Articles of reference request

Best fitting in a curve of the form $Ax^B+C$

I am trying to fit data of the form $(x_i,y_i)$, $i=1,\ldots,n$, in a curve of the form $y=Ax^B+C$, where $B\in (0,1)$. All three constants $A,B,C$ are to be determined optimally (no particular norm for the moment). Also, the $x_i$’s are positive integers, if that helps. Is there any standard method to attack such a problem?

Good applications of modular forms on $SL_2(\mathbb{Z})$

I’ve just read some materials of modular forms on $SL_2(\mathbb{Z})$, and find some interesting application. Deal with Ramanujan $\tau$ function. I saw it in Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?. And I think is really a good example. Prove Ramanujan conjecture. I have read the proof of Ramanujan conjecture in Ahlgren S, Boylan M. Arithmetic […]

Describe the invariant bilinear maps on the linear group

Apologies if this is a stupid question ; it is at least a natural question. Let $V$ be a finite dimensional space over $\mathbb R$ or $\mathbb C$. Denote by ${\mathcal L}(V)$ the vector space of all endomorphisms of $V$ ; it has dimension $n^2$, where $n={\sf dim}(V)$. Also, denote by ${\cal B}(V)$ the space […]

Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $

This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (=”$\eta$”) in terms of matrixoperations I asked myself, what we get, if we generalize the idea of the alternating signs to cofactors from the complex unit-circle. $$ \zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $$ With this the usual […]

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ submanifold. Then the Hausdorff dimension of $\Sigma$ is $n$ and, for any relatively open set $U \subset \Sigma$ $$\mathcal{H}^n(U)= \int_U \mathrm{dvol}$$ where $\mathrm{dvol}$ denotes the usual volume […]

How to expand a representation

If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there any conditions when this is possible? Thanks a lot edit: I am interesting above all in the case $G=D_n$ dihedral, […]

General solution to expressions, without calculating exact roots (A generalization of Newton's identities)

Consider the following equations: $$A_1^1=\sum_iy_i=y_1+y_2+\ldots+y_m=a_1$$ $$A_2^1=\sum_{i_1,i_2}y_{i_1}y_{i_2}=a_2\,\,,i_1< i_2$$ $$A_3^1=\sum_{i_1,i_2,i_3}y_{i_1}y_{i_2}y_{i_3}=a_3\,\,,i_1< i_2< i_3$$ $$\vdots$$ $$A_{m-1}^1=\sum_{i_1,\ldots,i_{m-1}}y_{i_1}\ldots y_{i_{m-1}}=a_{m-1}\,\,,i_1< \ldots< i_{m-1}$$ $$A_m^1=y_{1}\ldots y_{{m}}=a_m$$ What is the general solution to the following expressions without computing exact $y_i$’s: $$A_1^n=\sum_iy_i^n=y_1^n+y_2^n+\ldots+y_m^n=?$$ $$A_2^n=\sum_{i_1,i_2}y_{i_1}^ny_{i_2}^n=?\,\,,i_1<i_2$$ $$A_3^n=\sum_{i_1,i_2,i_3}y_{i_1}^ny_{i_2}^ny_{i_3}^n=?\,\,,i_1< i_2< i_3$$ $$\vdots$$ $$A_{m-1}^n=\sum_{i_1,\ldots,i_{m-1}}y_{i_1}^n\ldots y_{i_{m-1}}^n=?\,\,,i_1< \ldots< i_{m-1}$$ $$A_m^n=y_{1}^n\ldots y_{{m}}^n=a_m^n$$ Does anyone know a reference containing the results? As an example, $m=3$,$n=3$: $$(\sum_iy_i)^3=y_1^3+y_2^3+y_3^3+3y_1^2y_2+3y_1^2y_3+3y_2^2y_1+3y_2^2y_3+3y_3^2y_1+3y_3^2y_1+6y_1y_2y_3$$ […]

Reference for principal bundles and related concepts

I am looking for a good reference for fibre bundles on differential manifolds, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could anyone advise me on this? I have looked at the book Fibre Bundles by Hausm√∂ller, but it isn’t quite what I want. Thanks in advance.

Galois theory reference request

I ran into the following description of Galois theory in Gelfand and Manin’s Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and field extensions. I wonder if there is any great self contained books that treats this topic thoroughly enough so that I can […]

Urysohn's metrization theorem and Borel image

The Urysohn’s metrization theorem states that for any second countable and regular $(X,\tau)$ there exists a topological embedding $f:X\to [0,1]^{\mathbb{N}}$. What puzzles me is not the theorem itself, but why (and if) $f$ can be chosen so that the image $f(X)$ is a Borel subset of $[0,1]^{\mathbb{N}}$. This is something that is not clear to […]