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https://mathoverflow.net/questions/257982 | 5 | While working on some research, I have encountered an infinite series and its improper integral analogue:
\begin{align}\sum\_{m=1}^{\infty}\frac1{\sqrt{m(m+1)(m+2)+\sqrt{m^3(m+2)^3}}}&=\frac12+\frac1{\sqrt{2}}, \\
\int\_0^{\infty}\frac{dx}{\sqrt{x(x+1)(x+2)+\sqrt{x^3(x+2)^3}}}&=2.\end{align}
The evaluations were gues... | https://mathoverflow.net/users/66131 | sum, integral of certain functions | For the integral, notice that the expression under the square root is
$$ x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)} = \frac12\,x(x+2)(\sqrt x+\sqrt{x+2})^2. $$
Consequently,
\begin{align\*}
\frac1{\sqrt{x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)}}}
&= \frac{\sqrt 2}{(\sqrt x+\sqrt{x+2}) \sqrt{x(x+2)}} \\
&= \frac1{\sqrt 2}\,\frac{\sqrt... | 17 | https://mathoverflow.net/users/9924 | 257983 | 116,463 |
https://mathoverflow.net/questions/257988 | 14 | Are the elements of the set $\{\zeta(2n+1)| n\in \mathbb{N}\}$ $\mathbb{Q}$-linearly independent?
| https://mathoverflow.net/users/97275 | values of $\zeta$ function are linearly independent? | People have been exerting steady effort to prove/disprove irrationality of the Riemann zeta values in your list. Of course, $\zeta(3)$ is known to be irrational due to Roger Apery. Such investigations, among others, motivated the question of linear independence. As Felipe commented, however, not much is known, apart fr... | 20 | https://mathoverflow.net/users/66131 | 257991 | 116,466 |
https://mathoverflow.net/questions/257997 | 1 | I'm trying to prove that the series below converges to 1 and I noticed it looked strikingly similar to a probability distribution I once saw. My question is twofold:
1. Can anyone identify the distribution? I can't seem to, for the life of me, remember. I'm 90% sure this is a probability distribution (or some form of... | https://mathoverflow.net/users/97808 | Proof of Convergence + Identifying Probability Distribution | 1) Presumably you're talking about the Poisson distribution, but I don't think that's useful.
2) Possibly more useful is that this is related to the Taylor series for the Lambert W function.
| 2 | https://mathoverflow.net/users/13650 | 257999 | 116,468 |
https://mathoverflow.net/questions/257998 | 4 | I am trying to explain the differences between the following concepts to someone and I realized I myself am super confused:
Continuous/discrete Markov Process
Continuous/Discrete Markov chains
Markov property : $P\{X\_n=i|X\_{n-1}=j,X\_{n-2}=k,...\}=P\{X\_n=i|X\_{n-1}=j\}$ ?
I used to think: Every process that ... | https://mathoverflow.net/users/102504 | Markov Process, Markov Chain | Before adding the Markov property, I think it helps to distinguish between [discrete time](https://en.wikipedia.org/wiki/Discrete-time_stochastic_process) and [continuous time](https://en.wikipedia.org/wiki/Continuous-time_stochastic_process) stochastic processes with discrete or continuous state spaces. (This classifi... | 2 | https://mathoverflow.net/users/64449 | 258005 | 116,472 |
https://mathoverflow.net/questions/258011 | 4 | **Question**
At around minute 55 in [this lecture](http://www.youtube.com/watch?v=qmCh9KwrQq8/ "this lecture") by Andre Joyal, he claims that the category $Fam(Set\_{\*})$ of families of pointed sets is equivalent to the category $[I\_{\sharp},Set]$ of copresheaves on the walking-arrow-equipped with a section $I\_{\s... | https://mathoverflow.net/users/84563 | Families of Pointed Sets | If $\{X\_i\}\_{i \in I}$ is a family of pointed sets, then there is a function of (ordinary) sets $X = \sum\_{i \in I} X\_i \to I$ which maps every $x \in X\_i$ to $i$, and this has a ready-made section $I \to X$ which sends $i \in I$ to the basepoint of the pointed set $X\_i$. This describes the object part of the fun... | 8 | https://mathoverflow.net/users/2926 | 258013 | 116,474 |
https://mathoverflow.net/questions/257973 | 8 | I am looking for a reference to the following fact which seems to be true and which is probably well-known (at least to experts in resolution of singularities):
Consider an isolated cyclic quotient singularity ${\mathbb C}^n/G$, where $G$ is a finite cyclic group acting on ${\mathbb C}^n$ by linear complex transforma... | https://mathoverflow.net/users/102829 | Resolution of an isolated cyclic quotient singularity | **Edit.** As Michael Entov points out, the original example (below with strikethrough) is incorrect. (Sorry!) Here is a corrected example. It is not always possible to find an equivariant resolution satisfying all of the hypotheses you list. The simplest example I know of is as follows. Let $G$ be $\mu\_5$, the group o... | 5 | https://mathoverflow.net/users/13265 | 258018 | 116,476 |
https://mathoverflow.net/questions/257879 | 2 |
>
> **Question.** Is there a more general set of equations that satisfy mean value properties, similar to the Laplacian and heat equation?
>
>
>
For example, finding some kernel $K(x,y)$ and a set $B(x,r)$ such that
$$u(x)=\int\_{B(x,r)} u(y)K(x,y)dy.$$
Even existence of such a $K$ and a $B$ is good enough ... | https://mathoverflow.net/users/99863 | reference: modified mean value property for more general pdes | It generalizes for many elliptic and parabolic pdes:
1)"Spherical Means for Pdes"
edited by K. Karl Karlovich Sabelfeld, I. A. Shalimopva
2)"Spherical and plane integral operators for PDEs : construction, analysis, and applications", Karl K. Sabelfeld, Irina A. Shalimova,
| 0 | https://mathoverflow.net/users/99863 | 258023 | 116,478 |
https://mathoverflow.net/questions/257919 | 7 | I would like to understand precisely the structure of unitary groups.
Let $F$ be a global number field, $E$ a quadratic extension of $F$, and $U$ a unitary group on $E$ (*i.e.* the group of automorphism preserving an hermitian form on an $E$-vector space), say of rank $n$ (*i.e.* the hermitian form is on a squared $... | https://mathoverflow.net/users/43737 | Understanding the structure of unitary groups | As you already seem to know, unitary groups are classified by separable quadratic field extensions (in fact one should really work with separable quadratic algebras, i.e. also $F\times F$, corresponding to the ''trivial'' unitary group $\mathrm{GL}\_n$).
Thus your questions can essentially be reduced to problems abou... | 3 | https://mathoverflow.net/users/5101 | 258030 | 116,479 |
https://mathoverflow.net/questions/258031 | 15 | I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}\_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\mathfrak{gl}\_1)\subset\ldots\subset\mathcal U(\mathfrak{gl}\_n),$$ denote the centers $Z\_i\subset\mathcal U(\mathfrak{... | https://mathoverflow.net/users/19864 | Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$ | 1-2) Both of these seem likely, but without doing more reference hunting than I have time for, I won't swear to it. For 2), it's better to think about the $GL\_{n-1}$ invariants in $U(\mathfrak{gl}\_n)$, which one can understand using the PBW theorem.
3) If you think about the combinatorics here, this just doesn't s... | 6 | https://mathoverflow.net/users/66 | 258041 | 116,481 |
https://mathoverflow.net/questions/257989 | 8 | [A recent episode of Star Talk Radio](https://www.startalkradio.net/show/exploring-exoplanets-seth-shostak-startalk-stars/ "Link to podcast episode") discussed among other things the unknown object(s) orbiting [Tabby's star](https://en.wikipedia.org/wiki/KIC_8462852 "Wikipedia entry") (aka "Alien mega structure discove... | https://mathoverflow.net/users/3041 | Infering shapes from overlap with a shifting circle | A convex set and a non-convex set can have the same light curve.
Let $r$ be the reflection about the $x$-axis. Then $r(S)$ has the same light curve as $S$. So, let $C$ be a convex set above the x-axis, and let $C = C\_1 \cup C\_2$. Then $C$ has the same light curve as $C\_1 \cup r(C\_2)$.
| 3 | https://mathoverflow.net/users/2954 | 258046 | 116,484 |
https://mathoverflow.net/questions/258029 | 0 | Are there numbers $k > 1$ and $c > 1$ such that:
1 ) $\theta(c) \geq c \left( 1-\frac{1}{5 \ln^2(c)} \right) $
2 ) $\frac{c}{1+\frac{1}{\ln^4(c)}} \leq p(\pi(c))$ where $p(n)$ is the $n$-th prime number and $\pi(n)$ is the prime counting function
3 ) $\pi(c) \leq \frac{c}{\ln (c)} (1+\frac{1}{\ln (c)}+\frac{2}{\l... | https://mathoverflow.net/users/95470 | Numbers related to the Riemann hypothesis | Proving such results falls into three parts. First you take an effective version of the prime number theorem, which implies all your desired bound for sufficient. Second you write a computer program (or use computations done by somebody else) to check your inequalities for small $x$. Finaly you bridge the gap between t... | 4 | https://mathoverflow.net/users/37555 | 258054 | 116,486 |
https://mathoverflow.net/questions/258052 | 14 | [Francis Brown Theorem](http://www.ihes.fr/~brown/Arbeitstatung.pdf) says that $\zeta(a\_{1},\dots ,a\_{r})$ the multi-zeta value of weight $N=a\_{1}+\dots +a\_{r}$ is a $\mathbb{Q}$-linear combination of elements of the set $S=\{\zeta(a\_{1},\dots, a\_{n})| \text{for any i, } a\_{i}\in\{2,3\} \text{ and } a\_{1}+\dots... | https://mathoverflow.net/users/97275 | Multizeta function values | The elements of $S$ are conjectured to be $\mathbb{Q}$-linearly independent, and so a basis for the $\mathbb{Q}$-linear span of the multiple zeta values.
This is what Francis Brown accomplished at the motivic level. Actually, his proof via motives goes through the direction that is currently missing on the level of ... | 18 | https://mathoverflow.net/users/26522 | 258061 | 116,487 |
https://mathoverflow.net/questions/258039 | 1 | Let $C$ be a category, $D$ be a Grothendieck topos, and suppose we have a fully faithful, left-exact functor $F:C\rightarrow D$. Let $Lan\_{y}F:PShv(C)\rightarrow D$ be the Yoneda extension of $F$. Since both $C,D$ are locally presentable and $Lan\_{y}F$ is cocontinuous, $Lan\_{y}F$ has a right adjoint, call it $G$. Is... | https://mathoverflow.net/users/84563 | Faithfulness of Right adjoint to Kan extension | Assuming $C$ is a small category, the right adjoint to $Lan\_y(F): PShv(C) = Set^{C^{op}} \to D$ is just the functor $D \to Set^{C^{op}}: d \mapsto \hom\_D(F-, d)$; this basic "[generalized nerve](https://ncatlab.org/nlab/show/nerve+and+realization)" construction traces back to the original 1958 paper by Kan where adjo... | 3 | https://mathoverflow.net/users/2926 | 258063 | 116,488 |
https://mathoverflow.net/questions/258057 | -3 | Throughout this question, I call 'L-function' any automorphic L-function belonging to the Selberg class.
Suppose $ (F\_i)\_{(i>0)} $ is a sequence of L-functions with $ F\_i $ of degree $ p\_i $ for all $ i $ and let $ a\_{p\_{i}} $ be the $ p\_i$ -th coefficient of the L-series defining $ F\_i $ .
Does the sequenc... | https://mathoverflow.net/users/13625 | Does this particular L-series built from L-functions of prime degree define an L-function? | Surely not.
First, you can't just patch things together at random to get an $L$-function.
Second, your "construction" doesn't limit the growth of the $a\_{p\_i}$'s, so they need not satisfy Ramanujan.
| 4 | https://mathoverflow.net/users/6518 | 258068 | 116,489 |
https://mathoverflow.net/questions/242788 | 9 | I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:
>
> A map of commutative algebras is flat if pushing it out along any other morphism is quasi-isomorphic to the derived pushout when the algebras are embedded into dg-algebras. H... | https://mathoverflow.net/users/69037 | Homotopical interpretation of flatness? | The derived tensor product is defined as a homotopy pushout of $M\leftarrow R\rightarrow N$ in commutative DGAs. Now, if $R$ is a commutative ring and $M$ and $N$ are $R$-algebras, then the homology groups $H^i(M\otimes^\mathbf{L}\_R N)=\operatorname{Tor}^R\_i(M,N)$. Therefore $M\otimes\_R^\mathbf{L} N$ and $M\otimes\_... | 3 | https://mathoverflow.net/users/102390 | 258070 | 116,490 |
https://mathoverflow.net/questions/257926 | 4 | First of all, I'm not sure about asking this question here. I guess the answer is in the literature, but for the moment I'm a bit confused... a pointer to a specific source would be very helpful! Thanks in advance.
---
Let $\mathcal S$ be a small category with the trivial topology (such that every presheaf is a s... | https://mathoverflow.net/users/68603 | Computing $\mathrm{Ext}$ with a projective resolution (in a topos) | The terminal (and initial) object in $\mathrm{Mod}(\mathcal{O})$ is 0. The unadorned phrase "sheaf cohomology of $B$" would typically be the derived version of the global sections functor $\mathrm{Hom}\_{\mathrm{Mod}(\mathcal{O})}(\mathcal{O},-)$, or equivalently $\mathrm{Hom}\_{\mathrm{Psh}(\mathcal{S})}(1,-)$, applie... | 2 | https://mathoverflow.net/users/2362 | 258073 | 116,492 |
https://mathoverflow.net/questions/258037 | 1 | Motivated by a recent MSE [question](https://math.stackexchange.com/q/2071683/9464) about the sequence of function $\cos(n!\pi x)$, I have read related several related questions:
* [On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.](https://mathoverflow.net/questions/34369/on-the-behaviour-of-sinn-pi-x-when... | https://mathoverflow.net/users/nan | Is the existence of $\lim_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ an open problem? | Weyl proved in 1916 (see Satz 21 on Page 348 in [his paper](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002266423)) that if a sequence of real numbers $\lambda\_1<\lambda\_2<\dots$ grows sufficiently rapidly (and $\lambda\_n=n!$ satisfies the precise constraint there), then for almost every real number $x$ (in t... | 5 | https://mathoverflow.net/users/11919 | 258078 | 116,495 |
https://mathoverflow.net/questions/258027 | 7 | Let $E\_n$ be the number of isomorphism classes of groups of even order at most $n$, let $G\_n$ be the number of isomorphism classes of groups of order at most $n$ and $T\_n$ be the number of isomorphism classes of $2$-groups of order less than $n$.
I think that it is an open problem if $\frac{T\_n}{G\_n}$ approaches... | https://mathoverflow.net/users/24478 | Does the percentage of groups of order at most $n$ of even order aproach $1$? | I think the following works:
Looking at the book [enumeration of finite groups](https://www.cambridge.org/vu/academic/subjects/mathematics/algebra/enumeration-finite-groups?localeText=Vanuatu&locale=bi_VU&query=),
We obtain that the number of groups of order $p^k$ is at least: $$\frac{p^{\frac{2m^3}{27}}}{p^{\frac{... | 5 | https://mathoverflow.net/users/24478 | 258080 | 116,496 |
https://mathoverflow.net/questions/258084 | -1 | Let $\Lambda$ be the von Mangoldt function.
I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have
$$
\sum\_{\substack{1 \leq n \leq X \\ n \equiv a\,(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)).
$$
I was wondering if someone could give me ... | https://mathoverflow.net/users/84272 | Reference for Siegel-Walfisz Theorem under GRH | The usual error term in this problem is not $O(X^{1/2} \log (qX))$ but $O(X^{1/2} \log^2(X))$. This version of the result is part of Corollary 13.8 in Montgomery-Vaughan: Multiplicative number theory I. For a more general result (also featuring the slightly weaker error term above) see Theorem 5.15 in Iwaniec-Kowalski:... | 2 | https://mathoverflow.net/users/11919 | 258087 | 116,500 |
https://mathoverflow.net/questions/258093 | 6 | Let $F$ be a finite field and $A$ be the algebra of symmetric polynomials on $x\_1,\dots,x\_n$ over $F$. What is known about the generators of $A$ except the elementary symmetric polynomials? In particular, given some power sum polynomials, how to determine whether they generate $A$?
| https://mathoverflow.net/users/26700 | Generators of the algebra of symmetric polynomials over finite fields | The algebra $A$ can only be generated by power sums if the characteristic of $F$ is greater than $n$. In this case, the power sums $p\_i = \sum\_{j=1}^n x\_j^i$ with $1 \le i \le n$ generate $A$, and any other generating system consisting of power sums needs to include $p\_1,\ldots,p\_n$.
These claims can be justifie... | 2 | https://mathoverflow.net/users/82616 | 258109 | 116,506 |
https://mathoverflow.net/questions/257828 | 6 | In the thesis "[Topology of Function Spaces](http://d-scholarship.pitt.edu/7814/)" by Andrew Marsh (Galway, 2000) it is claimed (page 23, before Definition 30) that the space $C(\omega\_1)$ is Lindelöf with respect to the compact open topology. For a proof the reader is refered to
S.P. Gul'ko. On properties of $\sig... | https://mathoverflow.net/users/58628 | Why is $C_k(\omega_1)$ Lindelöf? | Here is (a bit lengthy and technical) proof of the Lindelof property of the function space $C\_k(\omega\_1)$. At first some notations.
For any function $f\in C\_k(\omega\_1)$ and a countable ordinal $\alpha$ let $\|f\|\_\alpha=\sup\_{x\in[0,\alpha]}|f(x)|$. Let also $\|f\|=\sup\_{x\in\omega\_1}|f(x)|$.
For every $f\i... | 4 | https://mathoverflow.net/users/61536 | 258117 | 116,510 |
https://mathoverflow.net/questions/258123 | 7 | Suppose $f(x)$ is continuous on $\mathbb{R}$, for $\forall \delta>0, \forall x\in\mathbb{R}, \lim\_{n\rightarrow\infty}f(x+n\delta)=+\infty$. Is it correct that $\lim\_{x\rightarrow+\infty}f(x)=+\infty$?
| https://mathoverflow.net/users/102742 | Continuous functions and infinity | Yes, this is true and well known. One of the references I know is the problem book of B. Makarov, M. Goluzina, A. Lodkin and A. Podkorytov (Selected problems in real analysis, Translations of Mathematical Monographs 107, AMS 1992), problem II.1.25 (it is slightly different, but essentially the same).
An idea is that ... | 6 | https://mathoverflow.net/users/4312 | 258125 | 116,515 |
https://mathoverflow.net/questions/258116 | 4 | Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{fix}(f) = \{x\in X: x = f(x)\},$$ and $$\text{Com}(f) = \{g\in\text{End}(X): g\circ f = f \circ g\}.$$
It is not hard to prove that if $2^{|\text{fix}(f)|} > |X|$ then $|\text{Com}(f)|> |X|$. D... | https://mathoverflow.net/users/8628 | Commuting functions and fixpoints | Let $X=\mathbf{N}$, $f:X\rightarrow X$ defined by $f(n)=0$ if $n$ even and $f(n)=1$ if $n$ odd. Then $f$ commutes with all elements of the set \begin{multline}
G:=\{g:X\rightarrow X\colon g(0)=0, g(1)=1, g(2k+1)=2m+1, g(2l)=2n, \\k,l,m,n\in X, k,l>0\}.
\end{multline}
Obviously, $|G|=2^{|X|}$, thus $|\text{Com}(f)|\geq ... | 2 | https://mathoverflow.net/users/26085 | 258127 | 116,517 |
https://mathoverflow.net/questions/258139 | 1 | Let $z=e^{2\pi i\theta}$ for a fixed real number $\theta$. It's known that if $\theta\not\in\mathbb{Q}$ (is irrational) then the set $S(\theta)=\{z^n: n\in\mathbb{N}\}$ is dense on the unit circle $\Gamma$.
I am (almost) sure the following is true and really "cute", but I couldn't remember a reference.
Can you help p... | https://mathoverflow.net/users/66131 | distance-set along the orbit of $e^{2\pi i\theta}$ | Tony van Ravenstein, The three gap theorem (Steinhaus conjecture), J Austral Math Soc 45 (1988) 360-370, available at <https://oeis.org/A000045/a000045_1.pdf>, references several proofs; perhaps the first was V. T. Sós, On the theory of diophantine approximations. I, Acta Math Acad Sci Hungar 8 (1957) 461-472.
| 6 | https://mathoverflow.net/users/3684 | 258146 | 116,527 |
https://mathoverflow.net/questions/258008 | 6 | Is there a reasonable definition for a category of categories equipped with a factorization system, say $({\bf C},(E,M))$ and functors $F\colon ({\bf C},(E,M))\to ({\bf D}, (E', M'))$ that preserve them? I see several possible choices, all unreasonable in some sense:
1. $F(E)\subseteq E'$, and $F(M)\subseteq M'$; thi... | https://mathoverflow.net/users/7952 | On the category of categories with factorization system | The main theorem of Korostenski and Tholen's [Factorization systems as Eilenberg-Moore algebras](http://www.sciencedirect.com/science/article/pii/002240499390171O) proves that categories with (orthogonal) factorization systems are precisely the normal pseudo algebras for a 2-monad on Cat. The 2-monad in this case is th... | 12 | https://mathoverflow.net/users/2181 | 258153 | 116,529 |
https://mathoverflow.net/questions/258132 | 22 | Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by many as a new axiom (or some stronger principle which implied it). This is because we intuitively think that if we canno... | https://mathoverflow.net/users/3199 | Platonic Truth and 1st Order Predicate Logic | The phenomenon accords more strongly with your philosophical explanation if you ask also that the sentences have complexity $\Pi^0\_1$. That is, the universal statement $\forall x\ \varphi(x)$ should have $\varphi(x)$ involving only bounded quantifiers, so that we can check $\varphi(x)$ for any particular $x$ in finite... | 19 | https://mathoverflow.net/users/1946 | 258160 | 116,532 |
https://mathoverflow.net/questions/258150 | 4 | For $f : [0;1] \to \mathbb{R}$, let $M\_f := \{x \in [0;1] \mid f(x)$ is a local strict maximum of $f\}$. It is easy to see that for any $f$, $M\_f$ is at most countable. It is also easy to see that there can be a continuous $f$ such that $M\_f$ is infinite.
The first question you can ask is the following: is there a ... | https://mathoverflow.net/users/102343 | Density of the max set of a non-differentiable function | The conjecture as stated, for *strict* maxima, is not correct. It becomes true, though, if possibly non-strict maxima are considered also.
All this and more is discussed in [Posey, Vaughan, Extrema and nowhere differentiable functions, Rocky Mountain J. Math. (1986).](https://projecteuclid.org/download/pdf_1/euclid.r... | 3 | https://mathoverflow.net/users/48839 | 258161 | 116,533 |
https://mathoverflow.net/questions/258163 | 9 | Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$
(i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q\_{i},\dot{q}\_{i},t)$, $i=1,2,...,n$.
It is well known from classical mechanics, that a sufficient condition for a Lagrangian dynamical system to be... | https://mathoverflow.net/users/85967 | When does a Lagrangian dynamical system have an equivalent Hamiltonian description? | Here's what I have done:
$\bullet$ Let the Lagrangian $L(q\_{i},\dot{q}\_{i},t)$, which under the **point transformations**
$$
\{q\_{i}\}\leftrightsquigarrow\{Q\_{i}\}
$$
given by the invertible relations $Q\_{i}=Q\_{i}\big(q\_{j},t\big)\Leftrightarrow q\_{j}=q\_{j}\big(Q\_{i},t\big)$, $\ \ i,j=1,2,...,n$, (i.e. ... | 4 | https://mathoverflow.net/users/85967 | 258180 | 116,545 |
https://mathoverflow.net/questions/245406 | 1 | I have a question about the derivative of a distance function.
Let $D \subset \mathbb{R}^{d}$ be a connected and unbounded open subset with smooth boundary. $B(z,r)$ denotes the open ball of radius $r>0$ centered at $z \in \bar{D}$. We define the following distance function $F$ on $\mathbb{R}^{d}$:
\begin{equation\*}... | https://mathoverflow.net/users/68463 | On the derivative of a distance function | Let $K$ be a compact set with smooth boundary. The distance function has gradient 1 everywhere where the gradient exists. The gradient exists in any $x$ there exists a unique $y \in \partial K$ boundary point minimizing the distance $d(x,y) = d(K,x)$. The proof is simple. Take the normal at $y$ and map a neighbourhood.... | 2 | https://mathoverflow.net/users/10834 | 258184 | 116,547 |
https://mathoverflow.net/questions/258176 | 1 | I got the following sum with which I want to prove one limit fact:
$$
f\_n(a) = \sum\limits\_{t=0}^{n-1} \binom{n-1}{t} (a^t)^{n-t}
$$
I want to prove that $f\_n(a) \to 1$ while $n \to \infty$ for $a\in [0,1)$ if true. (I plotted and it looks like it were true). As you see this is kind of binomial sum, but instead ... | https://mathoverflow.net/users/97885 | Almost binomial sum limit | Splitting your sum as $\sum\_{t=0}^{n-1}=1+\sum\_{t=1}^{n-1}$, it suffices to show that $\sum\_{t=1}^{n-1}\binom{n-1}ta^{t(n-t)}\rightarrow0$ as $n\rightarrow\infty$; provided $0<a<1$.
To illustrate our proposed method, assume $0<a<\frac12$. Since $t(n-t)\leq n-1$, we estimate
$$0\leq\sum\_{t=1}^{n-1}\binom{n-1}ta^{t... | 2 | https://mathoverflow.net/users/66131 | 258186 | 116,548 |
https://mathoverflow.net/questions/258195 | 8 | The following question arose from a discussion about the definability of bases of $\mathbb{R}$ as a $\mathbb{Q}$-vector space.
(ZF without AC) something we can note is that the existence of a (definable) well-ordering of $\mathbb{R}$ is easily seen to be equivalent to that of a (definable) well-ordered basis of $\math... | https://mathoverflow.net/users/102343 | Relation between well-orderings of $\mathbb{R}$, and bases over $\mathbb{Q}$ | The problem is *generally* open. However, recently Liuzhen Wu, Liang Yu, Ralf Schindler and Mariam Beriashvili posted a preprint in which they prove the consistency of the existence of a Hamel basis and $\Bbb R$ cannot be well-ordered. Specifically, they show there is such a basis in Cohen's first model.
This can be ... | 10 | https://mathoverflow.net/users/7206 | 258198 | 116,550 |
https://mathoverflow.net/questions/258151 | 2 | Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau\_x$ be the time to reach $x$ and condition on the event $\tau\_M < \tau\_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V\_k$ be the number of visits to $k$ up to time $\tau\_M$. What is $\mathbb E V\_k$?
All we ne... | https://mathoverflow.net/users/52896 | Local time for conditioned simple random walk | To second Anthony Quas's answer, here's a simple way to calculate $\mathbb{E}V\_k$. Notice that conditional on $\tau\_M<\tau\_0$ your walk will visit $k$ at least once. Once we are at $k$, the number of consecutive visits is a positive geometric random variable, since the walk is a Markov process. The parameter is $1-\... | 3 | https://mathoverflow.net/users/47484 | 258199 | 116,551 |
https://mathoverflow.net/questions/258192 | 2 | In the study of manifolds with positive sectional curvature, I guess the following statement is true:
>
> Let $M$ be a manifold with positive (but not necessarily constant) sectional curvature. Then $M\times M$ admits a Riemannian metric with non-negative sectional curvature such that there exists a direction with ... | https://mathoverflow.net/users/90655 | Manifolds with positive sectional curvature | Perhaps you are thinking of a conjecture of Heinz Hopf: no Riemannian metric on $S^2 \times S^2$ has positive sectional curvature. This conjecture is still open, as far as I can tell.
For more information see [Manifold with a quasi-positive curvature](https://mathoverflow.net/questions/144451/manifold-with-a-quasi-po... | 4 | https://mathoverflow.net/users/13268 | 258201 | 116,553 |
https://mathoverflow.net/questions/258170 | 6 | The *diameter* of a bounded set is the supremum of the distances between any two points of the set, and the *circumradius* is the infimum of the radii of balls containing the set. Obviously, the diameter is never greater than twice the circumradius.
A hyperplane passing through the center of the unit cube $[0,1]^n$ c... | https://mathoverflow.net/users/36904 | Splitting the $n$-cube into two small congruent convex halves | Answer to (a): We consider first $n$ odd and the hypercube $[-1,1]^n$. Let $(a\_1,\dots,a\_n)\in \mathbb{R}^{n}$ be a normal to the hyperplane. W.L.O.G we can assume that $0\leq a\_1 \dots \leq a\_n=1$. Consider the edge $(1,-1,\dots,1,-1,x), x\in [-1,1]$. The hyperplane will cut this edge at $x=\frac{a\_1-a\_2+\dots+a... | 4 | https://mathoverflow.net/users/100908 | 258203 | 116,554 |
https://mathoverflow.net/questions/258209 | 2 | Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{Com}(f) = \{g\in\text{End}(X): g\circ f = f \circ g\}.$$
Is there $f\in \text{End}(X)$ such that $\text{Com}(f) = \{\text{id}\_X, f\}$?
If not, what is $\min\{|\text{Com}(f)|:f\in\text{End}(... | https://mathoverflow.net/users/8628 | Functions $f:X\to X$ that commute with "almost no others" | Nope: since $f^2$ commutes with $f$, we should have either $f^2=\mathrm{id}\_X$ or $f^2=f$, that is $f$ is either idempotent or involutive. But on an infinite set it is easy to see these always have infinite commutators.
| 5 | https://mathoverflow.net/users/6101 | 258210 | 116,557 |
https://mathoverflow.net/questions/257597 | 6 | I am using this definition:
>
> An *algebra of functions on a finite quantum group* $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
>
>
>
I have the following (very well known --- folklore --- result)
>
> *(Classification Theorem)*
>
>
> Let $A$ be the algebra of functions ... | https://mathoverflow.net/users/35482 | Commutative and Cocommutative Quantum Groups | I took a quick look into Timmermann's book "An invitation to Quantum Groups".
It refers to:
Saad Baaj; Georges Skandalis
*Unitaires multiplicatifs et dualité pour les produits croisés de C\*-algèbres*
Annales scientifiques de l'École Normale Supérieure (1993)
Volume: 26, Issue: 4, page 425-488
ISSN: 0012-9593
The... | 2 | https://mathoverflow.net/users/10718 | 258225 | 116,564 |
https://mathoverflow.net/questions/258226 | 2 | As an algebraist, I have some strong intuitions about what it means for an algebraic result to be true. In particular, my intuition would lead me to believe that if I cannot construct a counter-example to a claim, then the claim must be true. This is what motivated [my previous question](https://mathoverflow.net/questi... | https://mathoverflow.net/users/3199 | Platonic Truth and 1st Order Logic - Take 2 | Yes, your theory is the same as the true arithmetic. In
particular, yes, it is consistent.
I claim that at stage $n$, your theory $T\_n$ consists of PA plus
the collection of true $\Pi^0\_n$ sentences (that is, true in the
standard model). This starts out true with $T\_0$. If $T\_n$ is like
that, then consider $T\_{n... | 5 | https://mathoverflow.net/users/1946 | 258229 | 116,566 |
https://mathoverflow.net/questions/258237 | 1 | Is there any characterization of ideals space or maximal ideals space of following Banach algebras?
1)$C^1{([0,1])}$
2) disc algebra
3)$A(SO(3))$
| https://mathoverflow.net/users/27066 | Ideals of Banach algebras | Just to answer the easy part of the question quickly
1) [0,1]
2) the closed unit disc
3) SO(3)
| 5 | https://mathoverflow.net/users/763 | 258240 | 116,570 |
https://mathoverflow.net/questions/251669 | 1 | There is this result characterizing irreducible square matrices (e.g., L. Hogben, *Handbook of Linear Algebra*):
>
> If $P$ is a non-negative square matrix, then $P$ is irreducible iff every eigenvector corresponding to the spectral radius $\rho(P)$ is a scalar multiple of a positive vector.
>
>
>
Is there any... | https://mathoverflow.net/users/78328 | Is there an infinite dimensional analogue of the characterisation of irreducible matrices | The characterisation of irreducible matrices as suggested in the question is not correct. The matrix
$$
P =
\begin{pmatrix}
1 & 0 \\
1 & 0
\end{pmatrix}
$$
is a counterexample: it has spectral radius $1$ and the eigenspace for this eigenvalue is spanned by the vector
$\begin{pmatrix}
1 \\ 1
\end{pmatrix}$;
yet, $P$... | 3 | https://mathoverflow.net/users/102946 | 258251 | 116,574 |
https://mathoverflow.net/questions/253701 | 1 | Let $X$ be a noetherian scheme $U \subset X$ an open subset with complement $Z = X- U$. Assume $Z$ is cut out by the ideal sheaf $\mathcal{I} \subset \mathcal{O}\_X$. We have exact sequences:
$$0 \to \mathcal{I}^n \to \mathcal{O}\_X \to \mathcal{O}\_X/\mathcal{I}^n \to 0$$
Here's what i'd like to be able to say:
... | https://mathoverflow.net/users/22810 | Relation between local cohomology and open immersions | This is true. The functor $RHom(\mathcal{O}\_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z\_n \to X$. Where $Z\_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In the limit we get a kind of $j^!$ too only now for the inclusion of the formal scheme $\widehat X\_Z$ (formal neighborho... | 1 | https://mathoverflow.net/users/22810 | 258260 | 116,576 |
https://mathoverflow.net/questions/258253 | 4 | **Question:**
Let $X$ be a topological space, $U\_\*\rightarrow X$ be an augmented simplicial space, and let $M\_n(U\_\*)$ be the n-th matching object computed in $sTop$ while $M\_n^X(U\_\*)$ denotes the $n$-th matching object computed in the category $s(Top \downarrow X)$ of simplicial spaces over $X$.
How does o... | https://mathoverflow.net/users/96885 | Matching objects and hypercovers in topology | This follows from the *construction* of $(\operatorname{cosk}\_m(X\_\bullet))\_n$ as a limit indexed by $(\Delta/[n])^{\operatorname{op}}\_{\leq m}$ (in the non-augmented case) or $(\Delta\_a/[n])^{\operatorname{op}}\_{\leq m}$ (in the augmented case). See e.g. [Tag 0183](http://stacks.math.columbia.edu/tag/0183) for a... | 3 | https://mathoverflow.net/users/82179 | 258266 | 116,580 |
https://mathoverflow.net/questions/258243 | 5 | It is well-known that each uncountable compact metrizable space $X$ contains a homeomorphic copy of the Cantor cube $\{0,1\}^\omega$. What about copies of Cantor cubes of larger weight?
**Problem.** Does every uncountable Dugundji compact space $X$ contain a topological copy of the Cantor cube $\{0,1\}^{\kappa}$ of w... | https://mathoverflow.net/users/61536 | Which compacta contain copies of Cantor cubes? | It was proved independently by Efimov and Gerlits that the answer is yes if $\kappa$ has uncountable cofinality. In fact they proved this for any dyadic $X$ (it is well known that Dugundji compacta are dyadic). Their theorem (see "Mappings and imbeddings of dyadic spaces" by Efimov or "On subspaces of dyadic compacta" ... | 6 | https://mathoverflow.net/users/17836 | 258268 | 116,581 |
https://mathoverflow.net/questions/258261 | 4 | Let us say that a measure $\mu$ on $\mathbb{R}^d$ is *locally doubling* if for each
$x\in\mathbb{R}^d$ there is a constant $C(x)$ such that for all $r>0$,
$\mu(B(x,2r)) \le C(x) \mu(B(x,r))$,
where $B(x,r)$ is the $r$-ball about $x$.
Additionally, $\mu$ is *bounded* if $\mu(\mathbb{R}^d)<\infty$.
>
> **Question.**... | https://mathoverflow.net/users/12518 | Locally doubling measures | The answer is **no**. In fact, something far more dramatic can happen: there exist Radon measures $\mu$ on $\mathbb{R}^d$ such that for $\mu$ almost all $x$, for any arbitrary Radon measure $\nu$, the normalized restriction of $\mu$ to $B(x,r)$ "looks like" $\nu$ (it is arbitrarily close in the weak$^\*$ topology).
T... | 7 | https://mathoverflow.net/users/11009 | 258273 | 116,583 |
https://mathoverflow.net/questions/258278 | 4 | I know that the Cantor set is countable dense homogeneous. My question is: if A,B,C,D are countable dense subsets of the Cantor set such that the pairs A and B and C and D are disjoint, there exists a homeomorphism f of the Cantor set such that f(A) = C and f(B) = D?
| https://mathoverflow.net/users/86469 | Is the Cantor set countable dense homogeneous in pairs? | The answer is yes, unless I'm missing something, via a back-and-forth argument.
For simplicity I'll view the Cantor set as the set of infinite binary sequences, $2^\omega$, and think of this as the set of paths through the infinite binary tree $2^{<\omega}$.
Any rooted-tree automorphism $\alpha$ of $2^{<\omega}$ in... | 5 | https://mathoverflow.net/users/8133 | 258280 | 116,584 |
https://mathoverflow.net/questions/258279 | 5 | Is there a set $X \subset \mathbb{R}^2$ such that every straight line in the plane is ordinary in relation to it? i.e. if $r$ is any straight line then $|r \cap X|=2$.
| https://mathoverflow.net/users/102962 | Can all lines in the euclidian plane be ordinary? | The answer is yes, by an argument using the axiom of choice.
There are exactly continuum many lines in the plane, and so by the
axiom of choice, we may enumerate them in a well-ordered sequence
of length continuum.
Let's build the set $X$ in stages, so that by stage $\alpha$ we've
included two points from all the l... | 12 | https://mathoverflow.net/users/1946 | 258287 | 116,585 |
https://mathoverflow.net/questions/258282 | 5 | In their paper, " On Interpretations of Arithmetic and Set Theory" (*Notre Dame Journal of Formal Logic*, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set Theory", Kaye and Wong make the following (interesting) remarks:
>
> It is fairly straightforward to see that $I{\Delta\_0}$ pro... | https://mathoverflow.net/users/20597 | Extensions of the Ackermann interpretation to nonstandard theories of arithmetic | Based on the comments, I think it's worth clarifying some points about the Ackermann interpretation.
The Ackermann interpretation is *definable*: it is a formula $\varphi(x, y)$, and - given a model $M\models PA$ - it gives us an interpreted structure $Ack(M)=(M, \varphi^M)$. This is true regardless of what $M$ is.
... | 6 | https://mathoverflow.net/users/8133 | 258289 | 116,587 |
https://mathoverflow.net/questions/258293 | 5 | A matrix $A\in M\_{4}(\mathbb{C})$ is called a simple tensor if $A=B\otimes C$ for two $2\times 2$ matrices $B,C$.
Assume that $X$ is a Hausdorff topological space.Assume that $f:X\to M\_{4}(\mathbb{C})$ is a continuous map such that $f(x)$ is a simple tensor, for every $x\in X$.
>
> Are there continous maps $g,h... | https://mathoverflow.net/users/36688 | Is a pointwise " simple tensor" valued continuous map a tensor product of two continuous maps? | The answer is 'no'. For example, let
$$
X = \{\ B\otimes C\ | \det(B)\det(C) = 1\ \}\subset \mathrm{SL}(4,\mathbb{C}).
$$
Then $X$ is isomorphic to the group $\mathrm{SO}(4,\mathbb{C})$ and hence $\pi\_1(X)\simeq \mathbb{Z}\_2$, while the set
$$
G = \{\ (B, C)\ |\ \det(B)\det(C) = 1\ \}\subset
\mathrm{GL}(2,\mathbb{C}... | 6 | https://mathoverflow.net/users/13972 | 258297 | 116,591 |
https://mathoverflow.net/questions/258269 | 14 | Recall that an elementary topos is a cartesian closed category with finite limits and a subobject classifier. A Grothendieck topos is a category equivalent to the category of sheaves on a site.
Are there examples of (co)complete elementary topoi that are not Grothendieck? On the "Cocomplete" side of things, such a ca... | https://mathoverflow.net/users/102957 | (Co)complete topoi that are not Grothendieck? | A classical example is $G$-$Set$ for a large group $G$. That this is a cocomplete elementary topos is not hard to see. Limits and colimits are formed at the underlying set level, and exponentials $Y^X$ are formed as usual as the set of functions $f: X \to Y$ with the $G$-action $(g, f) \mapsto g f$ defined by $g f: x \... | 18 | https://mathoverflow.net/users/2926 | 258301 | 116,594 |
https://mathoverflow.net/questions/190962 | 2 | A cycle $C$ in a connected graph $G$ is called *dominating* if its complement $V(G)-V(C)$ is an independent set. H.J. Veldman proved in 1983 ([Disc. Math. v.43, 281-96](http://dx.doi.org/10.1016/0012-365X(83)90165-6)) a general result that in particular implies that a connected $2K\_2$-free $G$ (i.e. $G$ does not conta... | https://mathoverflow.net/users/11100 | finding dominating cycles in $2K_2$-free graphs | We have written our argument up here: <https://arxiv.org/abs/1412.0514>
| 0 | https://mathoverflow.net/users/11100 | 258304 | 116,596 |
https://mathoverflow.net/questions/258202 | 7 | Let $\Gamma$ be a finite subgroup of SU(2) and consider the quotient of $S^3$ by $\Gamma$ via its left action. Pick a simply connected compact Lie group $G$ and take a flat connection on this quotient. Or equivalently, choose a homomorphism $\Gamma \to G$. The Chern-Simons action functional should assign a value in $\m... | https://mathoverflow.net/users/5420 | Value of the Chern-Simons functional for flat connections on $S^3/\Gamma$ | For $G=SU(N)$, there is a paper [*SU(n)–Chern–Simons invariants of Seifert fibered 3–manifolds*](http://www.worldscientific.com/doi/abs/10.1142/S0129167X98000130) (Int. J. Math., 09, 295-330 (1998))
| 6 | https://mathoverflow.net/users/102135 | 258315 | 116,600 |
https://mathoverflow.net/questions/258310 | 5 | Obviously the question in the title alone doesn't make sense so I'll develop on the context and then I'll ask my question :
Studying $AD$ (axiom of determinacy) I had to prove that $AD$ and $AC$ are incompatible (mod $ZF$). So to prove this I show that under $AC$, there are some undetermined games, and in order to prov... | https://mathoverflow.net/users/102300 | How "strong" is the existence of a non trivial ultrafilter on $\omega$? | It seems difficult to make the notion of "general choice principle" precise, but I would guess that the principle "every infinite set admits a nonprincipal ultrafilter" would qualify. If so, then it answers your second question. It obviously implies the existence of a nonprincipal ultrafilter on $\omega$, and, unless I... | 9 | https://mathoverflow.net/users/6794 | 258316 | 116,601 |
https://mathoverflow.net/questions/258303 | 3 | Given a quasiisomorphism of DGAs $f:A\rightarrow B$ and a DG-module $M$ over $A$.
Is the canonical chain map
\[M\rightarrow B\otimes\_A M \qquad m\mapsto 1\otimes m\]
an isomorphism on homology?
| https://mathoverflow.net/users/3969 | Induction along a quasiisomorphism of DGAs | This is an elaboration on the L's comment. Let me use the cochain convention, so that the differential has degree +1.
Let $B=\mathbb{Z}/2$ (concentrated in degree 0) and let $A = \mathbb{Z}[e]/(e^2,de-2)$, where $|e|=-1$. Then the canonical projection $A\rightarrow B$ is a quasiisomorphism.
Let $M$ be the acyclic c... | 2 | https://mathoverflow.net/users/3969 | 258325 | 116,604 |
https://mathoverflow.net/questions/255909 | 3 | For a matrix $M\in\mathbb{R}^{n\times n}\_{\geq 0}$ with nonnegative entries, we define $m$ as the smallest positive integer such that all the entries of $M^m$ are strictly positive (if there is one).
What is, as a function of $n$, the maximum possible (finite) value of $m$ over all possible choices of $M$?
**My a... | https://mathoverflow.net/users/11504 | Vanishing zeroes in matrix powers | This is a combination of the [answer Gerry Myerson gave on MSE](https://math.stackexchange.com/questions/450090/if-p-is-a-regular-transition-probability-matrix-then-pn2-has-no-zero-ele), the paper linked there, and the comments here.
The largest possible minimum $m$ is $(n-1)^2+1 = n^2-2n+2$. This was proved by Wiela... | 3 | https://mathoverflow.net/users/2954 | 258333 | 116,608 |
https://mathoverflow.net/questions/256108 | 0 | Continuous backpack with multiple choice items is the problem where you need to collect items by one from each of distinct sets and associate them with rational numbers so that their sum of weights will not be greater than K and sum of costs will not be less than B.
*Computers and Intractability: A Guide to the Theor... | https://mathoverflow.net/users/50375 | Continuous backpack with multiple choice items. NP prove | I have found the prove of NP-completeness at <http://www.orsj.or.jp/~archive/pdf/e_mag/Vol.23_01_028.pdf> P. 30-32
| 0 | https://mathoverflow.net/users/50375 | 258334 | 116,609 |
https://mathoverflow.net/questions/258342 | 4 | Let us consider $M\subseteq V$ a transitive class such that $M^\kappa\subseteq M$ for some cardinal $\kappa$. Now take $\mathbb{P}$ a $\kappa^+$-cc (or $\leq\kappa-$distrivutive) forcing notion in $V$. Then it is the case that, provided that $G\subset \mathbb{P}$ is $V$-generic, then $V[G]$ also thinks that $M[G]^\kapp... | https://mathoverflow.net/users/102990 | Closure properties of transitive classes when forcing with $\kappa^+$-cc forcing | First, let me give an incorrect argument:
Working in $V[G]$, suppose $X\_\eta$ ($\eta\in\kappa$) is a $\kappa$-sequence of *sets* in $M[G]$. Then we get a $\kappa$-sequence of *names* $\nu\_\eta$ in $M$ which correspond to those sets: $\nu\_\eta[G]=X\_\eta$. Since $M^\kappa\subseteq M$, we have $(\nu\_\eta)\_{\eta\in... | 6 | https://mathoverflow.net/users/8133 | 258344 | 116,613 |
https://mathoverflow.net/questions/258324 | 1 | Flat space in polar coordinates takes the form
$$ds^2=dr^2+r^2d\phi^2$$
To avoid a conical singularity at the origin, we must impose that $\phi$ is periodic with period $2\pi$.
Now consider the following generalization:
$$ds^2=dr^2+r^2d\phi^2+r^2d\psi^2$$
My basic question is how to interpret such a metric geometric... | https://mathoverflow.net/users/64029 | Regularity of a generalized polar coordinate metric with two angles | This is the cone $x^2+y^2=u^2+v^2$ with metric induced from $\mathbb{R}^4$ with coordinates $(x,y,u,v)$. It is clearly not smooth. You just take usual polar coordinates on $(x,y)$ and $(u,v)$ separately and then set the magnitudes equal.
| 2 | https://mathoverflow.net/users/13268 | 258345 | 116,614 |
https://mathoverflow.net/questions/258353 | 0 | Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$.
We want to show that the biconditional $k = 1 \iff q = 5$ holds.
~~It suffices to prove one direction, as the implication $q = 5 \implies k = 1$ was proved by [Iannucci (Lemma 12, page 873)](http://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-0... | https://mathoverflow.net/users/10365 | Proving $k = 1 \implies q = 5$, if $q^k n^2$ is an odd perfect number with Euler prime $q$ | The equivalence :
>
> $(k=1 \lor q> 5) \iff (q=5 \implies k=1)$
>
>
>
is not correct on its own, except if you have some other information (I don't know the problem, I'm simply evaluating the "logical" points of the proof); that's because $(q=5 \implies k=1)$ is itself equivalent to $(q\neq 5 \lor k=1)$ which... | 3 | https://mathoverflow.net/users/102300 | 258355 | 116,616 |
https://mathoverflow.net/questions/258349 | 9 | Let $f$ be a primitive form of an even weight $k$ and level $N\geq 1.$ By the theory of Hecke operators, $$\lambda\_f(n)=\frac{\hat{f}(n)}{n^{\frac{k-1}{2}}}$$ is a real number. Studying the distribution of signs of Hecke eigenvalues has become an active area of research in recent past. Many authors investigated the pr... | https://mathoverflow.net/users/76102 | Why is it interesting to study the sign distribution of Hecke eigenvalues? | Kowalski, Lau, Soundararajan and Wu have a very nice explanation of why one might want to study the sign changes of Fourier coefficients of primitive cusp forms in the introduction of the paper [On modular signs](https://arxiv.org/abs/0911.2269).
The idea is that the Fourier coefficients of a primitive cusp form are ... | 8 | https://mathoverflow.net/users/nan | 258358 | 116,618 |
https://mathoverflow.net/questions/258354 | 4 | Let $K$ be an algebraically closed field with characteristic $p>0$, and let $f(t,x)\in K[t,x]$ be a polynomial separable in $x$. Denote:
\begin{equation} \Lambda = \bigcup\_{i\in \mathbb{N}} K((t^\frac{1}{i}))
\end{equation}
I want to prove that $f(t,x)$ as a polynomial in $K[x]$ has a root in $\Lambda$.
If the chara... | https://mathoverflow.net/users/86475 | Puiseux decomposition over a field with positive characteristic | The comment of Felipe Voloch does not seem very useful without an example or reference. In fact, if $K$ is the algebraic closure of $\mathbb{F}\_p$ then the equation $tx^p-tx-1$ has no solution in $\Lambda$, an observation of Chevalley. For a "generalized Puiseux field" that does contain an algebraic closure of $K((t))... | 11 | https://mathoverflow.net/users/2807 | 258359 | 116,619 |
https://mathoverflow.net/questions/258283 | 5 | How to study the decomposition of a square matrix into a product of sparse matrices?
There are no restrictions on the number of matrices in the product, but the fewer the better.
| https://mathoverflow.net/users/16739 | Decomposing a matrix into a product of sparse matrices | Given an invertible $n \times n$ matrix $\mathrm A$, we perform [Gaussian elimination](https://en.wikipedia.org/wiki/Gaussian_elimination) until we obtain a (nonsingular) diagonal matrix. In other words, we left-multiply $\mathrm A$ by [permutation matrices](https://en.wikipedia.org/wiki/Permutation_matrix) (whose inve... | 10 | https://mathoverflow.net/users/91764 | 258366 | 116,622 |
https://mathoverflow.net/questions/258317 | 3 | Let $M$ denote a monoid and suppose we're given a function $[-] : M \rightarrow M$ satisfying $[a[b]c] = [abc].$ Then:
>
> **Proposition 0.** $[-]$ is idempotent.
>
>
>
*Proof.* Take $a=c=1$).
>
> **Proposition 1.** The set of fixed points of $[-]$ becomes a monoid with identity $[1]$ and multiplication $a... | https://mathoverflow.net/users/26080 | What do we call functions satisfying $[a[b]c] = [abc]$? | Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot b]=[[a] \cdot b].$$
Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. There are lo... | 6 | https://mathoverflow.net/users/98306 | 258368 | 116,623 |
https://mathoverflow.net/questions/258367 | 3 | Let $\delta > 0$ be small and $\chi\_{[-\delta, \delta]}(t)$ be a characteristic function on the interval $[-\delta, \delta]$. I am interested in a trigonometric polynomial $S$ such that
$$
\chi\_{[-\delta, \delta]}(t) \leq S(t)
$$
such that and $S(t) = 0$ if $|t| > 2 \delta$ with some control on the coefficients of $S... | https://mathoverflow.net/users/84272 | A "nice" trigonometric polynomial approximation of a characteristic function | Pick any $f$ that bounds above the characteristic function of $[-\delta, \delta]$. For example the sine kernel $(\frac{\sin x}{x})^2$ or a higher power conveniently rescaled to fit your requirements. Then make your upper bound $f(t)$ periodic by considering $F(t) = \sum\_{n} f(t + n)$. Then make sure that your $f$ has ... | 2 | https://mathoverflow.net/users/102997 | 258370 | 116,624 |
https://mathoverflow.net/questions/258369 | 4 | Let $G$ be a connected, reductive group defined over a local field $F$. Let $P$ be a parabolic subgroup of $G$ which is defined over $F$, and let $N = \mathscr R\_u(P)$ be the unipotent radical of $P$. Then $N(F)$ is a locally compact group which I understand should be unimodular. Why is this?
As an example, consider... | https://mathoverflow.net/users/38145 | Why is the unipotent radical of a parabolic subgroup unimodular? | The slickest way to handle all such matters is to exploit the underlying algebraic structure so as to avoid any need to compute anything (and to argue in a manner which works uniformly for groups of points over all local compact fields or adele rings).
Let $G$ be *any* smooth group scheme of relative dimension $d> 0... | 12 | https://mathoverflow.net/users/81332 | 258378 | 116,630 |
https://mathoverflow.net/questions/258275 | 0 | Consider the Hilbert scheme of curves in $P^3$ with genus $g$ and degree $d$, $H\_{g,d}$. Is this rational for some $g$ and $d$?
Edit: For which $(g,d)$ is this rational?
| https://mathoverflow.net/users/41873 | Rationality of Hilbert schemes? | [expanding on my comments above]
$H\_{g,d}$ is rational for some but not all $(g,d)$.
A rational example is $H\_{1,4}$: a quartic elliptic curve in
${\bf P}^3$ is the complete intersection of two quadrics, so
$H\_{1,4}$ is just the Grassmanian of $2$-dimensional subspaces of the
$10$-dimensional space of quadrics. ... | 9 | https://mathoverflow.net/users/14830 | 258379 | 116,631 |
https://mathoverflow.net/questions/258383 | 4 | **Definition:** Let $MH(n)$ be the maximal number of perfect matchings (1-regular graphs) on $n$ vertices where the union of any two perfect matchings is a Hamiltonian cycle.
**Question:** Is it true that $MH(N)=n-1$ for infinitely many $n$?
**Additional information:** Needless to say that the problem makes sense o... | https://mathoverflow.net/users/38267 | Maximal number of perfect matchings that pairwise form a Hamiltonian cycle | Assume that $n-1$ is an odd prime. Arrange the vertices as follows: $n-1$ of them lie on the circle and form a regular $(n-1)$-gon, and the $n$th vertex is the center. Now, each of the matchings consists of one radius and all the chords perpendicular to this radius.
Indeed, take any two matchings and remove the two r... | 4 | https://mathoverflow.net/users/17581 | 258387 | 116,635 |
https://mathoverflow.net/questions/258397 | 1 |
$
\newcommand{\essSup}{\mathop{\rm sup\_{ess}}\nolimits}
$
What can be concluded from the fact, that $f: X\ni x\mapsto f(x)\in [a,b]\subset\mathbb{R}\setminus\lbrace{-\infty,+\infty\rbrace} $ has the following property:
$$ f(x\_0) = \lim\_{\epsilon\to 0} \essSup\left(f\left(x\in X\cap U\_\epsilon\left(x\_0\rig... | https://mathoverflow.net/users/31310 | Implications of a Regularity Condition for Functions | Example: the characteristic function of a fat Cantor set (a compact nowhere dense set of positive Lebesgue measure): it is upper semicontinuous, so the equality holds; it has a positive set of discontinuities, thus it is not Riemann integrable.
| 3 | https://mathoverflow.net/users/6101 | 258399 | 116,636 |
https://mathoverflow.net/questions/232819 | 20 | (Sorry for the length of the question, I'm trying to communicate what is bothering me as thoroughly as possible)
In the construction of modular tensor categories (MTC) from ground zero, we put structures one by one:
* Tensor Product Structure $\to$ **Monoidal Categories**
* Dual Objects $\to$ **Rigid Monoidal Categ... | https://mathoverflow.net/users/74936 | Modular Tensor Categories: Reasoning behind the axioms | Interesting question! As far as I know, there are at least two secretly equivalent answers.
You somehow already gave the first one: a modular tensor category is the same as a modular functor (though the precise statement is quite subtle, see the beautiful introduction to this paper: <https://arxiv.org/abs/1509.06811>... | 21 | https://mathoverflow.net/users/13552 | 258400 | 116,637 |
https://mathoverflow.net/questions/258363 | 12 | Many papers in random matrix theory make passing references to duality between eigenvalue statistics of the GOE and GSE, for which the most concrete reference I can find is <https://arxiv.org/pdf/math-ph/0206011.pdf> (page 4 has the relevant discussion). Though I'm somewhat unclear on the actual matrix integral specifi... | https://mathoverflow.net/users/76764 | GOE/GSE duality and Bott periodicity | The entire set of correspondences can be read off from this table:
Listed are the 10 symmetric spaces and for each space in the left column the dual space is shown in the right column, as explained in [On duality and negative dimensions in the theory of Lie group... | 8 | https://mathoverflow.net/users/11260 | 258403 | 116,638 |
https://mathoverflow.net/questions/258357 | 8 | Let $L$ be a $p$-adic field $G = GL\_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G\_1\times \ldots \times G\_r$, for $G\_i \cong GL\_{n\_i}(L)$.
The following are classical:
1). If $\rho\_i$ are discrete series representations of $G\_i$, $\rho$ is the $M$ represe... | https://mathoverflow.net/users/30726 | How does Jacquet's "Generic Representations" classify tempered representations? | Do you assume that $\pi$ is irreducible ? You can find a proof in the book of David Renard, available on his webpage, p.343, VII.2.6.
| 5 | https://mathoverflow.net/users/92193 | 258409 | 116,642 |
https://mathoverflow.net/questions/258395 | 13 | If $X$ is an infinite set and $f:X\to X$, do we have $$\big|\{g:X\to X: g\circ f =f \circ g\}\big|\geq |X| ?$$
| https://mathoverflow.net/users/8628 | Cardinality of the set of functions commuting with $f:X\to X$ | I think I can show this in general $X$, heavily borrowing from Goldstern's constructions.
Lemma 1: Let $\alpha$ be the supremum of numbers of parents of elements of $X$. If, for each $x\_1,x\_2 \in X$, there is some $n\_1,n\_2$ with $f^{n\_1}(x\_1) = f^{n\_2}(x\_2)$, and $X$ is uncountably infinite, then $\alpha = |X... | 7 | https://mathoverflow.net/users/18060 | 258416 | 116,644 |
https://mathoverflow.net/questions/258412 | 4 | Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X\_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X\_1, X\_1] \times \cdots \times [-X\_n, X\_n]$. Could someone point me to some references related to this. Thank you very much!
| https://mathoverflow.net/users/84272 | Counting number of points on a lattice in a hypercube | After applying a suitable invertible linear transformation on $\mathbb{R}^n$, the lattice $\Lambda$ becomes $\mathbb{Z}^n$, and the box $[-X\_1, X\_1] \times \cdots \times [-X\_n, X\_n]$ becomes a parallelotope $\mathcal{R}$. So the task becomes to estimate the cardinality of $\mathcal{R}\cap\mathbb{Z}^n$.
Now, by t... | 7 | https://mathoverflow.net/users/11919 | 258417 | 116,645 |
https://mathoverflow.net/questions/258418 | 7 | There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one)
$$
\sum\_\lambda s\_\lambda(x)=\prod\_{i}\frac{1}{1-x\_i}\prod\_{i<j}\frac{1}{1-x\_ix\_j}
$$
and
$$
\sum\_\lambda s\_\lambda(x) s\_\lambda(y)=\prod\_{i,j}\frac{1}{1-x\_iy\_j}.
... | https://mathoverflow.net/users/3840 | Sum of the ratios of Schur functions | I am skeptical of any "nice" structure while $x$ and $y$ remain so free. It might be somewhat reasonable to check things out under certain specializations.
You may find some interesting quotients, after specializations on $x$ and $y$, in Richard Stanley's *Enumerative Combinatorics*, Vol. 2, Exercises 7.30 and 7.32.
... | 3 | https://mathoverflow.net/users/66131 | 258423 | 116,648 |
https://mathoverflow.net/questions/258422 | 3 | Let $F$ be an algebraic closed field of char(F)=0 and let $(a,b)\in F^2$ be a non-zero vector. Suppose I have two polynomials $f,g\in F[x]$ with $\deg f\neq \deg g$ and $1\leq \deg f, \deg g$. I want to show that
\begin{equation}
\gcd(f(x)-f(y)-a, g(x)-g(y)-b)=1.
\end{equation}
For instance, by Eisenstein criterion, i... | https://mathoverflow.net/users/91357 | greatest common divisor of polynomials with two variables | Not a complete argument but here are some ideas. Suppose $f$ has degree $d$ and let $c\_1,\ldots,c\_{d-1}$ be the zeros of $f'(x)$. Consider the curve $X: f(x)-f(y)-a$ in the projective plane. It can only be singular at points of the form $(c\_i,c\_j)$ with $a=f(c\_i)-f(c\_j)$. So if $a$ is not one of those $(d-1)(d-2)... | 3 | https://mathoverflow.net/users/2290 | 258426 | 116,650 |
https://mathoverflow.net/questions/258454 | 1 | I haven't taken any number theory courses but out of curiosity I learned about [Dirichlet's approximation theorem](https://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem). Afterwards, it occurred to me to define the following function $f$ using 'optimal' diophantine approximants:
$\forall \alpha \in \mathbb{... | https://mathoverflow.net/users/56328 | Beyond Dirichlet's approximation theorem | Since $\alpha$ is irrational, the [equidistribution theorem](https://en.wikipedia.org/wiki/Equidistribution_theorem) says that the fractional parts $\{q\alpha\}$ are uniformly distributed in $[0,1)$. In particular, a positive proportion of $q$ satisfy $\{q\alpha\} \in [\frac13,\frac12]$. For these $n$, we have $|\alpha... | 5 | https://mathoverflow.net/users/5091 | 258455 | 116,659 |
https://mathoverflow.net/questions/258459 | 31 | The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$.
>
> What happens if we replace $\mathbf{R}$ by $\mathbf{Q}\_p$?
>
>
>
It is well-known that the theorem fails if we replace... | https://mathoverflow.net/users/6506 | The Sylvester-Gallai theorem over $p$-adic fields | If $n \geq 3$ and $K$ is a field of characteristic not dividing $n$, containing a primitive $n$-th root of unity $\zeta$, then the $3n$ points of the form $(1:-\zeta^a:0)$, $(0:1:-\zeta^b)$, $(-\zeta^c:0:1)$ are a Sylvester-Gallai configuration. In particular, taking $n=p-1$, this gives an SG configuration over $\mathb... | 31 | https://mathoverflow.net/users/297 | 258473 | 116,665 |
https://mathoverflow.net/questions/258469 | 12 | Yesterday one my friend told about recent book of Hiroshi Toda, where the computations of 3-torsion in homotopy groups of spheres are given up to a very high stem (about 75). This book is published with a rare publisher and is overlooked by MathSciNet and ZentralblattMath. Does anybody know the explicit reference: titl... | https://mathoverflow.net/users/103052 | Toda's book on homotopy groups of spheres | If I've got the right reference, this appears to be listed as:
* "Unstable 3-primary homotopy groups of spheres, Study of Econoinformatics (Vol. 29), Himeji: Himeji Dokkyo University" in the book "Gottlieb and Whitehead Center Groups of Spheres, Projective and Moore Spaces" by Marek Golasiński and Juno Mukai,
* "Unst... | 13 | https://mathoverflow.net/users/360 | 258475 | 116,667 |
https://mathoverflow.net/questions/258389 | 1 | I have taken an introductory course on measure theory where I learned about the Borel-Cantelli theorem but I wonder whether there is a lebesgue integrable version. Given an uncountable collection of independent events $E\_{t \in \mathbb{R}\_+}$,
$$ \int\_0^{\infty} P(E\_t) dt <\infty \implies P( E\_t\quad i.o. )=0\t... | https://mathoverflow.net/users/56328 | Integrable version of the Borel-Cantelli theorem? | The second Borel-Cantelli lemma has the additional condition that the events are *mutually independent*. This requirement becomes problematic for an uncountable index set. For example, suppose that $\{E\_t\}\_{0\le t\le 1}$ is a collection of mutally independent events such that $(\omega,t)\mapsto 1\_{E\_t}(\omega)$ is... | 4 | https://mathoverflow.net/users/42851 | 258479 | 116,669 |
https://mathoverflow.net/questions/258463 | 17 | I was reading about the $\Omega$-conjecture and have thought of a refutation of it, which seems too simple to not have been noticed since the $\Omega$-conjecture has been around, so i'm skeptical and want to see whether anyone can spot a flaw in it.
I will assume there is a proper class of hyper-huge cardinals. This ... | https://mathoverflow.net/users/102873 | What is the error in this disproof of the $\Omega$-conjecture? | Woodin's theorem says that assuming the $\Omega$ Conjecture and the existence of a proper class of Woodin cardinals, the set $\mathcal V\_\Omega$ of $\Pi\_2$ sentences that hold in every universe of the generic multiverse is lightface definable over $H\_{\delta^+}$ where $\delta$ is the least Woodin cardinal. You claim... | 15 | https://mathoverflow.net/users/102684 | 258529 | 116,680 |
https://mathoverflow.net/questions/258350 | 2 | A matrix $M$ of order $n$ is an [MDS](https://en.wikipedia.org/wiki/MDS_matrix)
(Maximum Distance Separable)
matrix if and only if every sub-matrix of $M$ is non-singular.
For a matrix of order $n$, we should obtain $\sum\_{i=1}^n \, {n \choose i }^2$ determinant to find out
that a matrix is MDS or not. So, when t... | https://mathoverflow.net/users/64181 | The probabilistic method to find out a matrix is MDS | Since my comment is long, I write it as an answer, but it is not a complete answer and just give some insight.
Firstly, based on the paper you mentioned and based on the applications of MDS matrices in cryptography, I think the elements of the matrix belong to the finite field.
When the size of a randomly chosen ma... | 3 | https://mathoverflow.net/users/19885 | 258531 | 116,681 |
https://mathoverflow.net/questions/258491 | 13 | Can anyone tell me what $H\_3(SL\_n(\mathbb{Z});\mathbb{Z})$ and $H\_3(SL\_n(\mathbb{F}\_p);\mathbb{Z})$ are? It is easy to find references for $H\_1$ and $H\_2$, but it turns out that I need $H\_3$ as well. All I care about are the stable values.
| https://mathoverflow.net/users/103063 | H_3 of SL(n,Z) and SL(n,F_p) | Summarizing the comments, the stable ($n\geq 3$) values of $H\_3(SL\_n;\mathbb Z)$ are
* $H\_3(SL\_\infty(\mathbb Z);\mathbb Z) = \mathbb Z/24$
* $H\_3(SL\_\infty(\mathbb F\_q);\mathbb Z) = \mathbb Z/(q^2-1)$
and can be found in Weibel's [The $K$-book](http://www.math.rutgers.edu/~weibel/Kbook.html).
Namely, one ... | 14 | https://mathoverflow.net/users/78 | 258533 | 116,683 |
https://mathoverflow.net/questions/258525 | 16 | Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\displaystyle \ f'= e^{\displaystyle {f}^{-1}}?$$
**Note 01:** $f' =\displaystyle\frac{df}{dx}$.
**Edit:** ${f}^{-1}$ is the in... | https://mathoverflow.net/users/51189 | How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$? | There is no such function. Since $f$ would have to map $\mathbb R$ *onto* $\mathbb R$ for the equation to make sense at all $x\in\mathbb R$, it follows that $f^{-1}(x)\to -\infty$ also as $x\to -\infty$, so $f'\to 0$. Thus $f(x)\ge x$, say, for all small enough $x$, hence $f^{-1}(x)\le x$ eventually, but then the equat... | 36 | https://mathoverflow.net/users/48839 | 258544 | 116,687 |
https://mathoverflow.net/questions/258573 | 2 | Let $G=(V,E)$ be an infinite simple, undirected graph. Let $\text{MC}(G)$ denote the set of maximal cliques in $G$. It is easy to see that the union of $\text{MC}(G)$ is $V$, so $\text{MC}(G)$ is a vertex cover of $G$.
If ${\cal C} \subseteq \text{MC}(G)$ is a vertex cover, is there a vertex cover ${\cal M}\subseteq ... | https://mathoverflow.net/users/8628 | Minimal coverings by maximal cliques | Nice question. The answer is no, not necessarily.
**Theorem.** There is a graph $G$ such that there is no minimal vertex
covering of it by maximal cliques. Indeed, in every vertex covering
$\cal C$ of $G$ by maximal cliques, every vertex appears in infinitely
many of the cliques in $\cal C$, and so one can omit any d... | 3 | https://mathoverflow.net/users/1946 | 258576 | 116,695 |
https://mathoverflow.net/questions/258555 | 1 | Let $G$ be a finite simple group in which there is no element of order $p^2$ for all primes $p\mid\vert G\vert$. Suppose that $H$ is a finite group whose number of nontrivial proper subgroups is as same as $G$ and there is a bijection $\phi$ from the set of nontrivial subgroups of $G$ to the set of nontrivial subgroups... | https://mathoverflow.net/users/97247 | A criterion for determining simple groups | The answer is "yes". Indeed, much more is true. Refer to the following paper:
*Michio Suzuki*, MR 39717 [**On the lattice of subgroups of finite groups**](http://www.ams.org/mathscinet-getitem?mr=39717), *Trans. Amer. Math. Soc.* **70** (1951), 345--371.
The bijection that is mentioned by the OP is a *lattice isomo... | 5 | https://mathoverflow.net/users/801 | 258580 | 116,696 |
https://mathoverflow.net/questions/258553 | 1 | Our following question is motivated by [this very interesting answer](https://mathoverflow.net/questions/258293/is-a-pointwise-simple-tensor-valued-continuous-map-a-tensor-product-of-two-co/258297#258297)
Assume that $A$ is a $C^{\*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all ... | https://mathoverflow.net/users/36688 | Continuous factors for invertible simple tensors | Well, you don't get "finite dimensional", since this is true for all infinite-dimensional commutative $C^\*$-algebras too.
We can suppose $A = C\_0(\Omega)$ for some locally compact Hausdorff $\Omega$. Then $X$ is identified with an appropriate subset of $C\_0(\Omega^2)$: those $x : \Omega^2 \to \mathbb{C}$ which can... | 2 | https://mathoverflow.net/users/4832 | 258590 | 116,699 |
https://mathoverflow.net/questions/258579 | 2 | I asked this question on Stackexchange, but I got no answer within 24 hour, so I ask it here.
Iwasawa's criterion can be stated in the two following forms :
1° Let $G$ be a perfect group acting faithfully and primitively on a set $X$ with a least two elements. Assume that there exists an element $x$ of $X$ such tha... | https://mathoverflow.net/users/82840 | Slight strengthening of Iwasawa's criterion | (Converting Mark Wildon's comment and mine into an answer.)
Your strong version of Iwasawa's Criterion is Theorem 1.2 of the following book:
*Donald E. Taylor*, MR 1189139 [**The geometry of the classical groups**](http://www.ams.org/mathscinet-getitem?mr=1189139), *Sigma Series in Pure Mathematics* ISBN: 3-88538-0... | 2 | https://mathoverflow.net/users/801 | 258592 | 116,700 |
https://mathoverflow.net/questions/257562 | 17 | **Notation**: Suppose $\mathbf{A}$ and $\mathbf{B}$ are positive definite matrices in $\mathbb{R}^{n\times n}$ such that $\mathbf{A} \succeq \mathbf{B}$ (Loewner order). Let $\mathcal{S}(n,k)$ be the set of all $k$-subsets of $\{1,2,\dots,n\}$. For any $\mathcal{Q} \subset [n] \triangleq \{1,2,...,n\}$, $\mathbf{M}\_\m... | https://mathoverflow.net/users/102641 | A determinant inequality | As suspected, the desired inequality actually holds for all hyperbolic polynomials; the inequality in the OP follows as corollary (Corollary 1) to Theorem 2 (which seems to be new).
We will need the following remarkable theorem to obtain our result.
>
>
> >
> > **Theorem 1 ([Bauschke, Güler, Lewis, Sendov, 200... | 6 | https://mathoverflow.net/users/8430 | 258595 | 116,703 |
https://mathoverflow.net/questions/258597 | 9 | Consider the following infinite game: two players, I and II, are alternating and choosing a descending sequence of subsets of $\mathbb R$ of cardinality $\frak c$, so I chooses a set $A\_1\subseteq\mathbb R$, II chooses a set $A\_2\subseteq A\_1$, I chooses $A\_3\subseteq A\_2$ etc., all having the size continuum. Let ... | https://mathoverflow.net/users/30186 | Choosing subsets of $\mathbb R$ of cardinality $\frak c$, who wins? | In ZFC, the player aiming for the empty set has a winning strategy in the game played on any infinite set, including the reals. Using the axiom of choice, we can well-order the set and thereby pretend that we are playing sets of ordinals. Now, whenever it is the empty-set player's turn, she should look at the order typ... | 13 | https://mathoverflow.net/users/1946 | 258598 | 116,704 |
https://mathoverflow.net/questions/258596 | 1 | Suppose $R$ is a (possibly noncommutative) ring. I was thinking of $R=S[x\_1,\ldots,x\_n]$ or $R=S[x\_1,x\_1^{-1},\ldots,x\_n,x\_n^{-1}]$ for $S$ some (possibly noncommutative) ring. Now, let $GL\_2(R)$ be the group of invertible matrices over $R$, and $E(R)$ the subgroup of $GL\_2(R)$ generated by matrices of the form... | https://mathoverflow.net/users/102861 | Under what condition can any $X\in GL_2(R)$ be reduced to a triangular matrix? | This an extended comment rather than an answer.
First of all, when working in the general linear group over a non-commutative ring, one should be very careful in translating the usual notions from the commutative setting. For example, consider a ring $R$ and two elements $x$ and $y$ such that $xy=1$ but $yx\neq1$. Th... | 3 | https://mathoverflow.net/users/5018 | 258599 | 116,705 |
https://mathoverflow.net/questions/258591 | 7 | Over $\mathbb{A}^n$, it is easy to construct D-modules by writing down an explicit linear system of PDE's and then writing a presentation of the associated D-module
$$
\mathcal{D}^n \xrightarrow{} \mathcal{D}^m \to \mathcal{M} \to 0
$$
Is there an analogous process for constructing D-modules over projective space as an... | https://mathoverflow.net/users/78824 | How can I construct D-modules over projective space using explicit differential equations? | You can construct $\mathcal{D}$ modules on projective space (or actually any space) by precisely the same process: write down a system of PDE's (using differential operators on the whole of $\mathbb{P}^n$) and consider the corresponding quotient. The key caveat is that on a general space, not every $\mathcal{D}$-module... | 7 | https://mathoverflow.net/users/66 | 258624 | 116,712 |
https://mathoverflow.net/questions/258625 | 1 | Denote the set of $n\times n$ [permutation matrices](http://mathworld.wolfram.com/PermutationMatrix.html) by $\mathfrak{S}\_n$. The ordinary transpose preserves this group.
Given $P\in\mathfrak{S}\_n$, construct the $n\times n$ matrix ${}^tP$ according to the rules:
(1) leave the 4 rims unchanged (1st row, 1st colu... | https://mathoverflow.net/users/66131 | transposing "unrimmed" permutations | The answer is $a\_n=8(n-2)!$ for $n\ge 4$.
Subdivide into cases depending on the values at corners and count. (6 cases give 1, 1 case gives 2 times $(n-2)!$)
| 5 | https://mathoverflow.net/users/5793 | 258628 | 116,714 |
https://mathoverflow.net/questions/258552 | -3 | Let $R$$\subset$$ S$ be commutative noetherian rings,and $I$ is an ideal of $S$.
We now that $I$ is a $R-$module.
Do we have $grade\_{R}(I)$ $\le$ $ grade\_{S}(I)$?
Thank you!
| https://mathoverflow.net/users/92487 | Depth or Grade of an ideal | This is false. Take any $R$-module $M$ of grade $>0$, and put $S=R\oplus M$, the product of any two elements of $M$ being $0$. Then $M$ is an ideal of $S$ and $\mathrm{grade}\_S(M)=0\ $ (because $\mathrm{Hom}\_S(S/M,S)\neq 0$).
| 0 | https://mathoverflow.net/users/40297 | 258632 | 116,715 |
https://mathoverflow.net/questions/69037 | 44 | This is an extension of [this](https://mathoverflow.net/questions/14739/how-can-i-define-the-product-of-two-ideals-categorically) question. Let $I,J$ be ideals of a ring $R$; every ring is commutative and unital here. Is it possible to define $R \to R/(I\*J)$ out of $R \to R/I$ and $R \to R/J$ in categorical terms with... | https://mathoverflow.net/users/2841 | Categorical definition of the ideal product within the category of rings | The unitalization approach can be made to work.
Let $C\_K = \{ (r,s) \in R \times R \mid r-s \in K \}$ be the congruence defined by an ideal $K$.
Then, we have three maps defined on $S = C\_I \otimes\_R C\_J$ :
* $\pi\_0 : S \to C\_I$ induced by the first projection $C\_J \to R$ and the identity on $C\_I$
* $\pi\... | 4 | https://mathoverflow.net/users/nan | 258633 | 116,716 |
https://mathoverflow.net/questions/258589 | 5 | Let $M$ be a connected closed surface (possibly with non-zero genus) and let $P\subset M$ be a nonempty finite set of points. Set $\dot{M} = M \setminus P$. Let $\pi : E \rightarrow \dot{M}$ be a complex vector bundle of rank $2$. Is this vector bundle trivial?
| https://mathoverflow.net/users/103113 | Is a complex vector bundle over a punctured closed surface trivial? | For any topological group $G$, there is a classifying space $BG$ and a principal $G$-bundle $EG \to BG$ called the universal principal $G$-bundle which is determined up to isomorphism by the fact that $EG$ is weakly contractible. On a paracompact topological space $X$, any principal $G$-bundle $P \to X$ admits a map $f... | 11 | https://mathoverflow.net/users/21564 | 258646 | 116,719 |
https://mathoverflow.net/questions/258448 | 13 | Motivated by [this MO question](https://mathoverflow.net/questions/258284/is-the-matrix-left2m-choose-2j-i-right-i-j-12m-1-nonsingular/258291#258291). Consider the two matrices $A\_n$ and $B\_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A\_n=\det B\_n... | https://mathoverflow.net/users/66131 | in search of a transformation between determinants | There is such a transformation, of the form predicted in [Linear transformation that preserves the determinant](https://mathoverflow.net/questions/522/linear-transformation-that-preserves-the-determinant?rq=1).
Denoting $R$ the involution matrix $e\_i\mapsto e\_{n+1-i}$, it turns out that the matrix $A$ has an $LU$-d... | 9 | https://mathoverflow.net/users/6101 | 258648 | 116,720 |
https://mathoverflow.net/questions/258662 | 0 | The question is already in the title.
It is known that any subgroup of a free group is free. My question is:
**Is a closed subgroup of a free profinite group is again a free profinite group ?**
| https://mathoverflow.net/users/82229 | Subgroup of free profinite group is free profinite? | No. The free profinite group $\widehat{\mathbb{Z}}$ on one generator is the direct product of the groups $\mathbb{Z}\_p$, $p$ prime.
Therefore each $\mathbb{Z}\_p$ is a closed subgroup of $\widehat{\mathbb{Z}}$, but is not free as a profinite group.
| 5 | https://mathoverflow.net/users/101929 | 258663 | 116,726 |
https://mathoverflow.net/questions/258668 | 2 | Let $S$ be a (unit) sphere in a Hilbert Space $H$ with $\dim H \ge 3$. Let $A \subset S$ have the following properties:
1. $A$ is connected;
2. The affine hull of $A$ is the whole space;
3. For every $x,y\in A$ there is a unitary operator $T:H\to H$ (or, alternatively, an isometry $T:S\to S$), such that $TA=A$, and s... | https://mathoverflow.net/users/53155 | Homogeneous subsets of the sphere | The answer is 'no'.
For example, let $H$ be the (5-dimensional) space of symmetric, traceless $3$-by-$3$ matrices, where the Hilbert inner product is $\langle x,y\rangle = \tfrac12\mathrm{tr}(xy)$. Let $A\subset S$ be the subset consisting of those matrices of unit norm for which $0$ is an eigenvalue. Then $A$ is co... | 7 | https://mathoverflow.net/users/13972 | 258669 | 116,728 |
https://mathoverflow.net/questions/258641 | 6 | Consider
$H\_\*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e\_i$ the generator of $H\_i(BZ/2;Z/2)$., we see that the $E\_2=E\_{\infty}$ term of the Bockstein spectral sequence
for $X\wedge Y$ is trivial, thus the permanent cycles are the image of $\beta$. As we have
$$\beta e\_{2j}=e\_{2j-1}$... | https://mathoverflow.net/users/43326 | Homology of the product of spaces with integer coefficients and the Massey products | Here's a very general form. Suppose that we have six chain complexes $A\_0, A\_1, A\_2, A\_{01}, A\_{12}, A\_{012}$, with bilinear "multiplication" pairings of chain complexes:
$$
\begin{align\*}
A\_0 \otimes A\_1 &\to A\_{01}\\
A\_1 \otimes A\_2 &\to A\_{12}\\
A\_{01} \otimes A\_2 &\to A\_{012}\\
A\_0 \otimes A\_{12} ... | 9 | https://mathoverflow.net/users/360 | 258671 | 116,729 |
https://mathoverflow.net/questions/258674 | 1 | Hilbert transform defines as follow:
>
> $$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$
>
>
> $$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$
>
>
> Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \hat f$ is the fourier transform of $f(x)$, and $\mathcal{F}^{-1}(F(\gamma))$ is the inverse fourier transform... | https://mathoverflow.net/users/76453 | Representation of Hilbert transform by a singular integral | This is covered in many classical references where the Hilbert transform is defined one way and the other derived (and vice-versa). Take a look at [this note, page 42-](http://www.cims.nyu.edu/~chou/notes/harmonic.pdf) and reverse the argument shown there. An excellent reference is:
Elias Stein, *Singular Integrals a... | 3 | https://mathoverflow.net/users/66131 | 258676 | 116,731 |
https://mathoverflow.net/questions/258619 | 2 | Let $f(x)=a\_nx^n+\cdots+a\_0$ be a polynomial of degree $n$. To judge whether $f(x)$ has a square factor, we only need to consider the resultant $Res(f,f')$.
My question is, if we want to judge whether $f(x)$ has a $d$-multi factor, what can we do? Of course we can consider the resultant $Res(f^{(d-2)},f^{(d-1)})$ ... | https://mathoverflow.net/users/37096 | How to judge whether a polynomial have a factor of multiplicity $d$ | Let's rehomogenize to a binary form $F(x\_1,x\_2)=x\_2^n f
\left(\frac{x\_1}{x\_2}\right)$. Then (assuming $a\_n\neq0 $) the question is about detecting a factorization $F=L^d G$ where $L$ is linear. Consider all multiindices $\alpha=(\alpha\_1,\alpha\_2)$ of length $|\alpha|=d-1$ and introduce two sets of $d$ indeterm... | 5 | https://mathoverflow.net/users/7410 | 258680 | 116,732 |
https://mathoverflow.net/questions/258630 | 4 | Recall that any [extensive category](https://ncatlab.org/nlab/show/extensive+category) can be canonically endowed the structure of a site via the extensive topology, which is the Grothendieck topology whose covering morphisms are the coproduct injections.
I've heard that ($Cat$-valued) stacks for the extensive topolo... | https://mathoverflow.net/users/102957 | Stacks for the extensive topology? | This a genral fact: the coproducts in an extensive category are disjoint so the sheaf/stack condition with respect to a cover by the coproduct injection exactly say that the coproduct is sent to a product. So if they are the only cover a sheaf or a stack is just a functor sending the coproduct to products.
| 2 | https://mathoverflow.net/users/22131 | 258689 | 116,735 |
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