All Questions

3
votes
0answers
154 views

If 2-manifolds are homeomorphic and smooth, are they diffeomorphic? [on hold]

Perhaps this question has already been asked on Mathoverflow. I mean this question in a global sense. A friend mentioned it to me today, and I started thinking about it. I'm not sure how to prove it. ...
0
votes
0answers
54 views

Does any one know what this problem is called?

Given two sets $A$ and $B$, members of $A$ have been split into subsets $S_i$ which maybe they have intersections with each other and union of them maybe not equal to $A$ determine whether there is a ...
-5
votes
0answers
21 views

Diffusion Equation [on hold]

Kindly give me suggestions on my following assignment of Simulations in Fluid Flow: Solve the following differential equation for transport of f(x,y,z,t) by MS Excel ∂f/∂t+Ux ∂f/∂x+Uy ∂f/∂z+Uz ...
-1
votes
0answers
26 views

A combinatorial and number theoretical problem [on hold]

Given N positive integers, not necessarily distinct, how many ways you can take 4 integers from the N numbers such that their GCD is 1. For example,N=10 and the positive integers are ...
-2
votes
0answers
27 views

Summation with 2 functions [on hold]

Solve for $L$ if you can please. I want to know how to solve $G(x_i,y_j)$ with double integration, where $G=(x-y)$. The equation is below as follows: $L=\sum\limits_{i=1}^2 $ $\sum\limits_{j=1}^2 ...
-2
votes
0answers
22 views

How to calculate a sequence in Maple [on hold]

Please how can I using Maple obtain the following sequence defined successively $$X_1=1,\quad X_k=\Big(\frac{1/k+\sum_{i+j=k}X_i X_j}{k^2}\Big)^{1/2}$$ here $i,j,k\in\mathbb {N}$ i wish to have ...
3
votes
2answers
111 views

Powers of finite simple groups [duplicate]

I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but ...
7
votes
1answer
103 views

Is the Amitsur-Levitzki identity essentially unique?

Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...
0
votes
2answers
113 views

Is elliptic curve point division defined over the field of real numbers?

An elliptic curve is defined over the field of real numbers: $y^2=x^3 + ax + b$ A point P and scalar n can be multiplied using a combination of point doubling and adding. What about point division? ...
17
votes
4answers
5k views

How did Bernoulli prove L'Hôpital's rule?

To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
7
votes
2answers
439 views

Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by $$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$ Is it possible to decompose this into two separate ...
-3
votes
0answers
21 views

Find the vector component of vector u orthogonal to vector a [on hold]

I have vector u = (-2, 3, 1) and vector a = (-2, 2, 2). How do I find the vector component of u orthogonal to a? I've done the cross product and I get (-4,-2,-2), but I am assuming that this is also ...
-2
votes
0answers
59 views

how to solve 3 6-degree polynomial equations for 3 variables?

I am a physicist and need to solve three $6$-order polynomial equations for $3$ unknowns $(p, q, r)$. Here is the system of equations looks like: $$\sum(A[n]*p^i*q^j*r^k) = 0,$$ ...
0
votes
0answers
25 views

Integrate Faddeeva function

I came across this integration in my studies. $\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$ It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
7
votes
7answers
475 views

What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things? Thank you.
5
votes
1answer
50 views

Injectivity of Rewrite Rule in a Free Lie Algebra

Let $L$ be a free Lie algebra (over $\mathbb{Q}$) on generators $x_1, x_2, \ldots, x_n$, and let $V_k$ be the subspace spanned by the $k$-fold brackets. Let $U_1 = \mathrm{span}\{ x_i | i< ...
2
votes
0answers
31 views

What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...
0
votes
1answer
108 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
1
vote
4answers
167 views

Continuity in banach space for non-linear maps

I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...
2
votes
1answer
64 views

A perfect $(n,k)$ shuffle function

Suppose you have a deck of $n$ cards; e.g., $n{=}12$: $$ (1,2,3,4,5,6,7,8,9,10,11,12) \;. $$ Cut the deck into $k$ equal-sized pieces, where $k|n$; e.g., for $k{=}4$, the $12$ cards are partitioned ...
0
votes
0answers
20 views

Parallel topologies on a Prüfer group with the trivial group topology as the only group topology contained in both

Let $p$ be a prime number. A homomorphism $f:\Bbb Z_{p^\infty}\to \Bbb T$ induces a group topology $\mathcal T_f$ on $\Bbb Z_{p^\infty}$ with a base of neighborhoods $\mathcal N_f$ of $0$. Are there ...
-8
votes
0answers
79 views

Mathdialog. Can this new ZFC axiomatic system be considered research level mathematics? [on hold]

This is a justification of my original question “An axiomatic system with a set of constants that form a complete ordered field”. It was closed because: "This question does not appear to be about ...
23
votes
11answers
3k views

What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series $$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...
50
votes
9answers
5k views

Why do Bernoulli numbers arise everywhere?

I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...
2
votes
1answer
285 views

Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...
2
votes
1answer
129 views

Groups with a unique composition series

Which finite groups $G$ have a unique composition series? I don't mean in the sense of the Jordan-Holder theorem, but rather actually unique. Some examples are the cyclic groups $C_{p^n}$ and the ...
9
votes
5answers
830 views

Defining a topology in the Power Set

I have the following question: Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$. If the ...
4
votes
1answer
145 views

What does an endomorphism in a triangulated category give rise to?

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...
1
vote
0answers
95 views

Finding an overgroup or a subgroup in PGL

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x ...
4
votes
1answer
280 views

When is a `1-form' with continuous coefficients exact?

Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in ...
7
votes
0answers
208 views

Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} ...
24
votes
1answer
505 views

“Nyldon words”: understanding a class of words factorizing the free monoid increasingly

BACKGROUND. Let me first introduce some classical definitions, which appear, e.g., in §5 of Lothaire's Combinatorics on Words, in §5.1 of Reutenauer's Free Lie algebras, and in §6.1 of Victor ...
3
votes
2answers
90 views

Relation between Turing degrees and functions computable with them

Suppose $A<_T B$ ($A$ is a set computable from $B$ but not vice versa). Is it always the case that there exists a $B$-computable function which eventually outgrows all $A$-computable functions? Of ...
4
votes
3answers
474 views

Rigid nilpotent Lie algebras

I am very interested in the Nicola Ciccoli's answer to a question formulated in math*overflow*:67717 My questions are: Can a nilpotent Lie bracket be rigid in the scheme of Lie brackets on ...
0
votes
0answers
46 views

Minimal surface dividing a simply connected region into two regions of equal volume [duplicate]

let $\Omega \subset R^3$ (not necessarily convex) be simply connected. The the surface $\Gamma$ with minimal area that divides $\Omega$ into two regions of equal volume has constant mean curvature and ...
4
votes
1answer
71 views

$L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} ...
0
votes
0answers
49 views

Galois group for 0-dimensional motives

$\newcommand{\M}{\mathcal{M}_0}$$\newcommand{\Q}{\mathbb{Q}}$ It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference ...
2
votes
1answer
557 views

Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem: Exponentiating Polynomial Root Problem (EPRP) Let $p(x)$ be a polynomial with $\deg(p) \geq ...
5
votes
1answer
142 views

Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$. Two possible ways to compute $T_p$ mod $p$ seem to be: A) ...
1
vote
2answers
138 views

Rational Conjugacy Classes of Finite Groups

Suppose $G$ is a finite group and $A$ is the set of all character values of $G$. By character values, I mean entries of the character table of $G$. Let $\Gamma = ...
-2
votes
1answer
62 views

how to reduce 3-colorable graph to this? [on hold]

suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best problem to ...
1
vote
0answers
78 views

E- and A-algorithms for finite arithmetic prime progressions and other sets

There is certain Eratosthenes spirit to my problem (See below). First of all I'd like to stress the mathematical aspect of my question. Also, my question does not amount to the divide and conquer ...
0
votes
1answer
157 views

Question on the partial differential equations in complex space

As is known most of the theory now developed for partial differential equations is in the real space, especially function space like Sobolev space, BMO space, $L^p$ space, etc. However is there some ...
1
vote
1answer
78 views

Piercing of subspaces in a projective space?

The "piercing subspace" problem may be stated as follows: There are given several subspaces in a projective space, rather non-intersecting. Find an additional subspace of a prescribed dimension that ...
0
votes
0answers
21 views

A fredholm index associated with two vector fields generating a 2 dimensional foliation

Let $M$ be a compact manifold and $X,Y$ be two independent vector fields on $M$ with $[X,Y]=0$. Let $\mathcal{F}$ be the 2 dimensional foliation associated with the distribution ...
4
votes
1answer
134 views

Decomposing representations of finite groups of Lie type via computer

This is related to my previous question here. Let me remind you what that question asked: Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for ...
2
votes
0answers
34 views

Restricted singular values of random matrix

Let $X \in \mathbb{R}^{p\times p}$ be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of $X$ follow the Marchenko-Pastur law. Now let's ...
-7
votes
0answers
66 views

Does anyone want to see the critical figure for n= 7? [on hold]

I watched Ronald Lewis Graham's youtube blurb for the "happy ending problem". it's about 5 minutes long. I was able to supply him with the positions of the points for the case: n=7. it looks like a ...
0
votes
1answer
97 views

totally disconnected sets and homeomorphisms

For every totally disconnected perfect subset S in the plane one finds a homeomorphism of the plane onto itself mapping S onto the ternary Cantor set. This is an exercise in a book by Engelking and ...
1
vote
1answer
50 views

Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...

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