feynman lectures on physics exercises volume 1

I need help concerning exercise 1 in the Feynman lectures on physics exercises book, volume 1, chapter1.
The question is: Estimate the number of air molecules per cm3 in ordinary air and in liquid air, knowing that ordinary air has a density of about 0.001 g/cm3. The only numerical data given in the chapter 1 of the Feynman lectures on physics course is that atoms are 2 x e-8 cm or 1 x e-8 cm. At the beginning of the exercises, Feynman states that precise numerical results are not needed.
Thank you for your help.

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Pronunciation exercises to learn nasalization

My eleven year old son is learning French in school. Picking up words, phrases, and grammar is easy for him, but he cannot pronounce nasal vowels. He just doesn’t know how to create those sounds.

What are easy and efficient methods to teach (a child) to pronounce French nasal vowels?

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Does leaving the fan switched on in the dojo reverse the effect of warm up exercises?

While learning Kalaripayattu in a city, it was not done in the traditional mud based basement, but in a school’s hall. So there were fans available. When we used to perspire profusely from intense warm-up’s we asked the guru if we could turn on the fan, but he advised us not to, saying it would negate all the effort we put into the warm-up’s because the fan would cool down the body.

Although it initially seemed to make sense, I later found it hard to believe, since the fan couldn’t possibly reduce blood circulation, body temperature, muscle and joint mobility to such an extent that it could make the warm-up pointless. I felt that as long as we continued with our exercises, it would be fine to leave the fan switched on at medium speed just to evaporate the perspiration and the stink it creates.

Is this an accurate assumption?

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Unable to use left arm, what exercises should I do?

Due to an injury I have a partial paralysis in my left arm, both shoulder and elbow.

This means I cannot do most of the upper body exercises. The only things I can do are single handed (dumbbell/machine) ones- biceps curl, triceps extension, shoulder press, chest press, etc.
I can grip stuff but I cannot lift any weight.

I want to focus on my core and legs. For the upper body, I want to bulk up a bit on shoulders, biceps and triceps so that cuts are visible but not too much as the asymmetry will look very ugly.

As per the BMI machine, I need to loose 4.9 kg fat and gain 2.5 kg muscle. (I’m 63 kg, 169cm)

I need to make up my exercise routine for 4 days a week, especially on specifically which core exercises I should do and how much cardio.

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Small to medium scale exercises about Haskell’s higher abstractions

Short version: Where can I find highly varying, small or medium-sized exercises and examples on the use, identification and instance declaration of Haskell’s abstractions such as applicatives, monads, monoids, foldables, transversables and monad transformers?

I have identified a serious lack of small to medium-sized exercises on Haskell’s “higher abstractions,” such as applicatives, monads, monoids, foldables, transversables and monad transformers. When studying, say, the use of higher-order functions such as foldr, many materials contain a good set of exercises, starting from simple ones and proceeding to varying levels of challenge. When it comes to the “higher abstractions,” the situation is, in my humble opinion, extremely poor, even after combing through many sources and trying to extract everything that is available.

To help the student understand and appreciate these abstractions I would expect to see highly varying, small to medium-sized exercises. These exercises should make the student, among other things,

  • use these abstractions
  • write instances of these abstractions
  • identify cases where these abstractions could be defined and used
  • identify cases where these abstractions can not be defined and used
  • compare the use of these abstractions to not using them.

As background information, my two favourite materials are Hutton’s Programming in Haskell (2nd ed.) for a solid basis and Allen’s and Moronuki’s Haskell Programming from First Principles for some of their material on more advanced topics.

While Hutton’s Programming in Haskell is a superb text for the basics, a good example of what is missing is its chapter “Foldables and friends” on monoids, foldables and transversables. The exercise set does not contain a single exercise that would make the student practice the use of these abstractions. It does not contain a single exercise of identifying the existence of these abstractions in some domain.

These abstractions are, in my opinion, an order of magnitude more difficult to assimilate than, say, higher-order functions. There is also at least an order of magnitude difference in the number of exercises available, but it is in the wrong direction.

Granted, Haskell Programming from First Principles does make you write a lot of instances of these abstractions in classic cases. Unfortunately it missed the other parts, and made me feel a bit like having proved a bunch of lemmas with very little idea of where those lemmas might be useful.

I think this is relevant for the Haskell community in general. If we can find pointers to good exercises, or examples that can be turned into exercises, I can create a public github repo and start collecting the material into one place.

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Stretching and relaxing exercises before playing piano

Do you have any good exercises to relax before playing the piano.

I’m interested specifically in relaxing the arms, shoulders and back.

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Complementary exercises to YOGA for releasing tension?

What exercises and exercise systems are the best complementary exercises to Yoga for releasing tension (immediate effect)

I am thinking of non-strenuous exercises like foam rolling.

With Yoga I mean Asana Yoga as in Hata, Ingyear or Bikram.

My current Yoga practice are Asanas on the floor, sometimes with a belt, sometimes rolling over a roll on my back (facing up)

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Calculating the complexity of an algorithm exercises

I am really bad at calculating correctly the complexity of a given algorithm. I would like to know if there is some book or online resources where I can find many exercises that ask to calculate the complexity of a given algorithm.

Thank you!

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Positive Square Exercises

I’m trying to do these problems, but I’m not sure how to start. Can someone help me figure out how I’m supposed to approach these? Thank you

Let a, b, c be positive constants. For all positive numbers x, y with
product c, find the minimum value of ax + by.

If a, b, c are real numbers not all equal, prove that:
a^2 + b^2 + c^2 > ab + ac + bc

Given any positive constant c, find the minimum value of x^4 + 2y^4
for positive numbers x and y having product xy = c.

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Solutions for exercises about Brownian motion in Continuous-time Stochastic Processes [on hold]

I need some help in solving the exercises about Brownian motion in Continuous-time Stochastic Processes. These are the formulas I’ve found so far:
$Var(W_s)=s $
$E[W_t] = 0 $
$E[W_sW_t] = min(s, t)$ for all s, t ≥ 0
$E[W^2] = t $
$⟨X,Y⟩=frac 14(⟨X+Y⟩−⟨X−Y⟩)$
$⟨W⟩_t=t$

But the solutions for the more complex exercises are still unclear. For example, let W={Wt:t≥0} be a Brownian motion and 0 < s < t. Find:
1) $Var(W^2_1)$
2) $Var(W_s+W_t)$
3) $E[W_{2t}(W_t−W_{t^2})]$
4) $E[W_s+W^2_{t^2}]$
5) $Cov(W_1+W_3,W_2−W_5)$
6) $⟨W+5,7−W⟩_t$

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