Universality in the class of separable Banach algebras

Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric.

  1. Is there are separable Banach algebra that contains isomorphic images of all separable Banach algebras?

  2. Is there a commutative separable Banach algebra that contains commutative separable Banach algebras?

The trick with bounded the distance between commuting projections (of arbitrary norm) does not work in either case.

Why is universality of CFG undecidable?

Let $text{ALL-CFG} = {left<Gright> mid Gtext{ is a CFG and } L(G) = Sigma^*}$.

I have understood the proof of ALL-CFG is undecidable, but I wonder why the following proof is not appropriate.

Let $C$ be a CFG. Then $bar{C}$ is a CFG by closure under complement. Since the emptiness of CFG is decidable, we can use it to decide whether $bar{C}$ is empty and therefore whether $C$ is universal.

This problem has confused me a lot. Really appreciate your help. Thanks!!!

boolean algebra – Universality of $(+, oplus)$ over $mathbb{Z}_2^n$

Let $mathbb{Z}_2^n$ be the field of bitvectors of length $n$ and define the xor operator $oplus$ and the addition operator $+$ over this field, with $+$ having the usual overflow semantics (take addition modulo $2^n$).

Is it possible to express any mapping $f: mathbb{Z}_2^n rightarrow mathbb{Z}_2^n$ entirely in terms of $oplus$ and $+$? It seems like it might be possible due to the nonlinearity of $+$.

cv.complex variables – a soft question about the zeta function of Ramanujan: analogous statements for a Lindelöf hypothesis and universality

I put this as a gentle question with the aim to get an answer from an informative point of view, because I am curious about it. You can, however, also add references that will be answered as a reference request, or add your conclusions / arguments. You consider it useful for your colleagues.

In the past I have known the following theories about the hypothesis known as Lindelöf and its relation to the Riemann Hypothesis (I have read them from different sources and for different audiences, but I find it difficult and I understand only excerpts) and on the other hand a phenomenon , which is known as the universality of Riemann's zeta function.

See the Wikipedia Lindelof hypothesis respectively. Zeta function universality,

I also know about the zeta function of the Ramanujan

$$ varphi (s) = sum_ {n = 1} ^ infty frac { tau (n)} {n ^ s} $$

or also known as Ramanujan tau Dirichlet series, see article Tau Dirichlet series from the encyclopedia Wolfram MathWorld. And from an informative point of view the articles (1) and Lecture X from (2).

Question. A) Does Ramanujan's zeta function fulfill a similar Lindelöf hypothesis? I ask if it is possible / possible to write a similar expression for Ramanujan's Dirichlet-L series instead of the Riemann zeta function. B) Also, I would like to know if the phenomenon of universality makes sense for the zeta function of Ramanujan (tell us a similar statement)
similar to the set of Voroni of 1975. Many thanks.

As I ask as curiosity, if the literature answer is a reference request, I try to search for it and read it from the literature. If my questions have no mathematical meaning or there is no motivation to study such things, please explain why.


(1) Carlos Julio Moreno, A necessary and sufficient condition for the Riemann hypothesis for Ramanujan's zeta functionIllinois J. Math., Vol. 18, Issue 1 (1974), 107-114.

(2) G.H. Hardy, Twelve lectures on topics from his life and work, AMS Chelsea Publishing (2002).

Universality of $ y ^ 4-x ^ 3 $ mod $ p $

For educational reasons, I was interested in the equation $ y ^ 4-x ^ 3 = a $ over $ mathbf F_p $,

To my surprise (maybe I'm naive) there is only one pair $ (p, a) = (13.7) $ for which there is no solution, at least for $ p leq 2000 $,

My question :

is $ (p, a) = (13.7) $ the only couple for that $ y ^ 4-x ^ 3 = a $ has no solution over $ mathbf F_p $ ?

stochastic processes – universality for scale-dependent systems?

Researchers who consider critical points in dynamic systems often think that these systems belong to universality classes, so that the behavior of the system at its critical point depends only on its universal class and not on its exact specification.

From what I have read and seen, I have the impression that universality is just considered relevant at the critical point, since only here is the system scalar-variable.

My question is, What if the large-scale behavior is controlled by another dynamic system so that the entire system is not skalvariant?

I feel that it should still make sense to talk about the behavior of the system as a whole if it is equal among different decisions of small-scale dynamics and gives a form of universality. However, I have not found a discussion about such a thing. I would like to know if the concept makes sense and what a good starting point to think about it.