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https://mathoverflow.net/questions/191016 | 8 | The starting point of this question is the fact that any retract of a $T\_2$-space is closed.
Let's say a topological space $(X,\tau)$ is $T\_{\textrm{rc}}$ if all retracts of $X$ are closed.
All $T\_{\textrm{rc}}$-spaces are $T\_1$, because singletons are always retracts, and a space is $T\_1$ if and only if all s... | https://mathoverflow.net/users/8628 | "All retracts are closed" as separation axiom | No. Let $X$ be a compactly generated Hausdorff space which fails to be locally compact at precisely one point. Now take the Alexandroff compactification of $X$, adding exactly one new point, whose neighborhoods have compact complement in $X$. The new space is not Hausdorff, but has the property that it is compact, and ... | 10 | https://mathoverflow.net/users/17029 | 191022 | 93,954 |
https://mathoverflow.net/questions/190911 | 16 | Let $g \in GL\_n(\mathbb{F}\_q)$. Is it true that we can always write $g = u\_1lu\_2$, where $u\_1$ and $u\_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices be unipotent.
This is equivalent to being able to write $g=u\_1lu\_2d$, where $u\_1$ and $u\_2$ are unipotent u... | https://mathoverflow.net/users/61372 | ULU Decomposition of a matrix | The keyword for this is "Gauss decomposition". It states that for a ring R of stable rank 1 one can indeed write any element g of GL(n,R) as a product of three upper and lower triangular matrices (not unipotent, though). So it holds not only for finite fields, but for any field and, more generally, semilocal ring, alon... | 13 | https://mathoverflow.net/users/5018 | 191027 | 93,956 |
https://mathoverflow.net/questions/191024 | 2 | Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q\_n)\_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists $C>0$ such that $\|Q\_n\| \leq C$ for all $n \in \mathbb{N}$.
Now suppose that for all $u \in L^p(\Omega)$, we have $Q\_n u ... | https://mathoverflow.net/users/16702 | Uniformly bounded operator family and pointwise convergence | Let us consider the case $p=1$. Let $u \in L^1(\mathbb{R})$ be a positive function.
Define $f\_n := \chi\_{[n,n+1]}$ and $Q\_nu := u + f\_n \star u$, where $\star$ denotes the convolution.
Notice that, for any such $u$, we have $\|Q\_nu - u\|\_{L^1(\mathbb{R}^n)} = \|u\|\_{L^1} > 0$ and that the sequence $(Q\_n)$ s... | 8 | https://mathoverflow.net/users/62629 | 191029 | 93,957 |
https://mathoverflow.net/questions/191010 | 7 | Skolemization is often used for eliminating existential quantifiers, which is often useful for proving theorems, especially in automated resolution theorem proving. Skolemization in first order predicate calculus is often based on a second order identity:
$\forall$x$\exists$y $\phi$(x,y) $\iff$ $\exists$f$\forall$x $... | https://mathoverflow.net/users/38331 | When does Skolemization require the axiom of choice? | Upon rereading the question, I sense there may be a terminological or conceptual confusion in play.
The terms “second-order logic” and “higher-order logic” are unfortunately used to denote two vastly different things:
1. Proper second/higher-order logic. This is a semantically defined system; its models consist of ... | 6 | https://mathoverflow.net/users/12705 | 191040 | 93,962 |
https://mathoverflow.net/questions/191049 | 6 | Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F\_p$, $P$ will often split, and I am interested in polynomials with the property that, for any given $p$, **all splitting factors in $\mathbb F\_p[x]$ have the same degree**. Call suc... | https://mathoverflow.net/users/29783 | Which criteria for "uniformly splitting" polynomials? | Yes, there are uniformly splitting polynomials of all degrees which are not of the form $\Phi\_{n}(x^{k})$. (For example, $f(x) = x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1$.)
The Chebotarev density theorem implies that if $f(x)$ is irreducible and $p$ does not divide the discriminant of $f$, then the factorization of $f(x)$ m... | 11 | https://mathoverflow.net/users/48142 | 191053 | 93,968 |
https://mathoverflow.net/questions/191054 | 0 | Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy
$$
F\: : \: I\times PM\to PM
$$
Defined by $F(s,y)(t):=y(st)$. This map is an homotopy between $f\_{0}\: : \: PX\to PX$ , $(f\_{0}(y))(t)=y(0)$ and the identity.
Let $K$ be a simplicial set. I woul... | https://mathoverflow.net/users/41970 | contracting homotopy on simplicial sets | It is easier to define the map you're looking for if you adjoint it over and define a map
$Map(\Delta[1],X)\to Map(\Delta[1],Map(\Delta[1],X)) = Map (\Delta[1]\times \Delta[1],X)$
Then this comes from precomposition of the map $\Delta[1]\times \Delta[1]\to \Delta[1]$ sending $(1,1)$ to $1$ and the other vertices to... | 5 | https://mathoverflow.net/users/43054 | 191056 | 93,969 |
https://mathoverflow.net/questions/191047 | 7 | Let $\vartheta(x)=\sum\_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$
Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing $\pi(x)>\operatorname{li}(x)$, and Kotnik gives a lower bound, but I don't know what has been proved on the more fundamental (?)... | https://mathoverflow.net/users/6043 | When is $\vartheta(x)>x$? [Skewes number analog] | Platt and Trudgian show in <http://arxiv.org/abs/1407.1914> that
$$
\theta(x)<x\quad\text{for}\quad x<1.39\cdot 10^{17}
$$
and there is an $x<\exp(727.951332668)<1.4\cdot 10^{316}$ for which $\theta(x)>x$.
| 13 | https://mathoverflow.net/users/6756 | 191060 | 93,972 |
https://mathoverflow.net/questions/191066 | 4 | I'm quite confident that there is the same number of models of $ZFC$ and of $ZF\neg C$, which means that there would exist a bijection between the former and the latter, by definition. However, I was wondering - can we somehow employ a forcing techniques (or some other tools) to find an explicit bijection between them?... | https://mathoverflow.net/users/30186 | Is there a "natural" bijection between models of $ZFC$ and $ZF\neg C$? | Well, the simple answer is that if there is a single model, there is a proper class of models, of each cardinality possible.
But let's instead restrict to transitive models. Then the answer is considerably different. It is consistent that all the transitive models of $\sf ZFC$ have the same height, say $\alpha$. Of c... | 11 | https://mathoverflow.net/users/7206 | 191067 | 93,975 |
https://mathoverflow.net/questions/190985 | 9 | Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?
| https://mathoverflow.net/users/38889 | Is a free group a product of f.g subgroups of infinite index? | It's actually easier to prove the stronger statement that there are infinitely many double cosets $H\backslash F/K$.
First, note that if $F'$ is a subgroup of finite index in $F$, with $H'$ and $K'$ its intersections with $H$ and $K$ respectively, then the natural map $H'\backslash F'/K'\to H\backslash F/K$ is finit... | 7 | https://mathoverflow.net/users/1463 | 191072 | 93,977 |
https://mathoverflow.net/questions/191069 | 6 | Consider the following infinite matrix: $A\_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for parameter $t$?
| https://mathoverflow.net/users/4312 | Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$ | See ["On the Hilbert matrix II"](http://www.ams.org/journals/proc/1958-009-04/S0002-9939-1958-0099599-2/) by Rosenblum
| 10 | https://mathoverflow.net/users/1847 | 191089 | 93,979 |
https://mathoverflow.net/questions/191064 | 1 | Let $\gamma\colon H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ be the linear trace map which has a right continuous inverse $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$.
Is the image of $\xi$ dense in $H^1(\Omega)$? i.e., for every $u \in H^1(\Omega)$, do there exist $w\_n \in H^{\frac 12}(\partial\Omega... | https://mathoverflow.net/users/60869 | Image of (right) inverse trace map $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$ dense in $H^1(\Omega)$? | First, $\xi$ cannot be surjective.
Consider two distinct functions in $H^1(\Omega)$ with the same boundary values; they cannot both be in the range of $\xi$.
This is a general property of quotient maps (remember that $H^{1/2}(\partial\Omega)=H^1(\Omega)/H^1\_0(\Omega)$).
But $\xi$ cannot even have a dense image.
Supp... | 2 | https://mathoverflow.net/users/55893 | 191097 | 93,982 |
https://mathoverflow.net/questions/190819 | 10 | Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times\_H 1$ or fixed points $X^H = \mathrm{Hom}\_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed the $G$-action.
What we have left instead is an action of $N\_H/H$, where $N\_H$ denotes the normaliser of $H$, the... | https://mathoverflow.net/users/nan | What is the universal property of quotienting a normaliser of the subgroup? | The action is the action of the (natural) automorphism group of the relevant functor in each case. This is easiest to see for the case of
$$X^H \cong \text{Hom}\_H(1, X) \cong \text{Hom}\_G(G/H, X)$$
since by the Yoneda lemma the automorphism group of this functor is the automorphism group of $G/H$ as a $G$-set, w... | 10 | https://mathoverflow.net/users/290 | 191104 | 93,985 |
https://mathoverflow.net/questions/191107 | 4 | Assuem we have a finite surjective map between two irreducible, separated schemes, $f:X \rightarrow Y$, and for a dense open $U \subset Y$ and for any $y \in U$, $|X\_y| =1$, then can we say $f$ is injective everywhere?
We can also assume that $Y$ is normal.
| https://mathoverflow.net/users/59151 | Injective map between two schemes | With some additional hypotheses this follows from Zariski's Main theorem.
>
> If $f \colon X \to Y$ is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of $Y$ is connected. [Hartshorne (1977, Corollary III.11.4)]
>
>
>
See also: <https://en.wiki... | 4 | https://mathoverflow.net/users/21815 | 191108 | 93,988 |
https://mathoverflow.net/questions/190999 | 13 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's *Esquisse* he claims that the action of
$$\Gal(\mathbb Q)\to\text{Out}(\pi\_1(M\_{1,1})=\text{Out}(\widehat{\SL(2,\Z)})$$
obtained from the homotopy exact sequence of the étale fundamental group contains, after abelia... | https://mathoverflow.net/users/48554 | Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$ | What Grothendieck is doing to get all the Galois representations is:
1) Take a finite index subgroup of $\pi\_1(\mathcal M\_{1,1})$ that is stable by the $\operatorname{Gal}(\overline{\mathbb Q}|\mathbb Q)$ action.
2) Take the maximal pro-$\ell$ quotient of its abelianization.
3) Look at the action of $\operatorn... | 11 | https://mathoverflow.net/users/18060 | 191119 | 93,992 |
https://mathoverflow.net/questions/191101 | -1 | Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum\_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
Where $\{x\}$ is the fractional part of $x$.
I find this problem in G.Tenenbaum's book in the page 118.
Gérald Tenenbau... | https://mathoverflow.net/users/64143 | Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ | Write the function $x\mapsto\{x\}-\frac{1}{2}$ as a Fourier series, and approximate this series by a smoothed finite sum using Vaaler's lemma. You obtain something of the form
$$
\sum\_{a<n\leq b}\left(\{f(n)\}-\frac{1}{2}\right) = \sum\_{a<n\leq b}\sum\_{k\leq K} a\_k e(k f(n)) + R,
$$
where $e(t)=e^{2\pi i t}$, the $... | 3 | https://mathoverflow.net/users/37555 | 191122 | 93,993 |
https://mathoverflow.net/questions/191116 | 21 | Mostly out of curiosity: Where do I find Joyal's letter to Grothendieck in which he defines a model structure on simplicial sheaves?
The question was already asked in [this MO post](https://mathoverflow.net/questions/5179/global-fibrations-of-simplicial-sheaves), but that particular part of the question has not been ... | https://mathoverflow.net/users/62434 | Joyal's letter to Grothendieck | The letter may be found on Georges Maltsiniotis' [webpage](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps.html) containing material related to Pursuing Stacks. (A [direct link](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf) to the pdf.)
| 27 | https://mathoverflow.net/users/57405 | 191123 | 93,994 |
https://mathoverflow.net/questions/191000 | 9 | For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: [What is entropy, really?](https://mathoverflow.net/questions/146463/what-is-entropy-really), [What is the role of the logarithm in Shannon's entropy? - Stats.SE](https://stats.stac... | https://mathoverflow.net/users/9093 | Higher moments of information and Renyi entropy | In
* H. Jürgensen, D. E. Matthews, "Entropy and Higher Moments of Information", Journal of Universal Computer Science vol 16, nr. 5 (2010)
which is available [here](http://www.jucs.org/jucs_16_5/entropy_and_higher_moments/jucs_16_05_0749_0794_juergensen.pdf), the authors introduce the higher moments of information... | 5 | https://mathoverflow.net/users/43088 | 191126 | 93,997 |
https://mathoverflow.net/questions/191088 | 6 | I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance.
>
> Question: I am seeking further exposition of the material in Chapter 4 or 5 of the text. I have prolonged some tableaus, but, I have doubts an... | https://mathoverflow.net/users/24854 | Beginners Guide to Cartan for Beginners | The following may help: Fueter tried to describe quaternionic analysis in the sense of $\mathcal A$-Analysis in the 20's or 30's, but only first order quaternionic polynomials were possible.
The working generalization is hypercomplex Analysis or Clifford Analysis, where one generalizes the Laplacian from complex Analys... | 3 | https://mathoverflow.net/users/26935 | 191127 | 93,998 |
https://mathoverflow.net/questions/191129 | 2 | I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any compatibility between the Frobenius algebra structures (e.g. trace map) of $A$ and $B$. In all the examples I know of (group algebra... | https://mathoverflow.net/users/29738 | Example of a Frobenius algebra that is not projective over a Frobenius subalgebra | You can take $A={\mathbb C}[x]/x^3$ and $B$ -- subalgebra generated by $x^2$.
| 15 | https://mathoverflow.net/users/4158 | 191131 | 93,999 |
https://mathoverflow.net/questions/191139 | 14 | Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph\_\alpha} = \aleph\_{F(\alpha)}$ for all ordinals $\alpha$. It is known of course that $F(\alpha) = \alpha+1$ is a complete resolution of GCH (in the positive) that ... | https://mathoverflow.net/users/17218 | Complete resolutions of GCH | One candidate answer scheme might be the following: if $F$ is any (sufficiently absolute) definable function on the class of regular alephs such that $\kappa < \lambda \Rightarrow F(\kappa) \leq F(\lambda)$ and $\operatorname{cf}(F(\kappa)) > \kappa$, then ZFC + $(\forall \kappa = \operatorname{cf}(\kappa))(2^\kappa = ... | 7 | https://mathoverflow.net/users/57583 | 191142 | 94,002 |
https://mathoverflow.net/questions/191133 | 15 | Suppose $B$ is a $d$-dimensional ball (for some $d \geq 1$) and $T$ is a homeomorphism from $B$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that $T^n(x)=x$ for all $x$ in $B$). Can we conclude that a nonperiodic point exists (that is, for some $x$ in $B$ it is the ... | https://mathoverflow.net/users/3621 | Nonperiodic points of homeomorphisms of a ball | Montgomery showed in 1938 that a "pointwise periodic" homeomorphism of a manifold without boundary is actually periodic (this is in Montgomery and Zippin, page 223, or thereabouts), so if your ball is a closed ball, the result you want follows from this by doubling, and if it's the open ball, you don't even need to dou... | 13 | https://mathoverflow.net/users/11142 | 191143 | 94,003 |
https://mathoverflow.net/questions/191136 | 2 | I'm looking for a reference of an isogeny fact that I've used many times but am having a hard time proving formally.
One can define the degree of an isogeny as the degree of extension fields of the function fields of two elliptic curves using the pullback.
One can also write an isogeny $\phi$ as a map $\phi((x,y))... | https://mathoverflow.net/users/51725 | Simple Isogeny Question | First, I think you have a typo. You say "One can define an isogeny as the degree ...", but you probably mean that "One can define the degree of an isogeny as the degree ...".
Second, an isogeny $\phi$ is a homomorphism, so it commutes with $[-1]$, so $\phi$ induces a map on the quotient $E/\pm1$, which is just the p... | 6 | https://mathoverflow.net/users/11926 | 191146 | 94,005 |
https://mathoverflow.net/questions/191110 | 5 | Suppose a system of polynomial equations with rational coefficients has a real solution. Does necessarily there exists a real solution with algebraic coordinates? What about the simplest case of one polynomial equation in two variables?
| https://mathoverflow.net/users/51663 | Real algebraic solution | The is an immediate application of the "real Nullstellensatz", in *exactly* the same way that one answers the analogous questions between $\overline{\mathbf{Q}}$ and $\mathbf{C}$ by using the usual Nullstellensatz (over $\overline{\mathbf{Q}}$!). I trust that the OP knows this latter application of the usual Nullstelle... | 7 | https://mathoverflow.net/users/61939 | 191153 | 94,007 |
https://mathoverflow.net/questions/191157 | 4 | As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of natural numbers. But on one hand we have Forcing Methods and Theory of Core Model to investigate about reals and the real li... | https://mathoverflow.net/users/38866 | Where do Set Theory and Number Theory meet together? | The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:
There exists a classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according
to their properties of approximation by algebraic numbers.
In the p... | 3 | https://mathoverflow.net/users/11115 | 191158 | 94,011 |
https://mathoverflow.net/questions/191162 | 2 | Let $a\_{1},\dots,a\_{n}$ be positive natural numbers ($n>2$) such that $a\_{i}\neq a\_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum\_{i=1}^n a\_{i}^{k} \text{ for some } k\in\mathbb{N} \right\} \right\rvert = \infty . $$
I considered $p > a\_{i},\,\forall i=1,\dots,... | https://mathoverflow.net/users/60815 | Cardinality of the prime divisor set of a k-power sum | Without loss of generality, we can assume that there is no prime dividing all $a\_i$. For each prime $p$, denote by $n\_p$ the number of those $a\_i$ not divisible by $p$, and let $v\_p$ be the power to which $p$ divides $n\_p$. (Notice that $n\_p>0$ by the assumption that no prime divides all $a\_i$; hence, $v\_p$ are... | 4 | https://mathoverflow.net/users/9924 | 191163 | 94,012 |
https://mathoverflow.net/questions/191152 | 6 | For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified covers of $X^\text{an}$.
If $X$ is instead a Deligne Mumford stack over Spec $\mathbb{C}$ corresponding to a complex orb... | https://mathoverflow.net/users/15242 | Is there a Riemann existence theorem for orbifolds? | The answer is yes. See Theorem 20.4 on p. 80 in Behrang Noohi's
<http://arxiv.org/pdf/math/0503247v1.pdf>
| 4 | https://mathoverflow.net/users/4333 | 191164 | 94,013 |
https://mathoverflow.net/questions/191168 | 1 | 1)Let $Cd\_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd\_{\geq 0}ga\to dg\_{\geq 0}Mod$ be the forgethfull functor, where $dg\_{\geq 0}Mod$ is the category of positively graded cochain modules over $\Bbbk$ (equipped with its ... | https://mathoverflow.net/users/41970 | Bounded dg algebra vs unbounded dg algebras | The free-forgetful adjunction still works in the non-negatively graded setting and induces a cofibrantly generated model category structure in caracteristic zero.
This follows actually from a very general statement about the transfer of such model structures for algebras over operads in a symmetric monoidal model cat... | 1 | https://mathoverflow.net/users/36625 | 191172 | 94,016 |
https://mathoverflow.net/questions/191175 | 4 | Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^\*$-algebra such that $A''=M$.
Let $p,q \in M\_{\infty}(A)$ be (Murray–von Neumann) equivalent projections.
Let $U\_{p,q} = \{u \in M \otimes B(H) \text{ partial isometry } \vert uu^\... | https://mathoverflow.net/users/34538 | Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra | In general the answer is no; here is one example. Let $A=\{\lambda I +K:\lambda\in \mathbb C, \ K \text{ compact }\}$ (the unitization of the compacts in $B(H)$). Then $M=B(H)$. Let $q$ be the identity and $p$ a projection of codimension one. Then every member of $U\_{p,q}$ is an isometry which is unitarily equivalent ... | 4 | https://mathoverflow.net/users/13360 | 191176 | 94,018 |
https://mathoverflow.net/questions/191184 | 4 | I was trying to generalize,
$$\sqrt[3]{\sum\_{k=1}^5\cos\big(\tfrac{2^k\cdot\,2\pi}{31}\big)}+\sqrt[3]{\sum\_{k=1}^5\cos\big(\tfrac{2^k\cdot\,6\pi}{31}\big)}+\sqrt[3]{\sum\_{k=1}^5\cos\big(\tfrac{2^k\cdot\,10\pi}{31}\big)} = -\sqrt[3]{\tfrac{-11+3\,\sqrt[3]{62}}{2}} \tag1$$
which is a special case of an identity of... | https://mathoverflow.net/users/12905 | Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$ | Let $\zeta = e^{2\pi i/p}$ be a primitive $p$th root of unity. Then
$2 \cos (2\pi k/p) = \zeta^k + \zeta^{-k}$. The Galois group of $\mathbb Q(\zeta)$
is isomorphic to $(\mathbb Z/p \mathbb Z)^\times$ and acts transitively on
the powers $\zeta^k$ with $1 \le k \le p-1$. What you want is that the Galois
action fixes the... | 14 | https://mathoverflow.net/users/21146 | 191186 | 94,021 |
https://mathoverflow.net/questions/191185 | 3 | Let $S\_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$.
Say that a subset $S \subseteq S\_n$
*square-represents* $S\_n^2$ if every
square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting
at most one copy of squares of elements of $S$.
*Example*. For $n=7$, the set $S$ of $|S|=5$ numbers
$\{1, 2, 3, ... | https://mathoverflow.net/users/6094 | Sets of squares representing all squares up to $n^2$ | We can have the size of $S$ as small as $c \ln(n)$ for some constant $c$, and we can do this in such a way that every element of $\{1, 2, \ldots, n^2 \}$ can be represented by adding or subtracting at most copy of a square of something in $S$. I will construct a sequence of sets $T\_{k}$ so that $T\_{k}$ has size $k$, ... | 9 | https://mathoverflow.net/users/48142 | 191188 | 94,023 |
https://mathoverflow.net/questions/191201 | 6 | I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL\_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle A,B\rangle\_e :=tr(AB^\*)$. Let $pr:SL\_2(\mathbb{R})\rightarrow SL\_2(\mathbb{R})/SO\_2(\mathbb{R})=\mathbb{H}^2$ be ... | https://mathoverflow.net/users/3969 | Do geodesics in SL2R map to geodesics in the hyperbolic plane? | Exactly horizontal geodesics (perpendicular to the isotropy subgroup of the point in hyperbolic space) project to geodesics. Others do not, they give rise to more general curves
(called ballistic curves [here](http://www.mat.univie.ac.at/~michor/orbits.pdf)).
Edit (twice)
============
To be specific, my remark app... | 6 | https://mathoverflow.net/users/26935 | 191212 | 94,030 |
https://mathoverflow.net/questions/188571 | 13 | Here is an issue that thoroughly confuses me. I hope I can express it in a way that is clear cut enough for this site.
Let $G$ be a real reductive Lie group and $\mathfrak{g}$ be the complexification of its Lie algebra. (We work with complexifications since we are interested in complex representations.) Let $P$ be a ... | https://mathoverflow.net/users/41139 | What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra? | I don't know whether this helps, but there is a nice description of the relation between the two modules in geometric terms. The principal series representation can be viewed as the space of smooth sections of the homogeneous line bundle $E$ over $G/P$ induced by the representation $\lambda$ of $P$. Then you can form t... | 11 | https://mathoverflow.net/users/64141 | 191217 | 94,032 |
https://mathoverflow.net/questions/190490 | 2 | Let $X$ be a projective scheme flat over a local Artinian ring $A$, the residue field of $A$ is algebraically closed, and the special fiber of $X$ (under the natural morphism from $X$ to $A$) is smooth, projective. Let $E$ be a locally free globally generated sheaf over $X$ of rank $r$. As far as I understand for a gen... | https://mathoverflow.net/users/46578 | Degeneracy locus and flatness over local Artinian ring | There are some necessary basic facts about flatness. First of all, let $R$ be a local Noetherian, flat $A$-algebra, and let $x\_1,\dots,x\_c \in \mathfrak{m}\_R$ be a collection of elements. Then, by the [Local Flatness Criterion](http://stacks.math.columbia.edu/tag/00MD), if the images $\overline{x}\_1,\dots,\overline... | 3 | https://mathoverflow.net/users/13265 | 191219 | 94,033 |
https://mathoverflow.net/questions/184719 | 12 | Let $\Sigma\_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi\_1(\Sigma\_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma\_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
Moreover, what is $[G,G]$; e.g. if $g=2$?
| https://mathoverflow.net/users/5259 | Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$ | Let me start by interpreting the question "What kind of surface is $S$?" in the case of a general connected oriented topological surface (without boundary). (I am considering only oriented surfaces just for simplicity of discussion.)
If $S$ had finite complexity, i.e., would be homeomorphic to the interior of a compac... | 5 | https://mathoverflow.net/users/21684 | 191226 | 94,035 |
https://mathoverflow.net/questions/191120 | 17 |
>
> **Edit:** According to the Gelfand duality between topological spaces and commutative $C^{\*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is $C\_{0}(\mathbb{R})$. One can think to stabilization of this question, that is $A=C\_{0}(\mathbb{R})\otimes \mathc... | https://mathoverflow.net/users/36688 | The letters of the word "ART" | Yes there are infinitely many, by a version of Mike Jury's idea. In fact there are uncountably many embeddable in $\mathbb R^2$. Take the union of two real curves:
the open one $y= x^{-1}\sin(1/x)$
the closed one $x= f(y)$ for $f$ a function that is $0$ on some closed set $S$ and negative elsewhere.
The closure ... | 5 | https://mathoverflow.net/users/18060 | 191231 | 94,037 |
https://mathoverflow.net/questions/191223 | 3 | Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra.
If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ in general?
If not in general, for what kind of pair $(g, g')$ or $\tau$, this is true?
| https://mathoverflow.net/users/56989 | Extension of an involutive automorphism | [REVISED]
The answer to the first question is NO. It's difficult as a rule to show that an involutive automorphism of $\mathfrak{g'}$ fails to extend to $\mathfrak{g}$,
so it may be better to consider first the possible automorphisms of a *simple* Lie algebra $\mathfrak{g}$ (up to conjugacy in the adjoint group). Thi... | 2 | https://mathoverflow.net/users/4231 | 191233 | 94,038 |
https://mathoverflow.net/questions/190981 | 12 | Let $M$ be a $0-1$ matrix.
Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where each step you take you step on another $1$.
Can every $0-1$ be converted to a matrix of one component by permutations... | https://mathoverflow.net/users/10035 | Connected components $0-1$ matrices | This is fedja's beautiful comment, posted as an answer for better visibility:
Not all matrices can be brought to one component form by exchanging rows/columns.
Consider large $n\times n$ matrices with all possible entries equal to $0,1$. By partitioning this into $3\times 3$ blocks, we see that the number of matric... | 11 | https://mathoverflow.net/users/48839 | 191240 | 94,044 |
https://mathoverflow.net/questions/191241 | 2 | Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that:
1. $G$ has no complete subtrees (the graph below any vertex of $G$ is not a complete binary tree).
2. There exists some $\epsilon > 0$ such that for any $n \in \mathbb{N}$ th... | https://mathoverflow.net/users/38889 | Removing subtrees | Yes. In fact you can take the tree corresponding to all sequences $ x$ of 0s and 1s such that the fraction of 1s is no more than 2/3.
| 4 | https://mathoverflow.net/users/4600 | 191242 | 94,045 |
https://mathoverflow.net/questions/191236 | 2 | Let $n$ and $k$ be two given numbers. The goal is to choose $n$ subsets from $\{1,2,...,n\}$ such that the union of any $k$ of these subsets is the set $\{1,2,...,n\}$ and the union of any $m < k$ of these subsets is not the entire set $\{1,2,...,n\}$.
If $n$ and $k$ are given numbers, then how many ways are there to... | https://mathoverflow.net/users/64181 | Combinations Question about the construction of some special sets | Suppose that $\mathscr{F} := \lbrace F\_1, \ldots, F\_m\rbrace$ is a family of subsets of $X := \lbrace 1, 2, \ldots, n\rbrace$ with the property that the union of any $k$ members of $\mathscr{F}$ is all of $X$ but the union of any $k-1$ members of $\mathscr{F}$ is *not* all of $X$.
Consider the $m\times n$ 0-1 matri... | 1 | https://mathoverflow.net/users/3106 | 191246 | 94,048 |
https://mathoverflow.net/questions/191222 | 34 | Some years back (before MathOverflow was born), Tom Leinster [asked](https://golem.ph.utexas.edu/category/2008/01/2toposes.html#c014438) an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for:
>
> Does there exist a category $C$ that admits an essentially surjective functor... | https://mathoverflow.net/users/2926 | Cantor's theorem for presheaves? | No such category exists. My original argument for this assumed local smallness and is below the break; here is a simpler argument that does not require local smallness (though it does basically use my original argument in the special case $\mathbf{C}=\mathbf{Set}$).
Let us take $\kappa$ to be an inaccessible cardinal... | 22 | https://mathoverflow.net/users/75 | 191258 | 94,054 |
https://mathoverflow.net/questions/191260 | 11 | Let
$$ Br\_3 := \langle \tau\_1,\tau\_2\ :\ \tau\_1 \tau\_2 \tau\_1 = \tau\_2 \tau\_1 \tau\_2 \rangle $$
be the braid group on three strands, and consider the surjection
$$\phi : Br\_3 \twoheadrightarrow SL\_2(\mathbb Z), \qquad
\tau\_1 \mapsto \begin{pmatrix} 1&0\\ 1&1\end{pmatrix}, \quad
\tau\_2 \mapsto \begin{pmatr... | https://mathoverflow.net/users/391 | Is there a "good" reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$? | This answer is essentially equivalent to Misha's and Dylan's, but phrased in terms of configuration spaces rather than mapping class groups. As you note, $Br\_3$ is the fundamental group of the configuration space of 3 points in the (complex) plane, a 3-complex dimensional space. One may take a subspace whose center of... | 12 | https://mathoverflow.net/users/1345 | 191263 | 94,056 |
https://mathoverflow.net/questions/191266 | 6 | Let $(X,\tau)$ be a topological space such that
$\tau\ne\{\emptyset\ X\}.\ $
We call an open cover $\mathcal{U}$ of $(X,\tau)$ *proper* if
$\ X\notin \mathcal{U}.\ $ Moreover we say that $(X,\tau)$ is
* *anti-compact* if it does not have a finite proper cover;
* *anti-paracompact* if for every proper cover $\mathcal... | https://mathoverflow.net/users/8628 | Anti-compactness | A space is anti-compact iff it has no proper covers consisting of two sets, or equivalently if the intersection of any two nonempty closed sets is nonempty. This is equivalent to the specialization order being directed downwards.
We can use this to prove any anti-compact space is anti-metacompact, so your three condi... | 7 | https://mathoverflow.net/users/75 | 191268 | 94,058 |
https://mathoverflow.net/questions/191253 | 2 | In 1985 [Perelli, Pintz & Salerno](http://link.springer.com/article/10.1007%2FBF01388653#page-1) proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the reduction of the lower bound on $\theta$?
I'm not fussed if so... | https://mathoverflow.net/users/49438 | Bombieri-Vinogradov in short intervals | As far as I know the Prime Number Theorem in the form $\pi(x+x^\theta)-\pi(x)\sim\frac{x^\theta}{\log x}$ is not proven for any fixed $\theta<\frac{7}{12}$. So I guess the best you could hope for would be $\theta=\frac{7}{12}-\omega(x)$, where $\omega\searrow0$, but even that would be pretty difficult.
| 2 | https://mathoverflow.net/users/37555 | 191270 | 94,059 |
https://mathoverflow.net/questions/191280 | 3 | Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^\* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is said to be *dominating* if for all $f\in \omega^\omega$ there is $g\in D$ such that $f <^\* g$. Set $$\frak{d} = \textrm{min}\{... | https://mathoverflow.net/users/8628 | Is it consistent that $\frak{d} < 2^{\aleph_0}$? | Theorem 5.1 in Eric van Douwen's paper "The integers and topology" (Handbook of Set-theoretic Topology) is an old reference for a positive answer to your question and will provide a fuller explanation of the proof mentioned by Guest1245. There are newer references too: see the relevant chapter in L. Halbeisen, *Combina... | 4 | https://mathoverflow.net/users/57583 | 191283 | 94,065 |
https://mathoverflow.net/questions/191225 | 10 | Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have that $\frac{|\phi(S)|}{|G|} \geq \epsilon$ (that is, the closure of $S$ in the profinite completion of $F$ has positive Haar... | https://mathoverflow.net/users/38889 | Can a positive measure subset of a free group be nowhere dense? | Yes, $S$ can be nowhere dense. The idea is to build a 'fat Cantor set' inside $F$.
Let $H\_0=F,H\_1,H\_2,\dots$ be finite-index subgroups of $F$ such that $H\_{i+1}\leq H\_i$ for each $i$ and such that $\bigcap\_{i=0}^\infty H\_i = \{1\}$. Start with $i\_0=0$ and $C\_0=\{H\_0\}$, and then iteratively do the following... | 2 | https://mathoverflow.net/users/20598 | 191288 | 94,069 |
https://mathoverflow.net/questions/191228 | 3 | Let $F$ be a finitely generated free group, $H\_1, \dots, H\_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon F \to G$ such that $$\frac{|\phi(H\_1 \cdots H\_n)|}{|G|} < \epsilon ?$$ Where $H\_1 \cdots H\_n = \{h\_1 \cdots h\_n : h... | https://mathoverflow.net/users/38889 | Measuring products of finitely generated subgroups of free groups | The answer is yes.
Let's prove this by induction on $n$. If $n=1$, then by M. Hall's theorem there is a finite index subgroup $K \leqslant F$ such that $H\_1 \subseteq K$ and $|F:K|>1/\epsilon$. Choose any finite index normal subgroup $N \lhd F$, which is contained in $K$, let $G=F/N$ and let $\phi:F \to G$ be the n... | 4 | https://mathoverflow.net/users/7644 | 191291 | 94,072 |
https://mathoverflow.net/questions/191262 | 18 | In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to the following paper:
>
> Sierpinski, W. **"L’axiome de M. Zermelo et son rôle dans la théorie des ensembles et l’an... | https://mathoverflow.net/users/7206 | Sierpinski's construction of a non-measurable set | Here's Sierpinski's argument: Let $h:[\mathbb{R}]^{\omega} \to \mathbb{R}$ be any injection. Define $f:\mathbb{R} \to \mathbb{R}$ by $f(x) = h(E\_x)$ where $E\_x$ is the set of all reals which are at a rational distance from $x$. Note that $x - y$ is rational iff $f(x) = f(y)$. Towards a contradiction, suppose $f$ is L... | 17 | https://mathoverflow.net/users/2689 | 191293 | 94,073 |
https://mathoverflow.net/questions/191055 | 3 | Let $\;P\_{0} \in OPS^{m}\_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic equation
$$
\frac{\partial u}{\partial t} = i\;A(t,x,Dx)\;u
$$
Egorov's theorem applies to the operator $P(t) = S(t)\;P... | https://mathoverflow.net/users/62513 | Application of Egorov's Theorem for Pseudodifferential Operators | If $S(t)$ is such that $S(0)=Id$ and
$$
\dot S=iA S,
$$
the operator $S$ is a Fourier integral operator which quantizes the canonical transformation $\chi$
given by the (non-autonomous) flow of $H\_a$, the Hamiltonian vector field of the principal symbol $a$ of $A$, which is assumed of real-principal type.
Now if $P\_0... | 2 | https://mathoverflow.net/users/21907 | 191299 | 94,075 |
https://mathoverflow.net/questions/191250 | 1 | Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl\_p(\mathbb CP^2)$, then for the Line bundle $L=-K\_X$, we have for [Futaki invariant](https://web.math.princeton.edu/~chil/Futaki.pdf) $Fut\_L\neq 0$, I want to see, what about other Line bundles, have we same result?
| https://mathoverflow.net/users/nan | Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles | My knowledge is by no means up-to-date, but in case it's useful for posterity, the situation for cohomogeneity-one almost-homogeneous spaces with two ends (which includes Hirzebruch surfaces, the blow-up of $\mathbf{P}^{n}$ along linear subspaces of complementary dimension, etc.) can be found in [*On existence of Kähle... | 2 | https://mathoverflow.net/users/36697 | 191304 | 94,078 |
https://mathoverflow.net/questions/191311 | 0 | I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known.
1. Let $\Lambda$ be an odd, unimodular matrix of signature $(1,T)$. In particular, there exists a basis in which the bilinear form $B$ of the lattice is given by
\begin{equation}
B = ... | https://mathoverflow.net/users/13731 | Lower bounds on the rank of a unimodular lattice, given the binlinear pairing of a subset of basis vectors | $b(n,k)\ge2k$ is a pretty good bound. You definitely have $b(n,k)\le2k+\mathrm{const}$ with a constant not very large (something like at most $8$, but you can make it better depending on $kn\bmod8$). Here is the argument.
The situation with **even** form is somewhat simpler and everything follows from
>
> MR0525... | 1 | https://mathoverflow.net/users/44953 | 191314 | 94,081 |
https://mathoverflow.net/questions/191313 | 1 | Let $G$ be a LCA group. It is well-known that to every closed subgroup $H$ of $G$ correspond a closed subgroup in the dual group $\widehat{G}$, namely the annihilator of $H$.
My question is this : is there a one-to-one correspondance between the closed subgroups of $G$ and the closed subgroups of the dual group $\wid... | https://mathoverflow.net/users/64234 | Is every closed subgroup of dual group an annihilator? | Each closed subgroup $i: H \to G$ (each regular mono, in category-speak) induces a quotient map of topological groups $\hat{i}: \hat{G} \to \hat{H}$ (a regular epi), whose kernel is the annihilator. There is a one-one correspondence between such regular monos and regular epis under the Pontryagin dual equivalence. Furt... | 3 | https://mathoverflow.net/users/2926 | 191315 | 94,082 |
https://mathoverflow.net/questions/191095 | 5 | I would be very happy to know about original references for the following results;
For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1)
(1) There exist uncountably many ergodic invariant probability measures.
(2) Atomic invariant measures are weak star dense in the set of a... | https://mathoverflow.net/users/20471 | invariant measures of the expanding maps on the circle | The proof of (2) is contained in the much more general theorem due to Sigmund. This is because the main result of Sigmund "Generic Properties Of Invariant Measures
for Axiom A-Diffeomorphisms" Inventiones Math. 11 (1970), pp. 99-109 applies. One may argue that Sigmund considers only toral automorphisms, but his reasoni... | 7 | https://mathoverflow.net/users/24676 | 191317 | 94,083 |
https://mathoverflow.net/questions/191306 | 11 | In addition to the Euler totient function, there are a great many generalizations and related functions which go by the "totient", usually with some name: Jordan, Lehmer\*, Schemmel, Nagell, Alder, Lucas, Stevens, Eugeni–Rizzi, Holden–Orrison–Vrable, Cohen, Menon, Garcia–Ligh, von Sterneck, etc.
Is there some unifyin... | https://mathoverflow.net/users/6043 | What is a totient? | *What is a totient?* Here is one [answer](http://www.hindawi.com/journals/ijmms/1996/129618/abs/) that might have the generality you seek:
>
> An arithmetical function is said to be a totient if it is the
> Dirichlet convolution between a completely multiplicative function and
> the inverse of a completely multip... | 12 | https://mathoverflow.net/users/11260 | 191318 | 94,084 |
https://mathoverflow.net/questions/191320 | 6 | I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some highlights?
| https://mathoverflow.net/users/nan | Survey paper on isoperimetry | There's been several articles in the comments that are "historical survey" articles. Its not totally clear if you're interested in "current research surveys," but if you are, here are several very nice ones:
Osserman's article: <http://www.ams.org/mathscinet-getitem?mr=500557> provides an excellent survey of older re... | 5 | https://mathoverflow.net/users/1540 | 191321 | 94,085 |
https://mathoverflow.net/questions/191322 | 8 |
>
> The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
> of points so that a line through every two of them passes through a
> third unless they are all on a single line.
>
>
>
Is there any modern research that generalizes this theorem or finds some unexpected relations betw... | https://mathoverflow.net/users/nan | Generalization of Sylvester-Gallai theorem | Yes! See the beautiful recent [paper](http://arxiv.org/pdf/1208.4714v3.pdf) of Ben Green and Terry Tao, which shows that for large $n$, any collection of $n$ points not all collinear will have at least $n/2$ ordinary lines.
| 13 | https://mathoverflow.net/users/38624 | 191325 | 94,086 |
https://mathoverflow.net/questions/191326 | 16 | This may be inappropriate for MO, but here goes: **if** I have understood the statement of the Erdős–Rado theorem correctly, then it contains as a special case the following result:
>
> if $\mu$ is an infinite cardinal then $(2^\mu)^+ \to (\mu^+)\_\mu^2$
>
>
>
that is, every $\mu$-colouring of the $2$-elements... | https://mathoverflow.net/users/763 | Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper? | It is also stated as Theorem 4.(i), I think, and again on pages 470 and 472 where the reference is given to earlier results.
For a recent easy-to-read presentation of the proof, see Theorem 5.1.4 in David Marker's *Introduction to Model Theory*.
| 10 | https://mathoverflow.net/users/57583 | 191328 | 94,088 |
https://mathoverflow.net/questions/191330 | 9 | Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent to the identity mod $N$). Let $\overline{\Gamma(N)}$ denote its closure in $\widehat{SL(2,\mathbb{Z})}$.
Can we describ... | https://mathoverflow.net/users/15242 | what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)? | It is a [result of Melnikov](http://www.ams.org/mathscinet-getitem?mr=466341) that the congruence kernel $ker\{ \widehat{SL\_2(\mathbb{Z})}\to SL\_2(\hat{\mathbb{Z}})\} \cong \hat{F}\_\omega$, the free profinite group on a countable number of generators.
| 18 | https://mathoverflow.net/users/1345 | 191331 | 94,089 |
https://mathoverflow.net/questions/191332 | 3 | I am reading part of Dipendra Prasad's paper found here: <http://arxiv.org/pdf/1306.2729v1.pdf>.
In it (in the middle of page 8) he writes that compactly induced representations are projective. Why is this true? In general, what objects are known to be projective in this category?
| https://mathoverflow.net/users/64244 | Why are compactly induced representations projective in the category of admissible representations? | This follows from Frobenius reciprocity for compact induction. Let $G$ be the group of rational points of a reductive $p$-adic group and $K$ be a compact open subgroup of $G$. Then if $\lambda$ is any smooth representationf of $K$, we denote by ${\rm ind}\_K^G \lambda$ the compactly induced representation. Then by Frob... | 6 | https://mathoverflow.net/users/4767 | 191338 | 94,091 |
https://mathoverflow.net/questions/191227 | 6 | It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we can prove it for polynomial case by using elementary method.
| https://mathoverflow.net/users/11966 | Is there any elementary proof of No wandering domain for polynomials | If by "elementary method", you mean a proof that avoids using techniques from quasiconformal methods, then the short answer is "no". Giving a different proof of the No Wandering Domains (NWD) theorem, which works without dynamical assumptions (e.g., hyperbolicity, non-recurrence etc) is a well-known open problem. This ... | 6 | https://mathoverflow.net/users/3651 | 191339 | 94,092 |
https://mathoverflow.net/questions/191307 | 14 | Given a set $S$, a function $M: S\times S \rightarrow S$ is a **mean** if it satisfies the properties:
1. $M(a,a)=a\qquad$ **(identity)**
2. $M(a,b)=M(b,a)\qquad$ **(commutativity)**.
and possibly
3. $M(M(a,b),M(a,c))=M(a,M(b,c))\qquad$ **(weak associativity)**
4. $M(M(a,b),M(c,d))=M(M(a,c),M(b,d))\qquad$ (**stro... | https://mathoverflow.net/users/2480 | Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent? | Define a *mean algebra* to be a set $S$ with an binary operation $M$ satisfying (1), (2), and (4). We can define $M(a,b,c,d)=M(M(a,b),M(c,d))$ and this will depend only on the multiset $\{a,b,c,d\}$. More generally, we can think of $M$ as an operation defined on multisets of size $2^n$ for any $n>0$ (and this is well-d... | 12 | https://mathoverflow.net/users/75 | 191343 | 94,093 |
https://mathoverflow.net/questions/191316 | 18 | (***Note:*** See also the $a^4+b^4+c^4 = 1$ version in this old [MSE post](https://math.stackexchange.com/questions/509526/).)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + r)^4 + (p-2q - r)^4 + (q + s)^4 + (q - s)^4 = (2p - 2q)^4... | https://mathoverflow.net/users/12905 | More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$? | Given a rational number $m$, let $C\_{m}$ be the intersection of two quadrics given by (3) and (4) in the original question. The way to search for points is to test the curve $C\_{m}$ to see if it has local points. I did this for all $m$ with height $\leq 193$ (since I knew I'd find points on $C\_{193/18}$). There are ... | 15 | https://mathoverflow.net/users/48142 | 191366 | 94,099 |
https://mathoverflow.net/questions/190968 | 5 | Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?"
More formally and generally, what I'm looking for is an algorithm to solve the following modification of the TSP.
Suppose you have a digraph $G = (V, A)$ with edge wei... | https://mathoverflow.net/users/64066 | Algorithm to minimally connect line segments in Euclidean plane | Another way to solve the problem might be to reduce it to a traditional version of the TSP and then use a free, off-the-shelf solver [like one of these](http://en.wikipedia.org/wiki/Travelling_salesman_problem#Free_software_for_solving_TSP). At the very least, it should be a relatively easy way of validating that your ... | 2 | https://mathoverflow.net/users/8938 | 191367 | 94,100 |
https://mathoverflow.net/questions/190609 | 3 | This should be both well-known and probably easy, but I was wondering if the following is known (and, if so, how to easily calculate the thing or where to read about how to calculate it):
what is $$\int\_{\mathrm{SU}(n)} \mathrm{tr}(U^k) dU?$$ (Here by "$dU$" I mean normalized Haar measure.)
Of course for $k$ not a... | https://mathoverflow.net/users/12138 | Moments of random special unitary matrices | I should have stared at the Weyl integration formula longer --- it's clear from that that, once $k\gg\_n 1$, the integral is zero. In fact [this article](http://ac.els-cdn.com/S0021869307003420/1-s2.0-S0021869307003420-main.pdf?_tid=485ff84e-8af5-11e4-895a-00000aab0f6c&acdnat=1419374861_b065384d2f6ee608801b6f5a9ee645ff... | 1 | https://mathoverflow.net/users/12138 | 191384 | 94,107 |
https://mathoverflow.net/questions/191391 | 2 | Let $0<a<b<c$ be distinct positive reals.
Define four different probability distributions:
$$\mathcal{P}\_{ab}:P\_{a,ab}=\frac{a}{a+b}=1-P\_{b,ab}$$
$$\mathcal{P}\_{bc}:P\_{b,bc}=\frac{b}{b+c}=1-P\_{c,bc}$$
$$\mathcal{P}\_{ca}:P\_{c,ca}=\frac{c}{c+a}=1-P\_{a,ca}$$
$$\mathcal{P}\_{abc}:P\_{a,abc}=\frac{a}{a+b+c},\mb... | https://mathoverflow.net/users/10035 | Entropy dominance |
>
> Does Shannon entropy of a random variable from distribution $\mathcal{P}\_{abc}$ dominate the other three for all $a,b,c\in\Bbb R^+$ such that $0<a<b<c$?
>
>
>
No.
On a sample space $\{A,B,C\}$ we can think of $\mathcal P\_{ab}$ as just $\mathcal P\_{abc}$ conditioned on the event that $C$ did not happen.
... | 2 | https://mathoverflow.net/users/4600 | 191394 | 94,110 |
https://mathoverflow.net/questions/191397 | 1 | Say you have positive $\{a\_i\}\_{i=1}^n$ and you have $p\_i=\frac{a\_i}{\sum\_{i=1}^na\_i}$, then assume you have a $C$ such that $C<2a\_n\ll\sum\_{i=1}^na\_i$ (that is $C$ is not very large), then define $q\_i=\frac{a\_i}{C+\sum\_{i=1}^na\_i}$ and $q\_{n+1}=\frac{C}{C+\sum\_{i=1}^na\_i}$. Does Shannon entropy of $q$ ... | https://mathoverflow.net/users/10035 | Entropy dominance of certain restricted sequenes | I'll show that **Yes**, if you rule out an event of sufficiently small probability then the entropy decreases.
I'll change the notation around a bit so that my $p\_1,\dots,p\_n$ correspond to your $q\_{n+1},\dots,q\_1$.
Suppose $\sum\_{i=1}^n p\_i=1$ and we are given that the event corresponding to $p\_1$ did not o... | 1 | https://mathoverflow.net/users/4600 | 191405 | 94,113 |
https://mathoverflow.net/questions/191377 | 3 | Consider the Chebyshev polynomial of the first kind $T\_n(x)$ and its factorization in $\mathbb F\_p$ for a given prime $p$. Most often, this factorization is not *uniform* (meaning that the irreducible factors have not all the same degree, as in [this](https://mathoverflow.net/questions/191049/which-criteria-for-unifo... | https://mathoverflow.net/users/29783 | Chebyshev polynomials factoring uniformly modulo all primes | Let $\zeta$ be a primitive $4n$th root of unity. Then $\alpha = (\zeta+\zeta^{-1})/2$ is a root of $T\_n$. $\alpha$ generates the real subfield of
the $4n$th cyclotomic field, of degree $\varphi(4n)/2$ over $\mathbb Q$
(where $\varphi$ is the Euler phi function). Now $\varphi(4n) \le 2n$, with
equality if and only if $... | 3 | https://mathoverflow.net/users/21146 | 191407 | 94,114 |
https://mathoverflow.net/questions/191403 | 2 | Supposing that $\Gamma$ is an infinite, discrete group and that $\beta\Gamma$ is the Stone-Cech compactification of $\Gamma$, the group structure of $\Gamma$ can be extended to a semigroup structure on $\beta\Gamma$ by means of its universal property, for which the right multiplication maps over $\beta\Gamma$ are all c... | https://mathoverflow.net/users/58786 | Non-idempotent ultrafilters in the Stone-Cech compactification | Let $\Gamma=\mathbb Z$, and consider the sequence of all odd numbers, viewed as a filter. Pick an ultrafilter containing this filter. Then this ultrafilter is not idempotent.
The reason is simple: odd + odd = even.
| 5 | https://mathoverflow.net/users/5690 | 191409 | 94,115 |
https://mathoverflow.net/questions/191390 | 4 | Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one parameter subgroup $\lambda:\mathbb C^\times\to H$. I would like to lift this to a one parameter subgroup $\tilde\lambda:\... | https://mathoverflow.net/users/9947 | Lifting one parameter subgroups of algebraic groups | This is true for quotients by smooth unipotent normal subgroups over any field whatsoever: if $f:G \rightarrow G'$ is a surjective homomorphism between group schemes of finite type over a field $k$ such that $\ker f$ is a unipotent smooth group (in particular, affine) then for any $k$-torus $T' \subset G'$ there exists... | 5 | https://mathoverflow.net/users/61939 | 191418 | 94,117 |
https://mathoverflow.net/questions/191220 | 6 | Let us consider the points
$$p\_1=[1:0:...:0],p\_2 = [0:1:...:0],...,p\_{n-2} =[0:...:0:1],\\
p\_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl\_{p\_1,...,p\_{n-1}}\mathbb{P}^{n-3}$.
Furthermore, consider
$$p\_1 = ([0:1],...,[0:1]), p\_2 = ([1:0],...,[1:0]), p\_3=([1:1],...,[1:1])\in (\mathbb{P}^... | https://mathoverflow.net/users/nan | Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$ | Here is an explicit way to construct a small modification.
Consider the points $p\_1,...,p\_{n-3}$. We have $n-3$ codimension two linear subspaces $H\_{i\_1,...,i\_{n-4}}^{n-5} = \left\langle p\_{i\_1},...,p\_{i\_{n-4}}\right\rangle$. For any choice of $i\_1,...,i\_{n-4}$ we define $\{j\_1,j\_2\} = \{0,...,n-3\}\setm... | 0 | https://mathoverflow.net/users/14514 | 191419 | 94,118 |
https://mathoverflow.net/questions/191382 | 6 | Let $X$ a smooth projective scheme over a field $k$. And let $THH(X)$ denotes the topological Hochschild homology of $X$. Recall that the spectra $THH(X)$ admits an action of the of circle $S^{1}$. Let $C\_{p^n}$ the cycle subgroup of $S^{1}$ with $p^{n}$ elements.
**Question 1** when the map from homotopy fixed poi... | https://mathoverflow.net/users/61328 | homotopy fixed points and fixed points | The map (which actually goes from $THH^{C\_p}$ to $THH^{hC\_p}$) usually not an equivalence in the $p$-complete setting, at least if your input is genuinely a ring. You can detect the difference using a mapping cone; the mapping cone of this map is equivalent to the mapping cone of a map from $THH$ to the so-called Tat... | 5 | https://mathoverflow.net/users/360 | 191420 | 94,119 |
https://mathoverflow.net/questions/191415 | 8 | According to the [periodic table of k-tuply monoidal n-categories](http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category), it should be the case that a tetracategory (= weak 4-category) with one object, one 1-morphism and one 2-morphism is effectively equivalent to a symmetric monoidal category. I understand geometr... | https://mathoverflow.net/users/799 | Why does a tetracategory with one object, one 1-morphism and one 2-morphism give a symmetric monoidal category | It will come from a compatibility between different ways of composing interchangers.
(I'm going to use = to mean iso/homotopy in a HoTT-like way throughout, for ease of notation. I will also confuse proofs and homotopies throughout.)
To get a better intuition, let's first think about the case of higher groupoids. A... | 9 | https://mathoverflow.net/users/22 | 191424 | 94,122 |
https://mathoverflow.net/questions/191352 | 0 | It is known, that $\phi := \frac{sqrt(5)-1}{2}$, is the number, that is hardest to approximate by rationals (cf e.g. the section **properties of the golden ratio $\phi$** here: <http://en.wikipedia.org/wiki/Continued_fraction#A_property_of_the_golden_ratio_.CF.86>).
In the section **Infinite continued fractions** on ... | https://mathoverflow.net/users/31310 | Comparing the Rational Approximability of Infinite Continued Fractions | A frequently used measurement is ``the largest partial quotient''. Almost all numbers (in the sense of measure theory) have unbounded partial quotients, but many interesting numbers do not (rationals and quadratic irrationals, for example). This is, in my experience, the most usable such gradation of approximability.
... | 2 | https://mathoverflow.net/users/935 | 191428 | 94,126 |
https://mathoverflow.net/questions/191401 | 5 | Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\ $ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two isometric (not a priori isomorphic) Banach spaces are isometrically isomorphic. Thus in a sense a metric structure may ... | https://mathoverflow.net/users/8385 | Continuity of central point operation | No, it's not:
>
> **THEOREM (Example)** There exists a compact central metric space for which the central point operation is not continuous.
>
>
>
**PROOF (Construction)** Consider the spherical distance. For any two points that are not antipodals, $S\_1$ is a singleton. For antipodal points, like the Nort... | 7 | https://mathoverflow.net/users/955 | 191431 | 94,128 |
https://mathoverflow.net/questions/190702 | 8 | **Question:** Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}\_g$ for some $g\geq 1$? Here $\mathcal{A}\_g$ is the moduli space of principally polarized abelian varieties.
**Motivation:** Basically, I am trying to understand some... | https://mathoverflow.net/users/4181 | Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$? | You can always map $(T,h)$ to the trivial Shimura datum and then this into the Siegel one. I assume this isn't what you want however. May I therefore modify the question to ask whether a zero dimensional variety can be *embedded* in a Siegel variety. I.e. You want to know if all zero dimensional Shimura varieties are o... | 3 | https://mathoverflow.net/users/8080 | 191441 | 94,130 |
https://mathoverflow.net/questions/191436 | 2 | Let $G$ be a finite simple group and let $C$ be a (non-trivial) conjugacy class of $G$. Let $H$ be a subgroup of $G$ such that $$|H\cap C| \geq \epsilon |C|.$$
Can one conclude that the index of $H$ in $G$ is bounded by a constant that depends on $\epsilon$, but not on $G$ or $C$?
This is not a particularly urgent q... | https://mathoverflow.net/users/38468 | Generating subgroups of large index by a large chunk of a conjugacy class | It looks to me as if this will not work with $3$-cycles in $G = A\_{n}.$ Suppose, for example, that $n =2m$ and take $H=A\_{m}.$ Then $G$ contains $\frac{n(n-1)(n-2)}{6}$ $3$-cycles, and $H$ contains $\frac{m(m-1)(m-2)}{6}$ $3$-cycles, so about $\frac{1}{8}$ of the $3$-cycles in $G.$ However, $[G:H] \to \infty$ as $m \... | 4 | https://mathoverflow.net/users/14450 | 191446 | 94,132 |
https://mathoverflow.net/questions/191438 | 1 | Consider a collection of positive integers $\{a\_i\}\_{i=1}^m$ and the distribution $p\_i=\frac{a\_i}{\sum\_{i=1}^ma\_i}$.
Similarly for the collection $\{a\_i\}\_{i=1}^{m+1}$ form the distribution $q\_i=\frac{a\_i}{\sum\_{i=1}^{m+1}a\_i}$, for the collection $\{a\_i\}\_{i=1}^{n}$ form the distribution $r\_i=\frac{a\... | https://mathoverflow.net/users/10035 | Entropy difference dominance of sequences | Here's some examplary evidence that maybe the answers are Yes.
* For Case 2, let $a\_{i+1}=2a\_i$ for all $i$ with $a\_1=1$.
Then the binary entropy is
$$
-\frac1{2^{n}-1} \sum\_{k=0}^{n-1} 2^k \log\_2\left(\frac{2^k}{2^{n}-1}\right)$$
$$
\approx-\frac1{2^{n}} \sum\_{k=0}^{n-1} 2^k \log\_2\left(\frac{2^k}{2^{n}}\righ... | 1 | https://mathoverflow.net/users/4600 | 191462 | 94,137 |
https://mathoverflow.net/questions/191458 | 2 | Let $Y(3)$ be the fine moduli space (say, over $\mathbb{C}$) representing elliptic curves equipped with a full level 3 structure. Abstractly, there are 24 such structures for any elliptic curve, but thanks to every elliptic curve having $[-1]$ as an automorphism, there are generically only 12 equivalence classes. Thus,... | https://mathoverflow.net/users/15242 | sanity check about a morphism from a stack to its coarse moduli space | If the map from $M\_{1, 1}$ to the $j$-line can be said to have a degree, that degree should be $\frac{1}{2}$, which makes everything work out. The reason is that its fibers are generically not a finite set but a finite groupoid, namely $\text{pt}/\mathbb{Z}\_2$ (corresponding to the $-1$ automorphism), which has group... | 9 | https://mathoverflow.net/users/290 | 191466 | 94,140 |
https://mathoverflow.net/questions/191469 | 3 | The "random" sample is obviously very, very skewed: If *you* would be asked to name a random conjecture, it probably will be a "famous" conjecture, and the longer a conjecture stands, the more famous it tends to be.
But that is not my question. Has anybody tried yet a reliable statistic on truely "random" conjecture... | https://mathoverflow.net/users/11504 | Longevity of "random" conjectures | In some areas of mathematics, there are published lists of unsolved problems.
Sometimes, progress surveys on these problems are published later.
One example I am familiar with is "Hayman's collection" in classical Function theory.
It started with a book by Hayman, Unsolved problems in Function theory, Athlone press, ... | 2 | https://mathoverflow.net/users/25510 | 191471 | 94,142 |
https://mathoverflow.net/questions/189201 | 1 | Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}\_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a\*b = \mathcal{F}(\mathcal{F}^{-1}(a).\mathcal{F}^{-1}(b))$ with $\mathcal{F}: \mathcal{P}\_{2,\pm} \to \mathcal{P}\_{2,\mp}$ the $1$-click rotation. ... | https://mathoverflow.net/users/34538 | Is there a Frobenius reciprocity for the coproduct? | Yes, but in general, it's an "ultra-weak" Frobenius reciprocity.
Let $a, b$ be positive operators. The notation $a \preceq b$ means that the support projection of $a$ is a subprojection of the support projection of $b$.
**Lemma:**
Let $p \in P\_{2,+}$ be a projection, then $e\_1 \preceq p \*\overline{p} $, and ... | 0 | https://mathoverflow.net/users/34538 | 191481 | 94,146 |
https://mathoverflow.net/questions/144246 | 10 | I proved the following facts by unenlightening calculations. Since the statements are quite clean, I think there should be a conceptual explanation for them, which my proof certainly is not.
Let $q$ be a prime power, and let $\mu\_{q+1}$ be the set of $(q+1)$-th roots of unity in the finite field $\mathbf{F}\_{q^2}$.... | https://mathoverflow.net/users/30412 | Seeking conceptual explanation of these nice bijections on roots of unity | Let $E$ be a curve defined by a singular Weierstrass equation over $\mathbb{F}\_q$, where the singularity is a node, say at the origin. Then, Silverman says (Arithmetic of Elliptic Curves, page 46) that $E$ may be written as
$$
E: y^2 + A\_1 xy - A\_2 x^2 - x^3 = 0,
$$
where $A\_1^2 + 4 A\_2$ is not zero.
If the two ta... | 10 | https://mathoverflow.net/users/30158 | 191488 | 94,148 |
https://mathoverflow.net/questions/191404 | 6 | The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in [this paper](http://arxiv.org/abs/math/9912012). It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^\*$. Let ${\bf i}=(i\_1, \ldots, i\_l)$. Then $\pi=(\gamma = \gamma\_0, \gamma\_1, \ldots, \gamma\_l=\delta)$ ... | https://mathoverflow.net/users/11877 | Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky | There is a bit of a mistake in the question. If a formula like (1) held, then tensor product multiplicities for $G\_2$ could only be at most 1.
On the other hand, Theorem 2.2 of the paper does give a polyhedral formula of the form:
$$ V(\lambda)\otimes V(\mu) = \oplus\_{(t\_1, t\_2, t\_3, t\_4, t\_5, t\_6) \in A} V(... | 8 | https://mathoverflow.net/users/438 | 191493 | 94,151 |
https://mathoverflow.net/questions/191495 | 21 | Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
| https://mathoverflow.net/users/nan | Collection of conjectures and open problems in graph theory | The largest section in the [open problem garden](http://www.openproblemgarden.org/) is about graph theory.
The book [Erdös on Graphs](http://math.ucsd.edu/~fan/epbook.html) with its [living version](http://www.math.ucsd.edu/~erdosproblems/) might be interesting as well.
| 18 | https://mathoverflow.net/users/12674 | 191496 | 94,152 |
https://mathoverflow.net/questions/191452 | 23 | How many elements of $\mathrm{SL}\_n(\mathbb{F}\_p)$ have all nonzero entries? Just the answer mod $p$ would be fine as well. This seems like it should be easy/in the literature but I couldn't find it.
| https://mathoverflow.net/users/12138 | Number of elements of "$\mathrm{SL}_n(\mathbb{F}_p^\times)$" mod $p$ | Mod $p$ it's $(-1)^{n+1} n!$.
Let's compute the number of points with determinant $1$ and all entries nonzero by inclusion-exclusion, modulo $p$. For each set of entries, we get a term for matrices in $SL \_n$ with those entries $0$. This is an affine hypersurface of degree $n$ in some affine space. By Warning's theo... | 36 | https://mathoverflow.net/users/18060 | 191498 | 94,153 |
https://mathoverflow.net/questions/191523 | 2 | Let $S$ be a $2$-dimensional sphere endowed with a flat metric with $3$ conical singularities of positive curvature. Typically, $S$ is a metric space you get when you glue two copies of the same triangle along its boundary.
I have two questions about simple (not self-intersecting, avoiding singular points), totally ... | https://mathoverflow.net/users/25511 | Geodesic paths on a flat sphere | 1. Yes. Take such a path. It intersects the opposite side of the triangle. Now take *two* such paths. Either they don't intersect, in which case one of the regions between them is a bigon - impossible in the Euclidean metric, or they do, in which case you still get a bigon (to the first point of intersection).
2. The c... | 2 | https://mathoverflow.net/users/11142 | 191527 | 94,163 |
https://mathoverflow.net/questions/191532 | 3 | It is easy to think up interesting, natural models of the algebraic theory presented as follows, such that in these models, $x^\dagger$ is always the multiplicative inverse of $x$.
>
> **Question.** What are some interesting, natural models in which $x^\dagger$ is not necessarily the multiplicative inverse?
>
>
>... | https://mathoverflow.net/users/26080 | Looking for interesting, natural models of this algebraic theory in which $x^\dagger$ is not always the multiplicative inverse of $x$ | **not a solution**
Anything that satisfies 1,2,3; then define $x^\dagger = 1$ for all $x$.
**added December 27**
Try this example...
$U := \mathbb Z \times \mathbb Z$.
* multiplication is performed by componentwise addition, $(a,b)(c,d) = (a+c,b+d)$
* the lattice operations are also performed componentwi... | 2 | https://mathoverflow.net/users/454 | 191534 | 94,164 |
https://mathoverflow.net/questions/191541 | 5 | This is probably simple but I'm stuck somewhere. I am trying to solve the calculus of variation problem that arise in an applied field: $$\min\_{f \in C^1} \int^1\_0 \int^1\_0 (x-y)^2f(x,y)dxdy$$ $$\text{s.th: } f\geq 0 \text{, } \int^1\_0 f(x,y)dy=1 \text{ and } \int^1\_0 y \partial\_x f(x,y) dy=0 \text{ }\forall\text... | https://mathoverflow.net/users/33640 | Calculus of variation | For the moment, fix $$c=\int\_0^1 yf(x,y)\,dy.$$ It is clear that we must have $0\le c\le 1$ to satisfy the constraints on $f$. We can write the integral to be minimized as $$\int\_0^1\Bigl(x^2-2cx+\int\_0^1 y^2f(x,y)\,dy\Bigr)\,dx.$$ Now we fix $x$ and minimize the inner integral. Since by Cauchy-Schwarz we have $$c^2... | 2 | https://mathoverflow.net/users/12120 | 191549 | 94,169 |
https://mathoverflow.net/questions/191514 | 2 | Be a set of numbers $v=(a\_1, a\_2, \ldots, a\_n)$
I want to form the following average vector $\mu = (\frac{\sum a\_i}{n}, \frac{\sum a\_i}{n}, \ldots, \frac{\sum a\_i}{n})$.
If I do it iteratively step by step, in each step we pick three components, $a\_i,a\_j$ and $a\_k$ that are not all equal, and we replace th... | https://mathoverflow.net/users/43557 | Mean of a vector | This operation decreases the variance of the set of numbers. If you include the minimum and the maximum, the operation decreases the variance by a factor bounded by some $c(k) \lt 1$ (we can take $c(k) = 1-1/(2k)$ though that is not sharp) so the vector converges to a constant if you repeatedly include the minimum and ... | 1 | https://mathoverflow.net/users/2954 | 191552 | 94,170 |
https://mathoverflow.net/questions/191554 | 4 | Suppose we have a closed 3-manifold $M$, not necessarily simply connected.
What can I say about the homotopy groups of $M \setminus \text{pt}$? ($M$ punctured by one point)
In particular, what assumptions do I need to ensure that $\pi\_2 (M \setminus \text{pt}) = 0$?
Thanks!
| https://mathoverflow.net/users/64341 | Punctured 3-manifold | If $\pi\_2(M\setminus\{p\})=0$ then $M$ is simply connected (and hence $S^3$ by the Poincare conjecture). To see this, consider the universal cover $q:U\to M$ and let $V=q^{-1}(M\setminus \{p\})$. Then $V$ is a cover of $M\setminus\{p\}$ (in fact, the universal cover) and $U\setminus V$ is a discrete set of cardinality... | 17 | https://mathoverflow.net/users/75 | 191555 | 94,171 |
https://mathoverflow.net/questions/189190 | 1 | Let $\mathcal{P}$ be an irreducible finite index-depth subfactor planar algebra. The $2$-boxes space $\mathcal{P}\_{2,+}$ is equipped with the coproduct $(a,b) \mapsto a\*b = \mathcal{F}(\mathcal{F}^{-1}(a).\mathcal{F}^{-1}(b))$ with $\mathcal{F}: \mathcal{P}\_{2,\pm} \to \mathcal{P}\_{2,\mp}$ the $1$-click rotation.
... | https://mathoverflow.net/users/34538 | Is the coproduct of central operators, also central? | It's obviously true if $\mathcal{P}\_{2,+}$ is abelian.
It's also true for the irreducible depth $2$ case:
There is a nice direct diagrammatic proof using the splitting ([[KLS]](http://link.springer.com/article/10.1007%2FBF02829677) thm 5.1 p39, relation (3)).
It's *false* in general:
As observed by [Vijay K... | 1 | https://mathoverflow.net/users/34538 | 191556 | 94,172 |
https://mathoverflow.net/questions/191510 | 4 | A (weak) composition of a positive integer $n$ into $k$ parts is an ordered sequence of nonnegative integers $(a\_1, a\_2, \ldots, a\_k)$ such that $ \sum\_{i=1}^k a\_i = n $. I am interested in the case when the parts are bounded: $a\_i\in\{0, 1, \ldots, j-1\}$. The number of such compositions satisfies the two-variab... | https://mathoverflow.net/users/64319 | Diagonal asymptotics of integer compositions | Up to a factor of $j^k$, What you're asking for is the probability $P$ that a k-step random walk with steps chosen uniformly from $S = \{0, 1, ..., j-1\}$ lands on $\lambda n$. This is the probability of return to the origin at time $k$ of the (typically biased) random walk with steps chosen uniformly from $S - \lambda... | 3 | https://mathoverflow.net/users/44291 | 191567 | 94,177 |
https://mathoverflow.net/questions/191568 | 0 | Likely a mistake, but got very large exceptional set in Vojta's
more general abc conjecture.
In [A more general abc conjecture](http://arxiv.org/abs/math/9806171), p. 7 Paul Vojta conjectures:
If $x\_0,\ldots x\_{n-1}$ are nonzero coprime integers satisfying $x\_0 + \cdots x\_{n-1}=0$
$$ \max\{|x\_0|,\ldots |x\_{... | https://mathoverflow.net/users/12481 | More on Vojta's exceptional set for a more general abc conjecture | Your construction is not a counterexample because your tuples $(x\_0,\ldots,x\_5)$
(obviously) satisfy the relation $x\_1^2 - 4x\_0x\_2 = 0$, so they all lie in the
proper Zariski closed subset defined by this equation.
| 4 | https://mathoverflow.net/users/21146 | 191569 | 94,178 |
https://mathoverflow.net/questions/191575 | 2 | A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to [Large cardinal axioms and Grothendieck universes](https://mathoverflow.net/questions/12804/large-cardinal-axioms-and-grothendieck-universes?rq=1) gives Tarski sets as an interesting example.
I should clarify since the termino... | https://mathoverflow.net/users/38783 | What is the consistency strength of a standard model of ZF versus a transitive model? | What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.)
If the latter, the [Mostowski collapse theorem](http://en.wikipedia.org/wiki/Mostowski_collapse_lemma) ... | 7 | https://mathoverflow.net/users/6085 | 191578 | 94,182 |
https://mathoverflow.net/questions/191574 | 1 | It is well-known that there is a computable pairing function $<\ >:\mathbb N^2\to \mathbb N$. Let $X$ be some reasonable class of countable ordinals ($\omega\_1^{CK}$, $\epsilon\_0$, $\omega^\omega$, or etc.). Is there any computable bijection from $X^2$ to $\mathbb N$? I would like to code pairs like $<\omega, 1>$ by ... | https://mathoverflow.net/users/57448 | Is there a pairing function from countable ordinals to $\mathbb N$? | Coding the pairs from a set $X$ of ordinals is exactly as easy (or hard) as coding the individual elements of $X$. That's because there are very easily computable pairing functions on the natural numbers.
If $X$ is small enough, like $\epsilon\_0$, then this can be done computably, in the sense that there are algorit... | 10 | https://mathoverflow.net/users/6794 | 191581 | 94,185 |
https://mathoverflow.net/questions/191585 | 3 | Consider a closed $3$-manifold $M$ and a knot $K$ in $M$.
Is it necessarily true that $\pi\_2 (M \setminus K) = 0$?
If not, are there any conditions on $M$ and/or $K$ to ensure the above 2nd homotopy group of the knot complement is trivial?
Thanks!
(Note: This is, of course, true when $M$ is simply connected --... | https://mathoverflow.net/users/64341 | Knots in 3-manifolds | EDIT - I've rewritten my previous answer in an attempt to remove everything except the answer to your question. All submanifolds are assumed to be smooth.
>
> Suppose that $M$ is a closed, connected, oriented, irreducible three-manifold (and $M$ is not the three-sphere). Suppose that $K$ is a knot in $M$. Then $\p... | 8 | https://mathoverflow.net/users/1650 | 191588 | 94,189 |
https://mathoverflow.net/questions/191494 | 13 | I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify.
(1)
The Hoory-Linial-Wigderson review on expanders in its definition 5.11 calls a d-regular graph to be Ramanujan if the second highest (adjacency?) eigenvalue is bou... | https://mathoverflow.net/users/36554 | What is a "Ramanujan Graph"? | Ramanujan graphs were first defined by Lubotzky, Phillips and Sarnak:
<http://math1.math.huji.ac.il/~alexlub/PAPERS/ramanujan%20graphs/ramanujanGraphs.pdf>
As you can see, they are $d$-regular and and all eigenvalues of the adjacency matrix, except for $\pm d$, are in $[-2\sqrt{d-1},2\sqrt{d-1}]$. This is equivalent ... | 20 | https://mathoverflow.net/users/14497 | 191591 | 94,190 |
https://mathoverflow.net/questions/191605 | 5 | Ajai Choudhry [showed](https://math.stackexchange.com/a/1079272/4781) that special cases of the elliptic curve,
$$x(x+a^2)(x+b^2)=y^2\tag1$$
can be used to prove that,
$$u\_1^7+u\_2^7+\dots + u\_9^7 = 0\tag2$$
has an infinite number of primitive integer solutions. Let $a,b$ be positive integers. Define non-tors... | https://mathoverflow.net/users/12905 | On the elliptic curve $x(x+a^2)(x+b^2) = y^2$ | Yes, if $E : y^{2} = x(x+a^{2})(x+b^{2})$ has a point $P$ whose $x$-coordinate is $au^{2}$ for some nonzero rational number $w$, it also has a point $Q$ whose $x$-coordinate is $bv^{2}$ for some nonzero rational number $v$. In fact, we can take $Q = P + R$, where $R = (ab,a^{2} b + ab^{2})$ is a point of order $4$ on $... | 5 | https://mathoverflow.net/users/48142 | 191610 | 94,195 |
https://mathoverflow.net/questions/191387 | 10 | The $n$-dimensional hypercube is the graph formed by $0$-$1$ sequences of length $n$ where two vertices are adjacent if they differ at only one place.
The weight of a sequence is the number of $1$'s in it.
For a (connected) subgraph of a hypercube we say that its *level-vector* is $(n\_1,\ldots,n\_k)$ if the number of ... | https://mathoverflow.net/users/955 | Are there non-trivial graphs that uniquely embed to hypercubes? | For your first question, I believe that only hypercubes can embed uniquely.
First note that $G$ has to be $n\_2$-regular, and $n\_1=1$: take any embedding, and use coordinate flips so that your favorite vertex $v$ of $G$ is embedded in $(0,0,0,...,0)$. Then $n\_1=1$, and all $n\_2$ vertices of weight $1$ are neighbor... | 3 | https://mathoverflow.net/users/12487 | 191620 | 94,199 |
https://mathoverflow.net/questions/191609 | 37 | In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to theoretical physics for working mathematicians? I have heard of Armin Wachter and Henning Hoeber's, but I don't know if... | https://mathoverflow.net/users/nan | Companion to theoretical physics for working mathematicians | If you allow such a comprehensive reference to re-introduce basic mathematics, then either as a layman or a working mathematician your prayers are answered by the following (he even prefaces by saying that his intended layman-audience must have some mathematical sophistication):
>
> **The Road to Reality: A Comple... | 42 | https://mathoverflow.net/users/12310 | 191625 | 94,201 |
https://mathoverflow.net/questions/191586 | 1 | Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ being the largest eigenvalue of $A(t)$ and $t$ denotes the time. If $A(t)$ changes over time with each time a random element in $A(t)$ is being increased by a random value, will the spectral radius be monotonously increasing over ti... | https://mathoverflow.net/users/45615 | Spectral radius of a time-varying matrix with strictly positive increment of the matrix's entry | Anthony and David are referring to the Perron-Frobenius Theorem, which easily proves the monotonicity you ask about. There is a precise quantitative result in my paper *Perturbations of Shifts of Finite Type*, SIAM J. Disc. Math. 2 (1989), 350-365:
Lemma 6: Let $A = [a\_{i,j}]$ be a real square matrix with simple eig... | 4 | https://mathoverflow.net/users/8112 | 191635 | 94,205 |
https://mathoverflow.net/questions/191643 | 5 | Let $R$ be a discrete valuation ring with the field of fractions $K$ and the residue characteristic $p$. If $K$ is of characteristic $0$, then a celebrated theorem of Tate says that the pullback functor $G\mapsto G\_K$ is fully faithful on the category of $p$-divisible groups over $R$. I suppose that this full faithful... | https://mathoverflow.net/users/63877 | The generic fiber pullback for $p$-divisible groups in characteristic $p$ | The question of whether it is still fully faithful when the characteristic of $K$ is $p$ is mentioned in expose IX, SGA 7. The answer is yes, see "Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic" by de Jong (Invent. math. 134, 301-333 (1998))
| 10 | https://mathoverflow.net/users/2384 | 191645 | 94,208 |
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