tldr: Non-Newtonian Calculus is a transformation of regular calculus which is ultimately equivalent, but offers a more ergonomic method of tackling certain problems. I’m hoping this post is a useful starting point for those who want to learn—once you do, NNC shows up everywhere.
Easing Into Product Calculus
In classical calculus, there are two main operations.
The integral finds area between function and axis. It is approximated via Reimann Sum: sample rectangles with the height of the function and add up their total area. You can see below how this appears very accurate. The skinnier the rectangles, the better the approximation. By taking the limit where they are infinitely skinny, this ceases to be an approximation.
$\int_a^b{f(x)dx} = \lim_{n\rightarrow \infty}\sum_{i=0}^{n}{f(x_i)\cdot \Delta x}$
Rather than adding, what if we multiplied the rectangles [A]?
$\sum \rightarrow \prod$
By making a simple alteration to the Reimann Sum, we find ourselves with a product integral. This is likely the simplest conceptual introduction into product calculus.
The derivative finds the rate of change of a function at one point. It is approximated by the slope of a line between two points on the curve. At the limit where these points are infinitely close, we get the slope of the tangent line, the exact derivative. The mathematical form of this concept is called the difference quotient:
$\frac{d}{dx}f(x)=\lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$
To get the product derivative, we'll switch up this limit in a similar way as with the integral, but this time in a bit more detail. Rather than a difference quotient, we will upgrade each of the operations and use a quotient root. It looks like this:
$f^*(x)=\lim_{\Delta x \rightarrow 0}(\frac{f(x+\Delta x)}{f(x)})^{\frac{1}{\Delta x}}$
By upgrade, I mean just that we’ve converted addition to multiplication and multiplication to exponentiation (note though that the addition at $f(x+\Delta x)$ didn’t change). Now let’s try to rearrange this equation:
$\ln{(f^*(x))}=\ln{((\frac{f(x+\Delta x)}{f(x)})^{\frac{1}{\Delta x}})}$
Simplifying with log rules we find that
$\ln{(f^*(x))}=\frac{1}{\Delta x}[\ln{(f(x+\Delta x)) - \ln{(f(x))}}]$
If the right hand side looks familiar, you're on the right track.
$\ln{(f^*(x))}=\frac{d}{dx}\ln{(f(x))}$
We'll apply the derivative
$\ln{(f^*(x))}=\frac{f'(x)}{f(x)}$
finally yielding
$f^*(x)=e^{\frac{f'(x)}{f(x)}}$.
What we've found is that the product derivative has a very simple relation to the normal derivative, and that product calculus is perfectly reducible to the classical version which people are used to.
Does this mean it has no use? Probably not. It's common for a reframing of a problem, though solving something equivalent, to provide a much clearer path to the answer. Consider the example where you have a deck of cards and want to know the probability that after drawing 4 of them, exactly 2 of those are diamonds, and 2 are spades. By using normal probability this is very unintuitive, but by simply counting the number of possibilities we can more easily see that there are $13!12!13!12!$ "successful" outcomes and $52! $ total outcomes, giving us a probability of $\frac{13!12!13!12!}{52!}$. Mathematics is all about finding these types of elegant views that render a previously confusing problem obvious. For this reason Non-Newtonian calculi are likely to offer non-trivial insights.
It seems possible that product calculus might provide a more intuitive explanation to certain classes of problems [C], and my instinct is that many of these exist within the fields of economics and finance, where most relevant measurements are in terms of percentage rather than raw difference.
Imagine you want to find total returns after a time period where the interest rate is not constant. For a constant interest rate, we would use
$A=P(1+\frac{r}{n})^{nt}$
All this is really doing is multiplying each consecutive value by the interest rate to achieve a new amount for the next time period, which is exactly the purpose of the product integral. For a variable interest rate, $r(t)$, we can do the following (where the integral is a “product integral”, as indicated by the $dt$ being a power):
$A=P\int_0^t{(1+r(t))^{dt}}$
While this is perfectly reducible to a normal integral [B] it is a much more intuitive way to consider the problem. As problems get more complex, perhaps there is value in taking this approach.
Generators
All we really did to move from classical to multiplicative calculus was replace addition with multiplication. And then multiplication, which is just repeated addition becomes repeated multiplication, or exponentiation. If we wanted to make a further generalization of different versions of calculus, it might make sense that each of them simply replaces addition with some different operator.
This is very nicely achieved using "generators." A generator is a simple function taking one number and returning another. It turns out that we can recreate the multiplicative calculus if all addition is replaced with the following
$a\dot{+} b = \alpha(\alpha^{-1}(a)+\alpha^{-1}(b))$
where $\alpha(a) = e^a$, because $\text{exp}(\ln(a)+\ln(b))=\text{exp}(\ln(a\cdot b))=a\cdot b$. If we define $\alpha$ differently, different types of calculus will be found.
Recall from above that we upgraded all of the operations in the difference quotient except the addition in $f(x+\Delta x)$. We actually have two degrees of freedom in creating a “calculus.” You can think of them as altering the “input” and “output.” If we change the input by $\Delta x$ (where the addition can be any generated operator, we wish to know how the output changes, and the definition of change is determined by which output generator we use. The multiplicative calculus that we have been discussing has an input generator which is just $\alpha(x)=x$, giving us normal addition.
From just these two generating functions you can specify the entirety of a calculus. The book Non-Newtonian Calculus, which as far as I know introduced the topic, gives an approachable explanation of how this works and catalogs a number of different generator combinations and their ramifications [D].
Applications
Just as the derivatives of a function can be used to construct a Taylor Series, we can approximate a function with its product derivatives. Again, all we have to is upgrade the operations, including going from normal to product differentiation [E], which yields:
$f(x)=\sum_{i=0}^{\infty}{f^{(i)}(x_0)\frac{(x-x_0)^i}{i!}} \rightarrow\prod_{i=0}^{\infty}{f^{(i)*}(x_0)^{(x-x_0)^i/i!}}$
Rather than a series of terms, we have a single term consisting of different bases and exponents. It is reducible to approximating the log of the function by taking the Taylor Series of the log of the function, but looks much simpler for certain classes of function. An interesting note is that you can cross the x-axis by using imaginary exponents.
The Black Scholes Equation, which governs prices in derivatives markets, is based off of Geometric Brownian Motion. Imagine a random walk, except if the distance were multiplied by 1/2 or 2 at random. One can show that the walk should eventually follow an exponential path.
$dS_t=\mu S_t\ dt \ +\ \sigma S_t \ dW_t $
If Brownian motion can more simply be formulated by a product-differential equation, could Black Scholes be more ergonomically derived?
Income elasticity of demand is measured as percent change over percent change, which means that elasticity in general is actually just the derivative in bigeometric calculus (input and output generators are $\alpha(x)=e^x$).
I am curious as to whether an analog to the Runge-Kutta methods could converge faster on certain differential equations with product derivatives.
In physics, Taylor Series are used with extreme frequency to simplify problems given some assumption about comparative size of variables. Non-Newtonian Taylor Series offer another path to simplification.
It seems also that pedagogy could be reformed quite a bit by introducing concepts from their more native calculus. The best example I have is that of calculating with changing interest rates, but I imagine that much of economics could be taught with fewer symbols once you know the right ones.
After More Research…
Turns out lots of people have used NNC!
Here's an awesome page on many uses of NNC: https://planetmath.org/nonnewtoniancalculus
With new calculi can be developed all of the usual tools of calculus, some of these including (and whether they have been developed):
- Calculus of variations
- Multiplicative calculus of variations is developed in Bashirov's paper Applications of Multiplicative Calculus
- Runge-Kutta Methods and other differential equation solvers
- Optimization methods such as gradient descent and whatever else exists
- Optimization/curve fitting with Taylor Series: https://link.springer.com/content/pdf/10.1007/s11075-013-9813-9.pdf
- Special relativity could be reformulated with a calculus that respects that rapidities add, but velocities do not.
- If the edge detector kernel is an approximation of the gradient operator, then perhaps you could come up with analog kernels for other calculi
- Non-Newtonian calculus was used by Xiaohong Gong, Yali Zhou, Hao Zhou, and Yinfei Zheng (all from Zhejiang University in Hangzhou, China) in an article on ultrasound imaging. [211] From that article: ”The proposed technique combines a new multiplicative gradient operator of non-Newtonian type with the traditional Canny operator to generate the initial edge map… Thus, the proposed method is very suitable for fast and accurate edge detection of medical ultrasound images.”
- I see no reason why this would help, as normal multiplication behaves well for computation, but you could build a neural net with different definitions of addition and multiplication, then using that calculus for backprop.
- Black-Scholes
- Negative-Likelihood loss more easily worked with using product calculus?
- A few additional applications here: https://hal.archives-ouvertes.fr/hal-00945788/document
- The definition of curvature changes depending on your calculus. Reimann Geometry could be redone with each NNC, figuring out first which surfaces have constant curvature
- What if you could, given a function, find on the fly the calculus, however complicated its definition might be, for which that function is special in some way (for example: it is constant, has constant derivative, etc…). Then that function would be like a frame of reference for all other calculation.
Notes
[A] This is a coarse statement. We aren't just multiplying the areas of the rectangles, but rather their height to the power of their width.
[B] The product integral is equal to $e^{\int\ln{(f(x))}dx}$.
[C] Gauss agrees: The following Carl Friedrich Gauss quotation is from Carl Friedrich Gauss:
”In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, indeed to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange’s calculus of variations, with my calculus of congruences, and with Mobius’ calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.” —Carl Friedrich Gauss
[D] For an even more accessible introduction, see http://mathcs.pugetsound.edu/~mspivey/ProdCalc.pdf
[E] I use $f^{(i)*}$ to indicate the $i$th product derivative of $f$.
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