An Essay on how Quanta Affect the Math behind the Universe

Heinrich Augusto Townsend Olaechea

An Essay on how Quanta Affect the Math behind the Universe

Heinrich Augusto Townsend Olaechea

2022

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This essay explains the concept of the thorn derivative. It gives an approximation of what the derivative of a discontinuous function is. However, not exactly.

An Essay on how Quanta Affect the Math behind the Universe The Foundations of Quantum Calculus by H.A Townsend Turbulence Before we start with turbulent flow, we need to study laminar flow. The equation defining laminar flow is 𝑑𝑓𝑛 𝑑𝑡 = 𝑑𝑓𝑔(𝑛) 𝑑𝑡 , which we can also write as 𝑓𝑛(𝑡) = 𝑓𝑔(𝑛)(𝑡) + 𝑐. This basically means the paths of each particle in a fluid are parallel to each other. A way of simulating this would be turning your tap on. This wouldn’t be exactly laminar flow, but very similar. The path of each particle is almost parallel to any other particle. This is very easy to model, and we can predict where each particle will go. Let’s study this situation. Imagine you shoot water in a straight line at an initial velocity of 2𝑣, where 𝑣 = 𝑚 𝑠 need to know the initial condition of each particle, but that is very hard to figure out, so let’s use some calculus. Let’s first imagine a 10x10 square. We can first fit four major areas of fluid. Over a very small time, figure out the average of where that water is flowing, and assign a vector. Figure 1 explains what we’re doing. . The water would move laminary. The first moment(let’s call it 𝑡𝑐), the water is going in a straight line, the second moment, it’s falling a little, and at the last moment, it will fall and hit the ground. 𝑑𝑓𝑛 𝑑𝑡 = 𝑑𝑓𝑔(𝑛) 𝑑𝑡 is true every moment until the water hits the ground. But as 𝑡𝑐 approaches a value near zero, this will more accurately resemble reality. “But in calculus, it approaches zero, not a value near zero?” we’ll explain that later, but it’s very important. Now, what is turbulent flow? Turbulent flow is where 𝑑𝑓𝑛 𝑑𝑡 = 𝑑𝑓𝑔(𝑛) 𝑑𝑡 is not true, very far from true(if it’s very close to true you can label it as laminar flow). Turbulent flow is what we call in physics; chaotic. This means that a small change in initial conditions will result in a completely different output. This means it’s very hard to model turbulent flow. First, you Figure 1-1 We can see a cage with a bunch of circles inside. These circle’s are the major areas of water, and we can now approximate the turbulent flow! Or can we? The areas can’t even move, so we’ll have to make some smaller areas. Figure 2-1 This is a decent approximation of what the fluid will do, but it’s still bad. To simulate what the fluid will do, we need to assign a vector to every particle in the fluid. But we can’t just eye out that. We need to include every force that the particle can experience, and take the average of the vectors. However, if the unit of measurement for the box is centimeters, there will probably be so much particles, that if we write an a for each particle there is(font size 1), we’d probably fill up the earth with papers. We’ll probably need quantum computers for mediocre estimates. But we can write lim 𝐴 = 𝐹, or as the area 𝑎 → 𝑎𝑝 approaches the area of the particles in the fluid, the approximation approaches what actually happens. We can also write = 𝐹𝑡 𝑎𝐹 , or the vector that we use to represent the movement of the particle is equal to the total force over the number of forces used to get the total force. But again, we get this value just above zero, in this case the area of the particle. Why is this happening, and is calculus broken? Yes and no, you’ll see why. A more General form of Calculus Calculus deals with values approaching zero, but why hasn’t that happened? No values approached zero, but a value near it. This is because of something called Quanta. About 120 years ago, German physicist Max Planck figured out that energy is not continuous. This Means that if you wanted to add energy to something, you needed to add it in little packets. Basically, you can’t take the derivative of energy, and because most stuff is reliant on energy, a part of the universe is not continuous. Something else that is discontinuous is space and time. There is a minimum time and a minimum distance possible. So time doesn’t pass continuously. This feels wrong, but it is true. British-American Physicist and Computer scientist Steven Wolfram’s theory supported this idea. That’s why everything approached values near zero. But if this is correct, then there is no continuous function in the universe, “But we’ve used that to calculate stuff for years, how will we fix it?” here I will introduce the Thorn function. Thorn is a list of Numbers. These numbers are the levels of energy in quanta. Basically, because energy is discontinuous, you can assign values in which energy increases. Energy can only increase by these values. So let’s list these values in thorn. If you can only increase energy by 1 and 2, then þ = (0, 1, 2, − 1, − 2). We also include 0 because energy sometimes doesn’t increase or decrease. We also add the negative values because energy can also decrease(but let’s just take that as a given and just write (0, þ 1, 2) later). And then we can calculate 𝑓 (𝑥). This means that we calculate a discontinuous version of 𝑓(𝑥) where we can only increase/decrease every Planck time(or not increase or decrease at all). A Planck time is the minimum time possible. And there you have it. You can also write þ(𝑓(𝑥)) instead of þ 𝑓 (𝑥), but that’s only effective when you write 𝑥 something like þ(2 ). However, there is a problem with this, what about derivatives? You can’t take the derivative of a discontinuous function, right? This is the point of the Thorn derivative. Basically, everything is zero. Just kidding! What we do to calculate Thorn derivatives is a bit complex. Before the first discontinuous point, you make a line segment between (0, 0) and (𝑓(𝑑1 + 𝑑𝑥) − 𝑓(𝑑1 − 𝑑𝑥)), where 𝑑1 is the first discontinuous point in the function . However, at the first discontinuous point, you calculate the same thing you did before 𝑓(𝑑1 + 𝑑𝑥) − 𝑓(𝑑1 − 𝑑𝑥), and set that as the derivative at the discontinuous point. At the next discontinuous point, you have to the same. And between those two points, you make a line segment connecting them. Do the same for all remaining discontinuous points, We can write this as þ𝑓 þ𝑥 (or the Thorn derivative of 𝑓(𝑥) with respect to 𝑥). We can also write 𝑛 þ 𝑓 (the nth Thorn Derivative of þ𝑓 þ𝑥 = {𝑥 < 𝑑1: 2 þ𝑥 𝑓(𝑥) with respect to 𝑥). The formal definition of a Thorn Derivative looks very complex, but it isn’t! The formal definition of a Thorn Derivative is: a huge mess. Just kidding! The formula is (0, 0)(𝑓(𝑑1 + 𝑑𝑥) − 𝑓(𝑑1 − 𝑑𝑥), 𝑥 < 𝑑𝑛: ((𝑓(𝑑𝑛−1 + 𝑑𝑥) − 𝑓(𝑑𝑛−1 − 𝑑𝑥)) (𝑓(𝑑𝑛 + 𝑑𝑥) − 𝑓(𝑑𝑛 − 𝑑𝑥))), 𝑥 = 𝑑𝑛: 𝑦 = 𝑓(𝑑𝑛 + 𝑑𝑥) − 𝑓(𝑑𝑛 − 𝑑𝑥)}. This is slightly more efficient than discrete derivatives(any lines above stuff mean segments, not division). Anyway, this is the formula explaining derivatives, not 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ . However, this gives us a continuous function, so to get a valid answer, we need to write þ( þ𝑓 þ𝑥 ). We can write Ƿ𝑓 þ𝑥 (Quantized Thorn Derivative) to shorten the equation. To get to our second formula in general calculus, we need another fact. As time approaches infinity, the universe becomes more continuous. We can write Ƿ𝑓 𝑑𝑓 lim þ𝑥 = 𝑑𝑥 . Now, what is the equation for 𝑡→∞ the more general integral(The Gamma Integral). The equation is: Γ𝑓(𝑥) { } = 𝑑𝑛 ⊂ (𝑑𝑛+1 − 𝑑𝑛)𝑓(𝑑𝑛) . Whenever an intersecting point is encountered, calculate the product between 𝑓(𝑑𝑛) and the distance between 𝑑𝑛 and 𝑑𝑛+1. But as last time, this sometimes gives us a continuous answer, so instead of writing thorn of the gamma integral of 𝑓(𝑥), we can write Γ𝑓(𝑥) þ𝑥. We can also write lim 𝑡→∞ Γ𝑓(𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥. Another rule for Gamma integrals is that you can only integrate when the lower and upper limits are quanta of time. We don’t have to make a new version of limits, but we have to add the rule that things can only approach quanta of any quantity. Same with summation and 𝚷 products. Hell, we didn’t even need to replace integrals, just how you calculate them. But why can derivatives not work in real life? This is because derivatives require continuity, however, other functions don’t, specially summation and 𝚷 products. This is the reason the formula for Thorn derivatives was extremely large. Our final formula is Γ𝑓(𝑥) þ𝑥. This is a variation of the −1 Ƿ 𝑓 −1 þ𝑥 = Fundamental Theorem of Calculus. Basic Differential Equations Before we get into more complex math, let’s start with differential equations. Before thorn derivatives, we wrote 𝐴 = 𝑑𝑉 𝑑𝑡 , but that isn’t accurate, it says that acceleration is undefined. So we can write 𝐴 = Ƿ𝑉 þ𝑡 , 𝑡. Why do we write , 𝑡? As we said before, quanta change over time, so we have to specify a certain time. However, the equation above is always true, no matter the time, so we write , 𝑡. After a (Quantized) Thorn Derivative(or Gamma Integral), we have to write a condition. Imperfect Geometries Spherical and Hyperbolic geometries were a breakthrough discovery in the 1800s. However, they’re not correct. There is no perfect sphere in the universe, except maybe black holes, but it’s the event horizon that is a perfect sphere. However, there are only certain velocities in which you can approach it. Anyway, every sphere is imperfect. This is because stuff is made of particles, and we don’t know the shape of a particle, but let’s assume they are cubes(in real life, they’re obviously not, but this let’s us simplify the process) and they are always touching. If we have a group of particles that make something like a sphere, you would need a lot of them to successfully replicate spherical geometry. The same happens with hyperbolic geometry. These types of geometries are called “Imperfect Geometries”. We can represent each of these geometries as: Shape(𝑃 = 𝑎, 𝑆 = 𝑏, 𝑆𝑝 = ”Shape”). What does this mean? Let’s start with spherical geometry since it’s simpler. The variable Shape is the approximate shape of the surface. Since we are in a spherical surface, The Shape is a circle. P represents the volume of the particle we’re using to model. Let’s set this as 1. The variable S stands for the volume of the surface. Let’s set this as 100(Note: This is equal to the volume of each particle times the number of particles). And 𝑆𝑝 represents the shape of the particle, which we’ll label as a cube. So this geometry is named 𝑆𝑝ℎ𝑒𝑟𝑒(𝑃 = 1, 𝑆 = 100, 𝑆𝑝 = 𝑆𝑞𝑢𝑎𝑟𝑒), and those properties label it as its unique geometry. However, some shapes probably don’t have a name, so just invent a name for them, use your notation. Exceptions? Many physics concepts rely on waves. An example is sound. However, because nothing is continuous, waves can only be represented as þ(ψ), where psi represents a wave. This means that waves aren’t continuous either. This is easy to prove since everything that behaves in a wave works in quanta, but what about probability waves? Does this violate the rule of quanta? No! If there is a probability to find a particle in any position, the limit to smallest length is the Planck Length. Nothing can violate the rule of quanta, so there are no exceptions to this rule. However, as said before, this rule does have a limit, 𝑡 = ∞. This equation makes every rule that quanta introduce isn’t valid. This means that the Planck length, Planck time, and other constants decrease. But if the equation for the Planck length is ħ𝐺 3 𝑐 , then one of those values has to decrease. The most likely of which is the gravitational constant. If you have read my last paper, this says that the mass of the dartion decreases. Does this have anything to do with {𝑡 = ∞: 𝑄 = 0}(as time approaches infinity, quanta approach zero), 𝑄 = κ𝐺(size of quanta is directly proportional to the gravitational force) is true, so there may be a relation, but it’s possible that kappa is a random variable with no pattern(or maybe with)? Anyhow, we can conclude that there are no exceptions to the law of quanta when time is less than infinity. But there is one more question, if the multiverse theory is correct, what is the rate of creation of universes? This is really hard(maybe impossible) to find out, however, if we call the rate of creation of universes φ, then þ(φ) = φ if the quanta rule is true in all universes. Defining the Thorn Function Until now we haven’t given a formal definition of the thorn function, so here it is: þ { } 𝑓 (𝑥) = 𝑥 = 𝑛𝑄𝑥: 𝑦 = 𝑓(𝑛𝑄𝑥) . When x is equal to a multiple of one of the quanta for x, the y axis is equal to f of that value until you reach the next multiple. This is inefficient though. Instead of writing 𝑄𝑥, let’s make a thorn list for every variable. This is much more efficient. We also have the inverse thorn function, whose equation is −þ þ 𝑓 (𝑥) = lim 𝑓 (𝑥). Here we used thorn þ𝑐 → 0 sub x instead of q sub x. Instead of − þ to represent the inverse function, let’s use ß. This is not beta by the way, it’s called an Eszett. Anyhow, let’s apply the thorn function to 𝑐𝑜𝑠(𝑥). Here is a graph of 𝑐𝑜𝑠(𝑥): use most of them, so it doesn’t really matter. Ƿα þ𝑡 Our first equation should be β = , 𝑡. This means that velocity is the quantized thorn derivative of position. Then we can write Γ β þ𝑡 = α, which is the same but in integral form. We can do pretty much the same for each (thorn) derivative of position. Now, what about forces? We know that the derivative of force is momentum, but because of the law of quanta, derivatives aren’t useful, so we use thorn derivatives. This means we need to write 𝑝= Ƿ𝐹 þ𝑡 . However, in my last theory, I used a ( similar equation 𝑑𝐹 𝑑𝑡 =( correct it, we must write 2 4 2 𝐸 +𝑐 𝑚 𝑐 2 4 2 𝐸 +𝑐 𝑚 𝑐 Ƿ𝐹 þ𝑡 ) )δ , so to , 𝑡= ( 2 4 2 𝐸 +𝑐 𝑚 𝑐 is just a fancy way of saying )δ. momentum(and delta stands for dartion). Now, let’s look at some conditional thorn derivatives. An example is First, let’s define thorn sub x with the equation þ𝑥 = ( −π 2 − 1, 0. 5). From 𝑥 = −π 2 to 𝑥 =− 1, 𝑦 = 0. From 𝑥 =− 1 to 𝑥 =− 0. 5, 𝑦 = 𝑠𝑖𝑛(1). From 𝑥 =− 0. 5 to 𝑥 = 0, 𝑦 = 𝑠𝑖𝑛(0. 5). You do the same but in opposite order for the positive numbers, and you’ve got the thorn function of 𝑐𝑜𝑠(𝑥) from − π 2 to π 2 . One of the most important rules of the thorn function is that you must know at least one thorn list to calculate them. Ƿ𝑝 þ𝑡 , (𝑐 = 0). This one is pretty obvious, right? Since if the speed of light is zero, and also the greatest velocity, movement would be impossible. However, if we look at the mathematical roots of it, we get that it should be undefined, so at the end; Ƿ𝐹 þ𝑡 , (𝑐 ≠ 0) = ( 2 4 2 𝐸 +𝑐 𝑚 𝑐 )δ should be used. However, we shouldn’t say that the theory is correct just because it says stuff. The answer when 𝑐 = 0 to the thorn derivative of momentum should be zero. This means there is an error in our math. So basically, 𝑝= 2 4 2 𝐸 +𝑐 𝑚 𝑐 , (𝑐 ≠ 0). Let’s try to represent the equation by using the original formula, Derivatives of Position, Forces, Energy, Etc. 𝐸 = (𝑚𝑐 ) + (𝑝𝑐) , we can easily see that everything cancels out, so we just cancel out the c below and the c above is just equal to Velocity, Acceleration and Jerk are the first 3 (thorn) derivatives of position, and they’re pretty much the only ones used in practice. Let’s write position as α, velocity as β, acceleration as γ and jerk as δ. Why? Just to make calculations simpler. We probably won’t zero, so 𝑝 = 𝐸 , or 𝑝 = 𝐸. However, 𝐸 = 0 , so 𝑝 = 0. Wait, isn’t momentum just mass times velocity? So, in a universe where the speed of light is zero, the equation 𝐸 = 𝑚𝑣 works. What a coincidence, right? The reason it works is that there cannot be velocities 2 2 2 2 2 higher than zero. And because 𝑣 = 0, let’s 2 write 𝐸 = 𝑚𝑣 just for fun. Partial Thorn Derivatives & Directional Thorn Derivatives We have defined what a quantized thorn derivative is, but we haven’t defined partial quantized thorn derivatives yet. We first need to look at the formula for a thorn derivative. We just need to change one thing, there are discontinuous points for x and discontinuous points for y. So when you’re taking the partial thorn derivative of a 3d graph, you replace 𝑓(𝑑𝑛 ± 𝑑𝑥) into 𝑓(𝑦𝑛, 𝑥𝑛 ± 𝑑𝑥), assuming you’re taking the derivative with respect to 𝑥( 𝑥𝑛 is the discontinuous point n when you go Γ𝑓(𝑥), You can do this with any dimension. Now, the formal definition of a directional thorn derivative. This is also really simple. Assuming you’re working on a 3d graph, you replace 𝑓(𝑑𝑛 ± 𝑑𝑥) with 𝑓(𝑣𝑛 ± 𝑑𝑣). 𝑣 is a vector, so we can represent it as 𝑎 + 𝑏𝑖 + 𝑐𝑗, a complex number. Let’s ignore that right now, though. First, 𝑎 = 0. So we’re left with 𝑏𝑖 + 𝑐𝑗. Instead of using i and j, let’s use 𝑥 and 𝑦. So 𝑏𝑥 + 𝑐𝑦, because 𝑎 = 0, let’s write 𝑎𝑥 + 𝑏𝑦. And we can also add the restriction |𝑎𝑥 + 𝑏𝑦| = 1. The magnitude of the vector has to be one. And there we have it. The definitions of a partial thorn derivative and a directional thorn derivative. But what about directional quantized Gammaintegrals and partial quantized gamma integrals. Starting with directional gamma integrals. This formula is much shorter, so I can fit it here. The formula for a gamma integral is Γ𝑓(𝑥) { } = 𝑑𝑛 ⊂ (𝑑𝑛+1 − 𝑑𝑛)𝑓(𝑑𝑛) . We can define a directional thorn derivative as } almost the same thing but with a vector. Now, for the partial gamma integral, we use Γ𝑓(𝑥), ∂𝑥 =Γ𝑓(𝑥), 𝐴(𝑥) ≠ 𝐴(𝐷): 𝐷 = 𝑐. In words: the partial gamma integral of 𝑓(𝑥) is just its normal gamma integral, but when another dimension than 𝑥(or respect of the partial gamma integral), it is counted as a constant. We need to turn all of this into the quantized form, so for the directional gamma integral, we write thorn v at the end. For the partial gamma integral, we write a comma, and then we write a thorn before the respect of the derivative. For the partial thorn derivative, we just write wynn f over ∂ and the respect of the derivative. And for directional thorn derivatives, just write , ∇𝑣 at the end. Double Gamma Integrals and Second Thorn Derivatives through the 𝑥 axis and 𝑦𝑛 is the same but with the 𝑦 axis). If you’re taking the partial thorn derivative with respect to 𝑦, you instead replace 𝑓(𝑑𝑛 ± 𝑑𝑥) with 𝑓(𝑦𝑛 ± 𝑑𝑦, 𝑥𝑛). { 𝑣 = 𝑣𝑛 ⊂ (𝑣𝑛+1 − 𝑣𝑛)𝑓(𝑣𝑛) . It’s A double integral is what you get when you try to calculate the area under a 3d curve. If you write a double integral as ∫ ∫ 𝑓(𝑥) 𝑑𝑥𝑑𝑦, you have to write double gamma integrals ΓΓ 𝑓(𝑥) 𝑑𝑥𝑑𝑦. How do you compute those? First, remember how we used little packets for gamma integrals? For double gamma integrals we use 2 sets of packets, 𝑑𝑥 and 𝑑𝑦. 𝑑𝑥 is a little packet for the 𝑥 axis, and 𝑑𝑦 is a little packet for the 𝑦 axis. With double gamma integrals, we use rectangular prisms instead of rectangles. How do we compute double gamma integrals? Assuming every function is discontinuous, we use a pretty similar formula, ΓΓ { } 𝑓(𝑥) = (𝑥, 𝑦)𝑛 ⊂ (𝑥𝑛+1 − 𝑥𝑛)(𝑦𝑛+1 − 𝑦𝑛)𝑓(𝑥𝑛, 𝑦𝑛) . In words; if a discontinuous point in 𝑥 is encountered, and it is also discontinuous in 𝑦, then the area under the function until the next discontinuous points is equal to the 𝑥 value of the next discontinuous point minus the 𝑥 value of the 𝑛th discontinuous point, do the same but with the 𝑦 axis, take the product of both, and then multiply all that by 𝑓(𝑥𝑛, 𝑦𝑛). So it’s more complex(get it?.. Anyway) than the normal gamma integral formula. We can also obtain the triple integral by adding a factor of 𝑧𝑛+1 − 𝑧𝑛. A quadruple integral by adding a factor of 𝑤𝑛+1 − 𝑤𝑛 to the triple integral. If 𝐼(𝑥) is ∫ 𝑥, then we can write 𝑛+1 𝐼 𝑛 (𝑥) = 𝐼 (𝑥)((𝑛 + 1)𝑑)𝑎+1 − ((𝑛 + 1)𝑑)𝑎 . In words: The a plus one’ integral is equal to the nth integral times (the n plus one’ discontinuous point in the n plus one’ dimension minus the nth discontinuous point in the n plus one’ dimension). This is a more general definition of the gamma integral. Now, nth thorn derivatives. To take an nth thorn derivative of 𝑓(𝑥), we just take the thorn derivative of the thorn derivative. As always, we quantize everything. It is a bit harder to work with thorn derivatives, since the formula is pretty long. That’s the reason I’m not including it a lot. However, it’s pretty easy to memorize what double thorn derivatives are. Gauß Notation Gauß Notation(named after mathematician Carl Friedrich Gauß) is an easier way of expressing formulas in Quantum Calculus. Why was it named after Gauß? I don’t know, he was on my mind the moment I named it. Anyway, it replaces long terms with functions. It mostly uses the letter Eszett to differentiate from variables and functions. Remember the other dimensions. For that we just write 𝑑(ß𝑑𝑥(𝑑𝑛), where 𝑑 stands for dimension. The Eszett function can be used anywhere, not only in intersecting points, and you can use something like ß1, you can use more than ß𝑑𝑥. With this in mind, we can shorten the thorn derivative equation to þ𝑓 þ𝑥 = {𝑥 < 𝑑1: (0, 0)(ß𝑑𝑥(𝑑1)), 𝑥 < 𝑑𝑛: (ß𝑑𝑥(𝑑𝑛−1))(ß𝑑𝑥(𝑑𝑛)), 𝑥 = 𝑑𝑛, 𝑦 = ß𝑑𝑥(𝑑𝑛). See how shorter it is? Gradients A gradient is the direction of greatest increase in a function. For functions with a 1d input, it’s just the derivative. However, for functions with an input of more than 2d, the gradient is a vector made out of the partial derivatives with respect to all the input dimensions. For a 2d input function, the gradient would be ∂𝑓 ∂𝑥 𝑖+ ∂𝑓 ∂𝑦 𝑗. If you’re used to other vector Ƿ𝑓 ∂𝑥 𝑖+ Ƿ𝑓 ∂𝑦 𝑗. notation, sorry! You’ll have to get used to it. Anyway, as we said, derivatives don’t work in real life, so we should write that gradient as Modeling Heat Flow Before all, we must insert this distribution(in Kelvin). (𝑓(𝑑𝑛 + 𝑑𝑥) − 𝑓(𝑑𝑛 − 𝑑𝑥) term? We can shorten that to ß𝑑𝑥(𝑑𝑛). So when we encounter something like (𝑓(𝑑𝑛−1 + 𝑑𝑥) − 𝑓(𝑑𝑛−1 − 𝑑𝑥))(𝑓(𝑑𝑛 + 𝑑𝑥) − 𝑓(𝑑𝑛 − 𝑑𝑥)), we can shorten it to the much shorter equation; ß𝑑𝑥(𝑑𝑛−1)ß𝑑𝑥(𝑑𝑛). Next we have the version of the Eszett function for This is a temperature distribution, but how will it change over time? We normally would use something similar to a Quantized Eszett Function. But let’s first dive into the deep physical roots of temperature. Temperature can be defined as the average kinetic energy in a body. As volume increases, temperature decreases, and vice versa. Now, what is the mathematical basis for this? First, let’s define what we need. Let’s say temperature changes every Planck Time. So to calculate the rate of change, we should simplify the distribution to only 3 points. We get this: . We can label each of these points 𝑇1, 𝑇2, 𝑇3, 𝑇4, 𝑇5, 𝑇6, 𝑇7, 𝑇8 and 𝑇9. We will try to calculate the change in 𝐹5. We can tell it’s pretty similar to last time’s example. We just Let’s set our goal in calculating the temperature change of the middle point. We can sort the points as 𝑇1, 𝑇2, and 𝑇3, and their temperatures 𝐹1, 𝐹2 and 𝐹3. We will be measuring the immediate change of 𝐹2. We can notice that the change will be somehow related to the average of 𝐹1 and 𝐹3 compared to 𝐹2. We can write ∆𝐹2 = κ( 𝐹1+𝐹3 2 − 𝐹2), where kappa is a proportionality constant. Anyway, let’s do this experiment in 2 dimensions. I’m not the best 3d designer, so I’m just gonna make a grid and assign a value to each square. use ∆𝐹5 = κ( 𝐹2+𝐹4+𝐹6+𝐹8 4 − 𝐹5). However, there is a catch to this. This value changes only once every Planck time. Conclusion Quantum calculus is based on the thorn derivative. A thorn derivative isn’t the same as a discrete derivative, in fact, it doesn’t work in places where discrete derivatives do. However, using thorn derivatives as approximations gives us a closer answer to what would the derivative look like as the function becomes continuous. It just converges faster. This could be related to physics as the derivative at some point in the future. In a few words; The thorn derivative is meant to be an approximation, not a replacement.

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