined to the class in Deligne cohomology $c^D\_1\in H^2\_D(X, \mathbf{Z}(1))\cong H^1(X, \ma... | 5 | https://mathoverflow.net/users/18512 | 224822 | 105,131 |
https://mathoverflow.net/questions/224773 | 8 | Suppose I have a polynomial function $f\in \mathbb{Z}[x\_1, \dotsc, x\_k],$ such that whenever $r\_1, \dotsc, r\_k$ are roots of a monic polynomial of degree $k$ with integer coefficients, we have $f(r\_1, \dotsc, r\_k) \in \mathbb{Z}.$ Is it true that $f$ is a symmetric function of its arguments?
Note: this was aske... | https://mathoverflow.net/users/11142 | polynomials and symmetric functions | Consider the minimal polynomial $P$ of $f$ over the fixed field
$\mathbb Q(\sigma\_1, \ldots, \sigma\_k) = \mathbb Q(x\_1, \ldots, x\_k)^{S\_k}$
(where the $\sigma\_j$ are the elementary symmetric polynomials), so
$P$ is irreducible and $P(f) = 0$. Since $f$ is a polynomial,
the coefficients of $P$ will actually be pol... | 9 | https://mathoverflow.net/users/21146 | 224823 | 105,132 |
https://mathoverflow.net/questions/224826 | 13 | There are *Counterexamples in Analysis* and *Counterexamples in Topology*. Is there any similar book for commutative algebra? I want to see some more (counter)examples for Atiyah and MacDonald's book. Let us say "zoo" of rings and modules.
| https://mathoverflow.net/users/83409 | Examples and Counterexamples in Commutative Algebra | The best I can think of is Harry Clayton Hutchins' "Examples of Commutative Rings" published by UChicago Press (he was a student of Kaplansky). The second part of the book is just a long list of rings satisfying strange properties ("this, this and this, but not that"). The one difference with counterexamples in top is ... | 10 | https://mathoverflow.net/users/70501 | 224827 | 105,133 |
https://mathoverflow.net/questions/224699 | 6 | Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends by derivations to a representation of $\mathfrak{g}$ in $S(\mathfrak{g})$. We identify $S(\mathfrak{g})$ and $\mathbb{C... | https://mathoverflow.net/users/83211 | Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions | You can do this in GAP. To check algebraic independence you can restrict to a Cartan subalgebra.
For type $G\_2$ with the minimal faithful representation we obtain $p\_1(sh\_1+th\_2)=-2(t^2-3st+3s^2)$ and $$p\_5(sh\_1+th\_2)=-4s^6+12s^5t-13s^4t^2+6s^3t^3-s^2t^4=-s^2(t-s)^2(t-2s)^2$$ from which you can already see the... | 5 | https://mathoverflow.net/users/26635 | 224832 | 105,135 |
https://mathoverflow.net/questions/224824 | 5 | How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation:
\begin{equation}
\left\{
\begin{array}{ccc}
x\_1^2 x\_2 \cdots x\_{p-1} x\_p+x\_1 &=& 1 \, ,\\
&&\\
x\_1 x\_2^2 x\_3 \cdot... | https://mathoverflow.net/users/64181 | Existence of solutions to a nonlinear algebraic equation | Note that no $x\_k$ can vanish. If you put $ u :=x\_1x\_2\dots x\_p $ the system gives inductively, for $k=1,\dots,p$:
$$\frac{1}{x\_1 x\_2\dots x\_k}=1+u+\dots+u^k . $$
In particular $u$ solves
$$\frac{1}{u}=1+u+\dots+u^p . $$
Incidentally, $u\neq 1$, so we can express the solutions to the systems simply as
$$x\_k... | 10 | https://mathoverflow.net/users/6101 | 224835 | 105,137 |
https://mathoverflow.net/questions/224813 | 3 | Let $M$ be a connected manifold with precisely $k$ ends $\epsilon\_1,...,\epsilon\_k$. Choose a collection $(U\_i)\_{i=1}^k$ of pairwise disjoint open $\epsilon\_i$-neighborhoods. Then I wonder how to prove that $N := M \setminus (\bigcup\_{i=1}^k U\_i)$ is a bounded subset of M, i.e, a set with compact closure.
Thou... | https://mathoverflow.net/users/78554 | Is the complement of the ends of a manifold bounded? | Assume that $M$ is a noncompact manifold. Then there exists a bijection
between both sets of ends.
Start with $e\in\lim\_{\longleftarrow}\pi\_0(M\setminus K)$. For each exhausting sequence $K\_1\subset K\_2\subset\dots$ of compact subsets with $\bigcup\_iK\_i=M$, let $U\_i$ be the connected component of $M\setminus K... | 1 | https://mathoverflow.net/users/70808 | 224837 | 105,138 |
https://mathoverflow.net/questions/224817 | 3 | Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in [this note](http://www.math.hawaii.edu/~jb/math618/os8uh.pdf)).
Let $n$ be the length of $\mathcal{L}$ and let $\mathcal{B}\_n$ be the boolean lattice of rank $n$.
*Qu... | https://mathoverflow.net/users/34538 | Can a length n distributive lattice be embedded into Bn? | A distributive lattice $L$ of length $n$ is isomorphic to the set of order ideals of an $n$-element poset $P$, ordered by inclusion. This is the Fundamental Theorem of Finite Distributive Lattices, first proved by Garrett Birkhoff. Thus $L$ imbeds into the boolean algebra of all subsets of $P$.
| 5 | https://mathoverflow.net/users/2807 | 224839 | 105,139 |
https://mathoverflow.net/questions/224460 | 11 | David Speyer commented the following [here](https://math.stackexchange.com/a/1540270/149792).
>
> I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$ and $c(t) \in \mathbb{C}[t]$ such that
> $$a(t)^3 +... | https://mathoverflow.net/users/nan | Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$ | This answer is basically a longer version of Felipe Voloch's, but maybe it will be useful. Both proofs take a class in $H^1(E)$, pull it back to $H^1(\mathbb{P}^1)$ and note that $H^1(\mathbb{P}^1)$ is zero to conclude that the map $\mathbb{P}^1 \to E$ was constant. The difference is what form of $H^1$ we work with.
... | 8 | https://mathoverflow.net/users/297 | 224849 | 105,143 |
https://mathoverflow.net/questions/214742 | 5 | I am working on a product in Morse-Bott homology which has led me to the following considerations and unanswered question. I would be very grateful if anyone could help.
Suppose $H:\mathbb{R}^n \to \mathbb{R}^n$ is a linear map which i symmetric when viewed as a matrix. The spectral theorem gives a decomposition
$\ma... | https://mathoverflow.net/users/47228 | Transversality in Morse theory, linear algebra version | Consider $H$, $g$ as symmetric bilinear forms, not as endomorphisms.
Then $g^{-1}H$ is an endomorphism that describes the gradient vector field.
Note that $\ker H=\ker g^{-1}H$ is a well-defined subspace.
Moreover, the dimensions $n\_\pm$ of the positive/negative eigenspaces are determined by $H$ alone.
Every pair of... | 0 | https://mathoverflow.net/users/70808 | 224850 | 105,144 |
https://mathoverflow.net/questions/224825 | 6 | Numerical evidence suggests that:
$$\displaystyle F(s):= \lim\_{N \to \infty}\, \ln^s\left(p\_N\right)\, \prod\_{n=1}^N \left(\dfrac{\left(p\_n-1\right)^s}{p\_n^s-1} -\frac{1}{p\_n^s}\right)$$
with $p\_n$ is the n-th prime number, converges for all $\Re(s) > \frac12$.
Note that for $s=1$ the function reduces to t... | https://mathoverflow.net/users/12489 | Does the limit of this product over primes converge for all $\Re(s) > \frac12$? | As you mention, $F(1)=e^{-\gamma}$, so
$$
\begin{align\*}
e^{s\gamma} F(s)=\frac{F(s)}{F(1)^s}&=\frac{\lim\_{x\to\infty} \log^s x \prod\_{p\leq x} \frac{p^s(p-1)^s-p^s+1}{p^s(p^s-1)}}{\lim\_{x\to\infty} \log^s x \prod\_{p\leq x} \frac{(p-1)^s}{p^s}}\\
&= \prod\_{p}\frac{p^s(p-1)^s-p^s+1}{(p^s-1)(p-1)^s}\\
&=\prod\_{p}\... | 8 | https://mathoverflow.net/users/5263 | 224856 | 105,146 |
https://mathoverflow.net/questions/224864 | 5 | An order-$n$ magic square is an $n \times n$ matrix over the numbers $\{1, ... ,n^2\}$, each appearing exactly once, whose row and column sums are all equal. Sometimes the sums of the diagonals are required to be equal too.
These objects have a rich history, and they are hugely popular in recreational math. I remembe... | https://mathoverflow.net/users/17599 | Asymptotic enumeration of magic squares | This has been asked before on MSE, [here](https://math.stackexchange.com/questions/887990/what-is-the-total-number-of-magic-squares-for-given-n) and [here](https://math.stackexchange.com/questions/1491378/number-of-magic-squares-and-their-applications) for example. According to the [OEIS](http://oeis.org/A006052), not ... | 4 | https://mathoverflow.net/users/3106 | 224871 | 105,149 |
https://mathoverflow.net/questions/224876 | 1 | Suppose $p,q$ are distinct primes with least quadratic non-residues $n\_p$ and $n\_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms of $n\_p$ and $n\_q$?
I had originally thought this would be a consequence of quadratic reciprocity and the Chinese ... | https://mathoverflow.net/users/74453 | Least simultaneous quadratic non-residue | Unless I'm mistaken, this is quite easy and only involves the multiplicative property of the quadratic residue.
If $n\_p=n\_q$ then we are done, because this very number is a simultaneous quadratic nonresidue. Otherwise we can without loss of generality suppose $n\_p<n\_q$. Then $(n\_p|p)=-1$ while $(n\_p|q)=1$. Also... | 4 | https://mathoverflow.net/users/37103 | 224878 | 105,151 |
https://mathoverflow.net/questions/224881 | 9 | Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, for example: <http://www.mast.queensu.ca/~murty/poly2.pdf>).
My question is the following. Suppose that $\deg f = d$, ... | https://mathoverflow.net/users/10898 | Polynomials which always assume perfect power values | Here is a more elementary argument than that of nfcd23. It only assumes that $f(n)$ is a $k$th power for *some* $k$ depending on $n$, which need not be assumed to divide $d$. Over $\mathbf{C}$, write
$$f(x) = C \prod (x - \alpha\_i)^{r\_i},$$
where the $\alpha\_i$ are distinct.
By Cebotarev, (or more simply, by Fr... | 7 | https://mathoverflow.net/users/83446 | 224896 | 105,154 |
https://mathoverflow.net/questions/223133 | 11 | An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the swatter tries to guess where the fly will be $k$ steps in the future.
Formally, the game is played in two stages:
(1) Pla... | https://mathoverflow.net/users/82583 | Pursuit-Evasion type game on graph ("Flyswatter game") | When the graph is just the integer lattice ${\bf Z}$, this problem was proposed by Isaacs, and solved independently by Dubins [1] and Karlin [2].
To paraphrase Math reviews, [1] and [2] deal with the following game: At each unit of time, the Evader may move either one unit of distance to the left or one to the right... | 4 | https://mathoverflow.net/users/7691 | 224897 | 105,155 |
https://mathoverflow.net/questions/84197 | 22 | A famous theorem of [Joyal and Tierney](https://books.google.se/books?id=NDbUCQAAQBAJ&lpg=PR8&ots=A6hV2iaYxQ&dq=joyal%20tierney%20an%20extension%20of%20the%20galois%20theory%20of%20grothendieck&pg=PP1#v=onepage&q&f=false) says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I ... | https://mathoverflow.net/users/15934 | Toposes (topoi) as classifying toposes of groupoids | Perhaps [these slides](https://www.dpmms.cam.ac.uk/~zll22/slides/2013-06-19-ToposesAsGroupoids.pdf) will be helpful. I'll try to explain what happens in your special case.
Let $M$ be a monoid and let $\mathcal{B} M$ be the topos of right $M$-sets. The points of $\mathcal{B} M$ are the *left* $M$-sets $P$ that satisfy... | 10 | https://mathoverflow.net/users/11640 | 224906 | 105,157 |
https://mathoverflow.net/questions/224903 | 4 | Fix any non-negative matrix $M \in \mathbb{R}\_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}\_{> 0}^m$.
With $nz(M) := \{(i,j) \ | \ M\_{i,j} > 0\}$ the index set of non-zero-entries in $M$, and $\mathbf{1}$ the all-ones-vector, define the set
... | https://mathoverflow.net/users/27164 | Affine hull of a set of non-negative matrices with fixed row-sums | (The first part of this is the same as @Fedor's answer; I just carried out his algorithm.) Each row $i$ can be written as $a\_i^{(1)}x\_i^{(1)}+a\_i^{(2)}x\_i^{(2)}$ where $x\_i^{(1)},x\_i^{(2)}$ are non-negative row vectors of sum $r\_i$ (formed by scaling the positive and negative entries in that row) and $a\_i^{(1)}... | 1 | https://mathoverflow.net/users/9025 | 224907 | 105,158 |
https://mathoverflow.net/questions/223670 | 4 | We consider the usual Riemannian metric on $S^{n}$. Its corresponding LC connection gives us a distribution on $TS^{n}$. Is this distribution completely nonintegrable?
In general, what type of curvature criterions can be used to prove that certain Levi Civita connection gives a completely non integrable distributatio... | https://mathoverflow.net/users/36688 | Is this distribution completely non integrable? | Here is a more complete answer to your question, which describes exactly which vectors in $TM$ can be joined by $L$-curves. (I changed $H$ to $L$ since I have to use $H$ for holonomy.)
Recall that, when $(M,g)$ is a Riemannian manifold (assumed connected and simply connected for simplicity), there is a unique horizon... | 2 | https://mathoverflow.net/users/13972 | 224911 | 105,159 |
https://mathoverflow.net/questions/224914 | 4 | We all know Hall's marriage theorem as following:
>
> A bipartite graph $G$ with bipartition $\{ A,B \}$ contains a matching of $A$ if and only if $|N(S)|\geq |S|$ for all $S\subseteq A$.
>
>
>
And I am thinking about a generalized theorem of it.
>
> A bipartite graph $G$ with bipartition $\{ A,B \}$ conta... | https://mathoverflow.net/users/nan | A generalized theorem of Hall's marriage theorem | Yes, it is right. Just consider $k$ copies of each vertex in $A$ and apply usual Hall theorem to the new graph.
| 10 | https://mathoverflow.net/users/4312 | 224916 | 105,160 |
https://mathoverflow.net/questions/224899 | 0 | For any set $X$ we set $[X]^2 = \big\{\{a,b\}: a, b\in X\text{ and } a\neq b\big\}$.
We say a simple undirected graph $G=(V,E)$ is an $n$-*clique graph* if there are $S\_1,\ldots,S\_n\subseteq V$ such that
1. $|S\_k| = n$ for all $k=1,\ldots, n$;
2. $V = \bigcup\_{k=1}^n S\_k$;
3. $i\neq k \in \{1,\ldots,n\}$ impli... | https://mathoverflow.net/users/8628 | Maximal induced cycles on $n$-clique graphs | I am not sure that I understand all conditions correctly, since the question looks too straightforward, but if yes, then there are no cycles for $n\leq 2$ and for $n>2$ I claim that $c(n)=n$.
At first, $c(n)\leq n$. Indeed, no two edges of our cycle $x\_1\dots x\_{c(n)}x\_1$ may belong to the same clique (if $c(n)\ge... | 2 | https://mathoverflow.net/users/4312 | 224920 | 105,163 |
https://mathoverflow.net/questions/224924 | 6 | Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result :
$$\sum\_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{1+ \pi\coth\pi... | https://mathoverflow.net/users/17164 | Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$ | To do $\sum\frac{1}{an^2+bn+c}$, factor the denominator and get a [digamma](https://en.wikipedia.org/wiki/Digamma_function) answer.
$$
\sum\_{n=0}^\infty \frac{1}{(x+p)(x+q)} = \frac{\psi(p)-\psi(q)}{p-q}
$$
And note that digamma of a rational can be evaluated in terms of logarithms and trig functions ([Gauss's digamma... | 16 | https://mathoverflow.net/users/454 | 224926 | 105,164 |
https://mathoverflow.net/questions/224838 | 4 | Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in the inequality.
Thanks.
| https://mathoverflow.net/users/77923 | concentration inequality for $d$-dimensional martingale | Bounds (not depending explicitly on the dimension) on the moments of the norm of martingales in arbitrary 2-smooth Banach spaces (which of course include all finite-dimensional Euclidean spaces) can be found in **[[1](http://epubs.siam.org/doi/abs/10.1137/S0040585X97T987417)]**; see also further references there.
As ... | 2 | https://mathoverflow.net/users/36721 | 224935 | 105,166 |
https://mathoverflow.net/questions/224910 | 5 | Given a smooth manifold $M,$ it is well known that any derivation of the algebra of smooth functions $C^{\infty}(M)$ can be seen (or it is associated to) a smooth vector field on $M.$ I am looking for a similar geometric meaning for a derivation $D: C^{\infty}(M)\rightarrow C^{\infty}(T^\*M)$ along $\rho^\*,$ where $\r... | https://mathoverflow.net/users/83458 | geometric interpretation of derivation between two algebras | In general, given a map of smooth manifolds $\phi:N\to M$, derivations $C^\infty M \to C^\infty N$ are in canonical one to one correspondence with sections of the pullback bundel $\phi^\* TM$ on $N$. Such a section is a smooth map that associates to each point $p\in N$ a tangent vector in $T\_{\phi(p)}M$. These are som... | 5 | https://mathoverflow.net/users/745 | 224949 | 105,173 |
https://mathoverflow.net/questions/224934 | 3 | A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition
\begin{equation}
gr\_F^pgr\_{\bar{F}}^qgr\_n^W(V)=0\qquad\text{if }n\neq p+q
\end{equation}
It is a result of Deligne that this is the... | https://mathoverflow.net/users/76748 | Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator? | I guess you're refering to the main result of Deligne "Structure de Hodge mixtes réelles". Given a real (resp. complex) variation of mixed Hodge structures consisting of a holomorphic bundle $V$, integrable connection $\nabla$, flat real sub bundle $W\_\bullet$, holomorphic sub bundles $F^\bullet$ (and antiholomorphic ... | 2 | https://mathoverflow.net/users/4144 | 224957 | 105,178 |
https://mathoverflow.net/questions/224956 | 25 | Drew Zemke recently posted a [preprint on arXiv](http://arxiv.org/abs/1511.04978) proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.
>
> **Simple Loop Conjecture:** Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M$ of a closed orientable surface $\Sigma$ into a closed 3-manifold $M$. If ... | https://mathoverflow.net/users/2051 | What are the implications of the simple loop conjecture? | I would motivate the simple loop conjecture as follows. (I'm fairly idiosyncratic about this; I fear I'm going to turn off many 3-manifold topologists.)
As well as understanding spaces, we want to understand the maps between them. One instance of this is that it would be extremely useful to have some sort of 'classif... | 19 | https://mathoverflow.net/users/1463 | 224958 | 105,179 |
https://mathoverflow.net/questions/224968 | 4 | Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?
| https://mathoverflow.net/users/nan | Are there other integer solutions to the equation $9x^3 -1 = y^3$ besides $(x,y) =(1,2)$ and $(0, -1)$? | Theorem 6.4.30 in Cohen's *Number Theory: Volume I* asserts: For each nonzero integer $d$, there is at most one pair of integers $(X,Y)$ with $Y\ne 0$ and $X^3+dY^3=1$. Apply this with $X=-y$, $Y=x$, and $d=9$, to see that that there are no more solutions. Theorem 6.4.30 is attributed to Skolem.
| 10 | https://mathoverflow.net/users/16510 | 224972 | 105,187 |
https://mathoverflow.net/questions/224977 | 1 | $s\_2(n)$ denotes the sum of the standard base-2 digits of $n$.
For a fixed odd number $k>1$, can we construct $n\in \mathbb{Z}^+$, to make $s\_2(nk)<s\_2(n)$?
To clarify, that's not $s\_2(nk) \lt s\_2(k)$. For example, if $k=7$, we can take $n=23=10111\_2$ and $nk=161=10100001\_2$.
| https://mathoverflow.net/users/nan | Construction of $n$ makes $s_2(nk)<s_2(n)$ | Yes, basically because $1/k$ has infinitely many $1$s in its binary expansion.
Let $n=\lceil 2^t/k \rceil$ for $t$ large enough, larger than the length of the period of $1/k$ times (the number of digits of $k$ plus $2$). Then $nk$ is just over $2^t$, $nk \lt 2^t+k$, so $s\_2(nk)$ is at most $1$ more than the number ... | 1 | https://mathoverflow.net/users/2954 | 224983 | 105,190 |
https://mathoverflow.net/questions/224995 | 3 | A primitive sequence $1<a\_1<\ldots<a\_k\leq n$ is a sequence of integers no one of which divides any other, investigated by Erdos, Behrend and others, over the last 80 years. In fact, $\max k=\lfloor \frac{n+1}{2}\rfloor,$ which can be shown by taking exactly one member from each chain in the divisibility poset, namel... | https://mathoverflow.net/users/17773 | Primitive sequence $a_i$ attaining Pillai's bound on $\sum_{i} 1/a_i$ | Fix $k$ and consider the set $E\_k$ of all numbers with exactly $k$ prime factors (counted with multiplicity). They definitely form a primitive sequence. Already for $k=1$ sum of reciprocals is unbounded. I think, for $k=[\log \log n]$ we should obtain lower bound like $\log n/\sqrt{\log\log n}$. It is natural to expec... | 7 | https://mathoverflow.net/users/4312 | 224997 | 105,194 |
https://mathoverflow.net/questions/224981 | 2 | If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \mathfrak{U}$. Let $\pi: X\rightarrow X/R$ denote the quotient map and put $(Y,\mathfrak{W})= (X/R,\mathfrak{U}/R)$.
$... | https://mathoverflow.net/users/83060 | The separated uniform space associated with $(X,\mathfrak{U})$ | Let $(X,{\frak U})$ be a uniform space. We set $R = \bigcap {\frak U}$. It is a standard exercise to show that $R$ is an equivalence relation.
So we look at the following set:
$${\frak U}/R := \{A\subseteq (X/R)\times (X/R): \exists M\in{\frak U}\big((\pi\times\pi)(M) \subseteq A\big)\}.$$
First we need to verify t... | 1 | https://mathoverflow.net/users/8628 | 224998 | 105,195 |
https://mathoverflow.net/questions/224993 | 3 | I asked this question [here](https://math.stackexchange.com/q/1548838/75923) on Math.SE but uptil now it was not answered. So I decided to give it a try here. Thank you in advance.
If a lattice $L$ is distributive then it can be shown that for $a,b,c\in L$: $$[a\wedge b=a\wedge c\text{ and }a\vee b=a\vee c]\implies b... | https://mathoverflow.net/users/40263 | Is this a sufficient condition for distributivity of a lattice? | Yes that's true. See page 67 of [this book](https://books.google.tk/books?id=Ll0JXd11SW0C&pg=PA67#v=onepage&q&f=false).
| 3 | https://mathoverflow.net/users/47958 | 225003 | 105,196 |
https://mathoverflow.net/questions/224898 | 16 | Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the sentence tends to either 0 or 1 as $n$ goes to infinity. It is known that testing, for a given sentence, whether the limit i... | https://mathoverflow.net/users/440 | Convergence rate of Fagin's 0-1 law for first-order properties of random graphs | For definiteness, I will consider undirected graphs without self-loops, or equivalently, structures with a symmetric irreflexive binary predicate $E(x,y)$. These choices do not really matter.
The question mentions an exponential lower bound. It can be optimized as follows:
$$\phi=\forall x\_0\dots\forall x\_{k-1}\,\e... | 10 | https://mathoverflow.net/users/12705 | 225005 | 105,197 |
https://mathoverflow.net/questions/225000 | 2 | I am searching for a result in the literature that I am sure must be known, but I just fail to find it.
Let us starts with a simple example:
Let $A, B\subset \mathbb{Z}$ be a finite sets of integers such that $|A|=|B|=2m+1$ and denote by $I$ the set $\{-m,...,m\}$. I want to show that for all $k,l \geq 0$ we have
$$... | https://mathoverflow.net/users/24494 | Intersections of translates of finite sets of integers | For the case $t=2$, letting $A\_1:=A$, $A\_2:=-B$, $A\_3:=[-k,k]$, and $A\_4:=[-l,l]$, the sum in the left-hand side counts the number of quadruples $(a\_1,a\_2,a\_3,a\_4)\in A\_1\times A\_2\times A\_3\times A\_4$ with $a\_1+a\_2+a\_3+a\_4=0$.
It is known that this number is maximized, over all quadruples $(A\_1,A\_2,A... | 2 | https://mathoverflow.net/users/9924 | 225009 | 105,199 |
https://mathoverflow.net/questions/225001 | 3 | Let $\omega$ be the hermitian form for an hermitian metric on a compact complex manifold. Can $\omega^{n-1}$ be $\partial {\bar{\partial}}$-exact? (We know that our hermitian metric must necessarily be balanced and non-Kahler.)
| https://mathoverflow.net/users/30172 | Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar{\partial}}$-exact? | Yes, this can happen. For a simple example, consider $M = \mathrm{SL}(2,\mathbb{C})/\Lambda$ where $\Lambda\subset \mathrm{SL}(2,\mathbb{C})$ is a discrete, co-compact lattice. Then $M$ is a compact complex $3$-manifold.
Let $\alpha\_1,\alpha\_2,\alpha\_3$ be a basis for the right-invariant holomorphic $1$-forms on $... | 5 | https://mathoverflow.net/users/13972 | 225010 | 105,200 |
https://mathoverflow.net/questions/225008 | 1 | Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int\_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse $\mathrm{erf}^{-1}: (-1,1) \to \mathbb{R}$.
Now I have two questions:
1) On <https://en.wikipedia.org/wiki/Error_fu... | https://mathoverflow.net/users/57982 | Inverse error function in Hardy space? | To obtain a function that maps $R\to(-1,1)$ one has to define the error function
as $f(x)=(2/\sqrt{\pi})\int\_0^x\exp(-x^2)dx$, not as you defined.
With this definition it is an entire function which has no critical points (that is $f'(z)\neq 0,\; z\in C$) and two finite asymptotic values, $\pm 1$. From the general t... | 5 | https://mathoverflow.net/users/25510 | 225014 | 105,201 |
https://mathoverflow.net/questions/224999 | 6 | Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi\_1(X) \to G$ to it. I know how to construct such a presentation out of a principal connection on $P$ by taking an appropriate quotient of the holonomy repre... | https://mathoverflow.net/users/22810 | What does "higher monodromy" tell us about a principal bundle | There is a monodromy of sorts for any topological bundle (or even fibration) $\pi \colon E \to X$ with fiber $F$. Let $\gamma \colon [0,1] \to X$ be a path and set
$F = \pi^{-1}(\gamma(0))$. The homotopy lifting property applied to the square
$$
\begin{array}{ccc}
F & \hookrightarrow & E \\
\downarrow & & \downarrow \... | 8 | https://mathoverflow.net/users/33141 | 225016 | 105,202 |
https://mathoverflow.net/questions/225002 | 7 | Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly hyperbolic systems (Axiom A, Markov maps of the interval), but I don't know much other examples.
| https://mathoverflow.net/users/80894 | List of Bernoulli chaotic systems | The most well-understood examples are the ones you mention: Axiom A diffeomorphisms and Markov maps of the interval, since these can be modeled by SFTs. Note that "Bernoulli" refers to a particular choice of invariant measure; the SRB measure for an Axiom A attractor (or the ACIP for a Markov interval map) is Bernoulli... | 6 | https://mathoverflow.net/users/5701 | 225018 | 105,203 |
https://mathoverflow.net/questions/225028 | 2 | It is known that the seventh coefficient of $\Phi\_{105}(x)$ is $-2$ and that's the first occurrence of a coefficient with absolute value greater than $1$ for a cyclotomic polynomial. When I did a quick check for the seventh coefficient of $\Phi\_n(x)$ where $n=105k$ with $\gcd(105,k)=1$ and $\mu(k)\neq 0$ they all cam... | https://mathoverflow.net/users/83515 | Cyclotomic polynomials with 7$^{th}$ coefficient greater than 1 in absolute value | $k=11$ is the smallest counterexample - the 7'th coefficient is 0. Here are the details:
We have the following identity:
$$\Phi\_n(x) = \prod\_{d \mid n} (1-x^d)^{\mu(n/d)},$$
valid for $n>1$.
If we are interested only in the first $m+1$ coefficients ($x^0$ to $x^{m}$), it suffices to look at the following product... | 11 | https://mathoverflow.net/users/31469 | 225036 | 105,212 |
https://mathoverflow.net/questions/224964 | 11 | If we have distinct primes $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = -1,$ there is a solution to $u^2 - pq v^2 = -1$ in integers and the fundamental unit of $O\_{\mathbb Q(\sqrt{pq})} $ has norm $-1.$ Stevenhagen attributes this to Dirichlet (1834).
There is no such result for $p \equiv q \equiv 1... | https://mathoverflow.net/users/3324 | Fundamental units with norm $-1$ in real quadratic fields | Stevenhagen "[The number of real quadratic fields having units of negative norm](http://projecteuclid.org/euclid.em/1048516217)" Exp Math 1993 makes the following analysis (last paragraph page 127):
Let $D>0$. Let $C$ be the narrow class group of $\mathbb{Q}(\sqrt{D})$ (ideals modulo principal ideals whose generator ... | 8 | https://mathoverflow.net/users/297 | 225037 | 105,213 |
https://mathoverflow.net/questions/225039 | 6 | I would like to know about problems on (finite) tensor categories. I have read Etingof´s notes from his course at MIT. I have a question:
There exists any reference where I can find an open problem about this topic?
Thank you very much for your attention.
| https://mathoverflow.net/users/83524 | Open questions on (finite) tensor categories | We made a [list of open problems](http://aimpl.org/fusioncat/) at an AIM conference a few years ago.
| 8 | https://mathoverflow.net/users/22 | 225040 | 105,215 |
https://mathoverflow.net/questions/184538 | 14 | Suppose I'd like to:
\begin{align}
\mathop{\text{min}}\_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& \mathbf{C}\mathbf{x} = \mathbf{b}
\end{align}
where all vector variables are known except $\mathbf{x}$, and $\mathbf{C}$ is full row rank.
If I ... | https://mathoverflow.net/users/23064 | Linearly constrained eigenvalue problem | I think Pushpendre's answer isn't quite right, but it gets you most of the way there. Getting rid of that pesky constant term is a bit tricky relative to the homogeneous case.
Let's take his suggested substitutions:
$$
\begin{array}{rl}
M&:=N^\top N\\
y&:=Nx\\
D&:=CN^{-1}\\
B&:=N^{-\top}AN^{-1}
\end{array}
$$
I don't... | 11 | https://mathoverflow.net/users/25311 | 225043 | 105,217 |
https://mathoverflow.net/questions/225055 | 12 | The degree 10 polynomial
$$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$
given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is the smallest known Salem number. Moreover, it is a folklore conjecture that $\beta$ is in fact the smallest Salem number.... | https://mathoverflow.net/users/10898 | Is Lehmer's polynomial solvable? | Lehmer's polynomial is symmetrical,
so $x + x^{-1} =: y$ satisfies a polynomial of half the degree.
It turns out that this is the quintic $y^5 + y^4 - 5y^3 - 5y^2 + 4y + 3 = 0$,
whose Galois group is the unsolvable $S\_5$ (for instance, it's irreducible
$\bmod 2$ and decomposes as $(y^2-2y-1)(y^3-2y^2+2y+2)$ $\bmod 5$... | 26 | https://mathoverflow.net/users/14830 | 225060 | 105,222 |
https://mathoverflow.net/questions/225062 | 2 | Suppose $X$ is a process given by -
$dX\_t = db\_t$
where $b\_t$ is a standard Brownian motion with its filtration $(\mathcal{F}\_t)$.
Suppose an agent earns a payoff given by
$V(x) = \mathbb{E} [\int\_0^\infty e^{-\int\_0^t r(X\_s)ds} dt|X\_0 =x] $
where $r(x) = \begin{cases} 3 & \text{ if } x \ge 0 \\
7 & ... | https://mathoverflow.net/users/78761 | Differentiability of value function | I think it's differentiable (everywhere).
Interchanging operations freely, we have
\begin{eqnarray\*}
\newcommand{\D}{\frac{\mathrm d}{{\mathrm d}x}}
\newcommand{\E}{\mathbb E}
V'(x) &=& \D \E\_x \int\_0^\infty \exp\left(-\left(\int\_0^t 3+4[W\_s<0]\mathrm ds\right)\right)\mathrm dt
\\
&=& \int\_0^\infty \E\_0 \D\exp\l... | 1 | https://mathoverflow.net/users/4600 | 225068 | 105,225 |
https://mathoverflow.net/questions/225057 | 3 | Let $A$ be a finite set of positive natural numbers with $n$ elements, $|A|=n$, with the property that all sums of two (not necessarily different) elements are distinct, or in the usual notation for sumsets, $|A+A|$= $1\over2$ $n (n+1)$.
I am looking for ways to choose the elements of $A$ (for a given $n$) so $\max(... | https://mathoverflow.net/users/50847 | Sumsets with distinct numbers, upper bound for maximum element | Set $A$, $|A|=n$, with $|A+A|=n(n+1)/2$, is known as a `Sidon set'. This is a subject of numerous studies. As for your specific question, $c\_1n^2<S(n)<c\_2 n^2$ for some absolute constants, but I do not know current records.
UPD: already know, from Lucia's answer.
If you do not care on constants, then:
1) lower e... | 10 | https://mathoverflow.net/users/4312 | 225072 | 105,227 |
https://mathoverflow.net/questions/224769 | 8 | From a proof that 2D Wightman CFT leads to a vertex algebra **[1]**:
Let
$$
Y(a,z):=\frac{1}{(1+z)^{2\Delta\_a}}\Phi\_a\left(i\frac{1-z}{1+z}\right),\quad\text{with}\quad |z|<1.
$$
Here $\Delta\_a\ge 0$ is conformal weight of $\Phi\_a$, and $\Phi\_a$ is the scalar field satisfying the usual Wightman axioms (*Poinca... | https://mathoverflow.net/users/69505 | Fourier series of a Wightman field | Using Marcel Bischoff's comment:
$$
Y(a,z)=\sum\_n a\_{(n)} z^{-n-1} \iff a\_{(m)}=\frac{1}{2\pi i} \oint\limits\_{|z|=1} Y(a,z)z^{m} \,\mathrm{d}z.
$$
Since the author of **[1]** starts with the Schwartz space $\mathscr{S}(\mathbb{R})$, we need to show that $Y(a,z)$ is analytic in $|z|<1$.
By definition,
$$
Y(a,z... | 2 | https://mathoverflow.net/users/69505 | 225078 | 105,228 |
https://mathoverflow.net/questions/225081 | 12 | Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell\_2(X)$ of all sequences $(x\_n)$ in $X$ with $\sum\_{n=1}^\infty \|x\_n\|^2<\infty$ is not isomorphic to the space $L\_2([0,1],X)$ of square integrable $X$-valued functions on $[0,1]$.
How can I find an example o... | https://mathoverflow.net/users/39421 | Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$ | You can take any Banach space $X$ for which the weak$^\ast$-dentability index $Dz(X)$ is strictly larger than the Szlenk index $Sz(X)$ (note that we have $Dz(X)\geq Sz(X)$ in general). The reasons for this are that:
1. $Sz(\ell\_2(Y))=Sz(Y)$ for any Banach space $Y$ (see Theorem 2.12 of Brooker, *Direct sums and the ... | 14 | https://mathoverflow.net/users/848 | 225086 | 105,229 |
https://mathoverflow.net/questions/225076 | 3 | A partition regular system is a linear system of equations of the form $A\cdot x=0$, which satisfies a Ramsey-type result (namely, that for each $r>0$ whenever we colour the integers in $r$ classes, there is a class which contains a monochromatic solution). The well-known Rado's theorem gives a characterization of such... | https://mathoverflow.net/users/46573 | Partition regular systems: do they have solution in (very dense) set of integers? | This doesn't have anything to do with partition regularity: There is such a constant $C(A)<1$ provided only that there exists at least one solution to $Ax=0$ in positive integers.
Indeed suppose $x = (x\_1,\dots,x\_m)$ is a solution. Then $jx = (jx\_1,\dots,jx\_m)$ is a solution for each $j\geq 1$. Now take a large i... | 5 | https://mathoverflow.net/users/20598 | 225097 | 105,233 |
https://mathoverflow.net/questions/225085 | 2 | It suffices to say that all circle bundles on compact Riemann surfaces admit the structure of a closed Sasakian 3-manifold. The question is, the converse of this statement and/or what are the sufficient conditions on a Sasakian 3-manifold that ensure it will be such a circle bundle?
| https://mathoverflow.net/users/38257 | Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface? | A complete topological classification is due to Geiges, and can be found in [this 2001 paper by Guilfoyle](http://arxiv.org/pdf/math/0102015v1.pdf). (the first Theorem in the paper).
| 2 | https://mathoverflow.net/users/11142 | 225100 | 105,236 |
https://mathoverflow.net/questions/225015 | 7 | Raynaud and Gruson proved a beautiful "flatification" theorem (5.2.2): If $S$ is a quasicompact, quasiseparated scheme, and $X$ is a finitely presented $S$-scheme, $M$ is an $\mathcal O\_X$-module of finite type, then there is a blowup $f : S' \rightarrow S$ such that the strict transform of $M$ along $f$ is flat over ... | https://mathoverflow.net/users/32 | Failure of universal flatification | I feel like there should be an example using matrix multiplication, but I tried several variations that did not work (as I commented above). The example below is less naural than matrix multiplication, but it illustrates the key issue.
Let $k$ be a field. Let $S$ be a $3$-dimensional vector space, $V \cong \text{Spec... | 4 | https://mathoverflow.net/users/13265 | 225106 | 105,239 |
https://mathoverflow.net/questions/225031 | 1 | Let $n$ be a natural number and let $S\_n$ be a square $[0,n] \times [0,n]$ in the plane.
We say that a partition $\mathcal{Q} = R\_1 \cup \cdots \cup R\_t$ of $S\_n$ is *simple* if each of the sets $R\_1, \ldots, R\_t$ is connected, has positive area and has diameter at most 1.
**Question.** Does there exist a col... | https://mathoverflow.net/users/83519 | Is it possible to cover all pairs of points at distance at most 1 by constant number of partitions into sets of diameter at most 1? | More generally, replace $S\_n$ by $D$, a union of a collection of closed balls of nonzero radius, and suppose $D$ has diameter greater than 1. Consider partitions of $D$ which are bounded, so each set in a partition has diameter at most 1. Let us further assume that $D$ contains two open balls with centers $x$ and $y$ ... | 0 | https://mathoverflow.net/users/3402 | 225116 | 105,244 |
https://mathoverflow.net/questions/81194 | 8 | I am attempting to solve the argument maximization problem
$$\arg\sup\_x \{ \langle x,l \rangle − f\_1(x)−f\_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f\_1$ and $f\_2$ are concave but difficult to evaluate but their convex conjugates $f^∗\_1$ and $f^∗\_2$ are easy to evaluate. We can further assume tha... | https://mathoverflow.net/users/19341 | Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality? | I ultimately did find a solution to this problem. First, you compute the solution $\bar{w}$ to
$$\bar{w} \in \arg\inf\_w f^\*\_1(\ell - w) + f\_2^\*(w).$$
A solution $\bar{x}$ to (1) can then be computed by choosing
$$\bar{x} \in \partial f\_1^\*(\ell-\bar{w}) \cap \partial f\_2^\*(\bar{w})$$
where $\partial ... | 2 | https://mathoverflow.net/users/19341 | 225118 | 105,246 |
https://mathoverflow.net/questions/225119 | 1 | For any vector $w\in\mathbb{R}^n,$ let $\|w\|:=\|w\|\_\infty = \max\_i |w\_i|.$ Let $w\geq 0$ mean that $w$ is non-negative in each co-ordinate.
For $0<\alpha<\frac 14,$ I am interested in the quantity
$$f(\alpha):=\sup\_{u,v\in\mathbb{R}^n, u,v\geq 0, n\geq 1} \frac{\|(1-\alpha)u-\alpha v\|+\|(1-\alpha)v-\alpha u\... | https://mathoverflow.net/users/7576 | Ratio of sums of $\infty$-norms of vectors | You can do no better than $1+\alpha$.
Lower bound: Take $u=(1,0,1)$ and $v=(0,1,1)$.
Upper bound, the nominator is bounded by $(1-\alpha)\|u\|+(1-\alpha)\|v\|+2\alpha(\|u\|+\|v\|)$.
| 1 | https://mathoverflow.net/users/37103 | 225123 | 105,247 |
https://mathoverflow.net/questions/225094 | 0 | A graph $\Gamma$ is called prime with respect to the Cartesian product if
$\Gamma=\Gamma\_1\square\Gamma\_2$ implies that $\Gamma\_1=K\_1$ or $\Gamma\_2=K\_1$, where $\square$ denote the Cartesian product.
Is there any classification of connected and vertex-transitive prime graphs with respect to Cartesian product? Is ... | https://mathoverflow.net/users/27831 | connected and vertex-transitive prime graphs with respect to Cartesian product | There almost certainly isn't a meaningful classification of vertex-transitive prime graphs. In fact, it's quite likely that almost all vertex-transitive graphs are prime.
| 2 | https://mathoverflow.net/users/22377 | 225127 | 105,250 |
https://mathoverflow.net/questions/225126 | 3 | Given the additive group of the module $\mathbb{Z}^\mathbb{N}$ and a total ordering of the group that is compatible with addition and where $\chi\_{\{n\}} > 0$ for all $n \in \mathbb{N}$, can we say for sure that $\chi\_\mathbb{N} > 0$?
By "compatible" I mean that, for all $a, b, c \in \mathbb{Z}^\mathbb{N}$, if $a \... | https://mathoverflow.net/users/14257 | Compatible total orderings of the group $\mathbb{Z}^\mathbb{N}$ | So I think using the Axiom of Choice, you can see the answer is no.
Use Hamel's basis theorem to obtain a basis of $\mathbb Q^{\mathbb N}$ including $\chi\_{n}$ for each $n$ and $-\chi\_{\mathbb N}$. Well order the basis elements and then say $a<b$ if the expansion in the Hamel basis of $a$ is lexicographically smaller... | 5 | https://mathoverflow.net/users/11054 | 225129 | 105,252 |
https://mathoverflow.net/questions/224424 | 3 |
>
> Is there a profinite group $G$, a continuous automorphism $\alpha$ of
> $G$ and a topologically finitely generated closed subgroup $H \leq G$
> such that $\alpha(H) \lneq H$ ?
>
>
>
Note that if an example exists, then $G$ is not topologically finite generated, and $\alpha$ is not given by conjugation by a... | https://mathoverflow.net/users/38889 | Wild automorphisms of profinite groups | No.
Let $K\_n = K\_n(G)$ be the intersection of all open normal subgroups of $G$ of index at most $n$. Then $\alpha(K\_n) = K\_n$. If we replace $G$ with $G/K\_n$, we still have $\alpha$ acting as an automorphism on $G/K\_n$ and for $n$ large enough, $\alpha(H)K\_n/K\_n$ is a proper subgroup of $HK\_n/K\_n$ (since $H... | 2 | https://mathoverflow.net/users/4053 | 225133 | 105,254 |
https://mathoverflow.net/questions/125967 | 6 | Consider a central extension
$$1 \longrightarrow \mathbb{Z} \longrightarrow G \longrightarrow Q \longrightarrow 1$$
with Euler class $\zeta \in H^2(Q;\mathbb{Z})$. Let $Q'$ be a normal subgroup of $Q$ and let $\zeta' \in H^2(Q';\mathbb{Z})$ be the restriction of $\zeta$. Finally, consider some $\zeta'' \in H^2(Q';\math... | https://mathoverflow.net/users/29685 | Which group extensions are normal? | As a set, $G' = Q' \times \mathbb{Z}$, with the group operation given by $(p,x)(q,y) = (pq, x + y + \zeta'(p,q))$, and $G'' = Q' \times n\mathbb{Z}$ (again, as a set) with group operation $(p,na)(q,nb) = (pq,n(a + b + \zeta''(p,q)) = (pq,na + nb + \zeta'(p,q))$.
Now, let $p \in Q'$, consider the element $(p,na) \in ... | 3 | https://mathoverflow.net/users/38434 | 225140 | 105,255 |
https://mathoverflow.net/questions/225137 | 1 | I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the following terminal value problem:
\begin{align}
& F\_t+\frac{1}{2}σ^2x^2F\_{xx}=1\\
& F(x,T)=(\ln(x))^4,\ x>0
\end{ali... | https://mathoverflow.net/users/41769 | using Feynman-Kac formula | The idea is to choose a stochastic process of the form
$$dX\_t=\mu(t,X\_t)dt+\sigma(t,X\_t)dW\_t $$
and consider the process $Y\_t=F(t,X\_t)$.
Applying Ito to $Y\_t$ gives
$$ dY\_t=\left(F\_t+\mu(t,X\_t)F\_x(t,X\_t)+\frac12\sigma(t,X\_t)^2F\_{xx}(t,X\_t)\right)dt+\sigma(t,X\_t)F\_x(t,X\_t)dWt $$
And then to choose... | 2 | https://mathoverflow.net/users/30889 | 225151 | 105,256 |
https://mathoverflow.net/questions/225139 | 1 | Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes W)$. I think that $\Lambda^3 V \otimes S^3 W$ and $S^3 V \otimes \Lambda^3 W$ are subspaces of $\Lambda^3(V \otimes W)$. It... | https://mathoverflow.net/users/11877 | Decompose $\Lambda^3(V \otimes W)$ | **EDIT:**
The group $GL(V)\times GL(W)$ acts naturally on $\wedge^n(V\otimes W)$ for any $n\in \mathbb{N}$ (in your case $n=3$). One can describe decomposition of this space into irreducible components. For example for $n=2$ the decomposition you described coincides with that one.
Any irreducible representation of $... | 2 | https://mathoverflow.net/users/16183 | 225153 | 105,257 |
https://mathoverflow.net/questions/225138 | 10 | Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence theorem, shouldn't this suffice to prove the consistency of first order arithmetic? Why is Gentzen's proof necessary?
| https://mathoverflow.net/users/83579 | Why is there a need for ordinal analysis? | The axioms of first-order arithmetic include the induction schema, which says that, for every formula $A(x)$ with free variable $x$, the conjunction of $A(0)$ and $\forall x\,(A(x)\rightarrow A(x+1))$ implies $\forall x\,A(x)$. This is, of course, a special case of the well-known and basic induction property of the nat... | 26 | https://mathoverflow.net/users/6794 | 225158 | 105,258 |
https://mathoverflow.net/questions/225169 | 4 | Recently I'm thinking about question below, but I can not prove or disprove it.
>
> Is it true that for every model $M\models I\Delta\_0$ there exists a
> model $M'\models PA$ such that $M'$ is end extension of $M$?
>
>
>
How can this statement be proved or disproved?
Thanks.
| https://mathoverflow.net/users/83598 | End Extension models of $I\Delta_0$ | It's not true, because perhaps the model of $I\Delta\_0$ satisfies $\neg\text{Con}(I\Delta\_0)$. Since PA proves the consistency of the bounded induction principles, there can be no model of PA that has such a proof in it.
| 11 | https://mathoverflow.net/users/1946 | 225171 | 105,262 |
https://mathoverflow.net/questions/225172 | 25 | Let $n$ be a natural number. For every group $G$ of order $n$, denote
$d(G)$ : The number of elements of the smallest generating set of $G$
>
> How large is the maximum possible value of $d(G)$ depending on $n$ ?
>
>
>
If $n$ is a cyclic number, we have $d(G)=1$ for every group of order $n$.
For $n=2p$ , $p... | https://mathoverflow.net/users/49398 | How large can the smallest generating set of a group $G$ of order $n$ be? | By a Theorem of Guralnick and Lucchini (which does require CFSG), if each Sylow subgroup of $G$ (ranging over all primes) can be generated by $r$ or fewer elements, then $G$ can be generated by $r+1$ or fewer elements. As noted in comments, if $G$ has a Sylow $p$-subgroup $P$ of order $p^{a}$, then $P$ can be generated... | 42 | https://mathoverflow.net/users/14450 | 225178 | 105,266 |
https://mathoverflow.net/questions/225182 | 0 | Is the following true? I cannot see a counterexample and it seems very intuitively clear, at least in the embedded case.
Claim:
Consider the set $S$ of closed immersed Riemann surfaces $\Sigma \subset (X,g)$ (I am particularly interested in spheres), with the magnitude of the mean curvature bounded from above by some... | https://mathoverflow.net/users/16877 | Diameter of immersed surfaces with bounded from above mean curvature | Assuming I am interpreting your question correctly, this is true when the ambient space is Euclidean and the submanifold is closed (i.e. compact and without boundary). To see this one may invoke a result of Topping that bounds the (intrinsic) diameter of a closed, immersed submanifold of dimension $m$ by the $L^{m-1}$ ... | 1 | https://mathoverflow.net/users/26801 | 225186 | 105,272 |
https://mathoverflow.net/questions/225187 | 4 | We consider the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ and the language $L=\{+, \cdot , \frac{d}{dx} , 0, 1\}$.
The ring consists of elements of the form $$\sum\_{i=0}^N \alpha\_i e^{\lambda\_i x}$$ where $\alpha\_i , \lambda\_i \in \mathbb{C}$.
In the language there is no symbol $e^x$.
W... | https://mathoverflow.net/users/52805 | Can we use this symbol? | Note that the transformation $x\mapsto x+t$ leaves all relations defined by your language invariant.
Namely,
1. $\sum{a\_i e^{q\_i x}}+\sum{b\_i e^{r\_i x}}=\sum{c\_i e^{s\_i x}}$ iff $\sum{a\_i e^{q\_i (x+t)}}+\sum{b\_i e^{r\_i (x+t)}}=\sum{c\_i e^{s\_i (x+t)}}$
2. $(\sum{a\_i e^{q\_i x}})\cdot(\sum{b\_i e^{r\_i x... | 4 | https://mathoverflow.net/users/20186 | 225200 | 105,274 |
https://mathoverflow.net/questions/225196 | 1 | I would guess this is some standard fact related to the zero-free region. But cannot find it in the textbooks I read.
| https://mathoverflow.net/users/21929 | Upper and lower bounds for $|L(1+it,\chi)|$ for complex primitive character $\chi$? | For starters, see Montgomery & Vaughan's "Multiplicative Number Theory", Theorem 11.3 and 11.4. For example, in the zero free region of 11.3,
$$
\frac{1}{L(s,\chi)} \ll \log(q(|t|+4)).
$$
| 5 | https://mathoverflow.net/users/6756 | 225201 | 105,275 |
https://mathoverflow.net/questions/225205 | 5 | I saw in the answer of this post:
[Is it true that all sphere bundles are boundaries of disk bundles?](https://mathoverflow.net/questions/74756/is-it-true-that-all-sphere-bundles-are-boundaries-of-disk-bundles)
that a $S^3$-bundle over $S^4$ bounds a disc bundle over $S^4$ iff $O(4)\rightarrow Diff(S^3)$ is a homot... | https://mathoverflow.net/users/48618 | 3-sphere bundles over 4-sphere bound smooth disc bundles | The map $O(4) \to \text{Diff}(S^3)$ being homotopy equivalence is *not* equivalent to the assertion that every smooth $S^3$-bundle bounds a disk bundle, but the much stronger statement that every smooth $S^3$-bundle can be linearized, i.e. arises as the unit bundle of some four-dimensional vector bundle (and this clear... | 8 | https://mathoverflow.net/users/14233 | 225207 | 105,279 |
https://mathoverflow.net/questions/225160 | 7 | Let $f\colon X\to Y$ be a surjective morphism of algebraic varieties, defined over an algebraically closed field.
We take a morphism $\psi\colon X\to Z$, where $Z$ in another algebraic variety, that satisfies the following condition: for all $x,x'\in X$, $f(x)=f(x')\Rightarrow \psi(x)=\psi(x')$.
This yields the exi... | https://mathoverflow.net/users/23758 | Pushing-forward morphisms | If $f$ is finite surjective and unramified, I think that $\psi'$ is always a morphism. In fact we can prove that $f$ is an effective epimorphism, which means that letting $p\_1,p\_2:X\times\_Y X\to X$ be the projections, any morphism $\psi:X\to Z$ such that $\psi p\_1=\psi p\_2$ factors uniquely through a morphism $\ps... | 5 | https://mathoverflow.net/users/17988 | 225214 | 105,282 |
https://mathoverflow.net/questions/223816 | 3 | From a comment on [this question](https://mathoverflow.net/questions/111770/cycling-through-the-zeta-garden-zeta-functions-for-graphs-cycle-index-polynomi?noredirect=1#comment550560_111770):
>
> @draks, there is a connection between the Chebyshev polynomials and the Faber polynomials (a.k.a. Shur polynomials), whic... | https://mathoverflow.net/users/11856 | Connection between the Chebyshev polynomials and the Faber polynomials | The characteristic polynomial and the traces of the powers of a square matrix $A$ are related by
$$\operatorname{det}(I-x \; A)=\exp\left(-\sum\_{m\geq 1} \frac{\operatorname{tr}(A^m) \; x^m}{m} \right)$$
$$=1 + \sum\_{k \ge 1} P\_k(b\_1,..,b\_k) \; x^k / k! = 1 + \sum\_{k \geq 1} d\_k \; x^k$$
where $b\_k=-\oper... | 1 | https://mathoverflow.net/users/12178 | 225222 | 105,285 |
https://mathoverflow.net/questions/225239 | 2 | Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1.
Let $N=\langle y,w \rangle \cong C\_p \times C\_{p^2}$ and $T=\langle x \rangle$. I wish to calculate $H^2(N,\mathbb{C^{\star}})^T$, the $T$-stable ... | https://mathoverflow.net/users/49668 | How to claculate the $T$-stable subgroup of second cohomology group | The action is trivial, because the only action of $C\_{p^2}$ on the abelian group $\mathbb{Z}/p$ is trivial. You can also see it more directly, by thinking of a two cocycle as giving a twisted group algebra. If we call the cocycle $\alpha$, then we have the algebra $\mathbb{C}^{\alpha}N$, with basis $u\_g$ and multipli... | 2 | https://mathoverflow.net/users/41644 | 225243 | 105,289 |
https://mathoverflow.net/questions/225217 | 5 | Let $H$ be a finite $2$-group. Let $N\_{4}(H)$ be the subgroup generated by fourth powers. Let $H\_{4}$ be the last term in the short exact sequence $1\rightarrow N\_4(H) \rightarrow H \rightarrow H\_{4} \rightarrow 1$. Suppose that every element of $H\_{4}$ has a lift to an element of order $4$, $2$ or $1$ in $H$. Is ... | https://mathoverflow.net/users/6084 | Generalization of a lemma of Livne | The answer is yes. First, notice that if $\phi:G\rightarrow G'$ is an epimorphism of 2-groups, then $\phi(N\_4(G)) = N\_4(G')$. Let now $H$ be the group in your statement. Assume that $N\_4(H)$ is nontrivial. Then it contains a maximal subgroup $N$ of index 2 which is normal in $G$ (this follows from considering the ac... | 2 | https://mathoverflow.net/users/41644 | 225248 | 105,291 |
https://mathoverflow.net/questions/225231 | 2 | Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives *either* $X = 0$, *or* $X=A$, *or* det$(X) = 0$ and det$(X-A)=0$. It is the third case ($X$ is neither equal to 0, nor equal... | https://mathoverflow.net/users/18526 | How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite? | For any matrix $A$, if you make $X$ by keeping one column of $A$ and filling up with zeros, it trivially holds. For $n=2,3$ it may become a little bit more interesting if you request $X$ and $Y:=A-X$ not to contain an all-zero vector as a row or column. But for $n\ge4$ you may choose each of $X$ and $Y$ with two identi... | 2 | https://mathoverflow.net/users/29783 | 225252 | 105,292 |
https://mathoverflow.net/questions/222729 | 5 | Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let
$$D(n)=\{(d\_{1},\dots,d\_{n})\in\mathbb{Z}\_{+}^{n}:d\_{1}\geq\dots\geq d\_{n},\ \sum\_{i=1}^{n}d\_{i}\ \text{is even},\ \sum\_{i=1}^{k}d\_{i}\leq k(k-1)+\sum\_{i=k+1}^{n}\min\{k,d\_{i}\}\ \text{for all}\ 1\leq k\le... | https://mathoverflow.net/users/39026 | Volume of the convex hull of the set of all graphic sequences of a given length | The polytope considered by Richard Stanley is the polytope of degree sequences.
Sergiy Kozerenko's question is about the polytope of degree partitions for which see paper R46 of Volume 13 (2006) of the Electronic Journal of combinatorics. It is shown there that the volume of the polytope of degree sequences is n! times... | 2 | https://mathoverflow.net/users/83640 | 225255 | 105,293 |
https://mathoverflow.net/questions/225220 | 4 | Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups?
A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}\_2$ with this property?
| https://mathoverflow.net/users/83622 | Finite groups of order $n$ having exactly $n$ subgroups | There are finite groups of order $n$ having exactly $n$ subgroups
for $n = 1$, $2$, $6$, $8$, $28$, $36$, $40$, $40$, $48$, $54$, $72$,
$\dots$, and this list is exhaustive for $n < 96$.
The structures of the groups of order $< 96$ which satisfy the condition
are as follows:
* $1$,
* ${\rm C}\_2$,
* ${\rm S}\_3$,
* ... | 7 | https://mathoverflow.net/users/28104 | 225261 | 105,295 |
https://mathoverflow.net/questions/224135 | 6 | Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite $R$-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$.
**Definition**: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$; denoted by $\operatorname{grade}(I,M)$.
$\operatorname{grade}(m,M)$ is denoted by... | https://mathoverflow.net/users/47763 | Assuming $\operatorname{depth}M\ge \operatorname{depth}N$, what can one say about $\operatorname{depth}M_p$ and $\operatorname{depth}N_p$? | Let $R$ be the localization of $k[x, y, z]$ at $(x, y, z)$ where $k$ is a field. Let $M = R/(x)$ and $N = R/(y)$. Then the depth of $M$ and $N$ is $2$.
1. If $\mathfrak p = (x, y)$, then the depth of $M\_{\mathfrak p}$ is 1 and the depth of $N\_{\mathfrak p}$ is $1$.
2. If $\mathfrak p = (x, z)$, then the depth of $M... | 8 | https://mathoverflow.net/users/60618 | 225264 | 105,298 |
https://mathoverflow.net/questions/225250 | 2 | Consider a chevalley group a field $K$, with the right chevalley basis. Let $\alpha$ be a root. Let $x\_{\alpha}(t)$ be the corresponding root space. Define $w\_{\alpha}(t)=x\_{\alpha}(t)x\_{-\alpha}(-t^{-1})x\_{\alpha}(t)$ then define $h\_{\alpha}(t)=w\_{\alpha}(t)w\_{\alpha}(1)^{-1}$. Then chevalley proves in his exp... | https://mathoverflow.net/users/69289 | Maximal torus of Chevalley group $Sp(4)$ | The question is not well-formulated (for instance, it's not clear what you mean by "the right chevalley basis", and Chevalley is a proper name). Most important, your third sentence doesn't make sense: "Let $x\_\alpha(t)$ be the corresponding root space." The elements here should be in a related matrix group, which gets... | 5 | https://mathoverflow.net/users/4231 | 225282 | 105,307 |
https://mathoverflow.net/questions/225279 | 2 | Let $X$ be a smooth projective variety of dimension $d$ over $\mathbb C$ and let $E$ be a zero-dimensional coherent sheaf on $X$. The dual sheaf $E^D=\mathscr Ext\_X^d(E,\omega\_X)$ is again zero-dimensional.
>
> *Question*. Do $E$ and $E^D$ have support of the same length?
>
>
>
In other words, if $Z,Z^D\subs... | https://mathoverflow.net/users/30827 | Support of 0-dimensional sheaf and its dual | I misread your question, so the comments above are answering a different question than you asked. I thought you were asking about the lengths of the sheaf and its dual sheaf, not the lengths of the supports. At any rate, this does follow from duality, since the dual of the dual equals the original module (Proposition 5... | 4 | https://mathoverflow.net/users/13265 | 225284 | 105,308 |
https://mathoverflow.net/questions/225251 | 3 | Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & 1 & 1 & 1 & 0 & 0 &1\\
1 & 1 & 1 & 1 & 1 & 0 & 0 &1\\
1 & 1 & 1 & 1 & 1 & 1 & 1 &0\\
\end{array}
\end{... | https://mathoverflow.net/users/78180 | Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues? | There are some quantifiers unclear in your question, but regardless of how
to read it, your assertion is false. -- The smallest counterexample with
blocks of pairwise distinct size all of whose eigenvalues are integers has
blocks of size $5$, $8$ and $12$, and set of eigenvalues $\{-10,-6,0,16\}$.
This can be found wi... | 6 | https://mathoverflow.net/users/28104 | 225289 | 105,310 |
https://mathoverflow.net/questions/225145 | 6 | I'm looking for a parameterization $(x\_1(u,v),x\_2(u,v),x\_3(u,v),x\_4(u,v))$ of a knotted sphere in $\mathbb R^4$. How might one go about finding such a parameterization?
| https://mathoverflow.net/users/66774 | Parameterization of a knotted surface? | If you know how to parameterize a nontrivial knot in $R^3$ (say, the [trefoil](https://mathoverflow.net/questions/91444/what-is-parameterization-of-the-trefoil-knot-surface-in-r%C2%B3?rq=1) knot), you can use Artin's "spinning" construction to parameterize a nontrivial knot in $R^4$ via map $S^2\to R^4$ written in sphe... | 10 | https://mathoverflow.net/users/21684 | 225305 | 105,314 |
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