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https://mathoverflow.net/questions/239844 | 1 | If $G$ is a finite, connected simple graph then is there an expression for the average geodesic length?
That is suppose I know two nodes $n\_1$ and $n\_2$, the number of edges in my graph and at those points and the number of vertices then is there a formula giving reasonable bounds on the geodesic connecting $n\_1$ ... | https://mathoverflow.net/users/36886 | Asymptotic formula for average geodesic length on graph? | There is no formula depending just on the number of vertices (or even the number of vertices plus number of edges), and for a $d$-regular graph, for example, the average could range anywhere from linear in the number of vertices to logarithmic - this has to do with the expansion properties of the graph.
| 2 | https://mathoverflow.net/users/11142 | 239850 | 110,645 |
https://mathoverflow.net/questions/230662 | 14 | This question is about the computation of $H\_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to this computation I have seen quotations to the following two papers:
* Stein "Surjective stability in dimension $0$ ... | https://mathoverflow.net/users/25544 | Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$ | I've been corresponding via email with the OP about this (it is a paper of mine that she got these citations from), and she asked me to post an answer summarizing what I told her. I apologize for the length of this answer -- this is really quite a long story. I also apologize for sometimes butchering people's names. I ... | 8 | https://mathoverflow.net/users/317 | 239854 | 110,646 |
https://mathoverflow.net/questions/239849 | 7 | Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?
| https://mathoverflow.net/users/90626 | Density of prime divisors of $a^n + b$ | This is only known conditional on the generalized Riemann hypothesis. A prime $p$ dividing the numerator of $a^{n} + b$ is more or less equivalent to the statement that the subgroup of $\mathbb{F}\_{p}^{\times}$ generated by $a$ contains $-b$.
This problem is addressed in the 2000 Journal of Number Theory paper by Mo... | 7 | https://mathoverflow.net/users/48142 | 239856 | 110,648 |
https://mathoverflow.net/questions/239852 | 2 | Let $B\_1$ and $B\_2$ be $C^\*$-algebras. Let $U\_1, \ldots, U\_n$ be some unitaries in $B\_1.$ We consider the operator system $S$ spanned by $U\_iU\_j^\*.$
Let $\phi: S \rightarrow B\_2.$
Given that the block matrix $[\phi(U\_iU\_j^\*)]\_{1 \leq i,j\leq n}$ is positive semidefinite, can we conclude that $\phi$ is... | https://mathoverflow.net/users/32470 | Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries | The answer is no :
Let $U\_1 = I$, where $I$ is the identity and $U\_2$ linear independent to $I$ such that $U\_2^\* + U\_2 \ge 0$ .
Then choose $\phi(U\_1) = I$ and $\phi(U\_2) = -I$ .
| 1 | https://mathoverflow.net/users/17261 | 239860 | 110,650 |
https://mathoverflow.net/questions/239858 | 11 | I would like to study elements of representation theory as I often encounter it when reading texts on harmonic analysis. I was therefore curious if someone could recommend a book for this.
When looking into the books that have "representation theory" as their title, it seems there is a variety of books written by peo... | https://mathoverflow.net/users/nan | Text for studying group representations in the context of (abstract) harmonic analysis | As you've seen, "representation theory" is a broad subject, and means different things to different people.
Varadarajan's delightful book "An introduction to harmonic analysis on semi-simple Lie groups" is one of the most readable books on "harmonic analysis" in the repn-theoretic sense, while taking analytic issues ... | 13 | https://mathoverflow.net/users/15629 | 239863 | 110,653 |
https://mathoverflow.net/questions/239874 | 5 | This may be an elementary question, but I'm having trouble coming up with an answer: Let $\tilde{N} = T^\*(G/B)$ be the Springer resolution of the nilpotent cone. Does it have finitely many $G$-orbits? If not, what's the "first" example where it doesn't?
The answer for $SL\_2$ is obviously yes (finitely many orbits).... | https://mathoverflow.net/users/6059 | $G$-orbits in Springer resolution (or, stabilizer actions on Springer fibers) | The question is equivalent to studying the $B$-orbits in $\mathfrak u$. Then Kashin proved that for $G$ simple the number of $B$-orbits in $\mathfrak u$ is finite iff $G$ is of type $A\_n$, $n\le4$ or $B\_2$. See
"Orbits of an adjoint and co-adjoint action of Borel subgroups of a semisimple algebraic group. Problems ... | 7 | https://mathoverflow.net/users/89948 | 239879 | 110,659 |
https://mathoverflow.net/questions/239884 | 4 | Let $G$ be Lie group and $K \subset G$ a closed subgroup, such that there exists a $v \in T(G/K)$ whose isotropy-group $G\_v$ is discrete (so iff $\dim G\_v =0$). Lets assume $g$ acts properly on $T(G/K)$
Now we endow $G/K$ with a $G$-invariant riemannian metric $< \cdot, \cdot >$ such that there exists a geodesic, w... | https://mathoverflow.net/users/75382 | geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$ | Perhaps I'm misunderstanding your question, but what about the following example?
Let $G = \mathrm{SO}(3)$ and let $K=\{e\}$ be the identity subgroup. Then $G/K = \mathrm{SO}(3)$ and $G\_v = K$ for all $v\in T\mathrm{SO}(3)$.
Now let $g$ be a metric on $\mathrm{SO}(3)$ that is invariant under the left action of $G... | 4 | https://mathoverflow.net/users/13972 | 239898 | 110,665 |
https://mathoverflow.net/questions/198621 | 3 | Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance [Robert Bryant's notes](http://www.math.duke.edu/~bryant/Spinors.pdf)), one realises that the key fact bringing all the phenomena of triality etc. is the following: that there exist some octonions $u\_1,... | https://mathoverflow.net/users/62367 | Octonions product: inversion in the right and identity in the left | Let $u\_1,...,u\_8$ be any eight perpendicular unit octonions. Then product $L\_{u\_1}L\_{\bar{u\_2}}...L\_{u\_7}L\_{\bar{u\_8}}=I$ and
$R\_{u\_1}R\_{\bar{u\_2}}...R\_{u\_7}R\_{\bar{u\_8}}=-I$ (each even element is conjugated).
Alternatively you can take seven perpendicular imaginary unit octonions, then you can skip... | 1 | https://mathoverflow.net/users/nan | 239899 | 110,666 |
https://mathoverflow.net/questions/239762 | 0 | Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the presented items $(X\_i)\_{i \leq n}$ the problem is to what $\sigma$-algebra filtration $(\natural\_i)\_{i \leq n}$ the best st... | https://mathoverflow.net/users/92127 | Finding the right σ-algebra. Question on uncertainty related to the secretary problem | I am not sure exactly what your question is. What do you mean by "the right filtration"? In each example, $\sigma(X\_1,\dots, X\_i)$ would do the job fine.
Maybe you are looking for the smallest sigma-algebra in each case. But in fact the forms given in your examples are not the smallest possible:
(1) for the case... | 0 | https://mathoverflow.net/users/5784 | 239906 | 110,669 |
https://mathoverflow.net/questions/239911 | 1 | Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where
What happens in the $p$-adic case? Is there sphere still a compact manifold? By sphere I mean:
$$ S^2 = \{ (x,y,z) \in \mathbb{Q}\_p^3: x^2 + y^2 + z^2 = 1 \}$$
In order... | https://mathoverflow.net/users/1358 | harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$? | First, the group $G=SO(3,\mathbb Q\_p)$ acts transitively on that quadric surface, by Witt's theorem. For $p\not=2$ this group is non-compact. The isotropy group of a point on the quadric is a subgroup $H$ isomorphic to $O(2)$, split or not depending on $p$ mod $4$. Then the harmonic analysis on $G/H$ can be addressed ... | 6 | https://mathoverflow.net/users/15629 | 239916 | 110,671 |
https://mathoverflow.net/questions/239883 | 1 | Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its [total graph](https://mathoverflow.net/questions/239806/total-chromatic-number-and-total-clique-number). The *Hadwiger number* $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K\_n$ is a minor of $G$.
Is there an example of a graph $G$ suc... | https://mathoverflow.net/users/8628 | Hadwiger number of total graph | The graph $T(G)$ contains as a subgraph $L(G)$, the [line graph](https://en.wikipedia.org/wiki/Line_graph) of $G$.
Abrams and I [constructed](https://ajc.maths.uq.edu.au/pdf/60/ajc_v60_p314.pdf) a $K\_{9k + 1}$ minor in the graph $L(K\_{5k+2})$.
| 2 | https://mathoverflow.net/users/30994 | 239917 | 110,672 |
https://mathoverflow.net/questions/239923 | 2 | Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K\_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree of $G$, do we have $\delta(G)\leq\eta(G)$?
| https://mathoverflow.net/users/8628 | Hadwiger number and minimal degree | No, icosahedron does not have $K\_5$ as a minor being planar graph.
| 9 | https://mathoverflow.net/users/4312 | 239925 | 110,677 |
https://mathoverflow.net/questions/239243 | 16 | we know [Lagrange's identity](https://en.wikipedia.org/wiki/Lagrange%27s_identity)
$$(a^2\_{1}+a^2\_{2}+a^2\_{3})(b^2\_{1}+b^2\_{2}+b^2\_{3})=(a\_{1}b\_{1}+a\_{2}b\_{2}+a\_{3}b\_{3})^2+\sum\_{i=1}^{2}\sum\_{j=i+1}^{3}(a\_{i}b\_{j}-a\_{j}b\_{i})^2$$
then we have Cauchy-Schwarz inequality
$$(a^2\_{1}+a^2\_{2}+a^2\_{3}... | https://mathoverflow.net/users/38620 | An inequality concerning Lagrange's identity | Without loss of generality, all the $a\_i$'s and $b\_i$'s are nonzero.
Let $\tilde d$ denote the difference between the left- and right-hand sides of the conjectured inequality $(\*)$, which then of course can be rewritten as $\tilde d\ge0$.
In the previous version of my answer, I rewrote $\tilde d$ in new variables,... | 15 | https://mathoverflow.net/users/36721 | 239931 | 110,681 |
https://mathoverflow.net/questions/239870 | 2 | Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$?
What if $2p+1$ is replaced by $2p-1$ and what is the conjectured density and what is known so far for both cases $2p+1$ and $2p-1$?
| https://mathoverflow.net/users/nan | Density of Sophie Germain $3\bmod 4$ primes | As with most questions about finding primes in linear forms, nothing is known. We expect that the number of Sophie Germain primes $\leq x$ equivalent to 3 modulo 4 is $$\sim \frac{2x}{\log^2 x} \prod\_{p > 2} (1-2/p) (1-1/p)^{-2},$$ and for the same estimate to hold with 3 replaced by 1. The same should hold true with ... | 4 | https://mathoverflow.net/users/50426 | 239932 | 110,682 |
https://mathoverflow.net/questions/239941 | 2 | I'm stuck trying to find the maximum entropy probability distribution taking into account a joint distribution.
Basically, I want to find the maximum entropy expression for $p(x,y)$ when the marginal distributions $p(x)$ and $p(y)$ are known.
| https://mathoverflow.net/users/92233 | Maximizing joint entropy? | Well $H(X,Y)\leq H(X)+H(Y)$ with equality if and only if $X$ and $Y$ are independent. So pick $p(x,y)=p(x)p(y).$
| 3 | https://mathoverflow.net/users/17773 | 239943 | 110,684 |
https://mathoverflow.net/questions/239942 | 2 | A systems $<f\_0,f\_1>$ is minimal if the set $\{h(x): h=f\_{i\_n}\circ f\_{i\_{n-1}}\circ...\circ f\_{i\_1}, i\_k \in \{0,1\},n>0\}$ is dense in $S^1$, for every $x\in S^1$.
Consider $f:S^1 \to S^1, f(x)=2x (mod \, 1)$ and $f\_{\epsilon}=f+ \epsilon$.
Is there an $\epsilon>0$ such that $<f,f\_{\epsilon}>$ is minimal?
... | https://mathoverflow.net/users/74049 | IFS maps on circle | Yes, for example, you could choose $\epsilon$ so that $\{2^k \epsilon \mid k = 1,2,3,\ldots\}$ is dense in $S^1$. By letting all but one of the $f\_{i\_k}$'s be $f$, we see that the orbit of $x$ under words of length $n$ contains $f^n(x) + 2^k\epsilon$ for $1 \le k \le n$.
| 2 | https://mathoverflow.net/users/68305 | 239944 | 110,685 |
https://mathoverflow.net/questions/239851 | 13 | In the paper Sato-Wakui "[COMPUTATIONS OF TURAEV-VIRO-OCNEANU INVARIANTS OF 3-MANIFOLDS FROM SUBFACTORS](http://arxiv.org/abs/math/0208242)" they compute certain Turaev-Viro-Ocneanu invariants of certain lens spaces. One of the results is that the generalized $E\_6$ subfactor for $\mathbb{Z}/p$ distinguishes the lens s... | https://mathoverflow.net/users/184 | Does the Turaev-Viro theory for the generalized $E_6$ subfactor for $\mathbb{Z}/7$ distinguish $L(7,1)$ and $L(7,2)$? | I think [this](http://www2.itc.kansai-u.ac.jp/~wakui/ILDT07wa.pdf) paper of Wakui says that the answer is "No". The Turaev-Viro invariants associated to the generalized $E6$ subfactors for $\mathbb{Z}/7$ don't seem to distinguish $L(7,1)$ and $L(7,2)$.
| 10 | https://mathoverflow.net/users/401 | 239975 | 110,691 |
https://mathoverflow.net/questions/239978 | 3 | If $A$ is a $C^\*$-algebra, there is a notion of Hilbert $A$-module (which is something like Hilbert space but the inner product takes values in $A$). The standard example is $H\_A:=\{(a\_n)\_{n=1}^{\infty}: \sum\_{n=1}^{\infty}a\_n^\*a\_n \ is \ norm \ convergent\}$ with $A$ valued inner product $(a,b):=\sum\_{n=1}^{\... | https://mathoverflow.net/users/24078 | Operators on Hilbert $C^*$-module and families of Fredholm operators | The answer involves two arguments (I guess that $Fred (\ell^2)$ does have the norm topology).
1. If $F: X \to Fred(\ell^2)$ is continuous, then there is another map $G: X \to Fred(\ell^2)$ such that $FG-1$ and $GF-1$ are maps into $\mathcal{K} (\ell^2)$.
2. If $T:X \to \mathcal{K}(\ell^2)$, then $T$, viewed as an $C... | 2 | https://mathoverflow.net/users/9928 | 239982 | 110,693 |
https://mathoverflow.net/questions/239993 | 10 | Many years ago, I considered the following non-linear differential equation:
$y=y''\cdot(1+y'^{2})^{-3/2}$
This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and the value of its curvature at the same point.
I almost convinced myself that there is no nontrivial solution to th... | https://mathoverflow.net/users/13625 | Techniques to solve a non-linear differential equation related to curvature | Well the standard techniques would take advantage of the fact that the equation doesn't explicitly involve the independent variable $x$ to integrate the equation once, thereby leading to the conservation law
$$
y^2 + 2(1+y'^2)^{-1/2} = r^2,\tag1
$$
where, $r>0$ is a constant. Note that we must have $|y| < r$
The rela... | 19 | https://mathoverflow.net/users/13972 | 239999 | 110,700 |
https://mathoverflow.net/questions/239989 | 3 | I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras".
>
> Let $0\to B\to E\to A\to 0$ be a short exact sequence of $C^\*$-algebras, in particular $A=\mathbb{C}$, $B=C\_0((0,1))$. Now there are four possible choices of $... | https://mathoverflow.net/users/nan | busby invariant of extensions of $C^*$-algebras | The idempotents in the corona algebra are very restricted, because they lift to functions$\newcommand{\Real}{{\bf R}}\newcommand{\veps}{\varepsilon}$ $f\in C\_b(\Real)$ such that $f^2-f\in C\_0(\Real)$.
Fix such an $f$. Note that if $z$ is a complex number such that $|z^2-z|= |z(z-1)|$ is small (say less than $\veps$... | 3 | https://mathoverflow.net/users/763 | 240008 | 110,703 |
https://mathoverflow.net/questions/240002 | 2 | Let $k\geq 4$. As usual, let $r\_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r\_k(n) \gg n^{\frac{k}{2}-1-\varepsilon}$ ? Is there an easy proof of this fact? What can we say about the cases $k=2,3$?
Many thanks !
| https://mathoverflow.net/users/66686 | Lower bound for the number of representations of integers as sum of squares | For $k=4$, your statement would be that $r\_4(n) \gg n^{1-\epsilon}$. This is false. [Jacobi's four-square theorem](https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem) can be stated as that $r\_4(n)/8$ is the sum of the divisors of $n$ that are not divisible by $4$. Let $n = 2^t m$ with $m$ odd and $t$ positi... | 3 | https://mathoverflow.net/users/2954 | 240017 | 110,707 |
https://mathoverflow.net/questions/240014 | 12 | Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue measure zero?
There are such sets in models in which every set of reals of cardinality less than the continuum is Leb... | https://mathoverflow.net/users/2554 | Witness to a failure of Fubini/Tonelli | No. Suppose $m:\mathcal{P}([0, 1]) \to [0, 1]$ is a total extension of Lebesgue measure. Let $A \subseteq [0, 1]^2$ be such that every vertical section is Lebesgue null.
Claim: $A$ is $m \otimes \mu$-null.
Proof: For each rational $e>0$, let $U\_{e, x} \subseteq [0, 1]$ be an open set of measure less than $e$ that ... | 10 | https://mathoverflow.net/users/2689 | 240019 | 110,708 |
https://mathoverflow.net/questions/239393 | 14 | Let $E \to F$ be a $G$-Galois extension of fields.
What is the simplest example where the natural map $K(E) \to K(F)^{hG}$ is not an equivalence on connective covers (i.e., where finite Galois descent fails)?
| https://mathoverflow.net/users/344 | Simplest example of failure of finite Galois descent in algebraic $K$-theory? | First of all, by the long exact sequence in homotopy, it suffices to give a $G$-Galois extension $E \subset E'$ such that the homotopy fiber of $K(E) \to K(E')^{hG}$ contains a class in degree $-1$. This means that the map $\pi\_0 K(E) \to \pi\_0 K(E')^{hG}$ is not surjective.
The spectrum $\mathrm{fib}( K(E) \to K(... | 8 | https://mathoverflow.net/users/344 | 240025 | 110,710 |
https://mathoverflow.net/questions/239674 | 7 | Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\sigma$-algebra on $Z$ that contains the Borel subsets of $Z$ and is closed under the [Suslin operation](https://en.wikipedi... | https://mathoverflow.net/users/49623 | Product of limit $\sigma$-algebras | Every member of your limit $\sigma$-algebra is both Lebesgue measurable and has the Baire property (for a proof, see section 29.B in Kechris book). A result of Mansfield and Rao implies that the universal analytic set in plane is not in the sigma algebra generated by rectangles with measurable (resp. Baire property) si... | 4 | https://mathoverflow.net/users/2689 | 240028 | 110,711 |
https://mathoverflow.net/questions/240039 | 6 | Let $n$ be a positive integer greater than $1$, and define the polynomial $$p\_{n}(x)=\sum\_{k=0}^{n}\dfrac{x^k}{k!}$$
>
> Is $p\_{n}(x)$ irreducible in $\mathbf{Q}[x]$?
>
>
>
I can show it when $n$ is a prime number, since
$$n!p\_{n}(x)=x^n+nx^{n-1}+n(n-1)x^{n-2}+\cdots+n!x+n!$$ using Eisenstein's criterion. ... | https://mathoverflow.net/users/38620 | Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$ | That follows from a theorem of Schur saying that any polynomial $\sum\_{k=0}^nc\_k\frac{x^k}{k!}$ with $c\_i\in\mathbf{Z}$, $c\_0,c\_n\in\{1,-1\}$, $n\ge 1$, is irreducible over $\mathbf{Q}$.
I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen I, Sitzungsberichte
Preuss. Akad. Wiss. Phys... | 20 | https://mathoverflow.net/users/89948 | 240042 | 110,716 |
https://mathoverflow.net/questions/240038 | 5 | Consider the compactified modular curve $X\_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:
1. Where do the cusps of $X\_0(p^2)\_{\mathbb{Q}}$ live? That is for which extension $K$ of $\mathbb{Q}$, $X\_0(p^2)(K)$ contains all the cusps?
2. Let $K$ be as above and $\mat... | https://mathoverflow.net/users/11392 | Regular minimal model of $X_0(p^2)$ | The cusp divisor of $X\_0(N)$ is given (at least in principle) by the “torsion” of the degenerate elliptic curves and (in principle) described as such in Katz-Mazur's book, *Arithmetic moduli of elliptic curves*, Annals of math. studies, vol. 108 (1985), Princeton Univ. Press.
An answer to question 2 (and much more) ... | 3 | https://mathoverflow.net/users/10696 | 240054 | 110,720 |
https://mathoverflow.net/questions/240068 | 3 | Is the dimension of the projective space $\mathbb{P}^{{n \choose k} -1}$ into which we embed the Grassmannian $G(k,n)$ of $k$-planes in $n$-space minimal? In other words, is the Grassmannian variety (ever) contained in a Plücker hyperplane section?
Sorry, I wasn't able to find a reference for this.
| https://mathoverflow.net/users/92003 | Is the Grassmannian contained in a Plücker hyperplane? | The Grassmannian in its Plücker embedding spans the space. The space $\mathbb{P}^{\binom{n}{k}-1}$ of alternating tensors is spanned by the simple wedges (also called decomposable) $v\_1 \wedge \dotsb \wedge v\_k$, i.e., elements of the Grassmannian. So, no, in this embedding the Grassmannian is not contained in any hy... | 9 | https://mathoverflow.net/users/88133 | 240073 | 110,722 |
https://mathoverflow.net/questions/240081 | 8 | Define $N\_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$.
Landau's Theorem says $N\_\Delta(T)$ is $\Omega(\log T)$ [1].
But perhaps that estimate is too crude. Is the main term of $N\_\Delta(T)$ known? Or are only strict upper and lower bounds known?
What type of machinery ... | https://mathoverflow.net/users/92331 | Main term in the number of sign changes of $\psi(x) - x$ | An asymptotic for the number of sign changes is not known, and indeed only a lower bound of $c \log T$ is known. There have been small improvements in the constant $c$ that is allowed here (see [Kaczorowski](http://matwbn.icm.edu.pl/ksiazki/aa/aa44/aa4446.pdf)). For recent work related to this, see [Montgomery and Vorh... | 10 | https://mathoverflow.net/users/38624 | 240082 | 110,723 |
https://mathoverflow.net/questions/240098 | 2 | Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.
My primary interest is the case where $n = 3$, as this is the answer I'm *really* after. I would've... | https://mathoverflow.net/users/92343 | How many lines of exactly n points can be placed in a discrete, square grid of size m x m? | It seems that this has been answered here by S. Mustonen: [PointsInGrid.pdf](http://www.survo.fi/papers/PointsInGrid.pdf)
There $f(n,m)$ is denoted by $L\_n(m)$ and a formula would be
$$L\_n(m)=\frac{1}{2}[f(m,n+1)-2f(m,n)+f(m,n-1)]$$
for $$f(m,k)=\sum\_{\substack{-m<kx<m\\-m<ky<m\\(x,y)=1}}(m-|kx|)(m-|ky|)$$
| 2 | https://mathoverflow.net/users/39495 | 240105 | 110,729 |
https://mathoverflow.net/questions/240085 | 37 | That's a vague question so allow me to tighten it up a bit.
I recently noticed that there is a [formal machine verified proof](http://arxiv.org/abs/1405.7012) of the Central Limit Theorem (CLT) implemented with [Isabelle](https://en.wikipedia.org/wiki/Isabelle_(proof_assistant)). This requires a substantial amount of... | https://mathoverflow.net/users/1233 | How much mathematics has been formally verified? | A couple of years ago I made a database of all the formalization files that I could find at that point, in several different systems. You can view it here:
<http://bim.shef.ac.uk/formal/list_formalizations.php>
I would guess that the proportion of a typical undergraduate curriculum that has been formalized in at le... | 46 | https://mathoverflow.net/users/10366 | 240119 | 110,731 |
https://mathoverflow.net/questions/240115 | 0 | In the Quadratic Reciprocity Law
$$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$.
Is there any relation between $x$ and $y$ in general?
| https://mathoverflow.net/users/10035 | On the quadratic reciprocity law? | It is not clear what relation you look for.
E.g., $x$ and $y$ may be viewed as reductions of the same residue $z$ modulo $q$ and $p$, respectively, where $z^2\equiv p+q\pmod{pq}$.
| 4 | https://mathoverflow.net/users/7076 | 240123 | 110,732 |
https://mathoverflow.net/questions/240069 | 4 | In his book 'Forcing with Random Variables and Proof Complexity' Jan Krajíček claims (p.154) that it is possible to break the RSA encryption with public key $(e,N)$ if one has has an integer $w \neq 0$ such that $e^w = 1 \mod(N)$ holds.
Can anybody explain how this is done or give a source for the claim?
PS: In th... | https://mathoverflow.net/users/92325 | Breaking the RSA encryption based on a $(e,N)$ given an integer $w \neq 0$ such that $e^w = 1 \mod(N)$? | I think you got confused by the somewhat peculiar notation. Krajíček actually writes on p. 155 that one can break the given instance of RSA using $w\ne0$ such that
$$g^w=1\pmod N.$$
Now, what is $g$? Well, on p. 154 we see: *By an RSA function based on such a pair $(g,N)$ we mean a function*
$$x<N\to g^x\bmod N.$$
So, ... | 5 | https://mathoverflow.net/users/12705 | 240124 | 110,733 |
https://mathoverflow.net/questions/240125 | 1 | Let $f \in \mathbb{Q}[x]$.
Let $S(f)$ denote the number of distinct linear factors
of $f(x)-f(y)$.
$S(f)$ is bounded by $\deg(f)$.
>
> Q1 Is $S(f)$ bounded by constant?
>
>
> Q2 Is it possible $S(f)>2$ for arbitrary large degree of $f$?
>
>
>
This is related to the number of ways of polynomial
represents... | https://mathoverflow.net/users/12481 | How many the distinct linear factors of $f(x)-f(y)$ can be for f in Q[x]? | $S(f) \leq 2$ for all nonconstant $f$.
By context, I understand that you mean factorization in $\mathbb{Q}[x,y]$. If you factor in $\mathbb{C}[x,y]$ then, of course, $x^n-y^n$ is a product of lines. (Daniel McLaury, in comments, mentions a third option, that $y$ might be a fixed integer. I don't know about this.)
A... | 15 | https://mathoverflow.net/users/297 | 240126 | 110,734 |
https://mathoverflow.net/questions/240089 | 17 | I am reading this paper of Rezk's <http://arxiv.org/abs/0901.3602> *A cartesian presentation of weak n-categories*, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in the old version) and now there is a new proof for the main result, which is 6.6.
Now, section 6.9 of the new proof ref... | https://mathoverflow.net/users/57280 | Errata on Rezk's paper | It looks like I completely missed this.
Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets): If $X$ is a simplicial set, and $\mathcal{P}$ a finite collection of subobjects of $X$ which is *closed under intersections*, and... | 30 | https://mathoverflow.net/users/437 | 240127 | 110,735 |
https://mathoverflow.net/questions/239981 | 6 | I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}\_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the curvature is not necessarily well defined):
$$ K\geq \int\_B \frac{\nabla \varphi \cdot \nabla \psi}{\sqrt{1+|\nabla \phi... | https://mathoverflow.net/users/13093 | Regularization by mean curvature flow | The statement you want (and a bit more I think) can be found in Lemma 3.8 by Metzger--Schulze in their [article](http://www.ams.org/mathscinet-getitem?mr=2401622) "No mass drop for mean curvature flow of mean convex hypersurfaces".
| 4 | https://mathoverflow.net/users/1540 | 240130 | 110,736 |
https://mathoverflow.net/questions/240129 | -1 | Let $n$ be postive integer,I conjecture
$$(1+2n)^n\ge 1^n+2^n+4^n+6^n+\cdots+(2n)^n \tag{1}$$
This problem when I solve this equation
$$(1+2n)^n=1^n+2^n+4^n+6^n+\cdots+(2n)^n\tag{2}$$
if this $(1)$ hold,then $(2)$ have only postive integer solution $n=1$
| https://mathoverflow.net/users/38620 | Prove this conjecture inequality 2 | $n=1$ is clear, let $n\geqslant 2$.
Divide by $(2n+1)^n$ and use the estimate $(1-\frac{k}{2n+1})^{n}<e^{-k/2}$ for $k=3,5,\dots$ (for proving this take logarithm, get equivalent inequalities $n\log(1-\frac{k}{2n+1})<-k/2$, $k/2<n(\frac{k}{2n+1}+\frac{k^2}{2 (2n+1)^2}+\dots)$, two first summands already suffice: $$n\... | 8 | https://mathoverflow.net/users/4312 | 240132 | 110,738 |
https://mathoverflow.net/questions/240134 | 0 | If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T\_{C}$. We know that $df: f^{\*}{T\_{Y}}\_{|C}\cong N\_{C}.$ In fact both of these budnles are trivial and isomorphic to $C\times T\_{y}$(Voisin pg.223). Can we give e... | https://mathoverflow.net/users/54760 | Normal bundle of a fiber of the family of curves | Take a vector in $TX$ which represents the vector in $N\_C$; it is unique up to adding something tangent to $C$. Now apply $f'(x)$ to it, a linear map to $T\_y Y$. At a generic point of $X$, the map $f$ is a linear projection in local holomorphic coordinates, $f(x,y)=y$ and the map $f'$ is just $f$ in those coordinates... | 2 | https://mathoverflow.net/users/13268 | 240137 | 110,739 |
https://mathoverflow.net/questions/240128 | 0 | Let $G\_1$ and $G\_2$ be two groups (of some kind, e.g. finite groups).
Let $M\_1, N\_1$ be $G\_1$-modules, and $M\_2, N\_2$ be $G\_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G\_1 \times G\_2$, $M = M\_1 \otimes M\_2$ and $N = N\_1 \otimes N\_2$.
Is $Ext\_G(M, N)$ related in some way to $Ext... | https://mathoverflow.net/users/76332 | $Ext$ functor over a product of groups | The answer is given by Kunneth formula. In your case for each pair of modules $N\_1$, $N\_2$ we can consider a functor from the category of representations of $G\_1\times G\_2$ to $\mathbb C$-vector spaces, sending $M$ to $\mathrm{Hom}\_{\mathbb C[G\_1\times G\_2]}(M, N\_1\otimes N\_2)$. $\mathrm{Ext}^\bullet\_{\mathbb... | 0 | https://mathoverflow.net/users/42606 | 240140 | 110,740 |
https://mathoverflow.net/questions/240138 | 6 | Fix a $(B,N)$ pair (Tits system) of a semisimple Lie group $G$. Let $u$ and $v$ be two Weyl group elements such that $l(uv)=l(u)+l(v)$. It is known that $BuvB=(BuB)(BvB)$ (see for example Humphreys's Linear Algebraic Groups Section 29.3 Lemma A). Let $x$ be an element of the Bruhat cell $BuvB$ and I would like to facto... | https://mathoverflow.net/users/74343 | Product of Bruhat Cells | Many thanks to Paul Garrett's comment above. Inspired by his comment I come up with a proof of the following equivalent statement.
Proposition. *If $x\in BwB$ where $w=s\_{i(1)}s\_{i(2)}\dots s\_{i(l)}$ is a reduced word for $w$, then there exist $x\_k\in Bs\_{i(k)}B$ such that $x=x\_1x\_2\dots x\_l$; further if $x=x... | 1 | https://mathoverflow.net/users/74343 | 240147 | 110,743 |
https://mathoverflow.net/questions/240133 | 3 | Let $A$ and $B$ be two self-adjoint, positive definite Compact operators on a Hilbert space $\mathcal{H}$. Further, let $A$ be trace class. Define $C\_n \equiv AB(\frac{I}{n} + BAB)^{-1}$. Does $\frac{tr(C\_nC\_n^{T})}{n}$ converge to $0$ as $n \to \infty$. The answer is affirmative when $A$ and $B$ commute as can be f... | https://mathoverflow.net/users/65922 | Convergence in trace | Yes, this is true. Denote $Q\_n=\sqrt{n}\sqrt{A}B$. We have to prove that $c\_n:=tr(\sqrt{A}Q\_n(I+Q\_n^\*Q\_n)^{-2}Q\_n^\*\sqrt{A})\to 0$. We have $c\_n=tr(AQ\_n(I+Q\_n^\*Q\_n)^{-2}Q\_n^\*)$. We need a
**Lemma.** For any compact operator $Q$ we have $Q(I+Q^\*Q)^{-2}Q^\*\leqslant \frac14I$. Moreover, for any vector $... | 4 | https://mathoverflow.net/users/4312 | 240149 | 110,744 |
https://mathoverflow.net/questions/226513 | 1 | The motivation of the question comes from a geometric problem: can we approximate a $C^{1,\alpha}$ set $\Omega$ with positive curvature (in distributional sense) from inside with $C^2$ sets with positive curvature? (the approximation is in the Hausdorff distance).
Translating the problem into a graph setting we arriv... | https://mathoverflow.net/users/13093 | Approximation with a more regular function and an inequality constraint | Looks like the answer to this question is affirmative. In fact the question is equivalent to the following one: [Regularization by mean curvature flow](https://mathoverflow.net/questions/239981/regularization-by-mean-curvature-flow)
Take a look at the article in the given answer to see the proof.
| 0 | https://mathoverflow.net/users/13093 | 240151 | 110,745 |
https://mathoverflow.net/questions/240142 | 8 | Let $(a\_1,a\_2,\dots, a\_n)$ be a sequence of non-negative integers.
**Q.** When does there exists a simple graph $G$ such that its number of $k$-cliques is $a\_k$ (that is $G$ has $a\_1$ vertices, $a\_2$ edges, $a\_3$ trinagles, etc)?
Probably, it is hopeless to get a complete description of such sequences, so ... | https://mathoverflow.net/users/40950 | Graphs with prescribed numbers of k-cliques | What you are asking for is possible $f$-vectors of [Clique Complexes](https://en.m.wikipedia.org/wiki/Clique_complex). The [Kruskal-Katona Theorem](https://en.m.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem) characterizes $f$-vectors of simplicial complexes; so, it applies here but is no longer a complete character... | 6 | https://mathoverflow.net/users/51668 | 240152 | 110,746 |
https://mathoverflow.net/questions/240156 | 3 | M.I. Kadec and A. Pełczyński proved that if $E$ is a subspace of $L\_{p}(p>2)$ isomorphic to $l\_{2}$, then $E$ is complemented in $L\_{p}$. My question is:
Is there a constant $C\_{p}$ depending only on $p$ such that every subspace of $L\_{p}(p>2)$ isomorphic to $l\_{2}$ is $C\_{p}$-complemented in $L\_{p}$?
Thank... | https://mathoverflow.net/users/41619 | On the complemented subspaces of $L_{p}(p>2)$ | The answer to the question, as it is stated, is "No", and can be shown as follows: It is known (follows, for example, from results of Sobczyk, Duke Math. J., 1941; or from the Lindenstrauss-Tzafriri characterization of $\ell\_2$) that we can find a sequence of finite-dimensional subspaces $R\_n\subset L\_p$ with going ... | 4 | https://mathoverflow.net/users/37822 | 240164 | 110,750 |
https://mathoverflow.net/questions/240113 | 2 | I'm seeking the definition of some function $f(n,m)$ which evaluates to the number of distinct sets of $n$ collinear points which are selected from an evenly-spaced two-dimensional grid of $m \times m$ points?
Collinear in this case means that the points in the set fall exactly on some line with arbitrary slope, that... | https://mathoverflow.net/users/92343 | How many distinct sets of n collinear points are there in an evenly-spaced two-dimensional grid of m x m points? | Let $L\_n(m)$ be defined as in [this answer](https://mathoverflow.net/q/240105). Then the function $f(n,m)$ questioned here (aware that it is different from $f$ in the linked answer) can be computed as
$$f(n,m) = \sum\_{k=n}^{m} \binom{k}{n}\cdot L\_k(m).$$
This formula easily follows from the observation that from... | 3 | https://mathoverflow.net/users/7076 | 240167 | 110,751 |
https://mathoverflow.net/questions/240157 | 7 | It is well-known that any rational number can be represented using a sum of distinct Egyptian fractions (that is, rational fractions of the form $1/n$ with $n\in\mathbb{N}$). This may be proven by establishing a greedy algorithm that constructs a sequence of such decompositions. For instance,
\begin{align}
1
&= \frac1... | https://mathoverflow.net/users/55904 | Groupoid cardinality and Egyptian fraction representations of 1 | First a notational issue: you shouldn't write $G$ for the one-object groupoid corresponding to $G$. A much better name for this groupoid is $BG$, or $\text{pt} / G$.
A natural way to write down a groupoid whose groupoid cardinality is $1$ is to write down a groupoid / homotopy quotient $X/G$ where $|X| = |G|$. In tu... | 8 | https://mathoverflow.net/users/290 | 240168 | 110,752 |
https://mathoverflow.net/questions/239881 | 10 | By some reason the [Journal](http://www.jangjeon.or.kr/menu1/menu1_sub1.html) mentioned in the title is no longer covered by the AMS Math. Reviews. On the MathSciNet web page it says:
Last Issue: 24, no. 1 2014
Indexed cover-to-cover
Status: No longer indexed
The journal, however, still exists and it is still... | https://mathoverflow.net/users/nan | What's wrong with Advanced Studies in Contemporary Mathematics (Kyungshang)? | There has been an explosion in new allegedly "scholarly" journals because they are profitable and many university libraries have lost control over their inventory due to the "big deal" they were trapped into a decade ago where they were lured by lower prices into relinguishing autonomy over subscription decisions, curr... | 17 | https://mathoverflow.net/users/28128 | 240177 | 110,754 |
https://mathoverflow.net/questions/240080 | 20 | Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E\_2(R)$ be the subgroup of $GL\_2(R)$ generated by the matrices that differ from the identity by a single off-diagonal element.
The question whether $SL\_2(R) = E\_2(R)$ is a problem of S. Bachmuth and H. Y. ... | https://mathoverflow.net/users/84349 | Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$? | This is only a long comment. I would think of $\mathbb{F}\_q[x,t^{\pm 1}]$ as the function field analogue of $\mathbb{Z}[t,t^{-1}]$. For the latter, it is known that $SL\_2\neq E\_2$ from the results in the following paper:
* S. Krstic and J. McCool. Free quotients of $SL\_2(R[x])$. Proc. Amer. Math. Soc. 125 (1997)... | 10 | https://mathoverflow.net/users/50846 | 240178 | 110,755 |
https://mathoverflow.net/questions/240170 | 5 | Let $X$ be a projective scheme over a noetherian ring, $\mathcal F$ an invertible sheaf on $X$, and $U$ an arbitrary open subset of $X$.
Is $\Gamma(U,\mathcal F)$ a $\Gamma(U,\mathcal O\_X)$-module of finite type?
| https://mathoverflow.net/users/92322 | Finite generation of global sections of an invertible sheaf on a quasi-projective scheme | **Edit.** There was an issue with my first example. The second example is fine (but unfortunately it does not work over an arbitrary field).
There is a more "conventional" example as well where $X$ is regular. Begin with $C$ a curve of genus $g\geq 1$. Let $\mathcal{L}$ be an invertible sheaf on $C$ that is algebrai... | 5 | https://mathoverflow.net/users/13265 | 240188 | 110,757 |
https://mathoverflow.net/questions/240205 | 1 | Let $p>2$. Following from M.I. Kadec and A. Pełczyński's results (Studia Math. 1962), R.J.Whitley (Trans. Amer. Math.Soc. 1964) observed that $L\_{p}$ is subprojective, that is, every infinite-dimensional closed subspace of $L\_{p}$ contains an infinite-dimensional closed subspace that is complemented in $L\_{p}$. What... | https://mathoverflow.net/users/41619 | Subprojectivity of $L_{p}(p>2)$ | Since the space $L\_p$ $(1\le p<\infty)$ is stable in the sense of Krivine and Maurey (Espaces de Banach stables, Israel J. Math. 39 (1981), no. 4, 273–295), you can combine their result in this paper and get that any subspace isomorphic to $\ell\_2$ in $L\_p$ contains a subspace which is $(1+\varepsilon)$-isometric to... | 2 | https://mathoverflow.net/users/37822 | 240216 | 110,761 |
https://mathoverflow.net/questions/240203 | 1 | Let $H$ be a bialgebra and $B$ an $H$-module. The cross product $B \rtimes H$ of $B$ and $H$ is $B \otimes H$ as a vector space and the multiplication in $B \rtimes H$ is defined as follows: $(a \otimes g)(b \otimes h) = a(g\_{(1)}.b) \otimes (g\_{(2)}h)$, where $a,b \in B$, $g, h \in H$, $\Delta(g) = g\_{(1)} \otimes ... | https://mathoverflow.net/users/11877 | Motivations of cross product | It's the Hopf-algebraic version of the semidirect product of groups. To see this, just consider the case you have groups $G, H, G\rtimes H$. The group rings (over, say, a field $k$) are Hopf algebras (with $\Delta(g)=g\otimes g$); compare $kG\rtimes kH$ and $k(G\rtimes H)$ and you will see they have the same multiplica... | 4 | https://mathoverflow.net/users/805 | 240218 | 110,762 |
https://mathoverflow.net/questions/240210 | 10 | Is there a closed subscheme $D$ in $\mathbb P^2\_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^2(\mathbb C) - D(\mathbb C)$$
algebraizes? (That is, is there a morphism $f:X\to \mathbb P^2\_{\mathbb C}- D$ whose ... | https://mathoverflow.net/users/92405 | Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize? | On page 73 of Kobayashi's book Hyperbolic Complex spaces he shows that if D is a certain configuration of 6 lines in the plane then its complement is complete hyperbolic and hyperbolically embedded in the projective plane.By theorem 6.3.24 page 290 of the same book your map from X to the complement extends meromorphica... | 11 | https://mathoverflow.net/users/4696 | 240230 | 110,765 |
https://mathoverflow.net/questions/240211 | 10 | A cardinal $\kappa$ is *real-valued measurable* if there is a probability measure on the $\sigma$-algebra of all subsets of $\kappa$ which is zero on singletons and additive on disjoint families of fewer than $\kappa$ subsets.
What if I weaken this to: $\kappa$ is uncountable and there is a finitely additive probabil... | https://mathoverflow.net/users/23141 | Real-valued measurable cardinals | Suppose $m:\mathcal{P}(\kappa) \to [0. 1]$ is a finitely additive measure whose null ideal $I$ is $\kappa$-additive. Then $I$ is an $\omega\_1$-saturated $\kappa$-additive ideal over $\kappa$. Solovay showed this implies that $\kappa$ is measurable in $L[I]$.
If we start with a measurable cardinal $\kappa$, and force... | 12 | https://mathoverflow.net/users/2689 | 240233 | 110,766 |
https://mathoverflow.net/questions/240229 | 11 | This is a question related to ideas raised in <http://arxiv.org/abs/1410.1224> and <http://arxiv.org/pdf/1405.7456.pdf>. Basically, the idea is the following:
>
> Suppose I have a first-order theory $T$. Under what conditions are there "few" models of $T$ across all possible forcing extensions of the universe?
>
>... | https://mathoverflow.net/users/8133 | When are generic models not too wild? | For an example of a super-stable theory failing the property, you can take infinitely many unary predicates (so $Th(2^\omega, U\_n: n \in \omega)$ where $U\_n(\eta)$ holds iff $\eta(n) = 1$). Then if $\nu$ and $\mu$ satisfy that if $\nu[G\_0]$ embeds into $\mu[G\_1]$ in $\mathbb{V}[G\_0 \times G\_1]$ then every quantif... | 5 | https://mathoverflow.net/users/26705 | 240234 | 110,767 |
https://mathoverflow.net/questions/240257 | 2 | Let $M$ be a finite $CW$-complex. Let $\Sigma\_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma\_k$ on $M$. Then we have a covering map
$$
f:M\to M/\Sigma\_k.
$$
From $f$, we have an induced map on the fundamental groups
$$
f\_\*:\pi\_1(M)\to \pi\_1(M/\Sigma\_k).
$$
From $f\... | https://mathoverflow.net/users/65800 | induced group actions and covering maps on Eilenberg-Maclane space | If $M$ is connected, then @MarkGrant's fibration sequence gives a long exact sequence on homotopy groups showing that $\pi\_1(M/\Sigma\_k)\to \Sigma\_k$ is surjective. Now apply the $K(-,1)$-functor and obtain a map $K(\pi\_1(M)/\Sigma\_k,1)\to B\Sigma\_k$. The corresponding $\Sigma\_k$-bundle over $K(\pi\_1(M)/\Sigma\... | 8 | https://mathoverflow.net/users/83633 | 240259 | 110,771 |
https://mathoverflow.net/questions/240237 | 2 | Given the expression
$$K\_{ik} := \frac{\partial}{\partial x\_k} \int\_{\mathscr X} \frac{y\_i-x\_i}{|y-x|^3} \mu(y) dy,$$
where $\mathscr X=\mathbb R^3$, how does one derive the expression
\begin{align}
K\_{kk} &= -\frac{4\pi}3\mu(x) + \int\_{y : |y-x| \le d} (\mu(y)-\mu(x))\left(\frac{3(y\_k-x\_k)^2}{r^5} -\frac 1{r... | https://mathoverflow.net/users/89456 | How can we obtain the $-\frac{4\pi}3\mu(x)$ term? | $$K\_{ik}=-\frac{\partial}{\partial x\_i}\int\frac{\partial}{\partial y\_k}\frac{1}{|y-x|}\mu(y)dy$$
$$\quad=\int\frac{\partial}{\partial y\_i}\frac{\partial}{\partial y\_k}\frac{1}{|y-x|}\mu(y)dy$$
now use that
$$\frac{\partial}{\partial y\_i}\frac{\partial}{\partial y\_k}\frac{1}{|y-x|}=-\frac{4\pi}{3}\delta(y-x)... | 3 | https://mathoverflow.net/users/11260 | 240264 | 110,772 |
https://mathoverflow.net/questions/240262 | 0 | Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C\_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ Then how does the rank of $C\_i$ change with $i$? I conjucture that, for $i$ large enough, there exist non-negative integers $d,s$ such that the following equality holds: $$\mathrm{rank}... | https://mathoverflow.net/users/50466 | How does the rank of $C_i$ change with $i$? | Your conjecture is true. Let us apply Gauss' elimination method with some changes.
First, apply it to the first $(A\; B)$, among others, you obtain several rows containing nonzero values only in the $B$-area, with ones at some positions; let the fragments of these rows in the $B$-area span some subspace $U\_1$.
N... | 2 | https://mathoverflow.net/users/17581 | 240274 | 110,774 |
https://mathoverflow.net/questions/240276 | 12 | [This pdf by David Joyce](http://aleph0.clarku.edu/~djoyce/numbers/dedekind.pdf) notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set.
The proof exploits the assumption that there exists a set $S$ of all things, and that a mathematical thing is an object of our tho... | https://mathoverflow.net/users/28128 | Has Dedekind's proof of existence of infinite sets been analyzed by historians? | See [pages 107](https://books.google.com/books?id=vrQLbbxGNMsC&pg=PA107) and following and [pages 244](https://books.google.com/books?id=vrQLbbxGNMsC&pg=PA244) and following of [Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics](https://books.google.com/books?id=vrQLbbxGNMsC) by José Ferr... | 11 | https://mathoverflow.net/users/11260 | 240277 | 110,775 |
https://mathoverflow.net/questions/240148 | 6 | $b^2$ swimmers are to be put into one of the teams $1,2,\dots,b$. A team $i$ has a value function $f\_i$, so that if they get swimmer $k$, they get value $f\_i(k)$. The value $f\_i(k)$ is randomized uniformly from $[0,1]$, independently of this value for other $i,k$. (So, there are $b^3$ different values in total.) The... | https://mathoverflow.net/users/92373 | Roughly equal number of swimmers in teams | This is not exactly a solution, but what seems a good way to approach the problem.
1) Consider for each player $k$ the team $I(k)$ to which he brings a maximal value
$$
v(k)=\max\_i f\_i(k) = f\_{I(k)}(k).
$$
BTW: note, that most probably for most of the players $v(k)$ is quite close to 1, as it is a maximum of $b\g... | 2 | https://mathoverflow.net/users/31371 | 240287 | 110,779 |
https://mathoverflow.net/questions/235726 | 13 | Consider the following series:
$$
S = \sum\_{\text{odd } n} \sum\_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})}
$$
From the physical context, one can argue that the series should converge to
$$
S = - \frac{\pi^2}{96},
$$
and numerically calculating the fir... | https://mathoverflow.net/users/nan | Summation of series involving $\sinh$ of a square root | Here is a solution that I have found while working on other lattice sums.
It utilizes a very simple result:
>
> Define $f$ by
> $$f(x)=\sum\_{n=0}^{\infty} (-1)^n (2n+1) e^{-\pi x (n+\frac12)^2}.$$
> Then
> $$\int\_0^{\infty} e^{-y x} f(x)dx=\operatorname{sech}\sqrt{\pi y}.\tag{$\star$}$$
>
>
>
The proof ... | 13 | https://mathoverflow.net/users/92444 | 240291 | 110,780 |
https://mathoverflow.net/questions/240297 | 2 | Let $M$ be an n-dimensional manifold endowed with a Riemannian metric. Suppose we have a coordinate chart, say $(U,\varphi)$ where $U\subset M$ and $\varphi\colon U\rightarrow {\mathbb R}^n$, and let $\partial\_1,\ldots\partial\_n$ denote the corresponding coordinate vector fields. Suppose $\partial\_n$ is bounded.
Is... | https://mathoverflow.net/users/80435 | Coordinates on Riemannian manifolds | Not always.
Let $M = \mathbb{R}^2 \setminus \mathbb{R}^+$ with the induced metric (plane with a slit). Let $U = M$ and $\varphi$ be the polar coordinate map $\varphi = (r,\theta)$. Let $r = x^1$ and $\theta = x^2$.
Your condition 1 requires that the level sets of $x^2$ in the primed coordinate system be the same a... | 3 | https://mathoverflow.net/users/3948 | 240304 | 110,783 |
https://mathoverflow.net/questions/240282 | 5 | Let $X$ be a separable completely metrizable space, let $\mathscr{B}(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathscr{P}(X)$ denote the space of all probability measures on $(X, \mathscr{B}(X))$.
Let $\tau$ denote the topology of setwise convergence on $\mathscr{P}(X)$, i.e. the smallest topology on $\... | https://mathoverflow.net/users/49623 | Metrizability of the space of probability measures endowed with the topology of setwise convergence | It's not metrizable.
The simple argument given in [this answer](https://mathoverflow.net/questions/207103/topology-of-setwise-convergence-of-measures) shows that the finitely supported measures are dense in $\tau$.
However, assuming $X$ is uncountable, they are not sequentially dense. Let $\mu\_n$ be any sequence ... | 5 | https://mathoverflow.net/users/4832 | 240309 | 110,785 |
https://mathoverflow.net/questions/240296 | 1 | Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX\_t \ = \ \sigma(X\_t) dB\_t \ + \ b(X\_t) dt$$
(or given by its generator). Is there any way to write down the transition kernels for the process $X\_t$ from the above SDE? A... | https://mathoverflow.net/users/76335 | Computing transition operators for Markov processes | **1) Explicit expressions for transition densities**
In the case of linear systems with additive noise of the form
$d X(t) = (A X\_t + b) \, d t + \sigma \, d B\_t$
it is possible to obtain an explicit solution, see
*Ioannis Karatzas and Steven E. Shreve*, MR 1241411 [**Brownian motion and stochastic calculus**... | 2 | https://mathoverflow.net/users/22157 | 240310 | 110,786 |
https://mathoverflow.net/questions/240317 | 11 | Let $\gamma$ be an $n$-dimensional complex representation of a finite group $G$ with character $\chi$ and let $e=c\_0, c\_1, ..., c\_{\ell}$ be a set of conjugacy class representatives for $G$. In the case where $\gamma$ is faithful, I recently obtained the formula $$ \frac{1}{|G|} \prod\_{i=1}^{\ell} (n-\chi(c\_i))$$ ... | https://mathoverflow.net/users/33089 | Why is this character expression an integer? | Yes, this is an observation of H. Blichfeldt ( I think J-P. Serre also attributes it to Minkowski), which has been rediscovered many times over the years. Letting $1$ denote the trivial character, note that
$\theta = \prod\_{i = 1}^{\ell} ( \chi - \chi(c\_{i})1) $ is an algebraic integer combination of characters of $... | 26 | https://mathoverflow.net/users/14450 | 240328 | 110,789 |
https://mathoverflow.net/questions/240341 | 3 | Call a topological space $\langle X,\mathscr{O}\rangle$ **regular** iff it is both $T\_0$ and $T\_3$: for every point $x\notin A$, where $A$ is a closed subsets of $X$, there are open and disjoint sets $V$ and $U$ such that $x\in V$ and $A\subseteq U$.
A $\langle X,\mathscr{O}\rangle$ space is **linear-based** iff in... | https://mathoverflow.net/users/22019 | An example of a regular but not linear-based topological space | An uncountable product of copies of the unit interval or of the discrete two element space. Indeed none of the points in this space has a linearly ordered local basis.
Let us have a look at the latter. Let $X:= \prod\_I \{0,1\}$, let $0\in X$ be the sequence which is zero everywhere and let $S\_i\subset X$ be the sub... | 6 | https://mathoverflow.net/users/3969 | 240345 | 110,795 |
https://mathoverflow.net/questions/240337 | 32 | Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum\_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It is easy to see that for cyclic groups, we have that $\nu(G) = \sigma\_1(|G|)$. What is significantly more surprising i... | https://mathoverflow.net/users/1703 | Is there a nice explanation for this curious fact about cyclic subgroups? | (Essentially the same answer as Neil Strickland's:) Since a cyclic group of order $n$ has $\varphi(n)$ generators, your sum equals
$$ \DeclareMathOperator{\ord}{ord}
\nu(G) = \sum\_{g\in G} \frac{ \ord(g) }{ \varphi(\ord(g)) } .
$$
For elements $g$ of $p$-power order, we get $\ord(g)/ \varphi( \ord(g) ) = p/(p-1)$ and... | 30 | https://mathoverflow.net/users/10266 | 240348 | 110,796 |
https://mathoverflow.net/questions/240261 | 6 | Consider two quadratic algebras based on the vector space $\mathbb{R}^3$ with basis $x,y,z$. The antisymmetric tensor algebra $\Lambda \mathbb{R}^3$ obviously has zero divisors, e.g. $(x)(xy)=0$, but the symmetric tensor algebra $S \mathbb{R}^3$ has not, as it is isomorphic to the polynomial functions on $x,y,z$. For $... | https://mathoverflow.net/users/29625 | When does a quadratic algebra have zero divisors? | Partial answer: when $V$ is 4-dimensional there is a commutative counter-example. In other words, there is a subset of relations $R \subseteq V \otimes V$ containing the commutators and such that the associated quadratic algebra has zero divisors but not in degree 1.
Let $V$ be spanned by $W,X,Y,Z$, and let $R \subse... | 2 | https://mathoverflow.net/users/37368 | 240356 | 110,799 |
https://mathoverflow.net/questions/240352 | 3 | Lie $g$ be a finite dimensional complex simple Lie algebra and $U\_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\lambda)$, where $\lambda$ is a dominant weight. Every simple finite dimensional $U\_q(g)$-module is also of the form $... | https://mathoverflow.net/users/11877 | Crystal basis for quantum groups and Lie algebras | (1) Yes, the crystal basis for $V(\lambda)$ is the specialization of the $q$-crystal basis at $q=1$. In fact, that's the only way to construct it.
(2) Again, the crystal graphs coincide by definition.
| 7 | https://mathoverflow.net/users/89948 | 240362 | 110,802 |
https://mathoverflow.net/questions/240191 | 0 | If $K$ is a number field, $E$ an elliptic curve and $p$ a prime, does the Selmer group
$$H^{1}\_{\operatorname{Sel}}\left(K,E\_{p^{n}}\right)$$
always inject into
$$\prod\_{q \text{ a nonarchimedean prime of }K}H^1 \left(K\_{q},E\_{p^{n}}\right)$$
(Excuse me if that question is stupid)
| https://mathoverflow.net/users/70751 | Is $H^{1}_{Sel}\left(K,E_{p^{n}}\right)\rightarrow\prod_{q \nmid \infty} H^1\left(K_{q},E_{p^{n}}\right)$ an injection? | The answer is "no" in general.
By the definition of the Selmer group, you can replace the target of the map by the product of $E(K\_q)/p^n E(K\_q)$. Now $E(K)/p^n E(K)$ is a subgroup of the Selmer group. So if your map were always injective, this would imply that a global point in $E(K)$ is divisible by $p^n$ if and... | 3 | https://mathoverflow.net/users/5015 | 240369 | 110,806 |
https://mathoverflow.net/questions/240227 | 7 | Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ is parallel to $\eta$ on $C$). Then one can easily show that $h(t)=u(f(t))$ is constant for every $C^1$ curve lying o... | https://mathoverflow.net/users/42326 | Level sets of weakly differentiable funtions | Some thoughts on your problem. If $u$ is so smooth that $Du$ has a well defined trace, e.g., $u\in H^s$ for $s>3/2$, then you could flatten the surface by a $C^1$ change of variables and reduce to the case $S=\{x\_1=0\}$. Then the assumption on $Du$ becomes $Du=(D\_1u,0,...,0)$ which means the derivatives of the trace ... | 0 | https://mathoverflow.net/users/7294 | 240375 | 110,808 |
https://mathoverflow.net/questions/17697 | 22 | The connectedness of the moduli space $\mathcal{M}\_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the line, which in turn can be shown to be connected by proving the transitivity of the the natural action of the braid gro... | https://mathoverflow.net/users/431 | A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$ | I was able to find the resources online (6 years after this question was asked):
* 1871 <https://eudml.org/doc/156527> Lüroth - 4 pages
* 1873 <https://eudml.org/doc/156610> Clebsch - 16 pages
* 1891 <https://eudml.org/doc/157563> Hurwitz - 61 pages
I think it's pretty clear Luroth was first, but Hurwitz developed ... | 8 | https://mathoverflow.net/users/1358 | 240377 | 110,809 |
https://mathoverflow.net/questions/240292 | 7 | From [Wikipedia](https://en.wikipedia.org/wiki/Abelian_group#Historical_remarks):
"**Abelian groups** were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using ... | https://mathoverflow.net/users/18583 | Historical reference request on Nilpotent groups | In 1870, the American mathematician, **Benjamin Pierce** first introduced the term ***nilpotent*** in the context of his work on the classification of Algebras. In Algebra, an element $x$ of a ring $R$ is said to be nilpotent if there exists some positive integer $n$ such that $x^{n}=0$.
In group theory, a nilpotent ... | 7 | https://mathoverflow.net/users/92488 | 240384 | 110,812 |
https://mathoverflow.net/questions/240383 | 4 | The [contact process](https://en.wikipedia.org/wiki/Contact_process_(mathematics)) is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here.
The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at each site $x\in\mathbb Z$ the dynamics are: $0\rightarr... | https://mathoverflow.net/users/68851 | What is the stationary distribution for the contact process on the half line? | For the first part of your question, you can do this by coupling:
Define two processes: the first is the contact process on $\{0,1,2,\ldots\}$
and the second is the contact process on $\{1,2,3,\ldots\}$ with the same parameters. Call the processes $\eta$ and $\eta'$ respectively.
They can be monotonically coupled: th... | 5 | https://mathoverflow.net/users/11054 | 240386 | 110,813 |
https://mathoverflow.net/questions/240388 | 9 | Given a prime number $p$ and a positive integer $k$. Consider integer-valued polynomials $f$ satisfying the property that $p | f(x) \Leftrightarrow p^k | x$.
**Question.** What is the smallest degree of such $f$?
I can construct such an $f\_k$ with degree $2p^{k-1} - 1$. The idea is to use induction with $f\_1 = ... | https://mathoverflow.net/users/90626 | $p | f(x)$ if and only if $p^k | x$. | Your construction is optimal, that is, the degree of $f$ is always at least $2p^{k-1}-1$. Note that if $m,x$ have $p$-base expansions $m=\sum\_{i}m\_i p^i$, $x=\sum\_{i}x\_i p^i$, we have $\binom{x}{m}\equiv \prod \binom{x\_i}{m\_i} \pmod p$ by [Lucas' theorem](https://en.wikipedia.org/wiki/Lucas%27_theorem), and any i... | 5 | https://mathoverflow.net/users/4312 | 240399 | 110,817 |
https://mathoverflow.net/questions/240145 | 3 | I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is maximized. Namely
$max\_{\pi,c}\;\mathbb{E}\bigg[\,U\_{1}\big(\int\_{0}^{T}c(t)\,dt\big) + U\_{2}\big(V^{\pi,c}(T)\big)\;\b... | https://mathoverflow.net/users/66399 | Optimal control / Portoflio optimization: Maximize expected utility of total consumption | The problem as stated has no solution except in the special case where it is optimal not to consume at all.
To see this, note that the payoff depends on the consumption rate process $c(t)$ only through the integral $\int\_0^T c(s) \, ds$. Assume that there exists some optimal solution $(\pi^\*, c^\*)$ with $c^\* \neq... | 2 | https://mathoverflow.net/users/20026 | 240403 | 110,819 |
https://mathoverflow.net/questions/240351 | 2 | This post is a *relative version* of [General bound for the number of subgroups of a finite group](https://mathoverflow.net/q/132675/34538)
Let $[H,G]$ be a interval of finite groups with $|G:H| = n$.
*Question:* What is a good upper-bound of $|[H,G]|$, as a function of $n$?
If $H=\{ e\}$, then the best possible... | https://mathoverflow.net/users/34538 | A good upper-bound for the cardinal of an interval of finite groups | A chain in $[H,G]$ has length $\leq\Omega(|G:H|)$, where $\Omega$ denotes the number of prime factors counted with multiplicity. If $H=H\_0<H\_1<\dots<H\_k=G$ is a maximal chain, then there are elements $g\_1, \ldots, g\_k$, such that $H\_i=\langle H\_{i-1}, g\_i\rangle$. If $g\_iH=g\_i'H$, then $\langle g\_i H\_{i-1}\... | 2 | https://mathoverflow.net/users/37555 | 240415 | 110,823 |
https://mathoverflow.net/questions/240176 | 1 | I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem.
For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$.
I have the following equality
$$\int\limits\_0^\infty g(\kappa, x\_0) \exp{\left[-\frac{\left(\xi +
\k... | https://mathoverflow.net/users/18929 | Is there a way to solve this integral equation? | There is no unique solution to the integral equation
$$\int\limits\_0^\infty g(\kappa, x\_0) \exp{\left[-\frac{\left(\xi +
\kappa \right)^2}{2\alpha\theta}\right]} d\kappa %
= \exp{\left[-\frac{\left(\xi - x\_0\right)^2}{2\alpha\theta}\right]},\;\;\xi=(\alpha\theta)^{1/\alpha}.$$
Any function of the form
$$g(\kappa,... | 2 | https://mathoverflow.net/users/11260 | 240423 | 110,826 |
https://mathoverflow.net/questions/240378 | 7 | Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (universal) bundle over $G$.
On Kapranov's "Coherent sheaves on Grasmann manifold" the following result are stated:
$H... | https://mathoverflow.net/users/91935 | Global section of universal bundle on Grassmanian | These are simple instances of the Bott-Borel-Weil theorem. For a complex semsimple group $G$ and a parabolic subgroup $P$ and a complex irreducible representation $W$ of $P$ consider the homogeneous vector bundle $G\times\_P W\to G/P$. In this situation the BBW theorem computes the cohomology of the shaef of local holo... | 5 | https://mathoverflow.net/users/64141 | 240424 | 110,827 |
https://mathoverflow.net/questions/240431 | 0 | A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any subgroup $K$ and $L$ such that $H\leq K \unlhd L$ implies that $L \leq N\_L(H)K$.
A subgroup is $H$ is pronormal in $G$ if for each $g \in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x = H^g$.
A theorem characterising pronorm... | https://mathoverflow.net/users/92488 | Relation between the Frattini property and pronormal subgroups of solvable groups | Personally I would try to persuade someone who does have access to the paper to e-mail it to me, but that might not be legal, so I shouldn't have said it.
I have not thought this out in detail, but I think the following approach will work.
Use induction on $|G|$. Let $N$ be a minimal normal subgroup of $G$, so $N$ ... | 2 | https://mathoverflow.net/users/35840 | 240433 | 110,831 |
https://mathoverflow.net/questions/240075 | 12 | I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am currently learning commutative algebra from Atiyah and Macdonald's engaging book on commutative algebra.
I would really like t... | https://mathoverflow.net/users/nan | Learning roadmap for algebraic number theory | Two good books for an introduction to global algebraic number theory (i.e., number fields) are:
Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger (Dover Books on Mathematics) May 19, 2008 by Pierre Samuel (less than $8 in paperback)
A Classical Introduction to Modern Number Theory (Gradu... | 4 | https://mathoverflow.net/users/11926 | 240436 | 110,832 |
https://mathoverflow.net/questions/240316 | 4 | I was wondering whether the Schwartz functions are also dense in
$$\{f \in L^2(\mathbb{R}^n); \int\_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int\_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$
where the norm is given by $$||f||\_{L^2}^2 = \int (1+|x|^2) |f(x)|^2 dx + \int (1+|\xi|^2) |\hat{f}(\xi)|^2 d \xi.$$
... | https://mathoverflow.net/users/92454 | Density argument with Schwartz functions? | We can do this by approximating the derivative of such an $f$ in $L^2$, as follows: Given $\epsilon>0$, take $L>0$ so large that $\int\_{|x|>L}(x^2|f|^2+|f'|^2)<\epsilon$.
Now approximate $f'$ in $L^2(-L,L)$ by an $h\in C\_0^{\infty}(-L,L)$, and put
$$
g(x)=f(-L)+\int\_{-L}^x h(t)\, dt .
$$
Finally, we modify $g$ on ... | 3 | https://mathoverflow.net/users/48839 | 240446 | 110,838 |
https://mathoverflow.net/questions/240454 | 8 | A pair of groups $(G,H)$ is called a symmetric pair if $H$ is the group of fixed points of an involutive automorphism of $G$, for example $(GL(2n,\mathbb{F}\_q),Sp(2n,\mathbb{F\_q}))$ is a symmetric pair with respect to the involution $\theta(A)=J(A^{-1})^tJ^{-1}$ where:
$$
J=\left(\begin{array}{cc} 0&id\_n\\-id\_n&0... | https://mathoverflow.net/users/92513 | Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair? | The symmetric space ${\rm GL}(2,{\mathbb F}\_q)/T$, where $T$ is the diagonal torus, is not a Gelfand pair : the Steinberg representation contains the trivial character of $T$ with multiplicity $2$.
Here is a proof that the multiplicity of this character is $\geqslant 2$. The Steinberg representation may be viewed a... | 9 | https://mathoverflow.net/users/4767 | 240459 | 110,843 |
https://mathoverflow.net/questions/185186 | 5 | In Fedosov's version of quantisation of functions on a symplectic manifold, the product is given in terms of a symplectic connection. I have looked through Fedosov's book in deformation quantisation, and can't find the second order term for the product of two functions (which will involve the curvature), though I can f... | https://mathoverflow.net/users/29625 | Second-order term of the Fedosov quantised product | Indeed neither Fedosov's book nor his original paper (*A Simple Geometrical Construction of Deformation Quantization*, J. Diff. Geom. **40** (1993) 213-238) have an explicit formula for the second order term of his star product. To my knowledge, the first place where the recursive formulas for the terms in Fedosov's st... | 2 | https://mathoverflow.net/users/11211 | 240461 | 110,844 |
https://mathoverflow.net/questions/240442 | 1 | According to Loday and Prashvili's paper, to defined universal enveloping algebra of Leibniz algebras we need $g^{r}$ and $g^{l}$ as two copies of the Leibniz algebra $g$. What does it mean by copies of $g$?
Another question is that, we know that the opposite algebra of left Leibniz algebra is right Leibniz algebra,... | https://mathoverflow.net/users/40491 | universal enveloping algebra of Leibniz algebra | For reference the paper mentioned in the question is [here](http://www-irma.u-strasbg.fr/~loday/PAPERS/93LodayPira(Leibniz).pdf). As Qfwfq points out in the comments we are taking two Leibniz algebras $\mathfrak g^l$ and $\mathfrak g^r$ both isomorphic to our original Leibniz algebra $\mathfrak g$. Note the superscript... | 2 | https://mathoverflow.net/users/51668 | 241472 | 110,849 |
https://mathoverflow.net/questions/241476 | 3 | According to [Wikipedia](https://en.wikipedia.org/wiki/Transitive_reduction#CITEREFAhoGareyUllman1972), *"If a given graph is a finite directed acyclic graph, its transitive reduction is unique"*
Here is what I think might be a counter-example:
Imagine a diamond-shaped DAG where
A -> B, A -> C, B -> D, C -> D
I ... | https://mathoverflow.net/users/91445 | Is transitive reduction for a direct acyclic graph really unique? | In the first reduction you mention, you are removing C -> D, and so your subgraph does not mention that D is reachable from C. So your subgraph is not a transitive reduction. As the Wikipedia article mentions, the transitively reduced digraph must have the same reachability relation as the given relation. The original ... | 3 | https://mathoverflow.net/users/88774 | 241478 | 110,851 |
https://mathoverflow.net/questions/238651 | 10 | Let $K$ be a commutative field and ${\rm M}\_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a\_{ij})\_{1\leqslant i,j\leqslant n}\in {\rm M}\_n (K)$, define: $A^{[k]} =(a\_{ij}^k )\_{1\leqslant i,j\leqslant n}$.
>
> Is the desc... | https://mathoverflow.net/users/4767 | Coefficient-wise powers of matrices. Reference wanted | This is problem 50.4 by Moubinool Omarjee from [volume 50](http://www.ilasic.org/IMAGE/IMAGES/image50.pdf) of The Bulletin of the International Linear Algebra Society, with solutions in [volume 51](http://www.ilasic.org/IMAGE/IMAGES/image51.pdf) by Eugene Herman and Bojan Kuzma, and further work by Roman Drnovsek in [W... | 17 | https://mathoverflow.net/users/11260 | 241487 | 110,852 |
https://mathoverflow.net/questions/241490 | 1 | I am not sure my question is research type, but I am sure I can find here an answer.
So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:
>
> **Theorem 4** (Characterization of $W^{1,\infty}$). Let $U$ be open and bounded, with $\partial U$ of class $C^1$. Then $u: U\t... | https://mathoverflow.net/users/13904 | Lipschitz functions and $W^{1,\infty}$ | You are right. This theorem and the extension theorem (the step you are referring to in the proof) holds for domains with Lipschitz boundary. You can find a proof of this in Evans and Gariepy's book on measure theory.
| 4 | https://mathoverflow.net/users/35800 | 241492 | 110,855 |
https://mathoverflow.net/questions/240407 | 3 | Let $f:X \to Y$ be a surjective, smooth projective morphism of noetherian schemes. Let $\mathcal{L}$ be an inverible sheaf on $X$ satisfying $f\_\*\mathcal{L}$ is locally free and $s \in H^0(\mathcal{L})$ be a global section of $\mathcal{L}$. Is the zero locus of $s$ (in $X$), flat over $Y$?
N.B. If necessary, one ca... | https://mathoverflow.net/users/43198 | Is the zero locus of a global section flat? | You already have trivial counterexamples for your statement, but perhaps you were thinking of a section whose zero locus is irreducible and dominates $Y$. It is false even with that additional assumption:
Let $Y$ be an arbitrary non-singular surface and $Z$ the blow-up up of a (closed) point on $Y$. Then by construct... | 7 | https://mathoverflow.net/users/10076 | 241496 | 110,856 |
https://mathoverflow.net/questions/241495 | 13 | Let $M$ be a smooth even-dimensional manifold.
1. Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S\_J$ associated to $J$ ?
(I've read this somewhere but I didn't see the actual construction).
Is there a way to caracterize that spin$^c$ structure ? (What I ... | https://mathoverflow.net/users/89425 | Spin^c structures on manifolds with almost complex structure | Assume that $M$ is oriented throughout. Recall that $M$ has a $\text{Spin}^c$ structure iff the third integral Stiefel-Whitney class $\beta w\_2 = W\_3 \in H^3(M, \mathbb{Z})$ is trivial. Actually more is true: $\text{Spin}^c$ structures on $M$ are in bijection with trivializations of $W\_3$, which are a torsor over $H... | 13 | https://mathoverflow.net/users/290 | 241500 | 110,858 |
https://mathoverflow.net/questions/240472 | 8 | Davis, Figiel, Johnson and Pełczyński's Factorization Theorem states that each weakly compact operator $T:X \to Y$ between Banach spaces $X$ and $Y$ factors through a reflexive Banach space $Z$. In addition, they prove that $Z$ can be constructed to have a Schauder basis if $Y$ has a shrinking basis or an unconditional... | https://mathoverflow.net/users/15388 | Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases | I think Q1 has a positive answer. To see this, review how the results are proved in DFJP. You have a weakly compact subset $W$ of $Y$ (the closure of the image of the unit ball of $X$ under a weakly compact operator) and you apply the factorization technique (which we now know is just a real interpolation method applie... | 5 | https://mathoverflow.net/users/2554 | 241512 | 110,863 |
https://mathoverflow.net/questions/241521 | 2 | Suppose $a\_j \in \mathbb{R}$, $b\_j \ge 0$, and $0 = t\_0 < t\_1 < \ldots < t\_J$ are time points. Let $W\_t$ be a standard Brownian motion. Is it possible to further simplify the expression
\begin{align\*}
\mathbb{P}\left(\bigcap\_{j=0}^{J-1}\left\{\sup\_{t \in [t\_j, t\_{j+1})}\left\{\left|W\_t + a\_j\right| - b\_j\... | https://mathoverflow.net/users/83593 | Probability Brownian motion lies between $2$ functions | This is the problem of Brownian motion between two *moving* absorbing boundaries. For a linear time dependence some analytical progress can be made, but for arbitrary time dependence no closed-form solution exists. Here are some pointers to the literature:
1. C. Park and F.J. Schuurmann, Evaluation of barrier-crossin... | 11 | https://mathoverflow.net/users/11260 | 241523 | 110,867 |
https://mathoverflow.net/questions/239592 | 3 | In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows:
$$E(f):=\frac{1}{2}\int\_M\|df\|^2dvol\_g\qquad f:(M,g)\to(N,h).$$
In many Books such as *Calculus of Variations and Harmonic Maps-Hajime Urakawa*, used the covariant derivation as a map $\Gamma(f^{-1}TN... | https://mathoverflow.net/users/90655 | Question on Harmonic maps between Riemannian manifolds | the reason is the directional of derivatives allow us to multiply a tensor field and then we can introduce orthogonal coordinate by the free tensors and use the harmonic maps of covariant derivation (the levi-civita connection type) to study manifolds (in differential geometry)!
see:
<https://en.wikipedia.org/wiki/... | 0 | https://mathoverflow.net/users/nan | 241528 | 110,868 |
https://mathoverflow.net/questions/240473 | 5 | In this question [Operators on Hilbert $C^\*$-module and families of Fredholm operators](https://mathoverflow.net/questions/239978/operators-on-hilbert-c-module-and-families-of-fredholm-operators) I asked about the relation between being a family of compact operators $F:X \to K(H)$ on Hilbert space $H=\ell^2$ and being... | https://mathoverflow.net/users/24078 | $C(X)$-compact operators and families of compact operators | The Hilbert module $H\_{C(X)}$ is the space of all continuous functions $X \to H$, and hence for each $s \in H\_{C(X)}$, we can talk about $s(x) \in H$, for each $x \in X$.
Assume that $F:H\_{C(X)} \to H\_{C(X)}$ is $C(X)$-linear. We get individual operators $F\_x: H \to H$, namely
$$F\_x (v):= (F\tilde{v})(x), $$
whe... | 3 | https://mathoverflow.net/users/9928 | 241531 | 110,869 |
https://mathoverflow.net/questions/241535 | 11 | Let $k$ be a perfect field.
Recall that an algebraic torus $T$ over $k$ is called *quasi-split* if there exists some finite étale $k$-algebra $A$ such that
$$T \cong \mathrm{R}\_{A/k} \mathbb{G}\_m.$$
A reductive group $G$ over $k$ is called *quasi-split* if it contains a Borel subgroup $B$ over $k$.
A priori, I ... | https://mathoverflow.net/users/5101 | Quasi-split tori and algebraic groups | A torus $T$ is quasi-split if its character group is a permutation representation for the Galois group. So a counterexample to your question is: let $G$ be the quasi-split group $SO(n+1,n-1)$, $n\ge2$, $k=\mathbb R$. The weight lattice is $\mathbb Z^n$ with Galois action
$$
(x\_1,\ldots,x\_{n-1},x\_n)\mapsto(x\_1,\ldot... | 13 | https://mathoverflow.net/users/89948 | 241537 | 110,870 |
https://mathoverflow.net/questions/241498 | 4 | Consider the category of finite-dimensional representations for the algebraic group $\mathrm{SL}(n)$ in characteristic $p$. I know very little about this but am told there is a highest weight category here with indecomposable tilting modules $T(\lambda)$ which control the category in some way. These have a filtration w... | https://mathoverflow.net/users/57454 | Tilting modules in positive characteristic | For the algebraic group $\mathrm{SL}(n)$ over a field of characteristic $p>0$ the (indecomposable) tilting modules are the indecomposable direct summands of
tensor products of tensor powers of the fundamental representations.
The fundamental representations themselves are the exterior powers of the
defining representa... | 4 | https://mathoverflow.net/users/4794 | 241540 | 110,871 |
https://mathoverflow.net/questions/240346 | 16 | The principal curvatures of a surface is denoted by $\kappa\_{1}, \kappa\_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa\_{1}, \kappa\_{2})$ is an intrinsically invariant quantity of all surfaces in $\mathbb{R}^{3}$(It is invariant under isometries of surfaces).
>
> Is it true to... | https://mathoverflow.net/users/36688 | Are there some intrinsic invariants of surfaces other than Gaussian curvature? | As others have pointed out, it's not hard to show that any function $F(\kappa\_1,\kappa\_2)$ that is intrinsic to the surface metric must be a function of $K = \kappa\_1\kappa\_2$, so that settles what one might call the 'lowest-order' case. However, there are certainly higher-order versions. For example, the expressio... | 15 | https://mathoverflow.net/users/13972 | 241544 | 110,873 |
https://mathoverflow.net/questions/241534 | 2 | I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant.
I want to know the exact number of non-negative integer solutions.
Also I want to know what are those solutions. How can I find them?
| https://mathoverflow.net/users/93562 | Non-negative integer solutions of x^2+y^3=n | As mentioned in the comments this is essentially the classic problem of finding integer points of the Mordell curve, and a lot of work has gone into it (for example towards bounding the number of solutions, see [this paper](http://arxiv.org/abs/1405.7022)).
If you want to get understand the basics of the process of f... | 3 | https://mathoverflow.net/users/43108 | 241546 | 110,874 |
https://mathoverflow.net/questions/241545 | 1 | I am coming across a paper ( Proposition $1.1$ from <http://www.sciencedirect.com/science/article/pii/0304414987901840> ) that claims the following fact which I don't understand why:
>
> On a probability space $(\Omega, \mathcal{F} , \mathbb{P})$ with a Brownian motion $(B\_t)\_{t \in [0,T]}$, let $(C\_t)\_{t \in [... | https://mathoverflow.net/users/62049 | Girsanov theorem and the density of a process | Short answer: it follows directly from the Radon-Nikodym theorem.
Longer answer: Let $\mu$ denote the law of $Y(t)$ under $\mathbb P\_{y\_0}$ and let $\nu$ denote the law of $Y(t)$ under $\mathbb Q\_{y\_0}$ for some initial condition $y\_0$.
From your application of Girsanov theorem, $\nu(A) = 0$ implies $\mu(A) = 0$... | 3 | https://mathoverflow.net/users/22157 | 241550 | 110,876 |
https://mathoverflow.net/questions/241554 | 9 | Let $\pi \colon X \to T$ be a flat projective morphism, and let $Y$ be a closed sub-scheme of $X$ which is flat over $T$. We can assume that everything is defined over the complex numbers, and $T$ is one dimensional, say smooth and affine.
I want to consider the blow-up $\eta \colon X' \to X$ with centre $Y$. I have ... | https://mathoverflow.net/users/48866 | Blow-up in family | No, that is not true. First the positive result. If you add the hypothesis that $X$ is integral, then $X'$ is integral (Proposition II.7.16, p. 166 of Hartshorne). Thus $X'$ is flat over $T$ (Proposition III.9.7, p. 257 of Hartshorne). Even in this case, typically $(X')\_t$ does not equal the blowing up of $X\_t$ at $Y... | 16 | https://mathoverflow.net/users/13265 | 241560 | 110,878 |
https://mathoverflow.net/questions/241562 | 6 | Thanks to [Choi-Schoen theorem](http://link.springer.com/article/10.1007%2FBF01388577), we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we replace $S^3$ by $S^n$ for $n\geq3$, still for surfaces (not hype... | https://mathoverflow.net/users/9253 | minimal surfaces in $S^n$ | Without embeddedness, the Choi--Schoen theorem is false.
For example, there is a huge family of rotationally symmetric immersed tori in $\mathbb{S}^3$ (the only embedded one is the Clifford torus, by Brendle's [solution](http://download.springer.com/static/pdf/626/art%253A10.1007%252Fs11511-013-0101-2.pdf?originUrl=... | 9 | https://mathoverflow.net/users/1540 | 241570 | 110,882 |
https://mathoverflow.net/questions/241579 | 14 | Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded coalgebra over $k$; write $\Omega BA$ for its cobar construction. There is a natural dga map $\Omega BA \to A$ which uses the fac... | https://mathoverflow.net/users/4649 | When is bar-cobar duality an equivalence? | What the references are saying is correct, and you are right. Yes, $\Omega BA \to A$ is always a quasi-isomorphism. No, $\Omega$ does not in general take quasi-isomorphisms to quasi-isomorphisms.
A sufficient condition for $\Omega$ transforming a DG-coalgebra morphism to a quasi-isomorphism of DG-algebras is a filter... | 17 | https://mathoverflow.net/users/2106 | 241582 | 110,886 |
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