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https://mathoverflow.net/questions/182064 | 8 | A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal:
(1) Извѣстія Физико-математического общества при Казанском университете
I am surprised by the difficulty of finding the papers to read, surprised because of just how well-know this material is today. The only work by... | https://mathoverflow.net/users/58777 | A. Markov's papers? | Obtaining old issues of this journal is difficult, though perhaps some American libraries have it,
and you can try Interlibrary Loan (ILL). Alternatively, there are two volumes
of collected papers
Markov's Collected papers, MR0050525 the papers you are looking for must be in this volume.
Our library has this, so you ca... | 7 | https://mathoverflow.net/users/25510 | 182087 | 91,232 |
https://mathoverflow.net/questions/182097 | 0 | In following [post](http://www.mathcounterexamples.net/a-field-that-can-be-ordered-in-two-distinct-ways/), I describe the "classical" example of $\mathbb{Q}(\sqrt{2})$ that can be ordered in two distinct ways.
More generally, if $(k,P)$ is an ordered field, $R$ a real closure of $(k,P)$ and $K/k$ an algebraic extensi... | https://mathoverflow.net/users/41060 | Ways to order an algebraic extension | Orders on $K$ extending $P$ are in a 1–1 correspondence with $k$-embeddings of $K$ in $R$.
In particular, there can be infinitely many only if $K/k$ has infinite degree; a simple example is $k=\mathbb Q$, $K=\mathbb Q(\{\sqrt p:p\text{ prime}\})$.
| 6 | https://mathoverflow.net/users/12705 | 182098 | 91,234 |
https://mathoverflow.net/questions/182061 | 3 | **Q1:** Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to $S^{n-1}$?
**Q2** Let $Z$ be a topological space. What kind of assumptions can you impose on $Z$ in order to have the... | https://mathoverflow.net/users/11765 | cartesian product rigidity for the punctured open disc | Q1: No, see the third-from-last theorem of
R. H. Bing, [The cartesian product of a certain nonmanifold and a line is $E^4$](http://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10160-3/home.html).
| 4 | https://mathoverflow.net/users/318 | 182104 | 91,236 |
https://mathoverflow.net/questions/182099 | 7 | Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$.
If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's explicit solution) that the model exhibits a phase transition when $L=M\to\infty$.
If instead we fix $L=1$ ($1$-dimensio... | https://mathoverflow.net/users/58793 | Ising model on lattices with (vertical side length) $\neq$ (horizontal side length) | Any increasing sequence $(\Lambda\_n)\_{n\geq 1}$ of finite subsets of $\mathbb{Z}^d$, $d\geq 2$, such that $\bigcup\_{n\geq 1} \Lambda\_n =\mathbb{Z}^d$ will do. All sequences $(\mu\_{\Lambda\_n}^+)\_{n\geq 1}$ of finite-volume Gibbs measures in $\Lambda\_n$ with $+$-boundary condition converge to the same infinite-vo... | 5 | https://mathoverflow.net/users/5709 | 182130 | 91,249 |
https://mathoverflow.net/questions/182131 | 3 | Good morning everybody.
I would like to know if anybody is aware of nontrivial results of the following form : *if a family $\mathcal I$ of subsets of $\mathbb N$ satisfies such and such assumption, then one may conclude that $\mathbb N$ can be partitioned into finitely many sets from $\mathcal I$.*
This is of cour... | https://mathoverflow.net/users/37371 | Existence of partitions | Let $\lambda$ denote the uniform measure on the powerset of $\mathbb N$ (also known as the Lebesgue measure and the fair-coin measure).
Let $\tau$ be the product topology of the discrete topology on $\{0,1\}$.
In order to be able to conclude that $\mathbb N$ can be partitioned into finitely many sets from $\mathcal I... | 1 | https://mathoverflow.net/users/4600 | 182134 | 91,250 |
https://mathoverflow.net/questions/182158 | 4 | Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and
$$G=\{P,2P,3P,4P,\cdots\}.$$
Is there an integral point $Q\in G?$
| https://mathoverflow.net/users/41032 | Is there an integral point in the group generated by an rational point? | I presume that (1) $E$ is meant to be in minimal form with integer coefficients,
else you can change coordinates to put $P$ and $nP$ anywhere, and that
(2) $P$ is not itself integral (i.e. $d>1$), else $Q=P$ works.
Then the answer is No, because $d^2$ and $d^3$ divide the denominators
of $x(nP)$ and $y(nP)$ for all $... | 10 | https://mathoverflow.net/users/14830 | 182173 | 91,261 |
https://mathoverflow.net/questions/182168 | 7 | The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" modular polynomials (the minimum polynomial of the modular function $\tau \mapsto j(\ell \tau)$ for a given prime $\ell$... | https://mathoverflow.net/users/48499 | Modular polynomials for elliptic curves point counting | Andrew Sutherland (in his paper [On the evaluation of modular polynomials](http://arxiv.org/abs/1202.3985),
where he documents the point counting record) refers to Section 7 of the paper by himself, Broker and Lauter ([Modular polynomials via isogeny volcanos](http://www.ams.org/journals/mcom/2012-81-278/S0025-5718-201... | 3 | https://mathoverflow.net/users/48142 | 182176 | 91,264 |
https://mathoverflow.net/questions/182181 | 17 | It is mentioned [here](https://mathoverflow.net/questions/65424/determinant-of-sum-of-positive-definite-matrices/172884#172884) that if $A, B, C\in M\_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from [this article](http://www.math.technion.ac.il/iic/ela/ela-a... | https://mathoverflow.net/users/29783 | Hlawka inequality for determinants of positive definite matrices | My previous answer mis-attributed a claim to the paper of Paksoy, Turmen, and Zhang (also cited in the OP). Their claim is indeed strictly weaker than the inequality conjectured by Wolfgang. The details are slightly messy to type up, but they follow from the following more general *operator Hlawka inequality*
>
> *... | 11 | https://mathoverflow.net/users/8430 | 182193 | 91,268 |
https://mathoverflow.net/questions/182192 | 2 | Suppose we are given two polynomial rings $R\_1$ and $R\_2$ by presenting their generators, $S\_1$ and $S\_2$, where $S\_i$ are finite set of $m\_i$ variables, $i.e.$, $S\_i\subset P[x\_1,x\_2,\cdots,x\_{m\_i}]$. $R\_i$ are generated from $S\_i$ by the usual addition, subtraction, and multiplication.
The goal is to ... | https://mathoverflow.net/users/4987 | Bound the degree of the generator of polynomial ring | Without homogeneity assumptions, the intersection need not be finitely generated. This is discussed in <http://arxiv.org/abs/1301.2730>. (See also <http://www.emis.de/journals/BAG/vol.43/no.2/b43h2bay.pdf> which is referred to in that arXiv preprint).
| 3 | https://mathoverflow.net/users/1306 | 182201 | 91,271 |
https://mathoverflow.net/questions/182198 | 1 | I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions:
(1) $\sum\_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$
(2) $f(n,M) \approx 1$ whenever $n\ll M$
The function is needed as a summation mollifier to ensure a piecewise function is continuo... | https://mathoverflow.net/users/35683 | Summation mollifier to ensure a certain alternating series has the correct value | I'm not sure what exactly "smoother" means in the context of a function defined on a discrete set, but maybe this will do you: $f(0,M)=1-2^{-(M+1)}$, $f(n,M)=1-3\times2^{-(M-n+2)}$ for $n\ge1$. In tabular form, $$\matrix{1/2&&&&\cr3/4&1/4&&&\cr7/8&5/8&1/4&&\cr15/16&13/16&5/8&1/4&\cr31/32&29/32&13/16&5/8&1/4\cr}$$ where... | 1 | https://mathoverflow.net/users/3684 | 182203 | 91,273 |
https://mathoverflow.net/questions/182150 | 3 | Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S\_\lambda$ be a Young subgroup of $S\_n$.
Further let $S\_\lambda\backslash S\_n/ S\_\lambda$ be the set of double-cosets. Now I would like to get the most general bound on the number of these cosets. All I could come up with is roughly $n^{k^2}$ but I w... | https://mathoverflow.net/users/58818 | Number of double cosets of a Young subgroup | It is a standard result that the number of double cosets of the pair
$(S\_\lambda, S\_\mu)$ is the number of matrices of nonnegative integers
with row sum vector $\lambda$ and column sum vector $\mu$. See for
instance Exercise 7.77 of *Enumerative Combinatorics*, vol. 2. This
makes it clear, for instance, why there are... | 13 | https://mathoverflow.net/users/2807 | 182207 | 91,276 |
https://mathoverflow.net/questions/179693 | 18 | (In this question, all rings and algebras are commutative with identity.)
I have a situation that boils down to the following data: a ring $R$, an $R$-algebra $A$ with a subalgebra $B$ such that $A$ and $B$ are free of finite rank as $R$-modules, and an $R$-algebra homomorphism $B\to R$. (**Update: If it helps, you c... | https://mathoverflow.net/users/1474 | If $B\subseteq A$ are free & finite rank $R$-algebras, is $R\to A \otimes_B R$ injective? | The answer to the question is "No": here is a counterexample due to Bas Edixhoven:
Let $k$ be a field and $R=k[t]/(t^2)$. Define $B = R[x,y]/(x^2,xy,y^2)$ and $A =R[x]/(x^2) \times R[y]/(y^2)$; the injection $B\hookrightarrow A$ sends $x\mapsto(x,0)$ and $y\mapsto(0,y)$. Then $B$ and $A$ are free $R$-modules of rank ... | 7 | https://mathoverflow.net/users/1474 | 182227 | 91,284 |
https://mathoverflow.net/questions/182148 | 1 | Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $c k$ regular graph on $v$ vertices that can be given an edge coloring with $c$ colors, such that the edges corresponding to each color form a $(v,k,\lambda,\mu)$ strongly regular graph?
For what parameters is the $c$-edge-col... | https://mathoverflow.net/users/12911 | Regular graphs with strongly regular edge colorings | Such objects are closely related to *amorphic association schemes*; these are association schemes for which any merging of classes is again an association scheme.Such an object often leads to an object in the question (and there are quite often many non-isomorphic examples).
Here is a [link](https://pure.uvt.nl/portal... | 1 | https://mathoverflow.net/users/11100 | 182239 | 91,287 |
https://mathoverflow.net/questions/182232 | 3 | Suppose you have got a double complex in an abelian category with objects $(A\_{rs},d\_{rs})$ such that $A\_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A\_{r\bullet},d\_{r\bullet})$ for $r > 0$ and the columns $(A\_{\bullet s},d\_{\bullet s})$ for $s > 0$ are exact.
Then a standard result sa... | https://mathoverflow.net/users/2082 | Standard homology result on double complexes | This is Proposition 3.9 in Osborne's Basic Homological Algebra. I expect you could also find it somewhere in pretty much any homological algebra text, though it might be mentioned in passing rather than stated explicitly as a result.
${}{}{}{}{}{}$
| 3 | https://mathoverflow.net/users/75 | 182241 | 91,289 |
https://mathoverflow.net/questions/182225 | 1 | Let $f(z)dz^2$ be a holomorphic quadratic differential on the punctured disk $\{0<|z|<1\}$, which gives rise to a Riemannian metric $g=|f(z)|\,|dz|^2$ and hence a volume form $\nu=|f(z)| dx\,dy$.
>
> **Problem:** prove that if $f$ has essential singularity at $0$, then
>
>
> (1) the total area of $\nu$ is infinit... | https://mathoverflow.net/users/17294 | Metric properties of a quadratic differential at an essential singularity | For statement 2, you have to specify whether $f$ is allowed to have zeros.
(If yes, this is a metric with isolated singularities, but can be complete.
If $f$ is free of zeros, it is always incomlete by a deep result of A. Huber MR0094452)
To prove statement 1, it is better to make the change of the variable $z=1/w$,
... | 3 | https://mathoverflow.net/users/25510 | 182247 | 91,292 |
https://mathoverflow.net/questions/182250 | 2 | If G is an infinite planar group (it means that it has a generating subset C such that Cay (S, C) is a planar graph) and H is a normal subgroup of it, I would be very grateful if somebody helps me and tell me "is G/H a planar group?"
| https://mathoverflow.net/users/58585 | quotient of planar groups | The dihedral groups can be viewed as the set of all functions of the form $x\mapsto\pm x+c$ acting either on $\mathbb{Z}$ or on $\mathbb{Z}/n\mathbb{Z}$. The images of the infinite dihedral group are either finite dihedral groups. Taking as generators $x\mapsto x+1$ and $x\mapsto -x$ you get for the dihedral group of o... | 2 | https://mathoverflow.net/users/37555 | 182263 | 91,299 |
https://mathoverflow.net/questions/182221 | 5 | I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A\_1,...,A\_k \in \mathbb R^{n\times n}$ and I'm observing that the spectrum of the product is the same whatever order of multiplication I choose, that is, for an arbitrary permutation $\pi$ ... | https://mathoverflow.net/users/58860 | Why is the spectrum of this matrix product invariant with respect to order of the multiplicants? | It seems that the claim is false when $n=2$ and $k=4$, because
$${\rm Tr} (A\_1A\_2A\_3A\_4)=a\_1a\_3+a\_2a\_4+\sigma\_3({\bf a})+\sigma\_4({\bf a})$$
is not symmetric in ${\bf a}=(a\_1,a\_2,a\_3,a\_4)$.
**Edit**: just let me add a classical result. Let $C\_1,\ldots,C\_r$ be the companion matrices of unitary polynomi... | 8 | https://mathoverflow.net/users/8799 | 182265 | 91,300 |
https://mathoverflow.net/questions/181771 | 3 | For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$.
Several of these algebraic characterizations, known as *positivstellensatz*, are listed [here](http://en.wikipedia.org/wiki/Posit... | https://mathoverflow.net/users/22389 | Handelman's positivstellensatz for symmetric matrix-valued polynomials | The obvious generalization is false: Consider $A=\begin{pmatrix} 1+x &y \\y & 1-x \end{pmatrix}$ which is a linear matrix polynomial. It defines a compact set in the plane, namely the unit disc. The algebra $\mathbb{R}[A]$ of all polynomial expressions in $A$ is contained in $\mathbb{R}(x,y)[A]$ which is a two dimensio... | 2 | https://mathoverflow.net/users/36563 | 182271 | 91,301 |
https://mathoverflow.net/questions/182270 | 3 | The Generalized Associativity Equation is given by
$$ F(G(x,y),z)=K(x,H(y,z)),$$
where the functions $F,G,H$ and $K$ are all from $\mathbb{R}^2$ to $\mathbb{R}$. In his book "Lectures on Functional Equations and their applications" (page 329), Aczel proves that, under some differentiability assumptions, the terms in t... | https://mathoverflow.net/users/18474 | Counterexample for the Generalized Associativity Equation | With no continuity assumptions, you can always find such $f,a,b,c$.
Namely, let $A, B, C$ be subsets of $\mathbb R$ with cardinality of the continuum
such that the function $(x,y,z) \to x+y+z$ is one-to-one on $A \times B \times C$.
Let $a$, $b$, $c$ be one-to-one functions from $\mathbb R$ onto $A, B, C$ respective... | 5 | https://mathoverflow.net/users/13650 | 182284 | 91,307 |
https://mathoverflow.net/questions/180173 | 51 | Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
By using the structure theory of connected Lie groups and a couple of Serre spectral sequences, I was quickly able to... | https://mathoverflow.net/users/5690 | $H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group | I try to give an argument without spectral sequences, not sure if this can be considered non-computational though. At least, there is a non-computational syllabus: *torsion classes in $H^4(BG,\mathbb{Z})$ would be characteristic classes of torsion bundles over $S^3$ but the latter have to be trivial.*
Now for a slig... | 37 | https://mathoverflow.net/users/50846 | 182292 | 91,309 |
https://mathoverflow.net/questions/182172 | 5 | Given a weight vector $w\in [0,1]^d$ such that $\sum w\_i=1$, the game goes as follows:
Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x\_i = \sum y\_i = 1$.
The utility (profit) for player $X$ is given by $$u\_X=\sum\_{i:x\_i>y\_i} x\_i\cdot w\_i$$
That is summing over all coordinates in w... | https://mathoverflow.net/users/47499 | What is the (mixed strategies) equilibrium of this game? | If you change the payoff to
$$\sum\_{i:x\_i>y\_i} w\_i $$
then this type of game is called a [Colonel Blotto game or a Blotto game](http://en.wikipedia.org/wiki/Blotto_games). It was studied by Borel. Your assumption that $\sum x\_i = \sum y\_i$ is not always a requirement.
In [a 1950 paper](http://www.rand.org/... | 6 | https://mathoverflow.net/users/2954 | 182301 | 91,313 |
https://mathoverflow.net/questions/182302 | 6 | If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T\_p$) of $f$ is $\lambda\_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum\_{x\le p\le 2x}|\lambda\_f(p)|^2?$$
To avoid Confusion $$(T\_pf)(z)=\frac{1}{\sqrt{p}}\left[\sum\_{b=0}^{p-1}f\left(\frac{z+b}... | https://mathoverflow.net/users/36735 | Lower bound of Hecke eigenvalues of Maass form | If $x$ is large enough, then Rankin-Selberg theory will show that $S(x) \gg x^{1-\varepsilon}$. However, if $x$ is not large enough, then it is unknown how to obtain a lower bound for $S(x)$. In particular, it is unknown how to show that $S(x) \neq 0$. A good starting point for this is Chapter 13 of Iwaniec's book, Top... | 9 | https://mathoverflow.net/users/2627 | 182309 | 91,316 |
https://mathoverflow.net/questions/182304 | 3 | Problem set up: I have a long list of variables, $v\_i$ (say about 200 total).
I am given a bunch of Boolean statements as follows:
$$\omega\_1\land \omega\_2\land \omega\_3\land \omega\_4\land \omega\_5 \land \ldots \land \omega\_n$$
Each $\omega\_k$ is a disjuction of mutually exclusive statements (for example:)
$$... | https://mathoverflow.net/users/57474 | What kind of SAT am I dealing with here? | In practice, the best thing to do might be to convert to CNF and hand it to a standard SAT solver. It may "look messy" to humans, but not to the SAT solver.
You might find a solver that accepts more general formats (though perhaps it will convert the problem internally to CNF). I know SMT solvers will accept assertions... | 6 | https://mathoverflow.net/users/13650 | 182313 | 91,319 |
https://mathoverflow.net/questions/182319 | 2 | I know that the automorphic representation can be defined only for reductive algebraic group.
What property of algebraic group makes it hinder to define for all algebraic group and what nice property of reductive enable us define automorphic representation to it?
And I also wondering whether unitary group is reduct... | https://mathoverflow.net/users/29422 | On the reductive group | It is not at all that automorphic representations can be defined only for reductive algebraic groups, but, rather, that the essential difficulties arise in that case... as opposed to (for example) abelian unipotent (e.g., additive groups). Indeed, "Jacobi modular forms" are modular forms (in effect) for parabolic subgr... | 5 | https://mathoverflow.net/users/15629 | 182324 | 91,327 |
https://mathoverflow.net/questions/182196 | 2 | **General Question:** If I have an IVP with periodic and continuous initial condition, which rules the accuracy of the scheme - the manner in which we approximate spatial derivative or the acuuracy of the ODE solving scheme.
**Specific Case**:
I'm solving
$u\_t = u\_{xx} + u$ ,
$x \in \mathbb{R} $, $t>0$, $u(t,x... | https://mathoverflow.net/users/42864 | IVP accuracy - scheme accuracy Vs. derivative accuracy? | In general, when solving a PDE numerically, both the spatial and the temporal discretization will (of course) contribute to the local truncation error and hence to the global error. In many equations it is often the case that the spatial error is dominant. Of course, the choice of particular discretizations matters.
... | 1 | https://mathoverflow.net/users/20507 | 182332 | 91,330 |
https://mathoverflow.net/questions/182335 | 3 | I have a followup to the following question: [Flatness of normalization](https://mathoverflow.net/questions/64776/flatness-of-normalization).
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in a finite separable extension of $K(X)$. Then is ... | https://mathoverflow.net/users/4181 | Flatness of Normalization of regular schemes | I guess not. Take an affine integral normal $X'$ which is not Cohen-Macaulay. Find a Noether normalization map $f:X'\to X=\mathbb{A}^n$. This is a finite map, and $X'$ is the normalization of $X$ inside its field of rational functions. If $f$ were flat, by duality for finite flat morphisms $X'$ would be Cohen-Macaulay.... | 6 | https://mathoverflow.net/users/3847 | 182337 | 91,332 |
https://mathoverflow.net/questions/182252 | 0 | What results are known about the construction of models for a theory $T$ of the logic BL$\forall$ for languages of higher cardinality? The construction for the countable case relies on
1) The fact that you can form sentences of arbitrary finite length.
2) The fact that there is an undecidable statement in $T\_{n}\s... | https://mathoverflow.net/users/39939 | Models of BL$\forall$ | I'm definitely not an expert, but you could be interested in [A Henkin-style proof of completeness for first-order algebraizable logics](http://www.carlesnoguera.cat/files/Cintula-Noguera-2013-HenkinProofAlgLogics.pdf) by Cintula and Noguera.
| 2 | https://mathoverflow.net/users/58913 | 182340 | 91,334 |
https://mathoverflow.net/questions/181207 | 3 | We know that for a graph $G=(V,E)$, minimum edge coloring is a coloring of
$E$, i.e., a partition of $E$ into disjoint sets $E\_1, E\_2, \dots, E\_k$ such
that, for $1 \leq i \leq k$, no two edges in $E\_i$ share a common endpoint
in $G$.
Now how can we write an Integer linear program (ILP) to solve the minimum
edge ... | https://mathoverflow.net/users/58327 | ILP for minimum edge coloring problem | If $c$ is an upper bound for the number of colours (in case of doubt use $|E|$), then you could use binary assigment variables $x\_{ie}$ for assigning colour $i$ to edge $e$. Then, for every two edges $e,f$ which share a node one can introduce a constraint $x\_{ie} + x\_{if} \leq 1$. Furthermore one introduces binary v... | 1 | https://mathoverflow.net/users/3816 | 182344 | 91,336 |
https://mathoverflow.net/questions/182329 | 5 | $\omega \in S\_n$ is an FPFI (fixed point free involution) (also called a matching) if $\omega^2=1$ and $\omega(i) \neq i$ for all $i$.
For $\omega \in S\_n$, a descent occurs at $i$ if $\omega(i+1) < \omega(i)$. For example, $(1 \, 3)(2 \, 4) \in S\_4$, when written as $3412$, has one descent at $i=2$.
I'm curious... | https://mathoverflow.net/users/58905 | Is there a bijection of permutations onto mathematical objects that preserve information about descents? | In general, to find maps transporting statistics in a well-behaved way, it is useful to try [FindStat](http://www.findstat.org). In the case at hand, go to
<http://www.findstat.org/StatisticsDatabase/St000021/>
(which is the statistic "number of descents of a permutation") and click on "Search for values". After a ... | 7 | https://mathoverflow.net/users/3032 | 182350 | 91,338 |
https://mathoverflow.net/questions/182334 | 2 | In other words what do we call a magma which is associative and has divisibility property but not existence of identity? Or a groupoid when it loses the identity property?
A reference on such objects would be very helpful.
There is a table of various generalizations of groups [here](http://en.wikipedia.org/wiki/Gro... | https://mathoverflow.net/users/47249 | Reference request for generalization of groups with out identity element? | I think such structures are called nonunital semigroups. See, for example,
<ftp://ftp.math.ethz.ch/EMIS/journals/MPRIA/2000/pa100i2/pdf/100210ai.pdf> (Non-Unital Semigroup Crossed Products, by N.S. Larsen) and <http://www.hindawi.com/journals/aaa/2014/463918/> (Solution of Several Functional Equations on Nonunital Semi... | 2 | https://mathoverflow.net/users/32389 | 182353 | 91,339 |
https://mathoverflow.net/questions/182233 | 3 | We know that any substitution can generate sequence, for example the Fibonacci substitution:
$\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of $\sigma$. ($\sigma(\omega)=\omega$). We can define an orbit space which is based on the fixed point of the substitution, f... | https://mathoverflow.net/users/58508 | Substitutions and Sturmian sequences | My understanding is that there are no general techniques for answering your question, even for some specific examples. Certainly this was the case a few years ago.
A relatively well known example is the Kolakoski sequence, see <http://mathworld.wolfram.com/KolakoskiSequence.html> . This can be generated by a length ... | 3 | https://mathoverflow.net/users/24586 | 182358 | 91,341 |
https://mathoverflow.net/questions/182359 | 14 | O.K, a massively naive question. I've never really studied any non-standard models of PA before. I was just wondering if there's ever been any attempt to use the kind of Boolean valued model theory familiar from set theory in the context of arithmetic. Obviously, the actual structure of the model would have to be very ... | https://mathoverflow.net/users/45570 | Boolean Valued Models of PA | For any complete Boolean algebra $\mathbb{B}$, we may form the class $V^{\mathbb{B}}$ of all $\mathbb{B}$ names, and define $\mathbb{B}$-valued truth $[\! [\varphi]\! ]\in\mathbb{B}$ in the usual set-theoretic manner. By restricting to names $\tau$ for which $[\! [\tau\in\check{\mathbb{N}}]\! ]=1$, we get a natural $\m... | 7 | https://mathoverflow.net/users/1946 | 182361 | 91,342 |
https://mathoverflow.net/questions/182351 | 3 | I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and Weyl transitively on a regular locally finite building, and its image in the automorphism group is closed in the compac... | https://mathoverflow.net/users/15482 | algebraic groups over non-archimedean local fields acting on buildings | For such a general assertion, independent of classification, you'd need some one of the F. Bruhat and J. Tits papers, but (since I do not have copies nearby) I could not point you to any precise location within them.
F. Bruhat and J. Tits, BN-paires de type affine et donnees radicielles, C.R. Acad. Sci. Paris serie ... | 4 | https://mathoverflow.net/users/15629 | 182373 | 91,346 |
https://mathoverflow.net/questions/182387 | 0 | Assume we have a box of size $n$, some items $X\_i, i \in N$ of unknown distribution, with expected size $\mu>0$ and variance $\sigma^2$. We want to randomly and greedily pack the box with the items until one item does not fit anymore. Let $I$ denote the number of packed items.
I would like to show that, for a functi... | https://mathoverflow.net/users/58940 | Expected number of packed items in box | If $S\_i$ is the sum of the first $i$ items chosen (I assume this is with replacement), then $S\_i$ has mean $\mu i$ and variance $\sigma^2 i$.
Now $a(n) \le I < b(n)$ iff $S\_{a(n)} \le n$ and $S\_{b(n)} > n$.
Use Chebyshev on those.
| 0 | https://mathoverflow.net/users/13650 | 182400 | 91,354 |
https://mathoverflow.net/questions/182401 | 3 | Let $\Lambda :=\{\lambda\_1, \dots, \lambda\_n\}$ be a set of $n$ distinct real numbers.
For a given $p \in \mathbb N$, consider further the set
$$I\_p := \{ \{i\_1, i\_2, \dots, i\_p\} : i\_j \in \{1, \dots,n\} \text{ for all } j=1, \dots, p\}.$$
It is not hard to see that $|I\_p| = \binom{n+p-1}{p}$. For exampl... | https://mathoverflow.net/users/nan | Upper bound for different sums having the same value | I answer the first question in Corollary 4.1
of my paper at <http://math.mit.edu/~rstan/pubs/pubfiles/42.pdf>. The
maximum number of sums that have the same value is the middle
coefficient of the $q$-binomial coefficient $\left[ n+p-1\atop p
\right]$. This is achieved by taking $\Lambda=\{ 1,2,\dots,n\}$ and
the sum to... | 6 | https://mathoverflow.net/users/2807 | 182408 | 91,356 |
https://mathoverflow.net/questions/182404 | 13 | Does there exist a sequence of graphs $\{ G\_n \}$ such that
1. $G\_n$ has $n$ vertices,
2. the number of edges of $G\_n$ is $O(n)$, and
3. the crossing number of $G\_n$ is $\Omega(n)$?
In particular, do random $k$-regular graphs satisfy this?
Motivation: the crossing number inequality gives a lower bound on the ... | https://mathoverflow.net/users/4558 | Do there exist sparse graphs with large crossing number? | Take the following graph: start with the complete graph $K\_5$, and replace every edge by $n/10$ paths of length $2$. The resulting graph has $n+5$ vertices, $2n$ edges, and crossing number $n^2/100$.
For random graphs with expected number of edges $e>10n$, [Pach and Tóth](http://www.renyi.hu/~pach/publications/13pro... | 18 | https://mathoverflow.net/users/24076 | 182419 | 91,361 |
https://mathoverflow.net/questions/168499 | 2 | In [this paper of Mimura and Toda](http://projecteuclid.org/euclid.kjm/1250524818), tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi\_3(SU(4))$ as $i\_\ast \varepsilon\_3$. Here $i\colon SU(2) \hookrightarrow SU(4)$ ... | https://mathoverflow.net/users/4177 | Generator of $\pi_3(SU(4))$ in Mimura-Toda | Assembling comments of Neil Strickland and Allen Knutson, we have that $\varepsilon\_3$ is just the standard generator $\iota\_3\colon S^3 \stackrel{\sim}{\to} SU(2)$, since the inclusion $SU(2) \to SU(4)$ induces an isomorphism on $\pi\_3$. This is shown using the long exact sequence in homotopy groups, and the fact $... | 1 | https://mathoverflow.net/users/4177 | 182429 | 91,365 |
https://mathoverflow.net/questions/182410 | 2 | Let $G$ be a group, $X$ a generating set of $G$. Suppose $X$ is $\operatorname{Aut}(G)$-invariant, i.e. $\sigma(X)\subseteq X$ for all $\sigma \in \operatorname{Aut}(G)$. When is the restriction homomorphism
$$ \begin{gathered}
\operatorname{Aut}(G) \to \operatorname{Sym}(X)\\
\sigma \mapsto \sigma|\_X
\end{gathered}$... | https://mathoverflow.net/users/36720 | When is Aut(G) the symmetric group of an Aut(G)-invariant generating set? | For $G$ abelian, the complete list is $C\_2$, $C\_3$, $C\_4$, $C\_6$ and $C\_2\times C\_2$.
Proof:
Suppose that $X$ contains a non-involution $x$. Then $x^{-1}\in X$ (since inversion in an automorphism of $G$) but then $\{x,x^{-1}\}$ is a block of size 2 for the action of Aut(G) on $X$ and thus $|X|=2$ and $X=\{x,x^{... | 2 | https://mathoverflow.net/users/22377 | 182431 | 91,367 |
https://mathoverflow.net/questions/182424 | 15 | Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that $$f(x\_1,\dots,x\_n)=-x\_1^2-x\_2^2-\dots-x^2\_{i}+x^2\_{i+1}+\dots+x\_n^2.$$
There is an interpretation of Morse functions as "generic" functions. More prec... | https://mathoverflow.net/users/58952 | Higher Cerf Theory | This is what catastrophe theory does, at least for small $n$, $n\leq 10$. Volume 1 of the book by Arnold, Gussein-Zade and Varchenko on singularities has a nice description of this theory; see especially Part 2 of that book.
| 5 | https://mathoverflow.net/users/20302 | 182449 | 91,373 |
https://mathoverflow.net/questions/182427 | 5 | The question may be a little naive (or even appear as a duplicate) as I guess the result is well known. I saw on the other thread [that](https://mathoverflow.net/questions/64195/when-is-a-finite-dimensional-real-or-complex-lie-group-not-a-matrix-group)
"
c) A solvable Lie group G is linear iff its commutator subgrou... | https://mathoverflow.net/users/18850 | Why a nilpotent Lie group must be a matrix group? | Another reference which was not yet mentioned, I think, is the article of M. Moskowitz, "Faithful Representations and a local property of Lie groups", Math. Z. $143$, 1975.
There the question is discussed when *all* analytic groups with a given Lie algebra $\mathfrak{g}$ have a faithful linear representation. He prove... | 5 | https://mathoverflow.net/users/32332 | 182452 | 91,374 |
https://mathoverflow.net/questions/182442 | 9 | Assume two $CW$ complexes $X,Y$ give two functors $h\_X=[-,X], h\_Y=[-,Y]$ on the homotopy category of $CW$ complexes whose restrictions to the full subcategory of finite $CW$ complexes are naturally equivalent. Does this imply that the two functors are naturally equivalent (i.e. that X, Y are homotopy equivalent)?
| https://mathoverflow.net/users/58951 | Are Brown representable functors determined by restriction to finite complexes? | The answer depends on which category of CW-complexes you have in mind. It is true in the category of based connected CW-complexes simply by the Whitehead's Theorem.
**EDIT:** The concern raised by Matthias in the comment above is addressed in Brown's paper (Theorem 2.8 of *Abstract Homotopy Theory*). This is only pro... | 12 | https://mathoverflow.net/users/12547 | 182456 | 91,375 |
https://mathoverflow.net/questions/182437 | 17 | If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; free as an algebra; generated in positive degree? generated in negative degree?
What about commutative algebras or Lie a... | https://mathoverflow.net/users/4639 | Is homology finitely generated as an algebra? | Another counterexample: let $A$ be the algebra $\mathbb{Q}[y,z]/(y^2) \otimes \bigwedge(x)$ with $x$ in degree 1, $y$ and $z$ in degree 2. Put a differential on this by $z \mapsto xy$. This is a commutative dga in characteristic 0 generated in positive degrees, but of course it's not free. Its homology is spanned by th... | 15 | https://mathoverflow.net/users/4194 | 182465 | 91,380 |
https://mathoverflow.net/questions/182444 | 9 | Consider a power series
$$
\sum\_{n=0}^{\infty}a\_nz^n
$$
where $a\_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that for $|z|<R$ the series converges absolutely; for $|z|>R$ it does not converge.
On the other hand, when $|z|=R$, th... | https://mathoverflow.net/users/48866 | Power series with funny behavior at the boundary | The series $$f(z) = \sum\_{n=1}^\infty \dfrac{z^{2^n}}{n}$$
converges almost everywhere on the unit circle by Carleson's theorem (it is the Fourier series of an $L^2$ function). However, it diverges on a dense set, including all the $2^k$'th roots of unity:
in fact at each of those points the real parts of the partial... | 18 | https://mathoverflow.net/users/13650 | 182467 | 91,381 |
https://mathoverflow.net/questions/182471 | 5 | The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|\_p)$ to $(\mathbb R^m, \| \cdot \|\_q)$ is given by
$$ \| A \|\_{p,q} = \sup\_{x\in\mathbb{R}^n\setminus \{0\}} \frac{\|Ax\|\_q}{\|x\|\_p}.$$
I would like to compute $\| \cdot \|\_{2,\infty}$. In this paper: [On the Calculation of the $l\_2\... | https://mathoverflow.net/users/38490 | On the induced matrix norm $\| \cdot \|_{2,\infty}$ | Computing such induced norms is a hard problem. For the case of $p=2$ and $q \ge 2$, have a look at [this paper by Barak et al.](http://arxiv.org/abs/1205.4484) to see how tricky the problem is.
Typically, for other than the nice cases of $1,2, \infty$ style, these norms are NP-hard to compute, with well-known result... | 4 | https://mathoverflow.net/users/8430 | 182472 | 91,382 |
https://mathoverflow.net/questions/182317 | 3 | Let $f(x\_1,\ldots,x\_n)\in\mathbf{Z}[x\_1,\ldots,x\_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}\_p$-scheme.
... | https://mathoverflow.net/users/11765 | smooth connected affine scheme over Z has good reduction almost everywhere | I will first respond to the added question and then, for the sake of (likely unnecessary) completeness, give a few more details for the sort of argument that Ari proposed.
Let $g\_1,\dotsc,g\_r,f$ be elements of $S = \mathbf{Z}[x\_1,\dotsc,x\_n]$. In the rephrased question, one assumes $Z(g\_1,\dotsc,g\_r,f;\overline... | 8 | https://mathoverflow.net/users/59003 | 182489 | 91,389 |
https://mathoverflow.net/questions/182379 | 2 | Let $N$ be the total space of the vector bundle $\mathscr O\_{\mathbb P^1}(-1)\oplus \mathscr O\_{\mathbb P^1}(-1)$ over $\mathbb P^1$, and let $C\_0\subset N$ be the zero section. Then $N$ is a quasi-projective Calabi-Yau threefold, and the moduli space of stable maps $\overline{\mathcal M}\_g(N,[dC\_0])$ is quasi-pro... | https://mathoverflow.net/users/30827 | On the generating series of degree $d>1$ Gromov-Witten invariants of the local $\mathbb P^1$ | The formula is given by
$$ \sum\_{g\geq 0} c(g,d) t^{2g-2} = \frac{1}{d}\left( 2\sin \left(\frac{dt}{2}\right)\right)^{-2}$$
It is in Faber-Pandharipande <http://arxiv.org/pdf/math/9810173v1.pdf>. See also equations (1) and (2) in <http://arxiv.org/pdf/math/9811140.pdf>.
By the way, the inclusion you write $\ov... | 3 | https://mathoverflow.net/users/9617 | 182507 | 91,400 |
https://mathoverflow.net/questions/182501 | 1 | As I understand it, *rooted* maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A *map* is a graph $G$ embedded on a surface $X$ (such that every face in $X \setminus G$ is homeomorphic to a disc), while a *root... | https://mathoverflow.net/users/1015 | How to understand a rooting of a dessin d'enfant? | For dessin d'enfants, I believe the orientation is superfluous - each edge goes between a black vertex and a white vertex, so picking an orientation is just picking one of those, whcih doesn't help uniformize anything.
A dessin d'enfants corresponds to a cover of $\mathbb P^1$ ramified over three points $(0,1,\infty)... | 3 | https://mathoverflow.net/users/18060 | 182515 | 91,403 |
https://mathoverflow.net/questions/182513 | 0 | Recently I proved the following result.
"If a holomorphic function $f$ maps the unit disc $\Delta$ into the unit disk $\Delta $ with $0<|f(0)|$ then $f$ doesn't vanish in the disk $D(0,|f(0)|)$. "
I would like to know whether this result is well-known or how it can be useful. I will appreciate any kind of help. Thanks.... | https://mathoverflow.net/users/59021 | Location of the zeros set of holomorphic function | The result is trivial and follows from Schwarz lemma: if $f(z\_1)=0$, then $$|f(\frac{z+z\_1}{1+z\bar z\_1})|\le |z|$$ so $|f(0)|\le |-z\_1|$.
| 2 | https://mathoverflow.net/users/58947 | 182523 | 91,404 |
https://mathoverflow.net/questions/182518 | 4 | The [change-making problem](http://en.wikipedia.org/wiki/Change-making_problem) asks how to make a certain sum of money using the fewest coins. With US coins `{1, 5, 10, 25}`, the greedy algorithm of selecting the largest coin at each step also uses the fewest coins.
With which currencies (sets of integers including ... | https://mathoverflow.net/users/59026 | When does the greedy change-making algorithm work? | That's a different question, Gerry.
Believe it or not, the answers are different if one is asking
(a) given N and a system of denominations D, is the greedy algorithm using D optimal for N?
and
(b) given a system of denominations D, is the greedy algorithm using D optimal for ALL N?
I think the latter problem is t... | 11 | https://mathoverflow.net/users/44797 | 182526 | 91,407 |
https://mathoverflow.net/questions/182159 | 7 | Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given [here](https://www2.bc.edu/~reederma/Hansen.pdf). A sketch of the proof is as follows:
1. The author claims that $\mathrm... | https://mathoverflow.net/users/14510 | When is the Ad (Adjoint Representation) Morphism a Closed Map | Let $G$ be a connected Lie group. Equivalences:
* (i) every linear representation of $G$ has a closed image
* (ii) $\mathrm{Hom}(G,\mathbf{R})=0$
* (iii) $G/\overline{[G,G]}$ is compact.
Here homomorphisms are meant continuous, and reps are in $\mathbf{GL}\_n(\mathbf{R})$ for $n$ not fixed, or equivalently in $\mat... | 5 | https://mathoverflow.net/users/14094 | 182530 | 91,410 |
https://mathoverflow.net/questions/182533 | 10 | Suppose we have a game between two players in which they take alternating turns. The game can have finite length, length $\omega$ or any transfinite number of steps (however, I'm not concerning games which are continuous). Every game has a *winning condition*, which can be interpreted as the set of all game histories p... | https://mathoverflow.net/users/30186 | Explicit examples of undetermined games | Here is an amusing concrete non-determined game, under the assumption that the [dependent choice](http://en.wikipedia.org/wiki/Axiom_of_dependent_choice) principle fails.
Assume DC fails. This means that there is a set $X$ and a binary relation $R$ on $X$, such for every $x\in X$ there is $y$ with $x\mathrel{R} y$, ... | 10 | https://mathoverflow.net/users/1946 | 182541 | 91,413 |
https://mathoverflow.net/questions/182520 | 3 | Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
| https://mathoverflow.net/users/47958 | Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods | Two non-discrete group topologies which together generate the discrete topology are sometimes called *transversal*. It is proved in **"On transversal group topologies"** by Dikranjan, Tkachenko and Yaschenko (see Theorem 3.13) that no totally bounded group topology on an infinite abelian group admits a transversal grou... | 2 | https://mathoverflow.net/users/17836 | 182542 | 91,414 |
https://mathoverflow.net/questions/159684 | 10 | What is the easiest (and what is the most elementary) way of proving
Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce
it to existence of non-trivial harmonic functions (locally) on 2-dimensional
Riemannian manifolds, but this seems to be non-trivial, too. I want to have a construction using 1-d... | https://mathoverflow.net/users/3377 | Newlander-Nirenberg in dimension 2 | I like the discussion (only possible in this dimension) which uses (1) the fact that Calderón—Zygmund operators which are smoothing of order one transform bounded measurable functions into continuous ones with $e\log1/e$ modulus of continuity and (2) Osgood's elementary theorem that this modulus of continuity is good e... | 10 | https://mathoverflow.net/users/59043 | 182551 | 91,419 |
https://mathoverflow.net/questions/182481 | 10 | I am reading the paper *Ramsey-like cardinals II* by Victoria Gitman and Philip Welch (Journal of Symbolic Logic, vol. 76, no. 2. pp. 541-560, 2011) and maybe I am missing something.
According to the deffinitions provided there:
An $M\_0$-ultrafilter $U\_0$ is $0$-good if its ultrapower $M\_1$ is well-founded.
$U... | https://mathoverflow.net/users/41274 | On the definition of the $\alpha$-iterable cardinals | The objection is correct. The argument of Theorem 4.1 uses that $M\_\alpha$ is well-founded, which may not be the case if $\beta$ is a limit, according to the definition of $\alpha$-iterable cardinals given in the paper. There are two ways to address this. It does make sense to redefine the limit stages of the hierarch... | 6 | https://mathoverflow.net/users/5984 | 182555 | 91,420 |
https://mathoverflow.net/questions/182576 | 1 | Assume we are given a smooth manifold $M$ and let $TM^{\otimes r}, r>1,$ be some
tensor power of its tangent bundle.
Is there any general observation/result
saying when there exists a manifold $N$ and a smooth map $f: M \rightarrow N$ such that the pull back bundle $f^\*TN$ is isomorphic to
$TM^{\otimes r}$?
This ... | https://mathoverflow.net/users/25643 | Classifying tensor powers of tangent bundles as pullbacks of appropriate tangent bundles | the idea is the folk fact that the universal space and universal stable \*\* bundle can be taken to be a manifold and its tangent bundle [see proof below ]
take r=3 and let M have dim d
then N should have dimension 3d
any [stable] bundle [like the tensor power] over a d dim space [like M] can be induced by a map in... | 5 | https://mathoverflow.net/users/59043 | 182578 | 91,429 |
https://mathoverflow.net/questions/182563 | -3 | Two Questions:
(1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q\_1\in Q$, to the Tangent Space $T\_{q\_0}Q$, locally, be a contraction mapping?
Or more generally, in terms of a Lipschitz constant,
(2) Given a flow on TQ and the canonical projection $\pi\_Q$, if $\pi... | https://mathoverflow.net/users/59046 | The logarith map as a contraction | I interpret the question to be intuitively
the map of a small piece of a manifold back to its tangent distorts distances
expanding for positive curvature and contracting for negative curvature
and neither for zero curvature
so the answer intuitively sought is : for negatively curved spaces the inverse of the e... | 8 | https://mathoverflow.net/users/59043 | 182580 | 91,430 |
https://mathoverflow.net/questions/182574 | 2 | I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4.
Let's consider a secuence of probability measures $\delta(\hat{y},n)$ which concentrates on $\{\frac{0}{p^n}, \dots, \frac{p^n-1}{p^n}\}$ for each... | https://mathoverflow.net/users/59053 | Weak Convergence to Lebesgue Measure | I think Dan is saying for a given $y$, you have a sequence of measures $\delta(\hat y,n)$ supported on the unit interval. Each one of these is invariant (he claims) is under some group of translations: that is a subgroup of [0,1) considered as a group (with mod 1 addition). The finite subgroups of [0,1) are just the mu... | 1 | https://mathoverflow.net/users/11054 | 182582 | 91,431 |
https://mathoverflow.net/questions/182591 | 2 | For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J\_n(\alpha)=0 \textrm{ and }\alpha<Cn \}$$ where $C>1$ is fixed.
Has anyone come across any techniques or prior results well suited to this quest... | https://mathoverflow.net/users/15148 | The asymptotic distribution of a subset of Bessel function zeroes | Let $\alpha=j\_{n,m}$ be the positive zeros of $J\_n(\alpha)$, numbered so that $j\_{n,m+1} > j\_{n,m}$. Let $m\_{n,\alpha}$ denote the largest $m$ such that $j\_{n,m} < \alpha$. It seems to me that you want to asymptotically estimate the sum
$$N\_\lambda = \sum\_{n<\lambda/C} m\_{n,Cn}.$$
Since the numbers $m\_{n,Cn}$... | 2 | https://mathoverflow.net/users/2622 | 182594 | 91,437 |
https://mathoverflow.net/questions/182567 | 2 | Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?
By "connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we choose all the "1" sites and all the "0" sites are not chosen. Then
```
00000
00100
00100
00110
00000
```
is con... | https://mathoverflow.net/users/40780 | enumeration of connected blocks in finite size square | This answer has two main points:
1. If you find a good upper bound in the case $j=1$, you will get a reasonable bound for small $j$.
2. You cannot get polynomial bounds with respect to $n,m,k$ for any $j\geq1$. (The hope for polynomial bounds was expressed in a comment.)
Let $N(n,m,k;j)$ denote the number of choice... | 2 | https://mathoverflow.net/users/55893 | 182603 | 91,440 |
https://mathoverflow.net/questions/182539 | 3 | I'm looking for information about minimal directed cuts (dicuts) in (connected) DAGs (directed acyclic graphs).
A dicut in a directed graph, is a cut $(P\_1,P\_2)$ in which all edges in $E(P\_1,P\_2)$ are in the same direction between the parts of the cut. Minimal is in relation to inclusion of the edge set $E(P\_1,P... | https://mathoverflow.net/users/42567 | Directed Minimal Cuts in a DAG | Take two full binary trees, one directed towards the root and one away from the root. Identify the leaves of the two trees, so that the two roots become the source and sink of a DAG. If the number of vertices altogether is $n$, the number of minimal cuts is more than $2^{n/3}$. So you can't generate them all in polynom... | 4 | https://mathoverflow.net/users/9025 | 182613 | 91,443 |
https://mathoverflow.net/questions/182630 | 2 | I'm coming back to this [question](https://mathoverflow.net/questions/182621/the-space-of-sequences-of-rationals-and-its-dimension). Is it possible to have "an explicit" linearly independent family of sequences of rationals with a cardinal equal to the continuum?
PS: sorry for the duplicate on the previous question.
... | https://mathoverflow.net/users/41060 | Linearly independent family of sequences of rationals with a cardinal equal to the continuum | Choose a bijection $\alpha:\mathbb{N}\to\mathbb{Q}$, and for each $x\in\mathbb{R}$ let
$$a(x)\_i=\begin{cases}0&\mbox{ if $\alpha(i)<x$}\\1&\mbox{ if $\alpha(i)\geq x$}\end{cases}.$$
Then the set of sequences $\{a(x):x\in\mathbb{R}\}$ is linearly independent.
| 5 | https://mathoverflow.net/users/22989 | 182634 | 91,451 |
https://mathoverflow.net/questions/182638 | 1 | Consider $P^{2}(\mathbb{C})$, the space of all lines through the origin in $\mathbb{C}^{3}$ (or $\mathbb{R}^3$ if that works better). Let $X\subset P^{2}(\mathbb{C})$ be a (nonempty) hypersurface (algebraic in some coordinates). Must it be the case that for some $2$-dimensional subspace $V$ of $\mathbb{C}^3$, $P^1(V)\s... | https://mathoverflow.net/users/16107 | Does every hypersurface in the projective plane contain a projective line? | Q1: No. Indeed it is easy to see that each two-dimensional subspace is defined by some linear equation, so any homogeneous polynomial which has no linear factors provides a counterexample, e.g. $x^2+y^2+z^2$.
Q2: No. You can easily count the dimension of the space of all hypersurfaces and the dimension of the space o... | 6 | https://mathoverflow.net/users/18060 | 182639 | 91,455 |
https://mathoverflow.net/questions/182092 | 5 | **Context:** Let $B\_n$ be the space of symmetric bilinear forms on $\mathbb{R}^n$ and $L\_n\subset B\_n$ be the subset of non-degenerate forms of Lorentzian signature $(-,+,\ldots,+)$. Let $T$ be a finite dimensional real vector space. Both $B\_n$ and $T$ carry linear representations of $GL^+(n,\mathbb{R})$, where $+$... | https://mathoverflow.net/users/2622 | Smooth and $GL(n)$-equivariant implies algebraic? | If I understand you correctly, the answer is 'no'.
Because the open set $L\_n\subset B\_n$ is an orbit of $\mathrm{GL}^+(n,\mathbb{R})$ under the natural representation of $\mathrm{GL}^+(n,\mathbb{R})$ on $B\_n$, it follows that, if $\rho:\mathrm{GL}^+(n,\mathbb{R})\to \mathrm{GL}(T)$ is the representation that defi... | 5 | https://mathoverflow.net/users/13972 | 182644 | 91,456 |
https://mathoverflow.net/questions/182641 | 17 | What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces?
(Possibly introducing stacks also)?
I'm looking for something which really gets the pictures across and helps build my beginners intuition. However it should be something that is not a totally trivial read either... | https://mathoverflow.net/users/36886 | Soft question: beginners reference to moduli spaces | Here are some fairly recent and general references I like:
1. The Handbook of Moduli <http://intlpress.com/site/pub/pages/books/items/00000399/index.html>
The article on logarithmic geometry by Abramovich et al. is great!
2. The Isaac Newton Institute had a wonderful school on moduli spaces. You can find the vid... | 10 | https://mathoverflow.net/users/3847 | 182645 | 91,457 |
https://mathoverflow.net/questions/182643 | 3 | Suppose there exists a subset of $\Bbb R$ which has cardinality $\omega\_1$. Is it then necessarilly true that for every collection of $\omega\_1$ subsets of $\Bbb R$ there exists a choice function?
I suppose that the answer for the above question is no, just like existence of countable subsets of $\Bbb R$ doesn't im... | https://mathoverflow.net/users/30186 | Does existence of $\omega_1$ subset of reals imply $\omega_1$ choice for subsets of reals? | No, since there is a surjection from $\Bbb R$ onto $\omega\_1$, there is always an injection from $\omega\_1$ into $\mathcal P(\Bbb R)$; but it's not difficult to arrange that there is no choice function for some sequence of subsets like that.
To see that the range of the injection need not have a choice function, si... | 2 | https://mathoverflow.net/users/7206 | 182649 | 91,461 |
https://mathoverflow.net/questions/180666 | 14 | As the title suggests, I was wondering if anyone can point me to any examples in the literature to flag complexes that are shellable but not vertex decomposable.
It is well-known that if a simplicial complex $\Delta$ is vertex decomposable, then $\Delta$ is also shellable. There are examples where the converse fails... | https://mathoverflow.net/users/58087 | Flag complexes that are shellable but not vertex decomposable | I'm very slow to respond, but have finally found some time to put your complex into GAP and examine it.
**Questions on $k$-decomposability on flag complexes**
There were two questions about $k$-decomposability and flag complexes that seemed natural, and that I didn't know the answer to.
1. Is every flag complex t... | 5 | https://mathoverflow.net/users/19729 | 182653 | 91,463 |
https://mathoverflow.net/questions/182642 | 2 | In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity.
What's the current status in mathematics/computer science of this infinite combinatorial optimization problem?
Given ... | https://mathoverflow.net/users/40780 | Combinatorial optimization problem involving infinite spin system | There is one framework in the literature that maybe useful in your problem, depending on the larger context of the question.
In the **supernumber** framework your question (1) is actually a lower dimensional problem than typically encountered in quantum field theories, the usual places you find supernumbers.
( The... | 1 | https://mathoverflow.net/users/58777 | 182658 | 91,465 |
https://mathoverflow.net/questions/182640 | 5 | suppose $T$ is a complete stats for a parameter $\theta$.
Is any function $f(T)$ again complete?
It sounds weird but the definition seems to confirm that $f(T)$ is indeed complete..
| https://mathoverflow.net/users/59120 | Is a function of complete statistics again complete? | Geometrically, completeness means something like this: if a vector $g(T)$ is orthogonal to the p.d.f. $f\_\theta$ of $T$ for each $\theta$,
$$\mathbb E\_\theta g(T) = \langle g(T),f\_\theta\rangle=0$$
then $g(T)=0$ i.e., the functions $f\_\theta$ for varying $\theta$ span the whole space of functions of $T$. So in a wa... | 3 | https://mathoverflow.net/users/4600 | 182661 | 91,466 |
https://mathoverflow.net/questions/182618 | 4 | Let $G$ be a Lie group, $N$ a closed connected normal subgroup. Let $BG$, $BN$, $B(G/N)$ be the classifying spaces of $G,N$ and $G/N$. Is there a fibration $BN\to BG\to B(G/N)$ ?
It seems that such a construction is used in Atiyah, Bott: *Yang-Mills equations on Riemann surfaces*, formula (9.2), but I can't see how i... | https://mathoverflow.net/users/37354 | A fibration of classifying spaces | This is an edited extract from a book in preparation (Bruner, Catanzaro, May)
tentatively titled Characteristic Classes and is therefore overlong for an
answer. This is similar to Denis Nardin's answer, but more bundle theoretic; he did refer to an old Memoir of mine, so I thought I'd give an answer. Let $N$ be a close... | 10 | https://mathoverflow.net/users/14447 | 182663 | 91,467 |
https://mathoverflow.net/questions/182664 | 5 | The following result seems to be frequently quoted:
Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). Then the fiberations on $B$ with fiber $K(\pi,n)$ are classified by the space $K(\pi, n+1)$. In other words, the pull-ba... | https://mathoverflow.net/users/37354 | Classifying space for fibrations with Eilenberg-MacLane space as fibers | There is a very careful analysis of this question in Lemma 3.4.2, page 57, of More Concise Algebraic Topology, by Kate Ponto and myself. Assuming that $E$ and $B$ are connected, a fibration $E\longrightarrow B$ with fiber $K(A,n)$ for an abelian group $A$ is a pullback of of the path space fibration over $K(A,n+1)$ if ... | 9 | https://mathoverflow.net/users/14447 | 182665 | 91,468 |
https://mathoverflow.net/questions/182625 | 3 | Is there any literature corresponding to one or two-parameter semigroups such that e.g. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$?
I have only seen the case where $X(t) \equiv X$ (i.e. there is only one Banach space).
This may be useful for PDE proble... | https://mathoverflow.net/users/58029 | $C_0$ semigroups on parameterized Banach spaces or moving domains | You will not find many things about such operator families in the literature, but what there is is for the non-autonomous case. See for example
* M. Geissert & A. Lunardi [Invariant Measures and Maximal L2
Regularity for Nonautonomous Ornstein-Uhlenbeck Equations.](http://www.math.unipr.it/~lunardi/Files/glOUrevised.... | 2 | https://mathoverflow.net/users/12898 | 182678 | 91,473 |
https://mathoverflow.net/questions/182685 | 4 | If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult\_U^\alpha(V)\}\_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding $j:L\longrightarrow L$, we have a (well-founded) ultrapower $Ult\_U(L)$ for a weakly amenable $L$-ultrafilter $U$, but is i... | https://mathoverflow.net/users/41274 | Iterated ultrapowers of L | By Kanamori, ``The higher infinite'', Theorem 21.1 (due to Kunen), the existence of $0^\sharp$ is equivalent to the existence of an iterable $L-$ultrafilter.
Also note that if $j: L \to L$ is a non-trivial elementary embedding, then $crit(j)\in I,$ and $j\restriction I: I \to I,$ and hence $j\restriction I$ induces $... | 7 | https://mathoverflow.net/users/11115 | 182686 | 91,476 |
https://mathoverflow.net/questions/182690 | 2 | Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T\_1,T\_2,\ldots,T\_n]}{(f\_1,\ldots,f\_n)}$ where $f\_i$ has coefficients in $\overline{\mathbb Q}$ for every $i=1,\ldots,n$. If $S'$ is a minimal model of $S$, then can w... | https://mathoverflow.net/users/47136 | Minimal model of a non-singular complex projective surface defined over $\overline{\mathbb Q}$ | I think so, yes.
1) If $V\_{/\overline{\mathbb{Q}}}$ be a smooth projective variety. Then on Neron-Severi groups we have $NS(V) = NS(V\_{/\mathbb{C}})$. I believe this follows from the existence of the Picard scheme: the component group of a group scheme does not change under extension from one algebraically closed ... | 4 | https://mathoverflow.net/users/1149 | 182696 | 91,479 |
https://mathoverflow.net/questions/182684 | 1 | Let $u:[0,2\pi)\to \mathbb{R}$ be the step function
$$u(x) = \begin{cases}
1 & \text{if } x \in [0,\pi), \\
0 & \text{if } x \in [\pi,2\pi)
\end{cases}$$
By a direct computation, one discovers that
\begin{equation}\tag{$\*$}
\|u-S\_N u\|\_{L^2} \leq C N^{-1/2},
\end{equation}
where $S\_N u $ is the truncated Fourier s... | https://mathoverflow.net/users/41123 | Fourier approximation error in L^2 for piecewise continuous functions | Such an assertion is close enough to being an "exercise" that there may not be a really clear "reference" for it... but such a result can be explained easily and shortly enough, I think:
Finite sums of (dilates of translates of) derivatives of $\cos x/2$ can be subtracted from a given (finitely-) piecewise smooth per... | 2 | https://mathoverflow.net/users/15629 | 182697 | 91,480 |
https://mathoverflow.net/questions/182692 | 1 | When does an orthomodular lattice $L$ of projections onto a given Hilbert space have a non-trivial centre $Z(L)$ and what can we generally say about the cardinality of $Z(L)$?
| https://mathoverflow.net/users/45570 | When does an orthomodular projection lattice have a non-trivial centre? | $ Z (L) $ contains the subspaces that are orthogonal or comparable to all the other subspaces in $ L $. So if the Hilbert space has finite dimension $ d $ you can get $2^d $ many elements in $ Z (L) $ with $ L $ of size $2^d $ also. For instance take $ L $ to consist of the coordinate planes in $ R^3$.
| 1 | https://mathoverflow.net/users/4600 | 182700 | 91,481 |
https://mathoverflow.net/questions/182673 | 4 | Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma\_1$ and $\gamma\_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic metric in $F$, there is a unique geodesic in each free homotopy class of closed curve. Get $\tilde\gamma$ denotes the geodesic... | https://mathoverflow.net/users/9485 | Angle between geodesics in hyperbolic surface | The answer in general is no, because there are simple counterexamples based on the Gauss-Bonnet theorem. On the other hand there is a sufficient condition which is of an easily checked combinatorial nature, based on W. Thurston's theory of train tracks.
We might as well assume that $\gamma\_1,\gamma\_2$ have minimal ... | 6 | https://mathoverflow.net/users/20787 | 182709 | 91,486 |
https://mathoverflow.net/questions/153884 | 12 | Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product
$$ p=b\_1 b\_2 \cdots b\_n$$
where each $b\_i\in A$.
Clearly $n-1$ multiplications suffice to compute $p$; can we do it with fewer?
Let $m=m(b\_1,...,b\_n)$ be the minimum number of multiplications required to compute $p$. M... | https://mathoverflow.net/users/8938 | Faster multiplication with a restricted set of multiplicands? | This question has already been studied in the literature under the term "word chains". Doing a search on that term will turn up some relevant papers, such as
<http://dl.acm.org/citation.cfm?id=33329>
<http://www.sciencedirect.com/science/article/pii/0020019089901671>
<http://dl.acm.org/citation.cfm?id=79917>
B... | 3 | https://mathoverflow.net/users/44797 | 182715 | 91,487 |
https://mathoverflow.net/questions/182719 | 20 | Let $(X,d)$ be a metric space and $x,y \in X$. Assume that for all $r > 0$ the balls $B\_r(x)$ and $B\_r(y)$ are isometric.
Is it true that there exists an isometry of $X$ sending $x$ to $y$?
| https://mathoverflow.net/users/29319 | If all balls at $x$ and $y$ are isometric is there an isometry sending $x$ to $y$? | No. Let $x$ and $y$ be connected by an edge and let's use the graph distance as our metric.
At $x$, connect paths of length $n$ for each $n\in\mathbb N$. At $y$, do the same, but also connect an infinite path.
| 36 | https://mathoverflow.net/users/4600 | 182722 | 91,489 |
https://mathoverflow.net/questions/182714 | 0 | I want to find longest sequence of distinct squares that $\alpha\_{\_1}$ + ... + $\alpha\_{\_n}$ is given number.
In particular I want to find largest square in that sequence.
I've tried use Lagrange's four-square theorem, and then Fermat's theorem on biggest squares, but it it's not correct solution.
Please help m... | https://mathoverflow.net/users/59153 | Longest sequence of sum of distinct squares | I'll assume you're talking about squares of positive integers (of course you can always include $0$ if you prefer, increasing the length by $1$).
Let $F(n)$ be the
length of the longest sequence representing $n$, or $-\infty$ if there is no such sequence. Finding $F(n)$ can be viewed as a
type of knapsack problem: i... | 3 | https://mathoverflow.net/users/13650 | 182728 | 91,492 |
https://mathoverflow.net/questions/182724 | 1 | I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y\_{1},y\_{2},y\_{3}\in\mathbb{R}$ and the following function
$$f\left(y\_{1},y\_{2},y\_{3}\right)=y\_{1}\mbox{artanh}(y\_{1}-y\_{3})+y\_{2}\mbox{artanh}(y\_{2}-y\_{1})+y\_{3}\mbox{artanh}(y\_{3}-y\_{2})$$
whe... | https://mathoverflow.net/users/15667 | investigating positivity/negativity of a function | This function can be either positive or negative, even if
$y\_1,y\_2,y\_3$ are close enough that the hyperbolic tangents are real
(see **Robert Israel**'s comment). Fix $\eta\_1,\eta\_2,\eta\_3$ such that
$$
\phi(\eta\_1,\eta\_2,\eta\_3) :=
{\mathop{\rm arctanh}} (\eta\_1 - \eta\_3)
+ {\mathop{\rm arctanh}} (\eta\_2 -... | 3 | https://mathoverflow.net/users/14830 | 182730 | 91,493 |
https://mathoverflow.net/questions/182705 | 7 | Let $Li\_s(z)$ denote the usual polylogarithm. The elementary functional equation $$Li\_{-n}(z)=(-1)^{n-1}Li\_{-n}(1/z)$$ holds for $n\geq 1$. I remember only that the proof used some reproducing property of the Stirling numbers of the second kind.
This functional equation is rather useful because, taking linear com... | https://mathoverflow.net/users/10980 | How to prove an elementary functional equation for polylogarithms? | Let's try to understand in which sense this equality
$$\sum\_{k=-\infty}^{\infty}k^nz^k=\left(z\frac{d}{dz}\right)^n\sum\_{k=-\infty}^{\infty}z^k=\left(z\frac{d}{dz}\right)^n 0=0$$
may be understood. Of course, there is no convergence in usual sense on the unit circle (because terms do not tend to 0). Actually, we do ... | 13 | https://mathoverflow.net/users/4312 | 182735 | 91,495 |
https://mathoverflow.net/questions/182354 | 1 | Can someone suggest a way to calculate the maximum with respect to $x \ge 1$ of:
$$f(x)=\frac{1}{x!} \frac{1}{1-c^{1/\binom{x+n-1}{n-1}}}.$$
The constants $c$ and $n$ are parameters such that $c \in (0,1)$ and $n \in \mathbb{N}$.
If it helps, we can assume $x \in \mathbb{N}$ but it is not necessary.
I've tried looki... | https://mathoverflow.net/users/41258 | maximizing a function involving factorial | In a quite large range of the parameters we can approximate the fraction by a Taylor series to obtain
$$
f(x) = \frac{1}{x!}\frac{1}{\frac{-\log c}{\binom{x+n-1}{n-1}} + \mathcal{O}\left(\frac{\log^2 c}{\binom{x+n-1}{n-1}^2}\right)}
= \left(1+ \mathcal{O}\left(\frac{\log^2 c}{\binom{x+n-1}{n-1}^2}\right)\right)\frac{\... | 4 | https://mathoverflow.net/users/37555 | 182739 | 91,497 |
https://mathoverflow.net/questions/182506 | 1 | Let $\eta$ be a closed convex subset of ${\mathbb R}^n$ of convex dimension $n$ (not necessarily compact) with nonempty boundary $\partial \eta$. Then $\eta$ is an $n$-dimensional topological manifold with boundary and therefore $\partial \eta$ of $\eta$ is an $(n-1)$-dimensional submanifold of ${\mathbb R}^n$. Suppose... | https://mathoverflow.net/users/17218 | Zariski closure of the boundary of a closed convex subset of ${\mathbb R}^n$ | The answer to the first question is indeed yes by standard results in semi-algebraic geometry. A reference is Lemma 2.4 and Proposition 2.7 in
<http://arxiv.org/pdf/1405.7822.pdf>
The answer to the second question might also be yes. Does it not follow from the fact that every semi-algebraic set can be triangulated? A... | 1 | https://mathoverflow.net/users/21778 | 182748 | 91,501 |
https://mathoverflow.net/questions/182752 | 42 | Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that:
$$\int\_\mathbb{R} \rho(x)\, dx = 1$$
and
$$\int\_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$
Fact: the density function which maximizes the entropy functional
$$S(\rho) = -\int\_\mathbb{R} \rho(x) \log \rho(x)\, d... | https://mathoverflow.net/users/4362 | Central limit theorem via maximal entropy | There's a 1985 article by Derriennic called "Entropie, theoremes limite et marches aleatoires" (entropy, limit theorems and random walks). In it there is a section where the connection between your observation that the Gaussian maximizes entropy (which is attributed to Shannon) and the central limit theorem is discusse... | 18 | https://mathoverflow.net/users/7631 | 182755 | 91,503 |
https://mathoverflow.net/questions/182764 | 7 | Suppose that $C$ is a bounded convex subset of $\mathbb{R}^n$ such that the optimum value, over $C$, of any linear functional is attained at some point of $C$. Does this imply that $C$ is closed? If so, is there a simple proof? If not, a counterexample would be nice.
| https://mathoverflow.net/users/59182 | Characterization of closed convex set | No, as a counterexample take a closed "stadium" in $\mathbf{R}^2$ (the convex hull of two half-circles, I hope the word stadium makes it clear) and remove from it one of the endpoints of the half-circles. This is somehow related to the difference between extreme points and exposed points.
| 14 | https://mathoverflow.net/users/908 | 182767 | 91,506 |
https://mathoverflow.net/questions/182777 | 0 | Let $G$ be a graph and $M\subseteq V(G)$ be a maximal independent set. Is there a coloring $c:V(G)\to\chi(G)$ such that $c$ is constant on $M$?
(The answer is positive for graphs with infinite chromatic number: Given any coloring $c:V(G)\to\chi(G)$ define a new coloring $c'$ by $c(m) = 0$ for $m\in M$ and $c'(x) = c(... | https://mathoverflow.net/users/8628 | Coloring maximal independent sets with 1 color | Pick some bipartite graph A,B and add a huge new independent set of vertices connecting half of them with all of A and half of them with all of B. The resulting graph is bipartite, but there is no 2-coloring of it which is constant on the new set.
| 1 | https://mathoverflow.net/users/38889 | 182780 | 91,511 |
https://mathoverflow.net/questions/182766 | 1 | Consider an antilinear involution, that is an antilinear map on a complex vector space, whose matrix $M$ obeys $MM^\*=1$ where the star denotes complex conjugation. Can we find a change of basis whose matrix $\Lambda$ would be such that $\Lambda^\* M \Lambda^{-1} = 1$?
By taking the real components of $M$ and $\Lamb... | https://mathoverflow.net/users/12873 | Can any antilinear involution be trivialized by a change of basis? | If you let $I$ denote multiplication by $\sqrt{-1}$, then the two operators $I$ and $M$ on your vector space (say, $V$) satisfy
$$
I^2 = -1,\qquad M^2 = 1,\qquad\text{and}\qquad IM=-MI.
$$
(The former since $M$ is an involution; the latter follows since $M$ is anti-linear.) The operators $1,I,M, IM$ span an algebra iso... | 4 | https://mathoverflow.net/users/13972 | 182783 | 91,513 |
https://mathoverflow.net/questions/182793 | 1 | Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of 10000 to 20000. Also all entries of the vectors are real numbers between $0$ and $1$, if that helps.
We now consider the... | https://mathoverflow.net/users/41452 | Can we implicitly fit a system of linear ODEs by reduced information? | Let me expand my comments into an answer.
The essential conclusion is that you cannot recover the matrix $A$ and the vector $b$ from your data, no matter how large you take $r$.
Finding $A$ and $b$ *might* be possible if the amount of time measurements is at least $n$; three is not enough.
Increase the amount of time m... | 1 | https://mathoverflow.net/users/55893 | 182805 | 91,522 |
https://mathoverflow.net/questions/182757 | 8 | Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by
$$ f(t) = f\_0 + tuN\_0, $$
where $f\_0$ is some initial immersion, $N\_0$ is the associated Gauss map, and $u: M \to \mathbb{R}$ is an arbitrary smooth function. One can easily work out, e.g., the first variation in the ... | https://mathoverflow.net/users/59176 | Variation of curvature with respect to immersion? | If you just calculate using the moving frame, you'll get the answer for the variation of the principal curvatures in a few lines:
$$
\delta\kappa\_i = \mathrm{Hess}(u)(e\_i,e\_i) + \kappa\_i^2\,u .
$$
Here, $\delta\kappa\_i$ is the first $t$-derivative of $\kappa\_i$ at $t=0$ (i.e., the 'first variation of $\kappa\_i$'... | 17 | https://mathoverflow.net/users/13972 | 182807 | 91,524 |
https://mathoverflow.net/questions/182815 | 8 | Maybe I'm just being a bit dense here, but this has me stumped right now.
A fairly well-know thm is the following: Let $X\_0$ be a compact metric space and $f:X\_0\to X\_0$ be continuous. For each $n\ge1$, let $X\_n=f(X\_{n-1})$, i.e., the range of the nth "iterate" of $f$. Since $X\_0,X\_1,\dots$ is a decreasing seq... | https://mathoverflow.net/users/10031 | Must a map on a compact space be surjective on $\cap_{n=1}^\infty f^n(X)$? | Don't pick a convergent subsequence: pick a point in $ \cap\_{n \in \mathbb{N}} \overline{\{ x\_n,x\_{n+1}, \dots \}}$ (the overline meaning closure) which is non empty as an intersection of decreasing family non empty compact set, and your argument will work.
| 8 | https://mathoverflow.net/users/22131 | 182816 | 91,528 |
https://mathoverflow.net/questions/182550 | 8 | The [Volume conjecture](http://en.wikipedia.org/wiki/Volume_conjecture) says that if $J\_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim\_{N \to \infty} \frac{ 2 \pi}N \left\vert J\_N(e^{2\pi i / N})\right\vert = vol(K)$$
Here we are assuming that the complement of $K$ is hyperbolic, and... | https://mathoverflow.net/users/2669 | Does the limit in the Volume conjecture converge? | No, this is unknown. There are heuristic arguments for convergence based on the stationary phase approximation, but as far as I know, no one has made the argument precise in general. The closest I know of to a proof of convergence is an upper bound on the limsup in terms of the crossing number given in Theorem 1.3 of [... | 5 | https://mathoverflow.net/users/1345 | 182824 | 91,531 |
https://mathoverflow.net/questions/180673 | 7 | Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:
>
> In many cases, the category $\cal{M}od\_A\left(\cal{C}\right)$ of $A$-modules in
> $\cal{C}$ inherits the structure of a symmetric m... | https://mathoverflow.net/users/40779 | Category of modules over commutative monoid in symmetric monoidal category | One needs that $C$ is cocomplete and that $\otimes$ preserves colimits in each variable (and then $\mathrm{Mod}\_C(A)$ will have the corresponding properties). More precisely, you only need that $C$ has reflexive coequalizers and that $\otimes$ preserves them in each variable. This has been known for decades, but the f... | 6 | https://mathoverflow.net/users/2841 | 182826 | 91,532 |
https://mathoverflow.net/questions/182823 | 43 | **Dold-Thom Theorem:** $$\pi\_i(SP(X))\cong\tilde{H}\_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a homology theory. I've also seen many uses of the theorem, to explain features about Eilenberg-MacLane spaces and ... | https://mathoverflow.net/users/12310 | Why the Dold-Thom theorem? | Imposing some reasonable conditions on our spaces (I think semilocally-simply-connected ought to do), one works through
**Exercise 1** $\mathbb{Z}[X]$, the free topological $\mathbb{Z}$-module continuously generated by a convenient space $X$ is an $E^\infty$ space; the maps $\mathbb{Z}[X] \to \mathbb{Z}[Y]$ induced b... | 9 | https://mathoverflow.net/users/35529 | 182832 | 91,535 |
https://mathoverflow.net/questions/182797 | 5 | Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. My question is: Is there a generalization of this formula for forests?
Let $f\_{n,k}$ denote the number of forests with $k$ connected components on $n$ vertices. For example, $f\_{n,1} = n^{n-2}$ by Cayley's formula and $f\_{n,n-1}... | https://mathoverflow.net/users/17599 | Is there a formula for the number of labeled forests with $k$ components on $n$ vertices? | A formula as a single sum is
$$f\_{n,k} = \binom nk \sum\_{i=0}^k \left(-\frac12\right)^i (k+i)\,i!\, \binom{k}{i}\binom{n-k}{i} n^{n-k-i-1}.$$
This formula can be found in J. W. Moon's *Counting Labelled Trees*, Theorem 4.1. He attributes it to A. Rényi, *Some remarks on the theory of trees*, Publications of the Math... | 6 | https://mathoverflow.net/users/10744 | 182836 | 91,538 |
https://mathoverflow.net/questions/180867 | 1 | I am considering the following correspondence:
Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto f^\*:Qcoh(V)\rightarrow Qcoh(U)\end{equation},I denoted this assignment by $Qcoh()$
Another direction: \begin{equation}... | https://mathoverflow.net/users/41650 | Are these two "FUNCTORS" adjoint? | Unfortunately, the question, as stated, doesn't make much sense. The Rosenberg spectrum (see [here](http://arxiv.org/abs/1310.5978) for a survey) is
* only defined for (Grothendieck) abelian categories,
* just a ringed space and not always a scheme,
* only functorial with respect to equivalences of categories (as far... | 6 | https://mathoverflow.net/users/2841 | 182839 | 91,541 |
https://mathoverflow.net/questions/182833 | 2 | Assume that $D$ is the unit ball in $\mathbb{R}^n$ and let $f\in C\_0^\infty(D)$. Let $a>0$ and let $F$ be the Fourier transform. Define $$g(x)= F^{-1} (|x|^{-2a} \cdot (F f)(x)).$$ My question is for which parameters $a$ we have $$g\in L^2(\mathbb{R}^n).$$
| https://mathoverflow.net/users/58947 | Fourier transform (their inverse) | $F(f)$ is analytic and tends to $0$ faster than any power because $f$ is $C\_0^\infty$.
$F$ and $F^{-1}$ map $L^2$ onto itself. So the question is when $|x|^{-2a}$ belongs
to $L^2$ locally (near the origin). Normally this happens when $a<n/4$.
This is necessary and sufficient unless $Ff(0)$ accidentally equals to $0$.
... | 2 | https://mathoverflow.net/users/25510 | 182840 | 91,542 |
https://mathoverflow.net/questions/182847 | 1 | Let the topological space X be the so-called "long line" — which is an uncountable linearly ordered set containing all the countable ordinal numbers as well as a copy of the open unit interval (0,1) between each countable ordinal number and its successor.
The topology of X is its order topology.
Let the topological spa... | https://mathoverflow.net/users/4423 | Questions about the dimension-and other properties-of a non-separable topological space | 1. Dimension is a local property (there is a possibility that I am mistaken, please confirm) and the long line is locally homeomorphic to the real line, so $X$ has dimension 1. By the same argument $Z$ has dimension 3.
2. Yes, and the open sets can even have pairwise disjoint closures. (More precisely, the closure of e... | 3 | https://mathoverflow.net/users/55893 | 182850 | 91,545 |
https://mathoverflow.net/questions/182852 | 6 | The classical Hausdorff-Young inequality states that
$$
\Vert \widehat{f} \Vert\_{p'} \leq \Vert f \Vert\_p \text{ for } 1 \leq p \leq 2.
$$
For $p=2$, we even have equality due to Plancherel.
If we additionally assume that $f \geq 0$, we also get
$$
\Vert \widehat{f} \Vert\_\infty = \widehat{f}(0) = \int f(x) ... | https://mathoverflow.net/users/59219 | Reverse Hausdorff Young for nonnegative functions | If $1<p<2$, then it is not possible to have the inequality
$$
\|f\|\_p \lesssim \|\widehat{f}\|\_{p'} \quad\quad\quad\quad\quad (1)
$$
for all $f\ge 0$. This follows from the existence of (positive) purely singular measures $\mu$ with $\widehat{\mu}\in L^{p'}$ (in fact, $\widehat{\mu}$ can have power decay). (I used th... | 2 | https://mathoverflow.net/users/48839 | 182856 | 91,547 |
https://mathoverflow.net/questions/182843 | 1 | Suppose I have a coordinate system $t\_1,\ldots t\_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t\_1 + t\_2)=t\_2$.
Given a finite dimensional vector space V, define LT(V):=span$\_\mathbb{C}\{LT(w)|w\in V\}$.
I want to find a basi... | https://mathoverflow.net/users/33518 | Is there a Gröbner basis analogue that exists for vector spaces? | Darij is right. Computing Groebner basis for a linear ideal is the same as computing the row echelon form (this can be seen from the Buchberger algorithm). And I think he meant "nonlinear" instead of "noncommutative".
| 6 | https://mathoverflow.net/users/57615 | 182867 | 91,553 |
https://mathoverflow.net/questions/182476 | 8 | In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, a\_1 \otimes \dotsc \otimes a\_n \mapsto \sum\_{\sigma \in \Sigma\_n} \mathrm{sgn}(\sigma) \cdot a\_{\sigma(1)} \otimes ... | https://mathoverflow.net/users/2841 | Deligne's exterior power | It is not. Here is a counterexample. (I have essentially copied the setting from <https://mathoverflow.net/a/87958/2530> , which in turn goes back to a PBW counterexample by P. M. Cohn.)
Let $k$ be the commutative ring $\mathbb F\_2 \left[\alpha,\beta,\gamma\right] / \left(\alpha^2,\beta^2,\gamma^2\right)$.
Let $M$... | 3 | https://mathoverflow.net/users/2530 | 182877 | 91,557 |
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