Recent content by Gavran

Let ## (x_1,y_1) ## be a point on the curve ## f(x)=4-x^2 ##, which is the closest point on the curve ## f(x) ## to the curve ## g(x)=(x-3)^2 ##, and let ## (x_1+a,y_1+b) ## be a point on the curve ## g(x)=(x-3)^2 ##, which is the closest point on the curve ## g(x) ## to the curve ## f(x) ##...

There are Figures 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19 in the same text (https://cdn.prod.website-files.com/5dc3329c429c1a9866f49d34/5e37131bc62330e7ed1cedb0_3phXfmrs_GATech.pdf pages 12, 13, 14, 15, 16, and 17), which demonstrate how transformer terminals are connected with windings.

It depends on how X1, X2, and X3 are interconnected with W4, W5, and W6. See again Figure 4-D7 (the third image on the right).

Yes, you are right. That means X1 lags H1 by 30° in group 1, and X1 lags H1 by 330° in group 11. Both examples are correct. See Figure 4 - D7 from https://cdn.prod.website-files.com/5dc3329c429c1a9866f49d34/5e37131bc62330e7ed1cedb0_3phXfmrs_GATech.pdf (page 7).

This is group 1. If H1 points at 12 o'clock, X1 will point at 1 o'clock. This is group 11. If H1 points at 12 o'clock, X1 will point at 11 o'clock.

The substitution ## y_1=10^y ## simplifies the equation $$ 10^ydy=\frac{dx}{x\ln10} $$ to the equation ## dy_1=dx/x ##. The general case can be obtained by replacing ## 10 ## in the substitution ## y_1=10^y ## with ## b ##, where ## b\gt0 ## and ## b\neq1 ##, while the equation ## dy_1=dx/x ##...

In the general case, the time that the ant needs to reach the end of the rope is $$ T=\frac cv(e^\frac v\alpha-1) $$, where ## c ## is the initial length of the rope, ## v ## is the rate at which the rope stretches, and ## \alpha ## is the speed of the ant relative to the rope. Clearly, any...

You can notice from the animation that the ant absolutely moves at an increasing speed, while the rope end absolutely moves at a constant speed. By the way, 8,9×1043421 years is a long period, and the ant will definitely die before it reaches the rope end.

It is good to see how problem-solving by humans can be enriched by artificial intelligence. It can also be seen that problem-solving by artificial intelligence can be enriched by humans. The concept of human-artificial intelligence is a mutual augmentation, and it cannot be accepted as a one-way...

I remember the thread posted by @chwala in 2024, which is used as a reference in the article. The geometric solution provided by artificial intelligence is also described in the article. In this case, we cannot use it for comparing human labor with artificial intelligence labor because nobody...

A veridical paradox is one of three types of paradoxes, which is what defines it as a paradox.

100 kg of potatoes consists of 99% water. After some water evaporates, the potatoes consist of 98% water. What is the new weight of the potatoes? The answer is 50 kg. The explanation is simple. 100 kg of potatoes consists of 99% water and 1% dry matter, so the weight of the dry matter is 1 kg...

This can also be shown in the following way. ## \begin{align} &\arctan\frac\alpha c+\arctan\alpha=\arctan c\nonumber\\ &-\arctan\frac{\alpha}{\alpha\beta}+\arctan\alpha=\arctan c\nonumber\\ &-\arctan\frac1\beta+\arctan\alpha=\arctan c\nonumber\\ &\arctan\alpha-(\frac\pi2-\arctan\beta)=\arctan...

The equation ## C=A+B ## is actually Euler's formula $$ \arctan\frac12+\arctan\frac13=\frac\pi4 $$, and there are three more formulas of this kind. Hermann's formula $$ 2\arctan\frac12-\arctan\frac17=\frac\pi4 $$ , Hutton's or Vega's formula $$ 2\arctan\frac13+\arctan\frac17=\frac\pi4 $$ , and...

There are four formulas of this kind. Euler's formula $$ \arctan\frac12+\arctan\frac13=\frac\pi4 $$ Hermann's formula $$ 2\arctan\frac12-\arctan\frac17=\frac\pi4 $$ Hutton's or Vega's formula $$ 2\arctan\frac13+\arctan\frac17=\frac\pi4 $$ Machin's formula $$...