Social | The Core Mathematics Of Science | Founders' Review Editorial

The quadratic function, a cornerstone of basic mathematics, finds its application in a plethora of disciplines, from physics and finance to social sciences. Its simplicity and versatility allow it to be employed in a wide array of contexts, from describing energy to quantifying the variance in vehicular traffic.

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Why is this so? The essence of information is an energy spike. Each photon of light, each electron, carries energy that we capture upon measurement. The author, in 2017, constructed a robust data pipeline using a thermal sensor, a thumb-sized ESP embedded processor, AWS Lambda, and the highly scalable DynamoDB. While the experiment did not yield groundbreaking meteorological insights, it did prove that minor temperature fluctuations could be used to detect the opening and closing of doors in a household, a method both cheaper and more reliable than other sensors.

Problem: We have established that the law of conservation of energy is equivalent to the laws of Pythagoras and Fermat-Wiles. Solution: These laws guarantee that two systems will be independent if and only if a²+b²=c². Consider the world as a series of images captured sequentially over time. These laws suggest that as our world transitions from one moment to the next, t²+0²=0+T², the sum of energy quanta remains constant. This is the law of conservation of energy. It can be applied to any sliced and diced subsystems.

How does this relate to social sciences? When measuring customer or taxpayer behavior, random variables will yield a mean and a variance. Understanding these is crucial for conducting social research.

The variance of two independent variables or systems will be the sum of the variances of each participating system. This is the same quadratic function as mentioned earlier. Any correlation between the two variances can be calculated as the difference between the variance of the entire system and the variances of each subsystem. σ²(X+Y)=σ²(X)+σ²(Y)+2*Covariance(X,Y).

These values can be translated directly to two energy-storing system vectors. Two systems will store energy in their mean vectors, which are stable, an independent variance of the first system, an independent variance of the second system, and double the covariance of the two together.

There are several implications. Energy is an effective descriptor for systems as these values are additive. Imagine each variance and covariance as the position of planets on an elliptic curve, or quanta of an electron in an atom. Energy is the simplest descriptor for educational and engineering purposes. Any variance can be likened to a fluctuating spring carrying energy.

The scientific method can be illustrated thus: Astronomers observed that the centuries-old theory of planets orbiting their star and marking an equal area of an elliptic curve around the focal point is largely accurate, albeit with slight inaccuracies. The light from planets and stars was found to be slightly distorted from this system.

This covariance between the close proximity of the star to the light and the elliptic curve required an explanation. The prevailing theory was that gravity distorts the path of light.

A simpler explanation for educational purposes is that our space drifts in time. Each particle has the same speed, c. Adding the particles to a mass is a linear function of m. The energy of such an object is m*c².

However, this energy is not constant. It comprises a large part that drifts together in time, and a small part that describes all sorts of energy quanta. m*c²=m*a²+m*b² for an Fe particle that travels in our space. This is three-dimensional, but any particle can pick up many energy quanta like electron paths, magnetism, etc.

Particles in our space can then be described as a standing wave with a covariance between two particles, for example, a hydrogen ion and its electron. Our space remains intact because its energy is negligible compared to the drifting in time. The energy trapped in our system is relatively small similar to the loss function becoming negligible when training a stable model.

The wavelengths of the standing waves are usually constant due to the need to stabilize after a while.

Scientists have observed that any energy absorbed is usually radiated as electromagnetic or gravitational waves. These cannot be captured by the particles. Such waves are fluctuations with potential measurements of variances that are small enough to remain in our space.

We also noted that some energy quanta reach an orbiting energy to become part of the system, capturing some energy. Slow electrons or neutrons can be absorbed by atoms in this manner.

Gravitational waves drift objects towards each other in a similar way when the energy of fluctuations resonates and gets captured into mass. Gravity can thus be explained as a force proportional to the area of the arc of the sphere around a mass that collects these waves. It is also quadratic.

The slight interference around stars can be explained either as a distorted space or simply as a drift of the carrier material of electromagnetic waves, like a river. A boat entering perpendicular to the flow of a river will be drifted downstream. Our model describes gravity as compressional waves drifting towards masses that can absorb them. This distorts the path of light waves close to such large masses like the sun.

Different systems radiate when an energy quantum reaches the escape energy. Quantum waves directly correlating to the structure of atoms release this way, creating heat or colors.

Every single system in science or social sciences can be identified by an unexplained quantum of energy or variance. The method will be very similar using the expressions above.

Let's use the example of Olympic participants as a summary. If you collect the statistics, then individual performance will be explained by the following quadratic functions. One is the result of genetics. The second will be the result of motivation and hard work. You will likely find variances in each, as some people choose other professions. Adding two independent parameters usually ends up as a normal distribution.

However, there will be a covariance in our case that can be explained by trainers trying to find and recruit talent from the next generation. People who are genetically fit to be swimmers will more likely be invited to such competitive teams.

There is another interesting situation. The author analyzed the style of managers. Some were favoring task management and some were favoring relationship management. It turns out that the level of adequate funding has a covariance with the management style. Managers with adequate funding will likely use bonuses to set tasks, and they will favor outsourcing when needed. Managers with less funding will likely favor relationship management to build stronger teams that stay longer. These styles can directly correlate to decisions between quantitative tightening and easing, and also to the proportions of fixed and variable costs in the contribution margin income statements. Such decisions directly affect tenures of employment as well.