Archive of posts with tag “calculus”
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Relationship between the Gini coefficient and the variance
Both the Gini coefficient and the variance are measures of statistical dispersion. We are then motivated to find the relationship between them. It turns out that there is a neat mathematical relationship between them. -
Defend our earth against aliens’ bullets!
The aliens intiated their attack to the earth! They shoot bullets with mass and speed from a far-awar planet. To defend, humans built a field that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid. -
This is what will happen after you get a haircut
Denote the length distribution of one’s hair to be , where is hair length, and is time. Considering that each hair may be lost naturally from time to time (there is a probability of for each hair to be lost within time range from to ) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE. -
Solving ODE by recursive integration
By recursively integrating according to from , we can get the solution of the ODE
with initial conditions as the limit of the sequence of functions. -
Hölder means inequality
The Hölder mean of with weights and a parameter is defined as , and the value at are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as increases. This property can be used to prove HM-GM-AM-QM inequalities. -
Solving linear homogeneous ODE with constant coefficients
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation… -
Thoughts on a middle school thermal physics problem
To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit , where is the initial temperature of the object, is the temperature of the hot water, is the heat capacity of the object, and is the heat capacity of the hot water. -
Generalization of Euler–Lagrange equation
We may generalize Euler–Lagrange equation to higher dimensional optimization problems: find a function defined inside a region to extremize a functional defined as an integral over that region, with the constraint that the value of the function is fixed on the boundary of the region. -
Simulating a mechanical system using rpg_core.js
Continuing my last work of simulating a mechanical system using RGSS3, I made a new version using rpg_core.js, the game scripting system shipped with RPG Maker MV. This version is live on web! -
Simulating a mechanical system using RGSS3
Hamiltonian mechanics gives us a good way to simulate mechanical systems as long as we can get its Hamiltonian and its initial conditions. I implemented this simulation in RGSS3, the game scripting system shipped with RPG Maker VX Ace.
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