parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/297320 | 2 | I am given $3$ points $(x\_i,y\_i,z\_i) \in \mathbb{R}^3\setminus \{\mathbf{0} \}$, for $i=1,2,3$, satisfying the following two polynomial equations (the first equation is actually not the intended one, see edit 1):
$$3+z\_1+z\_2+z\_3-\frac14(z\_1x\_2x\_3 + z\_2x\_3x\_1 + z\_3x\_1x\_2) - \frac14(z\_1y\_2y\_3 + z\_2y\... | https://mathoverflow.net/users/81645 | How to show that the origin is not in the convex hull, in this problem? | I actually answered my question in edit 1. I apologize to Iosif Pinelis who had found a counterexample to my original version of the question, but after correcting my question (which I got via a calculation), the answer follows easily from Cauchy-Schwarz (see edit 1). This post is probably confusing to readers, but I h... | 0 | https://mathoverflow.net/users/81645 | 299896 | 131,553 |
https://mathoverflow.net/questions/299903 | 0 | Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation
$$u\_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$
where $f$ and $1-c$ are compactly supported in $\Omega$. Suppose $u(x,t)=0$ on $\Gamma \times (0,\infty)$. Can one guarantee that $u(x,t)... | https://mathoverflow.net/users/42326 | Unique continuation for the wave equation | Try $u(x,t) = x\_1 t$ with $\Gamma = \{x:\; x\_1 = 0\}$.
| 0 | https://mathoverflow.net/users/13650 | 299907 | 131,555 |
https://mathoverflow.net/questions/299910 | 6 | I am trying to understand cocycle description of gerbes as in <https://arxiv.org/pdf/math/0611317.pdf>.
Let $\mathcal{P}$ be a gerbe on a topological space $X$ i.e., $\mathcal{P}$ is a stack over groupoids that is both locally non empty and locally connected.
As $\mathcal{P}$ is locally nonempty, we can choose an ... | https://mathoverflow.net/users/118688 | Cocycle description of gerbes | It might help to divide this process into finer steps. We choose our open cover so that the groupoids $\mathcal{P}(U\_i)$ and $\mathcal{P}(U\_i \cap U\_j)$ are connected and nonempty. Choose objects $x\_i \in \mathcal{P}(U\_i)$ and $x\_{i,j} \in \mathcal{P}(U\_i \cap U\_j)$, and let $G\_i = \operatorname{Aut}(x\_i)$ an... | 4 | https://mathoverflow.net/users/121 | 299917 | 131,557 |
https://mathoverflow.net/questions/299748 | 7 | Let's start from a little bit far.
Basic probability theory - [chain rule](https://en.wikipedia.org/wiki/Chain_rule_(probability)) reads:
$$ P(AB) = P(A)P(B|A)$$
**Example:** consider n+m balls, where n - white balls, m - black balls,
consider A - first chosen ball is white, B - second chosen ball is also white.... | https://mathoverflow.net/users/10446 | Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$ | Let me begin with a sketch of an answer to your question two, namely giving an interpretation of the cross-multiplied equality $$\text{Gr}(2,n)\mathbb{P}^{n+m-1}\mathbb{P}^{n+m-2}=\text{Gr}(2,n+m)\mathbb{P}^{n-1}\mathbb{P}^{n-2}.$$
I claim this equality holds in the [Grothendieck ring of stacks](https://arxiv.org/pdf/0... | 14 | https://mathoverflow.net/users/6950 | 299925 | 131,559 |
https://mathoverflow.net/questions/299688 | 8 | What would be an example of a connected finite group scheme over a field $k$ that is not solvable? Here $k$ is algebraically closed.
Let $\operatorname{GL}\_n$ be the general linear group scheme over a field of characteristic $p>0$. Now consider the $H=\operatorname{ker}{F^r:\operatorname{GL}\_n \to \operatorname{GL}... | https://mathoverflow.net/users/124202 | Example of a connected finite group scheme which is not solvable | The connected finite kernel $H$ is not solvable, provided that $n>2$ or $p>2$, see the edit below.
$\def\eps{\varepsilon} \def\m{\mathfrak{m}}$Suppose by contradiction that $H$ is solvable and the $m$-th derived group of $H$ is trivial (where $[H,H]$ is the *first* derived group). It implies that for any $k$-algebra ... | 9 | https://mathoverflow.net/users/39304 | 299926 | 131,560 |
https://mathoverflow.net/questions/299928 | 5 | As we know the figure-8 ($4\_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}\_3)$. The resulting complete hyperbolic manifold has a cusp corresponding to the torus boundary.
On the other hand, the "conformal boundary" (bounda... | https://mathoverflow.net/users/49556 | Conformal boundary and cusp of figure-8 complement | The group of parabolic isometries fixing a point at infinity is isomorphic to ${\mathbb C}$. (Because it acts simply transitively on a horosphere $H$.) The discrete group $\Gamma$ intersects this stabilizer of the parabolic fixed point (i.e., the stabilizer of $H$) in a lattice $\Lambda\subset{\mathbb C}$. The quotient... | 3 | https://mathoverflow.net/users/39082 | 299932 | 131,562 |
https://mathoverflow.net/questions/299921 | 2 | Let $X$ be a locally compact, Hausdorff, topological space and denote by $\mathcal B\_+(X)$ the collection of all Borel-measurable functions from $X$ to $[0,+\infty]$ (extended positive reals).
Suppose that we are given a map
$$
I:\mathcal B\_+(X) \to [0,+\infty]
$$
which is positively homogeneous and satisfies
... | https://mathoverflow.net/users/97532 | Daniell integral vs. Borel measure | Here is an answer for $X$ metrizable and $\sigma$-compact. I assume $I$ is finite on compactly supported continuous functions.
Recall that a locally compact metrizable $\sigma$-compact space is Polish (separable and completely metrizable). Moreover, there is an increasing sequence of compact sets $\langle K\_n\rangl... | 1 | https://mathoverflow.net/users/35357 | 299933 | 131,563 |
https://mathoverflow.net/questions/299318 | 17 | In the following, we work with vectors and matrices whose entries are rational numbers. Inequalities between such vectors are understood to be coordinatewise: e.g., two vectors $a = \left(a\_1,a\_2,\ldots,a\_n\right)^T$ and $b = \left(b\_1,b\_2,\ldots,b\_n\right)^T$ are said to satisfy $a > b$ if and only if each $i \i... | https://mathoverflow.net/users/2530 | Constructive proof of a rational version of Perron-Frobenius? | Yes, both Theorem 1 and Theorem 2 have constructive proofs.
In the following, I will work with rational numbers, but the same arguments
work for any ordered field. (Note that in constructive logic, fields are
automatically discrete -- i.e., any two elements of a field are either equal
or not. Therefore, I don't think... | 5 | https://mathoverflow.net/users/2530 | 299937 | 131,564 |
https://mathoverflow.net/questions/263630 | 5 | A famous theorem of Drozd says that every finite dimensional hereditary algebra is either of tame or wild representation type. I am interested in infinite dimensional hereditary algebras. Is there a known dichotomy result for this situation? Are extra assumptions necessary?
| https://mathoverflow.net/users/41283 | Tame-Wild Dichotomy theory for infinite dimensional Hereditary algebras | This question appears to have been posted a long time ago, but I shall answer for the benefit of future searchers. This question (in a more general setting) has been addressed in a very recent [paper](https://arxiv.org/abs/1803.00173) by Iovanov. I have not gone through the paper in detail yet, as I have just stumbled ... | 4 | https://mathoverflow.net/users/95742 | 299941 | 131,565 |
https://mathoverflow.net/questions/299784 | 1 | Is there a simple $C^\*$ algebra $A$, not isomorphic to $M\_2(\mathbb{C})$, such that for every commutator element $x=ab-ba$, $x^2$ is an scalar element?
| https://mathoverflow.net/users/36688 | Simple $C^*$ algebras whose all commutator elements have scalar square | The answer is no:
>
> A simple C\*-algebra in which $[x,y]^2$ is central for all $x,y$ is isomorphic to $M\_n(\mathbf{C})$ for $n\in\{1,2\}$.
>
>
>
Let $A$ be a C\*-algebra. Let $H$ be an irreducible representation of $A$. By irreducibility its commutant is reduced to scalars, and hence its double commutant is... | 5 | https://mathoverflow.net/users/14094 | 299943 | 131,566 |
https://mathoverflow.net/questions/299938 | -3 | Is there a known result to the effect that it cannot be the case that for some natural $n$, there is a formula of length $n$ such that all cardinals can be defined by a formula whose length is shorter than $n$?
I'm speaking in the milieu of some fragment of a standard set theory extending first order logic with ident... | https://mathoverflow.net/users/95347 | Can there be an upper bound on definability of cardinal numbers in ZF? | Yes. This is an easy consequence of the reflection theorem. The point is that we can reason about definability with respect to a small collection of formulas within a model.
Put all formulas of length $\leq n$ (up to renaming variables) in a finite set $S$. Let $V\_\alpha$ be such that for all $\phi(x) \in S$, and al... | 8 | https://mathoverflow.net/users/11145 | 299944 | 131,567 |
https://mathoverflow.net/questions/299036 | 9 | Given a polynomial $f(x)\in \mathbb{Z}[x]$ of degree at least 2 and a positive integer $t$, does there always exist infinitely many primes $p$ such that the range of $f(x)$ modulo $p$ does not contain the $t$ consecutive numbers $s+1,\ldots, s+t$ for some integer $s$ (where $s$ can depend on $p$)?
This is known to b... | https://mathoverflow.net/users/81443 | Consecutive polynomial non-residues modulo a prime | Let $d = \deg f$. Consider the covering of $\mathbb A^1\_\mathbb Q$, with coordinate $s$, defined by $f(x\_1) =s+1 ,\dots, f(x\_t) =s +t$. The monodromy representation of this covering gives a map from the etale fundamental group of $\mathbb A^1$ minus the finite set of critical values to $\prod\_{i=1}^t S\_d$.
The ... | 4 | https://mathoverflow.net/users/18060 | 299946 | 131,568 |
https://mathoverflow.net/questions/299951 | 6 | I would like to have a good reference where the results in
"Inclusion relations between transitive compact transformation groups" <https://mathscinet.ams.org/mathscinet-getitem?mr=27:3740>
can be found in English/French/Spanish.
I am particularly interested about decompositions of compact simple Lie groups. I ha... | https://mathoverflow.net/users/69159 | Where to find the results of Onishchik? | This paper was translated to English and appears in *[Fifteen papers on algebra](https://bookstore.ams.org/trans2-50/).* American Mathematical Society Translations. Series 2. Vol. 50; American Mathematical Society, Providence, R.I. 1966; [MR189949](https://mathscinet.ams.org/mathscinet-getitem?mr=0189949).
This part... | 7 | https://mathoverflow.net/users/8250 | 299952 | 131,570 |
https://mathoverflow.net/questions/299901 | 2 | Suppose that $X$ is a metric space and let $C\_k(X)$ denote the space of real functions on $X$ with the topology of uniform convergence on compact sets. Then $C\_k(X)$ is a topological vector space. Let $C\_k(X)^\*$ denote the strong dual space. For each $x\in X$ the point evaluation functionals $\delta\_x$ belong to $... | https://mathoverflow.net/users/124321 | Boundedness of Dirac deltas | $S \subseteq C\_k(X)^\*$ is bounded in the strong dual topology if and only if $\sup \_{\omega \in S, f \in B} |\omega (f)| < +\infty$ for every bounded $B \subseteq C\_k(X)$.
$B \subseteq C\_k (X)$ is bounded if and only if $\sup \_{f \in B, x \in K} |f(x)| < +\infty$ for every $K \subseteq X$ compact.
Call your s... | 3 | https://mathoverflow.net/users/54780 | 299961 | 131,573 |
https://mathoverflow.net/questions/299962 | 2 | Q1. Does there exist a separable Banach space $X$ satisfying in the following property?
1- $X^\*$ is non separable.
2- For every countable subset $F\subset X^\*$ there exists $0\neq x\_F\in X$ such that
$f(x\_F)=0$ for all $f\in F$.
Q2. If it is impossible, what about if we replace $X$ by a separable topologic... | https://mathoverflow.net/users/84390 | A particular separation example | Suppose that $X$ is a separable Banach space. Then $X^\*$ is weak\*-separable, so take a countable weak\*-dense set $F$ in $X^\*$. Consequently, [$F$ is total](https://math.stackexchange.com/questions/1859334/g-is-dense-in-x-in-weak-sense-if-and-only-if-g-is-total-set), hence the only element on which all elements of $... | 4 | https://mathoverflow.net/users/15129 | 299965 | 131,575 |
https://mathoverflow.net/questions/299955 | 1 | The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\mathcal{L}$ be an invertible sheaf on $\mathcal{C}$ of relative degree $0$ (i.e., for all $t \in \mathbb{A}^n$, $\deg(\ma... | https://mathoverflow.net/users/43198 | Variation of global sections of line bundles | For a general smooth base $S$ it is not true that $T$ is regular. Assume even that $\mathcal{C} = C \times S$ for a fixed smooth curve $C$. The line bundle $\mathcal{L}$ then defines a morphism $f \colon S \to Pic^0(C)$ and then $T = f^{-1}(\{0\})$. Conversely, given a morphism $f$ recovers back the line bundle $\mathc... | 2 | https://mathoverflow.net/users/4428 | 299968 | 131,576 |
https://mathoverflow.net/questions/299900 | 9 | Let $V\_\lambda$ be the irreducible representation of $sl\_{n}(\mathfrak{C})$ with highest weight $\lambda$. There are well known formulas for the decomposition of $V\_\lambda^{\otimes^k}= V\_\lambda\otimes V\_\lambda\otimes \cdots\otimes V\_\lambda$ into irreducble representations using Littelwood-Richardson, or Litte... | https://mathoverflow.net/users/48969 | Symmetric Powers for Lie Algebras | For $sl\_2$ one has formulas. If $V\_n=S^n(\mathbb{C}^2)$, then the multiplicity of the irreducible $V\_k$ inside $S^m(V\_n)$ is
$$
M\_{m,n,k}=\left[m,n,\frac{mn-k}{2}\right]-\left[m,n,\frac{mn-k}{2}-1\right]
$$
where $[m,n,w]$ is the number of integer partitions of $w$ with at most $n$ parts of length $\le m$.
This is... | 7 | https://mathoverflow.net/users/7410 | 299971 | 131,578 |
https://mathoverflow.net/questions/299967 | 8 | I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that for every $v\in A$ there exists $x\_v\in \mathbb R^2$ such that $\mu(x\_v+v\mathbb R)>0$.
(A weaker requirement on $\... | https://mathoverflow.net/users/5628 | Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms? | **Such a measure cannot exist.** Suppose to the contrary that we have an uncountable family of lines $\ell$ such that $\mu(\ell)>0$. Then there is $\epsilon>0$ and an infinite family of lines $\{\ell\_i\}\_i$ with $\mu(\ell\_i)\geq\epsilon$. These lines intersect at countably many points. Since the measure has no atoms... | 12 | https://mathoverflow.net/users/121665 | 299973 | 131,579 |
https://mathoverflow.net/questions/299981 | 2 | The falling factorials are $(x)\_n=x(x-1)\cdots(x-n+1)$ with $(x)\_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by
$$Ef(x)=f(x+1) \qquad \text{and} \qquad \delta f(x)=f(x+1)-f(x).$$
Let $\lambda=(\lambda\_1,\dots,\lambda\_k)\vdash n$ be a partition of a positive integer $n... | https://mathoverflow.net/users/66131 | These polynomials are always either even or odd | We first note that $\Psi\_\lambda$ is either symmetric or odd-symmetric in $E$. In other words, if we think of $\Psi\_\lambda$ as a polynomial $P$ applied to the operator $E$ (of degree $n - 1$), then $P(t)=\pm t^{n-1} P(\frac{1}{t})$ (plus if the number of elements $|\lambda|$of the partition is odd, minus if even). T... | 5 | https://mathoverflow.net/users/44191 | 299985 | 131,582 |
https://mathoverflow.net/questions/299947 | 3 | Let $G$ be a finite or compact group and $\rho: G \to \mathrm{U}(d)$ a $d$-dimensional unitary representation of $G$. If $\rho$ is irreducible then the following seems to be true:
$$
\mathrm{span}\_\mathbb{C} \{\rho(g) : g \in G\} = \mathbb{C}^{d \times d}
$$
i.e., the complex linear span of $\rho(g)$, taken over all $... | https://mathoverflow.net/users/37211 | Is the linear span of irrep matrices a complete matrix basis? | This is known as Burnside's theorem. Nowadays people formulate it as any algebra of matrices over an algebraically closed field acting irreduciblly is the whole matrix algebra.
| 5 | https://mathoverflow.net/users/15934 | 299995 | 131,587 |
https://mathoverflow.net/questions/300005 | 5 | Is there an example of a smooth complex projective variety and an $S^{2}$-bundle over it which is not diffeomorphic to a complex projective variety?
| https://mathoverflow.net/users/99732 | $S^{2}$-bundles over complex projective varieties | Yes. Let $E\to T^2$ be the $S^2$-bundle over the torus $T^2$ obtained by gluing the trivial bundle over $[0,1]\times[0,1]$ by the identity map in the first coordinate and a reflection in the second coordinate. The torus admits the structure of a smooth complex projective variety, but $E$ isn't orientable, so cannot be ... | 3 | https://mathoverflow.net/users/97265 | 300008 | 131,590 |
https://mathoverflow.net/questions/278883 | 18 | The $q,t$-Kostka polynomials $K\_{\lambda,\mu}(q,t)$ appear as the change of basis coefficients between Macdonald polynomials $H\_\mu(x;q,t)$ and Schur functions $s\_\lambda(x)$:
$$H\_\mu(x;q,t)=\sum\_{\lambda\vdash|\mu|}K\_{\lambda,\mu}(q,t)s\_\lambda(x)$$
Macdonald conjectured that $K(q,t)\in\mathbb{N}[q,t]$ and this... | https://mathoverflow.net/users/57454 | What is currently known or conjectured about q,t-Kostka polynomials? | To answer some of your questions - note that it suffices to find the Schur-expansion of certain LLT polynomials, in order to figure out the $qt$-Kostka polynomials. There IS a combinatorial description of LLT polynomials, it is quite easy in fact. However, from the definition, it is not clear that they are Schur positi... | 4 | https://mathoverflow.net/users/1056 | 300023 | 131,595 |
https://mathoverflow.net/questions/65142 | 3 | Background
----------
Working on a quantum mechanics problem, I've stumbled on the problem of maximizing the functional
$$\int\_{A} \varphi\_m \varphi\_n$$
in the limit of large $m$ and $n$, given that $n \gg m$. The function $\varphi\_m$ is the $m$-th Hermite function, defined as
$$\varphi\_m(x) = (2^m m! \sqrt{\pi}... | https://mathoverflow.net/users/9211 | Asymptotic form of $L^1$-norm of Hermite functions | Results for the $L^p$ norms of general orthogonal polynomials have been given in Ref. [1].
In the case of the orthonormal Hermite polynomials $H\_n$, it was given in [1] for $L^{2p}$ norms, $0<p<4/3$, and then in [2], for any $0<p$. Let
$$N\_n(p)=\int\_{\mathbb R}|H\_n(x)e^{-x^2/2}|^{2p}dx.$$
Then, for $p<4/3$, as $... | 2 | https://mathoverflow.net/users/89429 | 300031 | 131,597 |
https://mathoverflow.net/questions/300024 | 8 | I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int\_{\mathbb R}f(x)e^{ix\xi}\,dx$) and where $f$ is an even function.
Looking here: [wiki](https://en.wikipedia.org/wiki/Fourier_transform#Di... | https://mathoverflow.net/users/38290 | About the Fourier transform of the logarithm function | As mentioned by Mateusz Kwaśnicki, the $1/|x|$ in the Fourier transform of the logarithm should be regularised in a "principal value" type of way, as explained for example in this [MSE posting](https://math.stackexchange.com/a/138013/87355). To check that everything works out, it helps to walk through a specific exampl... | 3 | https://mathoverflow.net/users/11260 | 300037 | 131,600 |
https://mathoverflow.net/questions/300032 | 0 | For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$).
I was wondering if this is also a necessary condition for this to happen, that is whether a form with this property must be either symmetric or antisymmetric.
| https://mathoverflow.net/users/124321 | Vanishing bilinear forms | Assume we are not in characteristic 2, then this is true. Let $x\_0 \in V$. Then the functional $\phi(x\_0,y)$ of $y$ and the functional $\phi(y,x\_0)$ have the same zero set. It follows that they are proporstional: $\phi(x\_0,y) = \lambda(x\_0)(y,x\_0)$ (or the opposite, but it will not change anything for the proof).... | 2 | https://mathoverflow.net/users/115052 | 300039 | 131,602 |
https://mathoverflow.net/questions/300029 | 3 | I am trying to evaluate the integral
$$
I\_k(x)=\int\_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt
$$
with $x$ tending to infinity.
In fact, I wish to have an estimate
$$
\sum\_{k=0}^\infty \frac{1}{\log^k x} \int\_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt = C+E(x),
$$
where $C$ is explicit and $E(x) \to 0$ as $x \to \infty$.... | https://mathoverflow.net/users/95838 | Evaluating the integral $\int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt$ | First of all, your series diverges for any $x>1$, because we always have
$$I\_k(x)\geq \int\_{\sqrt{x}}^x \log^k t\sqrt{t-1}t^{-2}dt\geq \frac{\sqrt{\sqrt{x}-1}}{x}\int\_{\sqrt{x}}^{x} \frac{\log^k t}{t} dt=\frac{\sqrt{\sqrt{x}-1}}{x}\frac{(\log^{k+1} x)(1-2^{-k-1})}{k+1}.$$
If you want to get the asymptotics for $... | 4 | https://mathoverflow.net/users/101078 | 300042 | 131,603 |
https://mathoverflow.net/questions/300040 | 0 | Given $h>0$, I would like to estimate the following summation by some function $f(N)$:
$$
S\_N=\sum\_{n=2}^{N} \frac{\log n}{\log^h(\log n)}=O(f(N)).
$$
Obviously, we see that
$$
S\_N>\sum\_{n=2}^{N} \log^\alpha n=O(N^\alpha\log^\alpha(N)), ~~\text{for every}~~0<\alpha<1,
$$
and
$$
S\_N<\sum\_{n=2}^{N} \log n=O(N\... | https://mathoverflow.net/users/45092 | Summation of $\log n/ \log(\log n)$ | I will answer the question in the title.
Put $f(x) = \sum\_{n \leq x} \log n$. We then have, by partial summation,
$\displaystyle \sum\_{n\leq x} \frac{\log n}{\log \log n} = \frac{1}{\log \log x} f(x) + \int\_1^x \frac{1}{t (\log \log t)^2 \log t} f(t)dt.$
We have an asymptotic formula for $f(x)$, namely
$\di... | 3 | https://mathoverflow.net/users/10898 | 300044 | 131,604 |
https://mathoverflow.net/questions/300034 | 3 | Klarner's theorem (<http://mathworld.wolfram.com/KlarnersTheorem.html>) says in a special case that you cannot tile a $10 \times 10$-board with $1\times 4$-tiles (that can also be rotated and used as $4 \times 1$-tiles). The reason is simple, for $1 \leq n,m \leq 10$, there are more pairs $(n,m)$ with $n+m=2$ modulo $4... | https://mathoverflow.net/users/8176 | Klarner's theorem | This is not exactly the same thing, but it is similar.
Let $\lambda = (10,10,\dotsc,10)$, and $\mu=(4,4,\dotsc,4)$.
The non-tileability implies that the irreducible character, $\chi^{\lambda}(\mu)$ is $0$. Furthermore, since $\mu$ has all entries equal, turns out that $|\chi^{\lambda}(\mu)|$ is an upper bound on the ... | 1 | https://mathoverflow.net/users/1056 | 300045 | 131,605 |
https://mathoverflow.net/questions/300022 | 11 | It is easy to see (using AC, of course) that there exist two sets $U\subset\mathbb{R}$ and $V\subset\mathbb{R}$ such that any real number $x$ can be represented as unique sum $x=u+v$, where $u\in U$ and $v\in V$.
There are $2^{2^\omega}$ such $(U,V)$ pairs.
The question given by my son: prove existence of pair $(U,V)... | https://mathoverflow.net/users/118366 | Existence of subset of reals such that any real number is unique sum of exactly two elements of the subset | The usual transfinite construction works. Let $\{r\_{\alpha} : \alpha < \mathfrak{c}\}$ list $\mathbb{R}$. Construct $\{U\_{\alpha}: \alpha < \mathfrak{c}\}$ by induction on $\alpha$ such that the following hold.
(a) For every $a, b, c, d \in U\_{\alpha}$, $a + b = c + d \implies \{a, b\} = \{c, d\}$.
(b) There are... | 9 | https://mathoverflow.net/users/124387 | 300047 | 131,606 |
https://mathoverflow.net/questions/272785 | 1 | In his book *Arithmeticity in the theory of automorphic forms* (<http://bookstore.ams.org/surv-82-s>) Shimura introduces at page 146 an operator $\Delta\_p^q$ which should act on nearly holomorphic modular forms (by raising weight and non-holomorphy degree).
Is this (a version of) the Shimura-Maass differential opera... | https://mathoverflow.net/users/98823 | (Reference) A Shimura operator acting on Hermitian modular forms | It turns out that the answer to my question was in a previous paper by Shimura, namely Differential operators, holomorphic projection, and singular forms (lemma 4.2, <https://projecteuclid.org/euclid.dmj/1077286742>).
The operator is (up to a constant factor)
$$\Delta\_q=\det(Y)^{\kappa-1-q}\det(d/dZ)\det(Y)^{q-\kapp... | 3 | https://mathoverflow.net/users/98823 | 300053 | 131,607 |
https://mathoverflow.net/questions/295003 | 4 | The paper I'm referring to is <https://projecteuclid.org/euclid.dmj/1077303203>. Here Shimura constructs an Hermitian Eisenstein series $E\_m(z,k,s,\psi,\mathfrak{b})$ for, in the case I am interested in, a variable $z$ in the Hermitian upper half plane ($m\times m$ complex matrices $z$ for which $i(z-\overline{z}^t)$ ... | https://mathoverflow.net/users/98823 | Main result of Shimura's On Eisenstein Series | The paper just states the fact that $E\_n(z,n+1,0)$ is in general non holomorphic but has a somewhat explicit form.
Nagaoka, in Note on Siegel-Eisenstein series of low weight (<https://link.springer.com/article/10.1007%2FBF02940802>, theorem 2.2), shows that for some values of $n$ the Eisenstein series in question is... | 0 | https://mathoverflow.net/users/98823 | 300054 | 131,608 |
https://mathoverflow.net/questions/300046 | 15 | (Prompted by reflection on [this old answer](https://mathoverflow.net/a/219595/2273), and its suggestion of the “harmlessness” of the axiom of regularity.)
In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as being “convenient, and harmless”, as follows. If $V = (V,\epsilon)$ is a mode... | https://mathoverflow.net/users/2273 | Does foundation/regularity have any categorical/structural consequences, in ZF? | Yes, the axiom of foundation has structuralist consequences.
Let $\phi$ be the assertion, "if every well-founded set is well-orderable, then every set is well-orderable."
This statement, I claim, can be made in a (technically) structuralist manner, since it is a statement about all nodes in a certain kind of digr... | 17 | https://mathoverflow.net/users/1946 | 300055 | 131,609 |
https://mathoverflow.net/questions/300049 | 8 | A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative sectional curvature the sum of all of its Betti numbers is at most $C(n)$.
On the other hand, for $n=2$ only sphere, tor... | https://mathoverflow.net/users/16183 | Homeomorphism/ homotopy types of non-negatively curved manifolds | As mentioned in comments, the first dimension where an infinite family of pairwise non-homeomorphic closed nonnegatively curved manifolds occurs is $3$ (the lens spaces). The question becomes more challenging for simply-connected manifolds. If (as expected) simply-connected nonnegatively curved manifolds are rationally... | 14 | https://mathoverflow.net/users/1573 | 300062 | 131,613 |
https://mathoverflow.net/questions/300063 | 10 | For a group $(G,\star)$, an element $x\in G$ is said to be *square* if there is $y\in G$ such that $x=y\star y$.
My question is: For which kinds of group $G$, can we decide whether $x\in G$ is a square or not?
As for the multiplicative group $(\mathbb Z/n\mathbb Z)^\*$ (where $n$ is a prime or the production of so... | https://mathoverflow.net/users/103866 | For which kinds of group $G$, can we identify a square element efficiently? | For a finite group $G$, there is a long-standing answer to this question using character theory. If $\chi$ is an irreducible complex character of $G$, then its Frobenius-Schur indicator $\nu(\chi)$ is defined by $\nu(\chi) = \frac{1}{|G|} \sum\_{g \in G} \chi(g^{2}).$ It is known that $\nu(\chi) \in \{0,1,-1\},$ and th... | 19 | https://mathoverflow.net/users/14450 | 300064 | 131,614 |
https://mathoverflow.net/questions/300072 | 7 | Let $F = \operatorname{GF}(2^n)$ be a finite field. Define a permutation $\phi:F \rightarrow F$ by the formula
$$
\phi(x) = x^{-1}, \ x\neq 0; \ \phi(0) =0.
$$
We say that a permutations $\psi$ of $F$ is linear if $\psi(x+y) = \psi(x)+\psi(y)$.
It is not hard to see that all automorphisms $\sigma\_k: x\rightarrow x^{... | https://mathoverflow.net/users/85489 | Linear permutations commuting with $x\rightarrow x^{-1}$ | The answer is **no**: a linear transformation of $F$ which commutes with $\phi$ is an automorphism of $F$.
This is a seemingly inelegant but simple argument.
Any map $\psi: F \rightarrow F$ can be represented uniquely as a polynomial $P\_\psi (x) = \sum a\_i x^i$ of degree less than $2^n$, with each $a\_i \in F$. I... | 16 | https://mathoverflow.net/users/44191 | 300079 | 131,617 |
https://mathoverflow.net/questions/299999 | 5 | Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.
>
> Let $A,B\in \mathcal{B}(F)^+:=\left\{T\in \mathcal{B}(F);\,\langle Tx, x\rangle\geq 0,\;\forall\;x\in F\;\right\}$, be such that $AB\neq 0$. I want to show that
> $$\sigma(AB)\neq\{0\}\;.$$
>... | https://mathoverflow.net/users/116483 | Spectrum of the product of operators | Use the fact that $\sigma(ST) \cup \{0\} = \sigma(TS) \cup \{0\}$. So if $A$ and $B$ are positive then, except possibly for the point $0$, $\sigma(AB)$ equals $\sigma(A^{1/2}BA^{1/2})$. If $AB \neq 0$ then the latter is a nonzero positive operator, and hence it has a nonzero element in its spectrum. Therefore so does $... | 5 | https://mathoverflow.net/users/23141 | 300080 | 131,618 |
https://mathoverflow.net/questions/300081 | 2 | Let $E/F$ be a quadratic extension of global fields. Denote by $J\_F$ the idèle group of $F$, by $P\_F$ the subgroup of principal idèles and by $N\_{E/F} : J\_E \to J\_F$ the norm map between idèles. In Timothy O'Meara's "Introduction to Quadratic Forms" it is shown that $[J\_F : P\_FN\_{E/F}J\_E] = 2$ (65:21), which i... | https://mathoverflow.net/users/123436 | $[J_F : P_FN_{E/F}J_E] = 2$ for quadratic extensions of global fields of characteristic 2? | Yes, this is true for separable extensions of global fields of arbitrary characteristic. More generally, Artin reciprocity gives an isomorphism $J\_F / P\_F {N\_{E/F} J\_E} \cong \operatorname{Gal}(E/F)^{\text{ab}}$, and in your case the latter is order 2. (In most expositions you'll see the left hand side written as $... | 5 | https://mathoverflow.net/users/422 | 300082 | 131,619 |
https://mathoverflow.net/questions/300108 | 2 | Let us consider the space of convergent sequences which is denoted by $c$. The space of all sequences $(x\_n)\in c$ with $\lim x\_n=0$ is also denoted by $c\_0$. Clearly $c\_0$ is a proper closed subspace of $c$ under the uniform topology.
These two Banach spaces are the same if and only they are the same as two com... | https://mathoverflow.net/users/84390 | $c_0$ is not isometrically isomorphic to $c$ | The (multiplicative) Banach–Mazur distance between $c$ and $c\_0$ is exactly 3:
>
> M. Cambern, [On mappings of sequence spaces](http://matwbn.icm.edu.pl/ksiazki/sm/sm30/sm3017.pdf), *Studia Math.* **30**. (1968), 73-77.
>
>
>
Let me take this opportunity to advertise a rather crazy conjecture (due to Pełczyńs... | 14 | https://mathoverflow.net/users/15129 | 300110 | 131,627 |
https://mathoverflow.net/questions/300083 | 9 | I suspect that the following matrix inequality is well known, but I can't find a reference or proof:
Given $n \times n$ symmetric matrices $A,B$ such that $I\_n \leq A,B$, is the following true?
$${Tr}\big[\big( \log A^{\frac{1}{2}} B A^{\frac{1}{2}}\big)^p\big] \geq {Tr}\big[\big( \log B \big)^p\big] + {Tr}\big[\big... | https://mathoverflow.net/users/17965 | Well known matrix inequality? | The desired inequality follows from an majorisation argument :
According to Bhatia Matrix Analysis Corollary III.4.6 we have :
$log \lambda(AB) \succ log \lambda^\downarrow(A) + log \lambda^\uparrow(B)$.
Using Corollary II.3.4 from the same book, we conclude that
$(log \lambda(AB))^p \succ\_w (log \lambda^\d... | 7 | https://mathoverflow.net/users/17261 | 300121 | 131,628 |
https://mathoverflow.net/questions/300119 | 6 | Let $K$ be a $p$-adic local field, for example $\mathbb{Q}\_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$($v\ge -1$) be the ramification groups in upper numbering, then is it true that $\bigcap\_{v=0}^\infty G^v=\{0\}$?
| https://mathoverflow.net/users/108486 | Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial? | Yes, at least if the upper ramification groups $G^\nu$ are defined as $\varprojlim\_L\mathrm{Gal}(L/K)^\nu$ for $L/K$ finite Galois (e.g. as in [1]). This makes sense because the upper-numbering is compatible with quotients. Then it follows from the fact that for each finite $L/K$, $\mathrm{Gal}(L/K)^\nu$ is trivial fo... | 7 | https://mathoverflow.net/users/2604 | 300123 | 131,629 |
https://mathoverflow.net/questions/233131 | 1 | Consider the Cauchy problem for the heat equation
$u\_t=\Delta u$, $u|\_{t=0}=\varphi$.
S. Täcklind showed its solution $u$ is unique in the class $|u|\le e^{|x|h(|x|)}$, $|x|>1$, iff $\int\_1^\infty \frac1{h(y)}\,dy=\infty$.
From the other hand if $|\varphi(x)|\le e^{cx^2}$ then at least for small enough $t>0$ the ... | https://mathoverflow.net/users/14551 | Integral representation of the Cauchy problem solution for the heat equation | In the book of Eidelman and Zhitarasu, Parabolic boundary value problems, Springer 1998, it is said that:
``using Tacklind's arguments in [76], it has been proved that if the initial function $\psi(x)$ satisfies the inequality $\psi(x)\geq \exp(|x|h(|x|))$, where $h$ is such that $\lim\_{r\to\infty} h(r)/r = \infty$,... | 1 | https://mathoverflow.net/users/124312 | 300129 | 131,630 |
https://mathoverflow.net/questions/300130 | 3 | If $G=(V,E)$ is a simple, undirected graph, we say $C\subseteq V$ is a *vertex cover* if for any $e\in E$ we have $e\cap C \neq \emptyset$.
Let $n>1$ be an integer. If $L(K\_n)$ denotes the [line graph](https://en.wikipedia.org/wiki/Line_graph) of the complete graph on $n$ vertices, what is the minimal cardinality th... | https://mathoverflow.net/users/8628 | Vertex-covering number of $L(K_n)$ | One can prove that a graph $G$ has a vertex cover of size $n - k$ if and only if the complement of $G$ has a clique of size $k$, by choosing complementary vertex sets which then fulfill the requirements.
Now the complement of $L(K\_n)$ is the well-known Kneser graph $K(n,2)$, by definition of [Kneser graphs](http://m... | 5 | https://mathoverflow.net/users/71028 | 300133 | 131,632 |
https://mathoverflow.net/questions/300104 | 2 |
>
> Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $e$ the idempotent such that $eA$ is the direct sum of all indecomposable projective-injective $A$-modules. Do we have $g=injdim(A/AeA)?$
>
>
>
I am mostly interested in the case when $eA$ is injective and faithful, which is also th... | https://mathoverflow.net/users/61949 | Injective dimension of $A/AeA$ | $AeA$ is a quotient of a direct sum of copies of $eA$, which is injective. Hence $AeA$ is a cosyzygy, and so $\text{Ext}^g(-,AeA)=0$.
The $\text{Ext}$ long exact sequence from the short exact sequence
$$0\to AeA\to A\to A/AeA\to 0$$
therefore ends with
$$0=\text{Ext}^g(-,AeA)\to\text{Ext}^g(-,A)\to\text{Ext}^g(-,A/Ae... | 1 | https://mathoverflow.net/users/22989 | 300135 | 131,633 |
https://mathoverflow.net/questions/299322 | 1 | I have a problem that can be described by the following equation:
\begin{equation}
\dfrac{d \left[\epsilon\_i\right]}{dt} = \left( \beta \mathbf{A} - \delta \mathbf{I} \right) \left[\epsilon\_i\right] - \alpha \mathbf{B} \left[\epsilon\_i^2\right],
\end{equation}
where $\left[\mathbf{\epsilon\_i}\right]$ is a vector wh... | https://mathoverflow.net/users/121398 | Solution of $\dfrac{d [\epsilon_i]}{dt} = (\beta \mathbf{A} -\delta \mathbf{I})\left[\epsilon_i\right] - \alpha \mathbf{B} \left[\epsilon_i^2\right]$ | If you are interested in conditions guaranteeing that **all** solutions starting at a point in the non-negative orthant $\mathbb{R}^n\_{+}$ remain there, the natural one is the following: At $(\epsilon\_1,\ldots,\epsilon\_n)$ the $j$-th coordinate, $F\_j$, of the vector field $F$ should be $\ge 0$ for those $j$ for whi... | 0 | https://mathoverflow.net/users/121784 | 300141 | 131,635 |
https://mathoverflow.net/questions/300143 | 2 | For any given integers $m,n\geq1$, let $\mathbb{R}[x\_1,\cdots,x\_n]\_m$ be the vector space of homogeneous polynomials of degree $m$ in $x\_1,\cdots,x\_n$ over the field of real numbers $\mathbb{R}$. Denote by $\mathbb{N}^n\_m$ the following set
$$\mathbb{N}^n\_m=\{(a\_1,\cdots,a\_n)\in \mathbb{Z}^n:a\_1,\cdots,a\_n\g... | https://mathoverflow.net/users/58096 | An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$ | It looks like you have rediscovered the Bargmann-Segal space! Take $n = 1$ for simplicity. Define $BS$ to be the set of analytic functions in $L^2(\mathbb{C},\mu)$ where $\mu$ is $\frac{1}{\pi}e^{-|z|^2}$ times Lebesgue measure. Thus the inner product of $f,g \in BS$ is $$\langle f,g\rangle = \frac{1}{\pi}\int\_{\mathb... | 5 | https://mathoverflow.net/users/23141 | 300144 | 131,636 |
https://mathoverflow.net/questions/299888 | 4 | Let $T\_1$ and $T\_2$ be two bounded linear operators in a complex banach space $X$.
>
> If $T\_1T\_2 = T\_2T\_1$, I want to know how to show that
> $$
> r(T\_1+T\_2) \leq r(T\_1) + r(T\_2),
> $$
> where $r(A)$ denotes the spectral radius of $A$, and it is given by
> $$
> r(A) = \inf\_{n>0} \|A^n\|^{1/n} = \lim... | https://mathoverflow.net/users/113054 | If $T_1T_2 = T_2T_1$, why $r(T_1 + T_2) \leq r(T_1) + r(T_2)$? | For simplicity, assume that $\|T\| \leqslant 1$ and $\|S\| \leqslant 1$. Write
$$ (T + S)^n = \sum\_{k = 0}^n \tbinom{n}{k} T^k S^{n - k} . $$
Fix $\varepsilon > 0$, and let $m$ be large enough, so that $\|T^j\| \le (r(T) + \varepsilon)^j$ and $\|S^j\| \le (r(S) + \varepsilon)^j$ for $j \ge m$. Then, if $n > 2 m$, we h... | 7 | https://mathoverflow.net/users/108637 | 300146 | 131,638 |
https://mathoverflow.net/questions/300106 | 5 | Steenbrink in his paper ["Limit of Hodge Structures"](https://link.springer.com/content/pdf/10.1007/BF01403146.pdf), (supplemented by the book "Mixed Hodge Structures" by Peters and Steenbrink) discuss the limit mixed Hodge structures for a fibration over the unit disc $\Delta \subset \mathbb{C}$, i.e. a fibration
$$\p... | https://mathoverflow.net/users/87910 | A question on Steenbrink's paper, limit of Hodge structures | If $\pi$ is semistable, then Fujisawa, *Limits of Hodge structures in several variables. Compositio (1999)*, does this. You might also look at some later papers by the same author for some refinements.
| 3 | https://mathoverflow.net/users/4144 | 300159 | 131,642 |
https://mathoverflow.net/questions/299698 | 8 | I am working with a certain Gauss hypergeometric function, $\_{2}F\_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.
It appears that $\left| \_{2}F\_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.
More generally, it seems that
$$
\left| \_{2}F\_{1}(a,b;c;1-z) \right| \geq 1
$$... | https://mathoverflow.net/users/124217 | lower bound for absolute value of a hypergeometric function | As I said, the problem is to show that the absolute value of
$$
z\mapsto\int\_0^\infty t^\alpha(t+1)^{-\beta}(t+z)^{-\gamma}\frac {dt}t
$$
attains its minimum in $\Omega=\{z:|z|\le 1, z\notin[-1,0]\}$ at $z=1$. Note that we are free to change the contour of integration to any curve that stays in the open angle bounded... | 4 | https://mathoverflow.net/users/1131 | 300169 | 131,645 |
https://mathoverflow.net/questions/299770 | 6 | In the [paper](http://math.mit.edu/classes/18.158/bourgain-restriction.pdf) of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a linear operator from $L^1(X,\mu)+L^2(X,\mu)$ to $L^\infty(Y,... | https://mathoverflow.net/users/47325 | Interpolation space between $L^1\cap L^2$ and $L^1$ | As requested, I post my comment as an answer (although this is not a true answer, just a possibly useful reference; feel free to edit it if this approach works out).
In Section 3 of the article [*Interpolation of sum and intersection spaces of $L^q$-type and applications to the Stokes problem in general unbounded dom... | 3 | https://mathoverflow.net/users/108637 | 300174 | 131,648 |
https://mathoverflow.net/questions/300134 | 18 | What is the value of this triple integral $$\int\limits\_0^{2\pi}\int\limits\_0^{2\pi}\int\limits\_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz?$$ It has to do with some Schwarz lemma.
| https://mathoverflow.net/users/124426 | An interesting triple integral | **Comment**
This is related to some papers (e.g. [this](https://carma.newcastle.edu.au/jon/walks2.pdf)) by P.M. Borwein et. al. on short random walks in the plane.
Let $X\_1, X\_2, \dots$ be i.i.d. random variables, uniformly distributed on the unit circle $|z|=1$ in the complex plane. Then
$$
X\_1+X\_2+X\_3
$$
is... | 25 | https://mathoverflow.net/users/454 | 300177 | 131,650 |
https://mathoverflow.net/questions/300067 | 2 | Is there any software that easily allows to make symbolic computations with involutions and homomorphisms? I need to define a product in an associative algebra with an (abstract) involution and involutive automorphism in terms of the original product, throw in some parameters, and then check if the new product satisfie... | https://mathoverflow.net/users/1234 | CAS implementing free algebras with involution | I've figured how to do this with MAGMA. I'm going to document it, in case anyone has the same problem in the future:
1) Define a parameter field:
>
> R$<$a1,...,an$>$:=FieldOfFractions(PolynomialRing(RationalField(),n));
>
>
>
where a1,...,an are the parameters and n is their number.
2) Define the free alg... | 2 | https://mathoverflow.net/users/1234 | 300201 | 131,658 |
https://mathoverflow.net/questions/300190 | 1 | I am not quite sure if this question is appropriate for this site as it might be not of a research level.
I am interested in the following ordinary differential equation on the real line
$$f’’(x)+(x-a)f(x)=0$$
where $a>0$.
>
> Is it possible to solve it explicitly? or using say hypergeometric functions? In any ca... | https://mathoverflow.net/users/16183 | Asymptotic behavior of a solution of an ODE | It is possible to solve it explicitly in terms of the Airy function.
Airy's equation in the standard form is
$$y''=xy.$$
Your equation is reduced to this by $x\mapsto-x$ followed by a shift of the independent variable.
Airy functions have been thoroughly studied and almost everything you want to know about them is know... | 3 | https://mathoverflow.net/users/25510 | 300204 | 131,660 |
https://mathoverflow.net/questions/284436 | 0 | Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $ZFC$ (this paraphrased from the Wikipedia entry, "Inaccessible cardinal") imply that, given a model $M$ of $ZFC$ in which... | https://mathoverflow.net/users/20597 | Forcing the existence of a weakly inaccessible cardinal in some strong set theory | *Before leaping into my rather long answer, the three-sentence response to your question is: since strengthening the theory only limits the class of models, you'll never make something forcible that was never forcible before by passing to a stronger theory. You can never force an inaccessible over a model of ZFC, so re... | 1 | https://mathoverflow.net/users/8133 | 300210 | 131,661 |
https://mathoverflow.net/questions/300214 | 6 | I think this is right but I want to check. The theory $\mathsf{WKL}^\*\_0$ is conservative over EFA for $\Pi^0\_2$ sentences. And the first order part of $\mathsf{WKL}^\*\_0$ is axiomatized by EFA plus the following formula scheme, known as the $\Sigma^0\_1$ bounding principle.
For every $\Sigma^0\_1$ formula $\varp... | https://mathoverflow.net/users/38783 | Am I counting quantifiers correctly? |
>
> To state the general rule: when counting quantifier changes, you ignore bounded quantifiers that come after all unbounded ones. But even a bounded quantifier, if it comes before an unbounded one, must be counted.
>
>
>
That's correct, at least when working in weak fragments of PA.
The complication is that ... | 7 | https://mathoverflow.net/users/8991 | 300225 | 131,666 |
https://mathoverflow.net/questions/300099 | 9 | Let $v\_1,...,v\_n\in \mathbb{R}^n$ be linearly independent. The parallelepiped defined by these vectors is $P(v\_1,...,v\_n)=\{\sum\_{i=1}^{n}\alpha\_i v\_i|~0\le\alpha\_i\le 1\}$. Observe that while the collection $v\_1,...,v\_n$ is reconstructible from $P(v\_1,...,v\_n)$ as a subset of $\mathbb{R}^n$, it is not reco... | https://mathoverflow.net/users/53155 | Parallelepiped is defined by the volumes of its faces | Here is a geometric proof of the result. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\be}{\boldsymbol{e}}$ As I comment at the end of the proof, this actually proves a stronger fact.
Define two equivalence relations"$\sim\_n$" and "$\approx\_n$" on the set of bases of $\bR^n$.
$$ (v\_1,\dotsc, v\_n)\sim\_n (w\_1,\d... | 2 | https://mathoverflow.net/users/20302 | 300226 | 131,667 |
https://mathoverflow.net/questions/300240 | 7 | I apologize in advance that this question must sound highly amateurish, but I am wondering if there is any connection between the formula <https://oeis.org/A127670> , which counts the number of fixed $n$-cell polycubes that are proper in $n-1$ dimensions, and Cayley's tree formula. The expression for the former is $$a\... | https://mathoverflow.net/users/70190 | Is there a natural relationship between OEIS A127670 and Cayley's tree formula? | Yes there is a connection. While $n^{n-2}$ counts the number of vertex labeled trees on $n$ vertices, the expression $2^n(n+1)^{n-2}$ counts the number of edge labeled trees on $n$ edges. There is a bijection between edge labeled trees on $n$ vertices and proper $(n-1)$-dimensional polycubes of size $n$. See lemma 2 (w... | 14 | https://mathoverflow.net/users/2384 | 300241 | 131,671 |
https://mathoverflow.net/questions/300251 | 0 | Let $\mathcal{X}$ be a regular, seperated Deligne-Mumford stack and $X$ be the coarse moduli scheme associated with $\mathcal{X}$. Then, is $X$ regular? I guess this is a basic fact but am not able to find any reference.
| https://mathoverflow.net/users/43198 | Properties inherited by the coarse moduli space | Not at all. Take for instance the quotient stack of $\mathbb{A}^2$ by $\pm 1$. The coarse moduli space is the quadratic cone.
| 1 | https://mathoverflow.net/users/4428 | 300252 | 131,673 |
https://mathoverflow.net/questions/300248 | 17 | Is $GL\_n(\mathbb{Q}\_p)=GL\_n(\mathbb{Z}\_p)GL\_n(\mathbb{Q})$? Generally, let $R$ be a discrete valuation ring and $K$ its fraction field. Let $\widehat{R}$ be the completion and $\widehat{K}$ the fraction field of $\widehat{R}$. Is $GL\_n(\widehat{K})=GL\_n(\widehat{R})GL\_n(K)$? We know it is true when $n=1$.
| https://mathoverflow.net/users/111816 | Is $GL_n(\mathbb{Q}_p)=GL_n(\mathbb{Z}_p)GL_n(\mathbb{Q})$? | Yes: more generally for every topological group $G$, dense subgroup $H$ and open subgroup $U$, we have $G=UH$.
This applies when $F$ is a valued field, $A$ an open subring (typically, elements of non-negative valuation), and $K$ any dense subfield of $F$, $G=\mathrm{GL}\_n(F)$, $H=\mathrm{GL}\_n(K)$, $U=\mathrm{GL}\_... | 26 | https://mathoverflow.net/users/14094 | 300262 | 131,674 |
https://mathoverflow.net/questions/300253 | 13 | We say that a $T\_2$-space $(X,\tau)$ has *homeomorphic open sets* if every non-empty open set $U\subseteq X$ endowed with the subspace topology is homeomorphic to $(X,\tau)$.
The rationals with the Euclidean topology are an example of a space with homemorphic open sets, as well as $\{0,1\}^\lambda$, where $\lambda$ ... | https://mathoverflow.net/users/8628 | $T_2$-spaces where all non-empty open sets are homeomorphic | The Baire space $\omega^\omega$ has homeomorphic open sets: it is an immediate consequence of the fact that it is the unique nonempty zero-dimensional Polish space, up to homeomorphism, every compact subset of whose has empty interior (theorem 7.7 in Kechris). In fact we actually have better:
>
> **Proposition.** F... | 19 | https://mathoverflow.net/users/120363 | 300263 | 131,675 |
https://mathoverflow.net/questions/300260 | 1 | Suppose $v$ is a (possibly time-dependent) vector field on a compact manifold $M$.
Its flow is a mapping $g: M \times \mathbb{R} \rightarrow M$, where $g$ satisfies the following conditions (written in the local chart):
$g(x, 0) = x$
$\frac{dg}{dt} (x, t) = v(g(x, t), t)$
I would like to find for a fixed $\epsi... | https://mathoverflow.net/users/124475 | Continuity of the differential flow under a perturbation of the vector field | Let me consider two autonomous vector fields $X\_1, X\_2$ on a compact smooth manifold $\mathcal M$ and assume that the Lipschitz condition is true for both of them. The flow $\psi\_j$
of $X\_j$ is defined on $\mathbb R\times \mathcal M$
by
$$
\dot \psi\_j(t,m)=X\_j(\psi\_j(t,m)), \quad \psi\_j(0,m)=m.
$$
Now we consid... | 1 | https://mathoverflow.net/users/21907 | 300284 | 131,680 |
https://mathoverflow.net/questions/300265 | 5 | In the **six-vertex model**, edges in a square lattice are oriented so that the in-degree of each vertex is exactly two. This gives six types of allowable vertices:
$$\begin{array}{cccccc}
\begin{array}{ccc}
& \uparrow & \\
\leftarrow & \bullet & \leftarrow \\
& \uparrow & \\
\end{array}
& \begin{array}{ccc}
& \... | https://mathoverflow.net/users/12261 | An extension of the Izergin-Korepin determinant to the eight-vertex model | This is essentially a long comment to accompany my answer, which is: not that I know of.
Before I explain why I think the eight-vertex model has received much less attention in this context (besides that it is technically *much* more involved) I should point out that the closely related 'elliptic solid-on-solid' (SO... | 4 | https://mathoverflow.net/users/45956 | 300295 | 131,683 |
https://mathoverflow.net/questions/299776 | 1 | Let $W$ be a symmetric $\alpha$-stable process with its generator
$-(-\Delta)^{\alpha/2}$ for some $\alpha \in (0, 2]$ under $\mathbb P$. Let $\mathbb P^x$ be the probability measure induced by a process
$$X(t) = x + t + W(t)$$
starting from $x$, and we set
$$\hat \zeta = \inf\{t>0: X(t) \notin (-1, 1)\}, \quad
\zet... | https://mathoverflow.net/users/5656 | Exit time of $\alpha$-stable process to an open set and to its closure | If $\alpha < 1$, then $X$ has bounded variation and positive drift, and therefore it does not *creep* downwards; see Theorem 7.11 in Kyprianou's book *Introductory Lectures on Fluctuations of Lévy Processes with Applications*. This means that $$\mathbb{P}^x(X(\tau\_{-1}) \ne -1) = 1,$$ for $x > -1$, where $$\tau\_{-1} ... | 1 | https://mathoverflow.net/users/108637 | 300300 | 131,685 |
https://mathoverflow.net/questions/300291 | 3 | Suppose $v\_1,\dots,v\_n \in \mathbb{R}^k$ are entry-wise non-negative (column) vectors with $k<n$. Let $r \leq k$ be the non-negative rank of the matrix $V = [v\_1 v\_2 \cdots v\_n]$ (i.e., the smallest $r$ such that $V$ can be written as the product $V= UW$ where $U \in \mathbb{R}^{k \times r}, W \in \mathbb{R}^{r \t... | https://mathoverflow.net/users/124495 | Size of a minimal non-negative conic basis | It is not true that $\text{rank}\_+^\*(V) \leq \text{rank}\_+(V) $. In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily columns of $V$) such that every column of $V$ is a conic combination of these vectors. Therefore, the opposite inequality $... | 2 | https://mathoverflow.net/users/2233 | 300301 | 131,686 |
https://mathoverflow.net/questions/300270 | 4 | The statement of the theorem is as follows:
Let $\rho$ be an irreducible two-dimensional representation of $G\_\mathbb{Q}=Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with Artin conductor $N$.
Suppose that $\epsilon = \det(\rho)$ is odd and $\rho$ satisfies condition $(A)$.
Let $L(s,\rho) = \sum a\_n n^{-s}$.
Then the funct... | https://mathoverflow.net/users/122640 | About the proof of Weil-Langlands theorem | I managed to confuse myself trying to answer your question, but I think if it does have an answer it's likely to be as follows. According to [1], the Artin root number of a Dirichlet character $\chi$ with conductor $f\_\chi$ is equal to
$$ W(\chi) = \frac{\tau(\chi)}{\sqrt{f\_\chi}i^a},$$
where $a=0,1$ is as in your qu... | 5 | https://mathoverflow.net/users/2604 | 300309 | 131,690 |
https://mathoverflow.net/questions/300272 | 1 | Given a field $k$ of null caracteristic, an associative/commutative/unital $k$-algebra $A$ and two free $k$-modules $M,N$, denote by $Hom\_k(M,A)$ the space of $k$-linear maps from $M$ to $A$.
I want to show that the following map is injective:
$Hom\_k(M,A)\otimes\_A Hom\_k(N,A) \longrightarrow Hom\_k(M \otimes\_k ... | https://mathoverflow.net/users/124484 | Tensor of Hom spaces over an algebra | For infinite-dimensional algebras this is not true.
As it is mentioned in the comments, the claim is equivalent to the injectivity of the coordinate-wise product map $u:A^I \otimes\_A A^J \to A^{I\times J}$ (where now the exponent stands for products, not coproducts).
For a counterexample, consider the commutative,... | 3 | https://mathoverflow.net/users/106723 | 300319 | 131,693 |
https://mathoverflow.net/questions/300290 | 3 | Bregman's Inequality gives an upper bound on the permanent of a given 0-1 matrix given only the row sums. Although there are now a number of different proofs I have not been able to find Bregman's original proof. In particular I have not been able to find the paper "Some properties of nonnegative matrices and their per... | https://mathoverflow.net/users/105971 | Reference Request: Bregman's Inequality | I went to the library and couldn't find it, but it turns out the citation in the OP and in [Wikipedia](https://en.wikipedia.org/wiki/Bregman%E2%80%93Minc_inequality) is mistaken, the volume number is 14 rather than 15 (page numbers 945–949 are correct); in any case, I located the article and scanned it; I'm not sure po... | 3 | https://mathoverflow.net/users/11260 | 300321 | 131,694 |
https://mathoverflow.net/questions/299832 | 6 | Following the paper by Ayala, Francis, and Tanaka: <https://arxiv.org/pdf/1409.0848.pdf>
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold and trivialized normal bundle.
Lets assume the monoidal category is dg-vector spaces with $\otimes$.
Corollary 0.3... | https://mathoverflow.net/users/124286 | Knot Factorization Homology inputs | Let me start with the second question:
2) The map $\alpha$ will indeed respect the degrees. You should think of it as a map between two $E\_2$-algebras in unbounded chain-complexes (whose domain happens to be concentrated in non-negative degrees, while its codomain is concentrated in non-positive degrees).
1) As a ... | 6 | https://mathoverflow.net/users/51164 | 300324 | 131,697 |
https://mathoverflow.net/questions/266457 | 3 | Let $B\subset\mathbb{R}^d$ be the Euclidean $d$-dimensional unit ball.
It is well-known that for any $x\_1,\ldots,x\_n\in B$, we have the following upper bound on the Rademacher complexity
$$ R\_n := \mathbb{E}\sup\_{w\in B}\sum\_{i=1}^n \sigma\_i(w\cdot x\_i)\le\sqrt n,
$$
where the expectation is over the Rademacher ... | https://mathoverflow.net/users/12518 | Rademacher average involving minima | Here is a writeup of the $O(\sqrt{k\log k})$ bound:
[https://www.cs.bgu.ac.il/~karyeh/rademacher-max-hyperplane.pdf](https://www.cs.bgu.ac.il/%7Ekaryeh/rademacher-max-hyperplane.pdf)
Update: 23-Jul-2018: Assuming the claims here are correct, the $\log k$ factor cannot be removed
<https://arxiv.org/abs/1807.07924>
U... | 4 | https://mathoverflow.net/users/12518 | 300331 | 131,700 |
https://mathoverflow.net/questions/300095 | 3 | Is it true to say that every nontrivial idempotent in the [Cuntz algebra $\mathcal{O}(n)$](https://en.wikipedia.org/wiki/Cuntz_algebra) is a commutator element?(Or a linear combination of commutator elements?)
| https://mathoverflow.net/users/36688 | Is every nontrivial idempotent in the Cuntz algebra, a commutator element? | There are a few things you could do to simplify this.
* Kaplansky's theorem ([Reference needed for: every idempotent in a C\*-algebra is similar to a hermitian one](https://mathoverflow.net/questions/16943/reference-needed-for-every-idempotent-in-a-c-algebra-is-similar-to-a-hermitian)) lets you show that every idemp... | 3 | https://mathoverflow.net/users/15438 | 300336 | 131,702 |
https://mathoverflow.net/questions/300332 | 5 | The [Leibniz integral rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule), in its multivariate form, deals with differentiation of the following sort:
$$ \frac{\partial}{\partial t} \int\_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{R}^d \, .$$
I am looking for a fully rigorous formulation of t... | https://mathoverflow.net/users/42864 | Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule) | This is probably just a more streamlined version of the proof in the article you linked to, mentioning somewhat less differential forms. I don't know a more elementary proof, but hope this is of use anyway.
I'll write $D\_t \subset \mathbb{R}^d$ for the domain of integration and I'll assume that the map $t\mapsto D\_... | 5 | https://mathoverflow.net/users/745 | 300351 | 131,706 |
https://mathoverflow.net/questions/300356 | 2 | Let $G= (V,E)$ be a simple, undirected graph. For $v\in V$ we set $N(v) = \{w\in V: \{v,w\} \in E\}$. Note that we always have $v\notin N(v)$. A function $f:V\to V$ is said to be a *neighborhood map* if $f(v)\in N(v)$ for all $v\in V$.
If $G$ is a graph such that very vertex has degree $\geq 2$, does there exist an i... | https://mathoverflow.net/users/8628 | Neighborhood maps for graphs $G$ with $\delta(G) \geq 2$ | Note that $K\_2$ and any cycle admit injective neighborhood maps.
More generally, a $(1,2)$-factor of a graph $G$ is a spanning subgraph $F$ where each vertex has degree $1$ or $2$. Such an $F$ is a disjoint union of cycles and paths. I will call a $(1,2)$-factor "short" if every vertex of degree $2$ (in $F$) belongs... | 1 | https://mathoverflow.net/users/25 | 300364 | 131,711 |
https://mathoverflow.net/questions/300352 | 3 | Let $X \subset \mathbb{P}^n$ be a reducible, projective subscheme. Assume that $X$ is reduced (meaning that every local ring is reduced i.e., does not contain nilpotent element). Denote by $S\_d$ the locus of degree $d$ hypersurfaces $H$ in $\mathbb{P}^n$ such that $X \cap H$ is singular. Is $S\_d$ irreducible for any ... | https://mathoverflow.net/users/32151 | Bertini-type theorem for reducible schemes | Take $X = X\_1\cup\dots\cup X\_k\subset\mathbb{P}^n$ be the union of $k$ distinct smooth hypersurfaces with $\deg(X\_i)\geq 2$ for $i = 1,\dots,k$, and $d = 1$. For any point $p\in X\_i$ then intersection $T\_pX\cap X$ is singular at $p$. Therefore, the dual hypersurface $X\_i^{\*}$ is a component of $S\_1$.
Now, if ... | 1 | https://mathoverflow.net/users/14514 | 300373 | 131,714 |
https://mathoverflow.net/questions/300374 | 8 | For a given topological group $G$ there are natural transformations $$K^\* \leftarrow K^\*\_G \overset a\to H^{\*\*}(EG \times\_G -;\mathbb Q)$$ from equivariant K-theory, the first forgetting the $G$-structure of a bundle and $a$ inducing from an equivariant bundle over $X$ a nonequivariant one over $EG \times\_G X$ a... | https://mathoverflow.net/users/5792 | Equivariant bundles invisible in K-theory and Borel cohomology | The simplest example of what you are looking for occurs when $G = S^1$ and $X=S^1/C\_6$, where $C\_6$ is the group of 6th roots of unity. Then the map
$$ K\_G(X) \rightarrow K(EG\times\_G X)$$
identifies with
$$ R(C\_6) \rightarrow K(BC\_6),$$
which has free abelian kernel of rank 2 (corresponding to the two conjugacy... | 11 | https://mathoverflow.net/users/102519 | 300377 | 131,716 |
https://mathoverflow.net/questions/300391 | 5 | I am looking for a closed form for the following integral
$$ I = \int\_0^\infty \mathrm{d} x \ x \ J\_0(ax) J\_0(bx) J\_1(cx) $$
which can be thought of as a particular case of the more general integral
$$ I(n\_1,n\_2,n\_3) = \int\_0^\infty \mathrm{d} x \ x \ J\_{n\_1}(ax) J\_{n\_2} (bx) J\_{n\_3} (cx) $$
i.e. ... | https://mathoverflow.net/users/19830 | An integral involving three Bessel functions | Without loss of generality we can set $c=1$; Mathematica returns a closed form for $a=b$.
For $|a|<1/2$ the result is
$$I= \int\_0^\infty \mathrm{d} x \ x \ J\_0(ax) J\_0(ax) J\_1(x)=\frac{4K(\alpha)}{\pi^2\sqrt{1-4a^2}}\left[2E(\alpha)-K(\alpha)\right]\qquad\qquad(\ast)$$
where $\alpha=\tfrac{1}{2}-\tfrac{1}{2}\sq... | 3 | https://mathoverflow.net/users/11260 | 300393 | 131,720 |
https://mathoverflow.net/questions/300318 | 6 | There is a well-known theorem for transporting a model category structure along a left adjoint $F:\mathcal{M}\to \mathcal{N}$ which is explained [here](https://mathoverflow.net/a/92548/24563) and which is due to Sjoerd Crans.
The difficult part is to check the third condition. By [Axiomatic homotopy theory for operad... | https://mathoverflow.net/users/24563 | Transporting a model category structure along a left adjoint | It suffices to dualize the proof of Theorem 2.2.1 in [Necessary and sufficient conditions for induced model structures](https://arxiv.org/abs/1509.08154). It uses indeed an argument coming from Quillen's book "Homotopical Algebra", II page 4.9 (the diagram in the bottom part of the page). And it is also necessary to us... | 2 | https://mathoverflow.net/users/24563 | 300397 | 131,721 |
https://mathoverflow.net/questions/139314 | 10 | Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question:
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds and let $f \in C^r(M \times N,P)$ for a given $r \in \mathbb{N}$.
Let $\hat{f}: M \to C^0(N,P)$ denote the adjoint ... | https://mathoverflow.net/users/13326 | Is there an "exponential law" for differentiable maps between smooth manifolds? | As far as I am aware, Kriegl and Michor's book deal only with the smooth case. To my knowledge the first full account on an exponential law for finite orders of differentiability was given in
Alzaareer, Schmeding: Differentiable mappings on products with different degrees of differentiability in the two factors, see... | 6 | https://mathoverflow.net/users/46510 | 300400 | 131,722 |
https://mathoverflow.net/questions/300350 | 7 | Assume that $1\le k \le n$ and let $\mathscr{Z}$ be the family of all subsets of $\{1,\ldots,n\}$ with at most $k$ elements. Pick a random element $X$ of $\mathscr{Z}$ (we consider the probablity distribution on $\mathscr{Z}$ is uniform, that is, each $X$ is chosen with probability $1/|\mathscr{Z}|$). What is the expec... | https://mathoverflow.net/users/24676 | Expected cardinality of a randomly chosen element of the family of subsets of $\{1,\ldots,n\}$ with at most $k$-elements | $\DeclareMathOperator\E{E}$As already noted by the other answers,
$$\E\xi^k\_n=\frac{\sum\_{i=0}^ki\binom ni}{\sum\_{i=0}^k\binom ni}.$$
One can then easily determine the asymptotics of $\E\xi\_n^{\lfloor n\delta\rfloor}$ for fixed $0\le\delta\le1$:
**Case 1: $0\le\delta<1/2$.** Then
$$k-\frac{\delta}{1-2\delta}\le\E... | 8 | https://mathoverflow.net/users/12705 | 300402 | 131,723 |
https://mathoverflow.net/questions/300407 | 20 | Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when considering the Laplacian on functions only, or on differential forms. feel free to answer about whichever).
Sometimes (for e... | https://mathoverflow.net/users/111049 | Can one hear the (topological) shape of a drum? | There are examples due to Ikeda of [isospectral Lens spaces which are not homotopy equivalent](http://www.numdam.org/item/ASENS_1980_4_13_3_303_0).
Likeliest the simplest examples are the compact connected 3-dimensional flat manifolds which are [a *tetracosm* and *didicosm*](https://msp.org/gt/2004/8-3/p07.xhtml). T... | 25 | https://mathoverflow.net/users/1345 | 300408 | 131,724 |
https://mathoverflow.net/questions/300404 | 9 | Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi\_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? More generally, one can consider pseudonatural transformations $U^n \Rightarrow U$, which are determined by a functor $... | https://mathoverflow.net/users/2362 | 2-natural operations on toposes | The "2-natural operations" $U^n \to U$ correspond to functors $\mathbf{FinSet}^n \to \mathbf{Set}$. (edit: As Simon points out, these correspond to the *finitary* functors $\mathbf{Set}^n \to \mathbf{Set}$.)
The 2-functor $U$ is birepresented by the [object classifier](https://ncatlab.org/nlab/show/classifying+topos+... | 10 | https://mathoverflow.net/users/57405 | 300409 | 131,725 |
https://mathoverflow.net/questions/300345 | 5 | Recently I'm learning the vertex modules. In the paper, there are a lot of abstract theory about the module theory,for instance the $C\_{2}-$cofinite conditions and associated variety. I hope to find some fundamental examples to get some intuition for these theory. As we all know, a vertex algebra is a module of itself... | https://mathoverflow.net/users/110479 | Some examples of vertex algebra modules | I don't know what paper you are reading, but you can find examples in most textbooks. For example, Frenkel and Ben-Zvi's book "Vertex algebras and algebraic curves" has a treatment of modules in chapter 5 that goes into some detail for the Heisenberg case. One thing they don't mention is that there are non-trivial self... | 3 | https://mathoverflow.net/users/121 | 300413 | 131,728 |
https://mathoverflow.net/questions/300414 | 3 | Let $\Omega$ be the set of all infinite binary sequences $(x\_i)\_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$.
Consider $0<\alpha<1$ and let $$K\_\alpha=\{(x\_i)\in\Omega:\lim\_{n\to\infty}\frac{1}{n}\sum\_{i=0}^{n-1}x\_i=\alpha\}.$$
Let $\mathcal{M}\_\sigma(\Omega)$ stand for... | https://mathoverflow.net/users/24676 | Measures maximizing entropy in a set of measures with fixed average for some observable | In the setting you describe, for each $\alpha \in (0,1)$ the $(1-\alpha,\alpha)$-Bernoulli measure is the unique measure achieving the maximum. The function $\alpha \mapsto \eta(\alpha)$ is the Legendre transform of the function $t\mapsto P(t\phi)$ where $\phi(x) = x\_0$ and $P$ is topological pressure. This is all par... | 5 | https://mathoverflow.net/users/5701 | 300420 | 131,730 |
https://mathoverflow.net/questions/300381 | 1 | I've found an argument that if valid suggests that there is always a prime $p > n$ that divides ${{x+n} \choose n}$ when $x > \pi(n)$.
As a math amateur, I am always doubtful about my results so I cannot claim that my argument is without mistake (for those interested in the argument, I have posted it [here](https://... | https://mathoverflow.net/users/15915 | Another question on strengthening the Sylvester-Schur Theorem | Gerry Myerson's observation should help you find the mistake in your argument. Even though there are many on MathOverflow who might find what I write below unsuitable for this forum, I will take the time to critique the linked proof. The argument against doing this on MathOverflow is that much of what is to be said is ... | 4 | https://mathoverflow.net/users/3402 | 300423 | 131,732 |
https://mathoverflow.net/questions/271371 | 3 | The pde
$$ Pf(x)=\sum \_{i,j=1}^{n}a\_{ij}(x){\frac {\partial ^{2}f}{\partial x\_{i}\,\partial x\_{j}}}(x)+\sum \_{i=1}^{n}b\_{i}(x){\frac {\partial f}{\partial x\_{i}}}(x)+c(x)f(x),$$
is said to be [semi-elliptic](https://en.wikipedia.org/wiki/Semi-elliptic_operator) when the matrix $(a\_{ij}(x))$ is *positive semid... | https://mathoverflow.net/users/99863 | Semi-elliptic pdes reference | I would have guessed that many do not pass through. One of the most basic things fails because for a divergence form operator $D\_i(a\_{ij}(x)D\_ju)$ you do not have
$$
\int |Du|^2 \leq \int a\_{ij}D\_iuD\_ju
$$
which is the basis of an energy estimate.
| 2 | https://mathoverflow.net/users/122587 | 300433 | 131,733 |
https://mathoverflow.net/questions/300378 | 3 | (a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint\_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a standard way to generalize this integral to more general surfaces (rectifiable varifolds, perhaps)?
(b) For the special ... | https://mathoverflow.net/users/102458 | total mean curvature for singular surface | The analogy between these two quantities goes back to Steiner ([Über parallele Flächen, 1840](https://biodiversitylibrary.org/page/11051416)). Both of them appear as coefficients of linear terms in the expansion of the area of the surface at distance $\epsilon$ from the given smooth/polyhedral surface.
Minkowski prov... | 2 | https://mathoverflow.net/users/98590 | 300434 | 131,734 |
https://mathoverflow.net/questions/300425 | 6 | Let $k$ be a finite field, $A$ a smooth $k$-algebra.
Does there exists a smooth algebra $B$ over the Witt vectors $W(k)$, such that $B/p\simeq A$? How is it constructed?
| https://mathoverflow.net/users/nan | Smooth algebras always lift | This follows from a result of Elkik in [R. Elkik *Solutions d’équations à coefficients dans un anneau hensélien* Annales scientifiques de l’É.N.S. 4e série, tome 6, no 4 (1973), p. 553-603.].
>
> **Theorem** (Theorem 6 in Section 4 on p. 580, changed notation to fit yours). Let $(R, J)$ be a noetherian henselian co... | 4 | https://mathoverflow.net/users/3847 | 300437 | 131,735 |
https://mathoverflow.net/questions/299859 | 3 | Let S be a Grothendieck site, the (either left or right adjoint to the) Grothendieck construction assigns to each groupoid fibration over S a presheaf valued in groupoids. The following feels it might be morally correct:
A prestack is a stack $\iff$ its corresponding presheaf of groupoids is a sheaf
Is this true? D... | https://mathoverflow.net/users/74739 | Stack descent to sheaf descent via Grothendieck construction? | Here's a variation which is true, when interpreted in a suitably non-strict / higher categorical sense (for example, "functor" means "pseudofunctor" below). I'm not sure on which side of the Grothendieck construction you prefer to define (pre)stacks and (pre)sheaves, so let's do both versions:
* Define a prestack on ... | 2 | https://mathoverflow.net/users/2362 | 300439 | 131,736 |
https://mathoverflow.net/questions/300368 | 4 | I have been working on solutions to $x^5+y^5+z^5=1$, and I found that the three solutions of $x^3+bx+\frac{1}{5b}$ satisfy that equation. Multiplying by $5b$: $5xb^2+5x^3b+1=0$, then solving for b yields:
$b=\frac{-5x^3+\sqrt{25x^6-20x}}{10x}$
Which leads me to my question... What are the rational solutions of the ... | https://mathoverflow.net/users/38744 | Solutions to diophantine equation | I probably put a little bit too much effort into this. The only rational point on this curve is $(0,0)$ (as well as the points at infinity $(1 : 5 : 0)$ and $(1 : -5 : 0)$).
There's a slightly non-obvious change of variables that will turn your equation into $y^{2} = x^{5} + 2000^{2}$. Using this form of the equation... | 4 | https://mathoverflow.net/users/48142 | 300443 | 131,738 |
https://mathoverflow.net/questions/282492 | 3 | I'm following [Sutton's](http://ufal.mff.cuni.cz/~straka/courses/npfl114/2016/sutton-bookdraft2016sep.pdf) book on Reinforcement Learning, and he casually states that "There is always at least one policy that is better than
or equal to all other policies" for a given finite MDP. This is trivially the case for determini... | https://mathoverflow.net/users/115250 | Proof of the existence of an optimal MDP with a stochastic reward signal? | Old question, but I will try to answer anyway: It depends.
If the search space only allows policies that have at least one state for which a non-deterministic action selection takes place, then no; trivially there is no (fully) deterministic policy that is better then the optimal stochastic policy.
If you allow al... | 1 | https://mathoverflow.net/users/124601 | 300447 | 131,740 |
https://mathoverflow.net/questions/300432 | 2 | Yet another question "I compute Bredon cohomology of something and I am not sure, whether it is correct".
So I am taking a sign representation $\sigma$ of cyclic group of order 4, $C\_4$. Then I compactify $\sigma$ to get $\mathbb{S}^\sigma$ with two fixed 0-cells and one 1-cell of type $C\_2$. Then I am taking $\und... | https://mathoverflow.net/users/123432 | Bredon cohomology of a sign representation for a cyclic group of order 4 | No, I don't think so - I think there are more Bredon $1$-cochains than that.
The orbit category $\mathcal{O}C\_4$ looks like
$$
C\_4/e \to C\_4/C\_2 \to C\_4/C\_4
$$
where the automorphism groups of the objects are $C\_4$, $C\_2$ and the trivial group, respectively. The Bredon cochains in degree $i$ are natural trans... | 3 | https://mathoverflow.net/users/8103 | 300452 | 131,743 |
https://mathoverflow.net/questions/300449 | 7 | The paper [The Tits alternative for $\operatorname{Out}(F\_n)$ I](https://arxiv.org/pdf/math/9712217.pdf) by Bestvina, Feighn and Handel and the paper [Automorphisms of free groups and Outer space](http://www.math.cornell.edu/~vogtmann/papers/Autosurvey/autosurvey.pdf) by Vogtmann both state that $\operatorname{Aut}(F\... | https://mathoverflow.net/users/104216 | $\operatorname{Out}(F_n)$ is not linear for $n > 3$ | There is an embedding $\text{Aut}(F\_{n-1}) \hookrightarrow \text{Out}(F\_n)$ for any $n \ge 2$, as follows.
First one embeds $\text{Aut}(F\_{n-1}) \hookrightarrow \text{Aut}(F\_n)$ by extending any automorphism of $F\_{n-1}$ to an automorphism of
$$F\_n \approx F\_{n-1} \* \mathbb{Z}
$$
using the identity automorp... | 17 | https://mathoverflow.net/users/20787 | 300455 | 131,744 |
https://mathoverflow.net/questions/300454 | 7 | The energy of a a simple graph $G$ is defined to be the sum of the absolute values of the eigenvalues of $G$. What is a good intuition of it?
| https://mathoverflow.net/users/34575 | Intuition of the energy of a graph | The name "energy" only makes physical sense for a bipartite graph, where the eigenvalues of the adjacency matrix come in pairs $\pm\lambda$. If graph represents a molecule, the adjacency matrix is the Hamiltonian in the socalled [tight-binding](https://en.wikipedia.org/wiki/Tight_binding) approximation. (The vertices a... | 11 | https://mathoverflow.net/users/11260 | 300456 | 131,745 |
https://mathoverflow.net/questions/300453 | 5 | What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?
I have a reference that says the group in question is $\mathbb{Q}\_p,$ the additive group of the quotient field of the $p$-adic integers. Can anyone provide a reasonable derivation of this result?
| https://mathoverflow.net/users/8027 | What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$? | Note that this equality is as topological groups, where Hom is endowed with uniform convergence on compact subsets.
Every such homomorphism is zero on $p^n$ for some $n$. Hence the given homomorphism group $G=\mathrm{Hom}(\mathbf{Q},\mathbf{Z}(p^{\infty}))$ can be written as $\bigcup G\_n$, where $G\_n$ is the set o... | 9 | https://mathoverflow.net/users/14094 | 300461 | 131,747 |
https://mathoverflow.net/questions/300466 | 46 | Is the Riemann zeta function surjective or does it miss one value?
| https://mathoverflow.net/users/122104 | Is the Riemann zeta function surjective? | The Riemann zeta function is surjective. First, $\zeta(1/z)$ is holomorphic in the punctured disk $0<|z|<1$. Looking at $z=(1/2+it)^{-1}$ with $t\to\infty$ reveals that $\zeta(1/z)$ has an essential singularity at $z=0$, hence $\zeta(s)$ misses at most one value. If $\zeta(s)=w$ then $\zeta(\overline{s})=\overline{w}$,... | 95 | https://mathoverflow.net/users/11919 | 300468 | 131,751 |
https://mathoverflow.net/questions/299919 | 4 | Let $H\_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials.
It is well-known that all the gaps between consecutive roots of $H\_k(x)$ are at least a multiple of $1 / \sqrt{k}$ (see, e.g., Szego, "Orthogonal Polynomials", page 130).
Is the same known to be true if we consi... | https://mathoverflow.net/users/3448 | Gaps between roots of consecutive Hermite polynomials | The answers are yes and yes.
Consider the Hermite Gauss functions : $\psi\_n(x)=e^{-x^2/2}H\_n(x) $, we have two properties:
$$\psi\_n''(x)+(2n+1-x^2)\psi\_n(x)=0 $$
and
$$\psi\_{n+1}=\psi\_n'(x) +x\psi\_n(x) $$
For $n$ large, we take a lenght $L>0$ which is very small (for example $L=1/\log{n}$). On any $[x\_0-L/2,... | 3 | https://mathoverflow.net/users/99045 | 300478 | 131,756 |
https://mathoverflow.net/questions/300490 | 3 | This question has a trivial starting point: If the open sets of $\mathbb{R}$ were closed under countable intersection, the Euclidean topology would be discrete because for all $x\in\mathbb{R}$ we have $\{x\} = \bigcap\big\{ (x-\frac{1}{n},x+\frac{1}{n}):n\in\mathbb{N}\setminus\{0\}\big\}$.
**Question.** Is there a co... | https://mathoverflow.net/users/8628 | Connected $T_2$-space such that the open sets are closed under countable intersection | In other words, you want every $G\_\delta$ to be open. I just learned that such spaces are called [**P-spaces**](https://www.encyclopediaofmath.org/index.php?title=P-space).
The following paper contains an example of a connected Hausdorff P-space (and yes it is uncountable).
>
> *Misra, Arvind K.*, [**A topologi... | 7 | https://mathoverflow.net/users/4832 | 300491 | 131,758 |
https://mathoverflow.net/questions/300474 | 1 | I'm looking for a reference that surveys (as generally as possible) the various notions of 'completion' for an ordered space and their relation to eachother.
In particular, I'd like a reference that covers notions like pseudo-convergent completion in relation to notions like Dedekind and Archimedean completion for or... | https://mathoverflow.net/users/92164 | Completion of an ordered space | For Dedekendian completions and (transfinite) Cantor completions and the fact that they preserve the value group, see the following classical works of COHEN and GOFFMAN.
COHEN, L. W. and GOFFMAN, C., A theory of transfinite convergence, Trans. Amer. Math. Soc. 66 (1949), 65–74.
COHEN, L. W. and GOFFMAN, C., The top... | 1 | https://mathoverflow.net/users/18939 | 300499 | 131,760 |
https://mathoverflow.net/questions/300277 | 4 | There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition [here](https://math.berkeley.edu/~ribet/Articles/luminy.pdf), more precisely see Theorem 2.1.
What is confusing me is that they don't seem to use the L-functions with... | https://mathoverflow.net/users/58001 | Kummer congruences for totally real number fields | I think that the point lies in the difference between a *primitive* and *imprimitive* $L$-function. Before entering the details, let me observe that Washington's definition of $p$-adic $L$-functions (as the one found in Washington's *Introduction to Cyclotomic fields*, **Theorem 5.11**) gives the interpolation
$$
L\_p(... | 1 | https://mathoverflow.net/users/18238 | 300504 | 131,761 |
https://mathoverflow.net/questions/300191 | 4 | From the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling" on page 11,
<https://arxiv.org/abs/1602.04251>
the authors states that on a curved manifold with a $spin\_{\mathbb{C}}$ structure, the $U(1)$-connection
$$D\_{\mu}^{(n)}=D\_{\mu}^{(0)}+inA\_{\mu}$$ is well-defined only for odd el... | https://mathoverflow.net/users/120604 | $spin_{\mathbb{C}}$ Connection and Charge Parity | I think this is how a physicst would treat spin\_c structures: Suppose $(M,g)\cong(\mathbb{R}^n,g\_{ij}\mathrm dx^i\mathrm dx^j)$ is a coordinate chart. We can then define $n$ complex $2^{\lceil\frac{n}{2}\rceil}$-"gamma matrices" $(\gamma\_{i\alpha}^\beta)\_{1\le i\le n,1\le \alpha,\beta\le 2^{\lceil\frac{n}{2}\rceil}... | 7 | https://mathoverflow.net/users/35687 | 300508 | 131,763 |
https://mathoverflow.net/questions/300486 | 7 | Let $M$ be a nonstandard model of $PA$.
>
> Q1. Is there any way to get a submodel $N\subset M$ such that $N\models Th\_{\Pi\_2}(PA)$, but $N\not\models PA$?
>
>
> Q2. Especially, what combinatorial principle can be used to construct a model $N\models Th\_{\Pi\_2}(PA)$ such that $N\not\models PA$? Or $Th\_{\Pi\_2... | https://mathoverflow.net/users/83598 | On models of $Th_{\Pi_2}(PA)$ | $\def\pa{\mathit{PA}}\def\rfn{\mathrm{RFN}}\def\pr{\mathrm{Pr}}\def\num#1{\ulcorner#1\urcorner}\def\Th{\mathrm{Th}}$ $\Th\_{\Pi\_2}(\pa)$ certainly does not prove $\pa$:
>
> **Theorem:** For any constant $k$, there is no consistent $\Pi\_k$-axiomatized theory $T$ such that $T\vdash\pa$.
>
>
> More generally, if $... | 8 | https://mathoverflow.net/users/12705 | 300514 | 131,764 |
https://mathoverflow.net/questions/300522 | 3 | Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function.
Q. Is it possible to find a sequence of simple functions $\{\phi\_n\}$ with $\lim\phi\_n(x)=x$ for every $x\in X$?
| https://mathoverflow.net/users/84390 | Approximation on separable topological space with size $\mathfrak{c}$ | No, take any $\mathbb{N} \subseteq X \subseteq \beta\mathbb{N}$ with $|X|=\mathfrak{c}$. If ${\phi\_n}$ is any sequence of functions with finite range (measurable or not) and $x\in X \setminus \bigcup\_n \phi\_n(X)$, then $\lim \phi\_n(x) \neq x$.
| 3 | https://mathoverflow.net/users/17836 | 300533 | 131,769 |
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