## ct.category theory – How to make an endomorphism of an LCA group invertible

Consider a pair $$(G,phi)$$ where $$G$$ is a (discrete) Abelian group and $$phicolon Gto G$$ is an endomorphism of $$G$$. There is a usual trick to construct a new pair $$(G’,phi’)$$ with the property that $$phi’$$ is an automorphism, and such that $$(G’,phi’)$$ is, in a certain sense, “as close as possible” to $$(G,phi)$$. Indeed, consider the following direct system:
$$Goverset{phi}{longrightarrow}Goverset{phi}{longrightarrow}Goverset{phi}{longrightarrow}Goverset{phi}{longrightarrow}cdots$$
Then one defines $$G’$$ as the colimit of the above system and lets $$phi’$$ be the map induced by $$phi$$.

Another, equivalent, approach is to see the pair $$(G,phi)$$ as a $$mathbb Z(X)$$-module $$G_phi$$ (where multiplication by $$X$$ corresponds to an application of $$phi$$), then $$(G’,phi’)$$ is the pair that corresponds with the $$mathbb Z(X)$$-module
$$G’_{phi’}:=G_{phi}otimes_{mathbb Z(X)}mathbb Z(X^{pm 1})$$
obtained by tensoring with $$mathbb Z(X^{pm1})$$ (the ring of Laurent polynomials). In this language it is easy to see that the obvious map $$varphicolon G_{phi}to G’_{phi’}$$ satisfies a universal property, in that any morphism of $$mathbb Z(X)$$-modules from $$G_phi$$ to a $$mathbb Z(X)$$-module $$H_psi$$ with $$psi$$ invertible, factors uniquely through $$varphi$$ (this makes the “as close as possible” in my second sentence precise).

Now the question is, What if the $$G$$ is a locally compact Abelian (always Hausdorff) topological group, and $$phi$$ is a continuous endomorphism? Can we do something similar to the above in this more general situation?

If it helps, I am happy to make some assumptions about $$phi$$, but I am looking for a construction that works when $$phi$$ is not necessarily open.

To conclude, let me precise that I am looking for functorial constructions that take as input the pair $$(G,phi)$$ (with $$G$$ LCA and $$phi$$ continuous), and give as output a pair $$(G’,phi’)$$ with $$G’$$ Hausdorff topological group, $$phi’$$ a topological automorphism (i.e., continuous, open and linear), that satisfy some kind of universal property as the one above.

## solution verification – A problem about measure theory, sigma algebra and Borel sigma algebra

Problem: Let $$mathcal{S}={(-b,b): bgeq 0}$$. Is $$sigma(mathcal{S})=mathfrak{B}o(mathbb{R})$$?

Notation: $$sigma(mathcal{S})$$ is the sigma algebra generated by $$mathcal{S}$$ and $$mathfrak{B}o(mathbb{R})$$ is the Borel sigma algebra.

My attempt: I think it’s true. My approach to this problem is as follows:

By definition in theory of sets, we need to prove two parts:

1. $$sigma(mathcal{S})subseteq mathfrak{B}o(mathbb{R})$$.
2. $$mathfrak{B}o(mathbb{R})subseteq sigma(mathcal{S})$$.

Now, for to prove $$boxed{1}$$, let’s define $$mathcal{O}(mathbb{R})=text{collection of all open sets of mathbb{R}}.$$
and since that by definition $$boxed{mathfrak{B}o(mathbb{R})=sigma(mathcal{O}(mathbb{R}))}$$, we can see that $$mathcal{S}subseteq mathcal{O}(mathbb{R}) implies sigma(mathcal{S}) subseteq sigma(mathcal{O}(mathbb{R})) quad text{and since that sigma(mathcal{O}(mathbb{R}))subseteq mathfrak{B}o(mathbb{R}) we have} quad sigma(mathcal{S})subseteq sigma(mathcal{O}(mathbb{R}))subseteq mathfrak{B}o(mathbb{R}).$$

Finally, for to prove $$boxed{2}$$, we need to remember that and every open set is the countable and disjoint union of open intervals, so we obtain $$mathcal{O}(mathbb{R})subseteq sigma(mathcal{S})$$, so $$mathfrak{B}o(mathbb{R})subseteq sigma(mathcal{S}).$$

Is it correct? any suggestion?
Thanks!

## elementary number theory – Find the hundredth place digit in 2008^2007^2006^…^2^1.

Find the hundredth place digit of 2008^2007^2006^……2^1.
Answer: Given M=2008^2007^……^2^1, let k1=2007^2006^…^2^1, k2=2006^2005^……^2^1 , k3= 2005^2004^……^2^1 .
Then M=2008^k1, k1=2007^k2, k2=2006^k3 . Now mod(M,1000)=mod(2008^k1,1000)=mod(8^k1,1000)…(1). Since 1000=8125,we find mod(8^k1,8) and mod(8^k1,125). Mod(8^k1,8)=0=>8^k1=8m1…(2), m1 being positive integer.
Since phi(125)=100,( phi= Euler’s phi function )mod(8^100,125)=1…(3).
Now mod(k1,100)=mod(2007^k2,100)=mod(7^k2,100)…(4).
Since phi(100)=40,mod(7^40,100)=1…(5). Now mod(k2,40)= mod(6^k3,40)=16,since k3>2. =>k2=40s+16=>mod(7^k2,100)=mod(7^16,100)=1…(6). From (4) mod(k1,100)=1=>k1=100t+1=>mod(8^k1,125)=mod(8,125) using (3).
So 8^k1=125m2+8=8m1,using (2). So8(m1-1)=125m2=>8 divides m2.
=> m2=8m3. So 8^k1=125
8m3+8=1000m3+8=>mod(8^k1,1000)=8.
From (1) it follows that hundredth place digit of M is 0.

## nt.number theory – a b c triples with bounded prime factors

(i) For any fixed $$B>0$$, are there only finitely many triples $$a,b,c$$ of coprime positive integers, such that $$a+b=c$$ and all prime factors of $$a,b,c$$ are at most $$B$$?

(ii) For which $$B$$ all such triples are known?

A positive answer to (i) would follow from the abc conjecture. For (ii), we may assume $$aleq b$$. There is one triple $$1+1=2$$ for $$B=2$$. For $$B=3$$, there are triples $$1+1=2$$, $$1+2=3$$, $$1+3=4$$, and $$1+8=9$$, and this list in complete. Indeed, the question reduces to equation $$1+2^n=3^m$$ or $$1+3^m=2^n$$. These equations has been solved by Gersonides in 1343. What about $$B=5$$?

## gr.group theory – Algorithm for Root System of Coxeter Group Generated by Permutations

Suppose we are given a group $$G$$ in terms of generators $$t_1, …, t_n$$ which are order 2 in $$S_m$$ (however we don’t assume anything other than that these elements generate $$G$$ and have order 2). Since $$G$$ is finite and generated by transpositions, it must have a root system. What is the best known algorithm for finding the root system?

## set theory – Transversal of an equivalence relation

Define a relation on $$mathbb{R}^2$$ by $$(a, b)sim(c, d)$$ if and
only if $$(c-a, d- b) in mathbb{Z}$$. Prove that $$sim$$ is an equivalence relation.
Identifying $$mathbb{R}^2$$ with the plane in the usual way, describe the most
natural transversal for $$sim$$ which you can find. What, if anything,
has this question to do with doughnuts?

I have already proven the equivalence relation, and know the solution of the second part, since it is provided, but I have problems with understanding it. I presume it’s asking about the equivalence classes?

Solution:

A natural transversal is $$I times I$$ where $$I = {r | r in mathbb{R}, 0 leq r < 1}$$. Join the top and bottom of $$I$$ to form a
tube, and bend the tube round to join the two circles as well. You
have the surface of a doughnut (a torus in mathematical language).

## set theory – Upper bound for constructibility orders of elements of the continuum

In a constructible universe (ZFC + V=L), is there any known upper bound for the constructibility orders of all elements of the continuum, i.e. some separately described ordinal $$alpha$$ such that we can prove $$mathcal{P}(omega)subset L_alpha$$ ? For example (under some large cardinal axiom), can it be proven that the first inaccessible cardinal is such an upper bound, or can this cardinal still fail at this ? I intuitively suspect undecidabilities in this matter but am no expert in the field. Thanks.

## rt.representation theory – Computing explicit matrix coefficients

I would like to understand in a more explicit way the Fell topology on unitary duals, that is to say the convergence of matrix coefficients of local representations.

If I consider a local representation $$pi_lambda$$ of a certain type (e.g. principal series for $$GL(2)$$ over a non-archimedean field), I would like to get an explicit dependence of the matrix coefficient $$langle pi_lambda(g)v, wrangle_lambda$$ in terms of the parameter $$lambda$$. This in particular involve explicit choices of a model for the representation and specific vectors: has this been done explicitly somewhere?

Ultimately, I want to see to what extent the Fell topology on representations corresponds to the usual topology on its (finite-dimensional) parameters.

## probability theory – Generalization of Black Scholes formula

I’m currently studying stochastic analysis and I’m struggling with the following exercise. (According to the autor, this formula appears when calculating the price in domestic currency of a call option on
a foreign stock with a strike in foreign currency. And it is some sort of generalization of the Black-Scholes formula.)

Let $$(X,Y)^Tsim LogN(mu,C)$$ and $$mathbb E(X)=mathbb E(Y)=1$$.

Define $$sigma:=bigglanglebegin{pmatrix}1\-1end{pmatrix},Ccdotbegin{pmatrix}1\-1end{pmatrix}biggrangle^{1/2}$$.

Show for all $$ageq0$$ and $$bin mathbb R$$ that

$$mathbb E((aX-bY)^+)=begin{cases}aPhi(d_1)-bPhi(d_2)&if a,b,sigma>0\(a-b)^+&otherwiseend{cases}$$

where $$d_{1,2}:=frac{1}{sigma}lnfrac{a}{b}pmfrac{sigma}{2}, for a,b,sigma>0$$.

$$Phi$$ is the cdf of the standard normal distribution.

The following hint is given:

Consider the event $$D={aX-bYgeq0}$$, define $$mathbb P_1(F)=mathbb E(X1_mathbb F)$$ and $$mathbb P_2(F)=mathbb E(Y1_mathbb F)$$ for all $$F in mathcal F$$. Show that $$Y/Xsim LogN(mpfrac{1}{2}sigma^2,sigma^2)$$ under $$mathbb P_{1,2}$$. Consider the boundary cases individually.

Let’s say I manage to show that $$Y/Xsim LogN(mpfrac{1}{2}sigma^2,sigma^2)$$ under $$mathbb P_{1,2}$$. How does this help? And what is meant by ‘boundary cases’?

## number theory – Cyclic sums of primes

Let $$G_k = p_1, p_2, dots p_k$$ be a set of consecutive primers. Define cyclic sum of order l = 1 .. k as:

$$S_1 = sum_{i}^{k}{p_i}$$;

$$S_2 = sum_{i}^{k}{p_i p_{i+1}} ; where ; p_{i+k} = p_{i+k mod k}$$;

$$dots$$

$$S_k = sum_{i}^{k}{prod_{l}^{k}{p_l}} = k prod_{i}^{k}{p_i}$$

Are there any known results, such as compact expressions, approximations, bounds and asymptotics for this type of sums?