0
votes
0answers
5 views

Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem), "Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...
1
vote
0answers
17 views

Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$

Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives: $$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} ...
1
vote
0answers
59 views

A convergence issue

Disclaimer: This could be a stupid question and could have a very simple answer which I am unable to see. Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space ...
9
votes
1answer
71 views

Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

I'm trying to understand the paper Arbarello, Enrico, Cornalba, Maurizio, Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Inst. Hautes Études Sci. Publ. Math. No. 88 ...
0
votes
0answers
33 views

Profinite Local Ring inside Polynomial Ring

This is a "technical" question that I came across in my research. Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
1
vote
1answer
53 views

Proof of existence of recursively inaccessible and Mahlo ordinals

As in title - I'm looking for a proof of the existence of a countable recursively inaccessible or recursively Mahlo ordinals, especially the first one. When looking for it in all the papers I stumbled ...
-2
votes
0answers
25 views

Show that there is a matris only $ A $ such that $ \varphi (t) = e ^ {tA} $. [on hold]

Let $ \varphi(t)$ of a matrix $n \times n$ functions $C^1$. If $\varphi(0)=I$ (identity) and $\varphi(t + s) = \varphi (t) + \varphi (s)$ for all $ t, s \in \Re $, show that there is a matris only $ A ...
-1
votes
0answers
15 views

Solution of ODE where A has not eigenvalue [on hold]

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.
-4
votes
0answers
38 views

Why is it true that write $\bar{G}=\bar{H}\bar{K}$ [on hold]

Let $G=HK$ and $N$ be a normal subgroup of $G$. Also let $\bar{G}=G/N$. Why is it true that write $\bar{G}=\bar{H}\bar{K}$
2
votes
0answers
19 views

Unfoldings of trajectories on the Veech triangle $V_4$

Let $V_4$ be the isosceles triangle with base angle $\pi/8$. $V_4$ is a Veech triangle, so the dynamics of billiards on it are very well understood. Above is the unfolding of $V_4$, with edge ...
-1
votes
0answers
35 views

Hessian Matrix and Kronecker Product

Given the following equation, $\Delta Y=J\Delta X+\frac{1}{2}H \Delta X \otimes \Delta X$ where $\Delta Y, \Delta X \in \mathbb{R}^{n}$, $J \in \mathbb{R}^{n \times n}$ is the Jacobian and $H \in ...
0
votes
0answers
50 views

Probability of differently loaded dice summing to a value

I have a real world problem that boils down to the following: I'm playing dice. I have $n \approx o(10)$ differently biased die. The probability of the $i^{th}$ die showing $x_i$ is given by ...
-3
votes
0answers
35 views

How to find the inverse function of a function like f: N x N -> N [on hold]

I need help on how to find the inverse of a function N x N -> N For example, if anybody could give me a step by step explanation how to find the inverse function of f(x,y)=3x-2y I would be very ...
2
votes
2answers
202 views

Is it meaningful to work on convergencies, integration, etc. on the Zariski topology? [on hold]

Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want ...
0
votes
0answers
19 views

Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.
9
votes
0answers
72 views

Besides F_q, for which rings R is K_i(R) completely known?

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$? Here, by "trivial examples" ...
2
votes
0answers
27 views

Measurability of functions with multiple parameters

For a formalisation of the Giry monad in a theorem prover, I think I require some notion of measurability of “curried” functions. I.e. I have measure spaces $A$, $B$, and $C$ and a function $f: A ...
2
votes
1answer
83 views

Which complete lattices arise as images of the Galois connections induced by binary relations?

Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois ...
1
vote
1answer
40 views

Iwasawa decomposition of the pseudo-orthogonal group

This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the ...
0
votes
0answers
15 views

Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...
0
votes
0answers
55 views

Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...
1
vote
1answer
97 views

Does normalization of projective varieties preserve very ampleness

Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any ...
16
votes
1answer
306 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
2
votes
1answer
88 views

Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...
0
votes
0answers
80 views

Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...
1
vote
0answers
22 views

Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that $$ \frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...
1
vote
0answers
15 views

TSP: Approximation Ratio of the Double Tree Heuristic after Diagonals have been Removed

In their article "Double-Tree Approximations for the Metric TSP: Is the Best One Good Enough?", Vladimir Deineko and Alexander Tiskin derive a lower bound for the approximation ratio of the ...
0
votes
0answers
61 views

Metrizable Coalgebras

A Coalgebra $C$ is called metrizable if there is a base $B$ for $C$(as a vector space) and a metric $d:B \times B \to \mathbb{R}$ on $B$ such that the linear extension $\tilde{d}: C\otimes C ...
-3
votes
0answers
71 views

Prove that (AxB)∩(CxD)=(A∩C)x(B∩D) [on hold]

Prove that $(A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D)$ where $\times$ represents the Cartesian product.
0
votes
1answer
155 views

Theorem with an example [on hold]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
0
votes
0answers
36 views

What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...
0
votes
0answers
32 views

Quick question about conjugate equivalence bimodules and inner products

let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, ...
2
votes
0answers
60 views

Diameter of $n$-unit-vector closed scribble

Suppose one creates a random, closed, likely self-crossing polygon from $n$ unit-length vectors arranged head-to-tail, randomly oriented except for the requirement that their sum is zero (so the ...
2
votes
2answers
108 views

Bound for the Frattini subgroup of a $p$-group

Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to ...
1
vote
0answers
55 views

Reference request: has this semilinear version of Navier Stokes been studied?

I have noticed that the Navier Stokes equations can be written as a semilinear symmetric first order system $$ u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u) $$ for a 9 by 1 vector $u$ containing the ...
3
votes
2answers
113 views

l-functions of calabi-yau varieties

This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...
7
votes
1answer
129 views

Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...
-2
votes
0answers
63 views

Example of flasque but non-soft sheaves? [on hold]

Does anyone have an interesting examples of a flasque but not soft $\mathscr{O}_X$-module over a ringed space? Of course with X being non-paracompact.
1
vote
0answers
67 views

How to show non-existence of elements in the intersection of two ideals?

Given l, k any two natural numbers, define $I_1 =\langle y^{l+2k}, (x+y)^{3l+2k} \rangle: x^{l+2k} + \langle y^{2l+3k}, (x+y)^{3l} \rangle: x^{l+2k}; $ $I_2 = \langle x^{l+k} \rangle.$ I want to ...
0
votes
0answers
48 views

Characteristic subgroups of the limit group

Let $\{ G_i \}_{i=1}^\infty$ be a direct spectrum of groups with respect to embeddings $\varphi_i:G_i \mapsto G_{i+1}$, $i \in \mathbb{N}$, and let $G$ be the limit group of this spectrum. Suppose ...
6
votes
1answer
401 views

Algebraic Closure of a Ring is Not a Ring?

I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is ...
4
votes
2answers
135 views

When distance nonincreasing map is an isometry

Let $f: M \to M$ be a distance nonincreasing map between a closed Riemannian manifold $M$ and $f$ is homotopic to the idendity map. Is it then $f$ an isometry?
9
votes
2answers
479 views

What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory. ...
1
vote
0answers
59 views

Chain homotopy of non-abelian category

How can one define the chain homotopy in non-abelian category? (The category I have in mind is the category of chain complexes of monoids.)
3
votes
1answer
149 views

Quotienting $SU(3)$ by $U(1)$?

As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
-3
votes
0answers
44 views

Product of Positive Intever Divisors of 6^16 equals 6^k [on hold]

Product of Positive Intever Divisors of 6^16 equals 6^k How would I find K? Don't give me the answer, just how to get it Thanks
1
vote
0answers
67 views

Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring

Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces $$R/\mathfrak{m}, ...
5
votes
0answers
72 views

Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$. We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...
0
votes
0answers
17 views

Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem \begin{align} \lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\ &r_i\geq ...
-5
votes
0answers
21 views

identifiability of a linear regression [on hold]

If we have a generative model $X_2=X_1a_1+\varepsilon$ where $\varepsilon \sim \mathcal{N}(0,\sigma_2^2)$ do we have $X_1=X_2a_2+\varepsilon '$ where $\varepsilon \sim \mathcal{N}(0,\sigma_1^2)$ ...

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