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https://mathoverflow.net/questions/359757 | 1 | Incidentally, I came across the following functional equation
$$f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$$
that is to hold for all $x,y\in \mathbb R$. Is there a neat way to find all solutions $f:\mathbb R\to \mathbb R$?
| https://mathoverflow.net/users/157747 | The functional equation $f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$ | Adding a constant to f does not change the property, so we may assume f(0)=0. Under this assumption, I claim that the property holds if and only if f is additive. The if part is obvious. For the only if part, we first set y=0 to obtain f(x)=2f(x/2)-f(0)=2f(x/2). Next, we set x+y=a, xy=b, and we obtain f(b)+f(a-b)=2f(a/... | 7 | https://mathoverflow.net/users/12120 | 359783 | 151,489 |
https://mathoverflow.net/questions/359772 | 5 | It is well-known in geometric analysis that one can use curve-shortening flow to prove the isoperimetric inequality (where the general result requires curve-shortening flow for non-convex curves).
I was wondering if it might be possible to also prove such an inequality using Ricci flow (given hypotheses on convexity ... | https://mathoverflow.net/users/119114 | Ricci flow proof of isoperimetric inequality | Anthony Manning proved that the [volume entropy decreases under volume normalized Ricci flow on surfaces of negative curvature](https://doi.org/10.1017/S0143385703000415). Question 4 at the end of his paper asks whether the Cheeger isoperimetric constant is a strictly increasing function of Ricci flow. So it looks like... | 6 | https://mathoverflow.net/users/1345 | 359784 | 151,490 |
https://mathoverflow.net/questions/359769 | 3 | Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f\_\*(L\otimes f^\*\mathcal{F})\cong f\_\*L\otimes\mathcal{F}$? I think it is a well-known result but I couldn't find a reference. You can assu... | https://mathoverflow.net/users/119065 | Projection formula for flat morphisms | Based on my comment, I constructed the following counterexample (which I believe is standard):
**Example.** Let $(E,O)$ be an elliptic curve, let $Y = E$ and $X = E \times E$, with $f \colon X \to Y$ the first coordinate projection. Let $\mathscr L = \mathcal O\_{E \times E}(\Delta - E \times O)$, and let $\mathscr F... | 4 | https://mathoverflow.net/users/82179 | 359786 | 151,491 |
https://mathoverflow.net/questions/359776 | 3 | Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}\_g$ and $\mathcal{A}\_g$ have natural structures as quasi-projective varieties. Mumford and Fogarty's book (Geometric Invariant Theory) is indicated as a reference for this statement. However, it is an advan... | https://mathoverflow.net/users/29836 | $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties | The GIT proof gives very nice compactifications of these spaces (and is the "right" way to do this), but they were known to be quasiprojective varieties long before GIT was developed.
The classical proofs depend on properties of theta functions. For $\mathcal{A}\_g$, it should be attributed to some combination of Sat... | 8 | https://mathoverflow.net/users/317 | 359794 | 151,492 |
https://mathoverflow.net/questions/359825 | 2 | Given two probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such that $f^\*\mu = \gamma$?
| https://mathoverflow.net/users/41638 | Image of probability measures under measurable mappings | There is a complete classification of probability spaces up to a measure-preserving
isomorphism.
Specifically, consider a category whose objects are triples
(X,Σ,μ), where X is a set, Σ is a σ-algebra
of measurable subsets on Σ,
and μ is a probability measure on (X,Σ).
If we want to have a nice description of morph... | 4 | https://mathoverflow.net/users/402 | 359832 | 151,502 |
https://mathoverflow.net/questions/359760 | 8 | Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact manifold. It was asked [here on MO](https://mathoverflow.net/questions/138535/are-smooth-functions-tame) if this space is ta... | https://mathoverflow.net/users/78745 | Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space? | Using Fourier series, $C^\infty(S^1)$ is ismorphic to the sequence space
$$s=\{(x\_k)\_{k\in \mathbb N}\in \mathbb C^{\mathbb N}: \sum\_{k=1}^\infty k^{2n} |x\_k|^2 <\infty \text{ for all $n\in\mathbb N$}\}$$
and this space is isomorphic to $\mathscr S(\mathbb R^d)$.
This is very classical, a modern treatment (with muc... | 10 | https://mathoverflow.net/users/21051 | 359833 | 151,503 |
https://mathoverflow.net/questions/359845 | 10 | By numerical computation it seems like, if $a\_0 < a\_1$:
$$
\begin{multline}
\log({a\_0}^2 + {a\_1}^2 + 2 a\_0 a\_1 \cos(\omega t)) = \log({a\_0}^2 + {a\_1}^2) \\
+ \frac{a\_0}{a\_1}\cos(\omega t)
- \frac{1}{2}\frac{{a\_0}^2}{{a\_1}^2}\cos(2\omega t)
+ \frac{1}{3}\frac{{a\_0}^3}{{a\_1}^3}\cos(3\omega t)
- \frac{1}{... | https://mathoverflow.net/users/157781 | Fourier series of $\log(a +b\cos(x))$? | Let's consider
$$
f(x) = \log (1+q^2+2q\cos x) = \log |1+qe^{ix}|^2 ,
$$
which differs from your function only by the additive constant $2\log a\_1$ if we take $q=a\_0/a\_1$. Since $|q|<1$, we can use the Taylor series of $\log(1+z)$ to write
$$
\begin{align}
f(x) = 2\,\textrm{Re}\; \log (1+qe^{ix}) = 2\,\textrm{Re}\su... | 17 | https://mathoverflow.net/users/48839 | 359849 | 151,509 |
https://mathoverflow.net/questions/359820 | -1 | Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus\_{j=1}^{n}I\_{j}$ be a morphism of $A$-modules, where each $I\_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we conclude that there is an injective envelope $g : X \rightarrow \bigoplus\_{t=1}^{m}I\_{j\_{t}}$? (Here $\{ j\_1, \ldots... | https://mathoverflow.net/users/156726 | Can we extract an injective envelope from a monomorphism? | Yes: given a monomorphism $f\colon X\to I$ with $I$ injective, as in the question, you can find a decomposition $I=I\_0\oplus I\_1$ such that $f=\begin{pmatrix}f\_0&0\end{pmatrix}$ and $f\_0\colon X\to I\_0$ is left minimal, so it is an injective envelope.
Getting this decomposition does not require any assumption on... | 2 | https://mathoverflow.net/users/21483 | 359850 | 151,510 |
https://mathoverflow.net/questions/359827 | 11 | I want to cite a MathSciNet review. Is there a standard format for that? (I couldn't find anything)
| https://mathoverflow.net/users/8588 | How to cite a MathSciNet review | Mathematical Reviews can be cited like any other journal, with a review number instead of volume and page numbers:
J. Smith. Review of the article “Regular doodads are widgits” by J. Doe. Mathematical Reviews **123456** (2009).
```
@article {review,
AUTHOR = {Smith, J.},
TITLE = {Review of the article ``{R... | 5 | https://mathoverflow.net/users/402 | 359853 | 151,512 |
https://mathoverflow.net/questions/357925 | 3 | Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H\_\*(A)=0$, i.e. $A$ is acyclic.
**Question:** Does this imply that the Hochschild homology $HH\_\*(A)$ also vanishes identically? If not, what hypotheses does one need to impose on $R$ and $A$? (In the case I am mostly interested i... | https://mathoverflow.net/users/155668 | Hochschild homology of acyclic complex | Hochschild homology is a derived invariant, and in particular a quasi-isomorphism invariant, since quasi-isomorphic algebras are derived invariant. Your algebra is non-unital, I assume, since $1$ is usually a non-trivial cycle in general.
In any case, however, you can consider the Hochschild cyclic complex of $A$, c... | 1 | https://mathoverflow.net/users/21326 | 359875 | 151,521 |
https://mathoverflow.net/questions/359730 | 2 | Im looking for a recurrence formula of type:
$$(\mu-\nu) x P\_\mu^\nu(x) + P\_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$
where $P\_\mu^\nu(x)$ is the Legendre function of the first kind (solution to the Legendre differential equation which is regular at the origin).
My goal is to rewrite the sum in one expressio... | https://mathoverflow.net/users/84558 | A recurrence formula for the Legendre function $P_\mu^\nu(x)$ | Let's take relation 14.10.3 from the [NIST Handbook](https://dlmf.nist.gov/14.10), which, after renaming $\mu \leftrightarrow \nu $ and shifting the new $\mu \rightarrow \mu -2$ reads
$$
(\mu -\nu ) P\_{\mu }^{\nu } (x) - (2\mu -1)x P\_{\mu -1}^{\nu } (x)
+ (\mu+\nu -1) P\_{\mu -2}^{\nu } (x) =0
$$
We can thus isolate ... | 2 | https://mathoverflow.net/users/134299 | 359881 | 151,524 |
https://mathoverflow.net/questions/359873 | 2 | Let $A\_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S\_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S\_{k}}$. Thus, $\mathcal{S\_k}$ has cardinality $N{\cdot}k$, with $k \in \{1,2,..,N\}$.
\begin{equation\*}
A\_{N,N} =
\begin{pmatrix}... | https://mathoverflow.net/users/157807 | Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S | Suppose that we selected $m$ random elements of $S\_k$. An element $s$ of $S\_k$ appears in the induced matrix iff (i) there is a selected element in the row of $s$ in $A$; and (ii) there is a selected element in the column of $s$ in $A$. Call such an element $s$ *lucky*, and so $X$ is the number of lucky elements.
U... | 2 | https://mathoverflow.net/users/7076 | 359884 | 151,525 |
https://mathoverflow.net/questions/359869 | 9 | **Question 1:** For which rings $R$ does there exist a small set $S \subseteq Mod\_R$ such that every module $M \in Mod\_R$ is a direct sum of modules in $S$?
Equivalenty, for which rings $R$ does there exist a cardinal $\kappa$ such that every module $M \in Mod\_R$ is a direct sum of modules generated by $\leq \kapp... | https://mathoverflow.net/users/2362 | Categories of modules generated under coproducts by a small set? | The rings satisfying your condition (for right modules) are the **right pure semisimple rings**. There are many equivalent conditions. You can find a lot of information in Section 4.5 of the book
*Prest, Mike*, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambrid... | 8 | https://mathoverflow.net/users/22989 | 359906 | 151,534 |
https://mathoverflow.net/questions/359766 | 4 | In Martin Hairer's 2013 paper "[Solving the KPZ equation](https://annals.math.princeton.edu/wp-content/uploads/annals-v178-n2-p04-s.pdf)", the process $X\_\epsilon^\bullet$ is defined as the stationary solution to
$$
\partial\_t X\_\epsilon^{\bullet} = \partial\_x^2 X\_\epsilon^{\bullet} + \Pi\_0^{\perp} \xi\_\epsilon
... | https://mathoverflow.net/users/48500 | How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper? | What I mean is that
$$
X\_\epsilon^\tau(t) = \int\_{-\infty}^t P\_{t-s} \Pi\_0^\perp (\partial\_x X\_\epsilon^{\tau\_1}(s)\, \partial\_x X\_\epsilon^{\tau\_2}(s))\,ds\;,
$$
where $P\_t$ denotes convolution with the heat kernel. The product appearing on the right is the usual pointwise product of two random variables wi... | 7 | https://mathoverflow.net/users/38566 | 359916 | 151,538 |
https://mathoverflow.net/questions/359911 | 5 | We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$.
My query is for every Banach space $X$ (need not be separable ) do there exist a Banach space $Y$ and an injective compact operator $T:X\to Y$?
| https://mathoverflow.net/users/41137 | Existence of injective compact operators | No, for cardinality reasons. The range of a compact operator is norm-separable hence has cardinality continuum (if non-zero). It is then enough to take $X$ to have bigger cardinality, for example, $X = \ell\_\infty^\*$. Then you have no chance of building such operators.
Another possibility for counterexamples comes ... | 12 | https://mathoverflow.net/users/15129 | 359919 | 151,539 |
https://mathoverflow.net/questions/359918 | 2 | **Background:** I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded interval in $\mathbb{R}$. I am mostly interested in the spaces where $q=2$. I found only one such embedding in the one ... | https://mathoverflow.net/users/117762 | Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension? | Here is a partial answer, which has to do with dual compact embeddings: If the embedding between (resonable) Banach spaces $X\subset\subset Y$ is compact then the dual embedding is compact too, $Y^\*\subset\subset X^\*$.
This is useful here since the space of Radon measures is the dual of continuous bounded functions... | 2 | https://mathoverflow.net/users/33741 | 359923 | 151,541 |
https://mathoverflow.net/questions/359895 | 6 | Let $n\geq 2$ and $x\_1,\ldots,x\_n > 0$ be such that $x\_1+\cdots+x\_n =1$. Is it true that there must exist a positive integer $k$ such that $$\{x\_1k\}+\cdots+\{x\_nk\} = n-1?$$
This looks closely related to the [density of the fractional part](https://math.stackexchange.com/questions/903142/for-an-irrational-numb... | https://mathoverflow.net/users/156705 | High sum of fractional parts | If the $x\_i$ are rational, take the lcm of the denominators and decrease it by $1$.
If they are not necessarily rational, act similarly: using, e.g., [Kronecker's theorem](https://en.wikipedia.org/wiki/Kronecker%27s_theorem), take a $k$ such that all the $kx\_i$ are sufficiently close to integers, and decrease that ... | 10 | https://mathoverflow.net/users/17581 | 359925 | 151,542 |
https://mathoverflow.net/questions/354076 | 7 | The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x\_1,\dots, x\_n)$ be the free graded-commutative algebra generated by $n$ elements of respective degrees $\lvert x\_i\rvert$. Then, the sign $\varepsilon(\sigma)$ of a permutation $\sigma$ on $(x\_1,\dotsc, x... | https://mathoverflow.net/users/144957 | Reason to apply the Koszul sign rule everywhere in graded contexts | A precise statement of the conventions (which Jesse is referring to) is that the authors are using the symmetric monoidal structure on graded vector spaces, where the braiding map,, $\tau: V \otimes W \to W \otimes V$, is given by $$v \otimes w \mapsto (-1)^{{\rm deg} ~w ~{\rm deg}~ v} w \otimes v.$$
Roughly, what it... | 17 | https://mathoverflow.net/users/52918 | 359960 | 151,554 |
https://mathoverflow.net/questions/359954 | 1 | If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, then $(2^G, \mu)$ is a Borel, essentially free, probability measure preserving action of $G$ on a standard Lebesgue space.
... | https://mathoverflow.net/users/62393 | Locally compact Polish groups acting on standard Lebesgue spaces | Yes - this is Proposition 1.2 of [Adams, Elliott and Giordano](https://www.ams.org/journals/tran/1994-344-02/S0002-9947-1994-1250814-5/) ([MathSciNet review](https://mathscinet.ams.org/mathscinet-getitem?mr=1250814)), Amenable actions of groups, Trans. Amer. Math. Soc. 344 (1994), 803-822.
| 2 | https://mathoverflow.net/users/8588 | 359964 | 151,555 |
https://mathoverflow.net/questions/356879 | 4 | When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E\_7(q))\cong Z\_2$ a square element in $2.E\_7(q)$?
A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is defined satisfying the following axioms:
(i) $[xy]$ is bilinear for $x, y\in L$.
(ii) $[xx]=0$ for $x\in L$.
... | https://mathoverflow.net/users/152963 | Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$? | The answer is always, yes. Note that there are three classes of involutions in the simply connected version of the algebraic group $E\_7$: the central involution $a$, an involution $t$ with centralizer of type $A\_1D\_6$, and the product $at$. If $a$ were not a square, then in the simple group $E\_7(q)$, we would only ... | 3 | https://mathoverflow.net/users/152674 | 359976 | 151,559 |
https://mathoverflow.net/questions/359856 | 7 | In [the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik](http://www-math.mit.edu/~etingof/egnobookfinal.pdf) on page 10 it says:
'Conversely, it is well known (and easy to show) that any exact faithful functor $F : \mathcal{C} \rightarrow \text{Vec}$ is represented by a uni... | https://mathoverflow.net/users/157078 | Any exact faithful functor is represented by a unique projective generator | Let $\mathcal C$ be the category of finite dimensional left modules over a finite dimensional ring $R$. Let $G: \mathcal C \to \mathrm{Vec}$ be an exact and faithful functor to finite dimensional vector spaces. We use $V^\*$ to denote the dual vector space. For motivation, notice that if we had a representing object $M... | 5 | https://mathoverflow.net/users/52918 | 359978 | 151,561 |
https://mathoverflow.net/questions/356800 | 3 | Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$.
Let $M$ be a finitely generated $kG$-module.
We denote the first syzygy of $M$ by $\Omega(M)$, i.e.
$\Omega(M):=\text{Ker}(p)$ where $P\stackrel{p}{\rightarrow} M$ is a minimal projective cover of $M$.
${}$
A $k... | https://mathoverflow.net/users/12826 | Can MAGMA compute almost projective $kG$-homomorphisms? | I cannot answer 100%, but I can tell you what I know is there, and maybe its enough with some tweaking. AR-sequences are not something I've needed to implement in Magma yet, so I've not grappled with this one.
Magma can compute projective covers and a syzygy, first off. Then it can compute $\texttt{AHom(A,B)}$, which... | 3 | https://mathoverflow.net/users/152674 | 359980 | 151,563 |
https://mathoverflow.net/questions/359965 | 8 | Following the conventions from Heil: "[A Basis Theory Primer](http://people.math.gatech.edu/~heil/papers/bases.pdf)" and Albiac, Kalton: "Topics in Banach Space Theory", we might define a *basis* of an (infinite-dimensional) normed space $V$ as a sequence $(e\_n)$ in $V$, such that for any $x \in V$ there is a unique s... | https://mathoverflow.net/users/114094 | Basis vs Schauder basis in normed spaces | You ask for an instructive example, so I'll be long winded.
Suppose $(x\_n)$ is basis for a normed space $(X,\|\cdot\|)$. The partial sum projections $S\_n$ are well defined but might be discontinuous. Define a new, larger, norm on $X$ by $|x| := \sup\_n \|S\_n x\|$. Then $(x\_n)$ is a Schauder basis for $(X,|\cdot|)... | 9 | https://mathoverflow.net/users/2554 | 359982 | 151,564 |
https://mathoverflow.net/questions/359894 | 8 | I would like to learn about perverse sheaves.
I will be grateful if someone could recommend me a book with the following structure.
1. Introduction to basic homotopy theory (derived category and t-structure)
2. Introduction to sheaves
3. Introduction to perverse sheaves
| https://mathoverflow.net/users/155635 | A recommendation for a book on perverse sheaves | Pramod Achar is working on a book on perverse sheaves and applications in representation theory. It's a great book!
**EDIT (2021):** The book has now been published by the AMS: [Perverse Sheaves and Applications to Representation Theory](https://bookstore.ams.org/surv-258/).
| 9 | https://mathoverflow.net/users/438 | 359985 | 151,565 |
https://mathoverflow.net/questions/359915 | 9 | My question concerns cofinal branches through [Kleene's $O$](https://en.wikipedia.org/wiki/Kleene%27s_O), which is a set of natural numbers and a computably enumerable relation $<\_O$ on this set that provides
ordinal denotations for any desired computable ordinal. For every number $n\in O$, the $<\_O$-predecessors of ... | https://mathoverflow.net/users/1946 | Does every cofinal branch through Kleene's O compute true arithmetic? | [Goncharov, Harizanov, Knight and Shore](http://pi.math.cornell.edu/~shore/papers/pdf/GHKSFinalVersion3.pdf) investigated the Turing degrees of $\Pi^1\_1$ cofinal branches (which they call "paths through $\mathcal{O}$"). They showed there is a $\Pi^1\_1$ cofinal branch which does not compute $\emptyset'$, so certainly ... | 10 | https://mathoverflow.net/users/32178 | 359987 | 151,566 |
https://mathoverflow.net/questions/359981 | 2 | The definitions of an ideal of an algebraic structure $A$ (as a substructure $I$ such that the product of $A$ and $I$ is a subset of $I$) do not involve associativity.
However, the definitions of a principal ideal I know (for a semigroup or a ring) assume associativity.
For example, the left principal ideal $S^1a$... | https://mathoverflow.net/users/148743 | Principal ideal of a non-associative magma | In a magma $M$, one can describe the 2-sided ideal generated by a subset $Y$ as follows: define by induction
$$M\_1=M,\;Y\_1=Y,\; M\_n=\bigcup\_{p,q\ge 1,p+q=n}M\_pM\_q,\;Y\_n=\bigcup\_{p,q\ge 1,p+q=n}(M\_pY\_q\cup Y\_pM\_q).$$
Then the 2-sided ideal generated by $Y$ is $Y\_\infty=\bigcup\_{n\ge 1} Y\_n$.
An alternat... | 2 | https://mathoverflow.net/users/14094 | 359991 | 151,567 |
https://mathoverflow.net/questions/359787 | 2 | Let $M$ be a smooth manifold with boundary $\partial M$. Let $Diff\_0(M)$ be the group of all diffeomorphisms homotopic to identity. According to [this article (Page 6, section "
Beyond mapping class group")](http://www.math.brown.edu/~mann/papers/RealizationProblems.pdf), the restriction of a diffeomorphism to the bou... | https://mathoverflow.net/users/9485 | Restriction of diffeomorphisms homotopic to identity to the boundary | As discussed in comments, $Diff\_0$ stands for the subgroup of the diffeomorphism group, consisting of diffeomorphisms isotopic (rather than homotopic) to the identity. With this in mind, the fact that the restriction map $\phi: Diff(M)\to Diff(\partial M)$ sends $Diff\_0(M)$ to $Diff\_0(\partial M)$ is clear. Let's pr... | 4 | https://mathoverflow.net/users/39654 | 359994 | 151,568 |
https://mathoverflow.net/questions/359858 | 5 | I am reading the presentation of Cuntz' proof of Bott periodicity for $C^\*$-algebras in Wegge-Olsen (Thm. 11.2.1). Here one considers the short exact sequence of $C^\*$-algebras
$$0 \longrightarrow \mathcal{T}\_0 \longrightarrow \mathcal{T} \stackrel{q}{\longrightarrow} \mathbb{C} \longrightarrow 0,$$
where $\mathcal{... | https://mathoverflow.net/users/16702 | Question on Cuntz' proof of Bott periodicity | The conclusion drawn in the book of Wegge-Olsen is wrong (explained below), but can, however, easily be tweaked to a correct proof. What is shown is that $j\circ q$ is homotopic to the identity on $\mathcal T$ and hence the same is true after tensoring with $A$. It follows that $K\_\ast(\mathcal T\_0 \otimes A) = 0$ fo... | 3 | https://mathoverflow.net/users/126109 | 359997 | 151,570 |
https://mathoverflow.net/questions/359522 | 0 | Consider a sequence of random observations $(O(t))\_{t\geq 1}$, with $O(t)=(D(t),J(t),Y(t))$. Denote $\mathcal{F}(t) := \sigma(O(1),\ldots,O(t))$, the filtration induced by the first $t$ observations.
$J(t)$ takes values in $\{1,\ldots,J\}$, for some integer $J \geq 2$, and should be understood as the index of a buck... | https://mathoverflow.net/users/100069 | Independence in a sequential problem with observations getting added to buckets | Let for all $t\geq 1$,$\mathcal{F}^-(t):=\sigma(\mathcal{F}(t-1), D(t), J(t)).$
The hypothesis in the third bullet point can be rephrased as $Y(t)|\mathcal{F}^-(t) \stackrel{d}{=} \widetilde{Y}(j,n(j,t)) | \widetilde{\mathcal{F}}(j,n(j,t)-1)$.
I prove the claim by induction. Fix $j$ and $t$. I show by induction that ... | 1 | https://mathoverflow.net/users/100069 | 360007 | 151,576 |
https://mathoverflow.net/questions/359931 | 2 | Let $(X,\mathscr{A},\mu)$ be a probability space and let $\{A\_1,\ldots,\}\subset\mathscr{A}$ be a countable family of sets with small measure: say $\mu(A\_i)\le\epsilon$. I am trying to show that one can find a countable (disjoint!) partition $\{B\_i\}$ of $X$ with the following property: Each $A\_i$ is covered by som... | https://mathoverflow.net/users/12518 | Covering families of sets by small-measure partitions | The answer is negative.
First we may always assume that there are only finitely many $B\_i$ -s: The sum of the measures of the $B\_i$-s converges so we may take the union of all but finitely many of them with measure of this union less that $\epsilon^2$ and replace this cofinite set of $B\_i$-s by their union.
Now le... | 1 | https://mathoverflow.net/users/6921 | 360009 | 151,578 |
https://mathoverflow.net/questions/78209 | 40 | Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth function $f: \mathbf R^n \to \mathbf R^{n-d}$, submersive at each point of $M$ and such that $M=f^{-1}(0)$.
Of course th... | https://mathoverflow.net/users/5239 | When is a submanifold of $\mathbf R^n$ given by global equations? | An example of a compact $16$-dimensional submanifold of $\mathbb{R}^{30}$ with trivial normal bundle that is not defined by such global equations may be found in [Akbulut-King, Submanifolds and homology of nonsingular real algebraic varieties, [Am J Math **107** 45 (1985)](https://dx.doi.org/10.2307/2374457), Theorem 5... | 10 | https://mathoverflow.net/users/2868 | 360014 | 151,581 |
https://mathoverflow.net/questions/360031 | 2 | In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X\_{n}(t)=nt1\_{[0,1/n]}(t)+(2-nt)1\_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the weak convergence of the finite dimensional distribution of a process $X\_{n}(t)$ does not imply the weak convergence ... | https://mathoverflow.net/users/nan | The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$ | The process given by
$$X\_n(t):=[nt1\_{[0,1/n]}(t)+(2-nt)1\_{(1/n,2/n]}(t)]Z,$$
where $Z\sim N(0,1)$, is a Gaussian process in $C[0,1]$ whose finite-dimensional distributions weakly converge to those of the zero process.
However, the process $X\_n(\cdot)$ does not converge in distribution to any process. Indeed, si... | 1 | https://mathoverflow.net/users/36721 | 360034 | 151,587 |
https://mathoverflow.net/questions/360028 | 0 | Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the conditional probability $$\mathbb{E}[f(X(t))|X(0)=x].\ \ \ (1)$$ I know that for a single Poisson process, its transition proba... | https://mathoverflow.net/users/nan | Find a conditional expectation of a difference of two independent Poisson process | First here, $E(f(X(t)|X(0)=x)$ is, not a conditional probability, but a conditional expectation. Second, the event $X(0)=x$ is certain to occur; therefore, conditioning on it does not affect the expectations or probabilities. So,
$$E(f(X(t)|X(0)=x)=Ef(X(t).$$
Finally,
$$Ef(X(t))=\sum\_{k=0}^\infty\sum\_{l=0}^\infty
f... | 1 | https://mathoverflow.net/users/36721 | 360036 | 151,588 |
https://mathoverflow.net/questions/360003 | 4 | Given an infinite dimensional Banach space, can we always find an infinite dimensional Banach space $Y$ and an injective bounded operator $T:X\to Y$ such that $T$ is not bounded below?
If $X^{\*}$ is $w^\*$-separable, then for every Banach space $Y$, there exists an injective compact operator $T: X\to Y$ (see Goldbe... | https://mathoverflow.net/users/41137 | Existence of an injective unbounded below operator | Yes. Using the injective property of $\ell\_\infty$, get an operator $S:X\to \ell\_\infty$ that is compact on some infinite dimensional subspace $X\_0$ of $X$. Let $Q: X \to X/X\_0$ be the quotient map. Define $T:X\to \ell\_\infty \oplus X/X\_0$ by $Tx = (Sx, Qx)$.
With a slightly different argument you can replace $... | 4 | https://mathoverflow.net/users/2554 | 360039 | 151,589 |
https://mathoverflow.net/questions/360038 | 2 | Is this statement true?
A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. That is, if $N>2s, $ then $|x|^{N-2s} v(x) \to l$ whenever $|x|\to \infty$ for some $l>0.$
| https://mathoverflow.net/users/127663 | On $s$-harmonic functions | Yes, it is. Although there are more elementary ways to prove this fact, I suppose the following argument is the shortest.
Every positive $s$-harmonic function in an arbitrary open set $D$ can be represented in terms of the Poisson kernel $P\_D(x, y)$ (with $y \in \mathbb{R}^N \setminus D\_M$) and the Martin kernel $M... | 3 | https://mathoverflow.net/users/108637 | 360042 | 151,590 |
https://mathoverflow.net/questions/360040 | 2 | Let $A\subset B$ be a finite extension of Dedekind domains. Let $0\neq b\in B$ and $0\neq a\in A$ such that $(a)=(b)\cap A$. In particular, we have $a=b\cdot c$ for some $c\in B$. Now for any $A$-linear map $\varphi:B\to A$ the map $x\mapsto\varphi(c\cdot x)$ maps the ideal $(b)$ into $(a)$. Thus we obtain an $A$-linea... | https://mathoverflow.net/users/36563 | Extension of Dedekind domains and their codifferent | Any $A$-module homomorphism $B/b\to A/a$ gives, by composing with $B\to B/b$, a homomorphism $B\to A/a$. Using the surjection $A\to A/a$ and the fact that $B$ is $A$-projective, the map lifts to give a homomorphism $B\to A$.
So, the only real assumption needed is that $B$ is $A$-projective.
| 3 | https://mathoverflow.net/users/9502 | 360044 | 151,591 |
https://mathoverflow.net/questions/360047 | 0 | Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$F(t,f(x)):=\dfrac{1}{t}\Big(\sum\_{k=0}^{\infty}\sum\_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\dfrac{t^{k+m}}{2^{k+m}k!m!}e^{-t}-f(x)\Big).$$ I want to compute the limit of $F(t,f(x))$ when $t\rightarrow 0$, but I don... | https://mathoverflow.net/users/nan | Analyze a complicated double summation | $$F(t,f(x))=\dfrac{1}{t}\Big(\sum\_{k=0}^{\infty}\sum\_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\dfrac{t^{k+m}}{2^{k+m}k!m!}e^{-t}-f(x)\Big)$$
only the terms $k=0,m=0$, $k=0,m=1$, $k=1,m=0$ survive in the limit $t\rightarrow 0$, the other terms vanishing at least linearly in $t$,
$$\lim\_{t\rightarrow 0}F(t,f(x))=-f(x)+\tfrac{... | 1 | https://mathoverflow.net/users/11260 | 360048 | 151,593 |
https://mathoverflow.net/questions/360035 | 10 | It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n roots is at a distance no more than 1 from at least one critical point. It has also been more than twenty years for its proof fo... | https://mathoverflow.net/users/128472 | Sendov's conjecture | A 2010 status report is given by D. Khavinson et al. in [Borcea's variance conjectures on the critical points of polynomials](https://arxiv.org/abs/1010.5167). A 2019 update is in [A note on a recent attempt to prove Sendov's conjecture](https://arxiv.org/abs/1903.04026), by N.A. Rather and Suhail Gulzar.
| 8 | https://mathoverflow.net/users/11260 | 360049 | 151,594 |
https://mathoverflow.net/questions/360062 | 0 | Quote:
"Chen's work mentioned in the discussion of the Goldbach conjecture also showed that every even number is the difference between a prime and a P2."
from: [link](https://primes.utm.edu/notes/conjectures/)
However I can't get this verified or find different sources stating the same.
so question is: is thi... | https://mathoverflow.net/users/157912 | p2 - p1 = 2n for every 2n | The two relevant papers for Chen's original proof are Chen, J.R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica. 16: 157–176. and Chen, J.R. (1966). "On the representation of a large even integer as the sum of a prime and the product of ... | 3 | https://mathoverflow.net/users/127690 | 360067 | 151,596 |
https://mathoverflow.net/questions/360063 | 5 | Let $\mathbb{N}$ denote the set of positive integers. For any prime $p\in\mathbb{N}$ let $p\mathbb{N} = \{np: n\in \mathbb{N}\}$. Is there a partition ${\cal P}$ of $\mathbb{N}\setminus\{1\}$ such that for all $B \in {\cal P}$ and every prime $p\in\mathbb{N}$ we have $|B \cap p\mathbb{N}|=1$?
| https://mathoverflow.net/users/8628 | Slicing up $\mathbb{N}\setminus\{1\}$ | Recursively define a sequence of $B$'s as follows. Initially, each is empty. At each step $n > 1$, place $n$ in the first $B$ that contains only elements coprime to $n$. Clearly, for each prime $p$, there is no $B$ that contains two distinct multiples of $p$. Now fix a prime $p$ and a natural number $N > 1$, and consid... | 4 | https://mathoverflow.net/users/2383 | 360069 | 151,597 |
https://mathoverflow.net/questions/359659 | 2 | Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$.
In my specific case, I have that $L=\mathcal{O}\_X(\Theta + D)$, where $\Theta$ is an ample divisor with $\chi(\Theta)=1$ and $D$ is an effective Cartier divisor.
I want to show that ... | https://mathoverflow.net/users/152522 | Polarization of an abelian variety made by the sum of two divisors | For any $P\in Pic^0(A)$ consider the map $|\Theta +P|\times |D-P|\to |\Theta +D|\cong \mathbb P ^2$. Since $D$ is effective, there is an abelian subvariety $T\subset Pic ^0(A)$ such that $|D-P'|\ne \emptyset$ for any $P'\in P+T$ and if $t=\dim T$ the $D^t\ne 0$ but $D^{t+1}=0$. If $t\geq 3$, then a general element in t... | 1 | https://mathoverflow.net/users/19369 | 360077 | 151,599 |
https://mathoverflow.net/questions/360075 | 6 | Let suppose we have a PDE on a manifold. I'm interested in the following question. How does the space of solutions of this PDE change when the topology of the manifold changes? For example in 2D we consider how the kernel of some partial operator changes under handle attachments. Do people study this type of problems?
... | https://mathoverflow.net/users/85466 | Solutions of PDE under changing topology | Yes, this has been studied intensively for quite a while. In particular, people who work in gauge theory (as applied to 4-manifolds) have studied the effect of surgeries on the solutions to the ASD Yang-Mills equations and Seiberg-Witten equations using essentially analytic techniques. An early result of this kind by D... | 10 | https://mathoverflow.net/users/3460 | 360079 | 151,600 |
https://mathoverflow.net/questions/360076 | 2 | Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous elliptic curves with CM in an imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. I'm not sure if this is standard terminology... | https://mathoverflow.net/users/105661 | Effective semi-group of a singular abelian surface | Below is a summary of the discussion in Lazarsfeld's *Positivity in algebraic geometry I*, Ex. 1.4.7, Lem. 1.5.4, and Rmk. 1.5.6.
>
> **Lemma.** *Let $D$ be an $\mathbf R$-divisor on an abelian surface $A$. Then the following are equivalent:*
>
>
> 1. *$D$ is nef;*
> 2. *$D$ is pseudo-effective;*
> 3. *$D^2 \geq ... | 2 | https://mathoverflow.net/users/82179 | 360086 | 151,603 |
https://mathoverflow.net/questions/23408 | 36 | The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann algebras;
(2) The category of hyperstonean spaces and hyperstonean maps;
(3) The [category](https://mathoverflow.net/questions/... | https://mathoverflow.net/users/402 | Reference for the Gelfand duality theorem for commutative von Neumann algebras | As shown in the paper [Gelfand-type duality for commutative von Neumann algebras](https://arxiv.org/abs/2005.05284), the following categories are equivalent.
* The category CSLEMS of compact strictly localizable enhanced measurable spaces,
whose objects are triples $(X,M,N)$, where $X$ is a set, $M$ is a σ-algebra of... | 5 | https://mathoverflow.net/users/402 | 360088 | 151,604 |
https://mathoverflow.net/questions/330799 | 16 | Let $\mathbf{H}P^\infty$ denote the infinite-dimensional quaternionic projective space. The inclusion of its bottom cell defines a map $S^4 \to \mathbf{H}P^\infty$. Does this extend to a map $\Omega S^5 \to \mathbf{H}P^\infty = BSU(2)$?
Since $\Omega S^5$ is the James construction on $S^4$, this question would be ver... | https://mathoverflow.net/users/102390 | Mapping a loop space to quaternionic projective space | [I should've posted this answer a long time ago, given that it was answered in the comments.] In order to extend the map $S^4\to \mathbf{H}P^\infty$ to a map from $J\_2(S^4)$, the composite of $[\iota\_4,\iota\_4]:S^7\to S^4$ with the inclusion of $S^4$ into $\mathbf{H}P^\infty$ must be null. This is not true: the Whit... | 2 | https://mathoverflow.net/users/102390 | 360094 | 151,606 |
https://mathoverflow.net/questions/357218 | 3 | I need to compute efficiently the sum
$$
\sum\_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor.
$$
We can do this in $O({\sqrt{n}})$ but I need a faster algorithm: for example, it would be fine an algorithm of complexity $O(\sqrt[3]{n})$ (cube root in time) or $O(\log n)$ whatever, but however less ... | https://mathoverflow.net/users/156063 | Efficient computation of $\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor$ | Following by comment of Alexey Kulikov we could split our sum in the next way:
$$\sum\_{i=1}^{[\sqrt{n}]} i^2\left [\frac{n}{i^2}\right ]=
\sum\_{[n/i^2]>[\sqrt[3]{n}]} i^2\left [\frac{n}{i^2}\right ]+\sum\_{[n/i^2]\leq [\sqrt[3]{n}]} i^2\left [\frac{n}{i^2}\right ]=$$
$$=\sum\_{i=1}^{\left [\sqrt{\frac{n}{[\sqrt[3]{n}... | 7 | https://mathoverflow.net/users/61438 | 360110 | 151,609 |
https://mathoverflow.net/questions/360105 | 0 | In [Nature Vol 580](https://www.nature.com/articles/d41586-020-00998-2), in an article about Shinichi Mochizuki's proposed proof of the abc-conjecture, there is a formulation saying:
>
> The conjecture roughly states that if a lot of small primes divide two numbers $a$ and $b$, then only a few, large ones divide th... | https://mathoverflow.net/users/57255 | A soft question on the ABC conjecture | If you are interested in the largest prime factor of $ab(a+b)$,
there is [xyz conjecture](https://arxiv.org/abs/0911.4147).
Smooth solutions to the abc equation: the xyz Conjecture
This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(... | 1 | https://mathoverflow.net/users/12481 | 360116 | 151,610 |
https://mathoverflow.net/questions/360097 | 4 | I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues explicitly for a class of hypergraphs. To start with, we can consider all the $0-1$ hypermatrices of order $2 \times 2 \times... | https://mathoverflow.net/users/33047 | Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor | As explained in a [previous MO question](https://mathoverflow.net/questions/319644/eigenvalue-and-eigenmatrix-of-a-3d-tensor-how-to-calculate-it), there is no unique generalization of the eigenvalue of an $n\times n$ matrix to an $n\times n\times n$ tensor. One approach is to construct a [higher-order singular value de... | 8 | https://mathoverflow.net/users/11260 | 360119 | 151,611 |
https://mathoverflow.net/questions/360090 | 1 | In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for this is contained in the answer by Jeremy Rickard here [Local property of split exact sequence](https://mathoverflow.net/q... | https://mathoverflow.net/users/135253 | Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring | Reiner's "Maximal Orders", Theorem 3.20, for instance.
| 3 | https://mathoverflow.net/users/86006 | 360126 | 151,615 |
https://mathoverflow.net/questions/360131 | 5 | I am looking for a digitalized version of paper by J.P. May and J. McClure *A reduction of Segal conjecture*, as I need it to understand some lemma from Kuhn's *Tate Cohomology and Periodic Localization of Polynomial Functors*. The paper was published in *Current Trends in Algebraic Topology*, Canadian Mathematical Soc... | https://mathoverflow.net/users/123432 | May-McClure "A reduction of Segal conjecture" | I found it on professor May's web site at <http://math.uchicago.edu/~may/PAPERS/42.pdf>
Since this link might disappear, it is also archived at the [Wayback Machine.](https://web.archive.org/web/20120608135619/http://math.uchicago.edu/~may/PAPERS/42.pdf)
| 4 | https://mathoverflow.net/users/11260 | 360136 | 151,618 |
https://mathoverflow.net/questions/360030 | 2 | In this [paper](https://arxiv.org/abs/1905.08047), definition 4.4.1 about supermanifold and definition 4.6.1 about graded manifold:
>
> **Definition 4.4.1:** An supermanifold $\mathcal{M}$ is a locally ringed space $(M,\mathcal O\_M)$ which is locally isomorphic to $(U, C^\infty(U)\otimes \wedge^\bullet V^\*)$ wher... | https://mathoverflow.net/users/133793 | $\mathbb Z$-graded manifold is isomorphic to the structure sheaf of supermanifold locally | A supermanifold is a graded manifold with only odd components, more precisely the equivalence is seen by setting $V\_{\bar{0}}=0$ and $V\_{\bar{1}} = V$ in your notation. In the opposite direction, a graded manifold is a supermanifold **only** if it has no even components, that is $V\_{\bar{0}} = 0$.
| 1 | https://mathoverflow.net/users/2622 | 360139 | 151,619 |
https://mathoverflow.net/questions/360135 | 5 | Let $A = \begin{bmatrix} A\_{11} & A\_{12} \\ A\_{21} & A\_{22} \end{bmatrix}$ be an invertible matrix where $A\_{11}$ is square. Let $A^{-1} =: B = \begin{bmatrix} B\_{11} & B\_{12} \\ B\_{21} & B\_{22} \end{bmatrix}$ and shape of $B\_{11}$ is same as $A\_{11}$.
What is known about the relationships between the rank... | https://mathoverflow.net/users/151406 | Rank of a block of an invertible matrix | That paper from Strang and Nguyen gives the answer: <https://epubs.siam.org/doi/abs/10.1137/S0036144503434381>
See "theorem 2.1: (nullity theorem). Complementary submatrices of a square matrix and its inverse have the same nullity". In your case the complementary submatrices are exactly $A\_{21}$ and $B\_{21}$, so th... | 6 | https://mathoverflow.net/users/157957 | 360143 | 151,620 |
https://mathoverflow.net/questions/360144 | 5 | Consider a dynamical system $(X, \mathcal{B}(X), \mu, T)$ where $(X, \mathcal{B}(X), \mu)$ is a measure space and $T$ is a measure-preserving, invertible transformation.
Then by the classical Birkhoff's ergodic theorem if $p\ge 1$, then for any $f\in L^p(X, \mu)$ the sequence
$$
\mathcal{M}\_N f(x):=\frac{1}{N}\sum\... | https://mathoverflow.net/users/157356 | Analog of the Birkhoff's ergodic theorem for the sequence of squares | No - the sequence of squares is **universally bad** which was proved by Buczolich and Mauldin. I will quote from Tom Ward's [review](https://mathscinet.ams.org/mathscinet-getitem?mr=2680392) of their paper *Divergent square averages*, Ann. of Math. (2) 171 (2010), no. 3, 1479–1530.
A consequence of J. Bourgain's work... | 5 | https://mathoverflow.net/users/8588 | 360145 | 151,621 |
https://mathoverflow.net/questions/360153 | 13 | Let $((x)):=x-\lfloor x \rfloor -1/2$, where $\lfloor x \rfloor $ denotes the greatest integer $\le x$.
Let $a,b,c,...$ denote arbitrary natural numbers. It is clear that
$$ \int\_0^a ((x/a)) dx =0.$$
A little exercise shows that
$$ \int\_0^{ab} ((x/a))((x/b)) dx =\frac{\gcd(a,b)^2}{12}.$$
It appears that
$$ \int\_0... | https://mathoverflow.net/users/12947 | Integrals of products of fractional parts | One place where I have seen this come up is in a paper of [Lemke Oliver and Soundararajan](https://arxiv.org/abs/1709.06168). Here such quantities appear naturally in computing moments of some remainder terms, and Proposition 3.1 gives some basic properties. The sums are very closely related to the integrals (see Propo... | 9 | https://mathoverflow.net/users/38624 | 360157 | 151,627 |
https://mathoverflow.net/questions/360167 | 0 | Given a real, invertible matrix $A$. For which vectors $b$ and $c$ is
$$
A^{-1} + bc^T
$$
similar to $A$? And is the rank-1 matrix $bc^T$ unique?
| https://mathoverflow.net/users/51478 | Similar to inverse plus rank 1 | Let's assume for simplicity that $A$ has distinct eigenvalues, so similarity to $A$
just means having the same characteristic polynomial.
The [Matrix determinant lemma](https://en.wikipedia.org/wiki/Matrix_determinant_lemma)
says that the characteristic polynomial
$$ \eqalign{\det(A^{-1} + b c^T - \lambda I) &= (1 + ... | 2 | https://mathoverflow.net/users/13650 | 360173 | 151,632 |
https://mathoverflow.net/questions/360118 | 6 | It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows:
*Obj:* $X \mapsto \pi\_{\leq 1}(X)$, where $\pi\_{\leq 1}(X)$ is the fundamental groupoid of $X$.
*Mor:* ($f:X \rightarrow Y) \mapsto F(f):\pi\_{\leq 1}... | https://mathoverflow.net/users/86313 | What is the geometric realization of the the nerve of a fundamental groupoid of a space? | The inclusion of groupoids into simplicial sets is fully faithful. Its left adjoint, $\Pi\_1$ is given by left Kan extension of the functor $\Delta\to \mathcal{Gpd}$ sending the n-simplex to the contractible groupoid with objects $\{0,...,n\}$.
The entirety of the data of the homotopy type of the space $X$ is contai... | 4 | https://mathoverflow.net/users/1353 | 360187 | 151,634 |
https://mathoverflow.net/questions/360208 | 0 | I was reading the following paper by Hubert Delange: <http://www.numdam.org/article/ASENS_1956_3_73_1_15_0.pdf> 1. In page 26, he provides a proof of Theorem b, the bulk of which relies on a result in some reference provided by him in footnote 5. However, despite my best attempts, I have been unable to find the aforeme... | https://mathoverflow.net/users/157984 | Reference needed for proof of a Tauberian theorem | As I don't speak French, this is partially a guess. But I think this is paper you're looking for. (Maybe somebody else will be able to suggest other references for the same result.)
Looking at the footnote 5 saying "Loc. cit., p. 235-236 et 239." and the footnote 4 saying "Ann. Se. Ec. Norm. Sup., (3), l. 74, 1934' p... | 2 | https://mathoverflow.net/users/8250 | 360212 | 151,641 |
https://mathoverflow.net/questions/360204 | 0 | I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" holds:
>
> Theorem: Suppose that $X: \Omega \rightarrow E$ and $Y: \Omega \rightarrow F$ are two random variables on a... | https://mathoverflow.net/users/157982 | Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$? | **Lemma.** Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space, $X$ a real-valued random variable on this space, and $\mathcal{C} \subset \mathcal{A}$ a $\sigma$-field. Let $F$ be the cumulative distribution function of the conditional law $\mathcal{L}(X \mid \mathcal{C})$. Let $\xi$ be a random variable with ... | 1 | https://mathoverflow.net/users/21339 | 360215 | 151,642 |
https://mathoverflow.net/questions/360214 | 1 | Let $\|\cdot\|\_F$ denote the Fröbenius norm on the set of $d\times d$ matrices. By restriction this induces a metric on $SO(n)$.
Let's make an observation.
Since $X\in SO(n)$ is a rotation matrix then it is an isometry hence if $\lambda$ is an eigenvalue of $A$ with corresponding eigenevector $x$ we have that
$$
\|... | https://mathoverflow.net/users/36886 | Bound between distance between Rotation Matrices | **First Point: The bound isn't sharp**
Consider the case where $n=2$. The every matrix in $SO(n)$ is of the form
$$
A\_{\theta} \triangleq \begin{pmatrix}
cos(\theta) & -sin(\theta)\\
sin(\theta) & cos(\theta),
\end{pmatrix}
$$
for some $\theta \in [0,2\pi]$ (note: fun easy proof of compactness of $SO(2)$). In partic... | 0 | https://mathoverflow.net/users/36886 | 360218 | 151,643 |
https://mathoverflow.net/questions/357999 | 0 | **Edit:**
Let $X$ be a strict [LB-space](https://en.wikipedia.org/wiki/LB-space) described by $\lim X\_n$ and suppose that $\{x\_n\}\_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x\_n$ must converge in some $X\_N$.
| https://mathoverflow.net/users/36886 | Convergence in LB-spaces | The result (even for LF-spaces) is due to J. Dieudonné and L. Schwartz
*La dualité dans les espaces (F) et (LF)*, Annales de l’institut Fourier, tome 1 (1949), p. 61-101, propositions 2 and 4.
(Proposition 2 says that the inductive limit topology induces on the ,,steps'' their original topologies, and proposition 4 s... | 1 | https://mathoverflow.net/users/21051 | 360221 | 151,644 |
https://mathoverflow.net/questions/360095 | 2 | Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the generator of a semigroup $T(t)$ with growth bound $\omega$, i.e., $\|T(t)\|\le Ce^{\omega t}$, we have $s(A) \le \omega$. ... | https://mathoverflow.net/users/114951 | Spectral representation of closed operators with finite spectral bound | I've looked it up now. The formula in question does indeed hold in the following sense:
**Theorem.** Let $(e^{tA})\_{t \in [0,\infty)}$ be a $C\_0$-semigroup on a complex Banach space $X$. Let $\omega \in \mathbb{R}$ be a real number that is strictly larger than the growth bound of our semigroup and let, for each $k ... | 4 | https://mathoverflow.net/users/102946 | 360230 | 151,648 |
https://mathoverflow.net/questions/360224 | 1 | Let $p=(p\_{1},p\_{2},p\_{3})\in\Delta$, with $\Delta:=\lbrace p\in(0,1)^{3}\ |\ p\_{1}+p\_{2}+p\_{3}=1 \rbrace$. I aim to prove (not knowing whether it is true though) that
\begin{equation}
p\_{1}^{p\_{3}-p\_{2}}p\_{2}^{p\_{1}-p\_{3}}p\_{3}^{p\_{2}-p\_{1}}\le1.
\end{equation}
Indeed, if at least two of the three numbe... | https://mathoverflow.net/users/85194 | Cyclic inequality for 2 dimensional simplex elements | I don't think your 'wlog' is correct. I think you can only assume 3 distinct, and not an order. If you agree with this, then rewriting as
\begin{equation}
p\_{1}^{p\_{3}}p\_{2}^{p\_{1}}p\_{3}^{p\_{2}}/
p\_{1}^{p\_{2}}p\_{2}^{p\_{3}}p\_{3}^{p\_{1}}
\end{equation}
if you call the numerator $f(p\_1,p\_2,p\_3)$ the the den... | 0 | https://mathoverflow.net/users/143907 | 360231 | 151,649 |
https://mathoverflow.net/questions/360222 | 4 | I am searching for the criterion stated above and also here: [The question about Kolmogorov tightness criterion](https://mathoverflow.net/questions/39098/the-question-about-kolmogorov-tightness-criterion).
It should state the following: If a sequence of stochastic processes $(X^n)$ fulfills:
$$\mathbb{E}[|X^n\_t-X^... | https://mathoverflow.net/users/157993 | Kolmogorov tightness criterion for stochastic processes | I assume that you want to show tightness in a space such as $C([0,1])$, as was the case in the question you link to. In fact, the approach of this answer will show tightness in $C^\beta([0,1])$ for every $\beta \in (0, \frac{\alpha - 1}{p})$.
Firstly, note that the statement you write cannot be sufficient for tightne... | 3 | https://mathoverflow.net/users/87850 | 360234 | 151,650 |
https://mathoverflow.net/questions/360219 | 2 | Let $G$ be an amenable group. I wonder whether it is true that the reduced group $C^\*$-algebra $C\_r^\*(G)$ is quasidiagonal.
| https://mathoverflow.net/users/153196 | Is the reduced group $C^*$-algebra quasidiagonal | This is true for countable discrete groups by the celebrated Quasidiagonality Theorem of Tikuisis-White-Winter (Quasidiagonality of nuclear C∗-algebras. Ann. of Math. (2) 185 (2017), no. 1, 229–284.)
| 7 | https://mathoverflow.net/users/126109 | 360235 | 151,651 |
https://mathoverflow.net/questions/359454 | 2 | The preprojective algebra of a module $M$ over a finite dimensional algebra $A$ is
defined as $P\_M:= \bigoplus\limits\_{n=0}^{\infty}{Hom\_A(M, \tau^{-n}(M))}$ with the canonical multiplication.
>
> Question 1: Is there an easy way to obtain quiver and relations of $P\_M$ in case it is finite dimensional? Are expl... | https://mathoverflow.net/users/61949 | Preprojective algebra of finite dimensional algebras | For Question 2: In QPA one can do the following using the latest uploaded extensions of QPA:
```
gap> A := NakayamaAlgebra( GF(2), [ 3, 2, 1 ] );
<GF(2)[<quiver with 3 vertices and 2 arrows>]>
gap> M := DirectSumOfQPAModules(IndecProjectiveModules(A));
<[ 1, 2, 3 ]>
gap> B := Preprojecti... | 2 | https://mathoverflow.net/users/130741 | 360239 | 151,653 |
https://mathoverflow.net/questions/360178 | 6 | Let $\ A\ $ be an arbitrary set. Let $\ |A|>1\ $ (to avoid triviality). Let each of the functions $\ f\_k:A^{\{1\ 2\ 3\}}\to A\ $ depend on all three arguments for $\ k=1\ 2\ 3,\ $ while each of the functions $\ g\_k:A^{\{1\ 2\ 3\}}\to A\ $ does not depend on $k$-th variable, for each $\ k=1\ 2\ 3.$
*I'll explain "de... | https://mathoverflow.net/users/110389 | Cartesian dissimilarity of a function $\ f:A^3\to A^3\ $ and its inverse | $\newcommand{\F}{\mathbb{F}}$
Your conjecture is false, I will construct a counterexample for all powers of two $2^n$ with $n \ge 2$.
Let us identify $A$ with $\F\_{2^n}$. An example from OP corresponding to the odd numbers, is a linear mapping. Our functions also would be linear. Let $G$ be a matrix constructed fro... | 4 | https://mathoverflow.net/users/104330 | 360245 | 151,654 |
https://mathoverflow.net/questions/360216 | 9 | Let $S\_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x\_1,x\_2,...,x\_n$ generated linearly (over any field of characteristic zero) by the monomials of the form $x\_i^2x\_jx\_k$ ($i,j,k$ are dist... | https://mathoverflow.net/users/37419 | Decomposing a polynomial ring into Specht Modules | The span of the monomials of the form $x\_i^2x\_jx\_k$ is the Young permutation module $M^{(n-3,2,1)}$. (*Proof.* Observe that $x\_1^2x\_2x\_3$ has stabiliser $\langle (2,3)\rangle \times S\_{\{4,\ldots,n\}}$, so the relevant Young subgroup is $S\_{n-3} \times S\_2 \times S\_1$.) Using Kostka numbers (equivalently, mul... | 6 | https://mathoverflow.net/users/7709 | 360253 | 151,657 |
https://mathoverflow.net/questions/360213 | 3 | $(c\_k)$ is real number sequence. It is known that $\lim\_{y\to 1^-}\sum\_{k=1}^{\infty} c\_k y^k $ exists. Moreover, $\lim\_{k\to \infty} kc\_k=0$ and $\sum\_{k=1}^{\infty}c\_ky^k$ exists for all $y \in [0,1)$.Suppose $\lim\_{n\to\infty}\frac{1}{n}\sum \_{k=1}^n kc\_k=0$
ٍٍٍٍٍٍٍٍٍٍٍٍٍSuppose that for every $y \in (... | https://mathoverflow.net/users/nan | Convergence of infinite series | Note that the following asymptotic expansion holds for $x\simeq0$
$$
(x+1)^k-1=kx+\mathcal{O}(x^2).
$$
Using the above with $x=1-y$ and recalling that $G(y)\simeq (1-y)^{-1}$, we get for $y\simeq 1$
$$
\sum\_{k=1}^{G(y)}c\_k(1-y^k)=(1-y)\sum\_{k=1}^{G(y)}k c\_k+\mathcal{O}\left((1-y)^2 \sum\_{k=1}^{G(y)} c\_k\right)=(... | 1 | https://mathoverflow.net/users/157356 | 360257 | 151,658 |
https://mathoverflow.net/questions/360252 | 2 | Let $(\Omega,\mathfrak{F})$ be a measurable space. When does there exist an injective measurable function $f:(\Omega,\mathfrak{F})\to (\mathbb{R}^n,B(\mathbb{R}^n))$ to some Euclidean space, here $B(\mathbb{R}^n)$ is the Borel $\sigma$-algebra.
Thoughts. Clearly, if $\Omega$ is a Riemannian manifold and $\mathfrak{F}... | https://mathoverflow.net/users/36886 | Existence of measurable "inclusion" into Euclidean space | (Basically the same as Michael's answer)
Theorem 6.5.7 of *Measure Theory* by V. Bogachev:
>
> **Theorem.** The following are equivalent:
>
>
> 1. $\mathfrak{F}$ is *countably separated* (Bogachev Definition 6.5.1 (ii)): there exists an at most countable collection of sets $F\_n \in \mathfrak{F}$ such that for ... | 4 | https://mathoverflow.net/users/4832 | 360269 | 151,664 |
https://mathoverflow.net/questions/360268 | 2 | For $n>1$ let $\omega=\sum\_{i=1}^n dx\_i\wedge dy\_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$.
We define the following distribution $D$ on $\mathbb{R}^{2n}\setminus\{0\}$:
For $Z\in \mathbb{R}^{2n}\setminus\{0\}$ we define $D\_Z=\{V\in \mathbb{R}^{2n}\mid \omega(V,Z... | https://mathoverflow.net/users/36688 | A metric naturally arise from the Euclidean symplectic structure? | Suppose we have a non-integrable distribution and a riemannian metric on a manifold. Then, one can define the metric using your construction, the resulting object is called sub-riemannian metric. It does not come from any riemannian or finslerian metric. Sub-riemannian metrics are important in the theory of nilpotent g... | 5 | https://mathoverflow.net/users/33286 | 360275 | 151,666 |
https://mathoverflow.net/questions/359616 | 3 | For finite-dimensional Lie algebras, see [this](https://mathoverflow.net/questions/77418/whats-the-lipschitz-constant-of-the-exponential-map-for-son-r/77456) for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to the infi... | https://mathoverflow.net/users/36886 | Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$? | There is no need to be so specific in the question, you can indeed answer this in general for $M$ a (paracompact, finite-dimensional) manifold.
It is well known, that in this setup $\mathrm{Diff}\_c(M)$ is an infinite-dimensional Lie group with Lie algebra $\mathfrak{X}\_c(M)$ (this was your space of $C^\infty\_c(M,M)... | 1 | https://mathoverflow.net/users/46510 | 360285 | 151,669 |
https://mathoverflow.net/questions/360290 | 13 | Is it known how much of ZFC is actually necessary for the basic, familiar constructions and theorems in sheaf theory, along the lines of section II.1 (and its exercises) in Hartshorne's "Algebraic Geometry" textbook?
I apologize for this strange question. Here is the motivation behind it: I am a mathematician who wor... | https://mathoverflow.net/users/nan | How strong a set theory is necessary for practical purposes in sheaf theory? | Colin McLarty has looked into this
>
> *The large structures of Grothendieck founded on finite order arithmetic*, Review of Symbolic Logic **13** issue 2 (2020) pp. 296--325, doi:[10.1017/S1755020319000340](https://doi.org/10.1017/S1755020319000340), arXiv:[1102.1773](https://arxiv.org/abs/1102.1773).
>
>
>
wi... | 13 | https://mathoverflow.net/users/4177 | 360296 | 151,673 |
https://mathoverflow.net/questions/360293 | 1 | Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C\_{0}$-semigroup. Suppose that in a $\varepsilon$-neighbourhood of the imaginary axis the operator $A$ does not have spectrum. Is it true that the resolvents $(A-i\omega I)^{-... | https://mathoverflow.net/users/85336 | Uniform boundedness of resolvents on the imaginary axis | This is essentially equivalent to asking whether the spectrum determines the growth bound. This is well known to be false. A pioneering counterexample is due to Zabczyk.
| 2 | https://mathoverflow.net/users/12120 | 360299 | 151,674 |
https://mathoverflow.net/questions/360260 | 4 | I want to calculate the fundamental group of the complement some collection of plane curves (specifically, two nodal cubics in a general position).
I've read about Severi problem (solved by Harris), which states that complement of every reduced irreducible nodal plane curve has an abelian fundamental group. I've unde... | https://mathoverflow.net/users/33286 | Fundamental group of the complement of the arrangement of plane nodal curves | As suggested, I write my comment as an answer: the result is actually due to Fulton (*On the fundamental group of the complement of a nodal curve*, Ann. of Math. 111, no. 2 (1980), 407-409), and it is valid with no irreducibility assumption. Fulton works with the *algebraic* fundamental group; the case of the topologic... | 6 | https://mathoverflow.net/users/40297 | 360305 | 151,675 |
https://mathoverflow.net/questions/360310 | 6 | I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ is a symmetric matrix with positive diagonals and the Gauss-Seidel method converges for all initial guess $x\_0$, is $A$... | https://mathoverflow.net/users/157997 | Is the matrix positive definite given the Gauss-Seidel method converges? | There is an interesting though partial answer to your question:
>
> Under your assumptions, it is not possible that the signature of $A$ be $(n-1,1)$ (exactly one negative eigenvalue).
>
>
>
Proof: With standard notations, $A=D-E-E^T$ where $E$ is strictly triangular and $D$ diagonal. By assumption, $D>0\_n$. ... | 5 | https://mathoverflow.net/users/8799 | 360313 | 151,679 |
https://mathoverflow.net/questions/360319 | 3 | This is a reference request, and soft question as companion.
I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in physics and that were in the literature recently, let's say in this decade (2009-2020), as remarkable advances.
For th... | https://mathoverflow.net/users/142929 | More important or relevant progress in discretizing hard problems in physics in last decade | I understand the question as a request for pointers in the literature to research in the discretization of spacetime. There are two reasons why this is an active research topic, a fundamental and a practical reason: Fundamentally, spacetime might be discrete at the smallest levels (Planck scale); practically, to simula... | 1 | https://mathoverflow.net/users/11260 | 360324 | 151,683 |
https://mathoverflow.net/questions/350183 | 1 | For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH\_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH\_k(X)\_{hom}$ denote $k$-cycles modulo homological equivalence.
I know that $CH\_k(X,\mathbb{Q}) = CH\_k(X)\otimes \mathbb{Q}$ (follows from flatness of $\mathbb{... | https://mathoverflow.net/users/90911 | Confusion regarding a definition of cycles | Yes, they are same. By definition $ CH\_k(X)\_{hom} :$= kernel of the cycle class map , $CH\_k(X)\to H\_{2k}(X, \mathbb{Z})$ also $CH\_k(X , \mathbb{Q})\_{hom}$ := kernel of the cycle class map $CH\_k(X,\mathbb{Q})\to H\_{2k}(X, \mathbb{Q})$. Now the assertion follows as $\mathbb{Q}$ is flat over $\mathbb{Z}$.
| 2 | https://mathoverflow.net/users/157738 | 360325 | 151,684 |
https://mathoverflow.net/questions/360332 | 2 | given a universe $\mathcal{U}$ of elements and a system $\mathcal{S}$ of weighted subsets of $\mathcal{U}$ whose union covers $\mathcal{U}$.
Assuming the existence of at least one subsytem $S\subseteq\mathcal{S}$ such that the disjoint union of its elements $\lbrace s\_1,\,\dots,\,s\_k\rbrace$ covers $\mathcal{U}$, w... | https://mathoverflow.net/users/31310 | Complexity of set-partition problems | The [exact cover problem](https://en.wikipedia.org/wiki/Exact_cover) being one of [Karp's
21 NP-complete problems](https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems), your optimization problem is NP-hard.
| 5 | https://mathoverflow.net/users/13650 | 360336 | 151,687 |
https://mathoverflow.net/questions/360339 | 4 | $\newcommand{\R}{\mathbb R}$
Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\ge f(x)-f(x-y)$. So, $f$ is midpoint-convex iff for any real $x$ and and any real $y>0$
$$\frac{f(x+y)-f(x)}y\ge\frac... | https://mathoverflow.net/users/36721 | Is there a non-convex function with non-decreasing average rate of change? | The standard example of a non-convex but midpoint-convex function is additive:
given a basis $B$ of $\mathbb R$ over the rationals $\mathbb Q$, choose one member $\alpha$ of $B$ and
if $x = \sum\_{\beta \in B} c\_\beta(x) \beta$ with $c\_\beta(x) \in \mathbb Q$ (only finitely many nonzero), take $f(x) = c\_\alpha(x)$. ... | 6 | https://mathoverflow.net/users/13650 | 360343 | 151,689 |
https://mathoverflow.net/questions/360171 | 15 | Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}\_{E\_\infty})^{\mathrm{op}}$, that is to say, a functor $X:(\mathbf{Aff}^{\mathrm{cn}})^{\mathrm{op}}\to \mathcal{S}$ satisfying fpqc descent. Then we can define its $\... | https://mathoverflow.net/users/1353 | When does QCoh have 'enough perfect complexes'? | Robert Thomason was the first person to draw attention to this question, before derived schemes and infinity categories. I believe that he proved that for a quasi-compact and quasi-separated scheme that $D\_{qc}=\textrm{Ind}(\textrm{Perf})$. For example, see [Thomason-Trobaugh](https://www.gwern.net/docs/math/1990-thom... | 10 | https://mathoverflow.net/users/4639 | 360345 | 151,690 |
https://mathoverflow.net/questions/360327 | 2 | I have tried to decompose this as following spans over the real field.
$V\_1=\operatorname{span} \langle x\_{ij}^2\rangle$
$V\_2=\operatorname{span} \langle x\_{ij}x\_{jk}\rangle$
$V\_3=\operatorname{span} \langle x\_{ij}x\_{kl}\rangle$
where the index $ij$ stands for the subset $\{i,j\}$ of size 2, thus the o... | https://mathoverflow.net/users/37419 | Decomposition of the homogeneous polynomial ring $\{\mathbb R[x_{ij}]_{1\le i,j\le n}\}$ of degree 2 into Specht modules | Write $W\_4$ for the $S\_4$ representation $$({\rm Ind}\_{S\_2 \times S\_2}^{S\_4} ({\rm triv} \otimes {\rm triv}))\_{S\_2}.$$
Then similarly to the answer to your previous question, you are considering $$Ind^{S\_{n}}\_{S\_4 \times S\_{n-4}} (W\_4 \otimes {\rm triv}),$$ which can be computed using the Pieri rule as soo... | 3 | https://mathoverflow.net/users/52918 | 360352 | 151,692 |
https://mathoverflow.net/questions/360243 | 2 | On the [Wikipedia page](https://en.wikipedia.org/wiki/Disintegration_theorem) there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to conditional expectation. Rigorously, how are the two related?
| https://mathoverflow.net/users/36886 | Disintegration, conditional probabilities, and conditional expectation | There's a nice discussion of these issues in "Conditionng as Disintegration" by Chang & Pollard: <https://onlinelibrary.wiley.com/doi/full/10.1111/1467-9574.00056>
| 3 | https://mathoverflow.net/users/42851 | 360357 | 151,694 |
https://mathoverflow.net/questions/359468 | 3 | One of the nice applications of decoupling is Bourgain’s record towards Lindelöf:
<https://arxiv.org/pdf/1408.5794.pdf>
Wooley has developed some techniques known as efficient congruencing which allow one to obtain estimates also derived from decoupling.
Lets call ’efficient boxing’ using Wooley’s efficient congr... | https://mathoverflow.net/users/nan | Efficient boxing for a mean value in the Bombieri Iwaniec method | It seems possible since (p-adic) Efficient Congruencing also delivered the Optimal Estimate for Vinogradov's Mean Value Theorems. Another reason that this may be possible is that, if I recall correctly, historically inefficient boxing was introduced by Vinogradov and called the "short intervals method". This was applie... | 2 | https://mathoverflow.net/users/38379 | 360360 | 151,697 |
https://mathoverflow.net/questions/360346 | 3 | I'm currently trying to understand the Brauer-Kuroda formula.
Although there are many recent papers on the formula but they seem to be purely algebraic.
They say that original analytic approach is much easier and short.
So I hope to find a reference on the original proof of Brauer-Kuroda formula using the formula... | https://mathoverflow.net/users/123226 | English reference for the Brauer-Kuroda formula | See Section VIII.7 in Fröhlich-Taylor, Algebraic Number Theory, Cambridge University Press.
| 5 | https://mathoverflow.net/users/3503 | 360366 | 151,699 |
https://mathoverflow.net/questions/360361 | 6 | For every integer $n > 0$, let $C\_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $0$'s everywhere else. For example, we have
$$C\_2 = \begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 3 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & ... | https://mathoverflow.net/users/158063 | Characteristic polynomial of checker matrix | Ok! Your conjecture is true.
Let $W$ be the space spanned by the eigenvectors for $\lambda \in \{-3, 3\}$ as described in my comments. Let $V$ be the subspace of $\mathbb{R}^{4n}$ consisting of vectors of the form
$$V = \{(a,b,a,b,a,b, \ldots, a, x, y, b, a, b, \ldots, a,b)\},$$
where the entries corresponding to... | 6 | https://mathoverflow.net/users/22512 | 360371 | 151,701 |
https://mathoverflow.net/questions/360376 | 3 | Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that
1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
2. $\tilde{Y}$ is a residual subset of $Y$ *i.e.* $\tilde{Y} = \bigcap\_{n=1}^\infty\tilde{Y}\_n$ for some collection $\{\tilde{Y}\_1,\tild... | https://mathoverflow.net/users/134135 | Does this strong form of being almost 1-to-1 imply injectivity? | No. Consider an irrational rotation $R$ of the circle (which I identify with [0,1)) by an angle $\alpha$. Let $\alpha<\beta<1$ be a point not lying in the orbit of 0 under $R$. Set $A\_1=[0,\alpha)$, $A\_2=[\alpha,\beta)$, $A\_3=[\beta,1)$ also $B\_1=A\_1$ and $B\_2=A\_2\cup A\_3$.
Consider the partitions $P=\{A\_1,A\_... | 5 | https://mathoverflow.net/users/11054 | 360379 | 151,705 |
https://mathoverflow.net/questions/360381 | 3 | I have the following sequence of holomorphic functions on $(f\_n(s))\_{n \geq 1}$ on the closed region $R:= \{ s \in \mathbb C : \Re(s) \geq 1\}$ by
$$f\_n(s) :=
\begin{cases}
\log(1+p^{-s})-p^{-s}\text{, if }n=p\text{ is prime}\\
0\text{, otherwise }
\end{cases}$$
(where we have taken the principal branch of the log... | https://mathoverflow.net/users/157984 | To show holomorphicity of a certain infinite series of functions | If $\Re(s) > 1/2+\epsilon$ for $\epsilon > 0$ we have $\log(1+p^{-s}) = \sum\_{j=1}^\infty p^{-sj}/j$ so
$|f\_p(s)| \le \sum\_{j=2}^\infty p^{-\Re(s)j}/j \le p^{-1-2\epsilon}/(1-p^{-1/2-\epsilon})$. Since $\sum\_p p^{-1-2\epsilon}$ converges, your series converges uniformly on $\{s: \Re(s) > 1/2 + \epsilon\}$, and the... | 7 | https://mathoverflow.net/users/13650 | 360382 | 151,707 |
https://mathoverflow.net/questions/360387 | 1 | This is not a research level question. But due to some reason I can't ask this question on Math Stack Exchange. So, I am asking this question here.
By definition we know that we can measure the multiplicity of a root of a function $f(z)$ as follow.
If the a root of some function $f(z)$ is $\alpha\_i$, then the mult... | https://mathoverflow.net/users/156029 | A question on multiplicity of complex polynomial | Your product on the right does not converge for any $z \neq 0.$ Never fear, [the Weierstrass factorization theorem](https://www.wikiwand.com/en/Weierstrass_factorization_theorem) is your friend.
| 1 | https://mathoverflow.net/users/11142 | 360390 | 151,709 |
https://mathoverflow.net/questions/360397 | 1 | It is well-known that the Toeplitz algebra $\mathcal{T}$ (that I view as concrete subalgebra of $\mathbb{B}(\ell^2(\mathbb{N})$) is the universal algebra generated by an isometry, that is, for any $C^\*$-algebra $A$ and an isometry $v \in A$, there exists a unique algebra $\mathcal{T} \rightarrow A$ such that $S \mapst... | https://mathoverflow.net/users/16702 | Proof of universality of Toeplitz algebra | The proof for Cuntz-Toeplitz algebras, i.e. algebras generated by some number of isometries with orthogonal ranges, is pretty $C^{\ast}$-algebraic. Let $s$ be the universal isometry and let $\mathcal{T}\_{u}$ be the universal $C^{\ast}$-algebra generated by an isometry. We want to show that the $\ast$-homomorphism $\Ph... | 2 | https://mathoverflow.net/users/24953 | 360401 | 151,714 |
https://mathoverflow.net/questions/360406 | 2 | Let $\lambda\_1>\lambda\_2>....>\lambda\_N$ be the ordered eigenvalues of Wishart matrix my objective is to find if it is possible the distribution of:
\begin{align}
s = \sum\limits\_{i = 1}^N {\frac{1}{{1 + a{\lambda \_i}}}}
\end{align}
where a is positive and $N\ge 2$.
| https://mathoverflow.net/users/144355 | distribution on the inverse Wishart matrix eigenvalues summation | For small $N$ explicit expressions are cumbersome. For $N\gg 1$ the distribution $P(s)$ is a Gaussian. The mean is given by integration with the Marchenko-Pastur distribution, the variance is given by integration with a formula given in [arXiv:9310010](https://arxiv.org/abs/cond-mat/9310010), Equation 17. Let me work t... | 4 | https://mathoverflow.net/users/11260 | 360408 | 151,716 |
https://mathoverflow.net/questions/85211 | 10 | In Sid Sackson's classic book *A Gamut of Games*, he introduces a game that he calls "Hold That Line." Briefly, it is an impartial pencil-and-paper game played on a finite grid of dots. The first player connects two dots with a horizontal, vertical, or diagonal line; thereafter, each player extends the given piecewise-... | https://mathoverflow.net/users/3106 | Has Sid Sackson's "Hold That Line" been analyzed? | Jindřich Michalik of Charles University has just proved that the player going first has a winning strategy for the 4 x 4 game. He also analyzes 2 x n and 3 x 3 games in a paper, "A Winning Strategy for Hold That Line" that will appear in *The Mathematical Intelligencer* later this year or early in 2021.
| 6 | https://mathoverflow.net/users/158094 | 360412 | 151,718 |
https://mathoverflow.net/questions/360279 | 2 | For $x=(x\_1,...,x\_n)\in \mathbb{R}^n$, let $Q\_x=(x\_1,x\_1+1)\times ...\times (x\_n,x\_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft [here](https://drive.google.com/open?id=1CDCUqG0aJhmrw35NazFTlzTR0fpccFcL)) the following
**Theorem.** Let $K\subset\mathbb{R}^n$.... | https://mathoverflow.net/users/53155 | Reference request: placing a set with respect to the integer grid | Based on Mathieu Baillif's comment we can actually prove a much more abstract
**Theorem.** Let $G$ be a Cech-complete topological group, let $U\_n$ be a sequence of non-empty open sets in $G$, and let $K\subset G$. Then there is $g\in G$ such that for every $n$ either $gK\cap U\_n\ne \varnothing$ or $gK\cap \overlin... | 0 | https://mathoverflow.net/users/53155 | 360415 | 151,719 |
https://mathoverflow.net/questions/360421 | 6 | Let $F$ be a field. Does the category $C\_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object?
This is, for instance, true if $F=\mathbb{F}\_{p}$ for some prime $p$: If $R$ is a local ring with residue field $\mathbb{F}\_{p}$, then any $x\in\mathbb{Z}\setminus(p)$ mus... | https://mathoverflow.net/users/70751 | Does the category of local rings with residue field $F$ have an initial object? | A typical approach is to ask for your universal property among complete DVRs, not among all local rings, because there you have a very nice positive result. Given a perfect field $k$ of characteristic $p$, the Witt ring $W(k)$ is initial among complete DVRs of characteristic $0$ equipped with an isomorphism of residue ... | 6 | https://mathoverflow.net/users/nan | 360435 | 151,727 |
https://mathoverflow.net/questions/360432 | 2 | Solving some problem parametrically, I got the following answer:
$$ \dfrac{5x}{4} + \sqrt{\dfrac{y^2}{4} - \dfrac{x^2}{16}} + \dfrac{1}{10} \sqrt{10x^2 + 9y^2} + \dfrac{1}{5} \sqrt{5x^2 + 16y^2} $$
And I had a question: can there be a rational answer for some $ x, y \in \mathbb {R\_+} $? Well, or what “maximally” s... | https://mathoverflow.net/users/119631 | The existence of rational points | Take $x=1$ and $y^2=z$. Then conditions are that $z/4-1/16$, $10+9z$ and $5+16z$ are squares. By multiplying these conditions we get the elliptic curve $u^2 = (z/4-1/16)(10+9z)(5+16z)$. It has torsion group $Z/2Z \times Z/4Z$ and rank $1$. A point $P$ of infinite order corresponds to $z=-1/3$. One of the torsion points... | 2 | https://mathoverflow.net/users/21337 | 360445 | 151,731 |
https://mathoverflow.net/questions/360436 | 2 | I read an article where it is said: $E\_1(\mathbb{Q}\_p)\cong \mathbb{Z}\_p$ where $E$ is an elliptic curve over $\mathbb{Q}\_p$ and $E\_1(\mathbb{Q}\_p)=\{P\in E(\mathbb{Q}\_p):\tilde{P}=\tilde{O}\}$.
The author says that the proof is in "Arithmetic of elliptic curves" by J. Silverman, at page 191, but there it is s... | https://mathoverflow.net/users/158108 | Why $E_1(\mathbb{Q}_p)\cong\mathbb{Z}_p$ | As RP says, there's a chapter in *The Arithmetic of Elliptic Curves* that discusses formal groups, and in particular the points of a formal group defined over a complete local ring. The specific result that you want is Chapter IV, Theorem 6.4(b), in the special case that $K=\mathbb Q\_p$ and $R=\mathbb Z\_p$ and $\math... | 5 | https://mathoverflow.net/users/11926 | 360459 | 151,736 |
https://mathoverflow.net/questions/332556 | 1 | Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions
$$X=\{u\in C^2(\bar B) \mid u|\_{\partial B}=0 \text{ and } \|\Delta u\|\_{L^2(B)}\leq 1\},$$
where $\Delta$ is the Laplacian.
>
> Is it true that the set of first derivatives $\{\frac{\partial u}{\partial x\_1}|\, u\in X\... | https://mathoverflow.net/users/16183 | Estimate on first derivatives given $L^2$-norm of Laplacian | Yes, this is true and the "proof" is as follows (I wouldn't really dare to call that a proof, though).
For any $u\in X$ write $f=-\Delta u$, and observe that trivially $u$ solves the elliptic PDE
$$
\left\{
\begin{array}{ll}
-\Delta u=f & \mbox{in }B\\
u=0 & \mbox{on }\partial B
\end{array}
\right..
$$
(This is a tauto... | 1 | https://mathoverflow.net/users/33741 | 360468 | 151,738 |
https://mathoverflow.net/questions/331489 | 2 |
>
> Let $\Omega$ be a compact manifold in $\mathbb R^2$. For $1 \leq p \lt
> 4/3$ can we claim that
>
>
> $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$
>
>
> with the first inclusion being compact and the second one being
> continuous?
>
>
> Note that $W^{-1,p}(\Omega)$ is identified with ... | https://mathoverflow.net/users/137472 | Question: can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$? | The answer is yes for both.
---------------------------
---
For the first compact embedding: The [Rellich-Kondrachov theorem](https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem) guarantees that in dimension $d$ and for $p<d$ we have $W^{1,p}\subset\subset L^q$ for all $1\leq q<p^\*=\frac{dp}{d-p}$. ... | 2 | https://mathoverflow.net/users/33741 | 360472 | 151,740 |
https://mathoverflow.net/questions/360461 | 2 | I was reading adelization of classical automorphic forms and learnt that each cusp form corresponds to an automorphic representation of $\operatorname{GL}\_n(\mathbb{A}\_\mathbb{Q})$. I understood the proof. But then I found a statement that there is a one-to-one correspondence between newforms of the congruence subgro... | https://mathoverflow.net/users/140336 | A question related to newform and irreducible cuspidal representation of $\operatorname{GL}_n$ | Newform theory for $\mathrm{GL}\_n$ was originally developed over non-archimedean local fields (at least for $n\geq 3$). The local statements readily yield their global adelic counterparts, since a cuspidal automorphic representation is uniquely a restricted tensor product of local admissible generic representations. I... | 5 | https://mathoverflow.net/users/11919 | 360474 | 151,742 |
https://mathoverflow.net/questions/360467 | 3 | Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous polynomial $f(x,y)=x^2+y^2\in k[x,y]$ . Then $f$ is irreducible in $k[x,y]$ , hence $R=k[x,y]/(f)$ is an integral domain... | https://mathoverflow.net/users/135253 | Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface | Your ring is Frobenius split. I also don't think you need the condition that $k$ does not contain a square root of $-1$.
*Proof.* Denoting by $\overline{k}$ the algebraic closure of $k$, we see that
$$\overline{R} := R \otimes\_k \overline{k} \simeq \frac{\overline{k}[x,y]}{x^2+y^2}$$
is $F$-pure by Fedder's criterio... | 4 | https://mathoverflow.net/users/33088 | 360479 | 151,744 |
https://mathoverflow.net/questions/360477 | 0 | Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude that $\int\_M\Delta u =0?$
| https://mathoverflow.net/users/94097 | Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero? | You have to say what is the meaning of $\Delta u$, and of $\int\Delta u$. For the integral to have a meaning, $u$ has to be a distribution and $\Delta u$ has to be a (signed, Radon) measure. Such distributions are called $\delta$-subharmonic functions.
Then $\int\_M\Delta u=0$
without any assumptions about $C^2$ or ... | 4 | https://mathoverflow.net/users/25510 | 360485 | 151,746 |
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