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https://mathoverflow.net/questions/226639 | -1 | Given a quasi-ordered set $(Q,\leq)$ the *interval topology* on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq x\}$ and $\uparrow x = \{y\in Q: y\geq x\}$.
Let $\mathbb{N}^\mathbb{N}$ denote the set of all functions $f:\mathbb{... | https://mathoverflow.net/users/8628 | Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$ | In view of the answer to [this question](https://mathoverflow.net/questions/226778/is-the-interval-topology-of-mathbbn-mathbbn-leq-connected/226783#226783), the answer is no, for $\tau\_i$ is connected, but $\tau$ is not.
| 1 | https://mathoverflow.net/users/37103 | 226800 | 105,886 |
https://mathoverflow.net/questions/211399 | 3 | In the paper [Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado,](http://arxiv.org/pdf/1410.2200.pdf) page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled configuration space to the section space of a $m$-sphere bundle
$$
\alpha: C(M;S^0)\to \... | https://mathoverflow.net/users/65800 | Configuration spaces of positive and negative particles | Using the homological stability results of Segal from the appendix of his paper "The topology of spaces of rational functions," one can see that the scanning map from $C\_k(M;S^0)$ to $\Gamma\_k(M;S^0)$ induces an isomorphism in homology groups $H\_i$ for $i \leq k/2$.
| 2 | https://mathoverflow.net/users/82566 | 226812 | 105,891 |
https://mathoverflow.net/questions/226804 | 15 | Wolstenholme's theorem is stated as follows:
if $p>3$ is a prime, then
\begin{align\*}
\sum\_{k=1}^{p-1}\frac{1}{k}\equiv 0 \pmod{p^2},\\
\sum\_{k=1}^{p-1}\frac{1}{k^2} \equiv 0 \pmod{p}.
\end{align\*}
It is also not hard to prove that
$$
\sum\_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 0 \pmod{p}.
$$
However, there are some ... | https://mathoverflow.net/users/6104 | Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$ | This goes way back to Emma Lehmer, see her [elementary paper](http://gradelle.educanet2.ch/christian.aebi/.ws_gen/14/Emma_Lehmer_1938.pdf) on Fermat quotients and Bernoulli numbers from 1938.
Assume $p\ge 7$. First, I reformulate your congruence. I replace
$$4\sum\_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum\_{k=1}^{p-... | 20 | https://mathoverflow.net/users/31469 | 226822 | 105,893 |
https://mathoverflow.net/questions/226647 | 5 | The title says it. I am looking for a good exposition on the K-theory of the curves $X\_{i}(N)$, $Y\_{i}(N)$, where $i\in\{0,1\}$.
I have some background in $K$-theory and also some background in modular curves.
| https://mathoverflow.net/users/70751 | Good references for K-theory of modular curves? | I wouldn't recommend Beilinson's 1985 paper as a general reference -- it's terrifyingly compressed, developing an entire new subject in a single short paper, and crashes through the necessary material on modular curves in a couple of sentences.
A much gentler reference would be Flach's 1992 Inventiones paper "A finit... | 7 | https://mathoverflow.net/users/2481 | 226830 | 105,895 |
https://mathoverflow.net/questions/226846 | 0 | I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of matrix $A$. Is the condition number one of spectral properties? What else might influence directly on solving $Ax = b$ vi... | https://mathoverflow.net/users/50309 | Is spectral properties a general term for condition number? | "Spectral properties" in general means "anything which is related to the eigenvalues (spectrum) of the matrix". This includes the condition number, but in general means much more.
The convergence of iterative methods is influenced deeply by the location of the eigenvalues of the system matrix, not only through the co... | 4 | https://mathoverflow.net/users/1898 | 226849 | 105,901 |
https://mathoverflow.net/questions/226854 | 4 | Given a conic $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ with integers and random coefficients, what is more probable? To find a rational point on the conic or not?
| https://mathoverflow.net/users/84475 | Conics, rational points and probability | For any given prime number $p > 2$ the probability that there is no $p$-adic point is $\ge c/p$ for some constant $c > 0$ indpendent of $p$. Since the sum over $1/p$ diverges, this implies that the `probability' (or rather, density) of conics with a rational point is zero.
EDIT: I assume that the `probability' is mea... | 11 | https://mathoverflow.net/users/21146 | 226857 | 105,903 |
https://mathoverflow.net/questions/226859 | 1 | Is there a commonly-accepted umbrella term for infinite-dimensional calculus problems where the goal is to compute an optimal geometric path between a pair of points? Three examples of this would be the [Brachistochrone problem](http://mathworld.wolfram.com/BrachistochroneProblem.html), the [Isoperimetric problem](http... | https://mathoverflow.net/users/70190 | The term for problems "like" Brachistocrone? | "*Calculus of variations*" seems an accepted umbrella term; at least, looking at the corresponding [Wikipedia entry](https://en.wikipedia.org/wiki/Calculus_of_variations), you'll recognize that most problems in this class are of the type you are looking for:
* The Catenary shape
* The Brachistochrone problem
* Isoper... | 6 | https://mathoverflow.net/users/11260 | 226861 | 105,905 |
https://mathoverflow.net/questions/226829 | 2 | Please refer to [this](http://snap.stanford.edu/class/cs224w-readings/erdos59random.pdf), it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the question:
>
> Define $N\_c=[\dfrac{1}{2}n\log n+cn]... | https://mathoverflow.net/users/66278 | Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1 | Let $n$ be large enough so that $N\_c=\lfloor\frac{1}{2}n\ln n+cn\rfloor\ge0$ (even when $c<0$). On the other hand, the condition that $s$ is a positive integer such that $s<n-\dfrac{2N\_c}{n}$ yields $N\_c\le M$. Thus, $N\_c\in\{0,\dots,M\}$, and so, the left-hand side of your inequalities is well defined -- assuming,... | 3 | https://mathoverflow.net/users/36721 | 226864 | 105,907 |
https://mathoverflow.net/questions/225583 | 3 | I require the following integral involving the modified Bessel functions of the first and second kinds of order one
$$I(a, b, c) = \int\_0^{\infty} \frac{\sin(ax)}{x} I\_1(bx) K\_1(cx) \mathrm{d}x, \quad\text{where} \quad c \ge b$$
For the case $b=c$, Mathematica gives the result
$$I(a, b) = \frac{1}{m} E(m) + \l... | https://mathoverflow.net/users/78657 | Definite integral with modified Bessel functions, trigonometric function and a power | One can use Mellin transforms to tackle this integral; in this case one obtains series involving hypergeometric functions. Here is a very brief summary of the process. I plan to complete the answer with more details, and with an asymptotic series for large $a$ soon.
Setting $$f(x) = \frac{\sin(ax)}{x}$$ and $$g(x) = ... | 3 | https://mathoverflow.net/users/8955 | 226882 | 105,913 |
https://mathoverflow.net/questions/226881 | 6 | Let $p\_n$ be the $n^{th}$ prime number. Suppose $E(F\_{p\_n})$ denotes an elliptic curve over the Galois field $GF(p\_n)$ which is defined by $y^2=x^3+ax+b$. Is the below claim true?
For each integer number $n>3$, there exist integer numbers $a$ and $b$ such that $\#E(F\_{p\_n})=p\_{n+1}$?
| https://mathoverflow.net/users/84430 | Elliptic curves and prime numbers | (Sorry, I misread the question at first.) The following result reduces your question to a problem of analytic number theory:
**Theorem** (Hasse-Deuring-Waterhouse): For a prime $p$ and $N \geq 1$ the following are equivalent:
(i) There is an elliptic curve $E\_{/\mathbb{F}\_p}$ such that $\# E(\mathbb{F}\_p) = N$.... | 14 | https://mathoverflow.net/users/1149 | 226883 | 105,914 |
https://mathoverflow.net/questions/226661 | 3 | Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer me to anything new or old?
| https://mathoverflow.net/users/5034 | Maximal size of minimal generating set | The maximum size of minimal generating sets of a finite group is studied recently by Lucchini in the following two papers:
Andrea Lucchini, [The largest size of a minimal generating set of a finite group](http://dx.doi.org/10.1007/s00013-013-0527-y), *Arch. Math. (Basel)* **101**(1) (2013), 1–8.
Andrea Lucchini, [M... | 4 | https://mathoverflow.net/users/40723 | 226886 | 105,915 |
https://mathoverflow.net/questions/226159 | 7 | This is a refinement of [an earlier question](https://mathoverflow.net/questions/226044/how-many-closed-measure-zero-sets-are-needed-to-cover-the-real-line).
This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
For the reader's convenience, I reproduce below the relevant parts... | https://mathoverflow.net/users/2415 | How many closed measure zero sets are needed to cover the real line, really? | Mathias model (i.e., the countable support iteration of length $\omega\_2$ of Mathias poset over a model of CH) satisfies $\mathrm{cov}(\mathcal{E})<\mathfrak{b}$ (recall that both $\mathfrak{d}$ and $\mathfrak{r}$ are above $\mathfrak{b}$). Refeer to Bartoszynski-Judah book Set Theory: *On the structure of the real li... | 4 | https://mathoverflow.net/users/67193 | 226888 | 105,916 |
https://mathoverflow.net/questions/226887 | 13 | My original question (posted in <https://math.stackexchange.com/questions/1584430/can-all-power-sets-be-limit-cardinals>) was:
Is it possible to create a model of ZFC, so that **the cardinality of each power set is a limit cardinal** (as opposed to GCH where they are always successor cardinals)?
Obviously, from Eas... | https://mathoverflow.net/users/59012 | When can Power Sets be Limit Cardinals? | The answer to your question is yes. In the Foreman-Woodin model [The generalized continuum hypothesis can fail everywhere](http://www.jstor.org/stable/2944324?origin=crossref), $2^\kappa$ is weakly inaccessible for all infinite cardinals $\kappa.$ To be more precise, Foreman and Woodin proved the following:
>
> **T... | 13 | https://mathoverflow.net/users/11115 | 226892 | 105,918 |
https://mathoverflow.net/questions/226855 | 2 | While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other subjects: for instance, one can argue that a vector space is a module over a field instead of making a "new" definition fo... | https://mathoverflow.net/users/48745 | Is there a reason for different nomenclature on Calculus of Variations? | Your question actually is quite well answered around page 10 of Giaquinta and Hildebrandt's *Calculus of Variations I: the Lagrangian formalism*. The upshot is that the correct phrase you are looking for is the (possibly nonlinear) Gateaux *differential*, and not the Frechet derivative, and that is for good reason
(wit... | 9 | https://mathoverflow.net/users/3948 | 226898 | 105,921 |
https://mathoverflow.net/questions/226871 | 11 | Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense to form the quotient curve $Y := X/G$, and $Y$ is $k$-smooth because it is normal. Is it true that the pullback of diff... | https://mathoverflow.net/users/63877 | Quotient of a smooth curve by a finite group and differentials | Yes, the formula holds, even when the action is not free. Here is the principle of a proof for the case of a tame action (which covers the characteristic $0$ case). Denote by $\pi:X\to Y=X/G$ the quotient morphism.
First consider the exact sequence of $G$-sheaves on $X$
$$ 0 \to \pi^\* \Omega^1\_{Y/k} \to \Omega^1\... | 7 | https://mathoverflow.net/users/11682 | 226909 | 105,927 |
https://mathoverflow.net/questions/226862 | 1 | Obtaining a non-trivial estimate for $\sum\_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle method (at least for the 'classical' proof I am aware of).
We know by Vaughn's identity that if $|\alpha - a/q| < 1/q^2... | https://mathoverflow.net/users/48408 | Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences | Bruedern, Granville, Perelli, Vaughan and Wooley, (Philos. Trans. Roy. Soc. London Ser. A, 356 (1998) 739 - 761) dealt with the sequence of $k$-free integers. Bruedern (in: Analytic Number Theory, Cambridge University Press 2009, 91-132) generalized this approach to almost periodic sequences, as obtained by converging ... | 2 | https://mathoverflow.net/users/37555 | 226913 | 105,931 |
https://mathoverflow.net/questions/226907 | 3 | A matrix $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{PSL}\_2(\mathbb{C})$
acts isometrically on the upper half-space model $\mathbb{H}^3$
via isometric extension of the Mobius transformation on $\widehat{\mathbb{C}}=\partial\mathbb{H}^3$,
which is defined by $X(z)=\dfrac{az+b}{cz+d}$.
$X$ is called loxodromic if... | https://mathoverflow.net/users/14835 | What does the trace of a loxodromic Mobius transformation tell us about how it rotates? | Complex translation length $\lambda$ is given by $tr X = 2\cosh \lambda,$ where $\Re \lambda > 0,$ and is the translation length, whilst the imaginary part is the rotation angle.
| 2 | https://mathoverflow.net/users/11142 | 226921 | 105,932 |
https://mathoverflow.net/questions/226922 | 2 | Let $B\_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W\_0^{1,2}(B\_0(1))$. For $r\in (0,1)$, define a function $f\_r(x):[0,1]\rightarrow \mathbb R$ by
\begin{equation}
f\_r(x):=
\begin{cases}
1,&\text{when} \ x\in(0,r],\\
\frac{1-x}{1-r},& \text{when} \ x\in(r,1].
\end{cases}
\end{equation}
Let $g\_r(x):B\... | https://mathoverflow.net/users/84068 | Convergence of energy of Sobolev functions near the boundary | Yes, this works. By computing the gradient with the product rule, this boils down to showing that
$$
\frac{1}{\delta^2} \int\_{1-\delta\le |x|\le 1} |h(x)|^2 \to 0
$$
as $\delta\to 0$ for $h\in W^{1,2}\_0$. If $h$ is also smooth, then, since $h(y)=0$ on $|y|=1$,
$$
h(x) = -\frac{x}{|x|}\cdot\int\_{|x|}^1 \nabla h(tx/|x... | 2 | https://mathoverflow.net/users/48839 | 226930 | 105,937 |
https://mathoverflow.net/questions/226939 | 14 | As mentioned in the title of this question, I want to be able to move from an intrinsic viewpoint of vector bundles, to an extrinsic viewpoint. For manifolds, this would be described via the Nash embedding theorem:
$$\text{n-dimensional Riemannian manifold $(M,g)$} \to \text{submanifold of $\mathbb{R}^{2n}$ with canoni... | https://mathoverflow.net/users/69531 | Is there an extrinsic-geometric viewpoint for connections? | The counterpart for connections of the Nash embedding is an easier result called the *[Narasimhan-Ramanan theorem](http://www.jstor.org/stable/2372896?origin=JSTOR-pdf&seq=1#page_scan_tab_contents)*,
>
> M.S. Narasimhan, S. Ramanan: [Existence of universal connections](http://www.jstor.org/stable/2372896?origin=JS... | 20 | https://mathoverflow.net/users/20302 | 226940 | 105,941 |
https://mathoverflow.net/questions/226951 | 4 | Let $\hat{R}\to R$ be a homomorphism of commutative unital rings and let
$\hat{M}$ be an $\hat{R}G$-module for a group $G$. Does the $R$-module isomorphism $$H^n(G,\hat{M}\otimes R)\cong H^n(G,\hat{M})\otimes R$$
hold, where $\hat{M}\otimes R$ is an $RG$-module by extension of scalars?
If true, this must be well kno... | https://mathoverflow.net/users/84521 | Behaviour of cohomology groups under extension of scalars | This isn't even true when $n = 0$. Let me write your morphism of rings as $f : R \to S$ because otherwise I'll keep thinking that $\hat{R}$ denotes some kind of completion. When $n = 0$ you want to know whether
$$\text{Hom}\_{R[G]}(R, M \otimes\_R S) \cong \text{Hom}\_{R[G]}(R, M) \otimes\_R S.$$
In other words, as... | 9 | https://mathoverflow.net/users/290 | 226956 | 105,947 |
https://mathoverflow.net/questions/226957 | 2 | I must be a terrible googling searcher but I cannot find a reference to the following inequality:
$$ \forall\_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$
I have just obtained this, it seems to have nice potential, and now I would appreciate a reference. (As a minimum, if this is new to ... | https://mathoverflow.net/users/8385 | A logarithmic cotangent inequality | Plug $x=\cot(\phi)$ and turn your inequality into
$$ x \ln x < \frac{x^2-1}{2},$$
where $x>1$. The RHS is the second order Taylor approximation of the LHS around $x=1$. Hence the inequality follows from concavity of the derivative of $x \ln x$, which is $1+\ln x$.
This argument has led me to the following elegant sol... | 8 | https://mathoverflow.net/users/31469 | 226960 | 105,950 |
https://mathoverflow.net/questions/226944 | 4 | Let $k$ be an algebraically closed field and let $X$ be a finite type $k$-scheme that is Cohen-Macaulay and equidimensional. Under these assumptions there is a relative dualizing sheaf $\omega\_{X/k}$ that is an $\mathcal{O}\_X$-module.
* Is $\omega\_{X/k}$ Cohen-Macaulay?
* Is $\omega\_{X/k}$ at least (S$\_1$)? (Eq... | https://mathoverflow.net/users/70964 | Is the realtive dualizing sheaf Cohen-Macaulay? | This is an immediate application of the behavior of dualizing complexes relative to finite morphisms (such as closed immersions).
To explain this, first
recall that if $f:Y \rightarrow Z$ is a finite morphism between noetherian schemes and $\omega$ is a dualizing complex on $Z$ then $f^{!}(\omega)$ is a dualizing c... | 4 | https://mathoverflow.net/users/81332 | 226975 | 105,955 |
https://mathoverflow.net/questions/226965 | 1 | Let $B\_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W\_0^{1,2}(B\_0(1))$, and $W^{1,2}(B\_0(1))\ni f\_i\to f$ in the sense of $L^2(B\_0(1))$-norm, as $i\to \infty$.
>
> **Question 1**: Can we find a sequence of positive numbers $\{\delta\_i\}$ (may depends on $\{f\_i\}$), going to $0$, s.t., $$\fra... | https://mathoverflow.net/users/84068 | Convergence of Sobolev functions near the boundary | I think Question 1 has a positive answer. Denote $B=:B\_0(1)$ and $A\_r:=\{r<\|x\|<1\} $ for $0<r<1$. For functions $f\in W^{1,2}\_0(B)$ we have a Poincaré inequality on $A\_r$ :
$$\int\_{A\_r} f^2dx\le \Big(\frac{1-r}{r}\Big)^2 \int\_{A\_r} |\nabla f|^2dx \ .$$
Now let $(f\_i)$ a sequence in $W^{1,2}(B)$ converging ... | 1 | https://mathoverflow.net/users/6101 | 226982 | 105,959 |
https://mathoverflow.net/questions/226983 | 6 | What does this statement describe? $X$ and $Y$ are matrices.
>
> The eigenvalues of $X$ in the metric of $Y$.
>
>
>
I've not seen this language used before in this fashion and I don't really know what taking the eigenvalues of a matrix in the metric of another matrix means.
Could someone decipher this for m... | https://mathoverflow.net/users/67077 | Eigenvalues of $X$ in the metric of $Y$ | It means the $\lambda$ such that $X - \lambda Y$ is not invertible. The usual eigenvalues are those where $Y$ is the identity matrix.
| 11 | https://mathoverflow.net/users/nan | 226984 | 105,960 |
https://mathoverflow.net/questions/92819 | 2 | Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.
If I do (relative) log mmp over a klt pair $(X,D)$ what I obtain is a klt pair $(X',D')$ as output.
If I do (relative) mmp over a terminal... | https://mathoverflow.net/users/1887 | Controlling singularities on log mmp | If $(X,D)$ is terminal and the stable base locus of $K\_X+D$ contains no components of the support of $D$, then any sequence of steps $f:X\to X'$ of the $K\_X+D$ MMP yields a terminal pair $(X',D'=f\_\*D)$. (Let $E$ be a divisor over $X'$ with discrepancy $a\_E(X',D')\leq 0$, then $a\_E(X,D)\leq a\_E(X',D')\leq 0$ so t... | 3 | https://mathoverflow.net/users/19369 | 226999 | 105,968 |
https://mathoverflow.net/questions/221848 | 2 | Since $n = \frac{n(n+1)}{2}-\frac{n(n-1)}{2}$, every natural number can be represented as the difference of two triangular numbers:
$ n = \frac{a(a+1)}{2}-\frac{b(b-1)}{2}$. Finding such a representation gives a factorization of $n = \frac{(a+b)\cdot(a-b+1)}{2}$.
A naive way of finding such representations would be to ... | https://mathoverflow.net/users/nan | Finding integer representation as difference of two triangular numbers | The representations of this type correspond one-for-one to odd divisors of $n$. So your request for a method for constructing such a representation without factoring seems to be hopeless: if you have a method for constructing such a representation it is automatically a method for factoring. See [Wikipedia: Polite numbe... | 3 | https://mathoverflow.net/users/440 | 227001 | 105,970 |
https://mathoverflow.net/questions/226974 | 5 | Let $E/\mathbb F\_{p^m}$ be an arbitrary elliptic curve over the Galois field $\mathbb F\_{p^m}$, and let $$[n]^{-1}(P)\cap E(\mathbb F\_{p^m})=\{Q\in E(\mathbb F\_{p^m})\mid nQ=P\}.$$ Also let $N=\#E(\mathbb F\_{p^m})$. Is the following claim true?
If $\gcd(n,N)=1$, then the only point in $[n]^{-1}(P)\cap E(\mathbb ... | https://mathoverflow.net/users/84430 | Division by $n$ in elliptic curves | Disclaimer: I am not speaking for the Magma group.
Even though Magma tries to provide the best (i.e., most efficient)
algorithms, it is very hard to make sure that all possible cases are
taken care of. From experiments with Magma's DivisionPoint function,
it looks like the shortcut you propose is not implemented.
I w... | 5 | https://mathoverflow.net/users/21146 | 227014 | 105,974 |
https://mathoverflow.net/questions/226994 | 2 | There is those one Q5 to Q7 in <https://en.wikipedia.org/wiki/Hilbert_system#Formal_deductions>
But I know the axioms of Boolean algebra were simplified to this <https://en.wikipedia.org/wiki/Wolfram_axiom>
I was wondering if similar researchs have been done on quantifiers?
| https://mathoverflow.net/users/84353 | Is there a simpler axiomatization for the quantifiers? | There is [The Epsilon Calculus](http://plato.stanford.edu/entries/epsilon-calculus/) developed by David Hilbert during the 20s.
It is based on the $ε$ *symbol* :
>
> if $A$ is a formula and $x$ is a variable, $εx \ A$ is a *term*
>
>
>
with the axiom (Hilbert's “transfinite axiom”) :
>
> $A(x) → A(εx A)$... | 3 | https://mathoverflow.net/users/42676 | 227022 | 105,978 |
https://mathoverflow.net/questions/226971 | 0 | This question has been asked here but there is no answer:
<https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y>
>
>
> >
> > Consider autonomous ODE $y' = f(y)\quad (1)$ which has an invariant set $M$
> > defined by the equations
> > $$h (y) = 0 \qquad (2) $$
... | https://mathoverflow.net/users/84530 | Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$ | Notice that
$$
\begin{split}\frac12\frac{d}{dt} \|h(y(t))\|^2
&=\frac12\frac{d}{dt}\bigl\langle h(y(t)),h(y(t))\bigr\rangle\\
&=\bigl\langle H(y(t))y'(t),h(y(t))\bigr\rangle\\
&=\bigl\langle H(y(t))f(y(t))−\gamma h(y(t)),h(y(t))\bigr\rangle\\
&=\bigl\langle H(y(t))f(y(t)),h(y(t))\bigr\rangle−\gamma \|h(y(t))\|^2\\
&\le... | 1 | https://mathoverflow.net/users/74138 | 227031 | 105,981 |
https://mathoverflow.net/questions/227023 | 5 | Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim\_{k\in\mathbb{Q}[\mu\_{p^\infty}]} \mathbb{Q}\left[\mu\_{p^\infty},k^{1/p^\infty}\right]$$
and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that $E\_p(K)=0$.
Question: Is there always a $\tau\in G$ so that
$$\operatorname{rank}\_{\mathbf{... | https://mathoverflow.net/users/70751 | Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$? | I think the answer is "yes" in the case that $E$ doesn't have complex multiplication. In that case, Serre's Open Image Theorem says that the image of Galois in open in $\mathrm{GL}(T\_{p}(E))$, and therefore, the image of $\mathrm{Gal}(\bar{\mathbb{Q}} / \mathbb{Q}(\mu\_{p^\infty}))$ is open in $\mathrm{SL}(T\_{p}(E))$... | 4 | https://mathoverflow.net/users/24757 | 227040 | 105,984 |
https://mathoverflow.net/questions/227036 | 4 | This question related to [this](https://math.stackexchange.com/q/1575992/230303) question in SE ,I would like to know how do I
evaluate this sum for $s$ is a complex variable :$$\sum\_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$$ .
**Edit01**:And I think the General complex solution of $$\sum\_{n=1}^{\infty} \frac{(... | https://mathoverflow.net/users/74330 | How do I evaluate this sum for $s$ is a complex variable :$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$? | *Mathematica* says that
$$\sum\_{n=1}^\infty \frac{(-1)^{n+1}}{n^k n!} =
\, \_kF\_k(1,\dotsc, 1;2,\dotsc,2;-1),$$ where the numbers of $1$s (and $2$s) are both equal to $k.$ This suggests to me that there is no closed form.
| 5 | https://mathoverflow.net/users/11142 | 227043 | 105,985 |
https://mathoverflow.net/questions/227050 | 4 | So I've recently read the infinite graph version of Kuratowski's theorem. It says that a graph $G$ is planar if and only if the following three conditions holds:
1. $|V(G)| \le |\mathbb{R}|$
2. $G$ has at most countably many vertex with degree at least 3
3. $G$ has neither $K\_{3,3}$ nor $K\_5$ subdivision
It is cl... | https://mathoverflow.net/users/nan | Generalisation of Kuratowski's theorem | I am not sure which direction gives you trouble, but the second condition is necessary, by the Moore-Young theorem, or a somewhat weaker version thereof, as discussed by Greg Kuperberg [here.](https://mathoverflow.net/questions/27244/how-many-tacks-fit-in-the-plane)
For the right direction, condition 2 says that ther... | 4 | https://mathoverflow.net/users/11142 | 227051 | 105,987 |
https://mathoverflow.net/questions/227049 | 10 | *I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it turns out this is a duplicate, I'll delete this question. Anyways, apologies in advance if this is too easy or is a dupl... | https://mathoverflow.net/users/8133 | Can an ultrapower be undone by forcing? | For set-forcing, the answer is no, see the following article
*Joel David Hamkins, Greg Kirmayer, and Norman Lewis Perlmutter*, [**Generalizations of the Kunen inconsistency**](http://dx.doi.org/10.1016/j.apal.2012.06.001), *Ann. Pure Appl. Logic* **163** (2012), no. 12, 1872--1890. (see also [arxiv.org/abs/1106.1951]... | 16 | https://mathoverflow.net/users/26705 | 227052 | 105,988 |
https://mathoverflow.net/questions/227016 | 5 | Let $n\ge 2$ be an integer and $C$ be a smooth projective curve of genus $g>n$. The $n$-the symmetric product $C(n)$ is a smooth variety of general type.
If $n=2$ then $C(2)$ is a minimal surface ad it is not hard to compute its numerical invariants. I would like to know whether $C(n)$ is minimal also for $n>2$ and wha... | https://mathoverflow.net/users/10610 | Numerical invariants of symmetric products of curves | **Claim:** If $C$ is a curve of genus $g \geq 2,$ the canonical class of $C^{(d)}$ is nef and big if and only if $1 \leq d \leq g-1.$
**Proof:** Let $\theta \in {\rm NS}(C^{(d)})$ be the class of the pullback of the theta-divisor of ${\rm Jac}(C)$ via the Abel map, and let $x \in {\rm NS}(C^{(d)})$ be the class of th... | 4 | https://mathoverflow.net/users/5496 | 227059 | 105,991 |
https://mathoverflow.net/questions/227024 | 4 | Suppose we consider primes of the form $p = 1 \text { mod } 4$, so that $p = a^2 + b^2$, $a$ and $b$ being integers. Considering only the first quadrant, all $(a,b)$ pairs will be of the form (odd,even) or (even,odd).
Now if we consider $q = (a + n)^2 + (b + n)^2 = p + 2na + 2nb + 2n^2$, $n$ an integer, then $q = 1 ... | https://mathoverflow.net/users/84560 | An elementary question about Gaussian primes | An infinite number of primes on a ray of slope $45^{\circ}$ would not imply close Gaussian primes any more than ordinary primes $0^{\circ}.$ I didn't read the paper you link, but $246$ is the current record for gaps between ordinary primes. You might enjoy reading about [Guassian moats](https://en.wikipedia.org/wiki/Ga... | 2 | https://mathoverflow.net/users/8008 | 227085 | 105,995 |
https://mathoverflow.net/questions/227090 | 10 | how to calculate the asymptotic growth rate of coefficients generating function $T(z)$ satisfied this identity
$T(z)=z+\frac{T(z)^3}{6}+\frac{T(z^2)T(z)}{2}+\frac{T(z^3)}{3}$
| https://mathoverflow.net/users/41522 | Asymptotic growth rate of coefficients of generating function | In order to analyze these type of equations one can use the methodology of Flajolet and Sedgewick (Analytic Combinatorics Book). More precisely, from the previous specification one gets that the first coefficients of $T(z)$ are
$$T(z)=z+z^3+z^5+2z^7+4z^9+8z^{11}+17z^{13}+39z^{15}+89z^{17}+211z^{19}$$
Observe now th... | 10 | https://mathoverflow.net/users/46573 | 227094 | 106,001 |
https://mathoverflow.net/questions/227035 | 1 | Let $ N\_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$.
In various papers one can read an estimate of the number of zeros over all L-functions:
$$ \sum\_{\chi \mod q} N\_\chi(\alpha,T) \ll ... | https://mathoverflow.net/users/1358 | How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences? | Firstly, a general comment: as understanding of a mathematical problem deepens, it is common (and even expected) for the most mathematically natural formulation of a given problem (or class of problems) to become "hardly recognisable" as arising from the formulation of the problem that historically motivated work in th... | 22 | https://mathoverflow.net/users/766 | 227099 | 106,002 |
https://mathoverflow.net/questions/227097 | 1 | Let $G$ be a $d$-regular graph, and $A$ be the incidence matrix of $G$. Also suppose $B$ is a reduced echelon form of $A$ such that computations are in $\mathbb F\_2$. Given matrix $B$, can we find matrix $A$?
If yes, how? and for arbitrary sparse matrix $A$ is this true?
And if no, can we use this method for const... | https://mathoverflow.net/users/84430 | Reduced echelon form of sparce matrices and constructing hash function | Assuming you mean the vertex-edge incidence matrix, your row-echelon form determines the cut-space of the graph. Hence it is is determined by the natural graphic matroid. If the graph is 3-connected, this matroid determines the graph. Otherwise it does not - look up "Whitney flip".
| 2 | https://mathoverflow.net/users/1266 | 227105 | 106,006 |
https://mathoverflow.net/questions/227101 | 2 | I'm trying to prove the sum of a sequence given by
$a\_{n+1} = \frac{nb-x}{(n+1)b} a\_n$
with $a\_1 = 1$. This gives the solution $a\_n = \frac{(-x/b)\_n}{n!}$. When trying to work out what this sums to, I looked at hypergeometric functions ${}\_1F\_0(-x/b;;1)$ to sum this, but this appears to be undefined.
I have ... | https://mathoverflow.net/users/66436 | Sum of difference equation involving hypergeometric functions 1F0 | The solution you give for $a\_{n+1} = \frac{nb-x}{(n+1)b} a\_n$ has a missing factor, I think it should read
$$a\_n=-\frac{b}{x}\frac{(-x/b)\_n}{n!}.$$
Then the sum over $n$ equals
$$\sum\_{n=1}^\infty a\_n =\frac{b}{x},$$
as follows from the generating function
$$\sum\_{n=0}^\infty (p)\_n\frac{z^n}{n!}=(1-z)^{-p}$$
... | 1 | https://mathoverflow.net/users/11260 | 227106 | 106,007 |
https://mathoverflow.net/questions/227081 | 2 | I have the following setting:
1.) A Galois extension of number fields $K\hookrightarrow L$, with $\operatorname{Gal}(L/K)=\mathbb{Z}\_{p}$. In my terminology, number field does not imply finiteness over $\mathbb{Q}$. Let $K\hookrightarrow {K}\_{{{p}^{n}}}\hookrightarrow L$ be such that $\operatorname{Gal}({K}\_{{{p}^... | https://mathoverflow.net/users/70751 | Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ is an Ab. var. and $m_n$ is strictly increasing | I suspect that the answer is "no". If you take an elliptic curve with supersingular reduction at $p$, $K$ a sufficiently carefully chosen imaginary quadratic field, and $L / K$ the anticyclotomic $\mathbf{Z}\_p$-extension, then you can rig things so that $E(L)$ is not finitely generated, because there are non-torsion H... | 2 | https://mathoverflow.net/users/2481 | 227110 | 106,011 |
https://mathoverflow.net/questions/227113 | 2 | Let $\ldots \to X\_n \to X\_{n-1} \to \ldots \to X\_0$ be etale maps between smooth projective curves of genera $g(X\_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know that the sets $X\_n(K)$ are finite.
>
> Is the set $\bigcup\_{n=0}^\infty X\_n(K)$ always finite?
>
>
>
| https://mathoverflow.net/users/3847 | Rational points on towers of curves | No. Start with a tower of Galois covers for which $X\_0(K) \ne \emptyset$. Then, inductively, replace each $X\_n$ by a twist such that the point from the previous layer lifts to a rational point on the twist. In this way, all $X\_n(K)$ are non-empty and their union is infinite.
| 8 | https://mathoverflow.net/users/2290 | 227114 | 106,014 |
https://mathoverflow.net/questions/227124 | 4 | First, I would like to say that I asked this question (a more general one actually) on [math.stackexchange.com](https://math.stackexchange.com/questions/1080591/statements-comparable-with-axiom-of-choice-in-zf).
Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC (assuming ZF). If w... | https://mathoverflow.net/users/61723 | Maximal chains and antichains of statements weaker than AC | It is not possible to prove there are no finite maximal antichains, because there are.
Paul Howard proved that Łoś's theorem and the Boolean Prime Ideal theorem imply in conjunction the Axiom of Choice. However there is a model in which every ultrafilter is principal, so Łoś's theorem holds trivially and therefore do... | 6 | https://mathoverflow.net/users/7206 | 227125 | 106,020 |
https://mathoverflow.net/questions/225864 | 3 | Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the Hausdorff distance.
>
>
> >
> > Does there exist a compact metrisable topological space $X$ and a Borel probability me... | https://mathoverflow.net/users/15570 | Is it possible for a random nowhere dense closed set to have a positive probability of hitting any given point? | Yes. As in the comments: take $X=\mathbb{S}^1$; and let $\nu$ be the law of the random set constructed by taking a positive-Lebesgue-measure Cantor set $K \subset \mathbb{S}^1$ and rotating $K$ through a random angle selected according to the uniform distribution.
| 1 | https://mathoverflow.net/users/15570 | 227127 | 106,022 |
https://mathoverflow.net/questions/227132 | 2 | Let $\xi^m$ and $\eta^n$ be vector bundles over a paracompact base space. Where can I find a reference to the Stiefel-Whitney classes of the tensor product $\xi^m \otimes \eta^n$ being computed as follows?
>
> There is a universal formula of the form$$w(\xi^m \otimes \eta^n) = p\_{m, n} (w\_1(\xi^m), \dots, w\_m(\x... | https://mathoverflow.net/users/nan | Stiefel-Whitney classes of tensor product $\xi^m \otimes \eta^n$, computation | The keyword is "splitting principle". The problem itself is a copy of an exercise from $\S7$ in Milnor-Stasheff, which has a nice hint.
For the detailed proof see Proposition 3.2.12 in this notes
<http://www.analg.ulg.ac.be/jps/rec/icc.pdf>
| 8 | https://mathoverflow.net/users/40950 | 227135 | 106,025 |
https://mathoverflow.net/questions/209539 | 6 | Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1\_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1\_{\mathrm{loc}}(\mathbb R^n)\cap L^1(\mathbb R^n,fdx)$. Is it true that
$$
f\nabla \log f = \nabla f,
$$
again in the sense of distributions?
Obviousl... | https://mathoverflow.net/users/16530 | Chain rule for weakly differentiable functions | I think it works by cutting-off as follows:
Fix $\epsilon>0$, let $f\_\epsilon=\min\{1/\epsilon,\max\{\epsilon,f\}\}$, and observe that $f\_\epsilon\to f$ in $L^1\_{loc}$.
Then obviously $\log f\_\epsilon=\min\{\log(1/\epsilon),\max\{\log(\epsilon),\log f\}\}$ and $\nabla \log f\_\epsilon =(\nabla\log f)\chi\_{[\epsi... | 1 | https://mathoverflow.net/users/33741 | 227146 | 106,028 |
https://mathoverflow.net/questions/227140 | 3 | It is well known (see [Derdzinski](https://eudml.org/doc/89617)) that for a Kaehler metric on a four-manifold, its self-dual Weyl curvature has only two distinct eigenvalues:
$$-\frac{R}{12},\ -\frac{R}{12},\ \frac{R}{6}.$$
I was wondering whether anti-self-dual Weyl curvature also has only two distinct eigenvalues (... | https://mathoverflow.net/users/51632 | A question on anti-self-dual Weyl curvature of Kaehler surfaces | The answer is 'no'. In fact, even for a Ricci-flat Kähler manifold $(M^4,J,g)$, the map $W\_-(x):\Lambda^2\_-(T\_xM)\to\Lambda^2\_-(T\_xM)$ can be any symmetric traceless linear map. Hence, the only constraint on the eigenvalues is that they sum to zero.
This fact was known to Élie Cartan already in 1926, although h... | 4 | https://mathoverflow.net/users/13972 | 227147 | 106,029 |
https://mathoverflow.net/questions/227150 | 3 | Let $G\_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle
$$
\gamma\_{k,N}: \mathbb{R}^k\longrightarrow E(\gamma\_{k,N})\longrightarrow G\_k(\mathbb{R}^N)
$$
where
\begin{eqnarray\*}
E(\gamma\_{k,N})&=&\{(V,v)\mid V\in G\_... | https://mathoverflow.net/users/41075 | self-Whitney sum of the canonical vector bundle on Grassmannians | Let me expand the comment above a bit. Consider the tautological bundle $\tau\to\mathbb C P^2$. It is complex and has total Chern class $c(\tau)=1+a$, where $a\in H^2(\mathbb C P^2)$ generates the cohomology ring of $\mathbb C P^2$. Viewed as a real bundle, it has $$p\_1(\tau\_{\mathbb R})=-c\_2(\tau\oplus\bar\tau)=-c\... | 2 | https://mathoverflow.net/users/70808 | 227152 | 106,031 |
https://mathoverflow.net/questions/225896 | 74 | Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my [previous question](https://mathoverflow.net/questions/218113/a-is-isomorphic-to-a-oplus-mathbbz2-but-not-to-a-oplus-mathbbz) about abelian groups: In fact, in any algebraic category we may ask if $A \c... | https://mathoverflow.net/users/2841 | $R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ | The answer to this quite beautiful question is that there *does exist* a commutative ring $R$ with $R\cong R[X,Y]$ but $R\not\cong R[X]$.
Let $F$ be a field, and take
$$
R=F[x\_i,y\_i,r\_i\ (i\geq 0)]
$$
subject to the relations
$$
\forall\ i\geq 0,\ r\_i=x\_i y\_i(x\_i+y\_i^2)(x\_i+y\_i^3)(x\_i+y\_{i+1}^4)r\_{i+1}.
... | 57 | https://mathoverflow.net/users/3199 | 227169 | 106,036 |
https://mathoverflow.net/questions/194552 | 1 | **Edit:** According to the essential comment of Alex Degtyarev, we revise the question as follows;
>
> Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $O(m)$, respectively. Is there an oriention preserving automorphism of $O(n+m)$ which restriction to $O(n)\oplus ... | https://mathoverflow.net/users/36688 | A (possible) equivalent relation on the space of vector bundles | EDIT: This is now a full answer, and the answer is "no".
Let $G$ be a structure group as in the question.
Let $\lambda$ and $\mu$ be automorphisms, then so is $\lambda\circ\mu$.
We write $E\sim\_\lambda F$ if $E$, $F$ have cocycles $g\_{\alpha\beta}$, $h\_{\alpha\beta}$ respectively such that $g\_{\alpha\beta}=\lambda\... | 2 | https://mathoverflow.net/users/70808 | 227175 | 106,037 |
https://mathoverflow.net/questions/227168 | 4 | I read the Lax-Milgram Theorem in the *Navier-Stokes Equations* by Temam:
>
> Let $X$ be a *separable* Hilbert space (norm $\|\cdot\|\_X$) and let
> $$
> a:X\times X\to\Bbb{R}
> $$
> be a bilinear continuous coercive form; that is, there exist $c,C>0$, such that for all $u,v\in X$, we have
> \begin{align}
> |a(... | https://mathoverflow.net/users/nan | Is Lax-Milgram true without the separability assumption? | A reference without the separability requirement: Elements of Nonlinear Analysis, by Michel Chipot, namely Theorem 3.2 in page 41.
Actually, you may have a look at Temam's book "Infinite-Dimensional Dynamical Systems in Mechanics and Physics", more precisely at Theorem II.2.1 in page 54. It also doesn't use separabil... | 1 | https://mathoverflow.net/users/74138 | 227185 | 106,042 |
https://mathoverflow.net/questions/227192 | -2 | I am very new to finite automata, and I came across an issue in my professors lecture slides which I think is wrong, and I'd wonder if any of you could confirm:
Alphabet: {1}
[Automata](https://i.stack.imgur.com/SZoEj.png)
Surely the accepting language is not this, and is rather
{x: x (iselementof) StarClosureAlp... | https://mathoverflow.net/users/84644 | Deterministic Finite Automata question | Yes, it should say "*the length of* $x$ is even," not "$x$ is even."
| 1 | https://mathoverflow.net/users/8133 | 227194 | 106,044 |
https://mathoverflow.net/questions/227196 | 4 | The goal of this question is to develop further the discussion
initiated in [Under which conditions is it possible to find points with same distances under bi-Lipschitz map](https://mathoverflow.net/questions/226446/). The mentioned question was closed because the author
stated the question somewhat vaguely, but I thin... | https://mathoverflow.net/users/37822 | Finitely isometrically persistent metric spaces | There is no such space with more than one point. Let $X$ be any metric space and suppose there exist $x,y,z \in X$ such that $$a = d(x,y) > {\rm max}(d(x,z), x(y,z)) = b.$$
That is, there is a triangle in which one side is longer than the other two. Define $\omega: [0,\infty) \to [0,\infty)$ by $$\omega(t) =
\begin{cas... | 6 | https://mathoverflow.net/users/23141 | 227198 | 106,047 |
https://mathoverflow.net/questions/227209 | 5 | I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder originally is $\zeta$: is it really Riemann's $\zeta$? I guess so, but then why did he choose that? Thinking about it, it is... | https://mathoverflow.net/users/6575 | The $\zeta$-word | Well, Riemann himself says "I denote this function by $\zeta(s)$" ("*Die Function [...] bezeichne ich durch $\zeta(s)$*"), so I would think the choice of which letter to use for this function was his.
first page of Riemann's [Über die Anzahl der Primzahlen unt... | 15 | https://mathoverflow.net/users/11260 | 227212 | 106,051 |
https://mathoverflow.net/questions/226896 | 8 | Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called *trivialising* (for this bundle), if after restricting this bundle to $Y$, it is a trivial bundle. In other words, $p: p^{-1}(Y)\to Y$ is trivial. $Y$ is called *maximally trivial*, if it is trivial and ther... | https://mathoverflow.net/users/40640 | Maximal trivialising subspace for a vector bundle | In addition to Mark's argument, a maximal trivialising $Y\subset X$ is also open, if we assume that $E$ is a $\Bbbk$-vector bundle with $\Bbbk=\mathbb R$ or $\mathbb C$. So every maximal trivialising subset is open and dense, but it is not (yet) clear that every trivialising subset is contained in a maximal one.
Assu... | 4 | https://mathoverflow.net/users/70808 | 227214 | 106,052 |
https://mathoverflow.net/questions/222162 | 11 | $\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL\_n^+$ (invertible matrices , $\det >0$) as an open Riemannian submanifold of $\mathbb{R}^{n^2}$.
We have two metrics on $GL\_n^+$ (in the sense of metric spaces); intrins... | https://mathoverflow.net/users/46290 | Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$? | In two dimensions the condition $ad-bc=0$ translates into $(a+d)^2-(a-d)^2-(b+c)^2+(b-c)^2=0$ or to simplify notation $x^2+y^2=z^2+w^2$. Intersecting this with the unit sphere in $\mathbb{R}^4$ gives a flat 2-torus which decomposes the $3$-sphere into two solid tori. One solid torus consists of matrices of positive det... | 3 | https://mathoverflow.net/users/28128 | 227240 | 106,060 |
https://mathoverflow.net/questions/227239 | 0 | A Green's function is defined as follows:
$$G(\omega) = \frac{1}{N}\mathrm{E}\big[ \mathrm{Tr}\frac{1}{I\omega - J} \big]$$, where $I$ is the $N$-dimensional identity and $E$ means expectation value with respect to the random matrix $J$.
It follows that
$$ G(\omega) = \frac{1}{N}\mathrm{E}\big[ \sum\_\lambda\frac{1}... | https://mathoverflow.net/users/48827 | Alternative formula of a Green's function for average density of eigenvalues of random matrix | I'll post an answer to spell out all the details.
You have $$G(ω)=\frac{1}{N}E\left[{\rm Tr}\frac{1}{Iω−J}\right]=\frac{1}{N}E\left[\sum\_\lambda\frac{1}{ω−\lambda}\right]$$
This can be written as $$G(ω)=\frac{1}{N}E\left[\int d^2z \sum\_\lambda\frac{\delta(z-\lambda)}{ω−z}\right],$$ where $\delta$ is the delta-fun... | 1 | https://mathoverflow.net/users/78061 | 227251 | 106,064 |
https://mathoverflow.net/questions/227248 | 3 | Let $x$ be a trace class operator on a Hilbert space $H$. Then $x$ induces unique normal functional on $B(H)$, which we denote it by $f\_x$.
Let us consider the polar decomposition $x=u|x|$ and $f\_x=v|f\_x|$.
>
> Question: Is $|f\_x|$ the corresponded functional of $|x|$ and $u=v$?
>
>
>
| https://mathoverflow.net/users/84390 | Polar decomposition | It depends on the conventions that you pick, but if you pick them correctly, you'll get what you want.
First, for a normal linear functional $f$ on $B(H)$, there is a unique partial isometry $v\in B(H)$ and positive normal linear functional $|f|$ such that $f=v|f|$ and $v^\*v$ is the support projection of $|f|$, wher... | 4 | https://mathoverflow.net/users/351 | 227254 | 106,066 |
https://mathoverflow.net/questions/226653 | 5 | Let $H$ be a non separable Hilbert space and $\Omega$ be a measurable space.
Naturally, we say that $f:\Omega\to B(H)$ is $w$-measurable if $f^{-1}(O)$ is measurable for any open set $O$ in the weak operator topology.
Question: Let $f$ and $g$ be two $w$-measurable functions on $\Omega$. Is the multiplication $fg$ ... | https://mathoverflow.net/users/84390 | Operator-valued measurable functions | The answer is no even for functions with values in $L\_\infty(\mu)$ (for the purpose of this question embedded in $B(H)$), as long as the cardinality of the space $L\_\infty(\mu)$ is larger than continuum. The following argument works for many reasonable topologies, including the w.o.t. in the question.
Let $\Omega$ ... | 3 | https://mathoverflow.net/users/83382 | 227255 | 106,067 |
https://mathoverflow.net/questions/225676 | 2 | Given a non-negative matrix $T$, how to find a orthogonal matrix $O$ minimising
$$ \left\lVert \, |O| - T \right\lVert\_F,$$
where $\lVert \cdot \lVert\_F$ denotes the Frobenius norm, and $| \cdot |$ denotes the entry wise absolute value.
If $|O|$ is replaced with $O$, then this is the classic Orthogonal Procrustes Pro... | https://mathoverflow.net/users/51478 | Orthogonal Procrustes problem with Absolute Values | Here are alternate encodings based on Sebastian's comments on my previous answer.
**Comment:** If you want to constrain the solutions found to a particular sign pattern matrix then an additional $mn$ linear inequalities added to the above formulation can do the job.
**Encoding 1:** Here is another encoding of the... | 2 | https://mathoverflow.net/users/29887 | 227266 | 106,073 |
https://mathoverflow.net/questions/227267 | 4 | The question I'm trying to answer is the following:
>
> Let $P \to X$ be a principal $G$-bundle (over a connected CW complex)
> satisfying that all pullbacks to spheres (of arbitrary dimension) are
> trivial. Is $P$ trivial? If not, what's the simplest example for a
> faliure of this?
>
>
>
The following "p... | https://mathoverflow.net/users/22810 | Principal bundles that can't be detected by spheres | If all maps are involved are pointed, your question is equivalent to the following: if $f : X \to BG$ is a map (where $X$ is connected) such that the induced map on $\pi\_{\bullet}$ is zero, is $f$ zero (nullhomotopic)? The answer is no, although it's easiest to give a counterexample if we replace $BG$ with an arbitrar... | 12 | https://mathoverflow.net/users/290 | 227268 | 106,074 |
https://mathoverflow.net/questions/227260 | 16 | Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r\_2(n)$, the number of ways of writing a natural number $n$ as the sum of two squares. My question is then, what happens if the cir... | https://mathoverflow.net/users/37103 | On (a generalization of) the Gauss Circle Problem | You can write a similar expression to the usual formula for the remainder term in the circle problem by using the Poisson summation formula. The shift of the center of the circle simply means that one gets a variant of the usual formula for the remainder term modified by suitable exponential terms.
Indeed Huxley has... | 15 | https://mathoverflow.net/users/38624 | 227270 | 106,075 |
https://mathoverflow.net/questions/227257 | 8 | Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph.
Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set of closed walks (some authors would write `closed paths') in $X$ *induced* by the elements of $R$ generates $H\_1(X)$, ... | https://mathoverflow.net/users/69681 | Does every generating set of the first homology group of a Cayley graph give rise to a presentation of its group? | Write $G=F/N$, where $F$ is free over $S$. Let $C\_F$ and $C\_G$ be the Cayley graphs of $F$ and $G$: then $C\_G=C\_F/N$ and in particular the fundamental group of $C\_G$ is naturally $N$ (note that we need the convention that if $1\in S$ then it yields a self-loop and if $S$ has elements of order 2 then they yield dou... | 4 | https://mathoverflow.net/users/14094 | 227271 | 106,076 |
https://mathoverflow.net/questions/227285 | 1 | Consider the following statement by Edward Nelson--this from the "Outline" of his 'proof' of the inconsistency of $PA$ (which Terry Tao found to contain an error):
>
> "The induction axiom schema of Peano arithmetic $\mathtt P$ is usually justified as follows. Assume the basis $\mathbb A\_x(0)$ and the induction st... | https://mathoverflow.net/users/20597 | Does mathematical induction presuppose the existence of a completed infinity? | This is quite a mouthful for a question but Peano Arithmetic does not seem to require infinity whereas Peano Axioms (second order) does seem to be equivalent to an axiom of infinity.
| 1 | https://mathoverflow.net/users/28128 | 227286 | 106,081 |
https://mathoverflow.net/questions/227234 | 2 | Given a quadric surface $Q$ over a field $F$ of characteristic $2$, assume it is irreducible and reduced, we say it is ruled, if $Q$ is birational to $C \times \mathbb{P}^1$ for some $C$.
A sufficient condition for such a quadric to be ruled is that there is a rational point (use the projection from this point).
My... | https://mathoverflow.net/users/84677 | necessary conditions for a quadric surface to be ruled (over a field of char 2) | The condition you state is not a necessary condition. You may find much more about these types of questions in Manin's "Cubic Forms".
First of all, over a finite field every quadric hypersurface has a rational point by Chevalley's theorem (or you can probably reduce this case to Wedderburn's earlier theorem). Thus, ... | 1 | https://mathoverflow.net/users/13265 | 227287 | 106,082 |
https://mathoverflow.net/questions/227238 | 40 | By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
* We are considering a first-order theory ($\mathsf{ZFC}$ or $\mathsf{PA}$) motivated as a first-order approximation to a second-order theory ... | https://mathoverflow.net/users/17064 | Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning? | A significant amount of the parallelism can be explained by the bi-interpretabiity of $PA$ with $ZF^{-\infty}$ ("finite set theory"), which is the theory obtained from $ZF$ by replacing the axiom of infinity by its negation, and adding the sentence asserting that every set has a transitive closure. For more detail and ... | 31 | https://mathoverflow.net/users/9269 | 227295 | 106,083 |
https://mathoverflow.net/questions/227298 | 4 | Let $S\_n$ be the set of all the $n\times n\ (0,1)$-matrices and divide $S\_n$ into two sets as follows:
$A\_n=\{M\in S\_n:$ there exist a row and a column of $M$ such that the sum of the row is equal to the sum of the column$\}$ and $B\_n=S\_n\setminus A\_n.$
I have two questions:
$(1)$Is there any estimation for ... | https://mathoverflow.net/users/58096 | A partition of the set of all $n\times n\ (0,1)$-matrices | For (2), consider a random matrix (each entry independently 0 or 1 with probability 1/2). See [this paper](http://arxiv.org/abs/1302.2446) (published in Journal of Combinatorics). By virtue of Corollary 2 it suffices to show that if $S\_1,\ldots,S\_n,T\_1,\ldots,T\_n$ are binomial random variables Bin$(n,1/2)$, indepen... | 7 | https://mathoverflow.net/users/9025 | 227304 | 106,085 |
https://mathoverflow.net/questions/227300 | 17 | In the paper *Quantifiers and Sheaves* by Lawvere, at the bottom of the second page, the author writes:
>
> "... the condition that every epi splits, which geometrically we would call 0-dimensionality and logically we would call the axiom of choice."
>
>
>
What is meant by this geometric interpretation of choi... | https://mathoverflow.net/users/69037 | Axiom of choice as zero dimensionality | Here are two possible motivating examples.
First, for any topos $\mathcal{E}$, if all epimorphisms in $\mathcal{E}$ split, then $\mathcal{E}$ is a boolean topos. In particular, for a topological space $X$, if all epimorphisms in $\mathbf{Sh} (X)$ split, then every open subset of $X$ is also closed (and vice versa), ... | 23 | https://mathoverflow.net/users/11640 | 227306 | 106,086 |
https://mathoverflow.net/questions/227269 | 6 | My question is the following:
Suppose $M$ is an $n \times n$ symmetric real matrix. I want to find an $n \times n$ symmetric real matrix X such that $|| X -M||\_F$ is minimized with the constraint that $U^T X U \succeq 0$ (positive semidefinite), where $U$ is an $n \times d$ matrix, $U^TU = I\_d$ and $n > d$. $|| \cd... | https://mathoverflow.net/users/61149 | Minimize Frobenius norm | Here's one approach to solve this problem.
Since $U$ is a thin matrix with orthogonal columns each of which has unit norm it spans a $d$ dimensional subspace within the $n$ dimensional ambient subspace. Now we can see that the constraint that $U^TXU \succeq 0$ is a looser constraint than $X \succeq 0$ since $X$ only ... | 3 | https://mathoverflow.net/users/29887 | 227315 | 106,088 |
https://mathoverflow.net/questions/227310 | 22 | In this [article](http://www.scottaaronson.com/blog/?p=710), Scott Aaronson talks about using Turing Machines for proving the Rosser Theorem.
What is the relationship between the numbering that Gödel used in his proof of incompleteness and Turing Machines?
| https://mathoverflow.net/users/84711 | What is the relationship between Turing Machines and Gödel's Incompleteness Theorem? | It's simple. If the halting problem is undecidable, then PA is not complete, since otherwise, you could solve the halting problem by searching for proofs in PA. And the same argument works for any sound computably axiomatizable theory $T$ able to express arithmetic. Given a Turing machine $M$ on input $i$, you formulat... | 22 | https://mathoverflow.net/users/1946 | 227316 | 106,089 |
https://mathoverflow.net/questions/227289 | 10 | Suppose that $C\subset \mathbb P^2$ is a plane projective curve (base field is $\mathbb C$) and $C^\*\subset (\mathbb P^2)^\*$ is its dual. What are the known examples in which $C$ is projectively (i.e., linearly) isomorphic to $C^\*$? Besides the non-degenerate conic, I am aware of the cuspidal cubic. Anything else?
... | https://mathoverflow.net/users/29992 | Self-dual plane curves | I believe that this is true for all plane curves parameterized by a monomial map $\mathbb{P}^1\to \mathbb{P}^2$. Denote by $[s,t]$ homogeneous coordinates on $\mathbb{P}^1$, and denote by $[u,v,w]$ homogeneous coordinates on $\mathbb{P}^2$. Up to permuting $u$, $v$, and $w$, every birational monomial map is given by $$... | 12 | https://mathoverflow.net/users/13265 | 227320 | 106,090 |
https://mathoverflow.net/questions/227293 | 4 | Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) $\mu^\star(\mathbf{N}^+)=1$;
(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;
(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\m... | https://mathoverflow.net/users/32898 | Superadditivity of the lower density | I will reuse the same trick as in the [answer](https://mathoverflow.net/a/226461/16537) to Paolo's other question to show that the answer to this is still in the negative.
Let $f$ and $g$ be two upper densities (in the sense of the OP), and let $\alpha \in [0,1]$ and $q \in [1,\infty[$. Then the function
$$h := ((1-\... | 2 | https://mathoverflow.net/users/16537 | 227324 | 106,092 |
https://mathoverflow.net/questions/227317 | 4 | I met with the following problem. Consider real manifold $M^{2n}$ with operator field $R$ (that is the tensor field of type $(1,1)$). We are to find a symmetric connection $\Gamma^k\_{ij}$ such that tensor $\nabla R$ is symmetric in lower indices.
We write an equation $\nabla R^k\_{i,j} - \nabla R^k\_{j,i} = 0$. Expa... | https://mathoverflow.net/users/84714 | Connection, compatible with type (1, 1) tensor field | **N.B.:** I'm fixing my answer, which was off for two reasons: First, I didn't correctly interpret the OP's notation. (Thanks, Sebastian, for pointing that out!) Second, I didn't check the case when the Jordan normal form of $R$ has blocks of size $2$ or more (i.e., multiple eigen*values* but not multiple eigen*vectors... | 6 | https://mathoverflow.net/users/13972 | 227330 | 106,097 |
https://mathoverflow.net/questions/227332 | 1 | Let $X$ be a surface and $Y$ be a curve over $\mathbb{C}$. Let $L$ and $L'$ be ample line bundles on $X$ and $Y$ respectively. Consider the product $X\times Y$. Let $p$ and $q$ be the projection from $X\times Y$ to $X$ and $Y$ respectively.
Then $E=:p^\*L\otimes q^\*L'$ is an ample line bundle on $X\times Y$. I want ... | https://mathoverflow.net/users/70211 | Intersection product of pull back under projection | $(p^\*D)^3=p^\*D^3=0$ and $(p^\*D)^2\cdot q^\*D'= D^2\deg (D')$, hence
$$ p^\*D\cdot (p^\*D+q^\*D')^2= 2 (p^\*D)^2\cdot q^\*D'=2D^2\deg (D')\ .$$Note that the additive notation (say, in the Chow ring) is better adapted for this kind of calculations.
| 2 | https://mathoverflow.net/users/40297 | 227334 | 106,098 |
https://mathoverflow.net/questions/224830 | 1 |
>
> **Theorem $1$(Burnside):** A simple nonabelian finite group can not have a conjugacy classes with prime power elements.
>
>
> **Theorem $2$:** A group of order $p^nq^m$ is solvable.
>
>
>
Theorem $1$ depends on character theory. Theorem $2$ is a direct consequences of Theorem $1$. Burnside proved Theorem $... | https://mathoverflow.net/users/47344 | The groups with nilpotent Hall $p'$ subgroup | It may be that Wielandt's proof that if a finite group $H$ has nilpotent Hall subgroups $A$ and $B$ with $H = AB$ is solvable assumes Burnside's result about
non-Abelian simple groups having no non-identity conjugacy class of prime power
order as a starting point ( so it can be assumed that both $|A|$ and $|B|$ have at... | 1 | https://mathoverflow.net/users/14450 | 227336 | 106,099 |
https://mathoverflow.net/questions/227299 | 3 | Let $X$ be a smooth projective variety of dimension $n$. Let $D$ be a smooth divisor of $X$. Let $i:D\hookrightarrow X$ be the inclusion. Let $H$ be an ample line bundle on $X$.
Let $E$ be a vector bundle of rank $r$ on $X$. Consider $E|\_D$. Suppose I know that $E|\_D$ is $\mu\_{i^\*H}$ semistable, does it imply tha... | https://mathoverflow.net/users/70211 | If the restriction of a vector bundle to a divisor is semi stable, then is the vector bundle itself semistable? | First of all, as Allen Knutson remarks, $X$ should have dimension at least two.
Now the Mehta-Ramanathan theorem tells you that if $\mathcal E$ is $H$-semistable and $D \in |mH|$ is general, with $m$ sufficiently large, then $\mathcal E \big|\_D$ will likewise be $H\big|\_D$-semistable.
But you are asking for the rev... | 7 | https://mathoverflow.net/users/44860 | 227347 | 106,102 |
https://mathoverflow.net/questions/227344 | 0 | Let $X$ be a Banach space and $1\leq p<\infty$. A bounded subset $K$ of $X$ is relatively weakly $p$-compact if $K$ is contained in $S(B\_{l\_{p^{\*}}})$for some operator $S$ from $l\_{p^{\*}}$ into $X$. Let $\mathcal{F}$ be the family of all relatively weakly $p$-compact subsets of $X$. For $K\in \mathcal{F}$, define ... | https://mathoverflow.net/users/41619 | The completeness of locally convex space generated by relatively weakly $p$-compact sets | I think that one can show that the space $(X^\*, \rho\_p^\*)$ is
complete on the following lines:
(1) Consider an arbitrary Cauchy net $x^\*\_\alpha\in X^\*$ in the
described topology. This assumption implies that the net is
pointwise convergent (I mean that $x\_\alpha^\*(x)$ are convergent
nets of scalars for all $x... | 2 | https://mathoverflow.net/users/37822 | 227357 | 106,105 |
https://mathoverflow.net/questions/227352 | 10 | Does there exist a **continuous** (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable $h(\alpha,\beta)$ is uniformly distributed on $[0,1]$ **independent** of $\alpha,\beta$?
**Clarification:** By inde... | https://mathoverflow.net/users/82510 | Constructing an independent uniform random variable from two independent ones | I think this works for a continuous $h$.
Let $f : \mathbb{R} \to [0,1]$ be the "triangle wave" function given on $[0,1]$ by
$$f(u) = \begin{cases}1-2u, & 0 \le u \le \frac{1}{2} \\ 2u-1, & \frac{1}{2} \le u \le 1 \end{cases}$$
and extended periodically. Note that for any $t \in [0,1]$ we have $$m(\{x \in [0,1] : f(x)... | 12 | https://mathoverflow.net/users/4832 | 227367 | 106,107 |
https://mathoverflow.net/questions/227314 | 7 | As I understand it from [Kostecki's notes](https://www.google.co.nz/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiOwN6d1YPKAhXkL6YKHQVLA_IQFggbMAA&url=http%3A%2F%2Fwww.fuw.edu.pl%2F~kostecki%2Fsdg.pdf&usg=AFQjCNHJONLnGglncX47CO1tLlAwrJ6_WA&sig2=rrk9jClR8Y6FXHZugM6wew), the $k$-jet $j^k f$ of a fun... | https://mathoverflow.net/users/56938 | Jets in synthetic differential geometry | It's correct that a jet $u\in J^k(M,N)$ can be defined as the equivalence class of maps $f:M\to N$ whose $k$th Taylor expansions agree at a point $x\in M$. A more "synthetic" way to think of $u$ is as a map from the $k$th infinitesimal nbhd of $x\in M$ to $N$. That's the approach Kock takes in his book *[Synthetic Geom... | 8 | https://mathoverflow.net/users/745 | 227371 | 106,109 |
https://mathoverflow.net/questions/227231 | 6 | Let us consider the short exact sequence of coherent sheaves on $\mathbb{P}^n$ $$0 \to \mathcal{O}\_{\mathbb P^n}(-1)^{r} \stackrel{N}{\longrightarrow} \mathcal{O}\_{\mathbb P^n}^{r} \longrightarrow \mathcal F \to 0,$$
where $r \geq 2$ and $N=\{L\_{ij}\}$ is a $r \times r$ matrix of linear forms in the variables $x\_0,... | https://mathoverflow.net/users/7460 | Global sections of coherent sheaves on determinantal hypersurfaces in $\mathbb{P}^n$ | This is meant to be an integration to Yusuf answer.
Consider a section
$$
\stackrel{\rightarrow}{\alpha}=
\begin{pmatrix}
\alpha\_1\\
\vdots\\
\alpha\_r
\end{pmatrix} \in {\mathbb C}^r\cong H^0({\mathbb P}^n,{\mathcal O}^r\_{{\mathbb P}^n}).
$$
Its image on $H^0(Y,{\mathcal F})$ vanishes in a point $p \in Y$, by def... | 2 | https://mathoverflow.net/users/46104 | 227373 | 106,111 |
https://mathoverflow.net/questions/227375 | 0 | Let $A$ be real square matrix.
Let $\mathcal{F}(A)$ be the set of real matrices $A'$ of the same size such that $A'\_{ii}=A\_{ii}$ for all $i$, and for all $i,j$, $A\_{ij}=0\Rightarrow A'\_{ij}=0 \land A\_{i,j} \ne 0 \implies A'\_{i,j} \ne 0$: in other words, obtained from $A$ by modifying nonzero nondiagonal entries... | https://mathoverflow.net/users/12481 | Making a real matrix positive definite by replacing nonzero and nondiagonal entries with arbitrary nonzero reals | The answer is quite boring... Necessary and sufficient conditions are $A\_{ii}>0$ for each $i$. In this case, let $B\_\varepsilon$ the matrix with entries
$$
(B\_\varepsilon)\_{ij} = \begin{cases}
A\_{ij} & i=j,\\
\varepsilon & i\neq j, A\_{ij}\neq 0,\\
0 & i\neq j, A\_{ij}=0.
\end{cases}
$$
The matrix $B\_0$ is diagon... | 3 | https://mathoverflow.net/users/1898 | 227377 | 106,113 |
https://mathoverflow.net/questions/227382 | 0 | Let $X$ be an abelian variety of dimension $n>2$. Let $L$ be a very ample line bundle on $X$. Is it possible to find two divisors $D\_1,D\_2\in |L|$ which do not intersect or intersect in codimension 3 or higher codimension? Is there a reference for such a result?
This should be possible I feel. Looking forward to th... | https://mathoverflow.net/users/70211 | Will any two linearly equivalent ample divisors on an abelian variety intersect? | Of course not.
If $L$ is very ample, $D\_1$ and $D\_2$ are two hyperplane sections for some embedding in a projective space. Therefore their intersection is at most codimension 2 in $X$, intersection of $X$ with a linear subspace of codimension 2. This has nothing to do with $X$ being an abelian variety.
| 4 | https://mathoverflow.net/users/46104 | 227383 | 106,115 |
https://mathoverflow.net/questions/227351 | 1 | Let $M$ be a manifold and $C\_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational homology module, were given in the paper [*On the homology of configuration spaces*](http://www3.nd.edu/~taylor/papers/cocs.... | https://mathoverflow.net/users/41075 | torsion part of the cohomology module of configuration spaces of manifolds | The following paper answers your question for the case $M=\mathbb{R}P^m$ and $n=2$:
*Carlos Domínguez, Jesús González, and Peter Landweber*, [**The integral cohomology of configuration spaces of pairs of points in real projective spaces**](http://dx.doi.org/10.1515/form.2011.145), *Forum Math.* **25** (2013), no. 6,... | 4 | https://mathoverflow.net/users/8103 | 227386 | 106,117 |
https://mathoverflow.net/questions/225769 | 7 | Good afternoon. I have a particular summation,
$$\zeta\_{n,k}(N)=\frac{k!}{N^{n+1-k}}\sum\_{j=0}^n\sum\_{i=0}^{N-1}\binom{n}{j}w\_N^{(j-k)i}$$
Here, the $w\_N$ is the root of unity $w\_N=e^\frac{2i\pi}{N}$. Given certain conditions on the $n$ and $k$, it is pretty easy to show that for positive integers $k, m, n, N$,... | https://mathoverflow.net/users/60457 | Simplifying Root of Unity Double Summation | I will elaborate on Fedor Petrov's comment.
Interchanging the order of summation and using the binomial theorem, we remain with
$$\frac{k!}{N^{n+1-k}}\sum\_{i=0}^{N-1} \omega\_N^{-ki} (\sum\_{j=0}^{n} \binom{n}{j} \omega\_N^{ji} (1+\omega\_N^i)^n)=$$
$$(\*)\frac{k!}{N^{n+1-k}}\sum\_{i=0}^{N-1} \omega\_N^{(N-k)i} (1+... | 1 | https://mathoverflow.net/users/31469 | 227396 | 106,123 |
https://mathoverflow.net/questions/227374 | 7 | suppose we have the following two sequences
$$\alpha\_k = (k-1)\left(1-\frac {1}{1+(k+1)l}\right) \quad , k \geq 2$$
$$\beta\_k = (k-1)\left(1+\frac {1}{1+(k-1)l}\right) \quad , k \geq 2$$
where $l$ is a positive constant
and define the sequence $c\_k$ recursively by:
$$c\_2 = - 1/\beta\_2 $$
$$c\_3 = 0 $$
$$c\_{k+... | https://mathoverflow.net/users/84747 | Asymptotics of a recursion | For the first sequence, $$d\_k=\frac{\alpha\_{2k-2}}{\beta\_{2k}}d\_{k-1}=\frac{k-3/2}{k+1/2-1/\ell}d\_{k-1}$$
so one has
$$d\_k=C\frac{\Gamma(k-1/2)}{\Gamma(k+3/2-1/\ell)}\ ,$$
the constant $C$ being determined by the initial condition $d\_1$, namely
$$C=d\_1\frac{\Gamma(5/2- 1/\ell)}{\Gamma(1/2)}\ . $$ Recall that $... | 6 | https://mathoverflow.net/users/6101 | 227399 | 106,124 |
https://mathoverflow.net/questions/227154 | 7 | Given a Banach space $X$ and a functional $f:X\rightarrow \mathbb R$, let
$$ X\_f := \{x\in X : f(x)\ge 0\} $$
("*functional*" means "*non-zero linear functional*"). Also, given a topological space $E$ and its topological subspace $A$, a retraction $r:E\rightarrow A$ is defined as a continuous map such that $r(x)=x... | https://mathoverflow.net/users/8385 | A Hilbert space characterization via retractions--a conjecture | I think that the desired result can be proved on the following
lines if the dimension is at least $3$.
(1) Consider such maps for $X\_f$ and $X\_{-f}$. Denote them $r$ and
$r'$ respectively. One can show that $f(r(x))=-f(x)$ for $x\in
X\backslash X\_f$. Similarly one can show that $f(r'(x))=-f(x)$ for
$x\in X\_f$. Le... | 4 | https://mathoverflow.net/users/37822 | 227400 | 106,125 |
https://mathoverflow.net/questions/227418 | 5 | Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite decreasing $E$-sequence. I am interested to know about the degree of illness of internally non-well-founded models in the l... | https://mathoverflow.net/users/38866 | Ill-founded models of set theory with well-founded ordinals |
>
> (I) is true (and therefore so is (II)), assuming that $ZF$ has a well-founded model $M\_\beta$ of ordinal height $\beta$. I will outline a construction that is meant to be carried out in *within* a model of $ZF$. It will produce the desired ill-founded model satisfying (II) when implemented within $M\_\beta$.
> ... | 9 | https://mathoverflow.net/users/9269 | 227423 | 106,133 |
https://mathoverflow.net/questions/227421 | 6 | Consider the permutations of $0,1,1,2,2,3,3.$ Each permutation is corresponding to a vertex in graph $G$. So, the graph $G$ has $630$ vertices.
Each vertex has exactly 6 neighbors. $P$ is connected $Q$ if $P$ can be obtained from $Q$ by swapping 0 with another element. For example,
0112233 is connected to 1012233, 11... | https://mathoverflow.net/users/78423 | Is this graph 3-colorable? | If I constructed the graph correctly, according to a program the chromatic
number is $4$, so the graph is not 3 colorable.
The program is:
<https://code.google.com/p/graphcol/>
Got the same result after converting the problem to SAT
and ran certified UNSAT solver.
The proof for unsatisfiability was only about 11M... | 12 | https://mathoverflow.net/users/12481 | 227424 | 106,134 |
https://mathoverflow.net/questions/227426 | 2 | Denote $p$ a prime number and $\mathbb Z \_p$ the ring of $p$-adic integers.
We have a canonical injective ring homomorphism $:\mathbb Z \rightarrow \mathbb Z\_p$ for all $p$. But $\mathbb Z$ is not the largest ring that maps into all $\mathbb Z\_p$.
Consider for instance the ring of all formal series
$$
F := \left\{... | https://mathoverflow.net/users/62593 | Rings that inject in all p-adic integers | The ring you are looking for is $\widehat{\mathbb{Z}}={\displaystyle\lim\_{\leftarrow}\mathbb{Z}/N\mathbb{Z}}$.
This has canonical maps to $\mathbb{Z}\_p={\displaystyle\lim\_{\leftarrow}\mathbb{Z}/p^k\mathbb{Z}}$ induced by taking $\mathbb{Z}/N\mathbb{Z}\twoheadrightarrow \mathbb{Z}/p^k\mathbb{Z}$ (where $p^k$ is the... | 10 | https://mathoverflow.net/users/250 | 227431 | 106,136 |
https://mathoverflow.net/questions/218113 | 91 | Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
| https://mathoverflow.net/users/2841 | $A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$ | Let $A$ be the additive group of bounded sequences of elements of $\mathbb{Z}[\sqrt{2}]$. Clearly $A\cong A\oplus\mathbb{Z}[\sqrt{2}]\cong A\oplus\mathbb{Z}^2$ as abelian groups, so we just need to show that $A\not\cong A\oplus\mathbb{Z}$.
Let $A\_i\cong\mathbb{Z}[\sqrt 2]$ be the subgroup of $A$ consisting of sequen... | 75 | https://mathoverflow.net/users/22989 | 227443 | 106,138 |
https://mathoverflow.net/questions/227435 | 17 | Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be smoothly embedded in $\mathbb R^{2n-1}$?
I know these cannot exist for $n=1$, i.e. $S^1$. If we ignore orientability, then if we take $n=2^r$, $\mathbb RP^n$ cannot be embedded in $\mathbb R^{2n-1}$.
W... | https://mathoverflow.net/users/33064 | Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$? | A closed smooth $n$-manifold embeds into $\mathbb R^{2n-1}$ if and only if the normal $(n-1)$th Stiefel-Whitney class vanishes. This is due to Hirsch-Haefliger in dimensions $\neq 4$ and to Fang in dimension $4$. Massey showed that if the normal $(n-1)$th Stiefel-Whitney class is nonzero, then $M$ is non-orientable and... | 33 | https://mathoverflow.net/users/1573 | 227444 | 106,139 |
https://mathoverflow.net/questions/227076 | 4 | (I have asked a similar question in [MSE](https://math.stackexchange.com/questions/1585550) four days ago, but did not receive any answers. I have therefore cross-posted it to this site, hoping to get some responses.)
An odd perfect number $N$ is said to be given in **Eulerian form** if $N = {q^k}{n^2}$ where $q$ is ... | https://mathoverflow.net/users/10365 | If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square? | There is no reason to believe that the non-Eulerian part is a 4th power. Moreover, Descartes spoof OPN shows us that such a guess is probably unmotivated and that there is no purely combinatorial way to prove that $n$ is a square. A proof of such a result would fundamentally require restriction to actual (rather than s... | 6 | https://mathoverflow.net/users/3199 | 227462 | 106,144 |
https://mathoverflow.net/questions/227461 | 0 | **Background:** The answer to a previous question I asked [here](https://mathoverflow.net/questions/224995/primitive-sequence-a-i-attaining-pillais-bound-on-sum-i-1-a-i) specified a construction to achieve Pillai's bound on reciprocal sums of primitive sequences. A primitive sequence $1<a\_1<\ldots<a\_k\leq n$ is a seq... | https://mathoverflow.net/users/17773 | Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals | Unfortunately, you have no chance whatsoever. Indeed, assume that $A$ is a subset of $\{1,\dots,n\}$ such that $[a',a'']\ge n+1$ for all $a'\ne a''$ in $A$. Then the numbers $ab: a\in A, 1\le b\le n/a$ are all different. Hence, taking the sum of their reciprocals, we obtain something like
$$
\sum\_{a\in A} a^{-1}\log(n... | 4 | https://mathoverflow.net/users/1131 | 227466 | 106,145 |
https://mathoverflow.net/questions/227477 | -2 | Let $G=(V,E)$ be a simple, undirected graph.
We call a partition ${\cal P}$ of a non-empty subset of $V$ a *Hadwiger partition* if
1. every block (member of ${\cal P}$) is non-empty and connected, and
2. if $x, y \in {\cal P}$ are distinct blocks then there are $v\in x$ and $w \in y$ such that $\{v,w\} \in E$.
Th... | https://mathoverflow.net/users/8628 | Hadwiger partitions where one block is always a singleton | The graph $G=2K\_n$ is a counterexample. Or, if $G$ is supposed to be connected, then $G=2K\_n+e\ $ (that's $2K\_n$ with an additional edge) is a counterexample for $n\ge3.$
| 2 | https://mathoverflow.net/users/43266 | 227483 | 106,154 |
https://mathoverflow.net/questions/227430 | 1 | Let $A$ be a finitely generated $k$-algebra, where $k$ is a field, let $I$ be an ideal in $A$, let $M$ be a finitely generated $A/I$-module, and let $M^{\prime}$ denote $M$ considered as an $A$-module. Let $B$ be a finitely generated $A$-algebra. Is it true (perhaps under some additional conditions on $A$ and $I$, thou... | https://mathoverflow.net/users/12395 | Base change for non-flat coherent sheaves and affine maps | There is an associativity identity for total derived tensor products, cf. [Stacks Project Tag 08YU](http://stacks.math.columbia.edu/tag/08YU). In your case, for the triple of rings, $$A\twoheadrightarrow A/I \xrightarrow{\text{Id}}A/I,$$ this gives an equivalence in the derived category, $$ M\otimes\_{A/I}^{\textbf{L}}... | 3 | https://mathoverflow.net/users/13265 | 227484 | 106,155 |
https://mathoverflow.net/questions/227354 | 3 | On pages 956-957 of [this paper](http://arxiv.org/pdf/math/9811185.pdf), it is established that for any two $v\_1, v\_2$ satisfying $v\_1^2 + 1 \equiv 0\operatorname{(mod} d\_1), v\_2^2 + 1\equiv 0\operatorname{(mod} d\_2)$, $$\left\lVert \frac{v\_1}{d\_1} - \frac{v\_2}{d\_2}\right\rVert > \frac{1}{4\sqrt{d\_1d\_2}}.$$... | https://mathoverflow.net/users/40983 | Well-spacing of the roots of a quadratic congruence | What you have written is not true:
$8^2 + 1 \equiv 0\operatorname{(mod} 13), 3^2 + 1\equiv 0\operatorname{(mod} 5)$ yet $$\left\lVert \frac{8}{13} - \frac35\right\rVert=\frac1{5\cdot 13}=\frac{1}{\sqrt{5 \cdot 13}\sqrt{5 \cdot 13}} <\frac{1}{8\sqrt{5 \cdot 13}}< \frac{1}{4\sqrt{5 \cdot 13}}.$$
The well written paper ... | 4 | https://mathoverflow.net/users/8008 | 227485 | 106,156 |
https://mathoverflow.net/questions/227498 | 1 | I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave:
*Let $M$ be a compact, orientable 3 manifold with non-empty boundary. Then $M$ can be embedded in $\mathbb S^3$. More precisely, $M$ is diffeomorphic to $\mathbb S^3 \setmin... | https://mathoverflow.net/users/78554 | On compact, orientable 3-manifolds with non-empty boundary | Even if you say the boundary has higher genus, it is still false.
You can't embed $\mathbb{RP}^2$ in $S^3$ but you can embed it into $\mathbb{RP}^3$ which is orientable. (Take the quotient of $S^3$ by the antipodal map. The equatorial $S^2$ becomes a one-sided projective plane.) Drilling out a torus in a ball from $... | 7 | https://mathoverflow.net/users/2954 | 227501 | 106,158 |
https://mathoverflow.net/questions/215536 | 4 | Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e\_i\}\;\;i\in \mathbb{N}$.
**Definition:** We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if there is a constant $M$ such that $Ind(T)\leq M$ for all operators $T \in W$ which are Fredholm. In the other words $W$ is a ... | https://mathoverflow.net/users/36688 | Fredholm subvector spaces of $B(\mathcal{H})$ | The statement is false for $n= 2$, as I'll show, with respect to both distances you are considering on $G(n,B(H))$ (this easily implies that it is also false for any $n>2$; on the other hand for $n=1$ the statement is true according to your definition of Fredholm subspace, as it reduces to the fact that Fredholm operat... | 1 | https://mathoverflow.net/users/6101 | 227503 | 106,159 |
https://mathoverflow.net/questions/227479 | 6 | Let $g(x)=(x-a)\mathbf 1\_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought as a first step to use this formula [here](https://en.wikipedia.org/wiki/Expected_value#General_definition):
>
> ...Using... | https://mathoverflow.net/users/48635 | Law of unconsious statistician: application in characteristic function | Formula (1) is correct for any random variable (r.v.) $X$ and any real $a$, even without assuming that $X$ is nonnegative and/or continuous. Indeed, by the Fubini--Tonelli theorem,
$$\int\_a^\infty (1-F(x))\,dx
=\int\_a^\infty P(X>x)\,dx =\int\_a^\infty dx \int\_{(x,\infty)}P(X\in du)
$$
$$=\int\_{\mathbb R}dx\,I\{x... | 3 | https://mathoverflow.net/users/36721 | 227516 | 106,164 |
https://mathoverflow.net/questions/227515 | 15 | Let $\mathbb{C}[S\_n]$ be the regular representation of the symmetric group $S\_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(n-1)!$) such that $V\otimes\mathbb{C}^n\cong \mathbb{C}[S\_n]$? If so, does $V$ admit any particularly nice descripti... | https://mathoverflow.net/users/10273 | factorization of the regular representation of the symmetric group | Let $H$ be a regular subgroup of $S\_n$, for instance a transitive cyclic subgroup of order $n$. Then the permutation module $V$ of the action on the coset space $S\_n/H$ has the requested property, as only the identity element of $S\_n$ lies in a conjugate of $H$ and fixes a point in the natural action at the same tim... | 8 | https://mathoverflow.net/users/18739 | 227519 | 106,165 |
https://mathoverflow.net/questions/227526 | -3 | Let $\cal F^k$ be a set of functions, each of class $C^k$,
i.e., both, for every function in $\cal F^k$:
* $k^{\textrm{th}}$ derivatives exist, and
* are continuous.
Let $D(\cal F^k)$ be the set of all those functions' derivatives,
up to the $k^\textrm{th}$ derivative.
My basic question is:
>
> ***Q0***. Wha... | https://mathoverflow.net/users/6094 | Does differentiation widen, or narrow, the class of functions? | The answer depends on the class you consider. If $F^k=C^k$, the class of all
functions with your properties, then $D(F^k)=C^{k-1}$,
and $C^k$ is a proper subset of $D(F^k)$. If $F^k$ is the class of all polynomials, then $D(F^k)=F^k$. If $F^k$ is the class of polynomials of degree at most $d$, then $D(F^k)$ is a proper... | 10 | https://mathoverflow.net/users/25510 | 227527 | 106,169 |
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