1.      The Classroom and the Rabbits

“Alright, settle down, you lot, settle down!” Bopa Rai’s voice, a gravelly rumble that could probably crack concrete, echoed through the classroom. The young cadets, still fidgeting in their seats, gradually quieted, their attention drawn to the old man at the front. His uniform, though crisp, bore the subtle marks of countless years of service, and his eyes, though framed by a network of wrinkles, held a sharp, intelligent glint.

“Today,” he began, leaning back against the whiteboard, “we’re not talking about drill formations or weapons maintenance. Today, we’re talking about something far more fundamental, something that underpins almost everything you’ll encounter in the world – from a simple interest calculation to the spread of a virus. We’re talking about e, and the beautiful, often maddening, dance of compound growth and decay.”

He smiled. “Now, I know what some of you are thinking: ‘Bopa Rai, are you going to put us to sleep with some dusty old math equations?’ Not today, cadets. Today, I’m going to tell you a story. A story that reminds me of a rather persistent rabbit.”

Bopa Rai straightened up. “Imagine, if you will, a vast, emerald-green field. And in this field, there’s a magnificent, plump carrot. Mr. Growth, our rabbit, loves carrots. Each hop covers half the remaining distance to the carrot. So: half, then half of what’s left, and so on.”

“Does he ever reach the carrot?”

No. He gets closer and closer, infinitesimally so, but never arrives. This is compound growth in motion.

“Now meet Ms. Decay,” Bopa Rai continued. “She’s fleeing a fox and hops half the remaining distance to her burrow each time. Again, she never quite reaches it. She’s approaching zero – decay.”

“This is the paradoxical beauty of e: self-referential change.”

He scrawled a single letter on the board: e. “It’s everywhere – population growth, cooling tea, flowing electricity, even in randomness.”

2.      Definitions and approximations of e

As a Limit (Bernoulli’s Original Insight)

2.1) Bernoulli’s limit (definition of e)
$$
e \;=\; \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
$$

% Examples
$$
\left(1+\frac{1}{100}\right)^{100} \approx 2.70481
$$
$$
\left(1+\frac{1}{1000}\right)^{1000} \approx 2.71692
$$
$$
\left(1+\frac{1}{1{,}000{,}000}\right)^{1{,}000{,}000} \approx 2.718280
$$

2.2) Natural log base via area (integral definition)
$$
\int_{1}^{e} \frac{1}{x}\,dx \;=\; 1
$$

2.3) Euler’s continued fraction for e
$$
e \;=\; 2 \;+\; \cfrac{1}{\,1
\;+\; \cfrac{1}{\,2
\;+\; \cfrac{1}{\,1
\;+\; \cfrac{1}{\,1
\;+\; \cfrac{1}{\,4 \;+\; \cdots}}}}}
$$

2.4) Euler’s series (factorial expansion)
$$
e \;=\; \sum_{n=0}^{\infty}\frac{1}{n!}
\;=\; 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots
$$

This series converges rapidly and is widely used for computing e.

$$ N(t)=N_0 e^{kt} \quad\text{and}\quad N(t)=N_0 e^{-kt} $$

Bopa Rai on Temperance, Truth, and the Biology of Staying Alive

Bopa: “Yogis say temperance is the key: less food, fewer bursts of reproduction; fewer bursts, fewer errors; fewer errors, less cancer. Perhaps more life. Perhaps less repair needed. But doubt everything. Ask questions. Truth has a carnality; lies have no legs.”

Life is a bargain between multiplication and maintenance.
When fuel is rich, the body raises its banners—grow, divide, replace. That is how skin heals and marrow refills the blood. But each division is a throw of the dice; most throws land clean, a few don’t. Given enough fuel and enough throws, a misbehaving lineage can get a head start. Temperance lowers the drums—fewer throws, more time for housekeeping—autophagy, proteostasis, quiet repair.

Most tissues are printing presses: they keep form by replacing matter. Adult size looks constant because birth ≈ death. Inside that steady shape, turnover is breath itself: blood to bile, protein to urea, bone laid down and taken up, waste cleared by the scavengers. The miracle is the truce.

Neurons are the exception. They are a library, not a press. Replacing them risks misplacing you. So they don’t divide; they maintain. Glia change the oil and clean the floor, synapses re-weight, circuits relearn. The library lasts while food, oxygen, and cleanup last.

Stoppages and gates keep the chessboard sane. Primitive stem cells split into lineages; each line gets rules—divide until capable, divide at most L times, stop when the neighbours say stop. When progenitors age or their niche fades, the replacement line slows, scavengers can’t keep up, and the person feels it—first as fatigue and fuzz, then as loss.

Temperance is not starvation. It’s a steering wheel: enough to live, not to spill. Fewer spikes of “grow now,” more windows of “repair now.” Doubt every promise that comes without a mechanism. Measure. Look for legs.

Backbone (continuous growth)
$$
N(t) = N_0\,e^{k t}.
$$

Doubling time and rate
$$
2^{\,t/T_d} = e^{k t},\qquad
k = \frac{\ln 2}{T_d},\qquad
T_d = \frac{\ln 2}{k}.
$$

Decay (mirror image)
$$
N(t) = N_0\,e^{-\lambda t},\qquad
T_{1/2} = \frac{\ln 2}{\lambda},\qquad
N(t)=N_0\,2^{-\,t/T_{1/2}}.
$$

World pushes back (capacity (K))
$$
\frac{dN}{dt} = r\,N!\left(1-\frac{N}{K}\right),
\qquad
N(t)=\frac{K}{1+\Big(\frac{K-N_0}{N_0}\Big)e^{-r t}}.
$$

Temperance as a dial on growth
Let resource level be (R(t)) (food/insulin–IGF/mTOR tone) and repair tone (A(t)) (autophagy/proteostasis).
A simple modulation is:
$$
k_{\text{eff}}(t) = k_0 + \alpha\,R(t) – \beta\,A(t),
\qquad \alpha,\beta>0.
$$
(Fewer spikes in (R), higher (A) → smaller (k_{\text{eff}}).)

Functional steady state (birth ≈ death)
Let (P) = progenitors, (D) = mature workers, (X) = senescent; total (N=P+D+X).
$$
\begin{aligned}
\frac{dP}{dt} &= \alpha_p P\Bigl(1-\frac{N}{K}\Bigr) – \beta P – \sigma P,\[4pt]
\frac{dD}{dt} &= \beta P – \delta D – \omega D,\[4pt]
\frac{dX}{dt} &= \omega D – \varphi X.
\end{aligned}
$$
At adult steady size, each derivative is (\approx 0).

Quality matters (effective function)
If senescent cells contribute fraction (\phi\in(0,1)):
$$
F(t) = D(t) + \phi\,X(t).
$$

Neurons as long-lived stock (library)
Treat neurons (N_{!\text{neu}}) as largely post-mitotic with slow attrition (\varepsilon \ll 1):
$$
\frac{dN_{!\text{neu}}}{dt} \approx -\,\varepsilon\,N_{!\text{neu}},
\qquad
\varepsilon \text{ rises if supply/clearance fail}.
$$

Half anchors (nice starting points)
Growth from a half-seed:
$$
N(t)=\tfrac12\,2^{\,t/T_d}=\tfrac12\,e^{(\ln 2)t/T_d}.
$$
Decay with (t=0) at half-life:
$$
N(t)=\frac{N_{\text{ref}}}{2}\,2^{-\,t/T_{1/2}}.
$$
Centered logistic (start at (K/2)):
$$
N(t)=\frac{K}{1+e^{-r t}},\qquad N(0)=\frac{K}{2}.
$$

1) Growth (one line)
$$
N(t)=N_0\,e^{k t}
$$
“More begets more” at rate (k).

Doubling time (how fast is ‘fast’?)
$$
k=\frac{\ln 2}{T_d}
$$
(If doubling time is (T_d), this is the matching (k).)

2) Decay (the mirror image)
$$
N(t)=N_0\,e^{-\lambda t}
$$

Half-life (the friendly clock)
$$
T_{1/2}=\frac{\ln 2}{\lambda}
\quad\Longleftrightarrow\quad
N(t)=N_0\,2^{-\,t/T_{1/2}}.
$$

3) The world pushes back (just the idea)
$$
\frac{dN}{dt}=r\,N!\left(1-\frac{N}{K}\right)
$$
Early it feels exponential; later it coasts toward the limit (K).

1) Bopa Rai on Temperance, Truth, and the Biology of Staying Alive (Words First)

Bopa: “Yogis say temperance is the key: less food, fewer bursts of reproduction; fewer bursts, fewer errors; fewer errors, less cancer. Perhaps more life. Perhaps less repair needed. But doubt everything. Ask questions. Truth has a carnality; lies have no legs.”

Life is a bargain between multiplication and maintenance. When fuel is rich, the body raises its banners—grow, divide, replace. That is how skin heals and marrow refills the blood. But each division is a throw of the dice; most throws land clean, a few don’t. Given enough fuel and enough throws, a misbehaving lineage can get a head start. Temperance lowers the drums—fewer throws, more time for housekeeping—autophagy, proteostasis, quiet repair.

Most tissues are printing presses: they keep form by replacing matter. Adult size looks constant because birth ≈ death. Inside that steady shape, turnover is breath itself: blood to bile, protein to urea, bone laid down and taken up, waste cleared by the scavengers. The miracle is the truce.

Neurons are the exception. They are a library, not a press. Replacing them risks misplacing you. So they don’t divide; they maintain. Glia change the oil and clean the floor, synapses re-weight, circuits relearn. The library lasts while food, oxygen, and cleanup last.

Stoppages and gates keep the chessboard sane. Primitive stem cells split into lineages; each line gets rules—divide until capable, divide at most L times, stop when the neighbours say stop. When progenitors age or their niche fades, the replacement line slows, scavengers can’t keep up, and the person feels it—first as fatigue and fuzz, then as loss.

Temperance is not starvation. It’s a steering wheel: enough to live, not to spill. Fewer spikes of “grow now,” more windows of “repair now.” Doubt every promise that comes without a mechanism. Measure. Look for legs.

---

Bopa finalising the long lecture: A student once asked me, with fear in his eyes, “The brain is the master, but it can be eaten by cancer. How can the master lose control of its own house?” A good question. To answer it, you must first ask: what part of the house is on fire?

You see, the brain is not one thing. It is a city, and in this city, there are two kinds of citizens.

First, there are the workers, the road-menders, the cleaners—the glia. They are the body’s printing presses, constantly running, replacing old parts, keeping the streets clean. Their job is turnover, maintenance, bustle. They live by the rule of division. But where there is constant reprinting, a page can be copied wrong, the ink can spill, and the press can jam and run wild. This is your glioma, your astrocytoma. It is the cancer of the busy helper, the cell that retains the ability to multiply and thus, the ability to make a fatal error in that multiplication.

But then there are the librarians. The neurons. They are not presses; they are the library itself. Each one holds a unique book—a memory, a skill, your mother’s face. You cannot reprint such a book without losing its soul. So, early in life, the neuron makes a profound bargain: it gives up the right to ever copy itself. In exchange, it receives a long, long life. It takes a vow of permanence, exiting the cycle of birth and rebirth that defines all other tissues.

A cell that has sworn off division cannot form a tumor. It is as simple as that. Cancer is the sin of uncontrolled multiplication, and the neuron is a monk that has taken a vow of non-proliferation.

You speak of telomeres, the fraying edges of copied pages. The printing press worries about this with every copy it makes, sometimes finding a devil’s trick to stop the fraying and run forever. The librarian does not worry, for his ancient book is never copied, only patched and mended with care. The ‘mismatch’ you see is not a secret code; it is the simple, profound difference between a workshop and a vault.

And what of the cancers of nerve cells in the very young, the neuroblastomas? That is the cancer of the construction site, not the finished library. It is a tragedy of the builders who are still dividing, before they have become librarians and taken their sacred vow.

So, the secret is this: a neuron cannot become a cancer for the same reason a memory cannot become a crowd. It has traded multiplication for meaning. It has chosen to be, rather than to become. And that is the corporeal truth of the matter. The self is held in the cells that cannot be replaced.

In the notes you have there are mathematical equatons those who want to understand mathematical modelling may come to me for detailed simplificaton of maths. Yoy may find some repetitions, they are to reverberate, create a conundrum in your memory “read them too”