Genady

Right, but I do it intentionally, before entering the water.

They did it in Russia, but not in Azerbaijan. It was entertaining to see them on TV. I scuba dove in a quary in PA in winter when most of the surface was frozen. But I wore dry suit with sweater and sweatpants. Very cozy.

In a field K the equality a4=a is satisfied for all a. Find the characteristic of the field K. My calculations: 2a = (2a)4 = 16a4 = 16a 14a=0 This allows for the characteristic to be 2 or 7. But, e.g., 34 = 81 = 4 (mod 7) rather than 3. So, the only answer is 2. 1. Is this derivation correct? 2. The hint in the textbook says to show that 2a=0. How to do that?

Thanks guys. I think it is important to clarify that my so called "cold water" is much warmer than 15°C.

No, I didn't consider this connection. Thank you. Interesting, but the connection to the water temperature is not clear to me. It could be a coincidence that the water happened to be cold, couldn't it? Maybe the plunging head-first did it. Or jumping itself, if he had an undiscovered brain aneurysm. Etc.

It might be so, while the other possibility is that inhaling and holding breath while entering water takes my mind away from the "pain" of being hit by the cold water. Yes, I've noticed it as well. Another "trick" that works for me is, while standing in the shallow, before putting the body in the water, I put my face in or simply splash the cold water on the face. The face does not feel much, but it somehow prepares the whole body, and the shock is diminished quite a lot.

Why does jumping into cold water feel less shocking when I inhale and hold breath while jumping in?

Consider the following proof: My question is, does it in fact use induction? It says, "Assume now that the theorem is true for k-1 elements," but I don't see this assumption being used in the proof to advance from k-1 to k elements, which would be an induction step. Actually, this assumption is not used at all, AFAICS.

Correct. Correct.

Just want to clarify that the class was in Duke University, NC, with American students. And that I was among the ones who understood and answered the question as was intended by the professor, i.e., it is visible sometimes.

This reminded me of another ambiguous term, ever. I remember (vaguely) an argument in a class I took many years ago, about a quiz question that was something like, Is the star A ever visible at the location B? Some students answered Yes because it is sometimes visible, while others answered No because it is sometimes invisible.

Thank you. It answers the question. Thank you. I will keep it in mind.

Nothing is wrong with them. I think it is as defined in textbooks. Yes, square root of 2 is algebraic. So, the question remains, is cosine of any fraction of pi algebraic? The other question is, was my English in the OP correct?

\(\cos(\frac {\pi} {4})=\frac {1} {\sqrt{2}}\) \(\cos(\frac {\pi} {5})=\frac {1+\sqrt{5}} {4}\) \(\cos(\frac {\pi} {6})=\frac {\sqrt{3}} {2}\) Is cosine of any fraction of π an algebraic number?

-4 - 4 = -8 -4 - 3 = -7 -4 - 2 = -6 -4 - 1 = -5 -4 - 0 = -4 -4 - -1 = -3 -4 - -2 = -2 -4 - -3 = -1 -4 - -4 = ?

I think, it is. Two observers measure distance between two events. Per the model, they count number of nodes in the graph on the path which connects two events. Do they count different number of nodes?

Does the graph distance depend on observer like the spatial distance does?

a=1.8954942670...

What is a justification of identifying the graph distance with a spatial distance? Is it a postulate of the model?

This is the same as the hyperlink in the OP: Anyway, is this model falsifiable?

Here is some: Claims in 'Duty to Warn' Letter to Harris Alleging Compromised Election Are Misleading | Snopes.com

Misconceptions.

They are entangled.

@dqqd, this analysis in Wikipedia is quite clear: Variable-mass system - Wikipedia

I disagree: the force has not been redefined. In the formula \(F=\frac {dP}{dt}\), \(P\) is the momentum of the entire system. If the system is composed of parts, \(P=P_1+P_2+...\), then \(F=\frac {d(P_1+P_2+...)}{dt}=m_1 \frac {dv_1}{dt}+v_1 \frac {dm_1}{dt}+m_2 \frac {dv_2}{dt}+v_2 \frac {dm_2}{dt}+...\), where no mass crosses the boundary of the composed system.