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https://mathoverflow.net/questions/172370 | 9 | Let $ABC$ be a triangle with incircle $\omega$. Let $A\_0,B\_0,C\_0$ be points outside $\omega$. The tangents from $A\_0$ to $\omega$ intersect $BC$ at $A\_1,A\_2$. Define $B\_1,B\_2$ and $C\_1,C\_2$ similarly. Is it true that $A\_1,A\_2,B\_1,B\_2,C\_1,C\_2$ lie on a conic $\Gamma$ if and only if $\Delta ABC$ and $\Del... | https://mathoverflow.net/users/50068 | $6$ points lie on a conic if and only if $ABC$ and $A_0B_0C_0$ are perspective | As noted by Todd this is a problem in projective geometry of conics. So we can use the duality principle and reformulate the "only if" part by the following statement (the other part is equivalent):
>
> Suppose $C$ and $D$ are two conics in the projective plane. For every point $A$ in $C$, we associate a line $T(A)... | 5 | https://mathoverflow.net/users/51663 | 172797 | 88,202 |
https://mathoverflow.net/questions/172738 | 4 | Let M be a manifold and $\Lambda(M)$ its free loop space, $A= C^\*(M)$ denotes the cochain algebra of $M$. We know that Hochschild chain model for the evaluation $ ev\_0: \Lambda(M) \rightarrow M$ is given by
$ A \hookrightarrow A \otimes T(s \bar{A})$, where $T(s \bar{A})$ denotes the free coalgebra generated by the ... | https://mathoverflow.net/users/27330 | Hochschild chain model for the evaluation map at half | You can't model this map using Hochschild chains since this information is too fine for this algebraic model to capture. However, you can model the homotopies between the evaluation maps.
You could, similarly, ask to model the map $ev\_{1/2}: PM \to M$, where $PM$ is the free path space, as a map from $A=C^\*(M)$ to... | 2 | https://mathoverflow.net/users/5450 | 172799 | 88,203 |
https://mathoverflow.net/questions/172639 | 4 | If $f\colon C \to D$ is a homomorphism of coalgebras and $\rho\colon V \to V\otimes C$ is a $C$-comodule, then $(1\otimes f)\rho\colon V \to V\otimes D$ is the comodule restricted to $D$. In e.g. Stephen Donkin's article [Hopf complements and injective comodules for algebraic groups](http://plms.oxfordjournals.org/cont... | https://mathoverflow.net/users/41840 | Is there a left-adjoint to the restriction of comodules? | If we only consider coalgebras over a fixed ground *field*, then Takeuchi's article [Morita theorems for categories of comodules](http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6208/1/jfs240306.pdf) seem to prove the existence of left-adjoints of the restriction.
In 1.4 he defines cohomorphisms of $C$-c... | 1 | https://mathoverflow.net/users/41840 | 172811 | 88,207 |
https://mathoverflow.net/questions/172815 | 4 | Let $C\_1$ and $C\_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C\_1 \rightarrow C\_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal functor $\Psi'$ which is naturally isomorphic to $\Psi$? If not, is this possible if we assume that the $C\_i$ are strict?... | https://mathoverflow.net/users/54442 | Strictifying strong monoidal functors | Take $C\_2$ to be a non-strict monoidal category, $C\_1$ to be its strictification and $\Psi$ to be the equivalence. Since $C\_2$ is not strict there's no strict monoidal equivalence between $C\_1$ and $C\_2$, so in particular $\Psi$ is not naturally isomorphic to a strict functor.
| 6 | https://mathoverflow.net/users/22 | 172816 | 88,208 |
https://mathoverflow.net/questions/169202 | 5 | Let $D = \text{diag}(1,d)\in M\_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma\_D\subset M\_{4}(\mathbb{Z})$ as:
$$\Gamma\_D := \left\lbrace R\in M\_{4}(\mathbb{Z}) \: | \: R
\left(\begin{matrix}
0 & D \\
-D & 0
\end{matrix}\right) R^t =
\left(\begin{matrix}
0 & ... | https://mathoverflow.net/users/14514 | Index of congruence modular subgroup of level (1,d) | Yes, there is a short exact sequence
$$ 1 \to \Gamma\_2(2,2d) \to \Gamma\_2(1,d) \to Sp(4,\mathbb{Z}/2\mathbb{Z}) \to 1. $$
One can obtain this sequence as follows.
Consider the group scheme $G := Sp(f)$ defined over $\mathbb{Z}$ where $f$ is the
symplectic form defined above, i.e. $f(u,v) = u^t\begin{pmatrix} 0 & ... | 3 | https://mathoverflow.net/users/54441 | 172817 | 88,209 |
https://mathoverflow.net/questions/172818 | 4 | Let manifold $S$ (connected, without boundary) have next property: for every submanifold $D \subset S$ (connected, compact, without boundary), every homeomorphism $f:D \to D$ extends to a homeomorphism $g: S \to S$.
What do we know about these manifolds? Can they have dimension 3, for example? For instance, we can a... | https://mathoverflow.net/users/54337 | Manifolds such that every homeomorphism of a submanifold to itself extends to the full manifold | I seem to have misunderstood the question. I left my original answer (which answers a different question) below, and added an answer to the actual question at the top.
**Answer to the actual question:** In dimensions 3 and higher, the answer is no for topological embeddings (as asked in the question). Consider the [A... | 8 | https://mathoverflow.net/users/3460 | 172820 | 88,210 |
https://mathoverflow.net/questions/168336 | 4 | For finite trees $T\_{1}$ and $T\_{2}$ labelled by elements of some infinite set $S$, (we may assume that $S=\mathbb{N}$ without loss of generality), we define an equality-preserving embedding $f$ to be an embedding of $T\_{1}$ into $T\_{2}$ in the usual graph-theoretical sense (i.e. a homeomorphism) with the extra con... | https://mathoverflow.net/users/49491 | Equality-preserving embeddings of finite trees | Somewhat surprisingly, the answer is in fact **no**, even for paths.
**Claim.** There is an infinite antichain of coloured paths under $\prec\_{ep}$.
**Proof.** Consider a path $P$ with $2k$ vertices, with vertices coloured from $[k]$. The first $k$ vertices of $P$ are coloured $1, \dots, k$, in that order, and th... | 3 | https://mathoverflow.net/users/2233 | 172823 | 88,213 |
https://mathoverflow.net/questions/172826 | 11 | I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}\_{i=1}^\infty$. The only points I have in this intersection at the moment are $2/3$, $(2/3)^3$, and $(2/3)^9$.
With the ternary representation of the Cantor set, the question ca... | https://mathoverflow.net/users/54446 | Cantor set intersecting a geometric sequence | This is an old unsolved problem. [Erdos](https://www.renyi.hu/~p_erdos/1979-22.pdf) conjectured (see the first problem in that paper) that for all $n\ge 9$ the ternary expansion of $2^n$ contains the ternary digit $2$ (this is equivalent to for every $n\ge 10$ the ternary expansion of $2^n$ contains a $1$). For recent ... | 11 | https://mathoverflow.net/users/38624 | 172830 | 88,215 |
https://mathoverflow.net/questions/172752 | 4 | A $Z^{\*}$ algebra is a $C^{\*}$ algebra which satisfies each of the following equivalent conditions:
1. All elements of $A$ are left zero divisor.
2. All elements are right zero divisor.
3. All elements are two sided zero divisors
4. All positive elements are two sided zero divisor.
The commutative(topological) in... | https://mathoverflow.net/users/36688 | A question on $Z^{*}$ algebras | Recall that a completely positive map $\phi\colon A\to B$ is said to be faithful if $\phi(a) \neq 0$ for all nonzero $a \in A^+$.
Lemma: If $\phi\_i\colon A\_i\to B\_i$ are faithful cp maps, then $\phi\_1\otimes\phi\_2\colon A\_1\otimes\_{\min}A\_2\to B\_1\otimes\_{\min}B\_2$ is faithful.
Proof: Since $\phi\_1\otim... | 6 | https://mathoverflow.net/users/7591 | 172834 | 88,217 |
https://mathoverflow.net/questions/172827 | 3 | Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X\_0=\{I \in X : \,\, A/I \textrm{ is finite dimensional } \mathbb{C} \textrm{-vector space}\}.$
Now consider $J= \bigcap\_{I \i... | https://mathoverflow.net/users/36563 | Intersection of Maximal Left Ideals with Finite Dimensional Quotient | This question is the same as asking when does $\mathbb C\Gamma$ have enough finite dimensional irreducible representations to separate points.
A necessary condition is that $\Gamma$ has enough finite dimensional irreps to separate points. Since a fg linear group is residually finite by a theorem of Malcev, having en... | 5 | https://mathoverflow.net/users/15934 | 172841 | 88,220 |
https://mathoverflow.net/questions/172833 | 3 | Inspired by [this question](https://mathoverflow.net/questions/172826/cantor-set-intersecting-a-geometric-sequence), is there some conjecture stating that
$$
\limsup\_{n \to \infty} \frac{d\_j(2^n)}{dc(2^n)} = \frac{1}{10}
$$
where $d\_j(m)$ counts the number of $j$s in the digits of $m$,
and $dc(m)$ is just the number... | https://mathoverflow.net/users/1056 | Normality property of powers of integers? | At least as of 2012 this should still be an open question. If it were known to be true, then the number
$$
.2\ 4\ 8\ 16\ 32\ 64\ \cdots
$$
would be known to be simply normal in base $10$ (of course a strengthening would give normality in base $10$). Pillai conjectured that this number is normal in 1939 and accordin... | 2 | https://mathoverflow.net/users/32036 | 172843 | 88,222 |
https://mathoverflow.net/questions/172819 | 7 | Let $V$ be a finite dimensional vector space over a field of characteristic $0$, and let $T(V)$ be the tensor algebra (also called the free associative algebra) on $V$. This is actually a Hopf algebra, where the coproduct is defined on words by splitting them into two pieces in all possible ways: $$\Delta(v\_{I})=\sum\... | https://mathoverflow.net/users/9417 | An identity in the free associative algebra | $\newcommand{\id}{\operatorname{id}}$ Just adding in a couple steps missing in the answer by "an eulerian idempotent":
Your map $\zeta$ can be rewritten as the logarithm of the map $\id : T(V) \to T(V)$ in the *convolution* algebra $\left(\operatorname{Hom}\left(T(V), T(V)\right), \star\right)$. (In fact, if we let $... | 5 | https://mathoverflow.net/users/2530 | 172844 | 88,223 |
https://mathoverflow.net/questions/172801 | 13 | I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $\mathrm{SL}(n,\mathbb{C})$:
$$\mathrm{Hom}(\pi\_1(M), \mathrm{SL}(n,{\mathbb C}))$$
It is known that volume and Chern-Simons invariant of representations are constant on connected components of the representation ... | https://mathoverflow.net/users/39082 | Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$ | These are difficult questions and very little in general is known about this. Mostly, what's known is ad hoc results for specific classes of manifolds (say, take some surgeries on 2-bridge knots...). You can find some references in answers to [this](https://mathoverflow.net/questions/88195/what-is-explicitly-known-abou... | 11 | https://mathoverflow.net/users/21684 | 172847 | 88,225 |
https://mathoverflow.net/questions/172838 | 14 | Metamathematics has a reasonably clear connotation,
enough to have [a Wikipedia page](http://en.wikipedia.org/wiki/Metamathematics),
with Gödel, Tarski, and Turing playing leading roles;
Kleene's book (*Introduction to Metamathematics* ([Amazon link](http://rads.stackoverflow.com/amzn/click/0923891579)));
Chaitin's art... | https://mathoverflow.net/users/6094 | Meta$^{n{-}th}$ mathematics | My opinion is that there is no crisp distinction between
mathematics, metamathematics and meta-metamathematics, and the
subjects thoroughly blend one into another in such a way that
prevents any coherent distinction.
Furthermore, even the categorization of particular topics as
mathematics or metamathematics has chang... | 36 | https://mathoverflow.net/users/1946 | 172848 | 88,226 |
https://mathoverflow.net/questions/172846 | 3 | Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite extension of $Q\_p$, $H^1(F,G)$ is finite.
When $F=\mathbb{F\_q}((t))$, it is known that if $G$ is connected reductive gr... | https://mathoverflow.net/users/5082 | The cardinality of first non-abelian Galois cohomology | Since the fppf cohomology group ${\rm{H}}^1(F, \alpha\_p) = F/F^p$ is visibly uncountable (where $p = {\rm{char}}(F) > 0$), perhaps you meant to assume $G$ is smooth (and then fppf cohomology coincides with etale cohomology, which in turn coincides with Galois cohomology as in the title of the question). The cohomology... | 5 | https://mathoverflow.net/users/52824 | 172852 | 88,228 |
https://mathoverflow.net/questions/172873 | 3 | Let $A$ be a commutative ring which is not an integral domain. I try to find a polynomial $P$ of $A[X]$ such that $d°P = 1$ and $P$ admits no root in any ring $B$ such that $A$ is a subring of $B$.
| https://mathoverflow.net/users/54471 | Polynomial without roots in a ring | Take a ring $A$ that is not an integral domain, and let $d\in A$ be a zero-divisor. Consider the polynomial $dx-1$. Since $d$ is a zero-divisor in $A$, it is a zero-divisor in every ring $B$ containing $A$ as a subring. In particular, $d$ is not a unit in any such $B$. Therefore, the polynomial $dx-1$ cannot have a roo... | 10 | https://mathoverflow.net/users/25358 | 172874 | 88,237 |
https://mathoverflow.net/questions/172869 | 5 | The question is whether the below is true.
$$\sum \_{k=0}^{s-1} \binom{n}{k}=\sum \_{k=1}^s 2^{k-1} \binom{n-k}{s-k}$$
Mathematica can simplify as follows, but it fails to Reduce[] or Solve[].
$$2^n=\binom{n}{s} \, \_2F\_1(1,s-n;s+1;-1)+\binom{n-1}{s-1} \, \_2F\_1(1,1-s;1-n;2)$$
| https://mathoverflow.net/users/nan | Combinatorics Problem: $\sum _{k=0}^{s-1} \binom{n}{k}=\sum _{k=1}^s 2^{k-1} \binom{n-k}{s-k}$ | A slightly less computational method is to note that both sides of the identity count the number of subsets of $\{1,\dots,n\}$ with fewer than $s$ elements. This is obvious for the left hand side. It's true for the right hand side because $2^{k-1}\pmatrix{n-k\\s-k}$ is the number of such subsets $S$ for which $k$ is mi... | 25 | https://mathoverflow.net/users/22989 | 172882 | 88,241 |
https://mathoverflow.net/questions/172855 | 7 | What is the maximum (absolute) value of the binomial coefficient
$\begin{pmatrix}x \\ k\end{pmatrix} = \frac{1}{k!}x(x-1)(x-2)\dotsb(x-k+1)$
for real $x$ in the interval $0 \leq x \leq k-1$?
| https://mathoverflow.net/users/33550 | Maximum value of the binomial coefficient as a polynomial | It's easy to see that the extremum in $(0,1)$ has the same magnitude as the one in $(k-1,k-2)$ and is more extreme than any of the other extrema. The extremum in $(0,1)$ occurs at $x\_0=(1+o(1))/\ln k$ (by looking at the derivative), but I'll only assume it is $\Theta(1/\ln k)$. Write $x\_0=z/\ln k$ and substitute this... | 11 | https://mathoverflow.net/users/9025 | 172887 | 88,244 |
https://mathoverflow.net/questions/172166 | 6 | One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers":
>
> Let $K$ be any discrete valuation field with ring of integers $O\_K$ ,
> uniformizer $\pi$, and residue field $k$ of characteristic $p \ge 0$.
> Let $A/K$ be an abelian variety of dimension $g$. Le... | https://mathoverflow.net/users/53135 | Understanding of Tamagawa numbers of hyperelliptic curve | (1) The answer is no for the first part of this question. This is explained in Sage's documentation you cited. The program genus2reduction only outputs the order of $\Phi(\overline{\mathbb F}\_p)$, while $c\_p$ is the order of $\Phi(\mathbb F\_p)$. The later is a subgroup of $\Phi(\overline{\mathbb F}\_p)$. I don't und... | 2 | https://mathoverflow.net/users/39387 | 172891 | 88,246 |
https://mathoverflow.net/questions/172860 | 4 | I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}\_p$ be the completion of the algebraic closure of $\mathbb{Q}\_p$.
Let $\log\_p$ denote the function on the unit ball around 1 (in $\mathbb{C}\_p$) given by the standard power series. I ... | https://mathoverflow.net/users/8846 | Iwasawa logarithm and analytic continuation | Naively, no. Look at the functions $\exp(z)$ and $\exp(z+\frac{z^p}{p})$ (in a neighborhood of the origin). The power series expansion of the latter about $z=0$ has a larger radius of convergence than the former, but there is no easy explanation of this in terms of singularities.
If there is a more subtle analysis th... | 4 | https://mathoverflow.net/users/1770 | 172894 | 88,249 |
https://mathoverflow.net/questions/172899 | 9 | Does anyone know the explicit formulation for the $q\_k$'s in, $$(x+D)^n=\sum\_{k=0}^n q\_k(x)D^k\ \ \ \ ?$$
I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that $$(x+D)H\_n(x)e^{-x^2/2} = 2n H\_{n-1}(x)e^{-x^2/2},$$ where $H\_n(x)$ are the Hermite polynomials. Hence, $$(x+D)^n H\_k(x)e^{-x^2/2} ... | https://mathoverflow.net/users/50388 | Differential operator simplification | (Updated Jan. 2 and 3, 2022)
I had forgotten this question by the time I wrote up last year in OEIS [A344678](https://oeis.org/A344678) a fairly complete characterization of the coefficients of the normal-ordering of the powers of $R = x+D$, the raising op for a family of Hermite polynomials.
For those interested i... | 5 | https://mathoverflow.net/users/12178 | 172900 | 88,252 |
https://mathoverflow.net/questions/172870 | 9 | Is there a simple proof that shows:
1. Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of
finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron.
2. Minkowski sum of two $\mathcal{H}$-polyhedrons is a $\mathcal{H}$-polyhedron.
I know a proof of (1.) based on Fourier-Motzkin elim... | https://mathoverflow.net/users/53059 | Linear transformation of a polyhedron | I'm not familiar with Fourier-Motzkin, so I don't know how different the following argument is from what one usually does, but it's direct and elementary (and constructive, it in principle produces the new constraints from the old ones).
The claim *is* trivial if $A\in\mathbb R^{n\times n}$ is invertible, and a gener... | 5 | https://mathoverflow.net/users/48839 | 172903 | 88,253 |
https://mathoverflow.net/questions/172906 | 7 | For three symmetric positive semidefinite matrices $A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases:
$$ \operatorname{tr} \left( A e^{B+C} \right) \leq \operatorname{tr} \left( A e^B e^C \right) $$
Note that if $A=I$ then this is the [Golden-Thompson inequality](https://... | https://mathoverflow.net/users/54487 | An extension of the Golden-Thompson inequality | The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not positive; unlike the trace of the product of two positive symmetric matrices, which is positive: ${\rm tr}\,e^B e^C={\rm tr}\,XX^t=\sum\_{... | 8 | https://mathoverflow.net/users/11260 | 172907 | 88,255 |
https://mathoverflow.net/questions/172911 | 10 | I raise this question following the reading of *Fifty challenging problems in probability with solutions*. One of the problem consists in computing the probability that the quadratic equation $x^2 + 2b x+c=0$ has complex roots. The "natural way" of doing it, is to suppose that the point $(b,c)$ is randomly chosen over ... | https://mathoverflow.net/users/41060 | Probability over a plane | well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, you could treat this problem as the special case $n=2$ of a classic problem in random-matrix theory: take an $n\times n$ real matrix $M$ with independent identical normal distributions of the matrix elements $M\_{nm}$; what is the probabil... | 12 | https://mathoverflow.net/users/11260 | 172917 | 88,259 |
https://mathoverflow.net/questions/172829 | 3 | I have a question concerning the following paper: "The fundamental domain of the tree of GL(2) over the function field of an elliptic curve" by Shuzo Takahashi. (1993, Duke Math. J., Vol. 72, No. 1). Everyone familiar with the paper may skip to the part after the line.
**Notation**:
$k$ an arbitrary field (not necess... | https://mathoverflow.net/users/54370 | Switching representatives of right coset in a paper on fundamental domain of tree of GL(2) | Thinking about my comment again, maybe my remark on necessity of using left multiplication with $\Gamma$ was too hasty.
First, some general comment on the coset change in the specified situation.
Assume that there is a matrix $A\in KZ$ such that
$$
\left(\begin{array}{cc}
t^3&bt^2+t^{-1}+lt\\0&1
\end{array}\right)=
... | 1 | https://mathoverflow.net/users/50846 | 172918 | 88,260 |
https://mathoverflow.net/questions/172924 | 5 | Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi\_r^S(S^0)$?
| https://mathoverflow.net/users/54479 | How nilpotent is the ring of stable homotopy groups of spheres? | Let $\alpha \in \pi\_s(S^0)$ for $s > 0$ be an element of positive degree in the stable stems. Then $\alpha$ has positive-dimensional filtration in the Adams-Novikov spectral sequence: in other words, it is annihilated by complex bordism.
The $E\_\infty$-page of the Adams-Novikov spectral sequence for the sphere is ... | 14 | https://mathoverflow.net/users/344 | 172925 | 88,263 |
https://mathoverflow.net/questions/172910 | 7 | Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let $V(p,\varepsilon)$ denote volume of the intersection of $X$ with the $\varepsilon$-ball centered at $p$, namely $2m$-dime... | https://mathoverflow.net/users/16183 | A geometric characterization of smooth points of a complex algebraic variety | The answer to all three of your questions is yes.See the book by E M Chirka titled
Complex Analytic Sets pages 189,190 and 120 .These questions are local so this is true on Kahler manifolds .
| 14 | https://mathoverflow.net/users/4696 | 172928 | 88,265 |
https://mathoverflow.net/questions/172741 | 9 | [No answers from stackexchange](https://math.stackexchange.com/questions/846395/combinatorics-and-symmetric-functions), so I'll try this here:
(The actual questions in this posting are at the bottom.)
Occasionally someone asks on stackexchange how to show that every nonempty finite set has just as many subsets of o... | https://mathoverflow.net/users/6316 | Combinatorics and symmetric functions | The algebraic significance of the operation $\diamond$ is that $(a+b)/(1+ab)$ is a formal group law. See *Enumerative Combinatorics*, vol. 1, 2nd ed., Exercise 1.163. As for $a\_1\diamond\cdots\diamond a\_n$, when $n\to \infty$ we get the symmetric function $(e\_1+e\_3+e\_5+\cdots)/(1+e\_2+e\_4+\cdots)$. Writing this a... | 11 | https://mathoverflow.net/users/2807 | 172932 | 88,267 |
https://mathoverflow.net/questions/172934 | 6 | Let $k$ be a finite field and $G$ a finite type smooth $k$-group scheme. Let $G^0$ and $\Gamma$ be the connected component of identity and the component group of $G$, so there is an exact sequence $1 \rightarrow G^0 \rightarrow G \rightarrow \Gamma \rightarrow 1$. Is the induced $H^1(k, G) \rightarrow H^1(k, \Gamma)$ s... | https://mathoverflow.net/users/53197 | Over a finite field, does a torsor under the component group of G lift to a torsor under G? | This is proved when $G$ is a linear algebraic group in Corollary III.2.4.3 of Serre's book on Galois cohomology (p. 135 in my version). I believe it is only stated here for linear algebraic groups as Serre is at that point looking at general fields of cohomological dimension at most $1$. In the special case of finite f... | 5 | https://mathoverflow.net/users/5101 | 172940 | 88,271 |
https://mathoverflow.net/questions/172941 | 7 | This question is related with Exercise III.5.4 (Page 230) of Hartshorne's Algebraic Geometry. Here I start with recalling the definition of Grothendieck group $K(X)$ of a noetherian scheme $X$, which is the quotient of free abelian group generated by all coherent sheaves on $X$, by the subgroup generated by all express... | https://mathoverflow.net/users/41541 | Grothendieck group $K(\mathbb A_k^n)$ (Hartshorne Exercise III.5.4) | Hailong's answer is of course exactly right, but there's also a sense in which it's overkill. That's because it invokes two facts:
**Fact I:** Any f.g. module over $R=k[x\_1,\ldots x\_r]$ has a finite resolution by projective $R$-modules.
**Fact II:** Any projective module over $R$ is free. (This is the Quillen/Sus... | 10 | https://mathoverflow.net/users/10503 | 172945 | 88,273 |
https://mathoverflow.net/questions/172740 | 5 | Let $S\_{g,b}$ be an oriented surface with $b$ boundary components and $S\_g^b$ be an oriented surface with $b$ punctures. Let $\mathrm{Mod}(S\_{g,b})$ and $\mathrm{Mod}(S\_g^b)$ their (orientation preserving) mapping class group fixing boundary/punctures pointwise. There is a short exact sequence:
$$1 \to \mathbb Z... | https://mathoverflow.net/users/41219 | Distorsion of subgroups of the mapping class group | Farb, Lubotzky, and Minsky proved in
B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups, Duke
Math. J. 106 (2001), no. 3, 581–597.
that all abelian subgroups of the mapping class group are undistorted.
| 6 | https://mathoverflow.net/users/317 | 172951 | 88,276 |
https://mathoverflow.net/questions/172955 | 14 | In the [OEIS entry](https://oeis.org/A000110) for Bell numbers, there appears a generating function
$$\sum\_{k=0}^\infty B\_k t^k = \sum\_{r=0}^\infty \prod\_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of reference for this formula. The contributor informs me that he discovered it by experimentat... | https://mathoverflow.net/users/9672 | Ordinary Generating Function for Bell Numbers | The proof is given, for example, in <http://www.sciencedirect.com/science/article/pii/S0097316503000141> (Bell numbers, their relatives, and algebraic differential equations, by Martin Klazar). Namely it is proved that the generating function $B(t)=\sum\limits\_{n=0}^\infty B\_nt^n$ satisfies the functional equation
$$... | 19 | https://mathoverflow.net/users/32389 | 172958 | 88,278 |
https://mathoverflow.net/questions/172960 | 2 | Let $G$ be the Suzuki group over the field with $q=2^{2m+1}$ elements, $m>0$. Then, by Theorem 3.10 from B. Huppert, N. Blackburn, Finite Group III, pp 192-193, or [wikipedia](http://en.wikipedia.org/wiki/Suzuki_groups), the group $G$ contains subgroups $F,A,B,$ and $C$ such that
$$\{A^x\setminus \{1\}, B^x\setminus ... | https://mathoverflow.net/users/38062 | On the Suzuki group | The [wikipedia page](https://en.wikipedia.org/wiki/Suzuki_groups#Conjugacy_classes) has a quite clear description of this partition, there are no unexplained parameters there.
| 2 | https://mathoverflow.net/users/11100 | 172963 | 88,279 |
https://mathoverflow.net/questions/172919 | 0 | Given an irreducible curve $C$ of degree $d$ in $\mathbb{P}^r$ and a general hyperplane $H\subset\mathbb{P}^r$, the uniform position theorem states that any $r$ points on the hyperplane section $H\cap C$ will be linearly independent. (Suppose we are working over the complex numbers)
One reference I have found so far is... | https://mathoverflow.net/users/21778 | Uniform position property and general hyperplanes | The answer to both questions is negative, as pointed out by aginesky in the comments. To complete his answer, let me give a simple concrete counterexample.
Consider a canonical curve $C$ of genus 4 with only one $g^1\_3$, so a complete intersection of an irreducible singular quadric $Q$ and a cubic $G$ in ${\mathbb P... | 0 | https://mathoverflow.net/users/46104 | 172972 | 88,285 |
https://mathoverflow.net/questions/172976 | 4 | I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations [for this operator](http://en.wikipedia.org/wiki/Weyl_law#Generalizations)
$H = -\Delta +V$. Now I was wondering whether there are any more recent overvie... | https://mathoverflow.net/users/nan | Asymptotic behavior of Schrödinger operators | This review from 2007 contains a very extensive reference list.
[Some bound state problems in quantum mechanics](http://www.math.uiuc.edu/~dirk/preprints/simonfest8.pdf) (Dirk Hundertmark).
*“Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday” (F. Gesztesy et al., eds.),... | 2 | https://mathoverflow.net/users/11260 | 172979 | 88,287 |
https://mathoverflow.net/questions/172959 | 6 | Let $k$ be a finitely generated field, $\ell$ a prime different from the characteristic of $k$, $S$ a $k$-variety, and $\mathcal{V}$ a lisse $\ell$-adic sheaf on $S$. Fix an algebraic closure $\bar{k}$ of $k$. Let $\bar{S}$ (resp. $\bar{\mathcal{V}}$) denote the pull-back of $S$ (resp. $\mathcal{V}$) to $\bar{k}$.
Ev... | https://mathoverflow.net/users/21815 | Does a lisse $\ell$-adic sheaf give rise to an affine group scheme? | I don't really know how to answer the question in a meaningful way, but let me make a few remarks about the parallel situation for Hodge theory. Suppose that
$S$ is a smooth complex algebraic variety, and that $V$ is a polarizable variation of Hodge structure over $S$ (e.g. $V= R^if\_\*\mathbb{Q}$ for a smooth projecti... | 5 | https://mathoverflow.net/users/4144 | 172986 | 88,289 |
https://mathoverflow.net/questions/172936 | 21 | Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y\_n = \frac{1}{\sqrt{n}}\sum\_{k=1}^{n} f(T^k x)$.
I've done some googling and found statements like "generic smooth functions $f$ do not obey a CLT f... | https://mathoverflow.net/users/1121 | Central Limit Theorem(s) for irrational rotation | The result depends on the approximation properties of $\alpha$.
Of course one has to assume $\int\_{S^1} f(z)dz=0$. A rotation by $\alpha$ has the effect that the $k$-th Fourier coefficient of $f$ is multiplied by $\exp(2\pi i \cdot k \alpha)$. Hence, the $k$-th Fourier coefficient (for $k \neq 0$) of $T\_n(f)$ is j... | 21 | https://mathoverflow.net/users/8176 | 172990 | 88,291 |
https://mathoverflow.net/questions/172973 | 4 | I had been reading an article by Spencer Bloch. There is a remark in this text which states the surjectivity of a particular map between cohomology groups without explaining further. I had been trying to work it out without much success. I elaborate on this map: Let $X$ be a smooth projective hypersurface of degree $d$... | https://mathoverflow.net/users/54369 | Surjectivity of certain cohomology groups on hypersurfaces of high degree | I assume that this is example 1.4 of Bloch's semi-regularity paper. The problem is that you've stated this in a way that makes it harder than it is. He says that you choose $Z$ *first* and then you choose $X\supset Z$ with $d\gg 0$.
So now it's just a matter of killing the appropriate cohomology group using Serre vani... | 4 | https://mathoverflow.net/users/4144 | 173000 | 88,295 |
https://mathoverflow.net/questions/172992 | 20 | Suppose $A$, $B$, are symmetric, real valued matrices and $B-A$ is positive-semidefinite, i.e. $A≼B$. Does that imply $e^A ≼ e^B$? Would love some intuition here.
I know for instance that $A≼cI \iff e^A ≼ e^cI$ and also for diagonal $A$ and $B$ the inequality is easy to show, but I'm not convinced that it translates ... | https://mathoverflow.net/users/54487 | Does the matrix exponential preserve the positive-semi-definite ordering? | To supplement Robert's counterexample, let me mention below some interesting facts about the matrix exponential, along with what may be regarded as the "correct" way of obtaining matrix exponential like operator inequalities.
Assume throughout that $A \ge B$ (in Löwner order).
The map $X \mapsto X^r$ for $0 \le r ... | 22 | https://mathoverflow.net/users/8430 | 173001 | 88,296 |
https://mathoverflow.net/questions/133636 | 3 | Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then
$$ \int\_{\Omega} f(r)\, dV \leq \int\_{\Omega^\star} f(r)\, dV \, ,$$
where $\Omega^\star$ is a geodesic ball with the same volume as... | https://mathoverflow.net/users/34942 | Computations with the distance function on a Riemannian manifold | This is sometimes called the ``bathtub principle'', and can be proved elementarily in a great generality (metric measure spaces).
Consider $\mu$ the measure on $[0,+\infty)$ obtained by pushing forward the volume measure restricted to $\Omega$ by the function $r$, and define similarly $\mu^\*$. Then since $|\Omega|=|... | 4 | https://mathoverflow.net/users/4961 | 173002 | 88,297 |
https://mathoverflow.net/questions/172966 | 0 | Let $X$ be an Abelian variety defined over a number field $K$, suppose that it has a good reduction over a fine place $\mathfrak{p}$ of $K$. Let $G\_{\mathfrak{p}}$ be the local Galois group for $K\_{\mathfrak{p}}$. Let $s$ be a Hodge cycle in $H^n\_{B}(X^{an}, \mathbb{Q})$, the Betti cohomology. By comparison theorem ... | https://mathoverflow.net/users/4504 | Absolute Hodge implies Galois invariant? | This is Proposition 2.9(b) of Deligne's 1982 notes.
| 3 | https://mathoverflow.net/users/54529 | 173004 | 88,298 |
https://mathoverflow.net/questions/173011 | -1 | Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a\_n)\_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has
$$f(x)=\sum\_{n\ge0}a\_nx^n.$$
Let $(P\_n)\_{n\in\mathbb N}$ be a sequence of polynomials that converges uniformly towards $f$ on ev... | https://mathoverflow.net/users/33128 | derivatives and uniformly convergence | No. Start out with any entire $f$ with $f(0)=0$, $f'(0)\not= 0$. Then modify the (natural) approximation of $f$ by its Taylor polynomials $p\_n$ as follows. Let $\varphi\_n\in C(\mathbb R)$, $0\le \varphi\_n\le 1$, $\varphi\_n(0)=0$ and $\varphi\_n(x)=1$ for $|x|>1/n$. Then $\varphi\_n p\_n$ is still uniformly close to... | 2 | https://mathoverflow.net/users/48839 | 173013 | 88,299 |
https://mathoverflow.net/questions/173021 | 5 | Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets.
Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
**Question1:** Is there a fixed constant $C$ such that the number of facets of $L(P)$ is bounded by $Ck$ ?
**Edit:**
**Question2:** Is there any bo... | https://mathoverflow.net/users/53059 | The number of facets of a polyhedron under linear transformation | No. Every bounded convex polyhedron $Q$ with $v$ vertices is an image of a simplex $P$ with $v$ vertices and $v$ facets by a linear transformation. However, $Q$ may have many facets. For example, the cross-polytopes (dual to hypercubes) have $2d$ vertices and $2^d$ facets, one for each orthant, and $2^d$ is not $O(d)$.... | 3 | https://mathoverflow.net/users/2954 | 173027 | 88,307 |
https://mathoverflow.net/questions/173031 | 1 | Let $A$ be an $n\times n$ symmetrix matrix, if $\forall i$,
$a\_{ii}\geq |a\_{ij}|,\forall j$
satisfies, can we say that $A$ is a positive semidefinite matrix? I tried to find a counter example, but failed.
| https://mathoverflow.net/users/49058 | Positive Semidefinite matrix | $$
A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}
$$
| 6 | https://mathoverflow.net/users/52964 | 173035 | 88,311 |
https://mathoverflow.net/questions/173037 | 4 | Let $\{ \xi \_a \}\_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space
$(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. Let $u$ be an independent of $\{ \xi \_a \}\_{a \in [0;1]}$ uniformly distributed on $[0;1]$ random variable. For $\ome... | https://mathoverflow.net/users/41071 | Uncountable family of random variables | $\alpha$ need not be a random variable.
The most natural choice for $(\Omega, \mathcal{F}, P)$ is product space: let $\Omega = [0,1]^{[0,1] \cup \{2\}}$ and for $A \subset [0,1] \cup 2$, let $\pi\_A : \Omega \to [0,1]^A$ be the projection map. (For $a \in [0,1] \cup 2$, we let $\pi\_a$ denote $\pi\_{\{a\}} : \Omega \... | 4 | https://mathoverflow.net/users/4832 | 173050 | 88,315 |
https://mathoverflow.net/questions/173049 | 4 | A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category **$(n+1)$-Cob** of $n$-manifolds and $(n+1)$-cobordisms to **FdVect**.
My question is about the symmetric monoidal structure of **$(n+1)$-Cob** and how it plays well with functoriality:
We consider our man... | https://mathoverflow.net/users/28145 | How do we handle the symmetry condition in nCob and TQFTs? | As Oscar has explained in comments, with the most common definitions it's just not true that $M \sqcup N$ is exactly the same as $N \sqcup M$. But even if you were working with some version of the category of sets where they were equal, this wouldn't be a problem because tensor functors don't have to be strict. In othe... | 12 | https://mathoverflow.net/users/22 | 173055 | 88,316 |
https://mathoverflow.net/questions/173060 | 11 | Many sources give an easy definition of a Toda bracket $\{f,g,h\}$ of appropriate maps $W \to X \to Y \to Z$ in spaces as a subset of the homotopy classes of maps $[\Sigma W, Z]$ (for example, Ravenel's *Complex Cobordism*.) However, the only real usages of Toda brackets that I can find are in the context of heavier li... | https://mathoverflow.net/users/54551 | Simplest example of non-trivial Toda bracket in spaces | I think that the simplest nontrivial case comes from the maps
$$ S^5 \xrightarrow{\Sigma^2\eta} S^4 \xrightarrow{2\iota} S^4 \xrightarrow{\Sigma\eta} S^3 $$
Both two-stage composites are trivial, so we get a Toda bracket which is a subset of $[S^6,S^3]$. This group (or at least the $2$-primary part) is cyclic of order ... | 12 | https://mathoverflow.net/users/10366 | 173063 | 88,318 |
https://mathoverflow.net/questions/172849 | 8 | For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite.
In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$,
$X\ge W\ge Y\ge 0$ and $X\ge Z \ge Y\ge 0$, then $$\det X+\det Y\ge \det W+\det Z.$$
I was thinking of relaxing the conditi... | https://mathoverflow.net/users/54458 | On an inequality among determinants | The answer is yes. I thank fedja for discussion.
WLOG, assume $X=I$ (we may do so by pre-post multiplying both sides by $\det X^{-1/2}$, with a standard continuity argument).
After this, we may further assume $Y=D$ to be diagonal (by unitary similarity).
So the question is equivalent to showing if $I+D\ge W+Z$... | 2 | https://mathoverflow.net/users/54458 | 173065 | 88,320 |
https://mathoverflow.net/questions/173081 | 0 | I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.
| https://mathoverflow.net/users/42804 | what is the first cohomology group of structure sheaf of grassmannian | Over the complex numbers, the Grassmanian $X = G(n,k)$ is simply-connected. Since this also a smooth projective variety, it is compact and Kaehler, so we have $0 =b\_1(X) = 2h^{1,0}(X)$, where $b\_1(X)$ denotes the first Betti number and $h^{1,0}(X) = \text{dim}\ H^1(\mathcal{O}\_X)$. This gives the vanishing that you ... | 3 | https://mathoverflow.net/users/24525 | 173083 | 88,327 |
https://mathoverflow.net/questions/173086 | 5 | (A related question is this
[On the fundamental group of hypersurfaces](https://mathoverflow.net/questions/51301/on-the-fundamental-group-of-hypersurfaces)).
Let $X$ be a simply connected projective complex manifold of dimension at least 3.
Let $Y\subset X$ be a smooth hypersurface. What is $\pi\_1(Y)$ then?
If $Y$... | https://mathoverflow.net/users/9833 | What is the fundamental group of a hypersurface? | I think the answer is no. Take any smooth projective surface $S$ in $\mathbb{P}^N$, and choose a generic projection to $\mathbb{P}^3$. The image is a singular surface $S'$ (with a double curve, some triple points etc.). There exists a birational morphism $P\rightarrow \mathbb{P^3}$ such that the strict transform of $S'... | 7 | https://mathoverflow.net/users/40297 | 173087 | 88,328 |
https://mathoverflow.net/questions/173090 | 16 | It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, what is the minimum amount cardinal numbers $A$ for which non-existence of cardinal $B$ with $A<B<2^A$ guarantees that ... | https://mathoverflow.net/users/30186 | How much of GCH do we need to guarantee well-ordering of continuum? | Yes, you are right. This is a theorem of Specker. If there are no intermediate cardinals between $A,\mathcal P(A)$ and $\mathcal{P(P(}A))$, then $A$ can be well-ordered.
You can find nice details in:
>
> Akihiro Kanamori, David Pincus, "**Does GCH imply AC locally?**, in "Paul Erdős and his mathematics, II (Budap... | 27 | https://mathoverflow.net/users/7206 | 173091 | 88,329 |
https://mathoverflow.net/questions/172519 | 4 | I am a graduate student just started learning dispersive PDE in MSRI's summer program. I roughly finished reading the paper by Klainerman and Machedon "ON THE UNIQUENESS OF SOLUTIONS TO THE
GROSS-PITAEVSKII HIERARCHY". However I do not really know much about the many body system problem in general. May I ask what shoul... | https://mathoverflow.net/users/18850 | What to read for many-body problems in 3D Schrodinger equation | Mean field theory is one of the most active subject concerning many body systems, and the papers you mentioned fit in this context. I think you should read this review by [Golse](http://arxiv.org/abs/1301.5494). However there are at least four different approaches to the problem. One, called coherent states method, int... | 2 | https://mathoverflow.net/users/50356 | 173098 | 88,333 |
https://mathoverflow.net/questions/122778 | 48 | Assuming the axiom of choice the following argument is simple, for infinite $A$ it holds: $$2\lt A\leq2^A\implies 2^A\leq A^A\leq 2^{A\times A}=2^A.$$
However without the axiom of choice this doesn't have to be true anymore. For example if $A$ is an amorphous set (infinite set that cannot be written as a disjoint uni... | https://mathoverflow.net/users/7206 | When does $A^A=2^A$ without the axiom of choice? | David Pincus in "[A note on the cardinal factorial](https://eudml.org/doc/215090)" (Fundamenta Mathematicae vol.98(1), pages 21-24(1978)) proves that $A^A=2^A$ does not imply the axiom of choice, therefore it does not characterise the sets for which $A=A\times A$. The counterexample is the model from his paper "[Cardin... | 28 | https://mathoverflow.net/users/17176 | 173102 | 88,334 |
https://mathoverflow.net/questions/173108 | 6 | If we are given compact complex manifold $X$ and a Kahler class $[\omega]$,
can we always find a positive definite representative $\omega \in [\omega]$ that is
real analytic?
| https://mathoverflow.net/users/4971 | Analytic representatives for Kahler classes | Yes. Run the Kähler-Ricci heat flow
$$
\frac{\mathrm{d}}{\mathrm{d}t}\left(\omega(t)\right) = -\mathrm{Ric}\bigl(\omega(t)\bigr)
$$
with initial condition $\omega(0) = \omega$. This will exist for some time interval $[0,T)$, and the $\omega\_t$ for $t>0$ will all be be real-analytic with respect to the natural real-a... | 10 | https://mathoverflow.net/users/13972 | 173113 | 88,339 |
https://mathoverflow.net/questions/173109 | 4 | I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi\_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the coefficient ring on purpose here). Now for the different powers of the same prime $p$ the cyclotomic polynomials are coprime when... | https://mathoverflow.net/users/34217 | Chinese Remainder Theorem backwards | Since $h\_k := \prod\_{i=1}^k \phi\_{p^i} = (X^{p^k}-1)/(X-1)$, you are trying to compute the exponent of the cokernel
of the inclusion
$$\mathbf{Z}[X]/(h\_k) \rightarrow \prod\_{i=1}^k \mathbf{Z}[\zeta\_{p^i}]$$
that is of finite index (since it becomes an isomorphism upon tensoring with $\mathbf{Q}$). We will prove ... | 6 | https://mathoverflow.net/users/52824 | 173119 | 88,343 |
https://mathoverflow.net/questions/173092 | 8 | Is there an "elementary" proof that $\alpha$-stable random variables only exist for $0 < \alpha \le 2$? By elementary I mean without using Fourier transforms. I'd be happiest with either a direct probabilistic argument, or a geometric argument, e.g., referring to embeddings of $L\_p$ spaces (as long as it doesn't depen... | https://mathoverflow.net/users/1044 | Nonexistence of stable random variables | I believe there to be a proof avoiding Fourier analysis in the literature given by Feller (1971) as part of Theorem VI.1.1.
The argument can be subdivided into two parts. The first part (which was already mentioned in the comments) states, that if an $\alpha$-stable random vector has a finite non-zero variance, then ... | 6 | https://mathoverflow.net/users/38547 | 173121 | 88,344 |
https://mathoverflow.net/questions/173085 | 4 | 1. How many contiguous zeros of zeta are known, to what height
2. How many contiguous primes are known, to what height
3. How many zeta zeros determine how many primes, to what exactness
For example, would knowing the first 1,000 zeta zeros pinpoint the location of the first 1,000 primes, exactly? (Assuming all zeros... | https://mathoverflow.net/users/10350 | Questions about the Riemann Zeta Function | You would need to know *all* the Riemann zeros to determine a single prime exactly, and vice versa. The primes and the zeros are on the opposite sides of a Fourier transform in Riemann's Explicit Formula, so the mathematical version of the Heisenberg Uncertainty Principle applies:
<http://en.wikipedia.org/wiki/Fourie... | 3 | https://mathoverflow.net/users/6756 | 173122 | 88,345 |
https://mathoverflow.net/questions/73210 | 2 | Let $K$ be a skew-field, infinite dimensional over its center $F$.
From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-algebras have finite dimension over their center).
There, a GPI (generalized polynomial identity) has coefficients from th... | https://mathoverflow.net/users/8338 | (Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity) | I recently came back to this question after leaving it aside for a few years, and this time my digging seems to have yielded an answer: Chen-Lian Chuang, "Differential identities with automorphisms and antiautomorphisms. I", Journal of Algebra 149, 371-404 (1992), proved the following theorem:
>
> A prime ring sati... | 2 | https://mathoverflow.net/users/8338 | 173140 | 88,349 |
https://mathoverflow.net/questions/173124 | 11 | Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a $\mathrm{O}(n)$-module via the diagonal action.
I am interested in the following specific submodules, which arise from Riemannian... | https://mathoverflow.net/users/17294 | Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry | As Qiaochu Yuan wrote, these irreducible decompositions are classical. You can read about them in Hermann Weyl's *The Classical Groups* (for example), and the questions you are asking are basically exercises in writing out what the standard methods tell you. (Of course, there are more recent treatments, which modern re... | 7 | https://mathoverflow.net/users/13972 | 173143 | 88,351 |
https://mathoverflow.net/questions/173126 | 0 | **Background**
I am reviewing some category theory, which I did not learn too well the first time around. One text I am using is Mac Lane's. Near the beginning of the chapter on adjunctions (pg 80),
he gives the basic definition of an adjunction $<F,G,\phi>$ ($G$ right-adjoint to $F$) in terms of the natural isomorp... | https://mathoverflow.net/users/nan | Basic category theory: Universality of adjunction unit is justified by Yoneda Proposition in Mac Lane's text | By Yoneda for every natural transformation $\tau \colon \mathbf C(c,-) \to F$, where $F \colon \mathbf C \to \mathbf{Set}$ and $\mathbf C(c,-)$ is the covariant $\hom$-functor, is a family of functions of the form
$$\tau\_x \colon \mathbf C(c,x) \to F(x)$$
such that $\tau\_x(f)=F(f)\circ \eta\_x$ for every $f \in \ma... | 2 | https://mathoverflow.net/users/14969 | 173144 | 88,352 |
https://mathoverflow.net/questions/173154 | 4 | Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some $a\in A$ such that $\widehat{a}|\_E\equiv 0$ and $\widehat{a}(x)=1$ (where $\widehat{a}$ is the Gelfand transform of $a... | https://mathoverflow.net/users/40551 | Regular commutative Banach algebras which are not closed under complex conjugate | I'm writing in a hurry and away from various sources, but there are known examples of proper uniform algebras (i.e. not C(X)) which are regular. By Stone-Weierstrass, these can't be closed under conjugation.
The first examples are usually attributed to McKissick (1963) but I have heard some specialists say that the c... | 5 | https://mathoverflow.net/users/763 | 173156 | 88,357 |
https://mathoverflow.net/questions/173134 | 5 | I have a probably very basic question about modules over symmetric ring spectra:
Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let $\varphi \colon M \to N$ be a morphism of $R$-module spectra, which is a stable equivalence of the underlying symmetric spect... | https://mathoverflow.net/users/3995 | endomorphisms of modules over symmetric ring spectra | Yes, this is true. This kind of property for a model category is a consequence of what is sometimes abusively called the "SM7" axiom for the enrichment:
* In $R$-modules, suppose that $A \to B$ is a cofibration and $C \to D$ is a fibration. Then the natural map of $R$-modules
$$
Hom\_R(B,C) \to Hom\_R(A,C) \times\_{H... | 7 | https://mathoverflow.net/users/360 | 173163 | 88,361 |
https://mathoverflow.net/questions/173088 | 19 | It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$
I am considering a possible extension of this result. Let $\mathbb{M}\_m(\mathbb{M}\_n)$ denote the set of $m\times m$ block matrices with each block the usual $n\times n$ matrix.
Let $\mathbf{A}=[A\_{i,j}]\_... | https://mathoverflow.net/users/54458 | A possible extension of a determinant inequality | The claim is true. We prove it using a few block matrix manipulations. Note, in the proofs below $A \ge 0$ means $A$ is (symmetric) positive semidefinite.
$\newcommand{\trace}{\text{trace}}$
>
> **Lemma** Let $X, Y \ge 0$. Then,
> \begin{equation\*}
> \otimes^k (X+Y) \ge \otimes^k X + \otimes^k Y.
> \end{equati... | 13 | https://mathoverflow.net/users/8430 | 173165 | 88,362 |
https://mathoverflow.net/questions/173158 | 10 | Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}\_{s = 1} L(E, s)$, a value conjecturally equal to $\mathrm{dim}\_{\mathbb{Q}} (E(\mathbb{Q}) \otimes\_{\mathbb{Z}} \mathbb{Q})$. There is a $c \in \mathbb{C}^\times$ such that $L(E, s) \... | https://mathoverflow.net/users/53197 | Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD? | Let me summarise the comments above that give a full answer (correct me if I am wrong).
1. The analytic continuation of $L(E,\bar{s})=\overline{L(E,s)}$ shows that $c\in\mathbb{R}$.
2. If $r=0$, the fact that $c>0$ is proven in [On the positivity of the central value of automorphic L-functions for GL(2)](http://proje... | 12 | https://mathoverflow.net/users/5015 | 173176 | 88,363 |
https://mathoverflow.net/questions/171950 | 7 | This is a follow-up to my question about nilpotent orbits [here](https://mathoverflow.net/questions/171677/) asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and representation theory, I'm curious about what has been written down explicitly ... | https://mathoverflow.net/users/4231 | Number of Richardson orbits in simple Lie algebras of types $E_n$? | Isn't this question a subset of the induction question (i.e. if we induce an orbit from a Levi subalgebra up to ${\mathfrak g}$, then what orbit do we get)? This was solved by Elashvili (for exceptional types) in 1979, and his computations were verified in a 2009 joint Elashvili-de Graaf paper ("Induced nilpotent orbit... | 4 | https://mathoverflow.net/users/26635 | 173186 | 88,370 |
https://mathoverflow.net/questions/173118 | 2 | Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with respect to $H$. However it might be possible that an $H$-stable vector bundle is no longer stable with respect to another ampl... | https://mathoverflow.net/users/4504 | The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof? | See the following paper by Adrian Langer: <http://arxiv.org/abs/0905.4600> and more precisely section 4.
I quote:
>
> In this section we show that strongly semistable torsion free sheaves with vanishing Chern classes
> are locally free and that they are strongly semistable with respect to all polarizations
>
> ... | 3 | https://mathoverflow.net/users/31051 | 173201 | 88,378 |
https://mathoverflow.net/questions/173214 | 4 | Given a hypergraph $H=(V,E)$ and a set $X\subseteq V$ of vertices, let $int(X)$ be the number of distinct intersections of edges with $X$, i.e.
$$int(X)=|\{S\subseteq X, \exists e\in E, e\cap X=S\}|.$$
$X$ is called *shattered* if $int(X)=2^{|X|}$, i.e. if $int(X)$ reaches its maximum feasible value.
>
> **Questi... | https://mathoverflow.net/users/54628 | Is it true that every hypergraph with a large "semi-shattered" set has large VC dimension? | This follows from the Sauer–Shelah lemma (mentioned in the Wikipedia article on VC-dimension linked in your question).
>
> **Theorem** Let $\mathcal F$ be a family of subsets of $\{1, 2, \ldots, n\}$. If $|\mathcal F| > \binom n 0 + \binom n 1 + \cdots \binom n k$, then $\mathcal F$ shatters a set of size $k+1$.
> ... | 4 | https://mathoverflow.net/users/25485 | 173218 | 88,384 |
https://mathoverflow.net/questions/173153 | 4 | Let $\mathbb{H}\_g$ denote Siegel space, and $M$ denote an order 4 element of the unitary subgroup $U(n)(\mathbb{R})$with $p$ eigenvalues equal to $i$, and $q$ eigenvalues equal to $-i$, $p+q=g$. Consider $M$ as an element of the symplectic group $SP\_{2g}(\mathbb{R})$ by identifying $U(n)(\mathbb{R})$ with a stabilize... | https://mathoverflow.net/users/4181 | Hermitian Symmetric Subspaces of Siegel Space | Use the bounded model (Harish-Chandra realization) of the Siegel upper half-space $\mathfrak H\_{p+q}$. Then $U(n)$ is "diagonalized" into blocks, so it's easy to see its action.
EDIT: as @jacob commented, probably my initial reaction was off by a sign, ... so, in greater detail: let
$$
M \;=\; \pmatrix{
i1\_p & 0 ... | 2 | https://mathoverflow.net/users/15629 | 173224 | 88,386 |
https://mathoverflow.net/questions/173208 | 2 | i am reading the article "Counter-example to global Torelli problem for irreducible symplectic manifolds" by Yoshinori Namikawa and i have two questions i can't answer:
1) he takes $T$ a complex torus, $Sym^3(T)$ the symmetric 3-product and $\alpha:Sym^3(T)\rightarrow T$ the sum map. Then he writes $K^2(T)$ for the g... | https://mathoverflow.net/users/54623 | Two questions about counter example for Torelli theorem for hyperkahler manifolds | Question 1.
It's not hard to see that $F$ is $\Sigma \times {\Bbb C} P^1$ and $\Sigma$ is a torus. Therefore the Albanese map is a projection to a torus.
Question 2.
If $f^\* \delta=-\delta$, then the effective cycle $f^\* \delta+\delta$ is homologous to 0. On a Kahler manifold this is impossible, because
an integr... | 3 | https://mathoverflow.net/users/3377 | 173232 | 88,387 |
https://mathoverflow.net/questions/173235 | 5 | This is just a reference request; I have no sharp mathematical question.
Inspired by the $(3+)$-year old MO question,
[In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?](https://mathoverflow.net/q/63158/6094),
I would like to ask:
>
> **Q1**. Are there are studies of
> knot polynomial invar... | https://mathoverflow.net/users/6094 | Knot invariants in 3-manifolds that are not $\mathbb{R}^3$ or $S^3$ or $B^3$? | For question 1) Yes analogs of knot invariants exist for any manifold regardless of prime decomposition or Thurston decomposition. For example, consider the Alexander polynomial and its categorification, knot Floer homology. These can both defined for general one cusped 3-manifolds. Computations of these invariants hav... | 7 | https://mathoverflow.net/users/27453 | 173240 | 88,389 |
https://mathoverflow.net/questions/173238 | 5 | Maybe this is a vague question. In Besse's book *Einstein manifolds*, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference between $\delta W^{\pm}=0$ and Einstein?
Are there examples of metrics with $\delta W^+=\delta W^-=0$ but not Einste... | https://mathoverflow.net/users/51632 | What is the difference between $\delta W^{\pm}=0$ and Einstein? | A very simple example of a non-Einstein manifold with harmonic Weyl tensor is the product $\mathbb{S}^2\times \mathbb{R}^2$ (with the usual metrics). In Besse's book there are more examples. The example above has also constant scalar curvature. So the assumptions $\delta W=0$ and constant scalar curvature are not suffi... | 8 | https://mathoverflow.net/users/51420 | 173241 | 88,390 |
https://mathoverflow.net/questions/173213 | 1 | Consider the concept of a module. This can be understood a multisorted algebraic theory $\mathsf{Mod}$ on two sorts, a scalarsort $S$ and a vectorsort $V$. An interpretation of $\mathsf{Mod}$ in $\mathbf{Top}$ assigns to the scalarsort a topological field, to the vectorsort a topological Abelian group, and to the scala... | https://mathoverflow.net/users/26080 | Is anyone talking about partial interpretations of theories? (Edited) | One common example where this arises is in the consideration of $\omega$-models of set theory. These are precisely the models that interpret their natural numbers as the standard natural numbers $\omega$, or $\mathbb{N}$. Thus, the class of $\omega$-models of a given set theory are essentially the partial interpretatio... | 2 | https://mathoverflow.net/users/1946 | 173244 | 88,392 |
https://mathoverflow.net/questions/173230 | 3 | Let $k$ be an algebraically closed non-Archimedean valued field with the value group $\mathbb R$, and let $X$ be a variety over $k$. Is it true that for any point $x \in X^{an}$ of the Berkovich analytification of a variety $X$ over $k$ there exists an affinoid neighbourhood $Sp A$ of $x$ such that $x$ is the unique po... | https://mathoverflow.net/users/2234 | is every point of a Berkovich space a Shilov point? | No. Saying that the point $x$ lies in the Shilov boundary of $\mathcal{M}(A)$ gives strong restrictions on the completed residue field $\mathscr{H}(x)$. In your case, its residue field $\widetilde{\mathscr{H}(x)}$ will be of transcendence $d$ over $\tilde{k}$, where $d$ is the dimension of the variety (see proposition ... | 4 | https://mathoverflow.net/users/4069 | 173249 | 88,395 |
https://mathoverflow.net/questions/173166 | 2 | Call a subset $H \subset S\_n$ "simple" if for every $q \in S\_n$, there is some $p \in H$ such that $pq$ is cyclic (i.e. consists of a single cycle of length $n$).
Is there some characterization known of either simple on non-simple subsets ? I'm interested in any computational techniques for implementing a fast test... | https://mathoverflow.net/users/54600 | When is the product of two permutations cyclic? | **LATER** I had a few thoughts which do not lead me to change the answer below but do make me less confident in it. A more general question is: Given two subsets $A,B$ of a group $G$ (in this case $G=S\_n$), is there an efficient way to decide if $AB=\{{ab\mid a\in A,b\in B\}}=G?$ In the case below $B$ is fixed to be t... | 1 | https://mathoverflow.net/users/8008 | 173251 | 88,397 |
https://mathoverflow.net/questions/173255 | 4 | Given universal set $U$. Is there any name of the collection of subsets of $U$ (call them quasi-open) satisfying the following axioms:
i) $\emptyset$ and $U$ are quasi-open;
ii) finite intersections of quasi-open sets are quasi-open;
iii) countable unions of quasi-open sets are quasi-open?
| https://mathoverflow.net/users/4312 | "countable" topology | This type of structure has sometimes been called a "$\sigma$-topology", especially in connection with ultraproducts. For instance the invariant Loeb measure on an ultrafinite group can be thought of as a sort of Haar measure on a $\sigma$-topological group. See for instance Section 2 of this paper of Bergelson and Tao:... | 8 | https://mathoverflow.net/users/20598 | 173257 | 88,399 |
https://mathoverflow.net/questions/173039 | 1 | What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function.
Thanks!
| https://mathoverflow.net/users/51469 | Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function | There is an explicit formula in the book:
A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev. INTEGRALS AND SERIES, Volume 4.
Direct Laplace Transforms. GORDON AND BREACH, 1992.
It is on the page 533 and is in terms of $\_{2}F\_{2}$ hypergeometric function. For special values of parameters for sure it can be simplified u... | 2 | https://mathoverflow.net/users/49208 | 173260 | 88,401 |
https://mathoverflow.net/questions/173231 | 4 | Let $U$ be an open subset of $\mathbb{R}^2$. For my purposes, we can assume that $U$ is just a rectangle. I have an infinitely differentiable map $M:U\to U$ that has a unique fixed point $p$ in $U$. Furthermore, the Jacobian of $M$ at $p$ only has eigenvalues with absolute value less than $1$, so I know that $p$ is a g... | https://mathoverflow.net/users/20838 | Jacobian has small eigenvalues everywhere in an open set. Does this imply globally attracting fixed point in that set? | No consider $U=\{(x,y)\in \mathbb{R}^2| |xy|<1/4\}$, $M(x,y)=(y^2,x^2)$ then $(0,0)$ is the only fixpoint but the orbit of $(1,0)$ does not converge to $(0,0)$.
| 4 | https://mathoverflow.net/users/35593 | 173269 | 88,405 |
https://mathoverflow.net/questions/173263 | 5 | Let $\kappa$ be a $\mu$-strongly compact cardinal, which means that there is an elementary embedding $j:V\rightarrow M$, with critical point $\kappa$ such that $M$ is well founded (even closed under $\kappa$ sequences) and there is $s\in M$ such that $j^{\prime\prime} \mu \subset s$ and $M\models |s|<j(\kappa)$.
What... | https://mathoverflow.net/users/41953 | Covering properties of strongly compact embedding | This is a very nice question (and indeed, I remember asking myself this question when I was a graduate student).
The answer in general is that no, you do not get any extra strength from having these small covering sets for strong compactness. Indeed, I claim that one can have such small covering sets for every regul... | 5 | https://mathoverflow.net/users/1946 | 173272 | 88,408 |
https://mathoverflow.net/questions/173239 | 5 | Let $\mathit{Mfd}$ denote the category of smooth manifolds. Let $W$ denote all projections of the form $$M \times \mathbb{R} \to M.$$ Let $\mathit{Mfd}\_W$ denote the Hammock localization of $\mathit{Mfd}$ at the class of maps $W$. Is the mapping complex $Map\_W\left(M,N\right)$ in $\mathit{Mfd}\_W$ between two manifol... | https://mathoverflow.net/users/4528 | Mapping complexes in the simplicial localization of the category of manifolds | Ok, so I have an argument:
First note that the homotopy coherent nerve of $\mathit{Mfd}\_W$ is equivalent to the quasicategory obtained by formally inverting $W$ in $N\left(\mathit{Mfd}\right)$- this holds in generality, as shown in a recent paper of Hinich (Proposition 2.2.1 of <http://arxiv.org/abs/1311.4128>). Let... | 3 | https://mathoverflow.net/users/4528 | 173275 | 88,410 |
https://mathoverflow.net/questions/173285 | 2 | I'm hoping for some help in nailing down a vague idea about independence. It starts with finding the expectation of a product of three RVs (or more, but I'll stick to three for now). These are not necessarily independent, but we know all the pairwise covariances. This is enough information to predict the expected value... | https://mathoverflow.net/users/30419 | higher-level independence of three or more correlated RVs | Higher order correlations are called joint cumulants or semi-invariants. They can be expressed in terms of joint moments. If you want one of the cumulants to be equal to zero, that gives you an algebraic equation on moments. A joint moment of Bernoulli r.v.'s equals the probability that the r.v.'s involved are all equa... | 4 | https://mathoverflow.net/users/2968 | 173294 | 88,418 |
https://mathoverflow.net/questions/173286 | 4 | Which finite simple groups have order N so that N+1 is a proper power?
As an example: the simple group of order $168=13^2-1$.
| https://mathoverflow.net/users/25762 | Orders of Finite Simple Groups | For abelian simple groups your question is merely a disguised form of
"enumerate the Mersenne primes".
The smallest examples for nonabelian simple groups are as follows:
* $|{\rm PSL}(2,7)| + 1 = 13^2$,
* $|{\rm A}\_6| + 1 = 19^2$,
* $|{\rm M}\_{11}| + 1 = 89^2$,
* $|{\rm PSU}(4,2)| + 1 = 161^2$,
* $|{\rm J}\_1| + 1 ... | 12 | https://mathoverflow.net/users/28104 | 173296 | 88,420 |
https://mathoverflow.net/questions/172538 | 20 | Let $\Gamma$ be a discrete group (though this could be asked for general locally compact groups) and consider the Banach $\*$-algebra $\ell^1(\Gamma)$. We have two natural $C^\*$-algebra completions: the reduced group $C^\*$-algebra $C^\*\_r(\Gamma)$ which is the closure of $\ell^1(\Gamma)$ acting by left translation o... | https://mathoverflow.net/users/406 | "Minimal" group C*-algebra? | The following is not an answer, but records some background and some related results which suggest that the original question might be hard to answer in the generality stated.
If $\Gamma$ is amenable,
$\newcommand{\Cst}{{\rm C}^\*}$
then the maximal and reduced $\Cst$ algebras coincide, and your question is equivalen... | 8 | https://mathoverflow.net/users/763 | 173306 | 88,426 |
https://mathoverflow.net/questions/173308 | 7 | Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial localization $L(C^D,W^D)$. Or I can first simplicially localize to get $L(C,W)$ and then form the $(\infty,1)$-functor category $L... | https://mathoverflow.net/users/49 | When does simplicial localization commute with functor categories? | For general $C$ (I will drop $W$ from the notation) this is not true even when $D$ is an infinite discrete category.
In the answer to [this similar question](https://mathoverflow.net/questions/139020/) I described how to construct a sequence of relative categories $C\_0, C\_1, C\_2, \ldots$ with objects $X\_i, Y\_i \... | 7 | https://mathoverflow.net/users/12547 | 173311 | 88,428 |
https://mathoverflow.net/questions/173148 | 7 | In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a *weakly closed* subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ implies that $g^{-1}Kg = K$.
The subgroup property I am interested in is this one: what kind of subgroup of $H$ is ... | https://mathoverflow.net/users/12610 | Subgroup property stronger than being characteristic | Instead of going via HNN extensions, one can work entirely with finite groups
to show that if $X \cong Y$ are subgroups of a finite group $H$, then there exists a finite group $G \supseteq H$ such that $X$ and $Y$ are conjugate in G.
The trick is to take $G$ to be the full symmetric group on he elements of $H$, with ... | 11 | https://mathoverflow.net/users/9694 | 173317 | 88,429 |
https://mathoverflow.net/questions/173338 | 1 | A hyperelliptic curve can be understood as the set of points satisfying an equation of the form
$$\displaystyle z^2 = f(x,y),$$
where $f(x,y)$ is a binary form of degree $d = 2g+2$. In this case, $g$ can be shown to be the genus of the curve.
It turns out that when $g=1$, the curve is an elliptic curve and in fact al... | https://mathoverflow.net/users/10898 | Non-hyperelliptic curves of genus at least two | Perhaps a couple of things should be pointed out.
1) All curves of genus $g=2$ are hyperelliptic.
2) Your equation $z^2=f(x, y)$ is not the equation of an hyperelliptic curve. If you take $f(x, y)$ to be a binary form of degree $2g+2$ then the (projective) equation of the curve is
$$ z^2 y^{2g} = f(x, y),$$
where ... | 7 | https://mathoverflow.net/users/nan | 173339 | 88,440 |
https://mathoverflow.net/questions/173329 | 0 | I have 3 neural networks processing 3 different vectors of values. Each NN processes a sample of it's vector and gives binary result (y/n) that is correct with given probability. All 3 NNs give answer to the same question. The task is to combine this results into single one (y/n) and figure out probability that it is c... | https://mathoverflow.net/users/54660 | Combine results with different veracity | Assume that the networks err independently and the corresponding probabilities are $(TP)\_j, (TN)\_j, (FP)\_j, (FN)\_j$ (false negative = the answer is yes and the network outputs no, etc.). Also assume that the price you pay for an error of each type is the same for both answers.
Suppose that you get YNY, say. This ... | 2 | https://mathoverflow.net/users/1131 | 173343 | 88,443 |
https://mathoverflow.net/questions/173284 | 1 | I am interested in a proof of the following fact :
Suppose that $X$ is a Riemann surface homeomorphic to the Riemann sphere. Then $X$ is *conformally equivalent* to the Riemann sphere.
Of course, this follows from the uniformization theorem which states that every simply connected Riemann surface is conformally equ... | https://mathoverflow.net/users/1162 | A special case of the uniformization theorem | An alternative is to simply use Riemann-Roch, which does not depend on the Uniformization Theorem.
R-R says that $\ell(D)-\ell(K-D) = \mathrm{deg}(D)-g+1$, where $\ell(D)$ is the dimension of the space of meromorphic functions $f$ on your surface $C$ such that $(f)+D\ge0$.
If $C$ is homeomorphic to the sphere, th... | 9 | https://mathoverflow.net/users/13972 | 173353 | 88,448 |
https://mathoverflow.net/questions/173177 | 1 | **EDIT:** Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume that for some $i$ the function $y\mapsto dim H^i(X\_y,\mathcal{F\_y})$ is constant on $Y$. Then the Grauert theorem sa... | https://mathoverflow.net/users/16183 | A generalization of the Grauert direct image theorem | **EDIT:** I checked in the book "Algebraic Methods in the global theory of complex spaces" by Banica and Stanasila mentioned in a comment by Ben A. I think it does contain the positive answer to my question.
The statement is explicitly formulated on p. 119 of the book during the proof of Theorem 3.4. It is the claim... | 1 | https://mathoverflow.net/users/16183 | 173354 | 88,449 |
https://mathoverflow.net/questions/172509 | 8 | The Bloch-Kato conjecture states that
$K\_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}\_l)$ for every $n,l$,while $l$ is invertible in $k$.
A important part in the proof of the Bloch-Kato conjecture is to proof the existence of norm varieties. In
M. Rost, Norm varieties and algebraic cobordism, Proc. of the Int.
Congr... | https://mathoverflow.net/users/51251 | Known norm varieties and the Bloch-Kato conjecture | In the paper of Rost you mentioned, there are not just examples of norm varieties - there is an outline of how to construct norm varieties in general. The preprint version of Rost's article can be found [following this link.](http://www.math.uni-bielefeld.de/~rost/data/nv-ac.pdf) Note that in that very same article, on... | 3 | https://mathoverflow.net/users/50846 | 173355 | 88,450 |
https://mathoverflow.net/questions/173337 | 1 | Let $f:\mathbb{P}^n\_1\dashrightarrow\mathbb{P}^n\_2$ be the standard Cremona transformation based on $p\_1,...,p\_{n+1}\in\mathbb{P}^n\_1$ and $q\_1,...,q\_{n+1}\in\mathbb{P}^n\_2$. That is, $f$ is the rational map associated to the linear system of hypersurfaces of degree $n$ having multiplicity $n-1$ at $p\_1,...,p\... | https://mathoverflow.net/users/nan | Cremona transformations | For simplicity of notation we may assume $\mathbb{P}^n\_1=\mathbb{P}^n\_2=\mathbb{P}^n$, and $q\_i=p\_i$. Let $P$ be the variety obtained by blowing up $\mathbb{P}^n$ at $p\_1,\ldots ,p\_{n+1}$, $E\_i$ the exceptional divisor above $p\_i$, $H$ the pull back of a hyperplane section. Then $f$ induces an involution of $P$... | 3 | https://mathoverflow.net/users/40297 | 173357 | 88,451 |
https://mathoverflow.net/questions/172496 | 1 | Let $H\_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms of geometric data.. I wonder in the case of type A, is there any classification in terms of concrete combinatoric?
... | https://mathoverflow.net/users/5082 | classification of irreducible finite dimensional representation of affine hecke algebra of type A | This is done in Orellana-Ram, `[Affine braids, Markov Traces and the category O](http://arxiv.org/abs/math/0401317)'. The answer is essentially the same as for the degenerate affine Hecke algebra.
| 1 | https://mathoverflow.net/users/4366 | 173368 | 88,455 |
https://mathoverflow.net/questions/173342 | 8 | I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but rather for classes of examples in which either a yes or no applies. I will repeat my question with more detail and example... | https://mathoverflow.net/users/78 | When is/isn't the monoidal unit compact projective? | Regarding schemes: on a scheme $X$, the functor of global sections is exact iff no quasicoherent sheaves have higher cohomology. By [Serre's criterion for affineness](http://amathew.wordpress.com/2012/08/01/serres-criterion-for-affineness-as-morita-theory/), for reasonable schemes this is equivalent to $X$ being affine... | 3 | https://mathoverflow.net/users/290 | 173379 | 88,461 |
https://mathoverflow.net/questions/173365 | 1 | Nisan's answer to this [question](https://cstheory.stackexchange.com/questions/562/how-to-produce-a-random-graph-that-does-not-have-a-hamiltonian-cycle/567#567) shows the Impossibility of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
... | https://mathoverflow.net/users/8784 | Implications of the impossibility of efficient sampling from random non-Hamiltonian graphs | You didn’t specify what exactly do you mean by efficient sampling. The following definition will work for my purposes. Let $L$ be a language, and assume for simplicity that $L\_n$ (the set of strings in $L$ of length $n$) is nonempty for every $n$. Then $f$ is a sampling function for $L$ if for some polynomial $p(n,m)$... | 1 | https://mathoverflow.net/users/12705 | 173380 | 88,462 |
https://mathoverflow.net/questions/173393 | 6 | Let $R$ be an associative ring. Let $K(R)$ be the category of chain complexes of $R$-modules and chain homotopy classes of maps between them, and let $D(R)$ be its localization with respect to acyclic complexes.
Is there a ring $S$ such that $K(R) = D(S)$? Is the question more likely to have a positive answer if I al... | https://mathoverflow.net/users/54695 | Is the homotopy category of a ring also the derived category of another ring? | Eric is right, $K(\mathbb{Z})$ has no generating set. This is Lemma E.3.2 in Neeman's *Triangulated Categories*. Presumably, the proof will apply to many other rings, but I can't comment on that.
| 9 | https://mathoverflow.net/users/12547 | 173409 | 88,469 |
https://mathoverflow.net/questions/173351 | 2 | We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x\_1,\cdots,x\_n \in X$ such that $M=[d(x\_i,x\_j)]\_{n\times n}$.
**Question.** For which metric spaces are the properties of distance matrices investigated?
| https://mathoverflow.net/users/53059 | Distance matrices | The first paper to read is "[Metric spaces and positive definite functions](http://datamin.ubbcluj.ro/semi/pdfs/Schoenberg.pdf)" by Schoenberg. (I first learned about this from Igor Rivin many years ago.) It follows from his main result + Corollary 1 that a (say, finite) metric space $(M,d)$ embeds isometrically in a H... | 4 | https://mathoverflow.net/users/21684 | 173410 | 88,470 |
https://mathoverflow.net/questions/165051 | 12 | I have seen a few techinques for proving that certain sets of real numbers *are* $\infty$-Borel ([definition](https://en.wikipedia.org/wiki/Infinity-Borel_set)) but it just occurred to me that I don't know of any way to prove that a set of real numbers is *not* $\infty$-Borel.
I'm afraid this may turn out to be a sil... | https://mathoverflow.net/users/1682 | Sets that are not $\infty$-Borel | Henle, Mathias and Woodin showed that if every set of reals is Ramsey, then forcing with $\mathcal{P}(\omega)/\mathrm{Fin}$ adds no new sets of ordinals. Any new set of reals in the extension (e.g., the ultrafilter given by the generic filter) would fail to be $\infty$-Borel in the extension. So I suppose that answers ... | 6 | https://mathoverflow.net/users/31807 | 173413 | 88,471 |
https://mathoverflow.net/questions/173412 | 1 | I want to calculate the sum
$$
\sum\_{i=1}^{n} \left\lfloor\frac{n}{i}\right\rfloor,
$$
and it seems to require to calculate the sum
$$
\sum\_{i=1}^{n} (n \bmod i).
$$
How can I get an $O(1)$ solution instead of the $O(n)$ solution?
| https://mathoverflow.net/users/41383 | How can I calculate $ \sum_{i=1}^{n} (n \bmod i) $ | You're probably not going to be able to compute $\sum\_{1 \leq i \leq n} n \bmod i$ efficiently. Let $f(n)$ denote this sum. This is sequence A004125 in Sloane's Online Encyclopedia of integer sequences (see <http://oeis.org/A004125> ). It is known and relatively simple to prove that $f(n) = n^2 - \sum\_{1 \leq i \leq ... | 7 | https://mathoverflow.net/users/44797 | 173421 | 88,474 |
https://mathoverflow.net/questions/173428 | -2 | This is more a puzzle than a research question,
a puzzle to me. Perhaps it is straightforward for others.
Imagine Repeatedly interpreting a number
expressed with the usual base-$10$ digits
as "digits" multiplying powers of $2$ rather than powers of $10$.
For example, interpret
$n=27$ as $2\cdot 2^1 + 7\cdot 2^0 = 11$... | https://mathoverflow.net/users/6094 | Decimal digits multiplied by powers of 2: leads to mod 8? | Just a simple proof of the formula for $2$ and $10$. The remainder modulo $8$ is unchanged under the operation $f$, since $2^k=10^k$ modulo $8$. On the other hand, you will never reach $0$ or $1$ if you did not start with them.
| 1 | https://mathoverflow.net/users/38468 | 173429 | 88,475 |
https://mathoverflow.net/questions/172520 | 14 | Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}\_p:=\{\xi\in T\_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in S^{2n-1},
$$
defines a *contact structure* on $S^{2n-1}$, i.e., a 1-codimensional completely non-integrable distribution (in ... | https://mathoverflow.net/users/22606 | Is there a "unique" homogeneous contact structure on odd-dimensional spheres? | Well, here is what I can say. Perhaps this will answer some of your questions about $S^{2n+1}$ at least.
Suppose that $G/H = S^{2n+1}$ where $n>0$ and that the action of $G$ on $S^{2n+1}$ is effective and preserves a contact structure on $S^{2n+1}$.
By a result of Montgomery ([*Simply connected homogeneous spaces*,... | 10 | https://mathoverflow.net/users/13972 | 173430 | 88,476 |
https://mathoverflow.net/questions/173422 | -2 | I want to calculate a pair of GPS coordinates(lat,long) that is at x meters N/S/E/W from a known point (lat\_old,long\_old).
I have found the Haversine formula
<http://upload.wikimedia.org/math/0/5/5/055b634f6fe6c8d370c9fa48613dd7f9.png>
for a for the distance between a two points and I have tried to isolate lat , b... | https://mathoverflow.net/users/54715 | Calculate GPS coordinates at x meters | I have to do a little guessing in answering your question:
I suppose, you are given WGS84 coordinates and an ellipsoid that approximates the shape of the earth, and you have the task of finding the WGS84 coordinates, that has an offset of x meters to the east and y meters to the north.
In that case you don't need... | 0 | https://mathoverflow.net/users/31310 | 173432 | 88,478 |
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