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https://mathoverflow.net/questions/101639 | 2 | I am reading Group theory and Physics by shlomo steinberg. On page no: 59 it says that two representations with same charecter functions are equivalent. I am unable to follow the argument.
I think it follows from the fact that decomposition of a representation of a group G over a vector space V into irreducible subre... | https://mathoverflow.net/users/24187 | Is decomposition of a representation of a group G over a vector space V into irreducible subrepresentations over invariant subspaces of V unique? | You are misunderstanding Sternberg's argument. The argument goes like this (with my notations):
Consider two representations $V$ and $W$ of $G$ with the same character.
For any representation $S$ of $G$, let $\chi\_S$ denote the character of $S$.
Take *any* decomposition $V=V\_1\oplus V\_2\oplus ...\oplus V\_k$ o... | 3 | https://mathoverflow.net/users/2530 | 101645 | 58,993 |
https://mathoverflow.net/questions/101619 | 3 | We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G\_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G\_\infty$ morphism from $HH^\cdot(A,A)$ to $C^\cdot(A,A)$, which makes Kontsevich's original $L\_\infty$ morphism as the $L\_\infty$ part of it.
... | https://mathoverflow.net/users/24965 | What is the definition of "the $L_\infty$ part of a $G_\infty$ morphism"? | A $G\_\infty$-morphism $\phi$ is determined by structure maps $\phi^{k\_1,\dots,k\_n}$, $n\geq1$, $k\_1,\dots,k\_n\geq1$.
The $L\_\infty$-part of $\phi$ is the $L\_\infty$-morphism $\ell$ with structure maps $\ell^k=\phi^{\overbrace{1,\dots,1}^{k~times}}$.
In order to make this precise you might have to put (de)s... | 3 | https://mathoverflow.net/users/7031 | 101656 | 59,002 |
https://mathoverflow.net/questions/101629 | 3 | Let $E$ be a normed (real) space which is not complete. Is it always possible to find $f$ a continuous bijective linear function from $E$ to $E$ such that $f^{-1}$ is not continuous?
| https://mathoverflow.net/users/41060 | Around Banach isomorphism theorem | No. If $E$ has codimension less than the continuum in a Banach space then such an $f$ must be open. This was proved by Saxon and Levin; see the proposition on page 95 of
[this paper](http://www.ams.org/journals/proc/1971-029-01/S0002-9939-1971-0280972-0/S0002-9939-1971-0280972-0.pdf)
which is
Saxon, Stephen; Levi... | 3 | https://mathoverflow.net/users/2554 | 101691 | 59,029 |
https://mathoverflow.net/questions/101719 | 8 | Let $M$ be a smooth manifold. The classical construction is the tangent bundle $TM$. What does the tangent groupoid $GM$ give me that this construction doesn't, and why is it useful in non-commutative geometry?
Heuristically, the tangent groupoid, which actually is a bundle too, thickens the tangent bundle with appro... | https://mathoverflow.net/users/22002 | Why is the Tangent Groupoid useful in non-commutative geometry? | The tangent groupoid can be used in constructing the index map and proving the Atiyah-Singer index theorem. This may illustrate its importance. Higson and Roe will write a book on it.
| 6 | https://mathoverflow.net/users/24965 | 101738 | 59,064 |
https://mathoverflow.net/questions/101746 | 1 | Given a rank-2 group $G= < a,b> $ .
Is it true and trivial that $ [G,G] = < [a,b], [b,a] > $ ?
Thanks !
| https://mathoverflow.net/users/20568 | Commutator Subgroup - Group Theory | No, the derived subgroup of the free group of rank 2 is infinitely generated (as every normal subgroup of infinite index), see W. Magnus, A. Karras, D. Solitar, Combinatorial group theory.
| 6 | https://mathoverflow.net/users/nan | 101747 | 59,069 |
https://mathoverflow.net/questions/101722 | 0 | Consider $Hilb\_{d,g}$ the Hilbert scheme of curves in $\mathbb{P}^3$ of degree $d$ and genus $g$. Is it true that if $L$ is an irreducible component of $Hilb\_{d,g}$
then there exists a curve $C \in L$ such that $C$ is irreducible? Is there some criterion under which this is true?
If the former statement is true, ca... | https://mathoverflow.net/users/9164 | Any irreducible component of the HIlbert scheme contains an irreducible element | The Hilbert scheme is pretty horribly behaved, and positive results of this nature are quite rare.
You probably mean to ask if there exists $C\in L$ such that $C$ is both reduced and irreducible. If you don't demand reducedness, the answer is yes. For suppose $H$ is a component of a Hilbert scheme of curves in $\math... | 10 | https://mathoverflow.net/users/7399 | 101759 | 59,078 |
https://mathoverflow.net/questions/101696 | 1 | The following question is a open question related to coding theory : What is the maximal size of a collection of $(\frac{n}{2} + 1)$-elements subsets of an n-element set such that each pair of subsets has at most $\frac{n}{2}-1$ elements in common ? We just have lower bound which is : $(\frac{1}{n} + O(\frac{1}{n^2})){... | https://mathoverflow.net/users/nan | Lower bound of the size of a collection of subsets with a intersecting property | The two problems are equivalent. Indeed, let $\mathcal{F}$ be the family of sets of size at least $n/2+1$ such that no two sets have $n/2$ elements in common. Clearly, such an $\mathcal{F}$ cannot contain two sets $S,T$ such that $S\subset T$. If there is a set $S\in \mathcal{F}$ of size greater than $n/2+1$, then repl... | 1 | https://mathoverflow.net/users/806 | 101760 | 59,079 |
https://mathoverflow.net/questions/101694 | 5 | We have a two sets of vectors ($\mathbb{C}^d$), $A=\{ v\_1, \ldots v\_n\}$ and $B=\{u\_1, \ldots u\_n\}$.
The question is if there is an efficient solution (polynomial in $n$) for checking whether $A$ and $B$ are related by an unitary rotation, i.e. if there exists an $U\in \text{U}(d)$ and a permutation $\sigma$, su... | https://mathoverflow.net/users/9093 | Sets of vectors related by a rotation | Since $d$ is small, you can do the following.
Choose a maximal linearly independed system $A'$ in $A$.
Consider all maps $A'\to B$ (since $|A'|\le d$, there are roughly $n^d$ of them).
For each check if it extends to a rotation.
| 4 | https://mathoverflow.net/users/1441 | 101761 | 59,080 |
https://mathoverflow.net/questions/101196 | 5 | Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are any general results concerning Kähler manifolds with this property?
| https://mathoverflow.net/users/14454 | Kähler potentials that depend only on geodesic distance | **Update:** I have had a little time to think about this further and have been able to show that if a smooth Kähler metric $g$ on a complex $n$-manifold $M$ has a potential $f$ that is a function of the distance from a point $p\in M$, then the metric must be locally rotationally symmetric about $p$, i.e., the group of ... | 6 | https://mathoverflow.net/users/13972 | 101766 | 59,082 |
https://mathoverflow.net/questions/101660 | 5 | *Fix notation*
Suppose that $Prf\_1(m, n)$ is the numerical relation that holds when $m$ numbers a $T$-proof of the sentence numbered $n$, according to scheme 1 for numbering wffs and sequences of wffs. Likewise $Prf\_2(m, n)$ is the relation that holds when $m$ numbers a $T$-proof of the sentence numbered $n$ accord... | https://mathoverflow.net/users/14111 | When are provability predicates provably equivalent? | $\DeclareMathOperator\prf{Prf}\DeclareMathOperator\con{Con}$
As for A, I don’t think there are any useful criteria known that would guarantee the provable equivalence of two proof predicates that would not beg the question.
As for B, no “derivability conditions” in the usual sense the word is used can do this, assumi... | 5 | https://mathoverflow.net/users/12705 | 101767 | 59,083 |
https://mathoverflow.net/questions/101676 | 4 | It is well-known that the theory of separably closed fields of some fixed positive characteristic and degree of imperfection is stable but not superstable. By a result of Cherlin and Shelah a superstable field is algebraically closed and many $\omega$-stable expansions of algebcaically fields are known ("coloured field... | https://mathoverflow.net/users/2234 | an example of a strictly superstable field | Hi Dima,
The expansion of the complex field by a predicate for the set of integer powers of 2 is an example. This follows from the results of Günaydin and Van den Dries in
"The Fields of Real and Complex Numbers with a Small Multiplicative Group"
<http://dx.doi.org/10.1017/S0024611506015747>
Their Corollary 6.... | 2 | https://mathoverflow.net/users/25002 | 101769 | 59,085 |
https://mathoverflow.net/questions/101773 | 0 | I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more general class.
**Example:** *Can one embed **[Petersen's graph](http://en.wikipedia.org/wiki/Petersen_graph)** in $\mathb... | https://mathoverflow.net/users/5628 | When do there exist locally regular embeddings of regular graphs? | Assume $\Gamma$ is a 3-regular graph in $\mathbb{R}^n$ and the angles at each vertex are $\tfrac23\pi$.
Then it is easy to see that the distance to the origin can not have a local maximum on $\Gamma$.
Therefore $\Gamma$ has to be infinite; in particular it can not be Petersen's graph.
| 4 | https://mathoverflow.net/users/1441 | 101777 | 59,089 |
https://mathoverflow.net/questions/101725 | 17 | I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a person, but I never see it capitalized.
For quick background: the ray class group for a modulus $\mathfrak{m}$ of a numb... | https://mathoverflow.net/users/1355 | What is the "ray" in ray class group? | There are many introductions to number theory, but few are as original (my term for what others would call weird) as Fueter's "Synthetische Zahlentheorie" published in 1925.
It starts with elementary number theory, and discusses the arithmetic of cyclotomic
fields up to the Dedekind zeta function and applications to qu... | 26 | https://mathoverflow.net/users/3503 | 101778 | 59,090 |
https://mathoverflow.net/questions/101762 | 8 | Let $X$ be a compact metric space and $f:X \to X$ a continuous map. We say that **$(X,f)$ is approximated from below** by a sequence of compact metric spaces $(X\_i)\_{i \geq 1}$ and a sequence of continuous transformations $(f\_i)\_{i \geq 1}$ on $X\_i$ if we have:
* $X\_i \subset X\_{i+1}$ for every $i \geq 1$;
* $... | https://mathoverflow.net/users/10518 | Aproximating dynamical systems by intrinsically ergodic systems | Every topologically transitive shift space, whether intrinsically ergodic or not, can be approximated from above by intrinsically ergodic systems.
Indeed, given a finite alphabet $A=\{1,2,\dots,p\}$ and a closed $\sigma$-invariant set $X\subset A^\mathbb{Z}$ (everything works just the same for one-sided shifts), let ... | 8 | https://mathoverflow.net/users/5701 | 101781 | 59,093 |
https://mathoverflow.net/questions/101057 | 12 | Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S\_t)\_{t\in[0,T]}$. Under the [no free lunch under vanishing risk](http://en.wikipedia.org/wiki/No_free_lunch_with_vanishing_risk) (NFLVR) assumption, the set $\mathcal P^\ast$ of equivalent martingale measures under which $S$... | https://mathoverflow.net/users/24820 | Compactness of the set of densities of equivalent martingale measures | The set $Z\_{\mathcal{P^\ast}}$ is *never* compact except in the case where it is a singleton (or empty). This is for the general case with $S=(S^1,S^2,\ldots,S^d)$ being an $\mathbb{R}^d$-valued semimartingale.
Let $\mathbb{Q}$ be an equivalent measure under which $S$ is a sigma-martingale, so $Z\_\mathbb{Q}\in\math... | 7 | https://mathoverflow.net/users/1004 | 101784 | 59,096 |
https://mathoverflow.net/questions/101785 | 1 | I'm studying Set Theory on my own and I have a question about a proof.
The book I am reading wants to prove $ZF \vdash (AC)^L$ in order to prove the relative consistency of AC from ZF. I have no problems with the proof of the following theorem:
1) $ZF \vdash (V = L)^L.$
Here comes the problem: my book is a bit conc... | https://mathoverflow.net/users/24883 | A proof of $ZF \vdash AC^L$ | If you write out in the obvious way the equivalence $WOT\iff AC$, it won't be $\Delta\_1$; the ony reason it's $\Delta\_1^{ZF}$ is that it's provable in ZF (hence provably equivalent to $0=0$).
My general impression of your argument is that you're making things unnecessarily complicated. Once you have that ZF proves ... | 6 | https://mathoverflow.net/users/6794 | 101789 | 59,098 |
https://mathoverflow.net/questions/101787 | 17 | Let $n$ be a natural number, and consider the discrete cube $2^{[n]} := \{ A: A \subset \{1,\ldots,n\}\}$ consisting of all subsets of the $n$-element set $[n] := \{1,\ldots,n\}$. Define a *downset* in $2^{[n]}$ to be a collection ${\mathcal D}$ of elements $A$ in $2^{[n]}$ with the property that if $A \in {\mathcal D}... | https://mathoverflow.net/users/766 | Optimal bounds for an alternating sum on a downset | Thanks for all the very quick responses,they were incredibly useful! Based on these responses, I think the conjecture is now settled in the affirmative, as follows.
For each n, let $F\_-(n)$ and $F\_+(n)$ be the minimal and maximal values of $\sum\_{A \in {\mathcal D}} (-1)^{|A|}$ respectively. The conjecture is that... | 9 | https://mathoverflow.net/users/766 | 101795 | 59,104 |
https://mathoverflow.net/questions/101327 | 2 | Is there any sort of Kneser Milnor decomposition in dimension d larger than or equal to 4?, I mean a family of d-1 dimensional spheres disconnecting the d-dimensional manifold into irreducible components and with some uniqueness?, If so, any reference so far?. Thanks.
| https://mathoverflow.net/users/24901 | Kneser Milnor decomposition in higher dimensions | Some results are available in dimension 4, although a "complete" answer is not known. As expected, the known results depend on the category: smooth/topological. Here are some references: first two papers by Matthias Kreck, Wolfgang Lück and Peter Teichner:
"*Counterexamples to the Kneser conjecture in dimension four*"... | 9 | https://mathoverflow.net/users/25011 | 101800 | 59,106 |
https://mathoverflow.net/questions/101801 | 12 | This works over the reals but not over the complex field. Consider the set of all $n\times n$ matrices $A$
such that
1. $A^2=A$
2.$A^T=A$
3. $\mathrm{Trace}(A)=1$
The first condition makes $A$ a projection to a subspace of $\mathbb{R}^n$. The second ensures that $A$ is *diagonalizable* so its eigenvalues are all 0 or... | https://mathoverflow.net/users/25013 | Projective spaces as affine varieties | The theorem that projective spaces are not affine varieties is a theorem over the complex numbers. As you note, your construction fails over the complex numbers, so there is no contradiction.
To give an even simpler example over the reals, $\mathbb {RP}^1=S^1$ is the vanishing set of $x^2+y^2-1=0$.
I think you're s... | 11 | https://mathoverflow.net/users/18060 | 101805 | 59,109 |
https://mathoverflow.net/questions/101808 | 10 | Assuming the standard conjectures (and whatever is needed in addition),
is there a nice proof of the Weil-conjectures written completely in the language of motives?
| https://mathoverflow.net/users/2837 | Motivic proof of Weil-conjectures? | See Theorem 5.6 in Kleiman's article "The Standard Conjectures", in the Motives volume (PSPM 55.1). I'm not quite sure what you mean by "written completely in the language of motives", so it might not be exactly what you are looking for.
I initially posted this as a comment, but it really should be an answer, so I'm... | 8 | https://mathoverflow.net/users/6753 | 101813 | 59,113 |
https://mathoverflow.net/questions/101817 | 3 | Let $X \to \mathbb P^n$ be a fiber bundle of algebraic varieties with $X$ an affine variety. What is the smallest dimension that $X$ can be?
An obvious lower bound is $n+1$.
An upper bound is $2n$, given by taking the complement of a generic effective divisor of bidegree $(1,1)$ in $\mathbb P^n \times \mathbb P^n$.... | https://mathoverflow.net/users/18060 | Dimension of Affine Bundles on Projective Space | Are you working over the complex numbers? Is $X$ smooth? If so, then I believe you can use the Leray spectral sequence and the Lefschetz hyperplane theorem to quickly deduce that $\text{dim}\_{\mathbb{C}}(X) \geq 2n$. Indeed, denote by $m$ the relative dimension of $X$ over $\mathbb{C}P^n$, so that the (complex) fiber ... | 4 | https://mathoverflow.net/users/13265 | 101820 | 59,116 |
https://mathoverflow.net/questions/101814 | 3 | ### Background
I am studying the paper "On the Green polynomials of classical groups" by Lusztig, in which he computes the values of the Deligne-Lusztig representation, corresponding to a Coxeter element of minimal length in a classical group, on unipotent elements. I am interested in computing the values of represen... | https://mathoverflow.net/users/2024 | Points on Deligne-Lusztig varieties: Interpreting Borels in relative position as flags with conditions | It might help if you specified what part of the translation your having trouble with (for example, it's unclear from what you wrote if you're comfortable with relative position).
For $SL\_n$, a Borel corresponds to a flag in affine $n$-space over an extension of your field (essentially, the one you need to diagonaliz... | 5 | https://mathoverflow.net/users/66 | 101825 | 59,119 |
https://mathoverflow.net/questions/101826 | 2 | Setup reminder: linear block error-correcting code is some linear subspace $C$ in $F\_2^N$. (Correcting error means to find a point $c \in C$ which is "nearest" to a given $r$ in $F\_2^N$, $r$ is signal with "errors").
Consider two codes which are given as images of some operators: $A: F\_2^k \to F\_2^L$,
$B: F\_2^L... | https://mathoverflow.net/users/10446 | Error correcting codes obtained as superposition of two codes e.g. CRC+Convolutional | You do know how to calculate the minimum distance (=free distance?) of a convolutional code? Cut the first edge from the zero state to the zero state (to disallow the all zeros word), and run Viterbi (but counting the weight, or distance to the all zero word) to the point, where all the states have surviving minimal pa... | 2 | https://mathoverflow.net/users/15503 | 101833 | 59,120 |
https://mathoverflow.net/questions/101831 | 0 | Dear MOs,
I am sorry if this problem is too elementary for someone. I just want to get confirmation.
Suppose $f\in L^1(R^d)$. Since almost all points are [Lebesgue points](http://en.wikipedia.org/wiki/Lebesgue_point) by the [Lebesgue differentiation theorem](http://en.wikipedia.org/wiki/Lebesgue_differentiation_theor... | https://mathoverflow.net/users/36814 | Pointwise limit at Lebesgue's point | Since $L^1(\mathbb{R}^d)$ really means equivalence classes of integrable functions, I am interpreting the question as follows. Given $f \in L^1(\mathbb{R}^d)$, does there exist $g$ which is equal to $f$ almost everywhere and such that for almost every $x \in \mathbb{R}^d$,
$$\lim\_{x' \to x} g(x') = g(x)? $$
Here is ... | 2 | https://mathoverflow.net/users/22052 | 101836 | 59,122 |
https://mathoverflow.net/questions/101830 | 10 | I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H\_n(X,{\mathbb Z}[\pi\_1 X])$ is not a finitely generated ${\mathbb Z}[\pi\_1 X]$-module.
Such an example would in particular give a finitely presented group $\pi... | https://mathoverflow.net/users/21095 | finite complex with non-finitely generated homology with local coefficients | As Ricardo points out in the comments, there's an error in my sketched calculation below. I also didn't notice the requirement that $X$ should be finite, so the natural $BK$ fails on two counts! However, it seems possible that a presentation complex for $K$ would do the job. Stallings shows that $\pi\_2$ of any complex... | 10 | https://mathoverflow.net/users/1463 | 101838 | 59,123 |
https://mathoverflow.net/questions/101809 | 15 | Let $R$ be the hyperfinite type $III\_1$ factor,
and let $Aut(R)$ be its group of automorphisms, equipped with the $u$-topology
(topology of pointwise convergence on the predual).
An automorphism $\alpha\in Aut(R)$ is called *inner* if it is of the form $\alpha(x)=uxu^\*$ for some unitary $u\in R$.
I've heard that in... | https://mathoverflow.net/users/5690 | Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor | It is relatively easy to give an explicit sequence of inner automorphisms that converges to the flip automorphism $\sigma$. First of all, realize $R$ as an infinite tensor product of matrix algebras $(R,\varphi)=\bigotimes\_n (M\_{k\_n}(\mathbb{C}),\varphi\_n)$. For every $n\in \mathbb{N}$, we find a unitary $u\_n\in M... | 8 | https://mathoverflow.net/users/2055 | 101839 | 59,124 |
https://mathoverflow.net/questions/101828 | 4 | In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?
| https://mathoverflow.net/users/10400 | Why limit of discrete series representation? | Here is the explanation I know, just for $SL\_2$.
The discrete series rep. have realizations in the Hardy spaces $H\_n$ which have the norm -
$$\|f\|\_ n ^2 = n\int\_{D}|f(z)|^2(1-|z|^{2})^{(n-1)}dxdy$$
notice this norm is scaled a bit differently than usual.
The limit of discrete series is realized inside $H\_2$ w... | 5 | https://mathoverflow.net/users/8857 | 101846 | 59,128 |
https://mathoverflow.net/questions/101841 | 2 | Let $SL\_2 (q)$ be group of all $2\times 2$ invertible matrices with unit determinant and $PGL\_2(q)$ is quotient group $GL\_2(q)/\{\text{scalar matrices over}\ q\}$.
| https://mathoverflow.net/users/25021 | Is $SL_2(q)$ isomorphic to $PGL_2(q)$? | The question itself is natural, but it's fairly elementary and has a clearcut answer in the literature on finite simple groups including the series of books by Gorenstein-Lyons-Solomon (and for small order groups the Atlas). It's easiest to understand what is going on from the algebraic group viewpoint, summarized with... | 9 | https://mathoverflow.net/users/4231 | 101849 | 59,129 |
https://mathoverflow.net/questions/101829 | 2 | I have a question during my intership. Given a convergent sequence of continuous et convex functions $\{f\_n(x)\}$ defined in $\mathbb{R}^M$. These functions are uniformly Lipschitz continuous which means that there exist a constant C such that
$$|f\_n(x)-f\_n(y)| \leq C|x-y|, \quad \forall x,y \ \textrm{ in } \ \mat... | https://mathoverflow.net/users/25005 | convergence of infimum | I think there are counterexamples: consider $f\_n(x)=\arctan(\frac{x}{n})$, then $f\_n(x)$ converge to $0$ and are uniformly Lipschitz continuous since there derivatives are smaller than $1$. However the inf of $\arctan(\frac{x}{n})$ is $-\frac{\pi}{2}$，which is not $0$.
| 1 | https://mathoverflow.net/users/24965 | 101851 | 59,130 |
https://mathoverflow.net/questions/101832 | 9 | Consider the following situation: $\Gamma\_0\leq\Gamma$ are both finitely generated groups and $\Gamma\_0$ has finite index in $\Gamma$. The restriction gives a well defined map between the character varieties of these groups:
$$\mathrm{Res}:M(\Gamma,\mathrm{GL}\_N(\mathbb{C}))\longrightarrow M(\Gamma\_0,\mathrm{GL}\_N... | https://mathoverflow.net/users/25017 | Character varieties of finitely generated groups | Claim. The restriction map is always proper, where the target group $G$ is the group of $K$-points of a reductive group over a local field $K$, e.g. $G=GL\_N({\mathbb C})$.
Proof. First, some generalities, details for which you can find, for instance, [here](http://arxiv.org/abs/1003.1111). Let $X$ be the symmetric s... | 11 | https://mathoverflow.net/users/21684 | 101860 | 59,134 |
https://mathoverflow.net/questions/101815 | 13 | Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate student was asking me about this, having dug a version out of an obscure research paper in the process of studying an unrelated... | https://mathoverflow.net/users/4231 | Convenient reference for subgroups of a finite semidirect product? | Usenko, Subgroups of semidirect products, Ukrainian Mathematical Journal, 1991, Volume 43, Numbers 7-8, Pages 982-988
| 6 | https://mathoverflow.net/users/5301 | 101865 | 59,139 |
https://mathoverflow.net/questions/101859 | 20 | I am currently reading Jean Bourgain's 1986 paper *A Szemerédi type theorem for sets of positive density in* $R^k$ and would appreciate some help in understanding a Fourier-analytic estimate used in that article. I suspect that my question is relatively elementary, but my knowledge of Fourier analysis is not very stron... | https://mathoverflow.net/users/1840 | A Fourier-analytic inequality used by Jean Bourgain | I assume you are referring to the argument in page 313 of Jean's paper [Link](https://doi.org/10.1007/BF02764959) . The point here is that the bound does not hold for all $t$, but for a single $t$ (out of $J$ possible choices $t\_1,\dots,t\_J$); note that Jean crucially refers in the paper to a "suitable" $t$ rather th... | 38 | https://mathoverflow.net/users/766 | 101871 | 59,143 |
https://mathoverflow.net/questions/101873 | 2 |
>
> Does there exist a graph $G$ which cannot be properly vertex-coloured with 3 colours (i.e. $G$ has chromatic number at least 4), such that for every graph $H$, if $H$ contains a triangle but there is no graph homomorphism from $G$ to $H$, then $H$ must contain at least as many vertices as $G$?
>
>
>
This que... | https://mathoverflow.net/users/7252 | A Ramsey-like lower bound? | I'm confused. Take $H$ to be the triangle. Since $G$ is not three-colourable, there is no map from $G$ to $H$, and $H$ contains a triangle.
| 2 | https://mathoverflow.net/users/14901 | 101891 | 59,153 |
https://mathoverflow.net/questions/100767 | 2 | Hi all,
What should I look for if I want to study existence/uniqueness of the system of PDEs:
$$u\_t -\Delta u + u\nabla \cdot v = f(u) \quad\text{on $\Gamma(t)$}$$
$$X\_t = \kappa N(X) + u \quad \text{for $\Gamma(t)$} \tag{MCF with forcing}$$
where $N$ is the normal vector, $\Gamma(t)$ is the surface parametrised ... | https://mathoverflow.net/users/23013 | Existence of PDE system (mean curvature flow coupled with surface PDE) | To the best of my knowledge, no theory has been done for this kind of problem.
One of the major problems is that the Laplace-Beltrami operator is no longer linear or monotone.
I am interested in similar problems though. Are you a PhD student?
| 0 | https://mathoverflow.net/users/25033 | 101897 | 59,157 |
https://mathoverflow.net/questions/101776 | 13 | The three altitudes of a triangle are concurrent -- this is true in all three constant curvature geometries (Euclidean, hyperbolic, spherical), but, as far as I know, the proofs are different in the three cases. Is there a "uniform" proof?
| https://mathoverflow.net/users/11142 | Altitudes of a triangle | The spherical and hyperbolic versions may be proved in a uniform way.
Consider the cross product $\times$ on $\mathbb{R^3}$ or on $\mathbb{R}^{2,1}$. If the vertices of the triangle are $a,b,c$ thought of as vectors in the unit sphere or hyperboloid, then the line through $a,b$ is perpendicular to $a\times b$, etc. ... | 11 | https://mathoverflow.net/users/1345 | 101901 | 59,158 |
https://mathoverflow.net/questions/101906 | 19 | Let $f: {\bf Q} \rightarrow {\bf Q}$ be a "${\bf Q}$-differentiable" function whose "${\bf Q}$-derivative" is constantly zero; that is, for all $x \in {\bf Q}$ and all $\epsilon > 0$ in ${\bf Q}$, there exists $\delta > 0$ in ${\bf Q}$ such that for all $y \in {\bf Q}$ with $0 < |x-y| < \delta$, $|(f(y)-f(x))/(y-x)| < ... | https://mathoverflow.net/users/3621 | functions from Q to itself with derivative zero | No, $f$ does not have to be locally constant. Let $a\_n$ be a sequence of irrationals that decreases to zero, define $f(x) = 0$ for $x \leq 0$, and let $f(x)$ be a (single) rational number in $(e^{-1/{a\_{n+1}}}, e^{-1/{a\_n}})$ for $a\_n < x < a\_{n-1}$. Voila!
| 20 | https://mathoverflow.net/users/23141 | 101907 | 59,162 |
https://mathoverflow.net/questions/101893 | 9 | *Motivation*: Loius Pojman mentions in *What Can We Know?* (2001) of a certain Carneades (ca. 214-129 B.C>) who must have been a "remarkable dialectician"because " in 155BC he was sent on a diplomatic mission to Rome and in his spare time he gave two lectures. On first day he eulogized justice, making a profound impres... | https://mathoverflow.net/users/20215 | Incidences of rigorous proofs used in legal proceedings | This is perhaps a borderline example. I heard this story from BCnrd a few years ago, so I hope it hasn't been too mangled in my head since then.
One day, Ken Ribet got a phone call in his office:
* "Is this Professor Ribet?"
* "Yeah."
* "Could you tell me what one tenth of one percent means?"
* "One part in a thous... | 19 | https://mathoverflow.net/users/121 | 101913 | 59,166 |
https://mathoverflow.net/questions/101916 | 2 | Recall in a $2$-category $X$, a $1$-cell $f:X\to Y$ is called an equivalence provided there exists a $1$-cell $g:Y\to X$ together with the data of a pair of isomorphisms $\eta\_X: gf \to \operatorname{id}\_X$ and $\eta\_Y: fg \to \operatorname{id}\_Y$.
Then suppose we impose the additional requirement that $f\ast \e... | https://mathoverflow.net/users/1353 | Is there a name for this type of equivalence in a 2-category? | This is well-known under the name [adjoint equivalence](http://ncatlab.org/nlab/show/adjoint+equivalence) (when you replace $\eta\_Y$ by $\eta\_Y^{-1}$ in your data). And yes, every equivalence can be modified to some adjoint equivalence (one may fix the unit, but has to modify the counit; or vice versa).
| 11 | https://mathoverflow.net/users/2841 | 101921 | 59,171 |
https://mathoverflow.net/questions/101922 | 13 | I know that no number less than 64 bits will fail the Miller-Rabin tests for all of the first 12 primes. That is, those 12 tests will provide a fully deterministic primality test for all 64 bit numbers. (See <http://oeis.org/A014233>). I also know that for 32 bit numbers, it suffices to apply the Miller-Rabin tests for... | https://mathoverflow.net/users/2165 | Smallest collection of bases for prime testing of 64 bit numbers? | According to <http://miller-rabin.appspot.com/>, the 7-element set {2, 325, 9375, 28178, 450775, 9780504, 1795265022} works for 64-bit integers.
| 12 | https://mathoverflow.net/users/12705 | 101936 | 59,181 |
https://mathoverflow.net/questions/101927 | 8 | Consider the following one-dimensional tiling problem. Each "tile" is a sequence of nonnegative integers. A "region" is also such a sequence. I can shift the "tiles", or reverse them. A tiling is a set of shifted and/or reversed tiles that add up to the region. For instance, if my tiles are these three:
1 2
1 1 1
... | https://mathoverflow.net/users/25038 | one-dimensional (sort of) tilings | Yes, the problem is NP-complete. Here is a reduction of 1-in-3 SAT to your problem.
Let $\{C\_i:i< n\}$ be the given collection of 3-clauses in variables $\{x\_j:j< m\}$. All tiles and the region will have length $n+m+2$, so no shift is possible. For each of the $2m$ literals $a$ (which is $x\_j$ or $\neg x\_j$), we ... | 9 | https://mathoverflow.net/users/12705 | 101939 | 59,183 |
https://mathoverflow.net/questions/101938 | 1 | Hello all,
Could you please help with the following problem? I have a set of two coupled ODE for $a$ and $b$ waves [wave is a general form of solution $a(z)=A(z)\exp(\imath \beta z)$]. The equations in question are as follows:
$a\_z(z) = \imath \beta a(z) + f[a(z),b(z)]$
$b\_z(z) = - \imath \beta b(z) + g[(a(z),b... | https://mathoverflow.net/users/25041 | Bidirectional ODE | This is a typical problem for time-dependent Schroedinger equation. Firstly, let us consider the unperturbed problem. We can write the free solutions as
$$
\tilde a=a\_0e^{i\beta z} \qquad \tilde b=b\_0e^{i\beta(L-z)}.
$$
Now, redefine $a$ and $b$ in you system as
$$
a(z)=\bar a(z)e^{i\beta z} \qquad b(z)=\bar b(z)e^... | 0 | https://mathoverflow.net/users/19520 | 101947 | 59,187 |
https://mathoverflow.net/questions/101946 | 3 | Google searches for "local ample cone" and "local Kähler cone" yield no results, but maybe there is a different term.
Let $\pi : \hat X \to X$ be a resolution of an isolated singularity on the (complex) projective variety $X$, with exceptional set $E$, which may have components of various dimensions. Assume that we c... | https://mathoverflow.net/users/22975 | Is there a notion of 'local ample/Kähler cone' for resolved singularities? | I think this should be called the *relatively ample cone* and its closure the *relatively nef cone*. Yes, this already exists and is sensible.
This dissertation might be somewhat useful (although things are done for smooth $X$ instead of singular $X$).
[UMich dissertation](http://deepblue.lib.umich.edu/bitstream/20... | 4 | https://mathoverflow.net/users/3521 | 101948 | 59,188 |
https://mathoverflow.net/questions/101892 | 5 | Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma\_n$ act on $X^{\otimes n}$ by permuting the factors (note that this action is far from being free). There is a natural map from the homotopy colimit of the action (which is the exten... | https://mathoverflow.net/users/11540 | When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action? | The equivalence (P) is a deep and subtle property of the
smash product of spectra in modern symmetric monoidal
models for the stable homotopy category. It is very
unlikely to hold in other contexts. It was first found
in EKMM [III.5.1] because the use of operads there visibly
builds it into the smash product. It was la... | 10 | https://mathoverflow.net/users/14447 | 101973 | 59,197 |
https://mathoverflow.net/questions/101978 | 1 | How to define zeta function for a curve over $\mathbb{Z}$ or $\mathbb{Q}$?
| https://mathoverflow.net/users/nan | How to define zeta function for curves over a number field | For any scheme $X$ of finite type over $\mathbf{Z}$ as a product of Euler factors:
$$\zeta(X,s) = \prod\_{x \in |X|}\frac{1}{1-|\kappa(x)|^{-s}}$$
One might also include the Euler factors at infinity.
For smooth projective varieties $X$ over $\mathbf{Q}$: Put $\bar{X} = X \times\_\mathbf{Q} \bar{\mathbf{Q}}$. For... | 2 | https://mathoverflow.net/users/nan | 101979 | 59,200 |
https://mathoverflow.net/questions/101954 | 5 | I have been playing around with the Möbius Function and primorials and I am finding results that I am not yet able to understand which I suspect are very elementary.
Here's the current result which is I am working through. Any help is greatly appreciated!
Let $p\_k$ be any prime. Let $x$ be any integer.
It seems ... | https://mathoverflow.net/users/15915 | A question about the Mobius Function | Note that
$$ \lfloor \frac{a \% p + b \% p}{p} \rfloor = \lfloor \frac{a+b}{p} \rfloor - \lfloor \frac{a}{p} \rfloor - \lfloor \frac{b}{p} \rfloor$$
so your expression can be written as $F\_{p\_k\#}(x+p\_k) - F\_{p\_k\#}(x) - F\_{p\_k\#}(p\_k)$, where
$$ F\_n(x) := \sum\_{i|n} \lfloor \frac{x}{i} \rfloor \mu(i).$... | 16 | https://mathoverflow.net/users/766 | 101985 | 59,202 |
https://mathoverflow.net/questions/101961 | 0 | Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X\_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0... | https://mathoverflow.net/users/25053 | Does homeomorphism preserves the family of cones? | Here is the counterexample, re-expressed. Let $S\_r$ be the sphere of radius $r \epsilon$.
Construct a homeomorphism $h : B \to B$ with the following properties:
* $h(S\_r) = S\_r$ for each $r \in [0,1]$
* $h(S\_{1/3} \cap X\_{C^\*}) = S\_{1/3} \cap X\_{C^\*}$
* $h(S\_{2/3} \cap X\_{C^\*}) \supset S\_{2/3} \cap (\m... | 1 | https://mathoverflow.net/users/20787 | 101988 | 59,204 |
https://mathoverflow.net/questions/101982 | 7 | Are there any good algebraic/algorithmic tools available to check if a given graph $H$ is a minor of $G$ from the adjacency matrix of $G$?
| https://mathoverflow.net/users/10035 | Graph minor check | There is a general implementation in [Sage](https://ask.sagemath.org/question/8112/graph-minor-code-too-slow-in-certain-situations-sage-46/). However, the algorithm runtime grows exponentially in the size of $H$. If you have a particular small $H$ in mind, there may be more efficient implementations available.
**EDIT... | 11 | https://mathoverflow.net/users/3106 | 101990 | 59,205 |
https://mathoverflow.net/questions/101984 | 1 | **Background:** In homological algebra, a *quasi-isomorphism* of chain complexes is a chain map$\phi:(C,d) \to (C',d')$ so that the induced map on homology $\phi\_\ast:H\_\ast(C,d) \to H\_\ast(C',d')$ is an isomorphism.
This is clearly a special instance of the following much more general phenomenon: given a functor ... | https://mathoverflow.net/users/18263 | Terminology generalizing "quasi-isomorphism" | I don't think this is completely standard, so if I were going to use it I would explain it first, but a natural possibility is "$\mathcal{F}$-isomorphism" (-monomorphism, -epimorphism).
Quasi-isomorphisms are in fact sometimes called $H\_\*$-isomorphisms, and similarly maps inducing isomorphisms on homotopy groups a... | 9 | https://mathoverflow.net/users/49 | 101992 | 59,207 |
https://mathoverflow.net/questions/101965 | 2 | For a vector $w$, let $T\_{w}$ be the translation by $w$.
I was told that the following observation about subsets of the plane was due to H. Hopf:
Let $X$ be a compact, path-connected subset of the plane. Then, if for some vector $v$, $X\cap T\_{v}[X]\neq \emptyset$, then for each positive natural number $n$, $X\ca... | https://mathoverflow.net/users/22088 | Hopf reference sought | Check out the Chord Theorem (Theorem 2B12) in Rolfsen's Knots and Links. Here is a link to the [Google book](http://books.google.com/books?id=s4eGEecSgHYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false).
| 3 | https://mathoverflow.net/users/14006 | 101995 | 59,209 |
https://mathoverflow.net/questions/101996 | 4 | It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum hypothesis, it is possible to construct a similar "universal order" of cardinal $\aleph\_1$. I have two questions : 1) Wher... | https://mathoverflow.net/users/17164 | Cantor theorem on orders | Concerning question 2:
a. It is consistent that $2^{\aleph\_0}= \aleph\_2$ and there is a universal order of size $\aleph\_1$.
b. It is consistent that $2^{\aleph\_0}= \aleph\_2$ and there is no universal
order of size $\aleph\_1$.
See Kojman+Shelah, JSL. [preprint](http://www.cs.bgu.ac.il/~kojman/UnivOrd.pdf)
... | 7 | https://mathoverflow.net/users/14915 | 101999 | 59,212 |
https://mathoverflow.net/questions/101941 | 6 | Let $M$ be a separable von Neumann algebra and let $A$ be a (von Neumann-)dense \*-subalgebra.
Suppose that $\alpha,\alpha\_1,\alpha\_2,\dots$ are automorphisms of $M$, such that for every $a \in A$,
$$ \alpha\_n(a) = \alpha(a) $$
for all $n$ sufficiently large.
Does it follow that $\alpha\_n$ converges to $\alpha$ in ... | https://mathoverflow.net/users/22052 | von Neumann automorphisms: does convergence on a dense algebra imply $u$-convergence? | No. For an example, consider $M=L^\infty([0,1])$ with the Lebesgue measure, take $A$ to be the functions that are piecewise constant on dyadic intervals and $\alpha\_n(f)=f\circ \phi\_n^{-1}$ where $\phi\_n(t)=k/2^n + (2^n t-k)^2/2^n$ if $t \in [k/2^n,(k+1)/2^n[$. In words, $\phi\_n$ acts as some fixed transformation (... | 6 | https://mathoverflow.net/users/10265 | 102002 | 59,214 |
https://mathoverflow.net/questions/101888 | 22 | In 1986 C. C. Hsiung published a paper "Nonexistence of a Complex Structure on the Six-Sphere" and in 1995 he even wrote a monograph "Almost Complex and Complex Structures" to further elaborate on his proof.
Yet answers to the 2009 [question on this site](https://mathoverflow.net/questions/1973/is-there-a-complex-struc... | https://mathoverflow.net/users/16504 | Hsiung on the Complex Structure of $S^6$ | While it's good to have a source, such as Datta's paper that points out the error, I find that his explanation of why the key equation is wrong is not as clear as it could be. In fact, with a little thought (requiring essentially no computation), it's clear why this equation must be wrong and what is wrong with the app... | 47 | https://mathoverflow.net/users/13972 | 102008 | 59,218 |
https://mathoverflow.net/questions/101964 | 4 | For a simply connected four-dimension manifold, we know the Freedmen's work.
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is smaller than N finite or infinite?
Or equivalently: Can we classify the bilinear forms of integer coefficients? Or what's th... | https://mathoverflow.net/users/25054 | The number of simply connected 4-dimension manifold | This was essentially answered by Agol in the comments. For any fixed second Betti number $N$, there are finitely many homeomorphism types, because there are finitely many isomorphism types of unimodular lattices of rank $N$.
As far as classification is concerned, the number of isomorphism types of definite unimodular... | 8 | https://mathoverflow.net/users/121 | 102021 | 59,222 |
https://mathoverflow.net/questions/101949 | 3 | So, my question is this:
given an affine manifold $X$, and a quotient manifold $Y$ of $X$, is $Y$ necessarily an affine manifold?
If it helps, I'm especially interested in the case that $X$ is not any affine manifold but the affine space of dimension $n$ and I also know that $Y$ is orientable, compact, complete and fla... | https://mathoverflow.net/users/25045 | when is a quotient manifold of an affine manifold affine? | With the 3rd edit, your question become a triviality: The affine space $A^n$ has canonical flat (i.e., zero curvature, zero torsion) linear connection $\nabla$: $\nabla\_{X\_i} X\_j=0$ for all coordinate vector fields. This is a calculus exercise to check that $\nabla$ is invariant under all affine transformations. The... | 4 | https://mathoverflow.net/users/21684 | 102036 | 59,228 |
https://mathoverflow.net/questions/102025 | 6 | Given $n$ unit vectors in $\mathbb{R}^n$ s.t. $0 \leq u\cdot v<1$ for all pair of distinct vectors $u,v$. These vectors span a $d$-dimensional subspace s.t. $d< n$. We conjecture that it is possible to partition the $n$ vectors into $d$ groups such that all the vectors within the same group are pairwise non-orthogonal.... | https://mathoverflow.net/users/39663 | Grouping vectors together | Kahn--Kalai's [counterexample to Borsuk's conjecture](http://arxiv.org/abs/math.MG/9307229) is a collection of vertices of the unit cube.
Imagine that this cube sits in a coordinate hyperplane $x\_1=s$ of $\mathbb R^{n+1}$, so that the origin projects to the center of the cube.
Project this cube centrally to the unit... | 5 | https://mathoverflow.net/users/1441 | 102043 | 59,233 |
https://mathoverflow.net/questions/102031 | 3 | how can i construct a strongly regular graph with parameter $(275,112,30,56)$(Mclaughlin Graph), (105,32,4,12)?
I need adjacency matrix of them?
I know they are unique.
| https://mathoverflow.net/users/22967 | Mclaughlin Graph | You can construct the McLaughlin graph using [GAP](http://www.gap-system.org/).
```
gap> LoadPackage("AtlasRep");;
gap> mcl:=Group(AtlasGenerators("McL",1).generators);;
gap> LoadPackage("grape");;
gap> Gr:=NullGraph(mcl);;
gap> AddEdgeOrbit(Gr,[1,2]);;
gap> VertexDegrees(Gr);
[ 112 ]
```
OK, so that's the 1st g... | 8 | https://mathoverflow.net/users/11100 | 102044 | 59,234 |
https://mathoverflow.net/questions/99505 | 15 | There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of [Cheeger, Muller and Schrader](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-92/issue-3/On-the-curvature-of-piecewise-flat-spa... | https://mathoverflow.net/users/18263 | Combinatorial analogues of curvature | Perhaps, the most general concept of curvature (measure) is that of *normal cycle*. This is an object $N^S$ naturally associated to a reasonably nice compact subset $\newcommand{\bR}{\mathbb{R}}$ $S\subset\bR^n$. PL sets are reasonabky nice and, more generally, the semialgebraic sets are nice. The compact smooth subman... | 6 | https://mathoverflow.net/users/20302 | 102045 | 59,235 |
https://mathoverflow.net/questions/100599 | 17 |
>
> Is the sequence $$w\_n=n! \int\_0^{1/2} \int\_{x\_1}^{2/3} \cdots\int\_{x\_{n-2}}^{\frac{n-1}{n}} \int\_{\frac{n}{n+1}}^1 dx\_n dx\_{n-1} \cdots dx\_1$$ increasing for $n\ge 3$?
>
>
>
This is a conjecture of F. Thomas Bruss and Marc Yor in a recent paper *Stochastic Processes with Proportional Increments and... | https://mathoverflow.net/users/21051 | The Bruss-Yor conjecture about an iterated integral | Update on the background of the problem:
The multiple integral in question is related to the last-arrival problem: A selector wants to pick, online, the last of an unknown number of items that arrive at independent times uniformly chosen in $[0,1]$. The number $w\_n$ is the probability of success (with $n$ items) for... | 6 | https://mathoverflow.net/users/14302 | 102052 | 59,238 |
https://mathoverflow.net/questions/101983 | 0 | Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a
\curlyvee b$ if only if there is a non-least element $c$ such that $c
\leqslant a \wedge c \leqslant b$.
I call a poset $\mathfrak{A}$ *separable* if and only if $\forall x
\in \mathfrak{A}: \left( x \curlyvee a \Leftrightarrow x \curlyvee b... | https://mathoverflow.net/users/4086 | A property of a product of posets | Here's a counterexample, taking advantage of the option to have or not have a 0 in a poset. Take the index set to be $\{1,2\}$. Take $\mathfrak A\_1$ to be an atomless Boolean algebra minus its 0 element. It is well-known and easy to see that this is separable; if $a\neq b$, then one of $a-b$ and $b-a$ in the Boolean a... | 4 | https://mathoverflow.net/users/6794 | 102058 | 59,241 |
https://mathoverflow.net/questions/64326 | 13 | I'm trying to understand the Weil representation and hope there are some experts around who can set me straight. Let $F$ be a non-Archimedean local field (I don't mind assuming that the characteristic of $F$ is not $2$ if it simplifies things; also the situation over $\mathbb{R}$ is similar but obviously I need a proof... | https://mathoverflow.net/users/3544 | metaplectic group does not split | I would like the credit to go to Peter Woit for suggesting Section I.6 of Stephen Kudla's "Notes on the Local Theta Correspondence," which contains a nice proof, but he posted this only as a comment and the bounty ends today.
The text of [Woit's comment](https://mathoverflow.net/questions/64326/metaplectic-group-does... | 2 | https://mathoverflow.net/users/3544 | 102064 | 59,243 |
https://mathoverflow.net/questions/102061 | 1 | Dear all,
Is there any possible way to construct a set $A \subseteq \mathbb{R}^n $ for which $ H^{n-1} (\partial A) > Leb^ + (A ) $?
Where $ H^{n-1} (\partial A) $ is the Hausdorff measure of the boundary of $A$ and:
$ Leb^{+} (A) = \lim\_{\epsilon \to 0 } \frac{ Leb(A\_ \epsilon) - Leb(A) }{\epsilon} $ ,
$A\_\eps... | https://mathoverflow.net/users/25080 | Different Measures On R2 | So, $A$ is just a "set"? Say $A$ is an open square minus a mid-line:
$$
A = \{ (x,y): 0 \lt x \lt 1 \text{ and (} 0 \lt y \lt 1/2 \text{ or } 1/2 \lt y \lt 1\text{)}\}
$$
Then $\partial A$ consists of 5 line-segments of length $1$ (the 4 sides and the mid-line), so $H^1(\partial A) = 5$. But $\mathrm{Leb}^+(A) = 4$, mi... | 3 | https://mathoverflow.net/users/454 | 102066 | 59,244 |
https://mathoverflow.net/questions/101872 | 7 | In classical functional analysis, one can construct a reproducing kernel Hilbert space by starting with a positive definite kernel, say
$K: [0,1]\times [0,1] \rightarrow \mathbb{R}$.
One then creates linear combinations of the form
$f(x) = \sum^n a\_i k(x\_i,x)$, together with an inner product
$\langle f,g \rangle = ... | https://mathoverflow.net/users/9564 | Is there a tropical analogue of a reproducing kernel Hilbert space? | See
1. G.L. Litvinov, V.P. Maslov and G.B. Shpiz. Idempotent functional analysis.
An algebraic approach // Mathematical Notes, v. 69, # 5, 2001, p. 696-729.
E-print [math.FA/0009128](https://arxiv.org/abs/math/0009128) (<http://ArXiv.org>).
2. G.L. Litvinov and G.B. Shpiz. Kernel theorems and nuclearity in
idempotent... | 10 | https://mathoverflow.net/users/25086 | 102072 | 59,247 |
https://mathoverflow.net/questions/102051 | 10 | Let us call the $\ell\_1$-product of intervals $[0,k\_1]\times...\times [0,k\_n]$ a *brick of size* $k\_1+...+k\_n$. Consider a tessellation $T$ of $\mathbb{R^n}$ by (shifted) bricks so that every point belongs to at most $n+1$ bricks and the $\ell\_1$-distance between any two disjoint bricks is at least 1 (thus ever... | https://mathoverflow.net/users/nan | Tessellating $\mathbb{R}^n$ by bricks. | Edit: Per Tapio's answer, $1 \times 2 \times 3 \times ... \times n$ bricks always suffice, so $s(n)\leq \left(\begin{array}{c} n+1 \\ 2 \end{array}\right)$. Consider the lattice in $\mathbb Z^n$ defined by the equations $x\_1 \equiv x\_2$ mod $2$, $x\_2 \equiv x\_3$ mod $3$, ..., $x\_{n-1} \equiv x\_n$ mod $n$. Place a... | 7 | https://mathoverflow.net/users/18060 | 102074 | 59,249 |
https://mathoverflow.net/questions/101898 | 8 | Suppose I have a functor $\mathcal{E} \to \mathcal{B}$ that is both a Grothendieck fibration and an opfibration, $\mathcal{B}$ is presentable, and the fibres $\mathcal{E}\_{b}$ for $b \in \mathcal{B}$ are all presentable. Are there any known conditions on this setup under which the category $\mathcal{E}$ will be presen... | https://mathoverflow.net/users/1100 | Presentability of the source of a fibration | In section 5.3 of *Accessible categories* by Makkai and Pare, they prove that if $\Phi : B^{\mathrm{op}} \to \mathrm{Cat}$ is a pseudofunctor such that
1. each category $\Phi(b)$ is accessible,
2. each functor $\Phi(\beta) : \Phi(b') \to \Phi(b)$ is accessible,
3. the category $B$ is $\kappa$-accessible, and
4. the p... | 8 | https://mathoverflow.net/users/49 | 102083 | 59,253 |
https://mathoverflow.net/questions/102082 | 0 | I am confused about a step in Hartshorne's proof (final part of Corollary 1.4) that an algebraic set $Y$ in affine n-space $\mathbb{A}^n$ having a prime ideal $I(Y)$ in the polynomial ring over $n$ variables $A$. The proof goes as follows:
>
> Let $\mathfrak{p}$ be a prime ideal, and suppose $Z(p) = Y\_1 \cup Y\_2$... | https://mathoverflow.net/users/25090 | If an algebraic set in affine n-space has a prime ideal then it is irreducible. (Hartshorne's Algebraic Geometry, Cor. 1.4) | If $P=I\cap J$ is prime, and if $P$ is not equal to $I$ or $J$, then choose $i\in I\setminus J$ and $j\in J\setminus I$. Then $ij\in P$, so $i\in P$ or $j\in P$. Contradiction either way. Thus $P=I$ or $P=J$.
| 5 | https://mathoverflow.net/users/10503 | 102085 | 59,255 |
https://mathoverflow.net/questions/100263 | 12 | Recall the definition of Heegaard Floer homology: $\Sigma\_g$ is a closed surface, and $\{\alpha\_1,\ldots,\alpha\_g\}$ and $\{\beta\_1,\ldots,\beta\_g\}$ are sets of attaching circles. Then Heegaard Floer homology is (more or less) the Lagrangian intersection Floer homology of $\mathbb T\_\alpha=\prod\_{i=1}^g\alpha\_... | https://mathoverflow.net/users/35353 | Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$? | There is a tacit assumption behind this question, which I don't think is justified: that the Abel-Jacobi images of the Heegaard tori $\mathbb{T}\_{\alpha}$ and $\mathbb{T}\_{\beta}$ are Lagrangian with respect to some reasonable symplectic form on the Jacobian torus.
One can make the Heegaard tori Lagrangian by usin... | 8 | https://mathoverflow.net/users/2356 | 102089 | 59,256 |
https://mathoverflow.net/questions/102108 | 0 | I am looking for a reference that describes how to decompose a tensor product of two finite dimensional simple modules for a reductive Lie algebra over $\mathbb{C}$.
In particular, I would like a reference that describes it along the same lines as the way it can be described for $gl\_n(\mathbb{C})$ and $sl\_n(\mathbb... | https://mathoverflow.net/users/4614 | Reference request: Tensor products of modules for reductive Lie algebras | This has been fully discussed in
[Decompose tensor product of type $G\_2$ Lie algebras.](https://mathoverflow.net/questions/85593)
| 2 | https://mathoverflow.net/users/3992 | 102117 | 59,269 |
https://mathoverflow.net/questions/102109 | 3 | I was browsing MO and stumbled upon [this post](https://mathoverflow.net/questions/70889/determinant-of-exterior-power), and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism $\mathrm{Gl}\_n(\Bbbk)\to\Bbbk^\times$ is a power of the ... | https://mathoverflow.net/users/9947 | A polynomial homomorphism from Gl to the group of units is a power of the determinant | Since ${\mathbb k}^\times$ is abelian, any group hom $\phi : GL\_n ({\mathbb k}) \rightarrow{\mathbb k}^\times$ factors through $\phi : GL\_n ({\mathbb k}) \rightarrow GL\_n({\mathbb k})/[GL\_n({\mathbb k}),GL\_n({\mathbb k})] \rightarrow{\mathbb k}^\times$. It is a nice easy exercise that $[GL\_n({\mathbb k}),GL\_n({\... | 6 | https://mathoverflow.net/users/5301 | 102120 | 59,271 |
https://mathoverflow.net/questions/102116 | 11 | Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle $(S^2E)^{\ast} \subset (E\otimes E)^{\ast}$. However, not every global section $s$ corresponds to a Riemannian metric, as the... | https://mathoverflow.net/users/21564 | Riemannian metrics as sections of a vector bundle | An orthogonal or hermitian structure on $E$ is a section of *fibre bundle*
(which is not a vector bundle). I will deal with the complex case.
The Lie algebra $\mathfrak{gl}(n)$
decomposes into
$$
\mathfrak{gl}(n)\simeq \mathfrak{u}(n)\oplus \textrm{Herm}\_n,
$$
where $\textrm{Herm}\_n$ is the vector space of Hermit... | 11 | https://mathoverflow.net/users/6278 | 102125 | 59,275 |
https://mathoverflow.net/questions/102100 | 4 | Let $G$ be a finite group and let $G$ act on a vector space $V$. Let $S(V)$ be the representation sphere. In the monoidal category of $RO(G)$-graded spectra $S(V)$ is invertible, so it's dualizable too.
Is $S(V)$ dualizable in the monoidal category of naive $G$-spectra?
| https://mathoverflow.net/users/25092 | Is a representation sphere dualizable inside naive G-spectra? | Let $G$ be of order $2$, and let $V$ be the nontrivial one-dimensional representation. I'll write $S(V)$ for the unit sphere in $V$ (which is just $G$); the one-point compactification of $V$ is then the unreduced suspension of $G$, which I'll call $S^V$. We can only form spectra from $G$-spaces with fixed basepoint, so... | 6 | https://mathoverflow.net/users/10366 | 102127 | 59,276 |
https://mathoverflow.net/questions/102111 | 2 | Is there any reference for equivariant Riemann-Roch formula: book, paper, notes or something? I want to compute the weight of the action of C^\* on the top wedge of cohomology group.
| https://mathoverflow.net/users/18498 | Reference for equivariant Riemann-Roch formula? | You also have some lecture notes on the web page of Michel Brion [here](http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html).
This [paper](http://www.math.jussieu.fr/~vergne/publications2/Remplacer/83zerosdunchampdevecteurs.pdf) of N. Berline and M. Vergne is well written (but is more "Lie Group theoretic" than the... | 4 | https://mathoverflow.net/users/25017 | 102134 | 59,282 |
https://mathoverflow.net/questions/102138 | 27 | This question has a very general part and a rather concrete part.
**General:**
When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some good ideas, but first one has to have a good set of tools at hand. Introductory books in algebraic topology provide a... | https://mathoverflow.net/users/18744 | A toolbox for algebraic topology | The subject is really way too big (as are so many others of course). I worry a lot about students not in Cambridge or Chicago or Stanford or other places where there are people with folklore at their fingertips. For spectral sequences as a tool, there is a lot to be said for McCleary's guide. Kate Ponto and I just publ... | 23 | https://mathoverflow.net/users/14447 | 102146 | 59,288 |
https://mathoverflow.net/questions/102143 | 5 | My field of research is coding theory and I am working on cyclic codes. During my research, I tackled an algebraic problem. After some simple definitions, I asked my question. I will appreciate any helpful answer and comment.
Let $F\_{2^{2m}}$ denote the finite field of ${2^{2m}}$ elements, where $... | https://mathoverflow.net/users/19929 | Special polynomials over finite fields | First, a polynomial $f$ is self-conjugate iff its coefficients belong to the fixed field of the Frobenius $x\mapsto x^{2^m}$, which is $F\_{2^m}$.
Second, an $f\in F\_{2^m}[x]$ which is irreducible over $F\_{2^{2m}}$ must have an odd degree $d$: the splitting field of $f$ is the unique degree $d$ extension of $F\_{2^... | 4 | https://mathoverflow.net/users/12705 | 102148 | 59,290 |
https://mathoverflow.net/questions/102142 | 3 | The generalized homological mirror symmetry conjecture states that for mirror dual models $(X\_E, w)$ and $(X\_E', w')$ , if $L$, a lattice polytope which is a Newton polytope of a nonsingular projective toric variety, is in $M\_{R}$ , then isomorphisms exist between:
$D^b(X\_E,w)$ and $DFS(X\_E',w')$, $D^b(X\_E',w')$ ... | https://mathoverflow.net/users/nan | Concerning the homological mirror symmetry conjecture | The conjecture has been solved for elliptic curves, abelian varieties, non-singular torus bundles over affine manifolds, and quartic surfaces. It remains to find a unification from algebraic geometry. A few references for this subject are the following:
Kontsevich, Maxim (1994), Homological algebra of mirror symmetry... | 1 | https://mathoverflow.net/users/nan | 102152 | 59,292 |
https://mathoverflow.net/questions/102144 | 0 | Is it well known that if $ H = -\bar{\Delta} + V$ (which is defined over $ L^2( \mathbb{R} ^n $ ) and $ lim\_{|x| \to \infty } = + \infty $, then $ H$ has compact resolvent?
Does someone know of any elegant way of proving this?
Thanks in advance
| https://mathoverflow.net/users/25080 | Schrodinger Operators with diverging Potential | If we impose some mild conditions on potential then it boils down to compact embeddings of Sobolev spaces. For example, one can assume that $V$ is bounded from below; in that case, for the sake of convenience, I shall consider nonnegative potentials. It is enough to prove compactness of resolvent just for one element o... | 2 | https://mathoverflow.net/users/24953 | 102162 | 59,296 |
https://mathoverflow.net/questions/102154 | 3 | I'm trying to classify compact manifolds $M^{16}$ with a metric which is locally conformal to a (local) metric with holonomy (included in) Spin(9)$\subset$SO(16). To do this, I would need a complete list of finite subgroups of Spin(9) acting freely on $S^{15}$.
Any hint on finite subgroups of Spin(9) (even not acting f... | https://mathoverflow.net/users/25099 | Finite subgroups of Spin(9) | Although there exists an algorithm which will list all finite subgroups of $Spin(9)$ I suspect it is not effective. For background and references see
[The finite subgroups of SU(n)](https://mathoverflow.net/questions/17072)
I don't know if imposing the condition that the group acts freely on $S^{15}$ improves the s... | 1 | https://mathoverflow.net/users/3992 | 102167 | 59,298 |
https://mathoverflow.net/questions/102166 | -2 | Let $X$ be a smooth projective variety, and let $E=\mathcal{O}\oplus \mathcal{O}(1)$ be a vector bundle of rank $2$. Then $L=\wedge ^{2} E $ is a line bundle on $X$. Is $L(-2)$ $\mathbb{Q}$-linear to an effective divisor?
| https://mathoverflow.net/users/24445 | Effectiveness of a wedged bundle | The answer is no already for $X=\mathbb{P}^1$. Indeed $c\_1(L)=c\_1(\mathcal{O}(1))$ and so $c\_1(L(-2))=c\_1(\mathcal{O}(-1))$. Therefore $L(-2)$ is linearly equivalent to the tautological bundle, which is not $\mathbb{Q}$-effective.
| 3 | https://mathoverflow.net/users/13168 | 102169 | 59,300 |
https://mathoverflow.net/questions/102119 | 1 | Hello everyone,
After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the [GRH](http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis) imposes the location of finding zeroes of a $L$-function with $\... | https://mathoverflow.net/users/24713 | Interplay between Riemann and Swinnerton-Dyer | In answer to question 1, there are certainly zeroes on the critical strip. A great way you can investigate this is to go to the [L-Functions and Modular Forms Database](http://www.lmfdb.org/), where you can view plots of the associated Hardy Z-functions associated to the Hasse-Weil L-function of any elliptic curve over... | 3 | https://mathoverflow.net/users/4872 | 102172 | 59,301 |
https://mathoverflow.net/questions/102165 | 1 | Let $\pi:X \longrightarrow C$ be a smooth projective morphism onto a smooth projective curve, and $F$ be a central fiber. If the Kodaira dimension $\kappa (F)$ is nonnegative, is $\pi\_{\ast} \mathcal O\_X (k K\_{X/C})$ nonzero for sufficiently divisible $k$? If it is, can anyone arrange an algebraic proof?
| https://mathoverflow.net/users/24445 | direct image of relative pluri-canonical bundle on a smooth fibration | I think one can argue as follows. (Let me know if I made a mistake!)
Choose $k$ large enough so that $kK\_F$ has a nonzero global section for some fibre $F$. For any fibre $F$, we have $K\_F = K\_{X/C}~\_{|F}$. So deformation invariance of plurigenera says that the function
$$h^0(X\_p, k K\_{X/C}~\_{|X\_p})$$
is c... | 2 | https://mathoverflow.net/users/nan | 102174 | 59,302 |
https://mathoverflow.net/questions/102173 | 4 | A graph manifold is a closed 3-manifold $M$ that admits a finite collection of disjoint embedded tori $\mathcal{T}$ so that $M \setminus \mathcal{T}$ is a disjoint union of Seifert fibred spaces (i.e. spaces admitting a foliation with a circle as fibre).
My interest in this class of manifolds lies in its characteriza... | https://mathoverflow.net/users/20557 | Second Homotopy Group of Graph Manifolds | The answer to your question is almost trivial: if the tori/Klein bottles are required to be $\pi\_1$-injective, then the manifold will be a $K(\pi,1)$, so $\pi\_2$ will vanish. Also, notice in the case of Seifert-fibered spaces, the universal cover is either $\mathbb{R}^3$ or $S^3$, and thus again $\pi\_2=0$.
However... | 11 | https://mathoverflow.net/users/1345 | 102181 | 59,304 |
https://mathoverflow.net/questions/102164 | 9 | I think this is not a research question, but in stackExchange remained unanswered.
Let $R$ be a finite commutative ring. For $n>1$ consider the full matrix ring $M\_n(R)$ . For a matrix $A\in M\_n(R)$ is true that the cardinality of the left annihilator (in $M\_n(R)$ ) of $A$ equals the cardinality of the right annhi... | https://mathoverflow.net/users/24864 | Annihilators in Matrix Rings | Let $k$ be a finite field, and let $R := k[X,Y] / (X^2,XY,Y^2)$. Then $R = k \oplus kx \oplus ky$ is a finite ring of order $|k|^3$ with maximal ideal $\mathfrak{m} := kx \oplus ky$ of square zero.
Now let $A = \begin{pmatrix} 0 & x \newline 0 & y \end{pmatrix} \in M\_2(R)$. Then
$\begin{pmatrix} 0 & x \newline 0 &... | 13 | https://mathoverflow.net/users/6827 | 102189 | 59,307 |
https://mathoverflow.net/questions/99497 | 8 | Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
>
> For each $\varepsilon>0$, we can find a compact subset $K$ of $X$ such that $\mathbb P(K)\geq 1-\varepsilon$.
>
>
>
The question is: is there a "nice" topological characterizat... | https://mathoverflow.net/users/17118 | Topological necessary and sufficient condition for tightness | You run into questions of set theory: If $X$ has the discrete topology (which is metric)
then your condition means: Every probability measure on the power set $2^X$ is discrete.
This is related to the 'measurability' of the cardinal $card(X)$. For general metric spaces
the maximal cardinality of discrete subsets is t... | 6 | https://mathoverflow.net/users/25106 | 102194 | 59,311 |
https://mathoverflow.net/questions/102079 | 3 | (This is a follow up to this [previous question on math.stackexchange.com.](https://math.stackexchange.com/questions/169649/average-time-to-find-a-duplicate-with-pairwise-independence))
Assume a process that samples uniformly at random from the range $[1,\ldots,n]$. I am interested in the time to find a duplicate giv... | https://mathoverflow.net/users/25089 | Bounds for duplicate finding with limited independence | For $k \ge 4$, the expected time until the first duplicate is $O(\sqrt{n})$. This leaves the case $k=3$ [Edit: The case of $k=3$ is resolved below with a construction with expected first duplicate at about $n/4$].
Let $D\_i$ be the number of duplicates in the first $i$ values. The expected time until the first duplic... | 2 | https://mathoverflow.net/users/2954 | 102214 | 59,322 |
https://mathoverflow.net/questions/72118 | 4 | For an application to analytic number theory, I'm considering the Fourier transform of test functions $h(x)$ on $\mathbb R$ with compact support, so the transforms $\hat h(y)$ are smooth. But I want to add a mild condition on $h$ that will force $\hat h(y) \ll 1/y^2$ as $y\to\infty$. Based on the examples I know, it lo... | https://mathoverflow.net/users/6756 | Fourier Transform: Smoothness and Decay | Your space is some sort of Besov space,
containing $B^{2+\epsilon}\_{1,\infty}$.
Using a Littlewood-Paley decomposition $1=\sum\_\nu\varphi\_\nu(\xi)$
with $\varphi\_0\in C^\infty\_c$, $\varphi\_\nu(\xi)=\varphi (\xi 2^{-\nu})$,
$\varphi\in C^\infty\_c$ supported in a ring
$1/2\le\vert\eta\vert\le 2$, a fonction $u... | 4 | https://mathoverflow.net/users/21907 | 102218 | 59,324 |
https://mathoverflow.net/questions/102156 | 2 | I have an elementary question about globally generation of a vector bundle. I would like to see why $\Omega\_{\mathbb{P}^n}(2H)$ is globally generated (it seems this is well-known among experts). Here $H$ is the hyperplane class of $\mathbb{P}^n$. In general how do we prove globally generation of a vector bundle (line ... | https://mathoverflow.net/users/25101 | Globally generation of $\Omega_{\mathbb{P}^n}(2H)$ | The projective space associated to a finite-dimensional vector space $V$ over a field $k$ is a universal pair $(\mathbb{P}V,\tilde{\gamma})$ of a $k$-scheme $\mathbb{P}V$ and a surjection of coherent sheaves
$$ \tilde{\gamma}:V^\vee \otimes\_k \mathcal{O}\_{\mathbb{P}V} \to \mathcal{O}\_{\mathbb{P}V}(1), $$
such that ... | 5 | https://mathoverflow.net/users/13265 | 102223 | 59,326 |
https://mathoverflow.net/questions/102247 | 10 | I may be missing an obvious example, but here goes...
Let $X$ denote a complex manifold of dimension $n$. If $X$ is Kähler, then the induced metric on any complex submanifold is also Kähler. If instead $X$ is non-Kähler, we can at least say that any coordinate neighborhood $(U,\varphi)$ inherits a Kähler metric from ... | https://mathoverflow.net/users/24525 | Kähler metric on a Zariski open subset of a non-Kähler manifold | One can consider the following example.
A Moishezon manifold $M$ is a compact connected complex manifold such that the field of meromorphic functions on $M$ has transcendence degree equal to the complex dimension of $M$. Complex algebraic varieties have this property, but the converse is not true if the dimension is ... | 16 | https://mathoverflow.net/users/7460 | 102252 | 59,343 |
https://mathoverflow.net/questions/102258 | 11 | Let $(X,\Sigma,\mu)$ be a measure space and consider a family of $\mu$-measurable functions $f\_i:X \to \mathbb{R}$ for $i$ lying in some index set $I$. Define $$f(x) = \inf\_{i \in I} f\_i(x)$$
I think every good analysis book mentions or proves that if $I$ is countable, then $f$ is also $\mu$-measurable. What is no... | https://mathoverflow.net/users/18263 | When is the infimum of an arbitrary family of measurable functions also measurable? | As you expected, the infimum of continuum many measurable functions need not be measurable, even in the case where $X$ is the real line with Lebesgue measure. In fact, if $A$ is any subset of $\mathbb R$ (in particular not necessarily measurable), its characteristic function is the infimum of at most continuum many mea... | 11 | https://mathoverflow.net/users/6794 | 102260 | 59,346 |
https://mathoverflow.net/questions/102256 | 1 | In the [wiki](http://en.wikipedia.org/wiki/Limit_%28music%29) section on prime limit tuning, one reads:
**`p-Limit Tuning. Given a prime number p, the subset of $Q^+$ consisting of those rational numbers x whose prime factorization has the form $ x=p_1^{\alpha_1}\ldots p_r^{\alpha_r}$ with $p_1, \ldots, p_r\leq p$ fo... | https://mathoverflow.net/users/15293 | are p-limits scales dense in the infinite musical scale of all rational frequencies? | I assume that you are everywhere working up to multiplication by a power of $2$ (or else the result is trivially false if $p = 2$ for example) and assuming that your scale has some nontrivial interval ratio $n$ in it which is not a power of $2$. Then this is straightforward.
**Lemma:** A subgroup of $\mathbb{R}$ is ... | 3 | https://mathoverflow.net/users/290 | 102265 | 59,350 |
https://mathoverflow.net/questions/102159 | 4 | In practice, both errors and erasures might be introduced in the channel. Could you point me to some good codes for correcting such combinations. Also what are their correction capabilities?
| https://mathoverflow.net/users/11105 | Good codes in practice for correcting combination of errors and erasures | The answer depends on several things.
If your channel (or receiver) produces erasures and errors, then the relevant metric is the Hamming metric, as a code with minimum distance $d$ can correct a combination of $t$ errors and $e$ erasures, iff $d>2t+e$. Therefore a code with good Hamming distance may be the way to go... | 5 | https://mathoverflow.net/users/15503 | 102272 | 59,355 |
https://mathoverflow.net/questions/102216 | 3 | It is easy to show that $Hom(\mathbb{Z}\_p,\mathbb{Z})=0$ (one just uses that each integer coprime to $p$ is a unit and therefore each image of such a homomorphism has infinitely many divisors). However, I wonder if this is true more generally:
*Question:* Given a commutative ring $R$ and a maximal ideal $m \subsete... | https://mathoverflow.net/users/17588 | The (algebraic) dual of a completion | Your example for $R=\mathbb{Z}$ (along with the proof) generalizes verbatim to the case $R$ a principal ideal domain and $m=(p)$ with a prime element $p$.
As another example let $R$ be a commutative Noetherian local domain that is not complete. Then also $Hom\_R(\hat{R},R)=0$. A concrete example is $R=k[x\_1,...,x\_... | 3 | https://mathoverflow.net/users/10194 | 102277 | 59,357 |
https://mathoverflow.net/questions/102253 | 3 | Let $f:\mathbb{R}\to \mathbb{R}$ be a function, then looking $f$ as a function between manifolds, $df:T\mathbb{R}=\mathbb{R}^2\to \mathbb{R}^2$ and $d^2f:TT\mathbb{R}=\mathbb{R}^4\to \mathbb{R}^4$ are given by
$$df(x,a)=(f(x),\frac{df}{dx}a)$$
$$d^2f((x,a),(b,c))= ((f(x),\frac{df}{dx}a),(\frac{df}{dx}b,\frac{d^2f}{dx... | https://mathoverflow.net/users/5259 | An elementary but confusing question in differential geomerty | It's not very clear to me which properties you expect from that object, but interpreting the question literally the answer is no: there is no natural way to associate to some vectors $v\_1,v\_2,\ldots,v\_k\in T\_x M$ and a function $f\in C^\infty(M)$ an object corresponding "only" to the partial derivative $\frac{\part... | 6 | https://mathoverflow.net/users/745 | 102278 | 59,358 |
https://mathoverflow.net/questions/102284 | 2 | Hello,
I am wondering, if we have a complete graph on $n$ vertices, and we have $k$ colours so that every edge of the graph is coloured with one of these colours, what is the least $n$ such that we will always be able to find a monochromatic cycle of length $m$?
It would be great to find a function F($k,m$) to give... | https://mathoverflow.net/users/25127 | Looking for monochromatic cycles in an edge-coloured clique | You in the realms of extensions of Ramsey Theory here, so general, precise answers may be a bit thin on the ground.
There are at least a couple of known results:
$2^{k} \leq F(k,3) \leq 3k!$
and
$F(k,4) \leq k^{2} + k + 1$
These lecture slides give a start:
<https://www.dpmms.cam.ac.uk/~dc340/Ramsey-course.ht... | 2 | https://mathoverflow.net/users/24952 | 102286 | 59,359 |
https://mathoverflow.net/questions/101823 | 1 | Let $S$ be a set of some size $n$. I'm interested in knowing about combinatorial designs that are approximately balanced incomplete block designs, that I want a collection of subsets $C$ of $S$ such that every pair of elements of $S$ appears in at least one element of $C$, all the elements of $C$ are approximately the ... | https://mathoverflow.net/users/24180 | Approximate version of a balanced incomplete block diagram | To expand on my comments above, an analogous problem is covering a graph by cliques.
Consider the relation on the set F of files as (a,b) is in the relation if the (distinct) files a and b are to be processed.
The result is like a directed graph with F as the set of vertices and the relation determining the edges. Now ... | 1 | https://mathoverflow.net/users/3568 | 102307 | 59,369 |
https://mathoverflow.net/questions/102241 | 4 | Chapter 21.2 of Friedlander and Iwaniec's *Opera de Cribro* begins (essentially) as follows:
Let
$$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We wan... | https://mathoverflow.net/users/1050 | 'Small' and 'large' discriminants of two-variable quadratic polynomials | Here are some naive comments. If $X, Y$ are large then the contribution of the quadratic terms swamps the other terms, so first let's concentrate on the quadratic terms
$$Q(X, Y) = a X^2 + b XY + c Y^2.$$
This can be written as
$$Q(X, Y) = \left[ \begin{array}{cc} X & Y \end{array} \right] \left[ \begin{array}{cc} a ... | 2 | https://mathoverflow.net/users/290 | 102312 | 59,374 |
https://mathoverflow.net/questions/102280 | 3 | In the paper by [Masbaum](https://arxiv.org/pdf/math/0211044.pdf), it was shown that the colored Jones polynomials for a twist knot $K\_p$ can be written as
\begin{eqnarray}
J\_{n}(K\_p;q)&=&\sum\_{k=0}^{\infty} {\cal C}\_{K\_p}(k)
\frac { \lbrace n-k\rbrace\lbrace n-k+1\rbrace \cdots \lbrace n+k\rbrace} {\lbrace n... | https://mathoverflow.net/users/17644 | On expressions of colored Jones polynomials | If you look at the paper by Garoufalidis and Le "[The colored Jones function is q-holonomic](https://arxiv.org/abs/math/0309214)", they prove that the colored Jones function is $q$-holonomic. Using techniques of Zeilberger, one can then verify such identities algorithmically. So you need only check the first few terms ... | 4 | https://mathoverflow.net/users/1345 | 102315 | 59,376 |
https://mathoverflow.net/questions/102316 | 12 | How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only understand $d\_2(h\_4)=h^2\_3h\_0$.
There seem to be two methods that are used or referenced in various texts, but I haven... | https://mathoverflow.net/users/24021 | Differentials in the Adams Spectral Sequence for spheres at the prime p=2 | In that range of dimensions one can cheat, as I did in my 1964 thesis. That is available on MathSciNet, and the differentials are penciled in on page A.2 (near the end). It was an easy exercise then to deduce the differentials algebraically as the only ones consistent with Toda's calculations of the homotopy groups sli... | 12 | https://mathoverflow.net/users/14447 | 102324 | 59,379 |
https://mathoverflow.net/questions/102323 | 6 | I can't seem to find a reference on the web that gives the $\mathbb{Z}$ cohomology of the Grassmann manifold of real n-planes in infinite dimensional Euclidean space and also the Bockstein maps associated with the coefficient sequence
$$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z/2Z} \to 0.$$
The real question is... | https://mathoverflow.net/users/25138 | Integer cohomology of the Grassman manifold of n planes in $R^\infty$ | I don't know if these have everything that you want, but see the following:
Brown, Edgar H., Jr.
The cohomology of BSOn and BOn with integer coefficients.
Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.
Feshbach, Mark The integral cohomology rings of the classifying spaces of O(n) and SO(n). Indiana Univ. Math. J... | 5 | https://mathoverflow.net/users/6646 | 102329 | 59,381 |
https://mathoverflow.net/questions/102313 | 17 | There is a well-known principle that one can recover classical mechanics from quantum mechanics in the limit as $\hbar$ goes to zero. I am looking for the strongest statement one can make concerning this principle. (Ideally, I would love to see something like: in the limit as $\hbar$ goes to zero, the position wavefunc... | https://mathoverflow.net/users/25136 | Classical limit of quantum mechanics | There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things... | 8 | https://mathoverflow.net/users/19520 | 102341 | 59,384 |
https://mathoverflow.net/questions/102338 | 2 | Hello,
Is it possible to reconstruct the Riemann zeta function given the precise location of its infinitely many zeroes?
Thanks
| https://mathoverflow.net/users/nan | Is it possible to reconstruct $\zeta$-function knowing its zeroes? | This is well known (Riemann could have writen it)
$$\zeta(s)=\frac{1}{2}\frac{\pi^{s/2}}{(s-1)\Gamma(1+s/2)}\prod\_{\Im\rho > 0}\Bigl\{
\Bigl(1-\frac{s}{\rho}\Bigr)\Bigl(1-\frac{s}{\overline{\rho}}\Bigr)\Bigr\}$$
Here $\rho$ runs through the non trivial zeros with positive imaginary part.
It is this what you call... | 9 | https://mathoverflow.net/users/7402 | 102342 | 59,385 |
https://mathoverflow.net/questions/99161 | 1 | Let $M$ be a compact manifold without boundary, $f:M\to M$ be a diffeomorphism.
Then $f$ is said to be (topologically) transitive if $\bigcup\_{\mathbb{Z}}f^nU$ is dense for every nonempty open set $U\subset M$.
Assume $f$ is transitive and $\lbrace U\_k:k\ge1\rbrace$ is a subbasis of the topology on $M$. Then defin... | https://mathoverflow.net/users/11028 | Volume of the set of transitive points of transitive diffeomorphisms | Dear Pengfei,
Such examples exist and they can be obtained by the so-called Anosov-Katok method (see, e.g., Theorem 5.1 of the article "Constructions in elliptic dynamics" <http://www.ams.org/mathscinet-getitem?mr=2104594> of B. Fayad and A. Katok).
Best,
Matheus
| 1 | https://mathoverflow.net/users/1568 | 102346 | 59,389 |
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