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Reference for the positive convergence probability to a stable point of a stochastic approximation algorithm

Consider a stochastic approximation process
$$ x_ {t + 1} = x_t + frac {1} {t} (g (x_t) + u_t) $$
Where $ (u_s) _s $ is a sequence of i.i.d. Shocks.
Accept $ g $ is Lipschitz, $ u_t $ has finite variance, and that
$ (x_s) _s $ is limited with probability one.
Let's also assume that
$$
C = {x in mathbb {R} colon g (x) = 0 }
$$

is finite.
Define the stable points in $ C $ than the points $ x $ so that $ g (x) (x-y) <0 $ to the $ y $ close enough to $ x $,
In this case it follows from the results of the stochastic approximation theory that $ x $ converges a.s. to a stable point in $ C $ (See, for example, Kushner and Yin, “ Stochastic Approximation and Recursive Algorithms and Applications & # 39; & # 39; Theorem 2.1, page 127).

Question: Do you know a reference that offers such conditions that $ x $ converges to any stable point in $ C $ with a strictly positive probability.

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Rings and Algebras – Example of an associative unital ring R with stable region 1 and Jac (R) = 0, which is not an exchange ring

Rings should be associative and unital, but not necessarily commutative.
Some definitions:

  • (Bass) A ring $ R $ should have a stable range $ 1 $ if for everyone $ a, b in R $whenever $ Ra + Rb = R $, then there is something $ x in R $ With $ a + xb $ be a unit.
  • (Warfield) A ring $ R $ is called an exchange ring if it has the property of finite exchange, or equivalent (Nicholson) if for all $ a in R $ There is an idemotent $ e in R $ With $ e in Ra $ and $ 1-e in R (1-a) $,

Examples: Semi-perfect rings have a stable area $ 1 $ and are exchanges.

My question is whether someone has an example of a ring $ R $ with a stable range $ 1 $ and Jacobson radical zeroisn't that exchange?

Version control – What is the best GIT workflow with CI / CD with submodules in a master to a test environment and a stable branch to a production environment?

To be honest, I've always tried to find good workflows for dealing with submodules, especially for the reason you pointed out:

For this to work, we have to change the submodules every time we merge masters into stable.

I'm not a fan of the need to coordinate a lot of mergers across a number of repos to implement the daily changes. Do the main repo and submodules often change in tandem? Are the submodules also used in other projects or are they only used by the project in the main review? Would you be ready to get rid of the submodules and buy only one repo? I think your workflow would be as clear as you described if you omitted the submodules.

There are situations where submodules are required. However, if you decide that they mean more work and complexity than value, you can roll the submodules into your main review while preserving the history. In the main repo, remove the submodules completely and confirm this. Then, in the repos of the submodule, move the content of the repos to a folder with the name of the submodule and confirm it. Finally, return to the main repo, add the repos of the sub-module remotely, and add what you referred to as remote / master to your main repo branch with the option --allow-unrelated-histories,