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https://mathoverflow.net/questions/413773 | 4 | Let $\overline{\mathbb{F}\_p}$ be the algebraic closure of $\mathbb{F}\_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}\_p\rightarrow \overline{\mathbb{F}\_p}$ induces a map $f: \mathbb{Z}\_p=W(\mathbb{F}\_p)\rightarrow W(\overline{\mathbb{F}\_p})$.
Is $f$ split ... | https://mathoverflow.net/users/163893 | Splitting the Witt vectors of $\overline{\mathbb{F}_p}$ | $W(\overline{\mathbb F\_p})$ is the completion of the direct limit of $W(\mathbb F\_{p^{n!}})$. The ring $W(\mathbb F\_{p^{n!}})$ is a free module of rank $n$ over $W(\mathbb F\_{p^{(n-1)!}})$, so the natural map $W(\mathbb F\_{p^{(n-1)!}}) \to W(\mathbb F\_{p^{n!}})$ has a splitting.
Combining all these splittings, ... | 7 | https://mathoverflow.net/users/18060 | 413774 | 168,785 |
https://mathoverflow.net/questions/413725 | 9 | Compare the following two results:
>
> Thm A) Let $A$ be a commutative $C^\*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $\*$-isomorphism from $A$ to $C(X)$, the $C^\*$-algebra of continuous functions on $X$.
>
>
>
>
> Thm B) Let $A$ be a commutative ri... | https://mathoverflow.net/users/131975 | Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)? | Yes, both Theorem A and Theorem B are special cases of a more general construction.
Denote by $R$ the category of commutative unital C\*-algebras or the category of commutative rings.
Denote by $R'$ the full subcategory of $R$ given by reduced objects in $R$, meaning the only nilpotent element is zero.
(All commutati... | 12 | https://mathoverflow.net/users/402 | 413776 | 168,786 |
https://mathoverflow.net/questions/413744 | 2 | Let $X$ be a semi-normal projective variety and $p: \widetilde{X} \to X$ be the normalization. Suppose that $\widetilde{X}$ is smooth and there exists two *smooth* divisors $D\_1, D\_2 \subset \widetilde{X}$ such that $D\_1 \cong D\_2 \cong X\_{\mathrm{sing}}$ and $p$ induces as isomorphism between $\widetilde{X} \back... | https://mathoverflow.net/users/38832 | When is the singularity of a semi-normal variety a double point singularity | I think this is true if we assume that the isomorphisms $D\_1 \cong X\_{\mathrm{sing}} \cong D\_2$ are also induced by $p$, at least in characteristic $0$ (though I think everything below is ok away from characteristic $2$). **Edit:** We also need to assume that the $D\_i$ are disjoint in $\widetilde{X}$. If $D\_1 \cup... | 3 | https://mathoverflow.net/users/12402 | 413780 | 168,789 |
https://mathoverflow.net/questions/413654 | 4 | Let $A$ be a von Neumann algebra and let $H$ be a (separable) Hilbert space.
It is known (see e.g., Section IV, Thm. 5.5 of Takesaki I) that there exists a Hilbert space $K$ such that $A \subset \mathbb{B}(K)$ such that any normal $\*$-homomorphism $\varphi : A \to \mathbb{B}(H)$ can be written as
$$ \varphi(a) = v^\* ... | https://mathoverflow.net/users/16702 | Families of representations of von Neumann algebras | **Theorem.**
The map $V(A,H)\to\operatorname{Hom}(A,\mathbb{B}(H))$ is open.
We write $\omega\_{\xi,\eta}$ for the linear functional $x\mapsto \langle x\xi,\eta\rangle$
and $\omega\_\xi$ for $\omega\_{\xi,\xi}$.
It is an elementary fact that if $\omega\_\xi=\omega\_{\eta}$ on a von Neumann algebra $A$,
then $a\xi\map... | 6 | https://mathoverflow.net/users/7591 | 413781 | 168,790 |
https://mathoverflow.net/questions/413778 | 3 | Let $G$ be a finite abelian group, $X$ and $Y$ be two non-empty subsets of $G$ of equal size. Suppose that for each irreducible character $\chi$ of $G$ we have $\sum\_{x\in X}\chi(x)=\sum\_{y\in Y}\chi(y)$. Is it true that $X=Y$ in general?
| https://mathoverflow.net/users/36341 | Equality of subsets of abelian groups | $\DeclareMathOperator\Irr{Irr}$You can see this from the fact that for abelian groups, irreducible characters form a $\mathbb{C}$-basis of the space of functions from $G$ to $\mathbb{C}$. These functions correspond bijectively to linear transformations from the group algebra $\mathbb{C}G$ to $\mathbb{C}$, and $\Irr(G)$... | 6 | https://mathoverflow.net/users/91692 | 413782 | 168,791 |
https://mathoverflow.net/questions/413788 | 1 | Given an increasing function $f:[0,\infty)\to[0,\infty)$, we can define $$F(x)=\int\_x^{x+1} f(t)dt,$$
which is continuous, increasing function satisfying $$f(x)\leq F(x)\leq f(x+1).$$
Question) For such $f$ can we construct a continuous increasing function $g:[0,\infty)\to[0,\infty)$ such that for all $x\in [0,\inft... | https://mathoverflow.net/users/184109 | Given an increasing function, need to construct a continuous increasing function equivalent to given function | Using the integer-part function, let
$$f(x) = x ^ {\lfloor x \rfloor +1}$$
Then there is no $g$ satisfying the given condition for this $f$.
For any such $g$, let $b = \lfloor c\_2/c\_1 \rfloor +2$. Then
\begin{align}
b-1<x<b \implies & g(x) \le c\_2 x^b\\
b<x<b+1 \implies & g(x) \ge c\_1x^{b+1}
\end{align}
By th... | 3 | https://mathoverflow.net/users/nan | 413792 | 168,794 |
https://mathoverflow.net/questions/413736 | 14 | The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =\_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot form the type of universe types, the usual univalence axiom is actually an axiom scheme, consisting of an instance of t... | https://mathoverflow.net/users/11640 | How to formulate the univalence axiom without universes? | One possibility along these lines is large eliminations for higher inductive types. For instance, here is a large elimination rule for the higher inductive interval type $\mathsf{I}$ with $0,1:\mathsf{I}$ and $\mathsf{seg}:\mathsf{Id}\_{\mathsf{I}}(0,1)$:
$$
\frac{\vdash A \,\mathsf{type} \quad \vdash B \,\mathsf{typ... | 8 | https://mathoverflow.net/users/49 | 413795 | 168,796 |
https://mathoverflow.net/questions/413770 | 12 | Please excuse that the following will be a somewhat soft question.
Let $(M,d)$ be a metric space and $X(\omega)$ a random variable on $M$ with distribution $\mu$.
Assume now that $M = \overline{B\_1^n(0)}$ the $n$-dimensional closed ball of radius $1$ around $0$ and that $\mu$ has a density funciton $f\_\mu$. Assum... | https://mathoverflow.net/users/127781 | Is there a homotopy/homology-theory for probability spaces? | One possible approach is via persistent homology. It is outlined in the paper "[Persistent homology for metric measure spaces, and robust statistics for hypothesis testing and confidence intervals](https://web.ma.utexas.edu/users/blumberg/metricmeasure1.pdf)" by Blumberg, Gal, Mandell and Pancia
In a nutshell, the id... | 10 | https://mathoverflow.net/users/6668 | 413798 | 168,797 |
https://mathoverflow.net/questions/413804 | 10 | The following question is about if it is compatible to add to $\sf ZF$ an axiom asserting the existence of a countable transitive model of $\sf ZF$ such that for every strictly increasing function $f$ on the ordinals, we have a transitive countable model of $\sf ZF $ satisfying: $$\forall\alpha>0:\beth\_\alpha = \aleph... | https://mathoverflow.net/users/95347 | Can GCH fail everywhere every way? | No. An early nontrivial constraint on the $\beth$ function comes from Kőnig's Theorem, that for all infinite $\kappa$, $\mathrm{cf}(2^\kappa)>\kappa$. This implies that we cannot have $\beth\_\alpha = \aleph\_{f(\alpha)}$ for all $\alpha$, when $f(1) = \omega$, nor when $f(\omega+1)$ is a cardinal below $\aleph\_\omega... | 21 | https://mathoverflow.net/users/11145 | 413807 | 168,800 |
https://mathoverflow.net/questions/413766 | 9 | $\newcommand\Sq{\mathit{Sq}}$Recall that a (graded) module $V^\ast$ over the Steenrod algebra $\mathcal A^\ast$ is said to be *unstable* if $\Sq^i v = 0$ for $i > |v|$. The motivating example, of course, is that if $V^\ast = H^\ast(X)$ for a space $X$ with its natural $\mathcal A^\ast$ structure, then $V^\ast$ is unsta... | https://mathoverflow.net/users/2362 | What is an unstable dual-Steenrod comodule? | Normally people think about Steenrod comodules as graded $\mathbb{Z}/2$-modules equipped with a graded coaction $\psi\colon M\_\*\to M\_\*[\xi\_1,\xi\_2,\dotsc]$. However, it is equivalent to consider ungraded modules with coaction $\psi\colon M\to M[\xi\_0^{\pm 1},\xi\_1,\xi\_2,\dotsc]$; the grading is recovered by th... | 15 | https://mathoverflow.net/users/10366 | 413808 | 168,801 |
https://mathoverflow.net/questions/413786 | 3 | It's known that if a compact Lie group $G$ acts freely on a compact manifold $M$, then the orbit space $M/G$ is a manifold. If we only assume that $G$ acts almost freely (i.e. $G\_x$ is finite for any $x\in M$ and there are only finitely many $x$ such that $G\_x$ is not trivial)，then can we deduce that $M/G$ is a orbif... | https://mathoverflow.net/users/356774 | Almost free Lie group action | I think the answer to the first question is yes and the answer to the second one is no:
Yes, the quotient is an orbifold. The action of the finite group $G\_x$ in a neighbourhood of $x$ can be linearized (at least if the action is by diffeomorphisms, I don't know about $C^0$ regularity), and the quotient $M/G$ is loc... | 4 | https://mathoverflow.net/users/173096 | 413810 | 168,802 |
https://mathoverflow.net/questions/413829 | 0 | Let $\nu : G \rightarrow H$ be a surjective group homomomorphism with kernel $N$, $H$ abelian, and $G$ finitely generated.
The rational abelianization of $N$, $H\_1(N)$ is a $\mathbb{C}[H]$-module, and we say that it is nilpotent if some power of the augmentation ideal $I\_H \subset \mathbb{C}[H] $ is in the annihila... | https://mathoverflow.net/users/170754 | Nullstellensatz and nilpotence of a module | If we call the annihilator by $J$, then the statement reads
>
> $J$ contains some power of the maximal ideal $I\_H$ if and only if the intersection of the set of prime ideals containing $J$ with the set of maximal ideals is contained in $\{I\_H\}$
>
>
>
which simplifies to
>
> $J$ contains some power of th... | 2 | https://mathoverflow.net/users/18060 | 413837 | 168,813 |
https://mathoverflow.net/questions/413772 | 3 | I have a triangulation of a surface without boundary in $\mathbb{R}^3$.
The triangulation gives a unit normal pointing outwards for each triangle. I need to find some point in the interior of the surface.
I will outline my general strategy. I let $V$ denote the set of vertices of the triangulation and compute a numbe... | https://mathoverflow.net/users/475087 | Finding a point inside a surface | Let me suggest another approach, conceptually simple but maybe not as easy
to implement as Matt F.'s algorithm.
Let $P$ be the polyhedron.
Let $F$ be any face of $P$ with centroid $p$, and $\vec{n}$ the outward normal vector to $F$.
Shoot a ray $r$ from outside of $P$ in direction $-\vec{n}$ through $p$.
Ignore all i... | 2 | https://mathoverflow.net/users/6094 | 413838 | 168,814 |
https://mathoverflow.net/questions/412879 | 5 | A Silver forcing "below $2^n$" is defined e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It is called *Infinitely equal forcing **EE*** there. In the same book, in Lemma 7.4.15 the authors show that *EE preserves p-points*... | https://mathoverflow.net/users/115702 | Silver-like forcing preserves p-points (reference request) | David Chodounský and Osvaldo Guzmán showed in arXiv:1703.02082, [There are no P-points in Silver extensions](https://arxiv.org/abs/1703.02082) that *There are no P-points in Silver extensions*. They prove that
>
> after adding a Silver real no ultrafilter from the ground model can be extended to a P-point, and this... | 6 | https://mathoverflow.net/users/14915 | 413843 | 168,816 |
https://mathoverflow.net/questions/408632 | 3 | Let $p, q \in \mathbb{Z}$.
Let $\operatorname{wt}(n)$ is [A000120](https://oeis.org/A000120), number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and
$$n=2^{t\_1}(1+2^{t\_2+1}(1+\dots(1+2^{t\_{wt(n)}+1}))\dots)$$
Then we have an integer sequence given by
\begin{align}
a(0, m)& = 1\\
a(n, m)& ... | https://mathoverflow.net/users/231922 | Sum with products turned into subsequences | Actually, these two conjectures are actually equivalent.
We need a lemma: Lucas' Theorem: $\binom{a}{b}$ is odd if and only if $a\&b=b$ where $\&$ is bitwise AND operation. Let $S\_j$ be the set of integers $i$ such that $i\&j=i$. Thus, if the first relation
$$a(n, -1) = \sum\limits\_{j=0}^{n}(-1)^{\operatorname{wt... | 3 | https://mathoverflow.net/users/170895 | 413850 | 168,818 |
https://mathoverflow.net/questions/409285 | 0 | Let $f(n)$ be [A153733](https://oeis.org/A153733), remove all trailing ones in binary representation of $n$. Here
\begin{align}
f(2n)& = 2n\\
f(2n+1)& = f(n)\\
\end{align}
Then we have an integer sequence given by
\begin{align}
a(0)& = 1\\
a(2n+1)& = a(n)\\
a(4n+2)& = 2a(n)\\
a(4n)& = 2a(2n)-a(n)
\end{align}
Here $a(... | https://mathoverflow.net/users/231922 | Modulo $2$ binomial transform of A124758 | The first relation $b(0)=0$ is trivial.
By the condition, $a(n)$ is the product of number of consecutive zeroes plus $1$, if we write $n$ in binary. For instance, if $n=1220$, we have $n=100 1100 0100$ in binary, thus $a(n)=3\times 4\times 3=36$. Notice that if $n=2^u\times v$ where $v$ is an odd number, we have $a(n... | 2 | https://mathoverflow.net/users/170895 | 413853 | 168,820 |
https://mathoverflow.net/questions/413667 | 11 | By definition, a *1-motive* over an algebraically closed field $k$ is the data
$$
M = [X\stackrel{u}{\to}G]
$$
where $X$ is a free abelian group of finite type, $G$ is a semi-abelian variety over $k$, and $u:X \to G(k)$ is a group morphism.
As far as I understand, this was one of the earlier attempts of coming up... | https://mathoverflow.net/users/nan | How should I think about 1-motives? | I don't fully understand the hostility to thinking of motives in terms of compatible systems. It's true that a motive is not just a compatible system of realizations (or else we would call it a "compatible system" and not a motive) but it's also true that studying compatible systems has always been a good way to get in... | 7 | https://mathoverflow.net/users/18060 | 413864 | 168,824 |
https://mathoverflow.net/questions/413858 | 0 | Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int\_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$.
Suppose that we know
>
> $$\frac{d}{dt} \alpha\int\_{\mathbb R} u(t,x) dx + A(t) \le \int\_{\mathbb R}u^2(t,x) dx.... | https://mathoverflow.net/users/nan | Gronwall-type inequality with nonlinearity | The stated inequality cannot hold: Let $\alpha=1$, take $u(t,x)= U$ if $x\in[0,1]$ and $0$ otherwise, and $A(t) = U^2$. Then all the $d/dt$ are 0, but the desired inequality,
$$U + tU^2 \le e^{t}U,$$
fails for $t=\log U$ and $U\to +\infty$.
| 0 | https://mathoverflow.net/users/141760 | 413870 | 168,825 |
https://mathoverflow.net/questions/412127 | 1 | If I had to partition the unit square $[0,1]\times[0,1]$ into $k^2$ rectangles such that the sum of their diagonals is minimum possible, I would simply choose the $k \times k$ grid of squares. Now suppose we also have a collection of $nk^2$ points in general position inside the unit square and impose the additional req... | https://mathoverflow.net/users/8028 | Partitioning unit square with equal frequency rectangles | Here is an answer inspired by redistricting and the [shortest split line](https://en.wikipedia.org/wiki/Gerrymandering#Shortest_splitline_algorithm) algorithm.
For any rectangle with $mn$ points, consider the $2m-2$ ways of dividing it horizontally or vertically into two rectangles with an integral multiple of $n$ po... | 1 | https://mathoverflow.net/users/nan | 413874 | 168,828 |
https://mathoverflow.net/questions/413849 | 4 | Let $X$ be an irreducible projective variety over $\mathbb{C}$ (note that I do not assume $X$ smooth) and let $ p : X \longrightarrow S$ be a projective surjective morphism. For any open $U \subset S$, I consider the natural map:
$$ \pi : \mathrm{Hilb}^2\_{U}(p^{-1}(U)) \longrightarrow S^2(p^{-1}(U)/U)),$$
where $\ma... | https://mathoverflow.net/users/37214 | irreducibility punctual Hilbert scheme of relative subschemes of length $2$ | If I understand the question correctly, the subset $\mathcal{H}\_U$ is the closure of the locus parametrizing two distinct points. Over this locus, the Hilbert-Chow morphism $\pi$ is an isomorphism so irreducibility of $\mathcal{H}\_U$ is equivalent to irreducibility of $S^2(p^{-1}(U)/U \setminus \Delta\_{p^{-1}(U)/U})... | 3 | https://mathoverflow.net/users/12402 | 413879 | 168,831 |
https://mathoverflow.net/questions/413739 | 8 | Let $X$ be a (smooth) manifold. [It's well known](https://math.stackexchange.com/questions/451223/the-stone-%C4%8Cech-compactification-of-a-space-by-the-maximal-ideals-of-the-ring-of/451318#451318) that its Stone-Cech compactification $\beta X$ is homeomorphic to $\operatorname{Specm}(C(X))$, with its Zariski topology.... | https://mathoverflow.net/users/131975 | Let $X$ be a manifold. Is it true that $\beta X\cong \operatorname{Specm}(C^\infty(X))$? | Indeed, $\operatorname{Specm}(C^\infty(X))$ is homeomorphic to $\beta X$ (through an explicit homeomorphism that will be described below). Essentially all the theory described in Gillman & Jerison's classic book *Rings of Continuous Functions* (1960) applies: for completeness of MathOverflow, let me recall how this wor... | 8 | https://mathoverflow.net/users/17064 | 413891 | 168,834 |
https://mathoverflow.net/questions/413888 | 1 | In an earlier [positing](https://mathoverflow.net/questions/413804/can-gch-fail-everywhere-every-way) to $\mathcal MO$, it appears that the answer to if the $\sf GCH$ can fail everywhere in *every way* is to the negative, this is the case in $\sf ZFC$, however it also appears that matters are more free in absence of $\... | https://mathoverflow.net/users/95347 | Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of ZF? | Here's a somewhat trivial answer.
Note that $V\_\alpha$, for an infinite $\alpha$, have a particularly nice set of properties which follow from the fact that $|V\_\alpha\times V\_\alpha|=|V\_\alpha|$.
Now, easily, if $\sf AC$ fails, we can find arbitrarily high such cardinals. Simply take $|V\_\alpha|+\aleph(V\_\al... | 3 | https://mathoverflow.net/users/7206 | 413899 | 168,836 |
https://mathoverflow.net/questions/413901 | 4 | Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not be the galois extension. So my question is, can we always find such cyclic extension $(L/L')$ of degree $k$ such that $(L/\... | https://mathoverflow.net/users/215016 | Common Galois extension over $\mathbb Q $ | If $k$ is odd, then yes. If $L'/\mathbb{Q}$ is a cyclic extension of degree $4$, choose an extension $M/\mathbb{Q}$ that is cyclic of degree $k$. Then the compositum $L'M/\mathbb{Q}$ will have ${\rm Gal}(L'M/\mathbb{Q}) \cong {\rm Gal}(L'/\mathbb{Q}) \times {\rm Gal}(M/\mathbb{Q})$ which will be cyclic of order $4k$.
... | 7 | https://mathoverflow.net/users/48142 | 413903 | 168,837 |
https://mathoverflow.net/questions/413893 | 0 | The Weyl group of $\frak{g}\_2$ is the dihedral $D\_6$. Let us denote its longest element by $w\_0$. How many reduced decompositions does $w\_0$ have?
| https://mathoverflow.net/users/378228 | Number of reduced decompositions of the dihedral group $D_6$ | The weak order of a dihedral group looks like a polygon (see e.g. Figure 3.1 in the book "Combinatorics of Coxeter groups" by Björner and Brenti). Hence there are 2 reduced decompositions of $w\_0$ (= maximal chains from bottom to top) in these cases.
| 2 | https://mathoverflow.net/users/25028 | 413907 | 168,838 |
https://mathoverflow.net/questions/413892 | 3 | Suppose that $V$ is a finite-dimensional real vector space and that $W\subseteq \operatorname{GL}(V)$
is a subgroup generated by reflections (elements $s$ of order $2$ whose locus of fixed points $H\_s$ is a hyperplane.)
Assume that $W$ contains only finitely many reflections. So the $H\_s$ divide $V$ into finitely m... | https://mathoverflow.net/users/8726 | Finiteness of a reflection group | There are two differences between what this question is asking and standard results that are easy to find in the literature: (a) We don't assume ahead of time that $W$ is finite, but only that it has finitely many reflections, and (b) We don't assume ahead of time that the reflections all fix some Euclidean metric. We ... | 2 | https://mathoverflow.net/users/5519 | 413913 | 168,840 |
https://mathoverflow.net/questions/357924 | -1 | The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates:
$$S\_{\mathbb{R}^2}=\int\_0^\infty 2\pi r dr=2\pi\left(\frac{\tau^2}2+\frac1{24}\right)=\pi\tau^2+\frac\pi{12}$$
where $\tau=\int\_0^\infty dx$.
The regu... | https://mathoverflow.net/users/10059 | Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$? | Apparently, the discrepancy comes from my use of non-natural definition of multiplication of divergent integrals. When using a more natural and intuitive [Levi-Civita field kind of construction](https://mathoverflow.net/q/413873/10059), the multiplication gives
$\int\_0^\infty dx\cdot \int\_0^\infty dx=\omega^2=2\int... | 1 | https://mathoverflow.net/users/10059 | 413914 | 168,841 |
https://mathoverflow.net/questions/413908 | 1 | I am new to complex analysis and polynomials and I am looking for tutorials/books/articles for Mahler measure$$M(p((z))= \left|a\_0\right| \prod\limits\_{i=1}^d\max\{1,|\alpha\_i|\}
$$ of univariate polynomials and special polynomials like cyclotomic polynomials, polynomials over finite fields, the bounds of Mahler me... | https://mathoverflow.net/users/158175 | Mahler measure literature | In addition to the references at Wikipedia, see James McKee and Chris Smyth, Around the Unit Circle – Mahler Measure, Integer Matrices and Roots of Unity, <https://link.springer.com/book/10.1007/978-3-030-80031-4> and Brunault, F., & Zudilin, W. (2020), Many Variations of Mahler Measures: A Lasting Symphony (Australian... | 3 | https://mathoverflow.net/users/3684 | 413921 | 168,844 |
https://mathoverflow.net/questions/413597 | 4 | ([This is an old MSE question from me](https://math.stackexchange.com/questions/3223033/a-question-on-a-possible-cyclic-sieving-phenomenon), which did not get any answer, and when looking back seems interesting to post it here:)
Let $G$ be a finite group. Consider the set $X\_G:=\cup\_{H\le G} G/H$, where the disjoin... | https://mathoverflow.net/users/165920 | A question on a possible cyclic sieving phenomenon? | Here is a sketch for the case of the cyclic group and $S$ containing only its generator.
Let $m$ be odd and $n=2^\ell m$. Then
$
(-1)\ast g^k H = \{ g^m e^{-1} | e \in g^k H \}
$
defines an action of $C\_2=\{+1, -1\}$ on the set of cosets of subgroups of the cyclic group $\langle g\rangle$ of order $n$:
$$
(-1)\ast... | 2 | https://mathoverflow.net/users/3032 | 413931 | 168,847 |
https://mathoverflow.net/questions/412667 | 6 | We know that there are non-standard models of arithmetic, and in such models there are non-standard proofs of standardly unprovable sentences. Now, we can imagine a version of representability relative to some non-standard model of some arithmetical theory such that the said function would be what I would call a "non-s... | https://mathoverflow.net/users/467143 | Relationship between non-standard computation and TM(oracle)? | **Edit:** I think the answers given so far do not completely address the original question nor Gro-Tsen's follow up questions. I believe I can address some of these points so I have greatly expanded my original answer. In brief, no reasonable notion of "computation according to a nonstandard model of arithmetic" seems ... | 4 | https://mathoverflow.net/users/147530 | 413942 | 168,849 |
https://mathoverflow.net/questions/413592 | 4 | To explain what we are looking for, let's have a quick review on some points in Fourier transform on periodic functions in both continuous and discrete cases. We emphasize that our attention is just concerned with the abelian groups $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Z\_n}$ and the unit circle $\textbf{T}$.
On $\te... | https://mathoverflow.net/users/84390 | The main topics (issues, problems) of the Fourier transform | Saccone considered the space of all functions on $\mathbb{T}$ with *uniformly convergent* Fourier series. Call this space $U$. There is a natural norm given by $$\|f\|\_U = \sup\_{n\in\mathbb{N}} \|S\_nf\|\_{L^{\infty}}$$ where $S\_nf$ is the $n$th partial sum of the Fourier series of $f$. $U$ is complete with this nor... | 3 | https://mathoverflow.net/users/164350 | 413943 | 168,850 |
https://mathoverflow.net/questions/413875 | 1 | I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1\_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in $H\_{ul}^\theta$. Then $B$ is pre-compact in
$$H^1\_\rho = \{ u: \mathbb R \to \mathbb R : u, \partial\_x u \in L^2\_{l... | https://mathoverflow.net/users/471464 | First derivative of cut off function | I constructed a function that satisfies what I want. Let $f(x) = e^{-1/x},\,\, x>0$ and vanishes everywhere. Now we define $\varphi\_\beta(x) = f(\beta +1 - |x|)$ and $\psi\_\beta (x)=f(|x|- \beta)$. Therefore the cut-off function is
$$\chi\_\beta(x) = \frac{\psi\_\beta (x)}{\psi\_\beta (x) + \varphi\_\beta (x)}.$$
... | 1 | https://mathoverflow.net/users/471464 | 413946 | 168,851 |
https://mathoverflow.net/questions/413922 | 3 | Consider the compact group $ G=\operatorname{SO}\_3(\mathbb{R}) $. The closed subgroups of $ G $ (other than the trivial group 1 and the whole group $ G $) are exactly $ O\_2$, $\operatorname{SO}\_2 $ and the finite groups $ C\_n$, $D\_{2n}$, $T \cong A\_4$, $O \cong S\_4$, $I \cong A\_5 $ (cyclic groups with $ n $ ele... | https://mathoverflow.net/users/387190 | Self-normalizing implies maximal for subgroup of compact Lie group | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Denote by $\U'(n)$ the normalizer of $\U(n)$ in $\mathrm{GL}\_{2n}(\mathbf{R})$. It is not hard to see that $\U(n)$ has index 2 in $\U'(n)$, which is generated by $\U(n)$ and by the coordinate-wise complex conjugation. Moreover, $\U'(n)$ is ... | 3 | https://mathoverflow.net/users/14094 | 413949 | 168,853 |
https://mathoverflow.net/questions/413933 | 3 | I would appreciate a reference to support this statement that
appears under the **Geodesic** entry of the
[CRC Encyclopedia of Mathematics](https://www.routledge.com/The-CRC-Encyclopedia-of-Mathematics-Third-Edition---3-Volume-Set/):
>
> "no matter how badly a sphere is distorted,
> there exists an infinite number ... | https://mathoverflow.net/users/6094 | Infinite number of closed geodesics on distorted sphere | Will Jagy answered my question:
>
> Bangert, Victor. "On the existence of closed geodesics on two-spheres." *International Journal of Mathematics* 4, no. 01 (1993): 1-10.
> [doi](https://doi.org/10.1142/S0129167X93000029).
>
>
>
"...one obtains the existence of infinitely many closed geodesics for every Rieman... | 3 | https://mathoverflow.net/users/6094 | 413956 | 168,854 |
https://mathoverflow.net/questions/413958 | 0 | Let $G$ be a simply connected Lie group. Is it true that any finite dimensional representation of its Lie algebra is the differential of a representation of $G$?
A reference would be helpful.
Sorry if the question is too basic.
| https://mathoverflow.net/users/16183 | Representations of simply connected Lie groups | A finite-dimensional representation of a Lie algebra is in particular a homomorphism of finite-dimensional Lie algebras. Hence your question is answered in the affirmative e.g. by Th. 3.27 in F. Warner's book "Foundations of differential geometry and Lie groups".
| 3 | https://mathoverflow.net/users/468624 | 413960 | 168,855 |
https://mathoverflow.net/questions/413924 | 12 | This question is certainly somewhat opinion-based, but hopefully not hopelessly so.
The granddaddy of all applications for an efficient period finding or factoring capability (e.g. Shor's algorithm) is obviously breaking the public-key cryptography systems that currently encrypt Internet traffic.
But if we could ev... | https://mathoverflow.net/users/95043 | Would efficient factoring have any *other* useful applications? | I'm not sure about "real world", but studies around [multiplicative functions](https://en.wikipedia.org/wiki/Multiplicative_function) (e.g., [aliquot sequences](https://en.wikipedia.org/wiki/Aliquot_sequence)) will definitely benefit from the availability of a fast factorization method. At very least it will allow to v... | 9 | https://mathoverflow.net/users/7076 | 413962 | 168,857 |
https://mathoverflow.net/questions/413939 | 2 | Let $X$ be a continuous time stochastic process, and denote by $\mathcal F\_t$ its natural filtration. We define $\mathcal F\_z = \mathcal F\_0$ for all $z \leq 0$.
$X$ is said to be *strongly predictable* if there exists some $r > 0$, and an $\mathcal F\_{t-r}$ adapted process $Y$ such that
$$\lim\_{T \to \infty} ... | https://mathoverflow.net/users/173490 | A martingale convergence theorem | $\newcommand{\F}{\mathcal{F}}$The answer is yes.
Indeed, by time rescaling, without loss of generality (wlog) $r=1$.
Take any random variable (r.v.) $Z$ with $EZ=0$, and let $Z'$ be an independent copy of $Z$. Then, by Jensen's inequality, for any real $z$ we have $E|Z|\le E|Z-Z'|=E|(Z-z)-(Z'-z)|\le2E|Z-z|$, so tha... | 3 | https://mathoverflow.net/users/36721 | 413966 | 168,859 |
https://mathoverflow.net/questions/410576 | 1 | Given a simply connected locally compact group $G$, is it true that $G$ admits enough finite dimensional representations (over any field and not necessarily continuous) to separate points in $G$, what about over $\mathbb{C}$ and we require the representations to be continuous?
Again, this question is a follow-up of [... | https://mathoverflow.net/users/128540 | Does every locally compact, simply connected group admit enough finite dimensional representations? | The connected Lie groups whose points are separated by the finite-dimensional complex representations are exactly the linear Lie groups, for instance by Th. 5.3 in [Beltiţă and Neeb - Finite-dimensional Lie subalgebras of algebras with continuous inversion](http://dx.doi.org/10.4064/sm185-3-3).
| 4 | https://mathoverflow.net/users/468624 | 413972 | 168,861 |
https://mathoverflow.net/questions/413969 | 4 | In the " A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit"
We have :
**Proposition** 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if and only if $\alpha$ is homotopic to $\beta$.
Proof. One direction is vacuous since an iso... | https://mathoverflow.net/users/155706 | About isotopy and homotopy | Once you have found an annulus $R \subset S$ whose two boundary components are $\alpha$ and $\beta$, by definition of "annulus" there exists a homeomorphism $H : S^1 \times [0,1] \to R$. The composition
$$S^1 \times [0,1] \xrightarrow{H} R \hookrightarrow S
$$
then defines an isotopy in $S$ from $\alpha$ to $\beta$.
... | 3 | https://mathoverflow.net/users/20787 | 413980 | 168,864 |
https://mathoverflow.net/questions/413992 | 8 | Let $X$ be a topological space (feel free to add some simplifying assumptions here, like “completely regular” provided at least the case of finite-dimensional manifolds is covered). Let $f,g \in C^\*(X)$ where $C^\*(X)$ denotes the ring of bounded continuous real-valued functions on $X$. Denote $f^\beta, g^\beta \colon... | https://mathoverflow.net/users/17064 | When are the zero sets of two continuous functions in the Stone-Čech compactification included in one another? | Lemma: Suppose that $X$ is a compact Hausdorff space. Let $f,g:X\rightarrow[0,\infty)$ be continuous functions. Then the following are equivalent:
1. $Z(f)\subseteq Z(g)$.
2. For all $\epsilon>0$, there exists a $\delta>0$ where $f^{-1}[0,\delta]\subseteq g^{-1}[0,\epsilon]$.
3. There exists a function $u:[0,\infty)\... | 7 | https://mathoverflow.net/users/22277 | 413997 | 168,868 |
https://mathoverflow.net/questions/413985 | 2 | I have a doubt that assails me.
The technique of gluing along edges between manifolds is generally considered in the topological context.
I don't know if there are other gluing techniques.
I was wondering if there were theorems regarding the possibility of obtaining smooth gluing between isometric manifolds to each oth... | https://mathoverflow.net/users/90594 | Isometry and gluing between smooth manifolds - some references | Let (Mi)i=1,2 be smooth manifolds with boundary, Ni ⊆ ∂Mi unions of connected components of the boundaries of M1 and M2, respectively, and φ : N1 → N2 a diffeomorphism. Then there exists a smooth structure on the space M1 ∪φ M2 that arises by gluing M1 to M2 along N1 ≃ N2. This structure is unique up to a diffeomorphis... | 1 | https://mathoverflow.net/users/39082 | 414003 | 168,869 |
https://mathoverflow.net/questions/412251 | 5 | Robert Bryant's answer to [Isometric embedding of SO(3) into an euclidean space](https://mathoverflow.net/questions/161295/isometric-embedding-of-so3-into-an-euclidean-space) mentions that there is an isometric embedding of the round tetrahedral space $ SO\_3/A\_4 $ into the round sphere $ S^6 $.
I like that fact. Dr... | https://mathoverflow.net/users/387190 | Embedding round manifolds into low dimensional spheres | Here are a couple more examples I whipped up with the help of MSE/MO. I think they are enough to constitute an answer.
As I mentioned in my question Robert Bryant inspired me with his example of
* The tetrahedral space $ SO\_3/A\_4 $ as the orbit of $ xyz $ with respect to the representation of $ SO\_3 $ on the sev... | 0 | https://mathoverflow.net/users/387190 | 414018 | 168,877 |
https://mathoverflow.net/questions/412777 | 2 | *Previously [asked and bountied at MSE](https://math.stackexchange.com/questions/4338046/what-are-the-symmetry-groups-of-exponentiation-only-terms):*
---
Let $\mathfrak{E}=(\mathbb{N};\mathit{exp})$ be the algebra in the sense of universal algebra consisting of the natural numbers with just exponentiation. To eac... | https://mathoverflow.net/users/8133 | Possible symmetry groups of power terms | Based on [an observation by MSE user Pilcrow](https://math.stackexchange.com/questions/4358618/do-cyclic-groups-appear-as-symmetry-groups-of-exponentiation-only-terms#comment9100944_4358618), it seems I've been overcomplicating this:
For simplicity, let "$[x\_1,x\_2,...,x\_k]$" be shorthand for the right-associating ... | 1 | https://mathoverflow.net/users/8133 | 414021 | 168,878 |
https://mathoverflow.net/questions/414017 | 0 | The following is a cross-post of [this](https://math.stackexchange.com/questions/4354605/automating-proofs-via-indicator-functions) question on math.SE, which did not attract any comment and may therefore be too research-oriented for math.SE.
---
It is a common technique in measure theory to prove something for i... | https://mathoverflow.net/users/136236 | Automating proofs via indicator functions? | I think, you are looking for a "monotone class argument", see the wikipedia entry for the [monotone class theorem](https://en.wikipedia.org/wiki/Monotone_class_theorem). In a nutshell, prove that the property holds for indicators and is preserved under finite sums, scalar multiplication and under monotonically increasi... | 2 | https://mathoverflow.net/users/9652 | 414030 | 168,879 |
https://mathoverflow.net/questions/414041 | 1 | Let $n\geq1$ be an integer. Take the matrix $M(n)$, with entries, $M\_{i,j}(n)=\sin\left(\frac{(i+j)\pi}2\right)$ if $i\neq j$ and $M\_{i,i}(n)=x\_i$.
I wish to ask (this question has been modified from my previous post):
>
> **QUESTION.** Is there any interpretation/meaning to the matrix $M(n)$ and its determina... | https://mathoverflow.net/users/66131 | Interpret this matrix and its determinant | Well,
\begin{align\*}
\sum\_{\substack{1\leq i\_1<\cdots<i\_{n-2}\leq n\\
\binom{n+1}2-(i\_1+\cdots+i\_{n-2})\,\equiv\, 1\,\, \text{mod}\, 2}}x\_{i\_1}\cdots x\_{i\_{n-2}}
&=x\_1\cdots x\_n\cdot \left(\sum\_{i+j\,\text{is odd}}(x\_ix\_j)^{-1}\right)\\
&=x\_1\cdots x\_n(x\_1^{-1}+x\_3^{-1}+\ldots)(x\_2^{-1}+x\_4^{-1}+\... | 3 | https://mathoverflow.net/users/4312 | 414052 | 168,884 |
https://mathoverflow.net/questions/414047 | 0 | A well known result in Ramsey theory is: If the set of positive integers is partitioned into a finite number of sets, then at least one of these sets will contain a solution to $x+y=z$
By "property" I mean arithmetic statements like "will contain a solution to $x+y=z$" and by "invariant under partition" I mean that w... | https://mathoverflow.net/users/103391 | References for properties which are invariant under partition of $\mathbb{Z}$ by a finite number of sets | What you are looking for is called [partition regularity](https://en.wikipedia.org/wiki/Partition_regularity), see the linked Wikipedia article for many examples of situations where this naturally occurs. The underlying set can be anything, it needn't be the set of integers $\mathbb Z$. Moreover, I would not consider t... | 4 | https://mathoverflow.net/users/14233 | 414055 | 168,885 |
https://mathoverflow.net/questions/414048 | 0 | Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it an absolute retract therein?
| https://mathoverflow.net/users/15129 | Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space? | $\beta\mathbb{N}$ is not a retract of a Tychonoff cube because it is not connected; it also not a retract of a Cantor cube, not even a continuous image, see problem 3.12.12 in Engelking's book.
It is an absolute retract for ED spaces: if it is embedded in the compact ED space $X$ then $\mathbb{N}$ is relatively discr... | 3 | https://mathoverflow.net/users/5903 | 414056 | 168,886 |
https://mathoverflow.net/questions/414032 | 0 | Let $X$ and $Y$ be topological spaces. By a simple function $\phi: X\to Y$ we mean a finite range Borel measurable function.
Q. Is the point-wise limit of a sequence of simple functions a Borel measurable function?
| https://mathoverflow.net/users/84390 | Is the point-wise limit of simple functions a measurable function? | The answer to your question is Yes provided the topology of $Y$ is such that for each non-empty open set $O\subset Y$ there is a strictly increasing sequence $(O\_k)$ of open sets:
$$
\overline O\_k\subset O\_{k+1}\subset O,\quad k=1,2,\ldots,
$$
with $\bigcup\_kO\_k=O$. For suppose $(f\_n)$ is a sequence of Borel func... | 4 | https://mathoverflow.net/users/42851 | 414058 | 168,887 |
https://mathoverflow.net/questions/414054 | 4 | Sheafification is needed in limits and colimits of condensed abelian groups? If I have a functor $T: i \mapsto T\_i$ from an index category to condensed abelian groups the limit and colimit of this functor are just $S \mapsto \lim\_i T\_i (S)$ and $S \mapsto \text{colim}\_i T\_i (S)$ or sheafification is needed for it ... | https://mathoverflow.net/users/130868 | Limits and colimits in the category of condensed abelian groups | I'll ignore the set-theoretic issues since I don't understand them well enough to say anything about them.
As with any site, the limits (and in particular, the kernel) of sheaves may be computed pointwise. On the other hand, the colomits are usually not computed pointwise (cokernels included). For example, an exact s... | 7 | https://mathoverflow.net/users/146933 | 414062 | 168,888 |
https://mathoverflow.net/questions/414011 | 2 | I want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$.
I can see in many places the definition of the Lie derivative with respect to a vector field, and I also understand that an e... | https://mathoverflow.net/users/42291 | Lie derivative on Lie group in the direction of an element of Lie algebra | What about the formula $(D\_X\varphi)(g)=\frac{\rm d}{{\rm d}t}\Bigl\vert\_{t=0}\varphi(g\exp\_G(tX))$ for $\varphi\in C^\infty(G)$, $g\in G$, and $X\in \mathfrak g$, where $G$ is a Lie group with its Lie algebra $\mathfrak g$? See for instance Eq. (5) in Ch. II of the book by S.Helgason, "Differential geometry, Lie gr... | 1 | https://mathoverflow.net/users/468624 | 414065 | 168,889 |
https://mathoverflow.net/questions/414067 | 1 | Recall that an ideal of a commutative ring is said to be a nil ideal if each of its elements is nilpotent. I am looking for a non-zero nil ideal of a commutative ring that is idempotent.
| https://mathoverflow.net/users/338309 | An example of a commutative ring with a non-zero nil ideal that is idempotent | Consider the ring $A:=k[X^{\mathbb{Q}\_{\geq 0}}] / X$ of polynomials with non-negative rational exponents with the relation $X^1=0$. Then the ideal $I:=\operatorname{span}\_k\{X^a \mid 0<a<1\}$ is nil, because every $X^a$ is nil. But it is also idempotent because $X^a=(X^{a/2})^2$.
| 7 | https://mathoverflow.net/users/3041 | 414071 | 168,891 |
https://mathoverflow.net/questions/412064 | 5 | This is in some sense a generalization of the [question](https://mathoverflow.net/questions/410387/decomposition-of-manifolds-with-toroidal-boundary) I asked some time ago. I am very sorry if this question is too basic for MathOverflow, but I just started learning about some more detailed things of the topology of 3-ma... | https://mathoverflow.net/users/259525 | "Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary | You can find the discussion of the characteristic compression body in Section 3.3 of [Bonahon’s survey.](https://www.math.csi.cuny.edu/abhijit/gt3m/bonahon-gs3m.pdf) See Theorem 3.7 for the irreducible case, which together with the uniqueness of connect sum in Theorem 3.1 I think gives the sort of generalization of th... | 5 | https://mathoverflow.net/users/1345 | 414077 | 168,893 |
https://mathoverflow.net/questions/414080 | 0 | We face many places to find the collision probability of two sets (or more) in my case the cryptographic hash functions. We can formalize as;
Given two sets of random variables $\mathbf{A}$ and $\mathbf{B}$ compromised $m$ and $n$ elements, respectively. Consider $t$ discrete values where each variable $\{A\_1, A\_2,... | https://mathoverflow.net/users/91106 | A good approximation for collision probability between (two) sets of random variables | We are assuming that $A\_1,\dots,A\_m,B\_1,\dots,B\_n$ are iid random variables each uniformly distributed over the set $[t]:=\{1,\dots,t\}$.
By the [Bonferroni inequalities](https://en.wikipedia.org/wiki/Boole%27s_inequality#Bonferroni_inequalities), for the probability of at least one collision
\begin{equation}
... | 3 | https://mathoverflow.net/users/36721 | 414086 | 168,895 |
https://mathoverflow.net/questions/414051 | 7 | Let $k=\mathbb{R}$ or $\mathbb{C}$ and let $A$ be a finite-dimensional $k$-algebra. If $A$ is simple, then the Skolem-Noether theorem says that any two algebra homomorphisms $f, g: A \to M\_n(k)$ are conjugate in $M\_n(k)$, i.e., there exists a matrix $X \in M\_n(k)$ such that for all $a \in A$, $f(a) = X g(a) X^{-1}$.... | https://mathoverflow.net/users/16702 | Are nearby subalgebras of matrix algebras conjugate? | I'll describe a 1-parameter family of nonisomorphic 4-dimensional subalgebras of $M\_4(K)$. Consider, for $t\in K^\*$, the matrices
$$X=\begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix},\;Y\_t=\begin{pmatrix} 0 & 1 & 1 & 0\\ 0 & 0 & 0 & t\\ 0 & 0 & 0 & -t \\ 0 & 0 & 0 & 0 \end... | 11 | https://mathoverflow.net/users/14094 | 414089 | 168,896 |
https://mathoverflow.net/questions/414074 | 2 | By 1 step breach of the GCH I mean the following: $$ 2^{\aleph\_{\alpha}} = \aleph\_{\alpha+2}$$
Now, it is known that there are more constrains on the cardinality of power sets at singlular cardinals that their cardinality is influenced by the behaviour of the continuum below them! The main theorem supporting that i... | https://mathoverflow.net/users/95347 | If GCH is breached the same way before a singular of uncountable cofinality, would that breach extend to that singular? | One can collapse $2^\lambda$ to have cardinality $\lambda^+$ without adding $\lambda$-sequences even if $\lambda$ is singular. Therefore one could start with $2^{\aleph\_\alpha} = \aleph\_{\alpha+2}$ for all $\alpha < \omega\_1$ by Foreman-Woodin and then force $2^{\aleph\_{\omega\_1}} = \aleph\_{\omega\_1+1}$ without ... | 5 | https://mathoverflow.net/users/102684 | 414091 | 168,898 |
https://mathoverflow.net/questions/414069 | 3 | Given a set of distinct real numbers $(x\_j)$ and non-zero complex $(c\_j)$, then the large sieve says that
$$\limsup\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2\leq \sum\_{j}|c\_j|^2.$$
Are there lower bounds saying that
$$\liminf\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}\... | https://mathoverflow.net/users/159298 | Are there general large sieve lower bounds? | If the $x\_j$'s are distinct modulo $1$ (which is the natural assumption), then
$$\lim\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2=\sum\_{j}|c\_j|^2.$$
Indeed, let us assume (without loss of generality) that the $x\_j$'s lie in $[0,1]$. Then
$$\sum\_{n=1}^{N}\left|\sum\_{j}c\_j... | 5 | https://mathoverflow.net/users/11919 | 414093 | 168,900 |
https://mathoverflow.net/questions/316326 | 3 | Let $$S(n) = \sum\_{p \le n} b(n-p),$$ where
$b(a)=1$ is $a$ is a sum of two squares of positive integers and $b(a)=0$ otherwise.
Trivially by PNT we have
$$S(n) \le \sum\_{p \le n}1 \ll \frac{n}{\log n}.$$
Could we do better or the above estimate is the best possible?
| https://mathoverflow.net/users/95838 | Sieve bound for the sum of two squares | One can do a bit better. For simpler presentation assume that we instead consider the function $b'$ that is the indicator function of integers all of whose prime divisors are $1 \mod 4$. We have $b'\leq b$ but $b'$ and $b$ agree on square-free odd integers and any proof for $b'$ can be adapted to $b$. Furthermore, we h... | 4 | https://mathoverflow.net/users/9232 | 414097 | 168,902 |
https://mathoverflow.net/questions/414098 | 0 | The article [Quantifiers and Quantification](https://plato.stanford.edu/entries/quantification/#QuaProModLog) in Stanford Encyclopedia of Philosophy gives the reference below, but the article is not in the Journal of Symbolic Logic, 35.
May someone help me with finding Kaplan's article?
Kaplan, D., 1970, “S5 with Q... | https://mathoverflow.net/users/37385 | Where is D. Kaplan's “S5 with Quantifiable Propositional Variables” published? | I find another more precise citation elsewhere online as:
David Kaplan. S5 with quantifiable propositional variables. *The Journal of Symbolic Logic*, 35(2):355, 1970.
It seems possible it occurs within these pages: <https://doi.org/10.2307/2270571> ("Meeting of the Association for Symbolic Logic").
| 3 | https://mathoverflow.net/users/25028 | 414100 | 168,903 |
https://mathoverflow.net/questions/414102 | 1 | Let $(\mathcal{X},\Sigma,P)$ be a Polish probability measure space, and $(\mathcal{X}^n,\Sigma^{\otimes n},P^n)$ be the product of its $n$ copies. Let $t: x^n \in \mathcal{X}^n \mapsto L\_{x^n} \in \mathcal{P}(\mathcal{X})$ be the empirical measure function, where $L\_{x^n}$ is the empirical measure of the $n$-length s... | https://mathoverflow.net/users/126001 | How fine is the Borel $\sigma$-algebra induced by the weak topology? | Yes, and you can even do the approximation by a single set. All the structure you need is that $t$ is a measurable function between Polish spaces.
The function $t$ is continuous and hence measurable, and the space $\mathcal{P}(\mathcal{X})$ is again Polish. Let $A$ satisfy the condition in 1. Then $t^{-1}(A)$ is a Bo... | 1 | https://mathoverflow.net/users/35357 | 414114 | 168,908 |
https://mathoverflow.net/questions/414113 | 3 | Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G\_\beta$ the stabilizer subgroup of $\beta$ by the cooadjoint action. Let $V$ be symplectic vector space on which $G\_\beta$ acts.
So, we have $V$ is symplectic, $G/G\_\... | https://mathoverflow.net/users/172459 | Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic | Your tangent space is too big. It has rank $\dim G + \dim V$ but it should have rank $\dim M = \dim G - \dim G\_\beta + \dim V$.
I think that the tangent space at $[g, v]$ is actually $T\_{g} (G/G\_\beta) \times T\_vV$. On the first factor you have a symplectic form $\omega\_\beta$ and on the second factor the symple... | 1 | https://mathoverflow.net/users/6818 | 414123 | 168,911 |
https://mathoverflow.net/questions/414124 | 11 | I am reading the following paper [1998(H.Hudzik)](https://www.jstor.org/stable/44152979) P.574
It reads using L'Hopital rule$$\liminf\_{u\to\infty} \frac{1/\varphi(1/u)}{\psi(u)}=\liminf\_{u\to\infty}\frac{\varphi'(u)}{\psi'(u)u^2[\varphi(1/u)]^2}.$$
That means we can apply L'Hopital for lower limits i.e. $$\liminf\_... | https://mathoverflow.net/users/147009 | L'Hopital rule for upper and lower limit? | The full L'Hopital rule says that
$$\liminf \frac{f'}{g'}\leq\liminf\frac{f}{g}\leq\limsup\frac{f}{g}\leq\limsup\frac{f'}{g'}.$$
So in the special case when the limit of $f'/g'$, exists, the limit of $f/g$
also exists and is equal to the limit of $f'/g'$.
This general rule is proved by integration.
| 23 | https://mathoverflow.net/users/25510 | 414138 | 168,918 |
https://mathoverflow.net/questions/414139 | 22 | There are [many models](https://arxiv.org/pdf/math/0610239.pdf) for $(\infty,1)$-categories: simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.
Doubtlessly the model most used to do higher category theory in is the model of quasi-categories, due to the work of Lurie ([Higher Topos T... | https://mathoverflow.net/users/475383 | Why are quasi-categories better than simplicial categories? | As a preface, I think that this question should be viewed as analogous to "what are the advantages of ZFC over type theory" or vice versa. We're talking about foundations -- in principle, it doesn't matter what foundations you use; you end up with an equivalent model-independent theory of $\infty$-categories.
The par... | 31 | https://mathoverflow.net/users/2362 | 414141 | 168,920 |
https://mathoverflow.net/questions/414135 | 6 | Let us consider $C(\mathbb{R})$, the space of continuous functions on the reals.
Q. Does there exist a sequence $\{f\_n\}$ in $C(\mathbb{R})$ such that for every $f\in C(\mathbb{R})$ one may find a subsequence $\{f\_{n\_k}\}$ of $\{f\_n\}$ pointwise converging to $f$, i.e., $f(x)=\lim f\_{n\_k}(x)$ for all $x$?
| https://mathoverflow.net/users/84390 | Some special sequence in $C(\mathbb{R})$ | Yes. Take the countable set $P$ of all polynomials with rational coefficients and enumerate it somehow so that $P=\{p\_1,p\_2,\dots\}$.
Given any $f\in C(\mathbb R)$, for each natural $k$ there exists some natural $n\_k$ such that $\sup\_{x\in[-k,k]}|p\_{n\_k}(x)-f(x)|<1/k$. Here without loss of generality we may ass... | 11 | https://mathoverflow.net/users/36721 | 414142 | 168,921 |
https://mathoverflow.net/questions/292936 | 5 | I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity.
More precisely: suppose $u$ satisfies weakly
$$
D\_i(a^{ij}D\_ju)=f
$$
on the $d$-dimensional ball of radius $R$, with $0$ Dirichlet boundary conditions, with the matrix $(a^{ij})\_{i,j=1..d}$ bounded f... | https://mathoverflow.net/users/100941 | Dependence of the Hölder exponent in De Giorgi-Nash-Moser | (By scaling we can take $\lambda=1$ for simplicity. I will also take $f=0$, and discuss just the interior estimates.)
The precise exponent is known in two dimensions (see Piccinini & Spagnolo 1972). It is $\Lambda^{-1/2}$. You can find the "worst" coefficient field in the obvious way: the diffusion matrix $a(x)$ has ... | 9 | https://mathoverflow.net/users/5678 | 414151 | 168,926 |
https://mathoverflow.net/questions/414150 | 2 | Let $S$ be a smooth projective surface. We denote $S^{[n]}$ the Hilbert scheme of artinian subschemes in $S$ of length $n$, which is a smooth projective variety of dimension $2n$ by Fogarty. Let $I\subset S^{[n]}\times S$ be the universal correspondance, and let $p: I\to S^{[n]}$ and $q: I\to S$ be projections. To be p... | https://mathoverflow.net/users/119184 | tangent bundle of Hilbert schemes of points on a projective surface | The tangent bundle on $S^{[n]}$ is not quite isomorphic to $(T\_S)^{[n]}$, rather by Theorem B of Stapleton's paper listed below there is an injection of the former into the latter.
*Stapleton, David*, [**Geometry and stability of tautological bundles on Hilbert schemes of points**](http://dx.doi.org/10.2140/ant.2016... | 4 | https://mathoverflow.net/users/6263 | 414152 | 168,927 |
https://mathoverflow.net/questions/414119 | 4 | For a non recursive $x \in 2^{\omega}$, define $C\_x = \{y \in 2^{\omega}: x \leq\_T y\}$. Note that $y \in C\_x$ iff $(\exists e)(\forall n)(\Phi^y\_e(n) = x(n))$ where $\Phi\_e$ is the $e$th Turing functional. So $C\_x$ is a $G\_{\delta \sigma}$-set (countable union of countable intersection of open sets). Can we sho... | https://mathoverflow.net/users/475361 | Borel ranks of Turing cones | Assume $x$ is noncomputable, as otherwise it isn't true. Fix a $G\_{\delta\sigma}$ set $\bigcup\_i \bigcap\_j A\_{i,j}$, where the $A\_{i,j}$ are open. We'll show that this is not the complement of $C\_x$ by building a real $y$ which is either in both sets or in neither. This will be a finite extension argument; I thin... | 3 | https://mathoverflow.net/users/32178 | 414155 | 168,929 |
https://mathoverflow.net/questions/414153 | 6 | The Pólya–Vinogradov inequality asserts that a non-principal Dirichlet character $\chi$ with modulus equal to $q$ satisfies
$$\displaystyle \left \lvert \sum\_{N < n < N+M} \chi(n) \right \rvert = O \left(\sqrt{q} \log q \right)$$
for all positive integers $N$, $M$.
My question concerns changing the summation con... | https://mathoverflow.net/users/10898 | Pólya–Vinogradov inequality for Eisenstein integers | No.
Such a bound would imply a similar bound on
$$\displaystyle \left \lvert \sum\_{N(z) = M} \left(\frac{z}{w} \right)\_3 \right \rvert.$$
If $M$ is a product of distinct primes $p\_1,\dots p\_n$ congruent to $1$ mod $3$, the norms of primes $\pi\_1,\dots,\pi\_n$ then $$\sum\_{N(z) = M} \left(\frac{z}{w} \right)\_... | 10 | https://mathoverflow.net/users/18060 | 414156 | 168,930 |
https://mathoverflow.net/questions/414116 | 7 | Let $X\to S$ be a family of smooth projective complex threefolds over a connected base $S$, could it happen that for some $a, b\in S(\mathbb{C})$, the fiber $X\_a$ is birational to $\mathbb{P}^3\_{\mathbb{C}}$, while the fiber $X\_b$ is not birational to $\mathbb{P}^3\_{\mathbb{C}}$?
| https://mathoverflow.net/users/nan | Is rationality a deformation invariant property for smooth threefolds? | I believe this is still an open question in dimension $3$.
There is an example due to Hassett, Kresch and Tschinkel ([Stable rationality in smooth families of threefolds](https://arxiv.org/abs/1802.06107)) of a smooth projective family of threefolds over a connected base where some fibers are *stably* rational and ot... | 6 | https://mathoverflow.net/users/12402 | 414157 | 168,931 |
https://mathoverflow.net/questions/414019 | 3 | $c\_{0}$, the space of the scalar sequence that converges to $0$ endowed with the sup norm, has two well-known bases: the unit vector basis $(e\_{n})\_{n}$, where $e\_{n}(k)=1$ if $k=n$ and $0$ otherwise, and the summing basis $(s\_{n})\_{n}$, where $s\_{n}=\sum\_{k=1}^{n}e\_{k}$. It is known that $c\_{0}$ has no bound... | https://mathoverflow.net/users/41619 | Bases in $c_{0}$ | You have answered your own question modulo a lemma that is basically obvious:
Lemma. Let $(x\_n)$ be a basis for a Banach space $X$ and let $Y=(\sum\_{n=0}^\infty E\_n)\_0$ be the $c\_0$-sum of $E\_n := \text{span} \{ x\_0,\dots,x\_n\}$. Let $(y\_k)$ be the obvious basis for $Y$ induced by concatenating the given bas... | 1 | https://mathoverflow.net/users/2554 | 414158 | 168,932 |
https://mathoverflow.net/questions/414159 | 1 | Let define $F(s)=\int\_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim\_{t\to 0}f(t)$ exists. Then we can get $\lim\_{t\to 0}f(t)=\lim\_{s\to\infty}sF(s)$.
Can we use upper limits or lower limits to replace all the above limits? i.e. we only know $\limsup\_{t\to 0}f(t)$ or $\liminf\_{t\to0}f(t)$ exist. Can we get ... | https://mathoverflow.net/users/147009 | Initial and final Theorem for upper and lower limits? | The answer is no.
Let $f(u)$ be defined piecewise. On intervals of the form $(2^{k},2^{k+1}]$, where $k\in \mathbb{Z}$, you set $f(u) = (-1)^k$. Then you have that $\limsup\_{t\to 0} f(t) = +1$ and $\liminf\_{t\to 0} f(t) = -1$.
The integral $s F(s) = \int\_0^\infty f(u/s) e^{-u} ~du$ is:
* Continuous in $s$
* Fo... | 2 | https://mathoverflow.net/users/3948 | 414162 | 168,934 |
https://mathoverflow.net/questions/410104 | 4 | Consider two independent continuous random walks on a graph $G$ with adjacency matrix $A$. I am interested in the probability that the two walkers will ever meet.
When the graph is a $k$-regular graph with $k=1,2$ then the probability is one, see this question for example: [What is the probability that two random wal... | https://mathoverflow.net/users/142153 | Probability that two walkers will meet on a graph | This is equivalent to the question whether two walks will necessarily meet infinitely often (with probability one) or not. This question has been studied in some detail, see in particular [2] where some general criteria are given.
[1] Krishnapur, Manjunath, and Yuval Peres. "Recurrent graphs where two independent ran... | 2 | https://mathoverflow.net/users/7691 | 414163 | 168,935 |
https://mathoverflow.net/questions/414081 | 0 | My question is about the compatibility and consistency between two definitions of cohomology in two books.
I asked this question about 10 days ago [on MathSE](https://math.stackexchange.com/questions/4351968/compatibility-and-consistency-between-two-definitions-of-cohomology-in-two-books)
and I set a bounty on it, bu... | https://mathoverflow.net/users/166540 | Coboundary operators, 1-cocycles and computing cohomology | Let me just put my comments as an answer: (i) To compute the cohomology groups $H^n(G,A)$, for $A$ a (left) $G$-module and where
$$H^n(G,A)=\text{Ext}^n\_{{\mathbb Z}G}({\mathbb Z},A)$$
one needs a projective resolution of the $G$-module ${\mathbb Z}$, say
$$\cdots\stackrel{d\_3}\rightarrow P\_2\stackrel{d\_2}\rightarr... | 4 | https://mathoverflow.net/users/3903 | 414166 | 168,936 |
https://mathoverflow.net/questions/412957 | 7 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^n $ is compact the isometry group $ \Iso(S^n,g) $ is also compact. And every compact group can be realized as the real p... | https://mathoverflow.net/users/387190 | Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold? | Maybe these arguments are of interest to you. It is known that for any compact symmetric space $M$ the tangent bundle $TM$ possesses a canonical structure of a complex
manifold. Multiplication by $-1$ on $TM$ is an antiholomorphic involution; its set of fixed points is $M$, when identified with the zero section
of $TM$... | 3 | https://mathoverflow.net/users/85592 | 414174 | 168,939 |
https://mathoverflow.net/questions/414083 | 12 | Roughly speaking, say that a logic $\mathcal{L}$ is **self-equivalence-defining** (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ unary relation symbols and an $\mathcal{L}[\Sigma']$-sentence $\eta$ such that the following are equivalent for all ... | https://mathoverflow.net/users/8133 | Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence? | (Working in ZFC.)
No (re $\mathcal{L}\_{\omega\_1,\omega}$). Suppose it is. Consider the signature $\Sigma$ with just one binary relation symbol $<$. Let $\Sigma',\eta$ witness SED-ness for $\Sigma$.
Let $\pi:M\to V\_\theta$ be elementary,with $\theta$ some sufficiently large limit ordinal, $M$ transitive, $M$ of c... | 13 | https://mathoverflow.net/users/160347 | 414179 | 168,940 |
https://mathoverflow.net/questions/414182 | 1 | I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching.
Fix $0<a<b<\infty$, and let $\mathcal{P}\_{[a,b]}$ be the set of all probability distributions on $[a,b]$. For each $p \in \mathcal{P}\_{[a,b]}$, let
\begin{align\*}
\mu(p) &= \int\_... | https://mathoverflow.net/users/15570 | What is the maximum possible coefficient of variation for data taking values within a specified range? | Indeed, for any random variable (r.v.) $X$ with values in $[a,b]$ and mean $\mu\in[a,b]$,
$$Var\,X\le Var\,X\_{a,b;\mu}=(b-\mu)(\mu-a), \tag{1}$$
where $X\_{a,b;\mu}$ is any r.v. with the unique distribution with mean $\mu$ supported on the two-point set $\{a,b\}$. So, the result that you conjectured follows, because
$... | 1 | https://mathoverflow.net/users/36721 | 414183 | 168,941 |
https://mathoverflow.net/questions/414096 | 4 | Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive leading coefficient and not of the form $(at+b)^2+c^2$ for some rationals $a,b,c$. Then standard sieve heuristics would s... | https://mathoverflow.net/users/9232 | How often is the value of a quadratic polynomial equal to a sum of two integer squares? | This problem has been addressed in a paper of Friedlander and Iwaniec in Acta Mathematica 1978, called [Quadratic polynomials and quadratic forms](https://projecteuclid.org/journals/acta-mathematica/volume-141/issue-none/Quadratic-polynomials-and-quadratic-forms/10.1007/BF02545740.full). Under general conditions they c... | 7 | https://mathoverflow.net/users/38624 | 414184 | 168,942 |
https://mathoverflow.net/questions/413906 | 9 | Iwaniec [1] proved that
$$
\pi(x+x^\theta)-\pi(x) < \frac{(2+\varepsilon)x^\theta}{\eta(\theta)\log x},\ x>x\_0(\varepsilon,\theta).
$$
with
$$
\eta(\theta)=\frac{15\theta-2}{9}.
$$
(Actually, he proves that a function $\eta(\theta)>\theta$ exists, and that this is an admissible choice. This choice gives nontrivial i... | https://mathoverflow.net/users/6043 | Primes between $x$ and $x+x^\theta$ | Montgomery (1) gives a list of 40 exponent pairs $(\kappa,\lambda)$ which can be plugged into Iwaniec's formula
$$
\eta(\theta)=\left(1+\frac{1-\lambda+2\kappa}{3-\lambda-\kappa/2}\right)\theta - \frac{\kappa}{3-\lambda-\kappa/2}
$$
to yield bounds for $0<\theta\le1/2.$ Of these, 36 are optimal in some interval; ad... | 6 | https://mathoverflow.net/users/6043 | 414189 | 168,944 |
https://mathoverflow.net/questions/414103 | 2 | Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true?
>
> Let $D\_{2n}$ be the denominator of $\frac{B\_{2n}}{4n}$ where $B\_n$ is
> the $n$-th [Bernoulli number](http://oeis.org/A006863). If $\gcd(a, D\_{2n}) = 1$ then
>
>
> $$ a^{2n} \equiv 1\pmod{D\_{2n}}.$$
>
>
>
This question ... | https://mathoverflow.net/users/23388 | Analogue of Fermat's little theorem for Bernoulli numbers | A proof is essentially given in Section 5.1 of [Notes on primitive lambda-roots](http://www.maths.qmul.ac.uk/%7Epjc/csgnotes/lambda.pdf) by P. J. Cameron and D. A. Preece.
| 5 | https://mathoverflow.net/users/7076 | 414205 | 168,947 |
https://mathoverflow.net/questions/414209 | 15 | **Motivation.** In a coin game, a player flips all their coins every turn, starting with just one coin. If the coins all land *heads* then the game stops; otherwise, the number of coins is doubled for the following turn. While the game may clearly terminate on any turn, there is in fact a positive probability that it w... | https://mathoverflow.net/users/8628 | Is $\prod_{i=1}^\infty (1-\frac{1}{2^{(2^i)}})$ transcendental? | I can prove that the number actually described by the word problem, which is $$ \prod\_{i=0}^{\infty} \left( 1- \frac{1}{2^{2^i}} \right),$$ is irrational, by a method similar to David Speyer's.
Expanding out the product, we get $$\sum\_{j=0}^{\infty} \frac{(-1)^{t\_j} }{2^j}= 1+\sum\_{j=1}^{\infty} \frac{(-1)^{t\_j}... | 18 | https://mathoverflow.net/users/18060 | 414214 | 168,952 |
https://mathoverflow.net/questions/414206 | 8 | It is well known that in $\mathbb{Z}\_2$-valued simplicial cohomology (and other cohomologies)
$$ Sq^1 = \beta\;,$$
where $Sq^1$ is the first Steenrod square and $\beta$ is the Bockstein homomorphism for the short exact sequence
$$ \mathbb{Z}\_2 \overset{\cdot 2}{\rightarrow} \mathbb{Z}\_4 \overset{\mod 2}{\rightarrow}... | https://mathoverflow.net/users/115363 | Analogue of Bockstein for crossed module extensions and higher Steenrod square | $\DeclareMathOperator{\Sq}{Sq}\newcommand{\Z}{\mathbb{Z}}$The short version is that every cohomology operation can be interpreted as a Bockstein operator for an "exact sequence" (read: fiber sequence) of grouplike $E\_\infty$ spaces.
Any cohomology operation $\delta:H^\*(-;A)\Rightarrow H^{\*+k}(-;B)$ such as $\Sq^i$... | 11 | https://mathoverflow.net/users/35687 | 414221 | 168,953 |
https://mathoverflow.net/questions/414186 | 8 | **Definition:** *Highly composite prime gap*
The three composite numbers between the consecutive primes $643$ and $647$ each have at least three distinct prime factors. This is the first occurrence of prime gap of length $> 1$ where each composite number in the gap has at least $3$ distinct prime factors. We call pri... | https://mathoverflow.net/users/23388 | Are there highly composite prime gaps? | Assuming the [prime tuples conjecture](https://en.wikipedia.org/wiki/Dickson%27s_conjecture), all of these questions have affirmative answers. For instance, one can use the Chinese remainder theorem to find $a,b$ such that the tuple
$$ an+b, \frac{an+b+1}{2^2 \times 5}, \frac{an+b+2}{3 \times 7}, \frac{an+b+3}{2 \times... | 12 | https://mathoverflow.net/users/766 | 414230 | 168,956 |
https://mathoverflow.net/questions/414224 | 0 | Let $S(x, y, m, n) = \sum\limits\_{i=0}^m \binom{n}{i}x^i y^{n-i}$, where $0 < m < n$. I want to derive the relation between $S(x, y, m, n)$ and $S(x, y, m, n-1)$.
Is there any formulas I can use?
| https://mathoverflow.net/users/475457 | Sum of the first m terms of the expansion $(x+y)^n$ | One can construct the relation using the identity
$$
\binom{n}{i} = \binom{n-1}{i} + \binom{n-1}{i-1}
$$
Then, writing the $i=0$ term separately,
\begin{eqnarray}
S(x,y,m,n) &=& y^n + \sum\_{i=1}^{m} \binom{n-1}{i} x^i y^{n-i}
+ \sum\_{i=1}^{m} \binom{n-1}{i-1} x^i y^{n-i} \\
&=& y^n + y\sum\_{i=1}^{m} \binom{n-1}{i} x... | 2 | https://mathoverflow.net/users/134299 | 414232 | 168,958 |
https://mathoverflow.net/questions/414233 | -2 | Consider the number of integer partitions $p(n)$ of $n$ whose generating function is
$$\sum\_{n\geq0}p(n)\,x^n=\prod\_{k\geq1}\frac1{1-x^k}.$$
Also, the number of partitions into distinct parts $Q(n)$ of $n$ whose genertaing function is
$$\sum\_{n\geq0}Q(n)x^n=\prod\_{k\geq1}(1+x^k).$$
Expand the ratio of these two gen... | https://mathoverflow.net/users/66131 | Congruence modulo 4 for a generating function leads to perfect squares? | $$\frac{1+x^k}{1-x^k}=(1+x^k)(1+x^k+x^{2k}+\dots)=1+2x^k+2x^{2k}+\dots$$
Multiplying all these together and looking at terms contributing to the coefficient of $x^n$, we see that the term will be contributing something divisible by $4$ unless it comes from multiplying a bunch of $1$s together with $2x^{k\cdot d}$ for $... | 6 | https://mathoverflow.net/users/30186 | 414235 | 168,959 |
https://mathoverflow.net/questions/361421 | 2 | (**UPDATED** for rapid decay considerations + new question)
In dimension 2, the Radon transform range theorem states that a *rapidly decaying* (Schwartz) function $g(t,\theta)$ can be represented as a Radon transform of some function $f(x,y)$ (i.e. $g=R[f]$) if and only if, for all integers $n\geq0$
$$P\_n(\theta) :=... | https://mathoverflow.net/users/158671 | Radon transform range theorem and radial functions | To answer Q1: There are trig identities at play. First, 0 is usually accepted under the definition of "homogeneous polynomial" (i.e., it's a polynomial whose coefficients are all zero) so there is no contradiction there. But for the case of the second moment being constant, we have the identity $\sin^2(\theta) + \cos^2... | 2 | https://mathoverflow.net/users/61024 | 414256 | 168,961 |
https://mathoverflow.net/questions/414261 | 0 | Let $k,m$ and $\rho$ be positive integers. In "Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II"
Mahler considered the complex integrals $A\_k(x)=\frac1{2i\pi}\int\_{\mathcal C\_0}\frac{e^{zx}\mathrm dz}{\prod\_{h=0}^m(z+k-h)^\rho}$ where $\mathcal C\_0$ is a circle centered in $0$ with radius les... | https://mathoverflow.net/users/33128 | It is obvious for Mahler, not for me! | These are just the details of prets's comment. Note that
\begin{align\*}
\frac{e^{zx}}{z^\rho}&=\sum\_{l\ge0}\frac{(zx)^l}{z^\rho l!}=
\sum\_{l\ge0}\frac{z^{l-\rho}x^l}{l!}
=\sum\_{l=0}^{\rho-1}\frac{z^{l-\rho}x^l}{l!}+\sum\_{l\ge\rho}\frac{z^{l-\rho}x^l}{l!}\\
&=\sum\_{l=0}^{\rho-1}\frac{z^{l-\rho}x^l}{l!}+\sum\_{n\ge... | 3 | https://mathoverflow.net/users/109085 | 414262 | 168,964 |
https://mathoverflow.net/questions/414271 | 4 | Let $P\_{H}(G, t)$ be the number of vertex colorings of a graph $G$ in $t$ colors that avoid having a monochromatic subgraph $H$. In particular, for $H$ given by a single edge we recover the usual chromatic polynomial $P\_{H}(G, t) = P(G, t)$.
Question: Are there easy proofs that $P\_{H}(G, t)$ is a polynomial for $t... | https://mathoverflow.net/users/141327 | Subgraph avoiding colorings | Fix a partition onto non-empty color classes (there are finitely many ways to do so, denote by $k$ the number of distinct color classes) without monochromatic $H$. After that, there is $t(t-1)\ldots(t-k+1)$ ways to assign colors. Sum up, you still get a polynomial.
| 6 | https://mathoverflow.net/users/4312 | 414273 | 168,968 |
https://mathoverflow.net/questions/413996 | 7 | Recall that a topological space $X$ is *scattered* if and only if every non-empty subset $Y$ of $X$ contains at least one point which is isolated in $Y$. Consider the statement: "Every scattered hereditarily separable space is countable".
In this [book](https://books.google.com.mx/books?id=k8uYhaIfHuAC&pg=PA229&lpg=P... | https://mathoverflow.net/users/146942 | Scattered hereditarily separable does not imply countable in ZFC | As clarified in the comments, the existence of a regular S-space is independent of ZFC, but a "real" $T\_2$ example can be constructed by taking a well-ordering of a set of reals in order type $\omega\_1$ and refining the Euclidean topology by declaring initial segments open. This example is $T\_2$ since it refines a $... | 8 | https://mathoverflow.net/users/121994 | 414276 | 168,969 |
https://mathoverflow.net/questions/414265 | 11 | $\DeclareMathOperator\PSU{PSU}$Let $ \PSU\_n $ be the projective unitary group. Let $ A\_m $ be the alternating group on $ m $ letters.
$ A\_5 $ is a maximal closed subgroup of $ PSU\_2 \cong SO\_3(\mathbb{R}) $.
The references in [The finite subgroups of SU(n)](https://mathoverflow.net/questions/17072/the-finite-s... | https://mathoverflow.net/users/387190 | Alternating subgroups of $\mathrm{SU}_n $ | The comment of @abx is part of a general picture. For $n >7$ the alternating group $A\_{n}$ has no non-trivial complex irreducible character of degree less than $n-1$, so that for $n > 7$, $A\_{n}$ is not isomorphic to any subgroup of ${\rm GL}(n-3, \mathbb{C}).$
Furthermore, if $n > 7$ is also even, then the double ... | 15 | https://mathoverflow.net/users/14450 | 414287 | 168,972 |
https://mathoverflow.net/questions/414269 | 0 | In this [question](https://mathoverflow.net/questions/412729/can-we-invoke-almost-supermartingale-theorem-for-deterministic-sequences), there is a proof for deterministic version of "Almost Supermartingale"
**Question: Can we extend [[1](https://mathoverflow.net/questions/412729/can-we-invoke-almost-supermartingale-t... | https://mathoverflow.net/users/123235 | Proof of yet another extension of deterministic variant of "(Almost) Supermartingale" convergence theorem | No. E.g., for $j=1,2$ and $k=0,1,\dots$, take $V\_j^k=2+(-1)^{j+k}$, $S\_{1,2}^k=0$, $U\_{1,2}^k=0$, $\alpha\_k=0$, and $\beta\_k=0$.
| 2 | https://mathoverflow.net/users/36721 | 414296 | 168,975 |
https://mathoverflow.net/questions/413805 | 10 | The Clausen-Scholze theory of condensed mathematics offers an abelian category with enough projective objects that embraces the study of arbitrary locally compact (and Hausdorff) groups. The behaviour of the tensor product is managed by restricting to a subcategory of solid abelian groups within the category of condens... | https://mathoverflow.net/users/124943 | Are condensed vector spaces over finite fields always solid? | Peter Scholze's comment gives a good answer to the main direct question and tells us that the condensed mod p vector space with basis a compact Hausdorff space S is solid if and only if S is finite. The advantages of solidification are going to become more apparent as we use the condensed maths more widely. Solidificat... | 1 | https://mathoverflow.net/users/124943 | 414301 | 168,976 |
https://mathoverflow.net/questions/412066 | 7 | Consider the map
$$f:\mathbb C^2\to\mathbb C^2$$
$$(x,y)\mapsto(x^2y,xy^2)$$
We can view $f$ as induced by the map of monoids $g:\mathbb Z^2\_{\geq 0}\to\mathbb Z^2\_{\geq 0}$ given by the matrix $(\begin{smallmatrix}2&1\cr 1&2\end{smallmatrix})$. Thus $f$ is a map of log schemes.
>
> Is $f$ log smooth?
>
>
>
... | https://mathoverflow.net/users/35353 | Is $(x^2y,xy^2)$ log smooth? | It is smooth. For a log map to have good "fiber bundle properties", one needs more than smoothness. Rather, one needs the relevant maps of monoids to be **exact** in the sense of Kato. A reference is Nakayama--Ogus "Relative rounding in toric and logarithmic geometry" [Mathscinet](https://mathscinet.ams.org/mathscinet-... | 0 | https://mathoverflow.net/users/35353 | 414307 | 168,977 |
https://mathoverflow.net/questions/414164 | 9 | Let $f\_i=f\_i(x\_1,x\_2,\ldots, x\_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda\_1,\lambda\_2, \ldots, \lambda\_n)$ consider the determinant
$$
D\_\lambda(f)=\left | \begin{array}{lllll}
f\_{\lambda\_1} & f\_{\lambda\_1+1} & f\_{\lambda\_1+2} & \ldots & f\_{\lambda\_1+... | https://mathoverflow.net/users/40637 | Determinant connection between Schur polynomials and power sum polynomials | I don't know a fully general result, but your pattern for partitions $\lambda$
of length $\leq n$ with $n$-th entry $\lambda\_{n}\geq n-1$ and with $n$
indeterminates persists:
>
> **Theorem 1.** Let $n$ be a positive integer. Let $\lambda=\left( \lambda
> \_{1},\lambda\_{2},\ldots,\lambda\_{n}\right) $ be an integ... | 4 | https://mathoverflow.net/users/2530 | 414316 | 168,981 |
https://mathoverflow.net/questions/414302 | 10 | Let $R$ be a complete local $\mathbf{Z}\_p$-algebra, for some prime $p$. In the 1970 paper [Group schemes of prime order](https://doi.org/10.24033/asens.1186) by Oort and Tate, they write down an explicit finite flat group scheme $G\_R(a, b)$ of rank $p$, for each pair of elements $a,b \in R$ satisfying $ab = p$, and s... | https://mathoverflow.net/users/2481 | Homomorphisms between Oort–Tate group schemes | Yes. The Tate-Oort description is an equivalence of categories between finite flat group schemes (over a $\Lambda$-scheme $S$ where $\Lambda$ is a certain ring decribed in the paper) and the category of triples $(L , a, b)$ where $L$ is an invertible sheaf over $S$ and $a\in \Gamma(S,L^{\otimes (p−1)})$, $b\in\Gamma(S,... | 8 | https://mathoverflow.net/users/17988 | 414318 | 168,982 |
https://mathoverflow.net/questions/414292 | 1 | Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.
Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we have a comodule map
$$
a:V\to\mathbb{C}[G]\otimes V.
$$
Let $v\in V$ be a non-zero vector; then we may choose a basis ... | https://mathoverflow.net/users/97652 | Koszul complex of equations defining a stabilizer | See arXiv:1411.7994, Proposition 1.28 for a description of the cohomology dg-algebra of a Koszul complex in the case when the zero locus has codimension smaller than the number of equations.
| 3 | https://mathoverflow.net/users/4428 | 414319 | 168,983 |
https://mathoverflow.net/questions/414242 | 5 | The following definition should be standard, but let me state it just in case there is some ambiguity:
If $\mathscr{L}$ is a set of subsets of a set $X$ that is closed under finite unions and intersections and contains $\varnothing,X$ (or more generally, if $\mathscr{L}$ is a distributive lattice with top and bottom ... | https://mathoverflow.net/users/17064 | Ultrafilters of closed sets | The construction you describe when $\mathscr{C}$ consists of all closed sets of $X$ is known as the Wallman compactification of $X$. I'll denote if $\omega(X)$. It is due to Wallman; *Lattices and topological spaces*, Ann. Math. 39 (1938) 112-126.
Of course some sort of techincal assumption is required.
>
> Let $... | 4 | https://mathoverflow.net/users/54788 | 414329 | 168,985 |
https://mathoverflow.net/questions/413705 | 1 | Let $(x\_{n})\_{n=1}^\infty$ be a bounded sequence in a Banach space $X$. We set
$$\textrm{ca}((x\_{n})\_{n=1}^\infty)=\inf\_{n}\sup\_{k,l\geq n}\|x\_{k}-x\_{l}\|.$$
Then $(x\_{n})\_{n=1}^\infty$ is norm-Cauchy if and only if $\textrm{ca}((x\_{n})\_{n=1}^\infty)=0$.
Let $X$ be a Banach space with a basis $(x\_{n})\_{... | https://mathoverflow.net/users/41619 | Quantifications of boundedly complete bases | This is an answer to Dongyang's query in the comments. I show $\text{bc}(x\_n)\ge 1$ if $(x\_n)$ is a monotone basis for $X$ that is not boundedly complete. Let $(x\_n^\*)$ be the functionals biorthogonal to $(x\_n)$ and let $Y$ be the closed linear span in $X^\*$ of $(x\_n^\*)$, so that $(x\_n^\*)$ is a monotone basis... | 0 | https://mathoverflow.net/users/2554 | 414331 | 168,986 |
https://mathoverflow.net/questions/414313 | 0 | If we know that there are two vectors $x,y\in\mathbb{R}^d$ satisfying
\begin{equation}
\|x\|\ge c\_1 \|y\|, \quad x^Ty\ge c\_2\|x\|^2,
\end{equation}
where $c\_1>0$ and $c\_2>0$ are some given constants, can we always find a positive definite matrix $M\in\mathbb{R}^{d\times d}$ such that
\begin{equation}
x=My?
\end{equ... | https://mathoverflow.net/users/178204 | Find a matrix transformation for 2 vectors with conditions | Yes. We only need $x^T y > 0$, which is implied by your conditions if neither x nor y is zero. Let $s$ be the positive square root of $x^T y$, and $A$ a $d\times d$ real matrix who's first column equals $\frac1s x$. Let the remaining columns be a basis for the codimension-$1$ space perpendicular to y. Let $e\_1$ be the... | 2 | https://mathoverflow.net/users/474034 | 414334 | 168,987 |
https://mathoverflow.net/questions/414320 | 8 | In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - **Why can't we identify those as the
trivial locales, and what's so great about considering locales
that have no points?**
>
> Anyway, here's the construction of... | https://mathoverflow.net/users/124549 | What's the point of a point-free locale? | A good answer to both questions is provided by the following [variant of the Gelfand duality for commutative von Neumann algebras](https://mathoverflow.net/questions/23408/reference-for-the-gelfand-duality-theorem-for-commutative-von-neumann-algebras/360088#360088),
which shows that the following categories are equival... | 15 | https://mathoverflow.net/users/402 | 414338 | 168,989 |
https://mathoverflow.net/questions/412686 | 11 | I almost expect the answer to this question to be no, but I can't find it explicitly said anywhere.
Given a formal group law $f$ of height $n$ over a perfect field $k$ of characteristic $p$, we can construct the Morava E-theory $E(n)$. The resulting cohomology theory is
$$E(n)^m (X) = MP^m(X)\otimes \_{MP(\*)} R$$
... | https://mathoverflow.net/users/170467 | Does the spectrum of Morava E-theory depend only on height? | Here's an argument that Eric Peterson and I came up with showing that the homotopy type of Morava $E$-theory only depends on the choice of perfect char $p$ field $k$ and the height $n<\infty$.
$\textbf{Lemma}$: Let $R$ be a ring with two Landweber exact formal group laws $e,f:\text{Laz}\rightarrow R$ and let $E$ and ... | 11 | https://mathoverflow.net/users/163893 | 414342 | 168,991 |
https://mathoverflow.net/questions/414293 | 8 | Let $\mathcal C=(\mathcal O, \mathcal M)$ be a category internal to topological spaces. Thus $\mathcal O$ and $\mathcal M$ are topological spaces: the space of objects and the space of morphisms respectively. These spaces come endowed with structure maps: $i \colon \mathcal O \to \mathcal M$, $s, t\colon \mathcal M\to ... | https://mathoverflow.net/users/6668 | What is the right notion of a functor from an internal topological category to topological spaces? | The definition you propose is that of a [$\mathsf{Top}$-internal diagram](https://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/internal+diagram) in $\mathcal{C}$. It comes from viewing categories as many-object monoids, and functors/presheaves on categories as the many-object generalisation of left or right modules ... | 13 | https://mathoverflow.net/users/130058 | 414345 | 168,992 |
https://mathoverflow.net/questions/414340 | 6 | Let $V$ be the vertices of a regular $p$-agon in the plane, where $p$ is prime, and let $C$ be a set. Given two maps $f,g:V\rightarrow C,$ I believe it is true that either
(a) $f=g\circ r$ for some rotation $r:V\rightarrow V,$ or
(b) there is $h:V\rightarrow C$ such that for all $v\in V,$ $h(v)\neq f(v),$ but for a... | https://mathoverflow.net/users/313687 | A question about colorings of the vertices of a p-agon, where p is prime | We may identify $V$ with the field $\mathbb{F}\_p$ of size $p$, then denoting $g\_k(x):=g(x+k)$ for $k\in \mathbb{F}\_p$ the questions reads as:
if $f(x)\not\equiv g\_k(x)$ for each $k\in \mathbb{F}\_p$, prove that there exists $h(x)$ such that $h(x)\ne f(x)$ for all $x$; but for each $k$ there exists $x(k)$ for whic... | 6 | https://mathoverflow.net/users/4312 | 414351 | 168,996 |
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