Splitting self-intersecting line at only one point using QGIS?

The analogy between zeta functions and trace formulas goes at least to Selberg, when he proved his famous trace formula for hyperbolic surfaces and the result turned out to resemble Weil's generalization of Riemann's explicit formula quite a bit.

Riemann-Weil explicit formula:

\begin{equation*} \begin{split} \sum_\gamma h(\gamma)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} h(r) \frac{\Gamma'}{\Gamma}\left(\frac{1}{4}+\frac{1}{2}ir \right)dr &+h\left(\frac{i}{2}\right)+h\left(-\frac{i}{2}\right)\\ &-g(0)\ln\pi-2\sum_{n=1}^\infty\frac{\Lambda(n)}{\sqrt{n}}g(\ln n) \end{split} \end{equation*} Selberg trace formula:

\begin{equation*} \begin{split} \sum_{n=0}^\infty h(r_n)=\frac{\mu(F)}{4\pi}\int_{-\infty}^{+\infty} rh(r) \tanh (\pi r)dr &+\sum_n \Lambda(n) g(\ln N(n)) \end{split} \end{equation*}

There's a big literature on this type of question, and the interplay of number theory, spectral analysis, mathematical physics... I recomend section 3 of Lagarias' survey "The Riemann Hypothesis: Arithmetic and Geometry" for references.

Connes approach started with

• "Formule de trace en géométrie non-commutative et hypothèse de Riemann" (1996)

And was completed (as far as I know) in

• "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function" (1998)

The main result says that given a global field $K$ and a character $\alpha=\prod_v \alpha$ of the space of adele classes $A/K$, and any adecuate test function $h$, we have:

$$\underbrace{\widehat{h}(0)+\widehat{h}(1)-\sum \widehat{h}(\mathcal{X},\rho)}_{\text{spectral side}}=\underbrace{\sum_v \int_{K_v^*}' \frac{h(u^{-1})}{|1-u|}d^*u}_{\text{arithmetic side}}$$

I know it doesn't directly address how to do this with Named Credentials, but I've historically used protected hierarchy custom settings for this in a managed package.

Being a hierarchy allowed for both default org level values and then user specific values if required.

It does require more work as the Named Credential won't do it for you. However, it gives you full control of the endpoint URL and the credentials that are passed to it.

As we know that in steady state capacitor acts as OC and inductor acts as SC.

This is true only for DC analysis, where "steady state" refers to values of voltage and current that are not changing at all.

But in steady state AC we deal with term like jXL or -jXC and uses phasor and other stuffs to determine circuit parameters like current and voltage. How it can happen when already all the values impedance values are either zero(in inductor) and infinite(in capacitor) ?

"Steady state" in AC analysis is a completely different concept. It means that the parameters of the excitation such as frequency and amplitude are not varying. It does NOT mean, however, that the instantaneous values of voltage and current are not changing, or that you can continue to model inductors and capacitors as shorts and opens. You need to evaluate their actual complex impedances at the excitation frequency. Those values are finite and nonzero, and they remain constant as long as the frequency doesn't change.