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https://mathoverflow.net/questions/133497 | 3 | Hello everyone,
Let $S\_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S\_g$ endowed with the topology of uniform convergence.
Let $h$ be in $\mathcal{A}$. Since $g \geq 2$, $S\_g$ is covered by $D^2 \simeq \mathbb{R}^2$, and... | https://mathoverflow.net/users/25511 | Random metrics on compact orientable surfaces | If you take any reasonable probability measure on $\mathcal{A}$, then with probability $1$ there should be a single point of maximum curvature ; then the isometry group of $\tilde h$ must preserve a $\Gamma$-orbit, namely the set of the maximum curvature points in the universal covering. You should also be able to sing... | 5 | https://mathoverflow.net/users/4961 | 133506 | 73,796 |
https://mathoverflow.net/questions/130486 | 3 | Let $X \subseteq \mathbf{R}^n$ be Zariski closed and absolutely irreducible, and let $U \subseteq \mathbf{R}^n$ be Euclidean open. My guess is that $X$ is the Zariski closure of $X \cap U$, if the latter is nonempty.
In the case of $X = \mathbf{R}^n$, [the proof as given in this answer](https://mathoverflow.net/quest... | https://mathoverflow.net/users/33946 | What is the Zariski closure of a locally closed set, when "locally" means the Euclidean topology? | This answer is distilled from the comments by Jérémy Blanc, François Brunault and xuhan.
The guess that $X$ is the Zariski closure of $X \cap U$, if the latter is nonempty, is false in general. As a counterexample, take the Zariski closed set in $\mathbf{R}^2$ given by the zero set of the equation
$x^2 + x^3 + y^2 ... | 2 | https://mathoverflow.net/users/33946 | 133514 | 73,801 |
https://mathoverflow.net/questions/133504 | 0 | What are necessary and sufficient conditions on a binary quadratic form $ax^2+bxy+cy^2$, with integer coefficients and solution set in integers, to be equivalent to $x^2-y^2$, and separately to $x^2+y^2$?
The conditions would be in terms of $a,b,c$ but can we say anything about the transformation matrix? Since we are... | https://mathoverflow.net/users/32575 | Classification of these Binary Quadratic Forms | For $x^2 + y^2$, the discriminant $b^2 - 4ac$ is $-4$, and the class number $h(-4)$ is $1$. So the condition is just that the discriminant matches up: $b^2 - 4ac = -4$, and the form is positive definite ($a, c > 0$). For $x^2 - y^2$, I think you just have to differentiate it from $2xy$, which also has discriminant $4$.... | 7 | https://mathoverflow.net/users/2698 | 133515 | 73,802 |
https://mathoverflow.net/questions/133461 | 7 | I need to graph real valued functions (for exposition and analysis).
The issue is: there are more independent variables so that the conventional graphing methods can't be used, and furthermore I don't want to slice the functions.
1. These functions are like $s=f\_1(x,y,z,t)$ and $s=f\_2(x,y,z,t,k)$
2. I also have vec... | https://mathoverflow.net/users/34859 | Visualizing functions with a number of independent variables | For time-dependent and three-dimensional data, there exist a number of established programs already; among the most popular free and open source ones are [Paraview](http://www.paraview.org/) and [VisIt](https://wci.llnl.gov/codes/visit). Both support a variety of plots such as isosurfaces, volume rendering or (for vect... | 4 | https://mathoverflow.net/users/30516 | 133524 | 73,808 |
https://mathoverflow.net/questions/133477 | 4 | Let $f : S \to X$ be a dominant morphism of smooth complex compact surfaces. Let $C \subset S$ be a smooth curve such that $df$, seen as a map from $T\_S \to T\_X$, is generically of rank $1$ along $C$ and that in a neighborhood of $C$, $df\_x$ is an isomorphism outside $C$. Suppose that $C$ is not contracted by $f$, i... | https://mathoverflow.net/users/24290 | Branch locus along a smooth curve | The question in essentially local on $X$, so the compacteness assumption is irrelevant.
The answer to both questions is no: consider $S\subset \mathbb C^3$ defined by $z^3+xz+y=0$ and the map to $X=\mathbb C^2$ given by $(x,y,z)\mapsto (x,y)$.
The ramification curve $C$ is smooth (given by $z^3+xz+y=3z^2+x=0$) whil... | 6 | https://mathoverflow.net/users/10610 | 133531 | 73,810 |
https://mathoverflow.net/questions/133511 | 4 | I have a very specific question about two series of GIT quotients.
Let $n\geq 2$ be an integer and let $k$ be another positive integer which is less than
$n$.
Then the first series of quotients is this: consider the Grassmannian $G(k,n)$ of $k$-planes
in $\mathbb C^n$. It has an action of an $n-1$-dimensional torus... | https://mathoverflow.net/users/3891 | Isomorphism between some GIT quotients | Yes, it's obvious (and due to Gel$'$fand-MacPherson). Start with $k\times n$ matrices and act with $GL(k) \times T^n$, then reduce in stages in either order.
Half my thesis was about the fact that the Gel$'$fand-Cetlin system on the Grassmannian can be carried over to an integrable system on the space $X(k,n)//PGL(k)... | 4 | https://mathoverflow.net/users/391 | 133535 | 73,812 |
https://mathoverflow.net/questions/133529 | 4 | Let $\Omega$ be the halting probability (see ([http://en.wikipedia.org/wiki/Chaitin's\_constant](http://en.wikipedia.org/wiki/Chaitin%27s_constant)) and R. Downey, and D. Hirschfeldt (2010), Algorithmic Randomness and Complexity for reference). If A is Martin-Löf random with respect to $\Omega$ (so A is 2-random), does... | https://mathoverflow.net/users/23835 | About infinite subset of halting probability and 1-random set | No. Since $A$ is ML-random relative to $\Omega$, by van Lambalgen's theorem it follows that $\Omega$ is ML-random relative to $A$. If $A$ could compute such a set $G = \{g\_0 < g\_1 < g\_2 < \dots \}$, then $A$ could form the ML-test $(V\_n)\_{n \in \omega}$, where $V\_n = \{ X : (\forall i < n )[X(g\_i) = 1]\}$. Note ... | 6 | https://mathoverflow.net/users/32178 | 133540 | 73,817 |
https://mathoverflow.net/questions/133544 | 5 | Is anyone aware of a result (or a counterexample) along the following lines: let $G$ be an algebraic group over $\mathbf Z$. Let $H$ be a finite group such that $H$ occurs as a subgroup of $G(\bar{\mathbf F}\_p)$ for all but finitely many primes $p$. Then $H$ also occurs as a subgroup of $G(\mathbf C)$.
I only really... | https://mathoverflow.net/users/34918 | Subgroups of algebraic groups | The functor of injective homomorphisms from $H$ to $G$ is represented by a scheme of finite type over $\mathbb Z$ (a locally closed subscheme of the product $G^H$). If this has points over $\overline{\mathbb F}\_p$ for infinitely many $p$, then it must dominate $\mathop{\rm Spec}\mathbb Z$, so it has points over $\math... | 10 | https://mathoverflow.net/users/4790 | 133549 | 73,822 |
https://mathoverflow.net/questions/133342 | 21 | There's a close relationship between curvature and the holonomy group; the holonomy theorem of Ambrose and Singer, for example. It seems to me that there should be an analogous result for torsion. I believe that torsion measures the extent to which certain geodesic parallelograms don't close. Is there a theorem that ma... | https://mathoverflow.net/users/14454 | Torsion and parallel transport | Here is another way to think of the relation between torsion and parallel transport, one that some may find more congenial than many of the other interpretations that have been proposed:
Start with a manifold $M$, and a connection $\nabla$ on $TM$. Consider the bundle $\hat TM=\mathbb{R}\oplus TM$ (where, by $\mathbb... | 28 | https://mathoverflow.net/users/13972 | 133576 | 73,835 |
https://mathoverflow.net/questions/72213 | 12 | Matrices $A$ and $B$ are *integrally equivalent* if there is an invertible integer matrix $L$ and $L^{-1}AL=B$. Suppose $f(t)$ is an integer polynomial with no repeated factors. Latimer and MacDuffee proved that the number of integral similarity classes of matrices with characteristic polynomial equal to $f(t)$ is equa... | https://mathoverflow.net/users/1266 | Ideal classes and integral similarity | A very belated answer to 1), but: I just saw that this is treated very nicely in Curtis and Reiner's *Representation Theory of Finite Groups and Associative Algebras*. Theorem 20.6 therein not only works with non-maximal orders $\mathbb{Z}[\theta]$ but even does the non-commutative case as well. And the argument is ver... | 9 | https://mathoverflow.net/users/1149 | 133577 | 73,836 |
https://mathoverflow.net/questions/133555 | 0 | Let C be a circle of radius 1 in the complex plane with n points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other n points. The goal is to provide an answer based on how densely the point are positioned on the circle.
The goal is to characterize this ... | https://mathoverflow.net/users/34919 | upper bound on product of distances from points on a circle | Taking logs, it seems you are interested in a bound on $$\sum\_1^n\log|a-e^{2\pi i\theta\_k}|$$ where $a$ is your given point on the unit circle. It is natural to estimate this by comparing the sum (or, better, the average, $(1/n)\sum\_1^n\log|a-e^{2\pi i\theta\_k}|$) to the integral, $$\int\_0^1\log|a-e^{2\pi i\theta}... | 0 | https://mathoverflow.net/users/3684 | 133579 | 73,837 |
https://mathoverflow.net/questions/133553 | 1 | We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J\_g^t M J\_g$ with $J\_g$ being the canonical almost complex structure - or involution whatever you prefer to call it.
The principal c... | https://mathoverflow.net/users/24392 | Reference on elements of finite order in principal congruence subgroups of symplectic groups | This is a well known argument (for any even $g \geq 2$, and for other integral matrix groups) . If $x$ is an element in the level $q$ congruence subgroup, then $\frac{x-I}{q}$ is an integral matrix, so has all its eigenvalues algebraic integers. If $\alpha$ is any eigenvalue of $x,$ then $\alpha$ is a root of unity as ... | 5 | https://mathoverflow.net/users/14450 | 133582 | 73,838 |
https://mathoverflow.net/questions/133586 | -2 | Given a nonsingular projective variety $X$ with a close subvariety $Y \subset X$, let the inclusion map be $i : Y \rightarrow X$. Let $A(X)$ and $A(Y)$ be the Chow ring of $X$ and $Y$ respectively, is the push forward map $i\_\* : A(Y) \rightarrow A(X)$ a ring homomorphism? (ref: Hartshorne AG, appendix A, top of pg 42... | https://mathoverflow.net/users/34924 | Is the Chow ring's push forward of inclusion map a ring homomorphism? | No. Try $\{0\} \hookrightarrow {\mathbb P}^1$.`
| 3 | https://mathoverflow.net/users/391 | 133587 | 73,841 |
https://mathoverflow.net/questions/133583 | 1 | One can of course give a power series for the derivative of RiemannZeta. But is there a "closed form", e.g. expressible in the field of functions generated by Zeta and Gamma and some constants?
| https://mathoverflow.net/users/12669 | derivative of Riemann zeta function | Perhaps this has what you're looking for?
<http://www.jstor.org/stable/2007806>
| 0 | https://mathoverflow.net/users/29979 | 133588 | 73,842 |
https://mathoverflow.net/questions/133567 | 6 | We know that if $f : X\to Y$ is a morphism between two affine varieties over an algebraically closed field $k$, then the function that assigns to each point of $X$ the dimension of the fiber it belongs to is upper semicontinuous on $X$.
Does anyone know of a simple counterexample when $X$ is not irreducible (but rema... | https://mathoverflow.net/users/3333 | Counter example of upper semicontinuity of global fiber dimension on the source | Let $X = (\mathbb{A}^2 \setminus \{x = 0\}) \coprod \mathbb{A}^1$, let $Y = \mathbb{A}^1$, and let $f$ be projection onto the first coordinate on the first component and the identity on the second. Then every point of $X$ lives in a one-dimensional fiber except the origin of the second component.
| 8 | https://mathoverflow.net/users/396 | 133590 | 73,843 |
https://mathoverflow.net/questions/133146 | 32 | Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$. Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and similarly for $Y \sqcup Y$). Then I'm pretty sure that we must have $X \cong Y$. This clearly holds if $X$ and $Y$ are con... | https://mathoverflow.net/users/34785 | Homeomorphisms and disjoint unions | The result you want is *false*. Counterexamples are given in
Yamamoto, Shuji and Yamashita, Atsushi,
A counterexample related to topological sums.
Proc. Amer. Math. Soc. 134 (2006), no. 12, 3715–3719.
These counterexamples are compact subsets of $\mathbb{R}^4$.
| 35 | https://mathoverflow.net/users/317 | 133596 | 73,844 |
https://mathoverflow.net/questions/133597 | 32 | The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By lexical ordering of finite formulas we see that the set of finite formulas is countable. So it is impossible to define ... | https://mathoverflow.net/users/nan | What would remain of current mathematics without axiom of power set? | This is a real question.
You're not the only one who find the power set axiom dubious. At the time of the great foundational controversies, Russell and Weyl both expressed a similar view. It is now known, from work of Weyl, Wang, Feferman, the reverse mathematics school, and others, that the vast bulk of mainstream m... | 45 | https://mathoverflow.net/users/23141 | 133598 | 73,845 |
https://mathoverflow.net/questions/133090 | 12 | As a preface to this question, this is my first time asking on Math overflow, and this seemed like the sort of question that would be acceptable here. However, I apologize if it is not.
A method for coding models from a Gale-Stewart game proceeds as follows: Our aim is to obtain a model of the form $L\_\gamma[a]$ suc... | https://mathoverflow.net/users/34726 | Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game | This problem is very much open.
Cheng Yong calls *Harrington's $\star$* the assumption that there is a real $x$ such that all $x$-admissible ordinals are $L$-cardinals. From the work of Yong we know that Second- and even Third-order arithmetic do not suffice to prove that Harrington's $\star$ implies the existence o... | 16 | https://mathoverflow.net/users/6085 | 133600 | 73,847 |
https://mathoverflow.net/questions/133527 | 4 | Is there any ellipse with nonzero rational minor and major axis lengths $a$ and $b$ such that the circumference of the ellipse is rational too? (or the weaker variant: ... such that the circumference is algebraic?)
Note that the surface area $S=\pi ab$ is of course always transcendental in such a situation.
| https://mathoverflow.net/users/33927 | any ellipse with rational axes and circumference? | No, there is no such ellipse. This is exactly theorem $6.5$ of Alan Baker's book TRANSCENDENTAL
NUMBER THEORY, as pointed out by Felipe Voloch.
Let $\omega$ be a primitive period of a $\wp$-function with algebraic
invariants $g\_2, g\_3$ and let $\eta=2\zeta(\omega /2)$ be the associated quasi-period of the
Weierstra... | 8 | https://mathoverflow.net/users/32332 | 133603 | 73,849 |
https://mathoverflow.net/questions/89179 | 2 | So I'm looking at a diffusion process with killing with a state- and time-dependent killing rate. This is described in Oksendal's Stochastic differential equations pages 143-145 "The Feynman-Kac Formula. Killing". Basically, you have a generator
$$
L f = -\sum\_i \frac{\partial }{\partial x\_i} A\_i(x,t) f(x) + \frac{1... | https://mathoverflow.net/users/19243 | Is the Feynman-Kac formula valid for a time-dependent potential | I was probably a bit confused because of the operators used. The operator shown in the question is the Kolmogorov forward operator. The backwards operator is the adjoint of $L$, given by
$$
L^\* f = \sum\_i A\_i(x,t) \frac{\partial f}{\partial x\_i} + \sum\_{i,j}\frac{B\_{i,j}(x,t)}{2}\frac{\partial^2 f}{\partial x\_i ... | 1 | https://mathoverflow.net/users/19243 | 133609 | 73,852 |
https://mathoverflow.net/questions/133145 | 11 | Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the form:
$$v:U \times \mathbb{R} \rightarrow \mathbb{R}^3: (x,y,z) \mapsto (0,0,w(x,y))$$
which satisfies:
$$\frac{\partial^2... | https://mathoverflow.net/users/33927 | Do circular pipes maximize flow rate? | The answer is positive, this is a consequence of a result by Talenti ["Elliptic Equations and Rearrangements", Annali SNS 3 (1976)]. Let $k>0$, $-\Delta u= k$ in $U$, $u=0$ on $\partial U$, and consider the ball $B$ having the same volume as $U$. If $v$ is the solution of $-\Delta v=k$ in $B$, $v=0$ on $\partial B$, th... | 6 | https://mathoverflow.net/users/34463 | 133622 | 73,855 |
https://mathoverflow.net/questions/133625 | 3 | I read that $O(M)$ is connected if and only if $M$ is not orientable. Is it true? If there is an answer, I would very much like to see a proof. Thank you.
| https://mathoverflow.net/users/34938 | When the orthonormal frame bundle is a connected manifold? | If you assume that $M$ itself is connected, then it is true.
First, if $M$ is orientable and connected, then $O(M)$ has (at least) two connected components, one for each orientation of frame, either agreeing with a specified orientation or not.
If $M$ is not orientable, then, using the Levi-Civita connection for th... | 6 | https://mathoverflow.net/users/13972 | 133626 | 73,856 |
https://mathoverflow.net/questions/133621 | 3 | Consider the (Deligne-Mumford compactification of the) moduli space of complex rational marked curves $\overline M\_{0;n}$. For each $i\in \{1,\ldots,n\}$ we can construct a line bundle $L\_i$ with a fiber given by cotangent space to the $i$-th marked point, and the divisor corresponding to $L\_i$ is the $\psi$-class.
... | https://mathoverflow.net/users/13921 | Dimension of the linear system of $\psi$-class on $\bar M_{0;n}$ | I think that this is done by Kapranov in "Veronese curves and Grothendiexk-Knudsen moduli space $M\_{0,n}$".
The projective dimension of this linear system should always be $n-3$ (so the linear dim is $n-2$) and the map is a birational morphism which is the inverse of a sequence of blow-ups.
| 2 | https://mathoverflow.net/users/4096 | 133637 | 73,861 |
https://mathoverflow.net/questions/133644 | 8 | I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-projective variety?
| https://mathoverflow.net/users/13441 | Fundamental groups of normal complex quasi-projective varieties | Yes, sure. Take any polycyclic group which is not virtually nilpotent (e.g.
upper triangular matrices with entries in $GL\_n(\mathbb{Z})$, $n\ge 2$). This cannot be the fundamental group of a normal quasiprojective variety by a theorem of Nori and myself [Solvable fundamental groups of algebraic varieties..., Composit... | 13 | https://mathoverflow.net/users/4144 | 133646 | 73,865 |
https://mathoverflow.net/questions/133564 | 5 | Let $k$ be a number field and let $I\_k$ denote the idele group of $k$. Let
$$|\cdot|: (x\_v) \mapsto \prod\_{v \in \Omega\_k} |x\_v|,$$
denote the adelic norm map.
If $I\_k^1$ denotes the kernel of this map, then we have a short exact sequence
$$1 \to I^1\_k \to I\_k \to \mathbb{R}\_{>0} \to 1. \qquad (\*)$$
Next, re... | https://mathoverflow.net/users/5101 | Does every equivalence class of Hecke characters contain a distinguished element? | Dear Daniel, the answer is yes.
An easy key lemma:
---
**Lemma**: Let $\alpha: \mathbb R^\ast\_+ \rightarrow \mathbb C^\ast$ be a continuous character,
and $n \geq 1$ an integer. Then there exists one and only one character
$\beta: \mathbb R^\ast\_+ \rightarrow \mathbb C^\ast$ such that $\beta^n = \alpha$.
... | 6 | https://mathoverflow.net/users/9317 | 133647 | 73,866 |
https://mathoverflow.net/questions/132145 | 8 | In a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a "nice" decomposition into circles (<http://en.wikipedia.org/wiki/Seifert_fiber_space>). That is, $M$ is decomposed into circles in a way such that $M$ has neighbourhoods which are "fibred as a solid tori would be, if these tori are given by... | https://mathoverflow.net/users/34473 | Seifert Fibrations and their associated Spectral Sequence | Indeed, you always get a fibration $M\to \hat B$, where $\hat B$ is the Haefliger's classifying space of the orbifold (the space whose cohomology is the orbifold cohomology of $B$). The fiber of this fibration is the principal leaf of your Seifert fibration.
One can write the corresponding spectral sequence. And inde... | 5 | https://mathoverflow.net/users/10086 | 133653 | 73,869 |
https://mathoverflow.net/questions/133616 | 8 | In a [paper of Casson and Gordon's](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=casson&s5=gordon&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=722728) "A loop theorem for du... | https://mathoverflow.net/users/1465 | A procedure to determine if an automorphism of a closed 2-manifold extends to an automorphism of a handlebody | This problem is solved in the paper "Algorithmic compression of surface automorphisms" by Casson and Long. They remark that their motivation is (partly) the paper of Casson and Gordon.
| 7 | https://mathoverflow.net/users/1650 | 133654 | 73,870 |
https://mathoverflow.net/questions/133659 | 12 | For a finite group $G$ with normal subgroup $H$, the induced representation $\text{Ind}\_H^G(1)$ decomposes as a sum of irreducibles with the multiplicities equal to the dimensions, because it is is the pullback of the right regular representation of $G/H$. For a subgroup which is not normal, this need not be true. For... | https://mathoverflow.net/users/6756 | Uncertainty principle on finite groups | Yes, it is. If for any irreducible $\rho$, $\langle\text{Ind}\_H^G(1),\rho\rangle$ is either 0 or dim $\rho$, then $H$ is the intersections of $\ker \rho$ for those $\rho$ for which this inner product is not 0,
and intersections of kernels of irreducible characters are precisely the normal subgroups of $G$.
Indeed, ... | 7 | https://mathoverflow.net/users/35416 | 133665 | 73,874 |
https://mathoverflow.net/questions/133662 | 12 | My current research involves locally compact groups and from time to time I am tempted to check whether certain notions and statements of geometric group theory of finitely generated groups are still meaningful and valid in the large scale geometry (or coarse geometry) of locally compact groups. But I don't even know i... | https://mathoverflow.net/users/nan | What are the best settings for the large scale geometry of locally compact groups? | I can give an example from my own papers which uses these ideas, a result of myself, Sageev, and Whyte in our paper "Quasi-actions on trees I: Bounded valence", MR1998479. This is a rigidity theorem for locally compact groups $G$ that contain a discrete cocompact finite rank free subgroup: $G$ must act continuously, pr... | 5 | https://mathoverflow.net/users/20787 | 133670 | 73,877 |
https://mathoverflow.net/questions/133660 | 2 | Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c+x \mid c\in C \}$, for some $x\in R^d$, is called a homothetic copy of $C$. The number $\lambda > 0$ is called the coefficient of homothety.
Let $C$ be a $d$-dimensional cube of side length 1. Now draw ... | https://mathoverflow.net/users/12844 | Covering the annulus of d-cube | $3^d-1$ and $3^d$, resp. Let $C=[0,1]^d$, Consider the $3^d$ points in $C$ all whose coordinates are from the set $\{0,\frac12,1\}$. No translated copy of $C'$ can cover two of these points, hence at least $3^d$ copies of $C'$ are required. On the other hand, a covering by $3^d$ copies of $C'$ (with the requirement tha... | 5 | https://mathoverflow.net/users/4354 | 133671 | 73,878 |
https://mathoverflow.net/questions/133673 | 12 | Turaev--Viro <http://www.ams.org/mathscinet-getitem?mr=1191386> defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\right|^2$ where $WRT(M)$ is the Witten--Reshetikhin--Turaev invariant of $M$. Both invariants depend on a choice of *qu... | https://mathoverflow.net/users/35353 | Are Turaev--Viro invariants secretly a discretized path integral? | Perhaps I asked this question a bit too quickly. An explanation exactly as I was looking for appears at the beginning of Section III of this paper: <http://arxiv.org/abs/hep-th/0401076>
| 6 | https://mathoverflow.net/users/35353 | 133676 | 73,879 |
https://mathoverflow.net/questions/132886 | 3 | Let $G$ be a locally compact (Hausdorff) group and let $H$ be a finite index subgroup of $G$. Can we say that $H$ has to be a closed subgroup of $G$? If it is not correct, do you know any counterexample?
Remember, we can always replace $H$ by a normal finite index subgroup (i.e. $N:=\bigcap\_{g\in G} gHg^{-1}$). This... | https://mathoverflow.net/users/nan | Finite index subgroups of locally compact groups | Here's a cw answer so as not to leave this unanswered. If $p$ is prime, $D$ is a countable abstract set, then
(1) the product $C^D$ has exactly $\aleph\_0$ closed subgroup of index $p$ (recall that for a finite index subgroup, closed=open), but
(2) it has $2^{2^{\aleph\_0}}$ abstract subgroups of index $p$.
The a... | 0 | https://mathoverflow.net/users/14094 | 133683 | 73,883 |
https://mathoverflow.net/questions/115446 | 8 | Suppose V is a finite-dimensional vector space and I have a linear subspace of its endomorphisms
$$W \subseteq \mbox{End}(V).$$
How can I easily check if every vector of $V$ is fixed by some element of $W$? I would also be interested in any nice conditions on $W$ that imply
$$\forall v \in V: \;\;\; v \in Wv,$$
even i... | https://mathoverflow.net/users/9068 | Subspaces of End(V) that can fix any vector | The question is closely related to the notion of algebraic reflexivity.
The (algebraic) reflexive closure of $W$ is the space of all operators $g \in End(V)$ such that $g(x) \in Wx$ for all $x \in V$.
The author asks when the identity operator belongs to the reflexive
closure of the operator space $W$.
I will cons... | 14 | https://mathoverflow.net/users/34951 | 133691 | 73,888 |
https://mathoverflow.net/questions/133652 | 0 | Suppose $X$ is a smooth projective complete intersection contained in the product $\mathbb{P}^n \times \mathbb{P}^m$, and call $X\_n$ and $X\_m$ the images of $X$ inside $\mathbb{P}^n$ and $\mathbb{P}^m$ via the two natural projections. Suppose moreover that I have the two Koszul complexes of $X\_n\subset \mathbb{P}^n$... | https://mathoverflow.net/users/4096 | Koszul complex of a variety inside a product | In general, you can't. By taking quotients, knowing the Koszul complex of $X$ tells you $X$. But there are many subvarieties of $X\_n \times X\_m$ that project surjectively onto $X\_n$ and $X\_m$. If $X\_n$ and $X\_m$ are both positive-dimensional, then the intersection of $X\_n \times X\_m$ with any hypersurface of bi... | 1 | https://mathoverflow.net/users/18060 | 133692 | 73,889 |
https://mathoverflow.net/questions/133554 | 5 | Hi.
Suppose we arrange all natural numbers in a matrix P defined as follows:
P[I][J] = The Jth number with I prime factors. So P looks something like:
1
[2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , ...](http://oeis.org/A000040)
[4 , 6 , 9 , 10 , 14 , 15 , 21 , 22 , 25 , 26 , 33 , 34... | https://mathoverflow.net/users/27456 | The Nth number with M prime factors | Each sequence of "$n$-almost-primes" opens with a string of even numbers that are twice the entries that begin the previous sequence. The first odd $n$-almost-prime is $3^n$. Thus the sequence the OP has observed, $2,4,7,13,22,38,63,\ldots$, is basically the sequence [A078843](http://oeis.org/A078843) $(3,5,8,14,23,39,... | 2 | https://mathoverflow.net/users/15837 | 133695 | 73,892 |
https://mathoverflow.net/questions/133675 | 1 | Assume we have a commutative diagram of functors
$$
\array{
X&\stackrel{f^\*}{\to}& Y\\
{\scriptstyle{g^\*}}\downarrow\phantom{m} & & \phantom{m}\downarrow{\scriptstyle{k^\*}}\\
Z&\stackrel{h^\*}{\to}& T
}
$$
(where the natural isomorphism filling the 2-cell is not explicitly written), and assume that all the functors ... | https://mathoverflow.net/users/8320 | A Beck-Chevalley type condition | The answer is *always*, provided you choose compatible 2-cells filling the first two diagrams. Probably, you didn't see it because you removed from notation these 2-cells. Let us give a name to the natural isomorphism for the first square:
$$\zeta\colon k^\ast f^\ast\cong h^\ast g^\ast$$
You need not demand the sec... | 5 | https://mathoverflow.net/users/12166 | 133701 | 73,894 |
https://mathoverflow.net/questions/133672 | 24 | Hello to everyone,
I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6.
At some point, he needs to show that in a presentable category $\mathcal C$ and a large enough set $S$ of morphisms in $\math... | https://mathoverflow.net/users/12773 | Is Lemma A.1.5.7 in Higher Topos Theory correct? | Looks like a typo. Condition $(4)$ should say that $B$ is downward closed under $\leq$, not under $\preceq$ (otherwise, $Y\_B$ is not defined).
| 42 | https://mathoverflow.net/users/7721 | 133703 | 73,895 |
https://mathoverflow.net/questions/133705 | 0 | I'm trying to multiply together a set of numbers.
However, the programs I have tried (Excel, R, FreeMat) do not allow numbers above around 300 digits. This limits me to sets of a few hundred numbers at most.
What program can I use that isn't limited by the number of digits? And what size numbers can I realistically... | https://mathoverflow.net/users/34952 | What program/language can I use to handle large numbers (over 10^300) | * Every computer algebra system for symbolic manipulations, for example [Sage](http://sagemath.org) or [Maxima](http://maxima.sourceforge.net/)
* Some programming languages have built-in support, for example Python, Haskell and Ruby
* [GMP](http://gmplib.org/) is a well-established library for C and C++
The size of n... | 2 | https://mathoverflow.net/users/33842 | 133706 | 73,897 |
https://mathoverflow.net/questions/133639 | 5 | Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the standard hamiltonian S^1-action (Diagonal action) to define the blow-up.
It should be possible to define blow-ups of genera... | https://mathoverflow.net/users/5259 | Symplectic blow-up | I don't know a place where this is written up explicitly, but it's not hard as soon as one has an appropriate standard model for a neighborhood of a symplectic manifold for use in the Weinstein Neighborhood theorem.
A model that I've found useful in other contexts (see also p. 12 of [this paper](http://arxiv.org/abs... | 6 | https://mathoverflow.net/users/424 | 133709 | 73,898 |
https://mathoverflow.net/questions/133667 | 1 | Are there explicit relations between hypergeometric functions over finite fields of different size?
In particular, if we consider the function 2F1(A,B,C|x) over Fp (i.e., with characters A,B and C defined over Fp), and the same hypergeometric function but over Fp^2 (i.e., with characters defined over Fp^2) or any other... | https://mathoverflow.net/users/13913 | Finite field hypergeometric functions. | Assuming by the hypergeometric function, you mean the trace of Frobenius the hypergeometric sheaves, there is a nice relation.
First I will mention the right way to move a character to a higher degree finite field. If $A$ is a character of $\mathbb F\_q^\times$, then the corresponding character of $\mathbb F\_{q^n}^... | 2 | https://mathoverflow.net/users/18060 | 133711 | 73,899 |
https://mathoverflow.net/questions/133712 | 2 | I was reading [a paper](http://www.ias.ac.in/j_archive/proca/11/2/73-80/viewpage.html) about solutions to $f(y) = P(m)$, where $f(y) \in \mathbb Z[y]$ and $P(m) = n(n + 1) \ldots (n + m - 1)$ is a product of $m$ consecutive integers.
On the first page, it is mentioned that it is not difficult to get the solutions in ... | https://mathoverflow.net/users/27598 | solutions to $f(y) = n(n + 1) \ldots (n + m - 1)$ | As to your first question, indeed, it is hard to construct explicit example, when solution exists unless $P(m)=f(y)$ is an identity. When the author claims, that for an irreducible polynomial it is easy to solve the problem, most likely, he intended to write that it is easy to get the condition on $m,$ when solution do... | 2 | https://mathoverflow.net/users/17503 | 133716 | 73,901 |
https://mathoverflow.net/questions/132408 | 9 | With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question:
>
> (**1**) Let's say we know that a finite simplicial complex $S$ is the barycentric subdivision of some other simplicial complex $K$. Can we find $K$ (up to simplicial isomorphism) from know... | https://mathoverflow.net/users/18263 | Can we invert barycentric subdivision? | [As requested by Vidit Nanda, I am reposting a slightly edited version of my comment above as an answer. Nevertheless, I hope someone will eventually give a satisfactory answer to this question.]
The following article is relevant for this question: "[*The insufficiency of barycentric subdivision*](http://projecteucli... | 3 | https://mathoverflow.net/users/21095 | 133725 | 73,906 |
https://mathoverflow.net/questions/133666 | 3 | Euler came up with following recurrence relation for the sum of divisors (refer to <http://arxiv.org/abs/math/0411587>)
$$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since $\sigma(p) = p+1$, where $p$ is a prime number, we can use the recurrence relation to verify if a number is prime. I... | https://mathoverflow.net/users/34928 | Finding primes using Euler's sum of divisors recurrence relation | Perhaps a summary of the comments is useful. Using Euler's formula for $\sigma(n)$ to test $n$ for primality
certainly is not efficient. The values $\sigma(n-1),\sigma(n-2),\sigma(n-5),\cdots $ are hard to
compute.
More precisely, in general computing $\sigma(n)$ is equivalent to factoring $n$ in the following sens... | 3 | https://mathoverflow.net/users/32332 | 133728 | 73,907 |
https://mathoverflow.net/questions/133730 | 6 | This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either.
Fourier series define an isometry $L^2(\mathbb{Z}) \rightarrow L^2(S^1), (a\_k)\_{k \in \mathbb{Z}} \mapsto \sum a\_k z^k \colon S^1 \rightarrow \mathbb{C}$ (all Hilbert... | https://mathoverflow.net/users/18256 | Fourier series representing a continuous function? | Maybe you are already aware of the following if-and-only-if condition, though I wouldn't say it counts as "easily checkable": let $(a\_k)$ be the coefficient sequence, let $s\_N$ denote the partial sums
$$
s\_N(z) = \sum\_{k=-N}^N a\_k z^k
$$
and let $\sigma\_N(z)$ denote the Cesaro means
$$
\sigma\_N(z)=\frac{1}{N+1}... | 10 | https://mathoverflow.net/users/13360 | 133737 | 73,912 |
https://mathoverflow.net/questions/133724 | 5 | I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the geodesics are even $C^{1, \beta}$ where $\beta = \alpha/(2-\alpha)$.
However, I did not find any results on the existen... | https://mathoverflow.net/users/16702 | Existence of Geodesics in continuous metrics | Ok, thank you Misha for the comments, let me try to fill out the hints you gave myself. I try to prove the following:
Let $g\_n$ be a sequence of complete smooth metrics that converge in $C^0$ against the continuous metric $g$. Let $p$, $q \in M$ such that they are not cutpoints of each other w.r.t. any $g\_n$. Let ... | 3 | https://mathoverflow.net/users/16702 | 133740 | 73,914 |
https://mathoverflow.net/questions/133738 | 1 | Let $V\_1, V\_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s\_1: M \rightarrow V\_1$
a smooth section transverse to the zero set and $s\_2: M \rightarrow V\_2$ a continuous section
such that $ s\_2 : s\_1^{-1}(0) \rightarrow V\_2$ is a smooth section, transverse to the zero
set. Let $p\in M$ b... | https://mathoverflow.net/users/4463 | A version of implicit function theorem when sections are not everywhere smooth? | Note that by your assumption $s\_2$ is only smooth along the submanifold $s^{-1}(0)$ of $M$ and might even jump in directions transverse to it. So it is easy to come up with a counterexample: $M=\mathbb R^2$, $s\_1(x,y)=x$, $s\_2(x,y) = y$ for $x\le 0$ and $=y+1$ for $x>0$.
But if you assume that $s\_2$ is strictly ... | 2 | https://mathoverflow.net/users/26935 | 133741 | 73,915 |
https://mathoverflow.net/questions/133650 | 2 | It is well known that if a smooth curve $C \subset \mathbb{P}^3$ has degree $ d \leq 6$. Then
$ g(C) \leq 4$ (Hartshorne pg 354). I know that the case $g=4$ correspond to the complete intersection of a quadratic and a cubic surface. Therefore, the ideal of the curve is
$$
I\_C = (f\_2(x\_0, x\_1,x\_2, x\_3),f\_3(x\_0,... | https://mathoverflow.net/users/16409 | explicity equations for curves in the projective space | As explained by J.C. Ottem, the curves of degree $6$ are contained in a cubic. Let us assume that the surface is smooth, so you can see the cubic surface as the blow-up of six points in $\mathbb{P}^2$ with no $3$ collinear and not all on the same conic.
The surface $X$ has Picard group generated by $L$, the preimage ... | 2 | https://mathoverflow.net/users/23758 | 133746 | 73,917 |
https://mathoverflow.net/questions/133755 | 2 | Let P be the Euclidean plane and let C be a compact and convex subset of P whose interior is non-empty.
Does there always exist a strictly increasing sequence of positive real numbers l(1),l(2),...,l(n),...
as well as a strictly decreasing sequence of positive real numbers e(1),e(2),...,e(n),... converging to
zero such... | https://mathoverflow.net/users/4423 | A question about how completely a rectifiable arc can fill a non-empty compact continuum that is the closure of its interior | No, the lengths must go to infinity.
If $s$ is a path of length $l$ in the plane and $\varepsilon>0$, then the area of the $\varepsilon$-neighborhood of $s$ is no greater than $20\varepsilon(l+\varepsilon)$. Indeed, $s$ can be divided into at most $l\varepsilon^{-1}+1$ subintervals of length at most $\varepsilon$. Pi... | 7 | https://mathoverflow.net/users/4354 | 133760 | 73,922 |
https://mathoverflow.net/questions/132904 | 3 | If the Gauss hypergeometric function
$F(1, 3/2, 5/2; z^2) = 3 [\tanh^{-1} (z) –z]/z^3$
what is the corresponding result for $F(1,15/8,23/8;z^8) $?
| https://mathoverflow.net/users/34704 | hyperbolic functions and Gauss hypergeometric series | There is a special form for hypergeometric functions of the form
$$ F(1,\beta,1+\beta,z) = \beta z^{-\beta}B(z,\beta,0) $$
when $\beta = p/q$ is a rational number (where $B$ is the incomplete beta function).
Write
$$ \frac{z^\beta}{\beta}F(1,\beta,1+\beta,z) = \sum\_{k\geq0} \frac{z^{\beta+k}}{\beta+k} = q\sum\_{k\ge... | 1 | https://mathoverflow.net/users/10423 | 133761 | 73,923 |
https://mathoverflow.net/questions/133772 | 0 | here's my question:
Let $V$ be a k-dimensional vector space over $\mathbb{R}$ and $q$ a quadratic form on $V$ of signature $(m,n)$ , $m+n=k$.
We have $W\subset V$ a positive (with respect to the quadratic form, of course) subspace of dimension $m-1$. Is it always possible to find a vector $c\in V$ such that $\langl... | https://mathoverflow.net/users/34980 | Positive subspaces of quadratic forms | Write $V= W + W^{\perp}$. Any vector $c$ can be decomposed into $c\_{W}+ c\_{W^\perp}$. $W+c$ is a positive subspace if and only if $c\_{W^\perp}$ is positive. The positive vectors in $W^{\perp}$, a quadratic space of signature $(1,n)$, form a (nonempty) solid bicone. So there is always such a vector, and the space of ... | 0 | https://mathoverflow.net/users/18060 | 133775 | 73,929 |
https://mathoverflow.net/questions/133785 | 2 | Consider a one-sided ( say, internal) neighborhood $U$ of the unit circle $S$ ( $U$ contains $S$) on the plane with a choice of smooth complex stricture $\tau$ on $U$.
By smoothness of $\tau$ on $U$ we mean that corresponding operator of almost-complex structure $J$( defined in the interior of $U$) as a matrix with r... | https://mathoverflow.net/users/34984 | What is the moduli space of germs of one-sided complex structures near the circle? | I think that, even if you include the boundary $S$, they are all equivalent. In saying so, I'm assuming that the final 'it' in your first sentence refers to the plane, not to $U$ and that you are considering smooth complex structures.
The reason is this: Take a sufficiently small open neighborhood $W$ of the closure... | 2 | https://mathoverflow.net/users/13972 | 133795 | 73,935 |
https://mathoverflow.net/questions/133803 | 4 | Let $(R, \mathfrak{m})$ be a noetherian local ring, and let $\hat R$ be its $\mathfrak{m}$-adic completion. Extension of scalars allows one to transform an $R$-scheme into an $\hat R$-scheme. Is this association surjective, up to isomorphism? More precisely, given a finite type scheme $X \to \mathrm{Spec} \ \hat R$, do... | https://mathoverflow.net/users/13410 | Schemes over a noetherian local ring and its completion | This is really just ayanta's example again: even with the condition on the generic fiber, the answer is still "no". Let $R$ be the local ring of $k[s,t]$ at the maximal ideal $\langle s,t\rangle$. Begin with $Y\_R$ being $\mathbb{P}^1\_R$ -- this is not the final scheme. For the base change $Y\_{\widehat{R}}$ of $Y\_R$... | 5 | https://mathoverflow.net/users/13265 | 133822 | 73,943 |
https://mathoverflow.net/questions/133750 | 13 | This question is inspired by a recent talk by [Matt Kahle](http://matthewkahle.org/) on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean space with the usual norm $\|\cdot\|$ and let $\mathcal{B}^k$ denote the $k$-fold product of t... | https://mathoverflow.net/users/18263 | What fraction of n-point sets in the unit ball have diameter smaller than 1? | I am certainly not the best person to answer this question, as I do not have much insight to share regarding how to approach this kind of problems. My only (fairly obvious) suggestion is to *estimate* the relevant quantities in any way possible. In this process, it can be very helpful to reduce the calculations to lowe... | 11 | https://mathoverflow.net/users/21095 | 133823 | 73,944 |
https://mathoverflow.net/questions/133758 | 5 | Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the equivariant cohomologies $H\_{T}(X;\mathbb{C})$ and $H\_{T}(X^T;\mathbb{C})$ using the existing localization theory. Yet, I ... | https://mathoverflow.net/users/25358 | Equivariant Cohomology of a Complex Projective Variety | Usually one uses a Białynicki-Birula decomposition derived from a general circle in $T$ to find cycles that give a basis of this free module. But this decomposition is less useful when the space is singular.
For a basic example, let $X$ be the triangle $\{ [x,y,z] : xyz = 0 \} \subset {\mathbb P}^2$. This space has ... | 5 | https://mathoverflow.net/users/391 | 133839 | 73,951 |
https://mathoverflow.net/questions/133840 | 8 | Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ?
Note that over a $p$-adic field, this is true for the Galois representations in the $p$-adic étale cohomology of $X$.
(With a bit more details: fix a field $K$ complete for a discrete valuation, wit... | https://mathoverflow.net/users/17988 | Potentially good, semi-stable reduction => good reduction ? | No. Take $K=\mathbb Q\_3$. Consider the projective genus $0$ curve $x^2+y^2+3z^2$. This has bad reduction, since it has no rational points. It is semistable, since after adjoining $i$ it has exactly that form. It has potentially good reduction, since all genus $0$ curves do.
So you at least need to say it has good r... | 9 | https://mathoverflow.net/users/18060 | 133851 | 73,955 |
https://mathoverflow.net/questions/133866 | 0 | If A and B are coprime integers, it seems that at least one of the exponents of the prime factors of AB(A+B) must be one or two.
<http://www.comparativetables.com/laeb.htm>
Is this interesting?
| https://mathoverflow.net/users/34958 | Generalisation of the Beal conjecture | This generalization is false and it fails infinitely often.
Some counterexamples:
$$ 271^3 + 2^3 3^5 73^3=919^3$$
$$ 3^4 29^3 89^3 + 7^3 11^3 167^3=2^7 5^4 353^3$$
In general consider the elliptic curve: $x^3 + y^3 = k$ where
k is 3-full integer, i.e. every exponent is at least 3.
It may have infinitely many rati... | 7 | https://mathoverflow.net/users/12481 | 133871 | 73,963 |
https://mathoverflow.net/questions/133872 | 1 | Is true that two birational projectives nonsingular curves have the same genus?
I know that for a non singular projective curve the genus is given by the formula
1/2(d-1)(d-2), where d is the degree of the curve.
Do i just can see it from this?
| https://mathoverflow.net/users/35018 | the genus of a nonsingular projective curve is a birational invariant | Two non-singular projective curves are birational if and only if they are isomorphic (a birational map between smooth projective varieties has indeterminacy points in codimension $\ge 2$).
The geometric genus of an irreducible projective curve $C$ is the arithmetic genus of its desingularisation $\tilde{C}$. Moreove... | 6 | https://mathoverflow.net/users/23758 | 133874 | 73,965 |
https://mathoverflow.net/questions/133847 | 10 | Let $A$ and $B$ be $n\times n$ real matrices.
When $n=2$, we have the equality
$$A\Big(\mbox{Trace}(B)A-\mbox{Trace}(A)B\Big) B=B\Big(\mbox{Trace}(B)A-\mbox{Trace}(A)B\Big) A.$$
1. Can we give an interpretation to this equality?
2. Are there similar equalities when $ n = 3,4,...$?
| https://mathoverflow.net/users/24060 | A strange matrix equality | Let us rewrite it using the commutators $[P,Q]=PQ-QP$, as follows:
$$
tr(B)[A^2,B]=tr(A)[A,B^2].
$$
Now, for matrices $X$ of size~$2$, we have $X^2=tr(X)X-det(X)I$ (a particular case of Cayley--Hamilton), so
$$
tr(B)[tr(A)A-det(A)I,B]=tr(A)tr(B)[A,B]=tr(A)[A,tr(B)B-det(B)I],
$$
since $I$ commutes with everything.
... | 13 | https://mathoverflow.net/users/1306 | 133876 | 73,966 |
https://mathoverflow.net/questions/133878 | 15 | Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at least from what I skimmed over.
| https://mathoverflow.net/users/13904 | Quintic polynomial solution by Jacobi Theta function. | Do you ask about solution to high degree polynomials? The following are some reference:
* Umemura H. (2007) Resolution of algebraic equations by theta constants. In: Tata Lectures on Theta II. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. doi:[10.1007/978-0-8176-4578-6\_18](https://doi.org/10.1007/978-0-8176-45... | 9 | https://mathoverflow.net/users/14024 | 133879 | 73,967 |
https://mathoverflow.net/questions/133877 | 5 | As we know, there are some open problems of formal languages. I am wondering if there is a somehow complete list of open problem of formal languages. If there isn't such a list, can we make it one as answer?
| https://mathoverflow.net/users/14024 | List of open problems of formal languages | Here is a link which describes some of the open problems:
* <https://cs.uwaterloo.ca/~shallit/Talks/open10r.pdf>
| 9 | https://mathoverflow.net/users/1483 | 133882 | 73,968 |
https://mathoverflow.net/questions/133835 | 3 | Say you have a quadratic algebra $A$ , that is, an algebra defined by a finite list of generators over a ground ring $k$ (either a field or a direct product of fields, which will be assumed $\mathbb{Z}/2$ for simplicity in this question) such that the corresponding ideal of relations $R$ is generated by its quadratic e... | https://mathoverflow.net/users/8041 | Thinking about the quadratic dual graphically | It seems to me like the concept you're searching for is duality of matroids. Your simplices exactly correspond to the dependent sets of the matroid given by your monomials, assuming I understand your definition correctly. Incidentally, it's topologically better to look at the sets of variables that *don't* fit into a r... | 6 | https://mathoverflow.net/users/66 | 133883 | 73,969 |
https://mathoverflow.net/questions/133841 | 3 | What is the inverse Mellin transform of (s-1/2)^k on the vertical line Re(s)=a where
0 < a <1 and k is a natural number?
| https://mathoverflow.net/users/21058 | Mellin Transform | In fact I was aiming at evaluating the expression:
$I\_k=\frac{1}{2\pi i}\int\_{a-i\infty}^{a+i\infty} (s-\frac{1}{2})^k x^{s-1} ds, 0 < a < 1, k\in N $ changing the variable $s-\frac{1}{2}=z$
one can rewrite the integral $I\_k$ as:
$I\_k=\frac{1}{2\pi i}\int\_{a-\frac{1}{2}-i\infty}^{a-\frac{1}{2}+i\infty} z^k... | 2 | https://mathoverflow.net/users/21058 | 133909 | 73,981 |
https://mathoverflow.net/questions/133910 | 7 | [In Terry Tao's notes on the Chowla conjecture](http://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture/), the said conjecture states that for any fixed integer $m>0$ and nonzero $(a\_1,\ldots,a\_m)\in\{0,1\}^m$,
$$
\lim\_{x\rightarrow\infty}\frac{1}{x}\sum\_{n\leq x}\mu(n+1)^{a\_1}\cd... | https://mathoverflow.net/users/29873 | Is there a stronger (but widely believed) version of the Chowla conjecture? | I imagine the "correct" conjecture is the Möbius $s$-tuples conjecture in the form that if $s \in \mathbb{N}$, $\alpha\_1, \ldots, \alpha\_s \in \mathbb{N}$ with at least one $\alpha\_i$ odd, and $d\_1, \ldots, d\_s \in \mathbb{Z}$ distinct, then for all $\varepsilon > 0$ we have that
$$\sum\_{n \leq x}{\mu(n + d\_1)^{... | 11 | https://mathoverflow.net/users/3803 | 133911 | 73,982 |
https://mathoverflow.net/questions/133917 | 0 | Let $\mathfrak{g}$ be a semi-simple finite dimensional Lie algebra.
Denote by $L(\lambda)$ an irreducible finite-dimensional $\mathfrak{g}$-module of highest weight $\lambda$. (I.e. $\lambda$ is integral dominant weight)
Suppose that the multiplicity of $L(\nu)$ in the tensor product $L(\lambda)\otimes L(\mu)$ is great... | https://mathoverflow.net/users/23060 | Does the nonvanishing of a Littlewood-Richardson coefficient implies comparability of highest weights? | What you want is not true. For example, take $\mathfrak{sl}\_3$ and $\lambda = \mu = \omega\_1$, the first dominant weight. Then the multiplicity of $\nu = \omega\_2$ is $1$. But
$$
\nu - \lambda = \omega\_2 - \omega\_1 = \frac13 \alpha\_2 - \frac13 \alpha\_1,
$$
so it is neither negative, nor positive.
On the other... | 1 | https://mathoverflow.net/users/4428 | 133924 | 73,988 |
https://mathoverflow.net/questions/133916 | 9 | Terence Tao [asked](https://mathoverflow.net/questions/86118/non-enumerative-proof-that-there-are-many-derangements) for a non-enumerative proof that a positive proportion of permutations are derangements and got a great [answer](https://mathoverflow.net/questions/86118/non-enumerative-proof-that-there-are-many-derange... | https://mathoverflow.net/users/2663 | Non-enumerative proof that there are many simple permutations? | I think the main idea of Brendan McKay's answer to Terence Tao's question still works. It's a bit more complicated (and less non-enumerative), but still "robust" in the sense of Tao's question.
First one should somehow "estimate away" those permutations that have an interval of length 3 or more. The proportion of su... | 6 | https://mathoverflow.net/users/14302 | 133926 | 73,990 |
https://mathoverflow.net/questions/133865 | 11 | Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.
---
**Definition 1:** A triplet $ (\Sigma,h,k) $ is called an **initial data set** if $ (\Sigma,h) $ is a Riemannian $ 3 $-manifold, $ k $ is a symmetric $ 2 $-form on $ \Sigma $, and both $ h $ and ... | https://mathoverflow.net/users/32467 | Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime? | Your construction won't work as stated. The problem is that $M\_p$ can be too small. Let me give a slightly silly example of why it doesn't work.
Cover $\mathbb{R}^3$ with the two local coordinate charts $U = \lbrace x\_1 < 1\rbrace$ and $V = \lbrace x\_1 > -1\rbrace$, and prescribe on it trivial initial data. An ad... | 7 | https://mathoverflow.net/users/3948 | 133939 | 73,993 |
https://mathoverflow.net/questions/133933 | 5 | Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. For definition of a hyperbolic knot we can avoid the real understanding of what a hyperbolic manifold is by simply sayin... | https://mathoverflow.net/users/27433 | Topological characterisation for a (closed irreducible) hyperbolic 3-manifold | A clear statement is the following:
>
> A compact 3-manifold $M$ is hyperbolic if and only if it has infinite fundamental group and does not contain any essential surface with $\chi \geqslant 0$.
>
>
>
You may remember that by saying that $M$ is hyperbolic unless there is some clear obstruction, and the obstru... | 14 | https://mathoverflow.net/users/6205 | 133940 | 73,994 |
https://mathoverflow.net/questions/133938 | 3 | In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not find any material online about this. Does anyone have reference on that?
| https://mathoverflow.net/users/1956 | Correspondence between fractal sets and trees | Fractals and trees can be connected through the machinery of CP-processes, which are a kind of discrete-time version of the scenery flow. This has been studied intensively in the last few years by my colleague Pablo Shmerkin and coauthors: I think that the paper "Local entropy averages and projections of fractal measur... | 5 | https://mathoverflow.net/users/1840 | 133942 | 73,995 |
https://mathoverflow.net/questions/133941 | 5 | Take a (unital) algebra map $f:A\to B$ between two unital C\* algebras - not necessarily star preserving. Under what circumstances is there a $b\in B$ so that $g(a)=b\ f(a)\ b^{-1}$ is a star algebra map? If not, is there another method of modifying $f$ to get a star algebra map?
| https://mathoverflow.net/users/29625 | when is an algebra map conjugate to a star algebra map | I will assume "algebra map" means "homomorphism." Certainly a necessary condition is that $f$ is bounded. When $B=B(H)$ this question (is every bounded homomorphism similar to a \*-homomorphism?) is known as the "Kadison Similarity Problem" and is still open. The answer is known to be affirmative when $A$ is a nuclear ... | 8 | https://mathoverflow.net/users/13360 | 133944 | 73,996 |
https://mathoverflow.net/questions/133937 | 0 | Hi everyone, first of all i must admit i'm very familiar with quadratic forms and positive subspaces, so i'm sorry if my question is too trivial. So, here's my problem:
Let $L$ be a real vector space of dimension 22 with quadratic form of signature $(3,19)$. Let $V\subset L$ be a positive space of dimension 2. The sp... | https://mathoverflow.net/users/35040 | Subspace generated by positive vectors | Any space with at least one positive vector $w$ is spanned by positive vectors: if $v$ is any vector then $v-\lambda w$ is positive for sufficiently large $\lambda$, and so $v=(v-\lambda w)+\lambda w$ is the sum of two positive vectors.
| 0 | https://mathoverflow.net/users/22989 | 133945 | 73,997 |
https://mathoverflow.net/questions/133974 | 9 | As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{\*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning its cup product structure. I assume this is also well-known, but I would appreciate a reference containing a precise stat... | https://mathoverflow.net/users/25358 | Reference for Ring Structure on Group Cohomology | See Example 3.41, p. 251 of [Hatcher's Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html). This example computes the cohomology ring of finite Lens spaces. Since the infinite Lens space is a $K(\mathbb{Z}/n,1)$, and is a increasing union of finite Lens spaces, the same computation holds for the in... | 10 | https://mathoverflow.net/users/1345 | 133976 | 74,009 |
https://mathoverflow.net/questions/133951 | 3 | I would like to know whether the following problem is decidable.
>
> Given the following system in $x \in [0,1]^n$
>
>
> $$x^T Q\_i x + r\_i = 0 \mbox{ for } i = 1, ..., k$$
>
>
> $$x^T Q\_j x + r\_j \neq 0 \mbox{ for } j = k+1, ..., t$$
>
>
> where $r\_i, r\_j \in [0,1]$ are rational constants and $Q\_i, Q\_... | https://mathoverflow.net/users/35046 | Is the feasibility of a system of non-convex quadratic equations and inequations decidable? | For any fixed $n,k,t$, the feasibility question is a first-order formula in the language of real-closed fields; it would have the components of $x$ as existentially quantified variables and the $r$'s and the entries of the $Q$'s as free variables. Tarski's quantifier elimination theorem for real-closed fields converts ... | 4 | https://mathoverflow.net/users/6794 | 133979 | 74,010 |
https://mathoverflow.net/questions/133981 | 0 | Let $X$ be a topological space, and $Y,Z$ be subspaces. My question is a bit vague and open-ended: when is it the case that, if $Y$ and $Z$ represent the same (nonzero) cycle in homology, then $Y$ and $Z$ have the same topology?
To be a bit more restrictive. Let $X$ be a smooth manifold. Is it always the case that tw... | https://mathoverflow.net/users/23434 | Subvarieties with different topology representing the same cycle | The answer to the first pair of questions is that there's no reason that such cycles should be homeomorphic. The question about algebraic subvarieties also has a negative answer, but presumably more restrictive questions might have more positive answers.
Let's stick to the case when the ambient space $X$ is an $n$-ma... | 1 | https://mathoverflow.net/users/3460 | 133984 | 74,013 |
https://mathoverflow.net/questions/133991 | 0 | Let $\mathcal{C}$ be a small finitely complete category equipped with a Grothendieck (pre)topology $\tau$. For $\ast$ the terminal category (one object, one morphism), denote by $i: \ast \to \mathcal{C}$ the inclusion of the terminal object into $\mathcal{C}$. Equip $\ast$ with the trivial topology in which the only co... | https://mathoverflow.net/users/1797 | Cocontinuous functor out of the terminal category | The functor $i$ does *not* have the cover lifting property in general. If it did, then every epimorphism $X \to 1$ in $\mathbf{Sh}(\mathcal{C}, \tau)$ would be an isomorphism, or equivalently, every $\tau$-cover of $1$ in $\mathcal{C}$ must contain a split epimorphism. (Check the definition of "cover lifting property" ... | 2 | https://mathoverflow.net/users/11640 | 133994 | 74,017 |
https://mathoverflow.net/questions/133954 | 4 | Let us first precise the question : let $T$ be a torus, $\alpha : T \to \mathbb{C}$ be an irreducible character. I am interested in the $T$-equivariant Euler class of the ($T$-equivariant) bundle $\xi\_\alpha:\mathbb{C}\to pt$ where $T$ acts on $\mathbb{C}$ via $\alpha$. $\alpha$ corresponds to an integral form $\alpha... | https://mathoverflow.net/users/21180 | Why is the equivariant Euler class a character ? | Well, there's going to be *some* map $T^\* \to H^2\_T(pt)$ taking a weight $\lambda$ to the equivariant Euler class of the corresponding line bundle over a point. If you add weights, that tensors the line bundles, so adds the first Chern classes; hence this is an additive map between these two free abelian groups. Sinc... | 5 | https://mathoverflow.net/users/391 | 134002 | 74,023 |
https://mathoverflow.net/questions/40196 | 2 | 1. How exactly do we put hyperbolic structures on a sphere with cone point singularities. Should I consider that sphere with cone points as an extended complex plane with punctures endowed with a suitable metric of curvature -1 ?
2. I am having trouble with the following questions when I was reading a paper : the paper... | https://mathoverflow.net/users/6953 | Questions about hyperbolic structures on a sphere with cone point singularities | Concerning your first question, if you're used to Riemann surfaces then one way to understand hyperbolic metrics with cone singularities on the sphere is through a version of the uniformization theorem: given a closed Riemann surface $S$ with marked points $p\_1, \cdots, p\_n$ and angles $\theta\_1,\cdots,\theta\_n\in ... | 5 | https://mathoverflow.net/users/9890 | 134014 | 74,027 |
https://mathoverflow.net/questions/134007 | 3 | Let $L$ be a Dedekind-complete Banach lattice. Let $\mathcal{B}$ be the family of nonempty norm-compact subsets of $L$ that are bounded from below. Endow $\mathcal{B}$ with the topology induced by the Hausdorff metric.
Consider the function $\psi\colon \mathcal{B}\to L$ defined by $\psi(A)= \inf{(A)}$ for all $A\in ... | https://mathoverflow.net/users/26674 | Continuity of lattice operations in Banach lattices | I guess finite dimensionality is needed for Q2. For example, in $\ell\_2$ consider the sets $X\_n = \{n^{-1/2} e\_j : 1 \le j \le n \}$. Then in the Hausdorff metric $X\_n \to \{0\}$ but the infimum of each $X\_n$ has norm one.
It looks like you need to restrict attention to compact sets that are in a common order i... | 2 | https://mathoverflow.net/users/2554 | 134016 | 74,028 |
https://mathoverflow.net/questions/133572 | 23 | I am wondering whether I should try to have some fun using proof systems. I have never used such a system, but I have some experiences in logic and programming. My question is: At which level of abstraction is it currently possible to perform formal proofs using proof assistants with reasonable effort?
Of course many... | https://mathoverflow.net/users/33842 | At which level is it currently possible to write formal proofs? | One can possibly make some headway by switching from a traditional formulation of a field of mathematics to a "synthetic" formulation, where the objects under study are not explicitly built out of smaller pieces, but are universally characterized axiomatically. The historical prototype of such an approach is "[syntheti... | 13 | https://mathoverflow.net/users/381 | 134018 | 74,029 |
https://mathoverflow.net/questions/134013 | 1 | Hello,
I'd like to know the square root of the following $n$ by $n$ matrix, for $n > 2$ and $r>0$:
$R\_{ii}=r+1$ for $i < n$
$R\_{ij}=r$ otherwise
The $2$ by $2$ case is given by
$\sqrt{R}=\frac{1}{d} \left[\begin{array}{cc} 1+r+\sqrt{r} & r \\\ r & r+\sqrt{r}\end{array}\right]$
where $d=\sqrt{1+2r+2\sqrt{r}}... | https://mathoverflow.net/users/nan | square root of a certain matrix | Let $P(a,b,c)$ be the $n\times n$ matrix where
$$
P(a,b,c)\_{ij} = \begin{cases}
a & \text{ if } i,j < n \\
b & \text{ if } i < n \text{ and } j = n \\
b & \text{ if } i = n \text{ and } j < n \\
c & \text{ if } i = j = n.
\end{cases}
$$
If I understand correctly, you want the square root of $I+P(r,r,0)$. You ... | 4 | https://mathoverflow.net/users/10366 | 134020 | 74,030 |
https://mathoverflow.net/questions/96782 | 51 | I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct.
---
This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest possible proof strategies?
I have vectors $v\_1, \ldots, v\_k$ in ${\bf R}^n$. Each of them has euclidean length ... | https://mathoverflow.net/users/23141 | vector balancing problem | [This paper](http://arxiv.org/pdf/1306.3969.pdf) contains a proposed solution to the problem. The acknowledgements suggest that MO might have facilitated the solution, should it be correct. (I'm speculating that Gil Kalai found out about Nik Weaver's formulation of the problem from this MO question.)
| 16 | https://mathoverflow.net/users/6269 | 134026 | 74,031 |
https://mathoverflow.net/questions/134025 | 5 | *The volume of a Riemannian metric on the projective plane is $2\pi$ and length of every non-contractible loop is greater than $\pi - \epsilon$ for some small, positive number $\epsilon$. Is this metric close to the canonical metric?*
The question is somewhat vague on purpose. I'm mostly interested in the best consta... | https://mathoverflow.net/users/21123 | Stability of Pu's isosystolic inequality | There is no Lipschitz or even Gromov-Hausdorff stability - just consider a round metric with long hairy tails of small area.
One can hope for stability with respect to intrinsic flat distance in the sense of Sormani-Wenger or some similar metric. This distance is basically Federer's flat distance between isometric im... | 5 | https://mathoverflow.net/users/4354 | 134036 | 74,034 |
https://mathoverflow.net/questions/134024 | 4 | Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ and $z + x \prec z + y$ for all $x,y,z \in G$ with $x \prec y$. It is not difficult to give a direct proof of the fact tha... | https://mathoverflow.net/users/16537 | Every abelian torsion-free group is strictly totally orderable (via the compactness theorem) | A compactness argument which Hodges may have had in mind can go as follows. Since a subgroup of a totally ordered group is also a totally ordered group, it suffices to *embed* the given abelian torsion-free group $G$ into a totally ordered group, i.e., to show that the theory of totally ordered abelian groups is consis... | 10 | https://mathoverflow.net/users/12705 | 134037 | 74,035 |
https://mathoverflow.net/questions/130391 | 4 | Does anyone know anything about the normal regular sequences in the quantum plane?
Here are the definitions:
Normal regular sequence: Let $R$ be a ring (not necessarily commutative). A sequence $x\_1, \ldots, x\_n$ is a normal regular sequence in $R$ if
1) $x\_1$ is a normal, regular element in $R$, and
2) $... | https://mathoverflow.net/users/24781 | Normal regular sequence in noncommutative algebras | Let $g \in k\_q[x,y]$ be normal with $q$ a primitive $\ell$th root of unity, $q \neq 1$. Then $g = \sum c\_i y^{m\_i} x^{n\_i}$ where $c\_i \in k^\times$ for all $i$ and $m\_i \cong m\_j\mod \ell$, $n\_i \cong n\_j\mod \ell$ for all $i,j$.
It's not hard to see that such elements are normal. Checking that these are al... | 1 | https://mathoverflow.net/users/12849 | 134046 | 74,038 |
https://mathoverflow.net/questions/133173 | 9 | By the **Arnold Conjecture**, I mean the following statement:
>
> Let $M$ be a closed symplectic manifold, and $\phi:M\to M$ a Hamiltonian symplectomorphism with nondegenerate fixed points. Then $ \# \operatorname{Fix} (\phi) \geq \dim H\_\bullet (M;\mathbb Q)$.
>
>
>
The proof is via defining the "Floer homol... | https://mathoverflow.net/users/35353 | How to prove Arnold Conjecture without using S^1 localization? | I was thinking about this on and of for a long time. However, after discussing it with several people (mostly Mohammed Abouzaid) I have arrived at the following explanation:
The PSS approach works without having to consider S^1 actions.
First of all it is very very important in this discussion **not** to ignore Nov... | 7 | https://mathoverflow.net/users/4500 | 134051 | 74,040 |
https://mathoverflow.net/questions/133565 | 5 | Imagine I have a string $s$ of length $L$ encoded over an alphabet of size $q$, e.g. $s = 000101$, where $L = 6$ & $q = 2$. For each of $T$ time intervals, $(t\_1, ..., t\_N) \in T$, I select a bit in the string with uniform probability, then randomly mutate or "flip" the bit to another of the $q$ characters in the alp... | https://mathoverflow.net/users/34922 | Keeping time by randomly drifting a $q$-ary string | $P(T = n) = \binom{L-1}{k-1} \sum\_{j=0}^{k-1} (-1)^j \binom{k - 1}{j} \left(\frac{k - j - 1}{L}\right)^{n-1}, \quad n \in \{k, k + 1, \ldots\}
$
k = $K(s\_f) - K(s\_0)$
Where K(s) is the Kolmogorov complexity of string s based on a particular description language;(the effect of changing languages is bounded ); Not... | 1 | https://mathoverflow.net/users/34859 | 134053 | 74,042 |
https://mathoverflow.net/questions/134029 | 3 | For $\Re s = 1/2$ numerical evidence suggest:
$$ \Re \zeta'(s)/\zeta(s) = 1/2 \log(\pi) - 1/2 \Re \psi(s/2) \qquad (1) $$
How this was found. Consider the symmetrized zeta function
$\zeta^\*(x)= \pi^{-x/2}\Gamma(x/2)\zeta(x)$.
Experimentally $\Re{ \zeta^\*{'}(1/2 + it)} $ vanishes.
Taking derivative and solving for... | https://mathoverflow.net/users/12481 | On the critical line $ \Re \zeta'(s)/\zeta(s) =? 1/2 \log(\pi) - 1/2 \Re \psi(s/2)$ ? | Yes, your formula is true (and no RH is needed),
Your symmetrized zeta-function is symmetrized such that the functional equation
$$
\zeta^\* (s)=\zeta^\* (1-s)
$$
holds. Thus we have that $\zeta^\* (s)-\zeta^\* (1-s)= 0 $ and on the critical line for $ s = 1/2+it $ this gives us that
$$
2 \Im \left( \zeta^\* ( ... | 5 | https://mathoverflow.net/users/10811 | 134061 | 74,044 |
https://mathoverflow.net/questions/127303 | 11 | Here are the two versions of Arnold's conjecture on Hamiltonian orbits:
>
> Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate Hamiltonian. Then the number of $1$-periodic orbits of the vector field $X\_H$ defined by $\omega(X\_H, \cdot) = dH$ is b... | https://mathoverflow.net/users/20391 | What is known about the strong Arnold conjecture? | Assume dim$(M) \geq 6$ and $M$ is simply connected.
By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C\_\* (M)$ to generate $H\_\* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator... | 3 | https://mathoverflow.net/users/4500 | 134067 | 74,047 |
https://mathoverflow.net/questions/133953 | 6 | Let $R$ be a complete discrete valuation ring with maximal ideal
$\mathfrak{p}$ and algebraically closed residue field $k$. Denote
the field of fractions of $R$ by $F$. Let $G$ be an affine flat
group scheme of finite type over $R$, and assume that the generic
fibre $G\_{F}$ is smooth. Since $G$ is not necessarily smoo... | https://mathoverflow.net/users/2381 | Group scheme over a DVR whose special fibre is the image of points under reduction mod p | Note that the question does not make sense until we know that the image of $\rho$ is closed, and such closedness is not obvious since not even construcibility is apparent at the outset (e.g., Chevalley's theorem on constructibility is not relevant here, as $\rho$ is in no sense an algebraic morphism). Also, it isn't sa... | 5 | https://mathoverflow.net/users/29283 | 134068 | 74,048 |
https://mathoverflow.net/questions/134066 | 22 | For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$ be defined as the supremum of all real numbers $\mu$ such that
$$ \left| \alpha-\frac{p}{q}\right| \le \frac{1}{q^\mu} $$ has infinitely many solutions $\frac{p}{q} \in \mathbb{Q}$, where $q \ge 1$.
It's known that if... | https://mathoverflow.net/users/26576 | Numbers with known finite irrationality measure greater than 2 | The answer is yes - see for example
>
>
> >
> >
> > >
> > > Yann Bugeaud *Diophantine approximation and Cantor sets* Math. Ann. (2008) **341**:677–684
> > >
> > >
> > >
> >
> >
> >
>
>
>
| 13 | https://mathoverflow.net/users/nan | 134069 | 74,049 |
https://mathoverflow.net/questions/134010 | 8 | There is a vast literature on embeddings of graphs into surfaces.
I am interested in embeddings of graphs that
belong to the given homotopy class. Here is the precise formulation.
I have two finite graphs $\Gamma, \Gamma'$ and a homotopy equivalence $f: \Gamma\to \Gamma'$.
I know that there exists an embedding $\... | https://mathoverflow.net/users/21684 | embeddings of graphs into surfaces | The answer in general is no. One obstruction is that a 4-valent vertex has three local resolutions as a pair of trivalent vertices, and only two of them may be realized in a given surface.
**Edit** I have simplified the counterexample below.
More precisely, let $\Gamma'$ be a graph that contains two simple closed c... | 9 | https://mathoverflow.net/users/6205 | 134072 | 74,052 |
https://mathoverflow.net/questions/134063 | 3 | We can map from set of coverings over $X$ to symmetric group $\mathfrak{S}\_n$ via monodromy (if we fix a loop at the basepoint). Also we can consider braid group $Br\_n(Y)$, allow strands pass through themselves and map $Br\_n \to \mathfrak{S}\_n$ such a way. Can we construct a category of "braided coverings" over $X$... | https://mathoverflow.net/users/35080 | Braided coverings and braided monodromy | Here is one possible interpretation of your question, I'm not sure it's what you're looking for.
An $n$-sheeted covering of $X$ is classified by a map $X\to B\mathfrak{S}\_n$. (On taking the induced map on fundamental groups, and picking an element of $\pi\_1(X)$, I believe you get the monodromy you describe).
Now... | 2 | https://mathoverflow.net/users/8103 | 134073 | 74,053 |
https://mathoverflow.net/questions/134074 | 7 | Can someone give me an example of a Banach Algebra which does not have an isometric representation in a Hilbert Space ?
(if possible, can you add a proof or a reference ? )
Thank you very much !
---
**Note added by YC**: this question has also been [asked on MSE](https://math.stackexchange.com/questions/42370... | https://mathoverflow.net/users/35085 | Banach Algebra Counterexample | Any Banach algebra which is not Arens regular cannot be embedded as a closed subalgebra of B(H), even if you allow for isomorphic embeddings that have closed range yet are not isometric.
If you are only interested in Banach $\ast$-algebras and isometric $\ast$-homomorphic embeddings, then it is easier to find example... | 6 | https://mathoverflow.net/users/763 | 134086 | 74,056 |
https://mathoverflow.net/questions/134042 | 4 | Let $\cal D$ be the set of all $D=1,2,3,\dots$ such that $D\equiv 0,1 \mod(4)$ and $D$ is not a square. For $D\in\cal D$ let $\varepsilon(D)$ denote the smallest number $\varepsilon=\frac{a+b\sqrt D}2$ which is $>1$ and satisfies $a,b\in\mathbb Z$, $a^2-b^2D=4$ (Pell's equation).
Let $\cal E$ denote the set of all such... | https://mathoverflow.net/users/nan | Differences between fundamental units | *Note:* The following answer assumes that the question means (all) fundamental units in its standard meaning as opposed to a litteral reading of the question in the body (see Impossible Hippopotamus's answer for details related to this, and also for a more complete answer in general).
For the second question, it is n... | 2 | https://mathoverflow.net/users/nan | 134090 | 74,059 |
https://mathoverflow.net/questions/134077 | 18 | A [measurable space](http://ncatlab.org/nlab/show/measurable+space) $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best of my knowledge, no language ([including Haskell](https://stackoverflo... | https://mathoverflow.net/users/238 | How do we express measurable spaces using type theory? | If you clarify your question, I can modify this answer to be better. (I am sure you won't like my answer. The best answer is this is not practical, except in the simplest of settings. Or you are interested in this from a logical perceptive, in which case you need to learn the background. Or you care about probability, ... | 13 | https://mathoverflow.net/users/12978 | 134097 | 74,062 |
https://mathoverflow.net/questions/134057 | 86 | For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with some weak forms of composition and associativity and .... It was recognized a bit later that it is worth recording two nu... | https://mathoverflow.net/users/78 | Is there an accepted definition of $(\infty,\infty)$ category? | One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories:
[arXiv:1112.0040](http://arxiv.org/abs/1112.0040)
(i.e. $(\infty,n)$-categories).
This axiomatization includes several variants of $(\infty,n)$-category for ... | 66 | https://mathoverflow.net/users/184 | 134099 | 74,064 |
https://mathoverflow.net/questions/134107 | 2 | Let $G$ be a discrete group and let $M$ be a $G$-module. Assume that I have a resolution
$$\cdots \rightarrow M\_1 \rightarrow M\_0 \rightarrow M \rightarrow 0$$
of $M$ by $G$-modules (with no further assumptions on the $M\_k$; certainly they do not need to be free or acyclic or anything). What is the relationship betw... | https://mathoverflow.net/users/35094 | Resolution of coefficient system in group homology | This is an example of the [hyperhomology](https://en.wikipedia.org/wiki/Hyperhomology) spectral sequences in the case of chain complexes of $G$-modules. See chapter 5.7 (especially proposition 5.7.6) of Weibel's book "*An introduction to homological algebra*".
More precisely, let $P\_{\ast\ast}$ be a [Cartan–Eilenber... | 3 | https://mathoverflow.net/users/21095 | 134111 | 74,069 |
https://mathoverflow.net/questions/112447 | 8 | A finite, two-player, nondegenerate, symmetric game is defined by a nondegenerate $n \times n$ *payoff matrix* $A$. If player 1 plays strategy $i$ and player 2 plays strategy $j$, then player 1's payoff is $A\_{ij}$ and player 2's payoff is $A\_{ji}$. It is well known that the problem of computing a symmetric Nash Equi... | https://mathoverflow.net/users/21816 | Can we efficiently compute a third Nash Equilibrium, given two? | In a two-by-two symmetric game, there are two possible symmetric equilibria in pure strategies. Suppose we know that these are both in fact equilibria. All this tells us is that
$A\_{11}>A\_{21}$ and $A\_{22}>A\_{12}$.
Then there is exactly one additional equilibrium, determined by the equation $${p\over 1-p}={A\_... | 3 | https://mathoverflow.net/users/10503 | 134112 | 74,070 |
https://mathoverflow.net/questions/129609 | 20 | There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. One long-studied example is chess. Some moves have inverses, and others do not. If we create a graph of all positions, it i... | https://mathoverflow.net/users/27933 | Discrete Morse theory and chess | The quick answer to your question is **no**, discrete Morse theory has not been used to study chess moves yet (unless this has been done in some very obscure journal). I would like to highlight a few likely reasons for this current state of affairs.
If I understand your question correctly, you are interested in the f... | 15 | https://mathoverflow.net/users/18263 | 134113 | 74,071 |
https://mathoverflow.net/questions/134122 | 1 | Let $G$ be a subgroup of $SL(n,q)$ such that $G$ is not a subgroup of $GL(n-1,q)$, where $q=p^\alpha$. If $G$ is a $C\_{pp}$ group, i.e. the centralizer of each $p$-element is a $p$-group, can we say that there exists a primitive prime divisor $r$ of $q^n-1$ or $q^{n-1}-1$ such that $r$ divides $|G|$?
We call $r$ a p... | https://mathoverflow.net/users/31045 | About the orders of subgroups of $SL(n,q)$ | I believe the answer is NO in general. For $n>3$ one obtains a counter-example by taking any maximal torus $T$ corresponding to a partition $\{k, n-k\}$ of $n$ such that $2\leq k \leq n-2$. Clearly $T$ is a $C\_{pp}$ group because it doesn't contain any $p$-elements. On the other hand it clearly doesn't preserve an $(n... | 3 | https://mathoverflow.net/users/801 | 134126 | 74,080 |
https://mathoverflow.net/questions/134130 | 3 | I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of index $p$).
I couldn't find the references online or in books of Y. Berkovich.
(Thanks in advance!)
| https://mathoverflow.net/users/6761 | On Groups of Maximal Class: Reference | I believe the study of $p$-groups of maximal class really kicked off with this paper:
>
> Blackburn, N.
> *On a special class of p-groups.*
> Acta Math. 100 1958 45–92.
>
>
>
These days the basic source is:
>
> Leedham-Green, C. R.; McKay, Susan (2002), *The structure of groups of prime power order*, Lon... | 3 | https://mathoverflow.net/users/801 | 134132 | 74,083 |
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