history - Tannis Esther reason

Why do we fast on Tannis Esther? In memory of what historical occurrence?

Answer

Some say the Jews fasted the day they fought, or the day before it. (So on Purim, more or less.)

Some say that the holiday commemorates "divrei hatzomos v'zaakasam", the matter of their fasts and cries; so we fast the day before Purim to commemorate that in the Purim story, Esther and the Jewish people all fasted and prayed before she approached the king. (Even though the actual fasting in the Purim story happened not in the Hebrew month of Adar in which Purim is celebrated, but 11 months earlier, in Nisan of the previous year).

Lastly, the Raavad in Sefer HaEshkol says the fast has nothing to do with any of this. It's that if people could eat the day before Purim, they might drink too, and once they're drinking ... well .. as Rabbi Hershel Welcher put it, "We want people to show up to Megillah reading, able to stand up on two feet!"

halacha - I heard not to say "Good Shabbos" after mincha. Is it true, and what defines mincha?

I heard from somewhere that, on Shabbos, one does not say "Good Shabbos" after mincha. I am not sure whether it's in a halacha sefer somewhere (I tried looking) or just a minhag.

First of all, is this true?

Second, what defines mincha? Is it after someone says mincha, or is it after mincha gedola or some different time (mincha ketana or the like)?

Answer

As @GershonGold mentioned in his answer, Mincha time of Shabbat is associated with the passing of Moshe, Yosef, and David. One of the ways we commemorate it is by not greeting someone by wishing them a Good Shabbat at Mincha time. [The Mishna Berurah S"K 6 on Shulchan Aruch (Orach Chaim 292:2) tells us that this is why we say Tzidkatcha after the Amidah in Mincha on Shabbat]

The Rema, in Shulchan Aruch Orach Chaim 292:2, says that for this reason the Ashkenazic custom is not to learn between Mincha and Maariv on Shabbat afternoon. The Mishna Berura (S"K 8) says that this only applies when Mincha is prayed close to the end of the day, otherwise it is not a problem. (He then goes on to describe the limitation of this practice, and the fact that we don't follow it these days).

From there we see that the time of mourning (which would prevent us from saying "Good Shabbat" to one another) is after Mincha, close to darkness.

---

All this is even more clear if we look at the other reason the Mishna Berurah (S"K 6) brings for saying Tzidkatcha by Mincha. We are saying Tzidduk HaDin (admitting the righteousness of G-d's Judgment) about the wicked people, who are going back to the torments of Gehenom when Shabbat ends, after there reprieve from the pain on Shabbat.

If so, it makes sense that the time of Judgment is close it when it gets dark and Shabbat will soon be over.

grammar - Where does the verbal form しとく come from?

In a manga I am currently reading, one of the character exclaims:

安心しな。秘密に しとく から。

The general meaning of the second part ("I'll keep it a secret") is quite obvious, and it seems that "しとく" (ostensibly derived from し, with a suffix appended) has more or less the same meaning as plain "する". But I am curious about the grammatical (or dialectal?) construct used here, its usage, nuance etc.

The closest to an explanation I could find, was this verb in WWWJDIC:

とく (v5k) to do something in readiness for, to get something (needful) done

... but I must admit I'm still at a loss as to why it would be used as an auxiliary here.

What is the exact nuance (and grammatical origin) of that "しとく" and does the "とく" suffix work with other verbs?

Answer

しとく comes from しておく, which in turn comes from して置く. The literal translation of して置く would be, "do it, and then put [the results]". Basically it describes the act of doing something and storing the result of that so that when that result becomes useful, you can use it.

EDIT:
This literal meaning changed overtime (I presume) and しておく became to mean "do something to prepare for something that might happen". In case of 秘密にしておく, the speaker is saying that he/she will keep it secret in case it turns out whatever thing they were discussing indeed needed to be kept secret. I guess it makes some sense, as you could think that the state of that thing being kept secret, is "stored" or "put" somewhere, and then it turns out that that state was useful, or something like that.

The nuance further changed (I presume!), and it became ok to use it almost as a softner. For example, a Japanese speaker might say 今日飲んどく？ instead of 今日飲みに行く？. Technically it still means to drink "just in case", but here it's just used to soften the speech. 秘密にしておく could also be a softner, depending on the context.

え、また泥酔して記憶なくなったの？　大丈夫！秘密にしとくから（笑）

　　　　-> Softner
今のところ会社の人には秘密にしておいてください
　　　　-> Let's keep it secret for the time being, in case it it wise to keep it as a secret.

しておく is frequently used with とりあえず, which translates to "for the time being".

今日は台風が来ているから、飛行機が飛ぶか分からないけど、取りあえず準備だけはしておこう。　　
今日は台風が来ているから、飛行機が飛ぶか分からないけど、取りあえず準備だけはしとこう。

Both are ok, while the latter is colloquial.

次いつ駐車できるかわからないから、ここでお昼ごはんにしとくか？

次いつ駐車できるかわからないので、ここでお昼ご飯にしておきましょうか？

Above is another example. しとく is more frank/colloquial form of しておく.

～おく can also work with other verbs. Here are some examples:

ここ、塗っとこうか？
ここ、塗っておきましょうか？

飯、食っとけ。
ご飯食べておいてください

EDIT2
Apparently it's even more complex than that. Here is a Japanese paper on ～ておく's meaning.

parshanut torah comment - Where was Adam when Chava cheated with the snake?

Where Adam was when Chava was sinning with the snake?

For me, it looks that Eden doesn't seem to be a huge place to walk around and Adam was so excited to have a woman by his side, also Adam and Chava didn't have anything important to do, after all, they weren't positively commanded on doing anything.

Answer

Bereishis Rabbah 19:3:

וְהֵיכָן הָיָה אָדָם בְּאוֹתָהּ שָׁעָה, אַבָּא בַּר קוֹרְיָיה אָמַר נִתְעַסֵּק בְּדֶרֶךְ אֶרֶץ וְיָשַׁן לוֹ. רַבָּנָן אַמְרֵי נְטָלוֹ הַקָּדוֹשׁ בָּרוּךְ הוּא וְהֶחֱזִירוֹ בְּכָל הָעוֹלָם כֻּלּוֹ, אָמַר לוֹ כָּאן בֵּית נֶטַע, כָּאן בֵּית זֶרַע, הֲדָא הוּא דִכְתִיב (ירמיה ב, ו): בְּאֶרֶץ לֹא עָבַר בָּהּ אִישׁ וְלֹא יָשַׁב אָדָם שָׁם, לֹא יָשַׁב אָדָם הָרִאשׁוֹן שָׁם.

So either Adam was sleeping after having had relations with Chava, or Hashem had sent him on a world tour to identify the various types of agricultural lands.

A couple of the assumptions in the OP are also questionable:

• 
  

  "Eden doesn't seem to be a huge place to walk around": (1) they were actually in the Garden, not in Eden itself. Those aren't the same - "a river flows out of Eden to water the Garden" (Bereishis 2:10), and Chazal point out that no human eye has ever seen Eden (Berachos 34b, Sanhedrin 99a). (2) Chazal also state (Pesachim 94a, Taanis 10a) that the entire world is 1/60 of the Garden, which in turn is 1/60 of Eden. True that such expressions aren't necessarily literal, it would still mean that the Garden (let alone Eden) is pretty big.

  
  


• 
  

  "Adam and Chava didn't have anything important to do, after all, they weren't positively commanded on doing anything": what about לעבדה ולשמרה? Whether you take those in the literal sense, or as referring to mitzvos,* that's definitely something they were commanded to do.

  
  

* A number of early acharonim (Alshich, Shaloh, etc.) quote a maamar Chazal לעבדה אלו מצות עשה ולשמרה אלו מצות לא תעשה, though I haven't found the original source. In Bereishis Rabbah 16:5 it does bring one opinion that they refer to working during the six weekdays and keeping Shabbos, and another that they refer to the korbanos.

halacha - Melachah on Shabbath - after Shabbath

If Melachah was done on Shabbath, can the product of that Melachah be enjoyed after Shabbath, and if so, under what criteria (does it depend on how long after Shabbath, or on the need, or something else)?

Answer

Based on Mishna Berura and Biur Halacha 318:

There are 3 variables which affect your answer:

1. Did the transgressor know he was violating shabbos,


2. Who wants to benefit, the transgressor or others,


3. Was the act Bibically or rabbinically prohibited.

There are two opinions about benefiting from a shabbos transgression-

Rabbi Meir: If the transgression was done knowingly (meizid), all may benefit from the act only after shabbos. If the transgression was done unknowingly (shogeg), all may benefit immediately

Rabbi Yehuda: We are stricter with the transgressor- If he did it knowingly, he may never benefit. If he did it unknowingly, he must wait until after shabbos. (All others are the same as R' Meir.)

There is a difference of opinion as to who we pasken like. The Shulchan Aruch by cooking (318:1) sides with Rabbi Yehuda. The Gra maintains that he only paskened so for Biblical prohibitions, but by rabbinic prohibitions, there is no penalty for an unknowing violation. Th Gra himself paskens like R' Meir.

If you are a Mishna Berura man:

All knowing violations, whether Biblical or rabbinic, are prohibited to the violator always and for everyone else until after shabbos- like R' Yehuda

Rabbinic, unknowing violations are permitted immediately to all. Biblical unknowing violations are still forbidden to the transgressor himself until after shabbos- like the Gra's version of the Shulcha Aruch

All unknowing violation, Biblical or rabbinic, in cases of necessity, are permitted immediately to all- like the Gra who paskens like R' Meir

Additionally:

In a case knowing case where a non-Jew was employed to violate the melacha, there may be an additional penalty after shabbos- the amount of time it took to perform the melacha- so that there is no nominal benefit.

In a case where there is an argument whether there is a violation, there is no penalty and all are permitted to benefit immediately.

halacha - Is it permitted for a male to shave his pubic hair nowadays?

Is it permitted for a male nowadays to shave his pubic hair for non-medical reasons? Perhaps this question could be phrased a different way: Assuming that the market for male-specific shaving products that are designed for body hair is an indication that it is currently normal for (non-Jewish) men to shave their body hair, would this in turn permit a Jew to shave his body hair since it has become a norm?

halacha - What to do after sh'mone esre before Sh'ma

Shulchan Aruch, OC 236:3:

If someone discovered a group that had already read [the evening] "Sh'ma" and is about to say amida [of maariv], then he should say amida with them and then say "Sh'ma" with its blessings.

• Is there any reason for him to say or hear "Bar'chu" before he says the blessings of "Sh'ma"? Any reason not to? Sourced answers only, please. (I know OC 69 discusses the somewhat similar rule of pores al "Sh'ma", for which one says "Bar'chu", but I'm looking for a source that explicitly says whether or not to say "Bar'chu" in the case discussed in OC 236:3.)


• The community will be saying kadish and "Alenu", which under normal circumstances a bystander would respond to. But our latecomer is between sh'mone esre and the blessings of "Sh'ma". Must he reply? Must he avoid replying? In general, is there any reason for him to avoid interruptions between sh'mone esre and the subsequent blessings of "Sh'ma"? What, for example, about chatting with his friend? (I know there's a general rule of avoiding delay before saying "Sh'ma" (Mishna B'rura 235:17), but I'm asking about cases where the general rule permits delay but there may be some rule specific to our latecomer's situation that bars interruption/delay.)

words - what is "kodesh"?

What does the word קֹדֶשׁ (sometimes קודש), kodesh, mean?

Its form is that of a noun, like אֹכֶל or בֹּקֶר or לֹבֶן. But קדושה is a noun, and (I'm pretty sure) it means "the state/quality of being קָדֹשׁ" (arguendo, "the state/quality of being holy"), so I find it unlikely that קֹדֶשׁ means the same. If it does, then what's the difference between it and קדושה?

And if קֹדֶשׁ means "something that is holy" — which I suspect, and which would explain phrases like קֹדֶשׁ קדשים (the room or the korbanos) — then how do we understand phrases like שַׁבַּת קֹדֶשׁ (literally then "the Shabas of something holy") and וּבְדִבְרֵי קָדְשְׁךָ (literally then "in your words of something holy")?

How do I find Rf values & length size of paper in chromatography?

A student is designing an experiment using paper chromatography. From previous results, she knows that the Rf values for two dyes that she wishes to investigate are 0.72 and 0.80.

It is proposed that the dyes be run on the same piece of paper and, for various reasons, that the sports should have a separation of a at least 1cm at the end of the procedure.

What is the minimum height of the chromatography paper that she should use in this experiment?

Can someone correct my working out. I don't know what the correct answer is.

• distance travelled by dye x = $x$


• distance travelled by dye y = $y$


• distance of solvent front = $p$


• $x/p=0.72$ therefore $x = p(0.72)$


• $y/p=0.8$, therefore $y = p(0.8)$


• $y-x=1$, therefore $y=1+x$


• new equation: $1+x=p(0.8)$ where $x=0(0.72)$, if we solve for $p$ we get $12.5$

so the piece of paper should be at-least 12.5 cm long.

Answer

$$\Delta R_f = 0.08$$ Let the separation distance be $\Delta r$ = 1 cm and let $l$ be the length of the paper in cm.

$$\Delta R_f = \frac{\Delta r\, \mathrm{cm}}{l\, \mathrm{cm}} $$

Solve for $l$, obtain $l$ = 12.5 cm.

parshanut torah comment - Why did Ya'akov work for Lavan for so long?

I'm having a bit of trouble understanding Or Hachaim's explanation on Breishit 29:18. Usually Jewish slaves work for only 6 years. Yet, Ya'akov offers to work for Lavan for 7 years as payment for marrying Rachel. Ohr Hachaim says that there is a "secret" behind the number 7. Despite that, esp. if Ya'akov had already suspected that Lavan was not trustworthy, why did he offer his work for so many years? (I may be missing something in Ohr Hachaim's explanation that would answer this.)

analytical chemistry - What is the relative size of the (M+2) peak?

The $(M+1)$ peak is often considered in the high-resolution mass spectra of organic molecules as it reveals the number of carbon atoms in the sample. In general, it is known that the ratio of the size of the $M$ to $(M+1)$ peaks is $98.9 : 1.1 \times n $ since the relative abundance in nature of $^{13}$C is $ 1.1$% for the mass spectrum of an organic molecule containing $n$ carbon atoms and no heteroatoms. It is mentioned that the $(M+2)$ peak is statistically insignificant on this site. However, I believe that only applies for organic molecules with a relatively small number of carbon atoms and this peak would become significant when considering the mass spectra of larger organics. Using simple mathematics, I derived that the ratio of the $M$ to $(M+2)$ peak is $98.9^{2} : 1.1^{2} \times _nC _2 $. I would like to verify if this is correct. If it is not, could someone then correct it by posting an answer?

Answer

You are correct on all accounts.

To a very good approximation, molecules can be thought as made of elements (with their respective isotope distributions) combining completely independently. You can think of it like rolling multiple die at once. This means that a simple multinomial distribution will describe this problem mathematically.

Let's start with something easy and consider the hypothetical molecule $\ce{C_5}$. Furthermore, let us consider that the only carbon isotopes with significant natural occurrence are $\ce{^{12}C}$ (98.9%) and $\ce{^{13}C}$ (1.1%). We can find all of the isotopic peaks and their relative abundances by then expanding the binomial $(0.989\times m[^{12}C] + 0.011\times m[^{13}C])^5$, where $m[^{12}C]$ and $m[^{13}C]$ denote the exact masses of the carbon-12 and carbon-13 isotopes, respectively. Expanding the binomial yields:

$\begin{equation} \begin{aligned} (0.989\times m[^{12}C] + 0.011\times m[^{13}C])^5 ={} & \ \ \ \ \ \binom {5} {0}(0.989\times m[^{12}C])^5 \\ & + \binom {5} {1}(0.989\times m[^{12}C])^4 \times (0.011\times m[^{13}C]) \\ & + \binom {5} {2}(0.989\times m[^{12}C])^3 \times (0.011\times m[^{13}C])^2 \\ & + \binom {5} {3}(0.989\times m[^{12}C])^2 \times (0.011\times m[^{13}C])^3 \\ & + \binom {5} {4}(0.989\times m[^{12}C]) \times (0.011\times m[^{13}C])^4 \\ & + \binom {5} {5} (0.011\times m[^{13}C])^5 \\ \end{aligned} \end{equation}$

Calculating the coefficients in each term:

$\begin{equation} \begin{aligned} (0.989\times m[^{12}C] + 0.011\times (m[^{13}C])^5 ={} & \ \ \ \ \ 0.946 \times (m[^{12}C])^5 \\ & + 0.0526\times (m[^{12}C])^4 \times m[^{13}C] \\ & + 0.00117 \times (m[^{12}C])^3 \times (m[^{13}C])^2 \\ & + 0.0000130 \times (m[^{12}C])^2 \times (m[^{13}C])^3 \\ & + 7.24×10^{-8}\times (m[^{12}C]) \times ( m[^{13}C])^4 \\ & + 1.61×10^{-10} \times m[^{13}C])^5 \\ \end{aligned} \end{equation}$

From the expansion, we see that 94.6% of all $\ce{C_5}$ molecules contain only carbon-12 (the lowest possible mass for the molecule), and almost all of the rest (5.3% out of the remaining 5.4%) is accounted for by molecules that contain a single carbon-13 atom. Only about 0.1% of $\ce{C_5}$ molecules contain two or more carbon-13 atoms.

---

But what happens in very large molecules? Intuitively, if there are many atoms, you would expect a higher chance of there being at least one less common isotope in the mix. Let's see the first few terms for the molecule $\ce{C_100}$:

$\begin{equation} \begin{aligned} (0.989\times m[^{12}C] + 0.011\times m[^{13}C])^{100} ={} & \ \ \ \ \ \binom {100} {0}(0.989\times m[^{12}C])^{100} \\ & + \binom {100} {1}(0.989\times m[^{12}C])^{99} \times (0.011\times m[^{13}C]) \\ & + \binom {100} {2}(0.989\times m[^{12}C])^{98} \times (0.011\times m[^{13}C])^2 \\ & + \ ...\\ \end{aligned} \end{equation}$

Calculating the coefficients:

$\begin{equation} \begin{aligned} (0.989\times m[^{12}C] + 0.011\times m[^{13}C])^{100} ={} & \ \ \ \ \ 0.331 \times (m[^{12}C])^{100} \\ & + 0.368 \times (m[^{12}C])^{99} \times (m[^{13}C]) \\ & + 0.203 \times (m[^{12}C])^{98} \times (m[^{13}C])^2 \\ & +\ ...\\ \end{aligned} \end{equation}$

Well that's interesting. Now only 33.1% of the molecules contain only carbon-12 atoms, and in fact more molecules contain exactly one carbon-13 atom, at 36.8% of the total. Even molecules with two carbon-13 atoms are quite abundant, at 20.3%.

Indeed, peaks containing rarer isotopes eventually dominate. For the huge molecule $\ce{C_10000}$, the strongest mass spectrum signal would come from molecules contaning 110 carbon-13 atoms, corresponding to 3.8% of the total, while a measly $9.2\times 10^{-47}\%$ of molecules contain only carbon-12. This happens because when $n$ is large, the term $\binom {n} {k}$ grows very quickly as $k$ rises from zero, overwhelming the increase in the exponent of the rarer isotope. You can see this behaviour quite nicely in this sequence of mass spectra of molecules with increasing size.

---

To calculate the specific $M/M+2$ ratio for a molecule containing only $n$ carbon atoms, all you need is to get the ratio for first and third terms in the binomial:

$\begin{equation} \begin{aligned} (0.989\times m[^{12}C] + 0.011\times m[^{13}C])^n ={} & \ \ \ \ \ \color{#0000ff}{ \binom {n} {0}(0.989\times m[^{12}C])^n} \\ & + \binom {n} {1}(0.989\times m[^{12}C])^{n-1} \times (0.011\times m[^{13}C]) \\ & + \color{#0000ff}{\binom {n} {2}(0.989\times m[^{12}C])^{n-2} \times (0.011\times m[^{13}C])^2} \\ & +\ ...\\ \end{aligned} \end{equation}$

The ratio is then:

$$\frac{\binom {n} {0}0.989^n}{\binom {n} {2}0.989^{n-2} \times 0.011^2}=\frac{2\times 0.989^2}{n(n-1)\times 0.011^2}$$

Technically this only holds if there are no other elements which contain multiple isotopes, though it will hold approximately if the other elements only have very rare alternate isotopes, such as hydrogen (99.98% hydrogen-1, 0.02% hydrogen-2).

---

As a last curiosity, all of the above extends to analysing more complicated molecules. For example, glucose ($\ce{C6H12O6}$) will have a mass spectrum exactly described by the expression:

$$(0.989\times m[^{12}C] + 0.011\times m[^{13}C])^6 \times (0.9998\times m[^{1}H] + 0.002\times m[^{2}H])^{12} \times (0.9976\times m[^{16}O] + 0.004\times m[^{17}O] + 0.020\times m[^{18}O])^6$$

Happy expanding!

grammar - し at the end of the sentence

I learned that shi is used to connect two or more sentences. But sometimes I hear it at the end of the sentence. What does it mean? Can you explain on these sentences?

1. 濃くなってきちゃったら、部分レーザー脱毛に通えばいいし。


2. 穴があいた後ろから好きな当て布してもいいし。


3. 「ねえ、マッギー」「わたしマギーだし…」


4. 簡単にYou're right.とかHe's right.で良いと思います。または、論争していて相手の言うことを認めるみたいなときはOK, you win!でいいし。


5. あんな人のことは忘れたし！


6. せっかく来たんだからゆっくりしてけし。

How the statements of R. El'azar Ben Azaria fits in the seder

What is the relevance of the Mishna amar rabbi El'azar ben Azaria to the haggada? Why mention this machlokes and why here why now? How do we practically gain from this mention?

clothing - Moshe's Mask and Purim?

Is there a connection between Purim masks and the mask Moshe begins wearing at the end of Parshat Ki Tisa? (see Exodus 34:29-35)

It seems too much of a coincidence that it's read near Purim for there to be none.

halacha - Making a Berachah on rabbinically prohibited foods

Should one refrain from making a Berachah on rabbinically prohibited foods, or does that rule only apply to biblically prohibited foods?

Answer

אכל דבר איסור, אף על פי שאינו אסור אלא מדרבנן, אין מזמנין עליו ואין מברכין עליו לא בתחלה ולא בסוף. (שולחן ערוך או"ח סימן קצו:א)‏
If one ate something prohibited, even if it was only prohibited rabbinically, one does not combine him to a zimmun, nor would he say a beginning or after blessing [on that food]. (Shulchan Aruch OC 196:1)

halacha - Does an ivory spoon make hot tea non-kosher?

Both the handles of cutlery and entire spoons have been made of elephant ivory, which is from a tusk (a tooth) of an elephant. Elephant is of course not a kosher animal. Presumably, then, these spoons and cutlery would render any food eaten with them (hot) non-kosher as well (depending on circumstances). Does anyone have any sources (or additional arguments) that say as much, or that say the opposite? (Of course, for practical halacha, consult your rabbi rather than relying on what you read here.)

Answer

Bones of "neveila" (improperly slaughtered animal) that have no marrow or moisture do not impart forbidden taste (based on Shulchan Aruch 99:1) because they are not fit for eating (Taz 99:1). The Taz's reasoning should apply to the bones, tusks and other inedible parts of a temeiah (forbidden animal) as well.

organic chemistry - Why are DCM and chloroform so resistant towards nucleophilic substitution?

In the book Organic Chemistry by J. Clayden, N. Greeves, S. Warren, and P. Wothers I found the following reasoning:

You may have wondered why it is that, while methyl chloride (chloromethane) is a reactive electrophile that takes part readily in substitution reactions, dichloromethane (DCM) is so unreactive that it can be used as a solvent in which substitution reactions of,other alkyl halides take place. You may think that this is a steric effect: Indeed, $\ce{Cl}$ is bigger than $\ce{H}$. But $\ce{CH2Cl2}$ is much less reactive as an electrophile than ethyl chloride or propyl chloride: there must be more to its unreactivity. And there is: Dichloromethane benefits from a sort of 'permanent anomeric effect.' One lone pair of each chlorine is always anti-periplanar to the other $\ce{C–Cl}$ bond so that there is always stabilization from this effect.

So, in MO-terms the situation would look something like this.

The reasoning looks plausible to me. The interaction between the free electron pair on $\ce{Cl}$ and the $\sigma^{*}$ orbital of the neighboring $\ce{C-Cl}$ bond, which would be the LUMO of DCM, lowers the energy of the free-electron-pair-orbital, thus stabilizing the compound and it raises the energy of the LUMO, thus making DCM less reactive towards nucleophiles.

But how important is this anomeric effect actually for explaining the unreactiveness of DCM toward nucleophiles, especially compared to the steric effect? And how important is the steric effect actually: Is the steric hindrance exerted by the second $\ce{Cl}$ in $\ce{CH2Cl2}$ really that much larger than the steric hindrance exerted by the methyl group in $\ce{CH3CH2Cl}$ (which is a good electrophile for $\mathrm{S_N2}$ reactions)?

I'm a little skeptical because if this anomeric effect was very important I would have expected that the $\ce{C-Cl}$ bond length in DCM would be slightly higher than in methyl chloride because there should be some transfer of electron density from the free electron pair into the antibonding $\sigma^{*}$ orbital, which should weaken the $\ce{C-Cl}$ bond. But the actual bond lengths don't show this. They show rather the contrary:

$$\begin{array}{c|c} \hline \text{Species} & \text{Average }\ce{C-Cl}\text{ bond length / Å} \\ \hline \ce{CH3Cl} & 1.783 \\ \ce{CH2Cl2} & 1.772 \\ \ce{CHCl3} & 1.767 \\ \ce{CCl4} & 1.766 \\ \hline \end{array} $$ (source: Wikipedia)

Now, I know that the stronger polarization of the $\ce{C}$ atomic orbitals in di-, tri-, and tetrachlorinated methane as compared to methyl chloride (due to the electronegativity of $\ce{Cl}$) should lead to an overall strengthening of the $\ce{C-Cl}$ bonds in those compounds. But I would have expected a trend that would show only a slight decrease (or maybe even an increase) of bond length when going from methyl chloride to DCM and then a more pronounced decrease when going from DCM to chloroform followed by a similar decrease when going from chloroform to tetrachloromethane. But instead the polarization effect seems to only slightly increase on adding more chlorine atoms.

Answer

In very short words you can say, that the anomeric effect is responsible for the lack of reactiveness. The electronic effect may very well be compensating for the the steric effect, that could come from the methyl moiety. In any way, most of the steric effects can often been seen as electronic effects in disguise.

I will analyse the bonding picture based on calculations at the density fitted density functional level of theory, with a fairly large basis set: DF-BP86/def2-TZVPP. As model compounds I have chosen chloromethane, dichloromethane, chloroform and chloroethane.

First of all let me state, that the bond lengths are a little larger at this level, however, the general trend for shortening can also be observed. In this sense, chloroethane behaves like chloromethane. An attempt to explain this will be given at the end of this article.

\begin{array}{lr}\hline \text{Compound} & \mathbf{d}(\ce{C-Cl})\\\hline \ce{ClCH3} & 1.797\\ \ce{Cl2CH2} & 1.786\\ \ce{Cl3CH} & 1.783\\\hline \ce{ClCH2CH3} & 1.797\\\hline \end{array}

In the canonical bonding picture, it is fairly obvious, that the electronic effects dominate and are responsible for the lack of reactivity. In other words, the lowest unoccupied molecular orbital is very well delocalised in the dichloromethane and chloroform case. This is effectively leaving no angle to attack the antibonding orbitals.


In the mono substituted cases there is a large coefficient at the carbon, where a nucleophile can readily attack.

One can also analyse the bonding situation in terms of localised orbitals. Here I make use of a Natural Bond Orbital (NBO) analysis, that transforms the canonical orbitals into hybrid orbitals, which all have an occupation of about two electrons. Due to the nature of the approach, it is no longer possible to speak of HOMO or LUMO, when analysing the orbitals. Due to the nature of the calculations, i.e. there are polarisation functions, the values do not necessarily add up to 100%. The deviation is so small, that it can be omitted.
The following table shows the composition (in $\%$)of the carbon chloro bond and anti bond. \begin{array}{lrr}\hline \text{Compound} &\sigma-\ce{C-Cl} & \sigma^*-\ce{C-Cl}\\\hline \ce{ClCH3} & 45\ce{C}(21s79p) 55\ce{Cl}(14s85p) & 55\ce{C}(21s79p) 45\ce{Cl}(14s85p)\\ \ce{Cl2CH2} & 46\ce{C}(22s77p) 54\ce{Cl}(14s85p) & 54\ce{C}(22s77p) 46\ce{Cl}(14s85p)\\ \ce{Cl3CH} & 48\ce{C}(24s76p) 52\ce{Cl}(14s86p) & 52\ce{C}(24s76p) 48\ce{Cl}(14s86p)\\\hline \ce{ClCH2CH3} & 44\ce{C}(19s81p) 56\ce{Cl}(14s85p) & 56\ce{C}(19s81p) 44\ce{Cl}(14s85p)\\\hline \end{array} As we go from mono- to di- to trisubstituted methane, the carbon contribution increases slightly, along with the percentage of $s$ character. More $s$ character usually means also a stronger bond, which often results in a shorter bond distance. Of course, delocalization will have a similar effect on its own.

The reason, why dichloromethane and chloroform are fairly unreactive versus nucleophiles, has already been pointed out in terms of localised bonding. But we can have a look at these orbitals as well.
In the case of chloromethane, the LUMO has more or less the same scope of the canonical orbital, with the highest contribution from the carbon. If we compare this antibonding orbital to an analogous orbital in dichloromethane or chloroform, we can expect the same form. We soon run into trouble, because of the localised $p$ lone pairs of chlorine. Not necessarily overlapping, but certainly in the way of the "backside" of the bonding orbital. In the case of chloroethane we can observe hyperconjugation. However, this effect is probably less strong, and from the canonical bonding picture we could also assume, that this increases the polarisation of the antibonding orbital in favour of carbon.
In the following pictures, occupied orbitals are coloured red and yellow, while virtual orbitals are coloured purple and orange.

(Note that in chloroform two lone pair orbitals are shown.)

Even though this article does not use the Valence Bond Approach, one can clearly see the qualitative manifestation of Bent's Rule (compare also: Utility of Bent's Rule - What can Bent's rule explain that other qualitative considerations cannot?). A higher $s$ character means a shorter bond. The lack of reactivity towards nucleophiles can be explained electronically with a delocalised LUMO. In terms of localised bonding, the lone pairs of any additional chlorine atom would provide sufficient electron density, to shield the backside attack on the carbon.

molecular structure - How does VSEPR theory explain the formation of different bond angles in PCl₅?

We know, that in the exited state, electrons of Phosphorus could go to higher states and form $\ce{dsp^3}$ hybridization. Therefore it will create five equal energy states of electrons. But my question is that, why are the angles of the bonds in $\ce{PCl5}$ 90 and 120 degree instead of equally distributed? How does VSEPR theory explain the difference in the angles?

Why do the five equal bonding pairs of electrons have a trigonal bipyramidal arrangement, with two different internal bond angles - 120° in the plane and 90° between the plane and the axial atoms?

particle と - What does と mean in 「ここにおいで」と?

As far as I know "koko ni oide" means "come here", but what does "to" add to the sentence "koko ni oide to". The context of the phrase is

今、 あなたの声が聴こえる
｢ここにおいで｣と

Answer

This と is a simple quotative-と, and this sentence is a typical example of a rhetoric device called 転置法 (hyperbaton) or 倒置法 (anastrophe). This is very common in Japanese poetry/slogans/lyrics because the grammatical role of a word is mainly expressed by the particle type rather than the word order.

Semantically this sentence is the same as 今、 ｢ここにおいで｣とあなたの声が聴こえる, where ここにおいで is the content of the voice which was heard by the speaker. The original sentence sounds more dramatic because of the reversed word order.

hashkafah philosophy - What exactly is the judgement of Rosh Hashanah?

In the Rosh Hashanah Musaf, it is implied that all human beings and countries are judged on this day whether they will have peace or war, life or death, and in the Unesaneh Tokef prayer, this is extended to many other forms of prosperity or suffering. It is also implied many places in the Gemara and Mishnah that the righteous are decreed for life and the wicked for death, for example:

אמר רבי כרוספדאי אמר רבי יוחנן: שלשה ספרים נפתחין בראש השנה, אחד של רשעים גמורין, ואחד של צדיקים גמורין, ואחד של בינוניים. צדיקים גמורין – נכתבין ונחתמין לאלתר לחיים, רשעים גמורין – נכתבין ונחתמין לאלתר למיתה, בינוניים - תלוויין ועומדין מראש השנה ועד יום הכפורים. זכו – נכתבין לחיים, לא זכו – נכתבין למיתה.

Three books are opened on Rosh Hashanah: one for the totally wicked, one for the totally righteous, and one for the intermediate. The totally righteous are inscribed and sealed immediately for life. The totally wicked are inscribed and sealed immediately for death. And the intermediate people are left dependent from Rosh Hashanah to Yom Kippur. If they merit they will be inscribed for life. And if they do not merit, they will be inscribed for death". (Rosh Hashanah 16b)

How can this be so?

• Weren't there many obviously wicked people or regimes that survived through more than one Rosh Hashanah (like the Egyptians who enslaved Yisrael, the Nazi regime, etc.)?


• Furthermore, how could it be that prosperity is determined on Rosh Hashanah but the rain, grain produce, and fruits are judged on the three holidays (Rosh Hashanah 16a)?

inorganic chemistry - Why doesn't hydrogen displace metals from their oxides?

As I understand it, a displacement reaction is where a more reactive element kicks out a less reactive element from a compound.

For example, chlorine is more reactive than iodine, so chlorine will displace iodine from potassium iodide, yielding potassium chloride and iodine.

If magnesium reacts with water you get magnesium oxide and hydrogen. But hydrogen is generally a lot more reactive than magnesium, so isn't this backwards? The above paragraph predicts that hydrogen would displace the magnesium from magnesium oxide.

So why does it actually happen the other way round?

Answer

If magnesium reacts with water you get magnesium oxide and hydrogen.

True. So you can't put out a magnesium fire with water. It just burns hotter.

$$ \ce{Mg(s) + H2O ->T[aqueous] MgO(s) + H2(g) ^}$$

But hydrogen is generally a lot more reactive than magnesium, so isn't this backwards?

No, in the above reaction you're reacting water with magnesium. The hydrogen is given off as a gas, though a tiny amount will stay dissolved in the water.

So why does it actually happen the other way round?

It will under suitable conditions. So with heat and under a pressurized flow of hydrogen gas the reverse reaction will occur.

$$ \ce{ MgO(s) + H2(g) ->T[heat, pressure] Mg(s) + H2O(g) ^} $$

meaning - How do Japanese understand "non-standard" words without looking at kanji?

This question seems so "obvious" that there might be a duplicate but could not find any.

Anyway, while I try to think how to better rephrase the title, let me explain briefly what I mean.

I'll give a simple but I think quite explanatory example. Suppose I'm watching a movie or an anime without subtitles. As an example let's take one of my childhood favorites: Hokuto no Ken.

At some point I hear: じんせいのシュウ。

My question is: how would even Japanese people understand that that: じんせい = 仁星?

I mean, I don't think 仁星 is a "real word" (at least not common for sure). So how would anyone be able to grasp the meaning of "star of benevolence" from the sound じんせい without an explanation or without looking at the kanji? The only reason I could guess it was because I already knew the story. But how would a native Japanese do?

There could be of course many more examples like this but I believe this one is simple an concise enough to express the whole point.

Answer

日本語が理解できると思いますので日本語で回答します。

一般に、また、私も、「じんせい」と聞いて「仁星」を思い浮かべることはできません。私は全く「仁星」という言葉を見たことも聞いたこともなく、自信をもって日本語には無いと思っています。 若者が「仁星」と理解した理由を想像しますと、唯一の可能性は、「しゅう」という言葉が「じんせい」とともに聞こえてきたからだと思います。
私には、その物語を知らないので「仁星」を想起できませんが、もし私が、「じんせい」と「しゅう」を同時に聞いたなら、「じんせいのしゅう」という言葉は日本語にはないので、「人生の終＊＊」か何かの言葉ではないかと考えるでしょう。

私には、あなたと似たような経験がありますのでお話しします。
私が大学生のとき、スイスからの留学生がいました。彼は日本語を勉強するために、ラジオの放送を録音して、それを何度も聞いていました。ある日、彼は私に、ある音が理解できないと言いました。それは「きんだい」でした。私は彼が録音したラジオ音声を聞かせてもらって、すぐにそれが「近大」、すなわち「近畿大学」だとわかりました。本当に瞬間にわかりました。そのスイス人は「なぜ？！」って、まるであなたのように本当にびっくりしていました。
私は、近畿大学の意味で「きんだい」という発音を生まれて何度聞いたか考えてみました。多分大変少なく、１０度ぐらいかも知れません。２０歳にもならない大学生である私が、１年に数回も聞く単語ではありません。それでも「きんだい」が近畿大学だとすぐにわかりました。 何故でしょうか。
今、手元にあるパソコンで「きんだい」を漢字に変換しますと、「近代」、「近大」、「金大」、「金台」と出てきました。最初の３つは分かりますが、最後の「金台」は漢字を見てもすぐには意味分かりません。多分、金色か金でできた台だろうと想像するだけです。 重要なのは最初の３つです。スイス人の録音は大学野球の実況かスポーツの解説でした。もう、「近大＝近畿大学」か「金大＝金沢大学」しかありません。近畿大学が金沢大学より多分スポーツでは有名だという基礎知識が私にはありました。すると、「きんだい」という音は「近大」しかありません。
では「近大」が「近畿大学」であることはどうしてわかったのでしょうか。多分、人間が「近代」「近大」「金大」という「言葉を覚えるということ」は、「その意味とともに覚える」のだと思います。ですから「きんだい」という音を「近大」と絞り込んだときには「近畿大学」が同時に頭の中にあるのだと思います。スイス人がびっくりしたのも納得できます。自分があんなに苦労したある音を聞いて、その大学生はその意味を言い当てるのに、１秒も必要としないのですから。
これは、私だけでなく、日本人なら誰でもできると思います。また、各々の国の人も、自分の国の言葉なら同じことができると思います。

さて、話は戻って「じんせいのシュウ」を考えてみます。
「じんせい」をかな漢字変換すると、「人生」「尽誠」「仁政」「靭性」「人世」「腎性」「仁成」「人性」「壬生」「神成」「仁正」が漢字として出てきます。一般に、人が言葉で何かを伝えたいときは、音で他人が理解できる範囲内の言葉を使いますので、上記の漢字の中では、「じんせい」は、「人生」と「仁政」しかありません。「人生のシュウ」「仁政のシュウ」。どちらも意味がはっきりしませんが、映画のストーリー（文脈）から、いずれかに絞り込めると思います。私は「シュウ」が何かわかりませんが、北斗の拳を知っている人には「人生のシュウ」「仁政のシュウ」のいずれであるかは分かると思います。「星」は「せい」の発音から比較的早く連想できますが、もし、この映画に惑星の話のようなものがあるのなら、「せい」が「星」であることは確実でしょう。
さて、「仁政」が、なぜ「仁星」になるのでしょう。全くの造語です。「仁」がbenevolenceの意味だとはすぐに分かりますが、ストーリー全体でbenevolenceという感じが伝わっておれば、たとえ造語だとしても「じんせい」という言葉を確定するのに迷っている映画の視聴者には、星の話が何となくちらついている状況下で、それが"star of benevolence"を意味する「仁星」という造語を頭の中で勝手に作り上げてもあまり気にならないと思います。

多分以上のような経緯で同じ映画を見ていた若者が「じんせいのシュウ」が「仁星のシュウ」だとしかも、ほぼ瞬時に理解できたものだと想像します。

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後日談

「仁星のシュウ」をインターネットで探すと、ここにありました。
拳士達の運命を司る宿命の星（複数）を宿星といい、その一つに仁星というのがあります。
その仁星は南斗白鷺拳のシュウがもつ宿星であって、「情け、思いやり」の星とあります。
これで、「仁星」も「シュウ」も北斗の拳に出てくる架空の固有名詞だと理解できます。

北斗の拳の物語を知っている人が、「じんせいのシュウ」と聞いたときにそれが「仁星のシュウ」だと思いつくのは、私が想定した頭の動きよりはるかに簡単そうに思えます。

漢字を見なくとも音声だけで何を意味しているかわかる例えとして、漢字を知らない幼児がテレビ番組の中で登場するキャラクターなどの固有名詞を音からだけでも簡単に想起できるのと同じことであって不思議なことでもなんでもないような気がします。

今は、なんだか、ネタが分からない方が良かったような気分です...。

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後日談その２

「きんだい」と「仁星のシュウ」を並べて見ていますと、ふと、外国語を覚える難しさ、あるいはその難しさを克服できるヒントのようなものが浮かび上がってきます。
まず、いずれもが固有名詞だということです。さらに「きんだい」の方は、省略形であるということです。

本当に有名な固有名詞は別ですが、一般に固有名詞は辞書には載っていません。載せたらキリがないからだと思います。更にその固有名詞が省略形の場合、ほとんど辞書で見つけることはできず、その意味を確認する方法は普通の単語に比べて極端に少ないと思います。また、実際の生きた教材を使える上級者は別にして、外国語の学習教材、あるいは教科書では、固有名詞はほとんど使われないのが実情です。多分、勝手に載せると差し障りがあるからでしょう。

実生活を通じて言葉を覚えるその国の子どもに対して、学習教材を通じて外国語を覚える学習者にとって、この固有名詞、更にその省略形の存在が障壁の一つとしてあるのではないでしょうか。 「きんだい」という発音が、固有名詞の「近大」であり、さらに「近畿大学」だという意味の連鎖を記憶する方法を手に入れることは、外国語の学習者にとって欠かせないもののように思います。

I think you can read Japanese, so I'll answer in Japanese.

In general, for myself included, when I hear "jinsei," I can't come up with the word 仁星. I've never seen nor heard the word 仁星 ever used, and I can confidently say it doesn't exist in Japanese. If I were to imagine a reason young people understand it as 仁星, I think the only possibility is that they've been hearing じんせい along with the word しゅう until now.

I don't know the story you're referring to, so I can't "remember" the word 仁星. But if I heard both じんせい and しゅう at the same time, I know that じんせいのしゅう isn't a word in Japanese, so I'd probably think it was something along the lines of 人生の終＊＊.

I have a similar experience to yours I'd like to relate.

When I was in college, we had an exchange student from Switzerland. In order to study Japanese, he was recording radio programs and listening to them several times over. One day, he told me he couldn't understand a certain sound. It was きんだい. When he played the recorded radio sound for me, I immediately knew that meant 近大, or 近畿大学. It honestly happened in an instant. The Swiss student went "How?!", very surprised, just like you are.

I thought about how many times in my life I've heard the sound きんだい to mean 近畿大学 in my life. Probably very little, around ten times or so. As a college student, I wasn't even twenty years old yet, so I might go an entire year without hearing the word. But I still immediately knew that きんだい referred to 近畿大学. Why was that?

I just typed きんだい into my computer now and converted it to kanji, and the words 近代, 近大, 金大, and 金台 appeared. I knew the first three, but I didn't know what 金台 meant right away just from looking at the kanji. I can only imagine it means a pedestal made of gold, or of a gold color. The first three are more important. The Swiss student's recording was a sports commentary, maybe of a live college baseball game. At this point, it could only be 近大 for 近畿大学 or 金大 for 金沢大学. I had the basic knowledge that 近畿大学 is probably more well-known for its sports than 金沢大学. That meant きんだい could only mean 近大. Then how did I know that 近大 meant 近畿大学? I think that when people learn the words 近代, 近大, and 金大, they're learning it along with their meanings. So by the time I narrowed the sound きんだい down to 近大, the word 近畿大学 came to mind at the same time. I could understand why the Swiss student was surprised. It hadn't even taken a second for a college student to guess right about a word's meaning when the student had been struggling so much with it.

I think anyone, not just Japanese people, can do this. And I think people from other countries can do the same thing with their own languages as well.

Anyway, back to the point, I thought about じんせいのシュウ.

When I convert じんせい from kana to kanji, I get 「人生」「尽誠」「仁政」「靭性」「人世」「腎性」「仁成」「人性」「壬生」「神成」「仁正」. In general, when someone wants to convey information using words, they'll use words that the other person can understand just by hearing, and of the above terms, only 人生 and 仁政 apply to that. 「人生のシュウ」 and 「仁政のシュウ」. I don't clearly understand the meaning of either, but given the movie's story, I think it can be narrowed down to one of those. I don't know what シュウ is, but those who know Hokuto no Ken will know it's either 「人生のシュウ」 or 「仁政のシュウ」. I can imagine the kanji 星 when I hear せい relatively quickly, but if this movie is a story about planets, then せい being 星 is probably certain.

Then why does 仁政 turn into 仁星? It's a completely made-up word. I know right away that 仁 means "benevolence." But if the story as a whole conveyed a feeling of benevolence, then even if it is a made-up word, people watching the movie confused as to exactly what じんせい means would think have been hearing the topic of stars come up once in a while, so they probably wouldn't think much of it if their brains came up with a made-up word 仁星 meaning "star of benevolence."

I would imagine similar circumstances led the young people watching the same movie to know じんせいのシュウ was 仁星のシュウ, and instantly, at that.

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Postscript

When I looked up 仁星のシュウ on the internet, I found this. The stars of destiny that control the fate of the 拳士達 are called 宿星, and one of those is called the 仁星. 仁星 is the 宿星 that Shuu, the user of Nanto Hakuroken has, and is the star of compassion and consideration. With that, I now know that both 仁星 and シュウ are fictional proper nouns that appear in Hokuto no Ken.

When someone who knows Hokuto no Ken's story hears じんせいのシュウ, it's probably much, much easier than how long it took me to guess that it was 仁星のシュウ.

In terms of knowing what something means just by hearing it and not seeing the kanji, I feel like it's the same as when a child who doesn't know kanji can easily recall character names and other proper nouns in TV shows, and it doesn't seem very strange to me at all.

Now, I kind of think I was better off not knowing what it came from...

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Postscript 2

When I look at きんだい and 仁星のシュウ　together, I'm struck by how hard foreign languages are, and possibly something like a hint that could overcome that difficulty.

First is that both are proper nouns. Second is that きんだい is also an abbreviation.

Proper nouns, very famous ones aside, generally won't be in a dictionary. If they started listing them, it would go on forever. And if that proper noun is an abbreviation, you won't be able to find it in most dictionaries, so I think there are far fewer ways to figure out its meaning compared to regular words. If we set aside advanced learners who can use authentic materials, foreign language learning materials and textbooks don't have many proper nouns in them. Putting them in would probably be inconvenient.

Children grow up learning the words of their country through everyday life, but for people learning foreign languages through study materials, I would think proper nouns--and abbreviated proper nouns--are one of the many obstacles to them. To me, it seems essential that foreign language learners find a way to memorize this "chain," where you hear the word きんだい, know it's the proper noun 近大, and also know it means 近畿大学.

self study - Unable to understand the derivation of the update equation for LMS

I am trying to follow the derivation of the Least Mean Square https://en.wikipedia.org/wiki/Least_mean_squares_filter#Proof but I cannot get the update rule. I am stuck in the following steps and shall appreciate help where I am going wrong. The transmitter generates a sequence of random input data $\{s_n\}$, each element of which belongs to an alphabet $\mathcal{A}$. The data sequence is sent through the channel whose output $y_n$ is observed by the receiver. The input/ output relationship of the channel can be written as: \begin{align} y_n &= \sum_{k=1}^L h_k s_{n-k} + v_n,\nonumber\\ &= \mathbf{h}^H \mathbf{s}_n + v_n. \tag{1} \end{align} where the additive noise $v_n$ is assumed to be stationary, Gaussian, and independent of the channel input $s_n$. We also assume that the channel is stationary, moving-average and has finite length. The output of the equalizer is: \begin{align} z_n &= \sum_{k=1}^L {w}_k y_{n-k},\nonumber\\ &=\mathbf{{w}}^H \mathbf{y}_n = \hat{s}_n \tag{2} \end{align} where the superscript $H$ denotes conjugate transpose, $\mathbf{y}_n =[y_n,y_{n-1},\ldots,y_{n-L+1}]^T$ is the signal to be equalized, $z_n$ is the output of the equalizer and $\mathbf{{w}} = [{w}_1,{w}_2,\ldots,{w}_L]^T$ is the vector of equalizer coefficients.

The cost function for LMS is $J = E[|e_n|^2]$ which is minimized. \begin{align} e_n &= s_n - \hat{s}_n,\\ &=s_n - \mathbf{{w}}^H \mathbf{y}_n \tag{3} \end{align} We need to take the gradient of the cost function and then equate it to zero to yield the equation for the equalizer coefficients. The gradient is \begin{align} \nabla J(\mathbf{w}) &= \nabla E[e_ne^*_n],\\ &=2E[\nabla(e_n)e^*_n] \tag{4} \end{align} Now, $\nabla(e_n) = \nabla(s_n - \mathbf{{w}}^H \mathbf{y}_n) = -\mathbf{y}_n$. Substituting this into Eq(5) and equating to zero we get, \begin{align} \nabla J(\mathbf{w}) &= \nabla E[e_ne^*_n],\\ &=2E[-\mathbf{y}_ne^*_n] \tag{5} \end{align} Using Eq(3), \begin{align} \nabla J(\mathbf{w}) =0,\\ 2E[-\mathbf{y}_ne^*_n] =0,\\ \mathbf{y}_ne^*_n = 0,\\ \mathbf{y}_n(s_n^* - \mathbf{{w}} \mathbf{y}_n^*) =0,\\ \mathbf{{w}} = {(\mathbf{y}_n \mathbf{y}_n^*)}^{-1}(\mathbf{y}_n s_n^*) \tag{6} \end{align}

I cannot understand how to plug in the $\mu$, the step size. This Eq(5) is completely different from the update equation :$\mathbf{{w}}_{n+1}=\mathbf{{w}}_n + \mu e_ns_{n}^*$

Answer

so with an LMS filter, we have a time-variant $N$-tap FIR filter:

$$ y[n] = \sum\limits_{k=0}^{N-1} h_n[k] \, x[n-k] $$

$x[n]$ is the input signal, $y[n]$ is the FIR output, and $h_n[k]$ are the FIR tap coefficients at the time of sample $n$.

with an LMS filter, we also have another input called the desired signal: $d[n]$. we want our LMS filter to adapt so that the output $y[n]$ follows the desired signal $d[n]$. from those two signals, we define an LMS error signal:

$$ e[n] \triangleq y[n] - d[n] $$

and this Least Mean Square adaptive filter will adjust its tap coefficients, $h_n[k]$, in such a manner as to eventually minimize the square of this error signal $e[n]$.

so the error signal is:

$$\begin{align} e[n] &= y[n] - d[n] \\ &= \sum\limits_{k=0}^{N-1} h_n[k] \, x[n-k] - d[n] \\ \end{align}$$

and the square of this error signal (which we want to lessen in an LMS filter) is:

$$ \big(e[n]\big)^2 = \left( \sum\limits_{k=0}^{N-1} h_n[k] \, x[n-k] - d[n] \right)^2 $$

now, one thing Bernard Widrow must have been thinking is that, since the inputs $x[n]$ and $d[n]$ are given, the only controls we can adjust to lessen this square error $e^2[n]$ are the $N$ tap coefficients $h_n[k]$ for $0 \le k \le N-1$. with $x[n]$ and $d[n]$ fixed, then $e^2[n]$ is a function, $f(\cdot)$, of the $N$ tap coefficients $h_n[k]$. so at time $n$ this function of $N$ variables looks like

$$\begin{align} e^2[n] & \triangleq f\big(h_n[0], h_n[1], h_n[2], ... h_n[N-1] \big) \\ &= \left( \sum\limits_{k=0}^{N-1} h_n[k] \, x[n-k] - d[n] \right)^2 \\ \end{align}$$

at time $n$, the derivative of $f(\cdot)$ with respect to the $m$-th tap coefficient is

$$\begin{align} \frac{\partial}{\partial h_n[m]} \, e^2[n] &= \frac{\partial}{\partial h_n[m]} \, f\big(h_n[0], h_n[1], h_n[2], ... h_n[N-1] \big) \\ &= \frac{\partial}{\partial h_n[m]} \, \left( \sum\limits_{k=0}^{N-1} h_n[k] \, x[n-k] - d[n] \right)^2 \\ &= 2 \, \left( \sum\limits_{k=0}^{N-1} h_n[k] \, x[n-k] - d[n] \right) \cdot x[n-m] \\ &= 2 \, e[n] \cdot x[n-m] \\ \end{align}$$

now from this we can set all of these derivatives to zero and set up $N$ equations and $N$ unknowns and try to solve the mess. i am sure Widrow did this. but it doesn't lead to a very cheap (in terms of computational cost) algorithm.

but for each $m$, where $0 \le m \le N-1$, the quantity

$$ \frac{\partial}{\partial h_n[m]} \, f\big(h_n[0], h_n[1], h_n[2], ... h_n[N-1] \big) = 2 e[n] x[n-m] $$

is one component of the vector which is the gradient of $f(\cdot)$.

for the moment, try to visualize this as a function of two variables: $f\big(h_n[0], h_n[1]\big)$. this is a surface above the point in a plane with coordinates $(h_n[0], h_n[1])$. the gradient

$$\begin{align} \left(\tfrac{\partial f(\cdot)}{\partial h_n[0]}, \ \tfrac{\partial f(\cdot)}{\partial h_n[1]}\right) &= \Big(2 e[n] x[n-0], \ 2 e[n] x[n-1] \Big) \\ &= 2 \, e[n] \Big(x[n], \ x[n-1] \Big) \\ \end{align} $$

is the 2-dim vector that points in the direction of the steepest increasing slope of this surface. so if $\tfrac{\mu}{2}$ is a tiny positive number, moving a tiny bit in the direction of the gradient should increase this function a tiny bit.

$$ f\big(h_n[0], \ h_n[1]\big) < f\big(h_n[0] + 2 e[n] x[n] \tfrac{\mu}{2}, \ h_n[1] + 2 e[n] x[n-1] \tfrac{\mu}{2} \big) $$

but $f(\cdot)$ is the square of error and we want to decrease it, so we move in the opposite direction of the gradient:

$$ f\big(h_n[0], \ h_n[1]\big) > f\big(h_n[0] - 2 e[n] x[n] \tfrac{\mu}{2}, \ h_n[1] - 2 e[n] x[n-1] \tfrac{\mu}{2} \big) $$

that's with two taps ($N=2$). extending the idea of this "surface" to more dimensions than two, we expect

$$ f\big(h_n[0]-e[n]x[n]\mu, ... ,h_n[k]-e[n]x[n-k]\mu, ... \big) < f\big(h_n[0], ... ,h_n[k], ... \big) $$

so, for a small stepsize $\mu$, the adaptation algorithm (or coefficient "update equation") is

$$ h_{n+1}[k] \ \leftarrow \ h_n[k] \ - \ (\mu \, e[n]) \, x[n-k] $$

that is the simple unnormalized LMS adaptive filter. you want $\mu$, also known as the adaptation gain, to be large enough so that the LMS filter adapts at an acceptably quick rate, but small enough so that it doesn't overshoot in the adaptation and become unstable.

because the stepsize is really $(\mu \, e[n]) x[n-k]$ and both $e[n]$ and $x[n-k]$ are expected to be proportional in magnitude to the mean $\big|x[n]\big|$, this means louder signals will cause the adaptation to happen faster than quieter signals. so from this, the Normalized LMS adaptive filter is defined to divide that term by some measure of the mean of $\big|x[n]\big|^2$.

so the Normalized LMS adaptation algorithm (or coefficient update) is

$$ h_{n+1}[k] \ \leftarrow \ h_n[k] \ - \ \frac{\mu \, e[n]}{\overline{x^2[n]} + \epsilon} \, x[n-k] $$

where $\overline{x^2[n]}$ is some measure of mean of $x^2[n]$ for the samples most recent to time $n$. you can simply square $x[n]$ and run that through a simple first-order IIR filter to get a measure of the mean:

$$ \overline{x^2[n]} \ \leftarrow \ p \, \overline{x^2[n-1]} \ + \ (1-p) x^2[n] $$

where $p$ is the pole of the one-pole IIR filter and $0 < (1-p) \ll 1$. the quantity $\epsilon > 0$ is a tiny amount to add to the denominator so that we never divide by zero, even if the input $x[n]$ goes to zero.

history - Was Gematria taken from other cultures?

The wikipedia entry of gematria starts by stating the following:

Gematria [...] is an Assyro-Babylonian-Greek system of code and numerology later adopted into Jewish culture that assigns numerical value to a word or phrase[...]

This statement was somewhat shocking to me, given the fact that I always believed this practice was of completely jewish origin.

Wikipedia supports this claims with the following information:

Although the term is Hebrew, it most likely derives from Greek geōmetriā, "geometry", which was used as a translation of gēmaṭriyā, though some scholars believe it to derive from Greek grammateia.

and

The first attested use of gematria occurs in an inscription of Assyrian ruler Sargon II (727–705 BC) stating that the king built the wall of Khorsabad 16,283 cubits long to correspond with the numerical value of his name. Gematria or isopsephy was borrowed into the Greek probably soon after their adoption of the Semitic writing system.

Maybe at first it doesn't seem like a big issue that we use this, possibly Hellenistic, method in Torah study. Until you find out there are multiple Halachot learned from gematria. Is there an opposing opinion about the origins of gematria? If not, how can we trust this method?

halacha - When do other punishments other than death penalty, exempt one from a fine?

I'm not well-versed in the laws of various punishments and how they interact in the g'mara. I've seen places -- most recently Ketubot 45, today's daf -- where it says that somebody is punished with lashes and also pays a fine. But I learned last week (Ketubot 38) that somebody who is liable to the death penalty does not also pay a fine, based on Shemot 21:22-23.

Is the death penalty special in that regard (because of the specific language in Sh'mot 21), or are there other punishments where there also cannot be an accompanying fine? And is it always true of the death penalty, or are there cases where one owes a fine before being executed?

I realize that we don't hand out most of these punishments today; I'm asking about the context of the g'mara, not current application.

Answer

When is one absolved from paying in event of another punishment? Hope this setup makes it easy. This is a summary of אלו נערות Kesubos chapter 3 for those doing Daf Hayomi.

These two punishments have to be incurred simultaneously e.g launching an arrow 4 amos (cubits) on Shabbos in the reshus harabim (public domain) and tearing clothes on its path (Kesubos 30b-31a).

There are 4 types of punishment enacted by Beis din
1. death
2. lashes
3. Mamon - payment of damages
4. Knas - payment of a fine

Death together with payment of damages,* one is exempt from paying the damage when liable to death penalty according to everyone as the questioner brought from Shemot 21:22-23.
Death together with lashes one is exempt from lashes when liable to death according to everyone. See Kesubos 37a-37b where Rabbi Meir and Rabbanan both agree.
Death with Knas (payment of fine) Rabah in the name of Rabbi Meir holds you are chayav both, but Reish Lakish in the name of Rabbi Meir holds you are absolved from Knas when Chayav death. Rabanan hold like Reish Lakish whom the psak follows. (Kesubos 34b)

Lashes together with payment of damages ** Rabbanan (whom the psak follows) and Ulla in the name of Rabbi Meir both hold one is absolved from payment of damages when liable to lashes (except chovel and eidim zommemin for Rabbi Yochanan, and also seducing ones sister for Ulla,where one pays damages and is absolved from lashes), Because of "Kedei Rishato"- Only one punishment can be enacted (Devarim 25,2). But Rabba in the name of Rabbi Meir says one is liable to both lashes and payment of damages (Kesubos 34a-34b see Tosfos "Sovar").
Lashes together with Knas (fine) Rabbanan say one is absolved from the fine when liable to lashes with the exception of Motzi Shem ra where one is liable to both lashes and a fine of 100 shekel (with Motzi shem ra Rabbi Yehuda says you are only liable for the fine but absolved from lashes) (see Kesubos 46a) however Rabbi Meir says you are liable to both lashes and payment of fine e.g eidim zomemin (Makkos 4a).

Mamon (payment of damages) and Knas (payment of a fine) are both liable simultaneously according to everyone see Kesubos 29b e.g gneiva, or naara shenispatsa.

.

*Note: We pasken if there is no death penalty - because there is no warning or no beis din with semicha like nowadays - one is still absolved from paying (Kesubos 35a tana devei Chizkiya).

**Note:We pasken like Rabbi Yochanan who says that if there is no lashes - in a case where there is no warning or no beis din with semicha like nowadays - one is chayav to pay for damages, but Reish Lakish says even without enforcing lashes one is absolved from payment (kesubos 35a-35b)

.

P.S since we hold that chayvei Kareisos get Malkus when warned in front of eidim (Makos 13a), Rabbi Nechunia Ben Hakana (Kesubos 29a-29b) is effectively saying the same as Rabanan that one is absolved from payment in such a case of kareis (for which one gets Malkus so does not pay damages). However where there is no warning so Kareis happens without lashes, Rabbi Nechunia Ben Hakana holds one is exempt from payment but Rabanan would hold one is chayav payment like Rabbi Yochanan*

word choice - 「電車が空く【すく】」 と「 電車が空く【あく】」

「電車が空く【すく】」と「電車が空く【あく】」はどう違いますか？
説明が難しい場合は、例文を教えていただけませんか。

minhag - Why fill the glass all the way to the top?

Why is it that when people make kiddush/havdallah they insist on filling up the cup all the way to the top, sometimes till it overflows just a little? Is this a documented minhag/custom etc...?

Answer

The Rema writes (Orach Chaim 183:2) that the cup used for a berachah (the context is talking about the cup of wine for birkas hamazon, but the same applies to kiddush, etc.) should be מלא על כל גדותיו, filled to the brim.

Though Taz (ibid. :4, end) points out that it is (or was, in his times) quite common to leave a space at the top. He questions the propriety of this, but concludes that perhaps it's okay in order to prevent wine from spilling on the food and ruining it.

organic chemistry - Does acetamide respond positively to iodoform test?

This answer explains why acetic acid does not give a positive iodoform test.

Does acetamide respond positively to the test?

Answer

From wikipedia article of haloform reaction:

Acetyl chloride and acetamide don't give this (iodoform or in general haloform) test.

Why?

Acetamide is not a ketone which is substantially more acidic than the amide.  It's about a million times too weak an acid at the alpha protons. Hydroxide is too weak a base. (Quora)

fft - Choosing the right overlap for a window function

On Wikipedia there is quite a list of window functions. However, when you are doing a lot of STFTs, you often have to have some overlap to get better results.

Generally an overlap factor of 2 (i.e., 50% overlap) works fine (I'm doing analysis & resynthesis), but for some windows it isn't good (I currently have this experience with the Blackman window).

I tried my best but I couldn't find a resource that would list the "good" overlap factors for common and less common windows.

Answer

Please refer to this paper on the optimum overlap percentage for the Blackman-Harris window, which is derived to be 66.1%. It has a lot of useful information on spectral analysis and windows.

mesorah tradition - Changes in Halacha

Is Halacha (specifically Halacha, not minhag) capable of changing, or is it an immutable constant? In other words, is Halacha an inflexible set of rules that never bend to societal needs, scientific findings, new readings/interpretations of written or oral law etc? Are there specific examples of Halacha directly changing?

Font used to create handwriting worksheets by primary teachers to teach Hiragana

I'm confused by all the different of fonts that can be used to learn to write Hiragana (e.g. the end loop of き may be closed or not). I understand that it is usually related to the difference between print and handwriting style, but after all my research I still don't know which handwriting font is really used by kids in schools when they learn Hiragana. I mean when teachers are creating handwriting worksheets - which font are they using ? Thanks for your advices.

Mechanism of initiation in contact explosives?

It's well known that undiluted nitroglycerin will explode when subjected to a physical shock. What is happening as a result of the physical shock that sets off the explosive reaction? In particular, why doesn't it explode spontaneously at room temperature without a physical shock?

Answer

My guess would be cavitation, presumably 'non-inertial cavitation':

Non-inertial cavitation is the process in which a bubble in a fluid is forced to oscillate in size or shape due to some form of energy input, such as an acoustic field. Such cavitation is often employed in ultrasonic cleaning baths and can also be observed in pumps, propellers, etc.

The energetics of cavitation are astonishingly out-sized with respect to the tiny size of bubbles, probably leading to more than enough energy to cross the initial activation barrier of the combustion reaction. As one example, the abstract here summarizes other reports' results indicating that the temperature inside collapsing cavitation bubbles may reach between 10,000 K and 100,000,000 K.

I don't have access to the full text to determine whether or not they met with success in attempting to ascertain cavitation's role in nitroglycerin initiation, but these authors apparently investigated the question almost fifty years ago. The abstract (references excised):

It is known that in some circumstances liquid explosives can be initiated with unexpected ease, and in other circumstances only with great diffculty.

Thus, Winning has shown that nitroglycerine (NG) free of gaseous inclusions and poured into a vessel so as to leave no wall surfaces free of liquid is not exploded even by the action of a fairly strong shock wave from a detonator immersed in the NG.

On the other hand, in handling liquid explosives there have been quite a few cases in which relatively weak vibrations or impacts have led to unexpected explosions, which have sometimes had serious consequences. For example, a British report describes an unfortunate accident that resulted from dropping a polyethylene bottle containing NG. Upon hitting the ground the NG exploded.

The initiation of explosion by “hot spots” resulting from the adiabatic compression of gaseous inclusions even before the arrival of the shock wave has been reliably demonstrated in numerous experiments. However, some cases of initiation of liquid explosives simply cannot be attributed to the heating of such gaseous inclusions, since in these cases the adiabatic compression temperatures of the gas are so small that it is not possible to talk of a “hot spot. “Such puzzling cases include, for example, the above-mentioned explosion of NG in a polyethylene bottle. In other experiments the role of gaseous inclusions has been completely eliminated by first subjecting the liquid explosive to a constant high pressure. This so reduced the degree of compression of the gaseous inclusions by a weak shock that strong heating of the gas in the bubbles, if any were present in the liquid explosives, was completely excluded. Nonetheless, there was no reduction in the sensitivity of the explosive to weak shocks.

In attempting to explain such puzzling cases it is usually pointed out that explosion can be initiated by cavitation, which may develop in a liquid even as a result of a weak impact or vibration. So far, however, no one has offered any direct experimental evidence of the possibility of cavitational initiation of explosion in liquid explosives. The object of our research was to fill that gap.

Clearly they published their results, so I figure the odds are good that they reached some sort of affirmative conclusion.

meaning - For what purposes is たろう used?

I am reading Fate Zero 1 by [虚淵]{うろぶち}玄{げん}, this book likes using archaisms and very precise vocabulary which is challenging.

Somewhere in the prologue there is this line:

その日{ひ}から、彼{かれ}は天秤{てんびん}の計{はか}り手{て}たろうと志{こころざし}を固{かた}めた。
He firmly conviced himself that from now on he will be a servant of the world's balance.

I looked up in a dictionary and I got this:

たろ・う
（ 連語 ）
〔過去の助動詞「た」の未然形に推量の助動詞「う」の付いたもの〕 活用語の連用形に接続する。ガ・ナ・バ・マ行五（四）段活用の動詞に付く場合には「だろう」となる。
① 過去の事柄やすでに完了し実現した事柄についての推量・想像などの意を表す。 「その時は，さぞ困っ－・うね」 「あの人の口から出たんじゃなかったら，僕も頭から信用しなかっ－・う」
②（多く上昇調のイントネーションを伴って）過去の事柄やすでに完了し実現した事柄について，相手に念を押したり同意を求めたりする気持ちを表す。 「お父さんもまだ小さかっ－・う。ちょっと怖かったよ」 「この間話し－・う，そんなことは考えない方がいいよ」

Which roughly translates to

たろう (Compound word)
(Auxiliary う showing supposition/guess that binds to the unrealis form of the auxiliary た) When sticking to a strong verb ending in ga, na, ba or ma, たろう becomes だろう.
① Shows guess about a situation of the past or a situation that is finished. (There is no doubt that that time it must have been troublesome) (If it was not him who said that I would have not believe it.)
② (Often with a raise of intonation) Try to convince about a past fact. (My father's still a child. That's freakin'.) (not sure about the translation: It's better not to think that this discussion has been a quiet one.)

So it's about auxiliary た and う. A quick look up leads to the fact that う is a deformation of auxialiary む according to goo.

む
（ 助動 ） （ ○ ・○ ・む（ん） ・む（ん） ・め ・○ ）
〔中古の半ば以降，発音が mu から m となり，さらに n に変わったので，後世「ん」とも書かれる〕
推量の助動詞。活用語の未然形に付く。
① 目前にないこと，まだ実現していないことについて予想し推量する意を表す。…であるだろう。…だろう。 「現（うつつ）にも夢にも我は思はずき古りたる君にここに逢はむとは／万葉集 2601」 「我が背子が国へましなばほととぎす鳴かむ五月（さつき）はさぶしけむかも／万葉集 3996」
② 話し手自身の意志や決意を表す。…するつもりだ。…するようにしよう。 「見れど飽かぬ吉野の川の常滑（とこなめ）の絶ゆる事なくまたかへり見む／万葉集 37」 「弓矢を取り立てむとすれども，手に力もなくなりて，萎えかかりたり／竹取」
③ 相手や第三者の行為を勧誘し，期待する意を表す。…してくれ。…してもらいたい。…すればよい。 「逢ひ難き君に逢へる夜（よ）ほととぎす他（あた）し時ゆは今こそ鳴かめ／万葉集 1947」 「若宮など生ひ出で給はば，さるべきついでもありなむ。命長くとこそ思ひ念ぜめ／源氏 桐壺」 「子といふものなくてありなん／徒然 6」
④ （連体形を用いて）実現していないことを仮定していうのに用いる。…であるようなものなら。…としたら。 「二人して打たむには，侍りなむや／枕草子 9」 「年五十になるまで，上手に至らざらん芸をば捨つべきなり／徒然 151」
⑤（連体形を用いて）実現が可能だったり予想されたりするとき，推量する形で婉曲に述べるのに用いる。 「恋しからむをりをり，取りいでて見給へ／竹取」 「これが花の咲かむ折は来むよ／更級」 〔 (1) 上代では，形容詞活用にはその古い未然形語尾「け」に付く。「大魚（おうお）よし鮪（しび）突く海人（あま）よ其（し）が離（あ）ればうら恋（こお）しけむ鮪突く鮪／古事記 下」 (2) 現代語でも文語調の文章の中に「ん」の形で用いられる。「幸多からんことを祈る」「政治家たらんとする者は」「あらん限りの力」「まさに出発せんとする時」〕

And た

た （ 助動 ） （ たろ ・○ ・た ・た ・たら ・○ ）
〔古語の完了の助動詞「たり」の連体形「たる」からの転。中世以降の語〕
動詞・形容詞・形容動詞および助動詞「れる・られる」「せる・させる」「ない」「たい」「らしい」「そうだ（様態）」「ようだ」「だ」「ます」「です」などの連用形に接続する。ただし，サ行以外の五段活用の動詞には，その音便の形に付く。また，ガ・ナ・バ・マの各行の五段活用の動詞に付く時は「だ」となる。
① 動作・作用が過去の事柄であることを表す。 「大昔，この辺一帯は海だった」 「去年，北海道に移った弟が，先月帰ってきた」
② 動作・作用が完了したことを表す。 「やっと手紙を書き終えました」 「飛行機は無事着陸した」 「日はすっかり沈んだ」
③ 物事が実現することを表す。 「何年ぶりかで当地方にも雪が降った」 「一番になった人には賞品をあげる」 「シャボン玉が屋根までとんだ」
④ 物事や事態の確認を表す。 「見ると，それは若いスマートな青年であった」 「国境の長いトンネルを抜けると雪国であった／雪国 康成」
⑤（連体形を用いて）動作・作用がすんで，その結果が状態として存在していることを表す。てある。ている。 「壁にかけた絵」 「弟の写した写真」 「とがった鉛筆」 「整った身なり」 ⑥（終止形を用いて）

㋐ 強い決意・断言や軽い命令などを表す。 「承知しました」 「わかった，わかった」 「邪魔になるから，そこをどいた」
㋑ 疑問・質問などをもちかけることを表す。 「今度の会合は何日でした」 「上りの列車は何時だった」
⑦ 仮定形「たら」は，接続助詞「ば」を伴わないで，それだけでも用いられる。
㋐ 仮定条件を表す。仮にそうであるならば。もしそうなったらば。 「雨が降ったら，中止にする」 「電話があったらメモしておいてくれ」 「その本を読んだら早く返してくれ」
㋑ 未来の確定条件を表す。 「春になったら暖かくなる」
㋒ 遠回しに命令する意を表す。主として女性が用いる。 「早くお帰りになったら」 「後片付けだけはしといたら」

I emphasized what I think is relevant. My question has a bit evolved since I started and I would like to know

• whether the first sense of たろう is equivalent to たでしょう?



• whether the second sense of たろう can be found in recent works (that do not try to sound ancient), if not what is an equivalent?

Answer

As @mirka said in the comment, this is from an archaic affirmative auxiliary verb たり (≒である), not the auxiliary verb た denoting past tense. And this う denotes someone's will (as in 学校に行こう). So basically it's a nuanced, literary version of 天秤の計り手であろう ("to be going to be a 計り手").

In modern usage, I think the nuance of this kind of (職業/身分) + たる is something like "behave properly as X", "a good X", "qualify as X", etc., as described in this entry:

《文語の断定の助動詞「たり」の連体形》
１ 資格を表す。…であるところの。「学生たるもの、勉強すべきである」

学生たるもの勉強すべき roughly means "A (good) student should study".

Xたろう is not very common, but one can say, for example, （困難な状況で）紳士たろうとする, which means "to try to behave properly as a gentleman".

て form - て形 to express 原因・理由, as opposed to から

Im having so much troubles trying to find a good explanation on how to use the て形 to express a cause, I saw that its more natural to use when the second part of the sentence is either a potential form or volitional, so I came up with the sentence :

喋らなくて練習できない。

Which sounded a bit weird to me and I had confirmation from a friend that it was better to use 【から】 here...

I have no trouble understanding this form when its used, but sometimes when I use it myself it doesnt feel right to my ear and often feel like I should use 【から】 instead of the て形

Here are two sentences :

車がなくて行けない。

車がないから行けない。

I asked him how he sees those two sentences and he said that the first sounded more like a condition than an explication of "why I can't go".

I really like to fully understand things but here I must admit I have no idea how this should be understood, like how japanese people understand those sentences and the nuance, and how you would translate them...

Thanks

grammar - How to list numbers of things

Where do you place the counter in Japanese when you want to specify how many of something there are, especially if you're listing multiple things?

I believe the counter usually goes after the particle associated with the noun:

I bought 3 pencils => 私は鉛筆{えんぴつ}を三本{さんぼん}買{か}いました。

But what do you do when there are multiple things you want to count? Are these sentences correct?

I bought two books and three pencils. => 私は本{ほん}を二冊{にさつ}と鉛筆{えんぴつ}を三本{さんぼん}買{か}いました。 ?

Two books and three pencils are on the desk. => 本{ほん}が二冊{にさつ}と鉛筆{えんぴつ}が三本{さんぼん}机{つくえ}の上{うえ}にあります。

Can you use the の particle to mean the same thing? Would it change the meaning or nuance?

私は二冊{にさつ}の本{ほん}と三本{さんぼん}の鉛筆{えんぴつ}を買{か}いました。 ?

Answer

In basically all informal daily conversations, the most natural way to express the numbers and items is:

Name of item + (particle) + number + counter + verb phrase

Natural: 「ビッグマック + を + ふたつ + [食]{た}べた。」 = "I ate two Big Macs."

Natural: 「みそラーメン +を + よっつ + [下]{くだ}さい。」 = "(Give us) four miso-ramens, please!"

Not very natural though grammatical: 「ふたつのビッグマックを食べた。」 & 「よっつのみそラーメンを下さい。」

What this means is that in Japanese, "number + counter" is most often used adverbially, modifying the verb that follows.

The not-so-natural versions will be understood by virtually all native speakers, but you will definitely sound foreign if you say those, using the quantity adjectivally as in English.

When you have to express an action involving a list of mutiple things, simply place them one by one.

「ビール（を）2本（と）、ラーメン（を）よっつ（と）、ギョウザ（を）みっつ下さい。」

= "Can we have 2 bottles of beer, 4 orders of ramen and 3 orders of pot stickers, please?"

In informal conversation at eateries, these particles in parentheses above are often omitted.

Finally, even though using the "number + counter + の + item name" is unnatural in everyday conversations as I stated above, it is used in literature, legal papers, business communication, etc. when the number needs to be emphasized for a reason.

organic chemistry - Is rotation of a mirror image allowed in order to determine chirality? Doesn't rotation alter the mirror image of the molecule?

So, chiral molecules are those which have non-superposable mirror images.

trans-1,2-Dimethylcyclopentane is chiral; 2-butanol is chiral; both don't superpose on their mirror image.

Then my book cited 2-propanol as achiral. How?

The mirror image doesn't seem to superpose with the molecule. How can it be achiral then?

Then the book provided a pic where the author showed how 2-propanol is achiral; he rotated the mirror image & then superposed! Done! 2-propanol is achiral!

But really how can rotation be allowed? After all the mirror image is a stereo-isomer; if it is rotated, wouldn't it be changed?

I'm not getting, why rotation is allowed before superposition. Not every-time, the mirror image is rotated before superposition.

Can anyone please explain how rotation is allowed before superposition?

Answer

As a rule of thumb, in order for a molecule to be chiral, it must have a chiral center. There are some exceptions. A chiral center is a generalized extension of an asymmetric carbon atom, which is a carbon atom bonded to four different entities.

Carbon-2 in 2-propanol is not a chiral center because it is only bonded to 3 unique substituent groups: ($\ce{-H}$), ($\ce{-OH}$), and 2 ($\ce{-CH3}$).

he rotated the mirror image & then superposed! Done!! 2-propanol is achiral!!

You seem to misunderstand. Mirror images of chiral molecules are non-superposable, and mirror images of achiral molecules are superposable (because they are the same molecule).

Rotating the molecule only means orienting a molecule differently in 3-D space, not changing anything about the molecule itself. I'd recommend playing around with a molecule building set to solidify this in your mind.

particles - 揺れる with に or で

I rarely have any problems with the に and で particles, but I have been a bit confused since seeing the word 揺れる in a few different sentences.

As far as I know it is ok to say either:

旗が風に揺れている

or

旗が風で揺れている

Where you could of course replace 旗 with another word like 木 or 電線.

My problem is that I can't tell the difference between these sentences.

To me, at the moment, they both mean "~ is swaying in the wind". But there must be some difference between these sentences that I am not seeing.

Can anybody tell me the difference in meaning when it comes to the に or で particles with the word 揺れる?

Any help would be appreciated.

Answer

風に揺れる sounds a bit poetic and literary to me. If I'm writing an essay, novel or poem, I'd write 風に揺れる. I think we usually say 風で揺れる in daily conversations.

elements - Melting and boiling point trend in Group II

The following picture shows the melting and boiling point trends down group II elements.

I have added question marks where the variability in data was rather disturbing (over two hundred degrees Celsius), or where the value is simply disputed. Nevertheless, this topic is not about the different results from measurements. Instead, it focuses on the "trend" in general.

For group I, we would say

• the number of electron shells increases,


• the increasing number of protons is unable to compensate for this fact,


• therefore, atomic radius increases and


• electronegativity, ionisation energies decrease,


• hence the metallic bond is weaker $\Rightarrow$ melting and boiling points decrease


• atomisation energy decreases.

Here are the atomic radii and ionisation energies of group II elements.

These (apart from radium) do seem to follow the same logic as in group I, however they do not correlate with the melting or boiling points. Some textbooks claim,

A good measure for the strength of a metallic bond is its atomisation energy.

Let us have a look.

Sure, this does agree better with the boiling and melting points. But note that it is actually not an explanation! Why do the atomisation energies behave in such a way?

This is where most resources either

1. ignore the problem altogether by not mentioning it,



2. give an explanation which is claimed by others to be faulty,


3. few acknowledge the issue and say no simple explanation is possible.

The faulty explanation

Beryllium and magnesium have crystal lattices that are hexagonal closest-packed (close-packed?). Therefore, they have their own trend. Then there are calcium and strontium which have face-centered cubic lattices. Finally, the pair barium and radium have body-centered cubic structures.

Others say this is not enough, because the differences should not be as significant, especially for magnesium but also for barium. Barium has a 8-coordinated structure but has a higher boiling point than 12-coordinated strontium!

It might also be stated that melting points themselves are not the best estimator of the strength of bonds. The bonds are not all broken, but rather rearranged and "made weaker" as a result.

The question

As of now, is there an explanation for melting and boiling point trends of group II elements? Is there consensus?

MATLAB: $tt fft$ and $tt ifft$ scaling

In MATLAB, the outputs of the fft and/or ifft functions often require additional processing before being considered for analysis.

I have heard many differing opinions on what is correct:

• 
  

  Scaling

  
  

  Mathworks states that fft and ifft functions are based on the following equations: \begin{align} X[k] &= \frac{1}{1} \cdot \sum_{n=1}^{N} x[n] \cdot e^{\frac{-j \cdot 2 \pi \cdot (k-1) \cdot (n-1)}{N}}, \quad\textrm{where}\quad 1\leq k\leq N\\ x[n] &= \frac{1}{N} \cdot \sum_{k=1}^{N} X[k] \cdot e^{\frac{+j \cdot 2 \pi \cdot (k-1) \cdot (n-1)}{N}},\quad \textrm{where}\quad 1 \leq n\leq N \end{align}

  
  
  


• 
  

  Scaling by signal length

  
  

  My peers typically scale the data by $\small \frac{1}{N}$ immediately after the processing the fft.
  (We do not consider the raw fft data before scaling.)

  
  

  
  %% Perform fft
  X_f = fft(x, n_sample, 1) / n_sample; 
  % fft must be normalized by the number of samples in the data.
  % This convention was set by the software developer (Mathworks).

  
  
  

  Is this correct?

  
  
  
  
  1. If so, why does the MATLAB ifft function expect that we have not scaled by $1/N$ already?
  
  
  2. Is there a MATLAB ifft function or function option which does not automatically scale by $1/N$?
  
  
  
  

  Alternatively, is there a better convention which we should be using in placing the $1/N$? For example, placing the $1/N$ in the fft rather than the ifft, or placing an $1/\sqrt{N}$ in both equations, instead of an $1/N$?

  
  


• 
  

  Scaling by sampling period

  
  

  I have heard that the fft and ifft functions assume that the sampling period $T_{\rm sampling} = 1/f_{\rm sampling} = 1$, and that for the functions to be true, the following would need to apply:

  
  
  

\begin{align} X[k] &= \frac{1}{T_{\rm sampling}} \cdot \sum_{n=1}^{N} x[n] \cdot e^{\frac{-j \cdot 2 \pi \cdot (k-1) \cdot (n-1)}{N}},\quad \textrm{where}\quad 1 \leq k \leq N\\ x[n] &= \frac{T_{\rm sampling}}{N} \cdot \sum_{k=1}^{N} X[k] \cdot e^{\frac{+j \cdot 2 \pi \cdot (k-1) \cdot (n-1)}{N}},\quad\textrm{where}\quad 1 \leq n \leq N \end{align}

See links:

• Link 1 (see comment to Matt Szelistowski by Dr Seis)


• Link 2 (see answer by Rick Rosson vs that of Dr Seis)


• Link 3 (see comment by Matt (Message: 7/16) and comment by Poorya (14/16)


• Link 4 (see pg. 10, slide [1,1])


• Link 5 (see pg. 8+9) [it seems he is using inverse convention for fft and ifft].

Is this true?

I'm particularly piqued because I cannot find any DFT or DTFT equations on Wikipedia which include the sampling period.

Answer

Whether or not to scale the forward FFT by 1/N depends on which result you want for further analysis: energy (preserving Parseval's identity), or amplitude (measuring height or volts, etc.).

If you want to measure or analyze energy, then don't scale by 1/N, and a longer sinusoid of the same amplitude will produce a larger FFT result, proportional to the greater energy of a longer signal.

Slightly more commonly, if you want to measure or analyze amplitudes, then to get a longer sinusoid (thus with more total energy at the exact same amplitude) to produce about the same FFT result as a shorter signal, you will need to scale down the FFT summation by an ratio proportional to the length. The ratio could be reference_length/N, which is sometimes 1/N if the system input gain is 1.0 for whatever dimensions or units, including the time interval dimensions, that you choose to use in your further analysis. You need to scale down proportionally because a DFT is a summation: the more you sum up similar items, the bigger the result.

So. Energy or amplitude. Which do you want?

Now if you scale down the forward FFT, then you should not scale the inverse so that IFFT(FFT(x)) == x. Or do vice versa.

The 1/sqrt(N) for scaling seems to me to be for either when one needs a formal symmetry for some proof, or when building some sort of hardware pipeline where the latency and/or number of arithmetic units/gates for the DFT and for the IDFT needs to be identical. But you get neither a good direct measurement of either energy or amplitude for any typical type of engineering analysis.

acid base - Why does HNO3 not give off H2 when reacting with Cu?

A dilute solution of which acid is most likely to produce a reduction product other than $\ce{H_{2}}$ when it reacts with a metal?

(A) $\ce{HF}$

(B) $\ce{HCl}$

(C) $\ce{HNO3} $

(D) $\ce{H2SO4}$

$\tiny{\text{Question from 2012 local chemistry olympiad}}$

The given answer is (C), $\ce{HNO3} $ but I don't understand why.

First off, what does it mean by reduction product -- is it the product that is formed when the acid is reduced in the redox reaction between the metal and the acid?

Second, why is the answer $\ce{HNO3} $?

Answer

Whenever any metal is added to an acid, the metal is oxidised to one of its positive ions, at the cost of either the acid which is reduced or water which is reduced to hydrogen. The product formed by reduction is what is referred to as the reduction product.

The reduction potential of $\ce{NO3-}$ is $0.8$ which makes the reduction of nitrate instead of hydrogen ion $\ce{H+}$ ($E^\circ=0.0$) more favourable. Thus it is the nitrate in nitric acid which is reduced instead of $\ce{H+}$ or $\ce{H2O}$. But at very low concentration of nitrate ion and low temperature, as explained by Nernst equation, the reduction potential decreases significantly and combined with overpotential effects, it is $\ce{H2}$ which is formed and not the reduction product of nitrate which is $\ce{NO2}$.

In the case of halic acids, the halogens cannot be further reduced. In the case of the sulfate anion, although it has a reduction potential of $0.17$, owing to overpotential effects, sulfate is not reduced, instead, water is reduced yielding $\ce{H2}$. At high concentration, the reduction potential rises sufficiently to overcome the overpotential effects, allowing the reduction of sulfate to form $\ce{SO2}$ at higher concentrations of sulfuric acid. Hence under normal conditions, only nitric acid gives a product other than hydrogen gas.

Reduction potential data

$$\ce{SO4^{2-} + 4H+ + 2e^- -> SO2 + 2H2O}(E^\circ=0.17)$$ $$\ce{NO3- + 2H+ + e- -> NO2 + H2O} (E^\circ=0.8)$$ $$\text{Nernst equation } E=E^\circ- \frac{RT}{nF}\ln Q$$ $Q$= reaction quotient, $E$=redox potential, $E^\circ$=Standard redox potential

EDIT
The oxidation of metal at the cost of reduction of another substance occurs whenever it is thermodynamically favourable. Usually, (except silver, copper) metals have negative reduction potentials, which makes their oxidation at the cost of hydrogen thermodynamically favourable as the $\Delta G<0$. This leads to a spontaneous redox reaction. This is not true if there is a kinetic or physical barrier to reaction. For example, Iron is rendered passive by nitric acid due to the formation of an oxide layer at the surface which prevents any further contact and hence stops the reaction.