Proximal function for logarithm of linear transformation

I would like to calculate the proximal function $Prox_{lambda f}(mathbf{x}) = argmin_{x>0} (f(mathbf{x}) + frac{1}{2 lambda}Vert mathbf{x} – mathbf{x}_0 lVert ^2)$ for function:

$$ f(mathbf{x}) = sum_{i}^m -b_i log(mathbf{a}_i mathbf{x}) $$

Where $mathbf{x} ge 0$ ($mathbf{x}$ is a vector with all non-negative elements)), $b_i ge 0 : forall i$ (b are non-negative scalars), $mathbf{A} = (mathbf{a}_0, mathbf{a}_1, …, mathbf{a}_m) $ is a matrix with all non-negative elements.

I am aware that this can be easily accomplished by iterative minimization methods based on the gradient of $f(mathbf{x})$, but I’m wondering if it has a nice closed-form solution.

I know that for scalar function $g(x) = -c : log(x)$, $Prox_{lambda g}(x) = frac{x + sqrt{x^2 + 4 lambda c}}{2}$, and that there are some rules for composing proximal functions – particularly for the case $mathbf{A} mathbf{A}^T = alpha I$, which is not the case here.

I would like to know if there are any properties that would allow to obtain a closed-form solution based on the solution for the scalar function case.

Matrix logarithm not in Lie algebra

In Hall’s Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, he defines the Lie algebra of a matrix Lie group $G$ as the set $mathfrak{g}$ of all matrices $X$ such that $e^{tX}in G$ for all $tinmathbb{R}$. Here the exponential of matrices is defined using the power series. Similarly, he defines the logarithm of a matrix $A$ using power series:
begin{equation}
log A=sum_{m=1}^{infty}(-1)^{m+1}frac{(A-I)^m}{m}.
end{equation}

It is known that this series converges when $||A-I||<1$, where $||cdot||$ is the Hilbert-Schmidt norm.

Now, in Exercise 3.7, we are asked the following question:

Given an $A$ in a matrix Lie group such that $||A-I||<1$ (so that the
series above converges), is it always true that $log A$ is in
$mathfrak{g}$? Prove or give a counterexample.

My idea is the following: We know that the exponential map $exp:mathfrak{g}to G$ is a local diffeomorphism between a small neighbourhood $U$ of $0$ in $mathfrak{g}$ and a small neighbourhood $V$ of $I$ in $G$. However, $V$ may be very small such that it may not contain some $A$ that satisfies $||A-I||<1$ (that is, although $A$ is already closed to $I$, it may still not in $V$). In this case, $log A$ may not necessarily inside $mathfrak{g}$. But then when I tried to find some counterexamples, they are all outside the radius $1$ ball of $I$ (i.e., these examples $A$ are such that $||A-I||>1$). Thus, I am lost again.

Any hint, suggestion, comment and answer are much appreciated.

Reference Request on logarithm derivative of L-functions

I’m looking for references that show almost all Dirichlet characters $chi mod q$ satisfy
$$|frac{L’}{L}(1+it, chi)|=o(log q)$$
where $tin mathbb{R}$ is fixed. I have been able to adapt a method of Granville-Soundararajan to show results in this direction, but I wonder if there were published results on this problem: the method of theirs gives much stronger results than what I need. So I want to ask about the existence of earlier researches on this question.

I’m aware of papers by Ihara-Murty-Shimura and perhaps another one by Ihara and Matsumoto that assumed GRH, the Generalized Riemann Hypothesis. However, in this case the question is made too easy since

$$|frac{L’}{L}(1+it)| ll loglog q,$$
as is mentioned in textbooks.

Asymptotic behavior of the Coulomb logarithm

In transport theory in plasma physics, there’s an important integral called the Coulomb logarithm, which relates to the scattering cross section off the Yukawa potential. It can be written as
$$
ell(Lambda) = int_0^infty cosleft(int_0^{u^*}left[1 – 2frac{u}{xi}expleft(-frac{xi}{Lambda u}right) – u^2right]^{-1/2}duright)^2 xi dxi,
$$

where $u^*$ is the largest $u$ value where the integrand is real.

Now, for a plasma, we usually have $Lambda gg 1$. So being the lazy mathematicians that we are, we instead use the Coulomb potential (the limit $Lambda rightarrow infty$), which is exactly solvable but the $xi$ integral diverges, then cut off the $xi$ integral at $Lambda$ and say “close enough”. This gives $ell(Lambda)approx ln Lambda$.

Calculating the above integral numerically indeed gives $ell(Lambda) sim ln Lambda$ as $Lambdarightarrowinfty$. But I’d like to be able to show this through analytically, and through a somewhat less handwave-y method. Unfortunately, I’m not really sure where to start–that integral is kind of a hot mess. Any ideas how to get to $ell(Lambda) sim ln Lambda$ from that?

Approximation of logarithm of standard normal CDF for x<0

Does anyone know of an approximation for the logarithm of the standard normal CDF for x<0?

I need to implement an algorithm that very quickly calculates it. The straightforward way, of course, is to first calculate the CDF (for which I can find suitably simple approximations on Wikipedia), and then to take the logarithm of that. Obviously I’d like to avoid the time-cost of having to calculate two special functions, not to mention that the tiny intermediate value (in the tail) rules out using fixed-point arithmetic which is much faster than floating-point arithmetic.

I know there are hundreds of approximations for all kinds of statistical functions, but the fact that this is the logarithm of one makes it harder to find one. I’d be very grateful if anyone could point me to one, or to a source where I might find one.

Proving continuity of Logarithm using $delta$-$epsilon$

Say we wanted to prove the continuity of the logarithm using the $delta – epsilon$ proof. For any log base $a>1$ Starting with $|log_a({x_1}) – log_a({x_2})|, We can find that if $frac{x_1}{x_2} < a^epsilon$ (and $frac{x_1}{x_2}), then $|log_a({x_1}) – log_a({x_2})| as desired. But for the $delta$ part to come in, we need it in the form of $|x_1 – x_2|$, not $frac{x_1}{x_2}$.

So then I thought that maybe if $|x_1 – x_2| < frac{x_1}{x_2} -1 < a^epsilon -1$ then we would have $|log_a({x_1}) – log_a({x_2})| which would mean that the function is continuous at $x_1$ and since it was arbitrary it is continuous at all x (right?).

The reason I think this is true is because we want the distance between $x_1$ and $x_2$ to be very small, so $a^epsilon -1$ (which we would call $delta$) would get really small as $epsilon$ gets small. Is there a better way to arrive at this value of $delta$? Assuming it is a valid answer

Discrete logarithm modulo powers of two

Given an odd integer $x$ and integer $k>1$, what is known about the set ${x^n mod 2^k | n in N}$, in particular it’s size and anything about the series of integers that will appear?

I know some obvious facts from the fact that the odd numbers form a group under multiplication under $2^k$, and I am taking subgroups of that group, so the length of the sequence will divide $2^{k-1}$.

If it is an easier problem, I am interested in generating a large set. For example, for (k=10), the largest set possible is 256, which can be generated by 256 different values.

In particular, there appears to be a pattern where there is $2^{k-2}$ maximal sets of size $2^{k-2}$.

Simplifying/Finding the natural log of two terms without logarithm laws.

If there’s a natural log of two terms, which I cannot simplify with the laws of logarithms, how should I simplify it?

e.g. ln(e^(6x) + 17)

The full equation could be something like this:
ln(e^(2x)) + ln(e^(6x) +17) = ln(50).

I know that I can simplify ln(e^(2x)) to just 2x, and I can evaluate ln(50) with a calculator. But I’m not sure how to simplify the ln(e^(6x) + 17).