0
votes
0answers
43 views

Find an analytic function [migrated]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
1
vote
1answer
206 views

What are the solutions for discrete integers b, d to $a^b \equiv c^d \pmod p$ where $p$ is a large prime number?

Is there a way to efficiently discover or choose the integers $b$, $d$ for the congruence relationship below where $p$ is a large prime number? Is there a name for this relationship? $$ a^{b} = c^{d} ...
0
votes
0answers
68 views

Do we have $\widetilde{K_0}(\mathbb{Z}[G])=Wh_{0}(G)$ for the general group

We clearly have that $\widetilde{K_0}(\mathbb{Z}[G])\subseteq Wh_{0}(G)$, and if can be seen from a theorem of Swan, we have that for a finite group, this is true. In the commutative case, this should ...
3
votes
0answers
101 views

Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} + ...
2
votes
1answer
156 views

If there exists a nontrivial vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat?

If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be ...
0
votes
0answers
61 views

primitive polynomial on $\mathbb{F}_2[x]$

For some reason I need some primitive polynomial $f$ on $\mathbb{F}_2[x]$ where $\deg f \in [1,10^4]$. (Especially for $\deg f = 10\pm \epsilon, 10^2\pm \epsilon, 10^3\pm \epsilon, 10^4 \pm ...
2
votes
0answers
192 views

Find the Range of Function

What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$, $$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$ where $b_k=\prod_{j\neq k}(z_j-z_k)$ for $1\leq k\leq m$ ?
0
votes
0answers
39 views

Factor of 2 In the Definition of Metric Contact Structure

In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation \begin{equation} d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right) ...
22
votes
3answers
1k views

Are the higher homotopy groups of the Hawaiian earring trivial?

The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not ...
5
votes
1answer
122 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 ...
11
votes
1answer
284 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} ...
0
votes
1answer
220 views

A convergence issue [Edited]

Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in ...
10
votes
1answer
230 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
72 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 ...
2
votes
1answer
111 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
38 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
21 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 $\varphi(t) = ve^{\mu t}$.
0
votes
0answers
57 views

An application of surreal numbers towards fast-growing ordinal functors?

The surreal numbers $\mathbb{SN}$ form a class of numbers introduced by J.H. Conway, which behave as an ordered field (even if technically it is not a set). In particular, Conway showed that every ...
-4
votes
0answers
53 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 $\bar{G}=\bar{H}\bar{K}$
5
votes
1answer
63 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 ...
-2
votes
0answers
56 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
86 views

Probability of differently loaded dice summing to a value [on hold]

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
45 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 ...
3
votes
2answers
696 views

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

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
31 views

Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.
11
votes
0answers
103 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
31 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
99 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
66 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
1answer
56 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
77 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 ...
2
votes
1answer
145 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 ...
22
votes
1answer
525 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
100 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
97 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 ...
2
votes
1answer
72 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
24 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
1answer
178 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
43 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
37 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
1answer
113 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
154 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
102 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
123 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 ...
9
votes
1answer
199 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 ...
0
votes
0answers
77 views

How to show non-existence of elements in the intersection of two ideals? [on hold]

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
56 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
432 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
166 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
512 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. ...

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