physical chemistry - What are the maximum number of electrons in each shell?

In my textbook, it says that the maximum number of electrons that can fit in any given shell is given by 2n². This would mean 2 electrons could fit in the first shell, 8 could fit in the second shell, 18 in the third shell, and 32 in the fourth shell.

However, I was previously taught that the maximum number of electrons in the first orbital is 2, 8 in the second orbital, 8 in the third shell, 18 in the fourth orbital, 18 in the fifth orbital, 32 in the sixth orbital. I am fairly sure that orbitals and shells are the same thing.

Which of these two methods is correct and should be used to find the number of electrons in an orbital?

I am in high school so please try to simplify your answer and use fairly basic terms.

fast days - How does prolonged (multiday) fasting work?

In Esther 4:16, Esther tells Mordecai to have the entire Jewish community fast for three straight days:

‘Go, gather together all the Jews that are present in Shushan, and fast ye for me, and neither eat nor drink three days, night or day; I also and my maidens will fast in like manner; and so will I go in unto the king, which is not according to the law; and if I perish, I perish.’

And in Jonah 3, there also seems to be a suggestion of some sort of prolonged fasting (though it's less explicit):

Now Nineveh was an exceeding great city, of three days’ journey. And Jonah began to enter into the city a day’s journey, and he proclaimed, and said: ‘Yet forty days, and Nineveh shall be overthrown.’ And the people of Nineveh believed G-d; and they proclaimed a fast, and […] And [the king] caused it to be proclaimed and published through Nineveh by the decree of the king and his nobles, saying: ‘Let neither man nor beast, herd nor flock, taste any thing; let them not feed, nor drink water; […] Who knoweth whether G-d will not turn and repent, and turn away from His fierce anger, that we perish not?’

How do such extended fasts work?

I've met many people who aren't able to completely fast for even 25 hours or so for Yom Kippur. Is it just a matter of motivation — that the people of Shushan and of Nineveh were more afraid during these events than we are on Yom Kippur? Or did Hashem lend them strength to continue fasting? Or something else?

Answer

I don't see (in my Lublin-style Mikraos G'dolos) any commentary note how many days the Nineveh folks fasted. I'd have to assume, absent other knowledge, that they fasted until the end of the day the fast was proclaimed, or maybe only the entire next day. (That's what I'd do if an announcement were made in the king's name that everyone must fast, with no duration specified.)

The Esther fast is of course harder to explain that way: it says it was to last "three days, night and day". But it was only three days: the Bavli, Y'vamos 121 amud 2, says that there was no miracle in their surviving this 72-hour fast.

Finding the z-transform of $h[n] = a^ncos(2pi frac{n}{F_s}f_0)$ for $n ≥ 0$ and zero for $n < 0$

So I'm trying to decide whether the cosine part is intended to be plugged in for $z$ or whether it is strictly part of $h[n]$. (the number a lies in the open unit disk)

I mean I was pretty sure it was all part of $h[n]$ but then upon performing the z-transform I get this rational function

$$\frac{1 - a\cos(2\pi\frac{f_0}{F_s})z^{-1}}{1-2a\cos(2\pi\frac{f_0}{F_s})z^{-1} + a^2z^{-2}}$$

The thing is then I'm supposed to evaluate the poles and zeros and if you just ignore the cosine parts you get this really nice rational expression which factors and simplifies down to $\displaystyle\frac{z}{z-a}$.

So that has gotten me thinking that maybe I'm not understanding things correctly and the cosine portion is supposed to be plugged in for $z$ or something. Can anyone clarify this for me?

history - When did the practice of Exodus 35:2 stop?

Exodus 35:2 JPS

ב שֵׁשֶׁת יָמִים, תֵּעָשֶׂה מְלָאכָה, וּבַיּוֹם הַשְּׁבִיעִי יִהְיֶה לָכֶם קֹדֶשׁ שַׁבַּת שַׁבָּתוֹן, לַיהוָה; כָּל-הָעֹשֶׂה בוֹ מְלָאכָה, יוּמָת.‏

Six days shall work be done, but on the seventh day there shall be to you a holy day, a sabbath of solemn rest to the LORD; whosoever doeth any work therein shall be put to death.

I assume that this is no longer practiced. When did it stop?

Answer

From myjewishlearning.com

According to the Mishnah (Sanhedrin 1:4) the death penalty could only be inflicted, after trial, by a court composed of twenty-three judges and there were four types of death penalties (Sanhedrin 7:1): stoning, burning, slaying (by the sword), and strangling. A bare reading of these and the other accounts in the tractate would seem to suggest a vast proliferation of the death penalty. Yet, throughout the Talmudic literature, this whole subject is viewed with unease, so much so that according to the rules stated in that literature the death penalty could hardly ever have been imposed...

...the right to impose the death penalty had been taken away from the Jewish courts by the Roman authorities. According to one report in the Talmud (Sanhedrin 41a) the power of the Jewish courts to the death penalty ceased around the year 30 BCE; according to another report (Sanhedrin 52b) it could only have been imposed while the Temple stood and must have come to an end not later than 70 CE when the Temple was destroyed.

The first opinion was saying that the Jewish court had the power to rule death penalties only if they were in their proper location, the Beis Hamikdash, see this note by the Soncino Talmud on Sanhedrin 41a.

"Deut. XVII, 10: And thou shalt do according to the tenor of the sentence which they shall declare unto thee, from that place; this implies that it is the place that conditions the authority of the Sanhedrin in respect of the death sentence. [J. Sanh. I, 1 has, 'the right to try capital cases was taken away from them, i.e., by the Romans. For a full discussion of the subject v. Juster. op. cit, II, 138ff.]"

The second opinion is saying that the power to rule death penalties comes from the Beis Hamikdash, once that was destroyed, regardless of where the court was, the power to give death penalties ceased.

It is important to note that death penalties were occurred very rarely throughout Jewish history.

As the myjewishlearning.com article continues

(Mishnah Makkot 1:10): "A Sanhedrin that puts a man to death once in seven years is called destructive. Rabbi Eliezer ben Azariah says: even once in seventy years.

eretz yisrael - Why did it take so long to conquer and divide the land?

We learn that Yehoshua took 7 years to conquer the land and another 7 to divide it. Why did both of these take so long?

• Conquest (at least from the details of the description in the text) took at most a couple of months, especially with the Divine assistance. (There is a Midrash Tanchuma that says he intentionally prolonged the conquest and was punished by losing 10 years of his life, however, it cannot be that he prolonged it for 7 years instead of a couple months - that'd be too gross of an error - there must have been something else.)


• Dividing land - what else had to be done besides the lottery, which I suspect hadn't lasted more than 1 day?

Thank you very much.

Answer

The division of the land included, among other steps:

• sending out commissioners to survey it (Josh. 18:4ff)


• ...and to evaluate the worth of the individual regions and plots of land, to make sure that the division would be equitable (Rashi to Num. 26:54)


• designating which roads would be private vs. public, depending on their grade (Eruvin 22b)


• assigning a plot to each adult male Jew living at that time, then redividing based on the identities of the Jews who had left Egypt (Bava Basra 117a and Rashbam there ד"ה אלא; Rashi to Num. 26:55)


• and perhaps only then casting the lots and asking the urim vetumim which tribe would end up in which part of the land.

So all of that can easily have taken quite a while.

nuances - What is the difference in terms of grammar between ～かける and ～っぱなし?

Spawned from What is the difference in terms of grammar between きり　and　っぱなし?; I started thinking about ～かける. Don't these essentially mean the same thing? I'm failing to see any difference except that maybe ～かける doesn't necessarily have a negative nuance to it.

• 食べかけたリンゴ　→　An apple I started eating
  


• 食べっぱなしのリンゴ　→　??? (is it even grammatical; or used?)


• やりかけた　→　Started but not finished (?)
  


• やりっぱなし　→　Unfinished, incomplete

Answer

As Chocolate suggests, one difference is that かけ implies that the activity is not completed, whereas ぱなし means that the activity is completed, and as a result, some negative situation (messed up, etc.) is left.

やりかけた 'had started working on something (but have not completed)'
やりっぱなし 'completed doing something, (and have not cleaned/put away the tools after it)'

食べかけたりんご 'an apple I have started eating (but have not eaten up)'

食べっぱなしのりんご is very strange because, if you complete eating it, there should not be anything to talk about regarding the situation of the apple that had disappeared. You can talk about a dish (that had the apple on it) not being put away, or a table on which such dish is left, but then, the sentence should be

食べっぱなしの皿
食べっぱなしの食卓

halacha - Can one be counted in a minyan if one has already davened?

Can one be counted in a minyan if one is present for Minchah/Ma’ariv but has already davened Minchah?

Related question: Should I alter my schedule to daven Minchah with a minyan?