history - Sherlock Holmes & the Maharal of Prague

What is the connection between the Maharal and SHerlock Holmes?

organic chemistry - Claisen Condensation to Ketone

Can someone explain (and preferably draw) the intermediates of this reaction?

I understand that it is a Claisen Reaction, and likely a condensation due to the heat, but I am not sure why there isn't an ester in the final product.

kohen priest - Why there was no railing on the 5m high Mizbeach?

While looking at drawings of the Temple's Altar, I was impressed by its dimensions: up to 5m (16ft) high and 15m (100ft) wide. Here's a pretty truthful depiction (courtesy of WIKI) that gives an idea of its size:

Here's another picture from Mesivtah©^{needs permission?} that shows the full altar from a different angle:

As we can see the whole area has no railings. I thought I was a safety freak, but the Mishnah describes that some careless Cohanim actually fell and were hurt (Yomah 2,2):

מעשה שהיו שניים שווים רצים ועולים בכבש, ודחף אחד מהן את חברו, ונפל ונשברה רגלו.

It once happened that two were even as they ran up the ramp, and one of them pushed his fellow who fell and broke his leg.

Interestingly, even after the incident they didn't think of adding the railings, they (the court) simply canceled the lot altogether.

Please look at the two Cohanim I marked with red arrows - they endanger their lives (or at least their service as Cohanim, because a broken limb can lead to disqualification). THere's also a 50cm Hasovev around the Mizbeah about 2.5m high (notice marks ב and י"ז)!

The top of the altar is called "גג" - "roof" and the Torah has a specific and explicit double commandment/prohibition of "making a railing for your roof" and "prevent personal injuries in one's house" (Deut. 22,8):

כִּי תִבְנֶה בַּיִת חָדָשׁ וְעָשִׂיתָ מַעֲקֶה לְגַגֶּךָ וְלֹא־תָשִׂים דָּמִים בְּבֵיתֶךָ כִּי־יִפֹּל הַנֹּפֵל מִמֶּנּוּ׃

When you build a new house, you shall make a parapet for your roof, so that you do not bring bloodguilt on your house if anyone should fall from it.

(My Rabbi offered a simple Tiruzt, as many Mitzvos don't hold in the Mikdash (Mezuza?), we can learn it from the Posuk says "בביתך" you're obligated but not in G-d's house :)

So why didn't the Mizbeach have railings or safety guards?

Answer

I asked this question to many of my friend Rabbis, and all were stumped but one - R' Shimon Greenfield. He came up with an interesting idea (my translation):

In the Temple, we face numerous contradictions, e.g. regarding Sacrifices on Shabbos, putting Mezuzos, etc. It appears that this question presents yet another opposition - between the structure of the Altar and other Mitzvos of the Torah.

As the structure and the measures of the Altar are prophesized, and they did not include additional parts, like railings, we could conclude that it wasn't the part of the divine plan and the structure [of the Altar] overrides other considerations.

algorithms - Chromatography Baseline Placement

I have been looking around on Google, and I have not been able to find an explanation of how to do baseline placement in chromatography. what are some algorithms for placing a baseline on a chromatogram, how do those algorithms work. (Referral to Python implementations are welcome).

Answer

In some physico-chemistry methods, baselines can be modelled by closed-form lineshapes. This is not typically the case for chromatography, and it can be even worse depending on the type of chromatography (gaz, liquid).

So you have to rely on some morphological properties that you believe valid for your type of signal. You can model the background by a given polynomial, assume derivability of the peaks, etc. And then fit, or optimize a cost function.

In a recent work, we have modeled gaussian peaks as sparse with sparse derivative. We add that peaks are typically positive. Then we suppose that the baseline is smooth enough to remain after the use of a low-pass filter. Finally, we consider Gaussian noise, yet it works with Poisson noise as well.

The BEADS (Baseline Estimation And Denoising w/ Sparsity) method is described, and compared with two other methods in "Chromatogram baseline estimation and denoising using sparsity (BEADS)", by Xiaoran Ning, Ivan W. Selesnick, Laurent Duval, in Chemometrics and Intelligent Laboratory Systems, December 2014, http://dx.doi.org/10.1016/j.chemolab.2014.09.014.

We have made the associated Matlab toolbox available in Matlab. Contributed versions in C++, R, and Python are made available through the companion BEADS page.

stability - Why is a C–D bond stronger than a C–H bond?

I have heard that the carbon–deuterium bond is stronger than the carbon–hydrogen bond. What are the possible reasons for it? Is this also the reason that C–H bonds participate more in hyperconjugation than C–D bonds? Please explain.

Answer

Physics is better able to answer "how" questions than "why" questions, but here goes. The quantum mechanical description of the C–D system versus the C–H system gives the former a lower zero-point energy, which is the minimum energy the quantum system can attain. A good conceptual model of this is to consider the C–X system as two masses connected by a spring. When X = D, the system will vibrate more slowly than when X = H simply due to mass.

Since energy is proportional to the frequency of vibration, the energy of the C–D system is less (lower). This table lists the C–D bond dissociation energy as $\pu{341.4 kJ/mol}$ and C–H as $\pu{338 kJ/mol}$. Since this is the energy to break the bond, the C–D bond is stronger.

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As far as your question about hyperconjugation; I'm surprised that C–H would have greater hyperconjugation than C–D, but it is not something that I know anything about. I guess I could speculate and wave my hands around and say that since the C–D bond is stronger, it keeps the electrons "closer" and so more confined to the sigma orbital, but that is just hot air.

Here's something which claims that the reason D does less hyperconjugation is because it vibrates less (see slide 15). So, it seems (if this is correct) that it's the smaller bond deformations that reduce the hyperconjugation for deuterated systems. Here is a pretty clear explanation of the kinetic isotope effect.

sources mekorot - Different Chanukah Blessing for Converts?

In 139:12, Kitzur writes that when a convert lights Chanukah candles, he should say "She-asah nissim l'Yisrael" instead of "She-asah nissim lavoteinu". What is his basis for this ruling? Isn't a halachically valid convert treated the same as a born Jew?

I can find several reasons to disagree with Kitzur, for example that "avoteinu" means "our fathers", i.e. the fathers of the Jewish nation, not "my fathers".

Answer

Rambam raises the possibility of a convert reciting such a variant blessing in a responsum to R. Ovadya the proselyte (ed. Blau #293) that references the Jews, rather than "our forefathers" and writes that he may recite such a variant. However, he emphasizes that he is equally entitled to recite the traditional formula that references our forefathers, (and he seems to prefer this latter option):

אבל שהוצאתנו ממצרים או שעשית נסים לאבותינו אם רצית לשנות ולומר שהוצאת את ישראל ממצרים ושעשית נסים עם ישראל אמור. ואם לא שנית אין בכך הפסד כלום מאחר שנכנסת תחת כנפי השכינה ונלוית אליו אין כאן הפרש בינינו ובינך. וכל הנסים שנעשו כאלו לנו ולך נעשו. הרי הוא אומר בישעיה ואל יאמר בן הנכר הנלוה אל ה' לאמר הבדל הבדילני /יבדילני/ ה' מעל עמו וגו', אין שום הפרש כלל בינינו ובינך לכל דבר. ודאי יש לך לברך אשר בחר בנו ואשר נתן לנו ואשר הנחילנו ואשר הבדילנו. שכבר בחר בך הבורא יתעלה והבדילך מן האומות ונתן לך התורה שהתורה לנו ולגרים שנ' הקהל חוקה אחת לכם ולגר הגר חוקת עולם לדורותיכם ככם כגר יהיה לפני י"י. תורה אחת ומשפט אחד יהיה לכם ולגר הגר אתכם. ודע כי אבותינו שיצאו ממצרים רובם עובדי ע"ז היו במצרים נתערבו בגוים ולמדו מעשיהם עד ששלח הקדוש ברוך הוא משה רבינו ע"ה רבן של כל הנביאים והבדילנו מן העמים והכניסנו תחת כנפי השכינה לנו ולכל הגרים ושם לכולנו חוקה אחת. ואל יהא יחוסך קל בעיניך אם אנו מתיחסים לאברהם יצחק ויעקב אתה מתיחס למי שאמר והיה העולם. וכך מפורש בישעיה זה יאמר לי"י אני וזה יקרא בשם יעקב וגו'. וכל מה שאמרנו לך בענין הברכות שלא תשנה כבר ראיה לזה ממסכת בכוריםו תמן תנינן הגר מביא ואינו קורא שאינו יכול לומר אשר נשבע י"י לאבותינו לתת לנו. וכשהוא מתפלל בינו לבין עצמו אומר אלהינו ואלהי אבות ישראל. וכשהוא מתפלל בבית הכנסת אומר אלהינו ואלהי אבותינוז זהו סתם משנה. והיא לר' מאיר ואינה הלכהח אלא כמו שנתפרש בירושלמי תמן אמרינן תני בשם ר' יהודה גר עצמו מביא וקורא מאי טעמיה כי אב המון גוים נתתיך לשעבר היית אבי לאברהם מיכאן ואילך אב לכל הבריות. ר' יהושע בן לוי אמר הלכה כר' יהודה. אתא עובדא קמיה דר' אבהו והורי כר' יהודה. הנה נתברר לך שיש לך לומר אשר נשבע י"י לאבותינו לתת לנו. ושאברהם אב לך ולנו ולכל הצדיקים ללכת בדרכיו והוא הדין לשאר הברכות והתפלות שלא תשנה כלום. וכתב משה ב"ר מימון זצ"ליב

This responsum is referenced by the Elya Rabba (676:2) who seems to rule accordingly.

However, R. Saadiah Halevi Mirkado (18th-19th cent.) notes in his Neve Tzedek (Hilkhot Megillah v'Hannukah 3:4) that Rambam rules in Hilkhot Ma'aser Sheni (11:17) that converts cannot recite the declaration on ma'aser, since it contains the words "the land that you gave to our forefathers". R. Mirkado, therefore seems to prefer that a convert similarly recite the variant formula for Hannuka candles.

It should be noted that even the Kitsur Shulhan Arukh (139:12) who prefers the variant formula concedes that if the traditional formula is recited, that is sufficient.

גר אומר שעשה נסים לישראל ואם אמר לאבותינו יצא

Eliyahu was answered as soon as he prayed. How does that prove the importance of mincha over other prayers?

The gemoro in Berachot 6b says,

Rabbi Chelbo said in the name of Rav Huna: one should always be careful about the prayer of Mincha, as Eliyahu was answered only at the time of the prayer of Mincha, as the pasuk says: “And it was at the time of the afternoon offering that Eliyahu approached and said, ‘answer me, Hashem, answer me.’”

Translation from here.

The gemoro almost gives the mistaken impression that Eliyahu started praying before mincha and was only answered at mincha.

But the narrative in Kings 1 18: 25 etc. seems to show that it was only at mincha that Eliyahu started his “operation” and uttered his prayer. That is, he was answered as soon as he prayed. See the text below slightly adapted from here.

25 And Elijah said to the prophets of the Baal, "Choose for yourselves the one bull and prepare it first since you are the majority, and call in the name of your deity, and fire place not."

26 They took the bull that he gave them and prepared [it]. And they called in the name of the Baal from the morning until noon, saying, "O Baal, answer us!" But there was no voice and no answer, and they hopped on the altar that they had made.

29 And as noon passed and they feined to prophesy until the time of the sacrifice of the [afternoon] offering, and there was no voice and no answer and no one was listening.

After that Eliyahu set up his altar and prayed.

If he would have set up the altar at another time, he might well have been answered then too!

Do we have to conclude that he was after all answered at mincha and therefore mincha is important despite the fact that he did not try at any other time?

See related What is special about praying Shacharis and Mariv (as opposed to Mincha)?

Answer

Check out the Rif (not the halachic commentary) on the Ein Yaakov. He asks this question and brings several answers.

One of the answers he gives is that Eliyahu may not have deserved to be answered at all, since he was offering a sacrifice outside of the Beit Hamikdash (even though it was permitted to him, it still was slightly connected to a sin - See Rashi Bereshit 35:11 where Eliyahu's offering of the sacrifice is referred to as "actions of the nations"). However, since he prayed at Mincha, the time when Yitzchak was being offered as a sacrifice, the merit of Mincha helped him out and his prayers were answered.

Hence we see the importance of the Mincha Prayer from the fact that Eliyahu was answered at Mincha.

word choice - Choosing the right prefix of the ko-so-a-do series

When talking to native speakers, I find that I'm often corrected for picking the wrong prefix when using words from the ko-so-a-do series (as in これ、そちら、あそこ、どんな、etc). My confusion mainly involves ko-so-a (the do prefix being simpler since it indicates a question).

What I've been taught is to choose based on a concrete or abstract idea of proximity, which is simple enough for the following particular series:

• これ: thing close to in-group



• それ: thing close to out-group


• あれ: thing distant from both in and out-group

which also works nicely for the その、この、あの series. But I find the following difficult:

Quotation of oneself or another person:

彼女は{そう、こう、ああ}言いました。　

Talking about location of oneself or another person:

Xは{ここ、そこ、あそこ}で旅行している間に、...

Emphasis of some property:

{こんな、そんな、あんな}に美味しい魚を食べたことがないって。

Those are maybe not the best/most natural example sentence, and the particular choice here is maybe not so relevant. Rather, I am interested in a more general description of how to make these types of choices in an abstract context where proximity is not well-defined.

Answer

The general rules for ko/so/a words in abstract situations are:

そ-words usually refer to what was said previously

こ-words are often used for matters of emotional importance to speaker
あ-words are used in personal statements to refer to remembered things

These are illustrated in the following examples:

港について船を降りた。そこで母が待っていた。

この曲は始めのメロディが好きだ。ここはなんど聞いても飽きない

子どものころ、近くの公園でよく遊んだ。あの公園はまだ残っているだろうか。

Note: The choice depends on the context (文脈), given in the examples above in the previous sentence. In your examples we cannot say which is the correct choice because you have not given the previous sentence.

Reference: JLPT 新完全マスターN３文法

Additional comment on other rules

I think the above rules are needed for more complicated reading. I have seen the other rules you mention in your comment used for conversations so perhaps these are best looked at in context of spoken Japanese. I find too many rules can be paralysing so my suggestion is to take a few “model examples” such as the following, and then try apply those rules:

A: あれ、持っているの？
B: あれ、あ、もっている

C: 昨日、マークさんに会いました
D: あの人はずいぶんやせましたね

E: 雨がよく降りますね。これは台風の影響ですよ

F: 風邪をひいて、頭が痛いんです
G: それはいけませんね

H: 昨日、車でびわ湖へ行って来たよ
J: その湖に魚がいたかい？

K: いつアメリカへ行ったんですか
M: あれは去年８月でした

Alternative rules [related general rules そ、こ、あ, given above]
(i) Distance can be physical or psychological.
(ii) So-words can be used for things just mentioned by one of the speakers. [そ]
(iii) Ko-words: The speaker feels closest to items described with ko-words [こ] and furthest from a-words [あ]
(iv) A-words can be used to refer to something someone said or did in the past but both speakers must have previous knowledge of it. [あ]

Application to above model conversations:

A&B: (iv)
C&D: (iv) - Both have previous knowledge of Mark
E: (iii) - E uses kore to refer to “this thing that I am talking about”
F&G: (ii)
H&J: (ii) – “J” does not have “previous knowledge” to recall because he was not there.
K&M: (iv)

Note: We have been focused on the application of ko/so/a words. The application of kore/koko/kochira/konna etc may also require practice.

References: Based on explanations and examples in "The Dictionary of Basic Jpse Grammar" and "An Introduction to Advanced Spoken Japanese".

Parameters of Gaussian Kernel in the Context of Image Convolution

Hi Everyone i am new at image processing. I copy code from this website, I have image $600 \times 480$ gray scale.

• What will be the value of standard deviation or $\sigma$?



• What will be the value of radius?


• What will be the size of kernel?

After I get the kernel how can I use the convolution method? Can use the same like in this website? I know its lot to ask.

grammar - Help on a specific usage of こそ

好きこそものの上手なれ。 We tend to be good at those things we like. / People become best at what they love the most.

How does the こそ work here? Is this use of こそ common outside of proverbs?

Answer

Edit in revision 4: If you are interested in this proverb, do not miss the answer by Matt to another question, where he cites an earlier form of the proverb.

Edit in revision 2: I rewrote the answer completely to give a more detailed explanation. If you prefer a shorter answer, see the old revision.

こそ signifies emphasis. In other words, it adds the meaning of exclusiveness (as sawa wrote in a comment to this answer). It is not archaic, and it is common to see こそ outside of proverbs or set phrases (see Paul Richter’s answer for examples). In this answer, I will try to explain 好きこそものの上手なれ, focusing on the role of こそ.

Without こそ, the sentence would be

好きはものの上手なり。 Liking makes mastery.

好き means the action of liking something, just as in modern Japanese. ものの[上手]{じょうず} is an archaic phrase which means a person who is very skilled at something. なり is a copula in classical Japanese. Therefore the literal meaning of the sentence above is roughly “Liking = A very skilled person,” but obviously this does not make sense because an action is not a person. I think that it means “Liking makes a very skilled person,” or in short, “Liking makes mastery,” but I do not know the exact reason this “action = person” construct can be used.

Now we add こそ and the sentence becomes

好きこそものの上手なれ。 It is liking that makes mastery.

好きこそ means “liking, and nothing else.” So the sentence means that liking and not anything else (such as aptitude) makes mastery. A similar meaning can be expressed in English using a cleft sentence “It is … that …” like the sentence above.

I changed the auxiliary verb なり at the end of sentence to なれ when I added こそ. As Chocolate pointed out in a comment, this is an instance of the grammatical phenomenon called 係り結び. In classical Japanese, when a phrase/clause ending with particle こそ modifies a conjugating word at the end of a sentence, that conjugating word takes 已然形 instead of 終止形. Similarly, when a phrase/clause ending with one of particles ぞ, なむ, や, and か modifies a conjugating word at the end of a sentence, that conjugating word takes 連体形. This phenomenon is called 係り結び.

purim torah in jest - Where did Moses's successor come from if he had no parents?

After Moses died, he was succeeded by יְהוֹשֻׁעַ בִּן נוּן (Joshua, son of none). If Joshua had no parents, where did he come from?

---

Answer

Everyone who was at the revelation at Sinai and was considered a convert, and a convert is considered a tinok shenolad, without relatives (see the discussion abou ger who can marry his sister who is a giyoress). After Sinai Hashem explicitly set up the family and tribal relationships with "return to your tents".

Yehoshua was thus able to get back his connection to the tribe of Ephraim, but he lived in the "tent of Moshe". He was not yet married and he no longer lived with his parents. Thus he had no "father".

Another explanation is Yehoshua Bin No-one.

Since he was training to become the leader he could not fall into the same "bin" as the rest of Bnai Yisrael. He had to remain objective.

Another possibility is Yehoshua Bin Noon

Just as Moshe Rabbeinu took care of Bnei Yisrael from early morning, Yehoshua could only handle matters for himself at midday. This may have been because Moshe Rabbeinu was getting ready for Mincha and it was too late for Shacharis. Thus, Yehoshua had a break at that time.

organic chemistry - Are methyl n-propyl ether and methyl iso-propyl ether metamers?

Are methyl n-propyl ether and methyl iso-propyl ether metamers?
It says so here.

So the difference must be in number of carbon atoms or the distribution?

Definition(http://dictionary.reference.com/browse/metamerism)

grammar - Nuance of ないようにしないと and a potential relative clause

I have a following line from a manga

せめてお願いをする / あの子たち以外には

できるだけ迷惑をかけないようにしないと

which of course means punctuation is lacking, hence / to indicate where a line break was, which might indicate a comma.

Context: This is being monologued by a person who intends to take something from one of "those girls" which is of course likely to involve all of them and get a fight started.

So here (and before a bit) she is I think deciding that she wants to do things as peacefully and quietly as possible so not to bother other people.

What I'm wondering is:

1) ないようにしないと. I know しないと is one of those "must" forms only with the negative state missing which is common. So that alone means "must do". I wonder what additional meaning is added with the inclusion of ように. With just ように I'd say that segment means "So that (people) other than those girls aren't troubled". Adding しないと seems to turn this just into "I must not bother other people as much as possible". I'm just then wondering why use ように at all? Couldn't をかけないと have conveyed the same? Or in other words, is there a more nuanced meaning added with that ように?

2) せめてお願いをする, does this describe the girls? Or is this maybe a separate sentence segment. Is this: "I wish at least, not to bother anyone except those girls as much as possible" Or is is something like: "I'll at least ask the girls (for what I need), and endeavor not to bother anyone else".

Sorry for the two-fer :D

Answer

ようにする means "to try to ～", "to behave in a ～ manner", "to lay out a situation so that ～" etc. For example, 会う is simply "to meet", but 会うようにする indirectly refers to some action/effort/thoughts that eventually leads to the result of 会う (e.g., making an appointment).

• 迷惑をかける
  to bother/trouble


• 迷惑をかけるようにする
  
  to try to bother; to behave in a troublesome manner


• 迷惑をかけないようにする
  to try not to bother; to behave in a trouble-less manner
  (or "to avoid doing anything that may end up bothering them")


• 迷惑をかけないようにしないと。
  I must try not to bother them.

お願いをする modifies あの子たち, but せめて ("at least, ...") adverbially modifies 迷惑をかけないようにしないと. If I could put a comma in this sentence, I would place it right after せめて.

electrochemistry - How to determine the value for n in the Gibbs free energy and redox potential equation?

I am having a problem with this equation for redox potentials

$$\Delta G = -nFE_\mathrm{cell}$$

In this equation I never am totally sure about what the value of $n$ should be, for example for the reaction shown below, would the $n$ be 2 electrons or 1 electron? Personally, I think it should be 2 electrons because that is the number of mol of electron under the simplest whole number ratio.

\begin{align} \ce{cyt $c$(Fe^3+) + e- &-> cyt $c$(Fe^2+)}\\ \ce{NADH &-> NAD+ + H+ + 2e- }\\ \end{align}

I think this is a different matter than in the already answered question "Does the relationship equation between standard cell potential and equilibrium constant violate potential's intensive properties?" Because in that Nernst equation the change in $n$ would be balanced by the change in the equilibrium constant. I assume that you are insinuating that there is one of the variables here which will correct for the increase in $n$. I don't see which would, as $E_\mathrm{cell}$ should be independent of quantity and $F$ is a constant.

Answer

Personally, I think it should be 2 electrons because that is the number of mol of electron under the simplest whole number ratio.

Yes, it is $\ce{2e-}$.

Assuming that the two reactions provided above are half-reactions, the $n$ has to be $2$.

$E_\mathrm{cell}$ is independent of quantity because it is an intensive property. It does not depend on the amount of matter present. The $n$ accounts for the number of moles of $\ce{e-}$, this is used to calculate $\Delta G$, which in turn depends on the amount of matter present.

Overall, one must understand that $E_\mathrm{cell}$ is just the difference in cell potentials of two half-reactions. Increasing or decreasing the amount of both does not matter.

image processing - Standard Deviation in Gaussian Blur

I have a function that performs gaussian blur on image for some specific $\sigma$ (the standard deviation).

It first computes kernel of size $\lceil 3\sigma \rceil$ and then performs convolution with that kernel.

However, I would like to specify blurring radius in pixels rather than $\sigma$.

I suppose that the blur radius (in pixels) is just $\sigma^{2}$ as this denotes variance of a random variable.

Is that right? Can the same thought be extended to 2D?

UPDATE:

The problem is that I need to do things like building a gaussian pyramid (successively blurred and downsampled image).

When the image gets downsampled to 1/2 of its width, I suppose I need a gaussian blur of radius 2 pixels ($\sigma=\sqrt{2}$ ?). And for 1/4 subsampling, I would need blur of 4 pixels ($\sigma=2 ?)$... But I am not sure about that...

Answer

The standard deviation $\sigma$ is itself the appropriate linear scale for a Gaussian. For example, in 1D the Gaussian is $f[x,\sigma] = f[x/\sigma] \propto e^{-(x/\sigma)^2}$, i.e. $\sigma$ has the same units as $x$. As Arrigo notes, these units can be pixel-units.

The 2 in the exponent of $\sigma^2$ does not have anything to do with the dimensionality: it's the same in 1D or 2D (or nD).

The use of a $t=\sigma^2$ to index the "scale" in scale-space is more for 1) mathematical convenience, and 2) connection to the time-scale of diffusion.

halacha - Al Hanisim or V'Al Hanisim

There seems to be different Nuschaos regarding Al Hanisim. Some say Al and some say V'Al. What is the difference and what do most Halachic authorities recommend.

Answer

"Al Hanissim" seems to be the older version (as found, for example, in Machzor Vitry and in various siddurim printed in the 16th and 17th centuries).

R' Zalman Hanau (Shaarei Tefillah, sec. 110) cites and agrees with an opinion that it should correctly be with a vav, though, since it's a continuation of the preceding list of things we thank Hashem for. Mishnah Berurah (682:1) concurs.

On the other hand, R' Yaakov Emden (Luach Eresh, sec. 175) defends the older version, on the grounds that in Hebrew it's perfectly okay to omit the vav in a series of items. Indeed, he argues that in Shemoneh Esrei it would be grammatically incorrect to add it, since "Al Hanissim" is really the beginning of a new thought (which then continues in the paragraph "Ve'al kulam").

I guess most halachic authorities would recommend sticking with your community's nusach.

word choice - What is the difference between 「変更」 and 「変化」?

I came across the word 「変化」 meaning "change, alteration". I already know the word 「変更」 that means "change" as well. What is the difference between both terms?

I found a related question where they discuss 変わる vs 変化, but they say nothing on 変更。

よろしくお願いします！

Answer

変更 specifically refers to a human-caused update/modification of a plan, project, document, destination, etc. 変化 refers to change in general.

As suru-verbs, 変更する is transitive, and 変化する is intransitive.

For example, you can say 季節の変化 but not 季節の変更. You can say 計画を変更する but not 計画を変化する.

sources mekorot - Each generation in which the Temple wasn't rebuilt it's as if it was destroyed anew

I've often heard the phrase "In each generation in which the Temple isn't rebuilt, it's as if it was destroyed anew". כל דור שלא נבנה בית המקדש בימיו, כאילו נחרב בימיו.

I'm trying to find the earliest source for this phrase. A search on Sefaria brought six late, mostly chassidish sources:

1. Shemiras HaLashon (Chofetz Chaim, quoting Chazal)


2. Pri HaAretz [1] and [2] (Rabbi Menachem Mendel of Vitebsk, quoting Razal)


3. Pele Yoetz (Rabbi Eliezer Papo)


4. Yismash Moshe (Rabbi Moshe Teitelbaum of Satmar)


5. Likkutei Halachos (Rabbi Nosson of Breslov)

I found in the Yerushalmi Yoma 1:1 the phrase:

כל דור שאינו נבנה בימיו מעלין עליו כאילו הוא החריבו

Any generation in which [the Temple] was not rebuilt in their days, They consider it upon them as if they destroyed it

This version is much harsher, blaming the generation itself. I'm looking specifically for the earliest source of the softer version quoted above, that it was as if it was destroyed.

inorganic chemistry - Potassium vs sodium bicarbonate in fire fighting

Why is it that potassium bicarbonate reduces flashback as opposed to sodium bicarbonate when fighting fires?

From what I understand, high heating of bicarbonate salts decomposes them into carbon dioxide, which helps suffocate the fire, and the corresponding metal hydroxide. Could it be potassium hydroxide that is interacting differently with the fire versus sodium hydroxide that causes this issue?

Answer

To answer this question, firstly we must understand both how fires are started, and how they are perpetuated. It is common knowledge that in order for fire to exist there must be heat, fuel, and an oxidizing agent (usually oxygen). Once started, a fire is then self perpetuated by a radical chain reaction. Removing any of these three ingredients or interrupting the radical chain reaction can be effective ways of extinguishing a fire. Both sodium and potassium bicarbonate are effective at performing several of these actions, though there are slight differences between them.

• Differences between $\ce{NaHCO3}$ and $\ce{KHCO3}$

It is widely believed that fireextinguishing powders can function as both energy-absorbing materials and solid surfaces on which free radicals can be destroyed. Heat may be absorbed by the heat capacity of the solid [$C_p(\ce{KHCO3})=90.05\:\mathrm{J\:mol^{-1}\:K^{-1}}$ vs. $C_p(\ce{NaHCO3})=87.61\:\mathrm{J\:mol^{-1}\:K^{-1}}$], the heat of fusion, the heat capacity of the liquid, heat of dissociation from breaking of chemical bonds [$\Delta H_\mathrm{decomp}(\ce{KHCO3})=1063.28\:\mathrm{kJ\:mol^{-1}}$ vs. $\Delta H_\mathrm{decomp}(\ce{NaHCO3})=995.72\:\mathrm{kJ\:mol^{-1}}$], and heat of vaporization. All of these contribute to the total endothermicity of the fireextinguishing powder.

From a chemical aspect, it has been found that potassium salts are more effective than sodium salts, and iodide anions are more effective than chloride anions. Presumably, there is a catalytic path for destruction of free radicals, such as $\ce{H}$, $\ce{O}$, and $\ce{OH}$, utilizing the potassium in the salts. It must be remembered that any powder that has a chemical fireextinguishing capability will also have a heat-absorbing (endothermic) capability.$^{[1]}$

• The Chain Radical Mechanism

Combustion of hydrocarbons is thought to be initiated by hydrogen atom abstraction (not proton abstraction) from the fuel to oxygen, to give a hydroperoxide radical ($\ce{HOO}$). This reacts further to give hydroperoxides, which break up to give hydroxyl radicals. There are a great variety of these processes that produce fuel radicals and oxidizing radicals. Oxidizing species include singlet oxygen, hydroxyl, monatomic oxygen, and hydroperoxyl. Such intermediates are short-lived and cannot be isolated.$^{[2]}$

• The Chain Radical Breaking Mechanism

The mechanism by which chemical inhibition occurs when certain powders are added to fuel-air flames is not completely understood. Solid particles may scavenge chain propagating species by surface adsorption or reaction; or alternatively, the solids may vaporize in the combustion zone to produce gaseous products that react with active species via homogeneous gas reactions.

Most investigators seem to favor the latter mechanism, although direct experimental verification is very lacking. If homogeneous gas reactions are, in fact, responsible for the inhibition, then for cases in which potassium salts are the inhibiting agents, the chemical reactions

$$\ce{KOH(g) + H->K(g) + H2O}\:\:\:\:\:\Delta H = -33.2\:\mathrm{kcal\:mol^{-1}}$$ $\hspace{78 mm}$and/or

$$\ce{KOH(g) + OH -> KO(g) + H2O}\:\:\:\:\:\Delta H = -1.7\:\mathrm{kcal\:mol^{-1}}^{[3]}$$

---

_{[1] Fire Extinguishing Powders}

_{[2] Oxygen Fueled Combustion Reaction Mechanism}

_{[3] Flame Inhibition by Potassium Compounds}

tefilla - What does cantillation do to prayer's status?

I recall seeing recommendations, in multiple Halachic discussions, that someone who's reciting a Scripturally-derived prayer alone that normally requires a minyan, e.g. the 13 Attributes of Mercy, one should say it with the trop. Apparently, this feature converts the recitation from prayer into Torah learning, so one isn't incorrectly subverting the minyan rule.

On the other hand, I also recall learning in school to recite the Keriat Shema with the trop as a standard practice.

• 
  

  Is my understanding of the function of trop in the first case correct?

  
  



• 
  

  Is it true that reciting Keriat Shema is best done with the trop?

  
  


• 
  

  If so, why doesn't the same function similarly de-prayerify the Keriat Shema?

  
  

Answer

The issue is not just whether something is or is not a prayer, but also whether it is a דבר שבקדושה.

The Beit Yosef brings (OC 565:5) in the name of Rashba that an individual (not in a minyan) shouldn't say the 13 Attributes in the context of בקשת רחמים (requesting mercy), since that is a דבר שבקדושה (learned from the gemara on Rosh Hashanah 17b). However, when said with trop¹, the 13 Attributes lose their status of דבר שבקדושה, and become permitted, the same way we read other Torah verses while praying alone. Such is the psak given in the Shulchan Aruch, ad loc.

In addition to not being a prayer as such, as mentioned in other answers, the shema is not a דבר שבקדושה (and we recite it while praying alone), hence according to Rashba, there's nothing lost by reciting the shema with trop.

Whether shema is ideally recited with trop or not seems to be a separate question, but in short, the Beit Yosef describes this (OC 61:25) as a relatively new practice, but that one should indeed use the trop. The Darkhei Moshe thinks it's unnecessary, and liable to be a distraction, but one may use the trop if one can without being distracted. Both these views are brought as psak in Shulchan Aruch, ad loc. See here for more.

---

¹ The language is actually קריאה בעלמא, which could be taken to mean reading without the intention of בקשת רחמים, and not imply that trop needs to be used (à la קריאת התורה). However, it seems to me that the best way to make sure that קריאה בעלמא isn't a request for mercy is to read it explicitly in a different context, namely as if it were קריאת התורה. In addition, the Darkhei Moshe (OC 565:5) cites the Terumat Hadeshen (siman 8) as agreeing with Rashba here, and the Terumat Hadeshen explicitly says to use trope, so the Darkhei Moshe clearly thinks that קריאה בעלמא means trop.

grammar - Who scolded whom in Aに怒られる?

I can see that someone got mad or someone made someone else mad. The things that make it difficult are に and the られる form of 怒る. What does this mean? How do you figure out who did what in these types of sentences?

Answer

It may first bear mentioning that 怒られる is usually used specifically to mean "got scolded by", rather than "to became the focus of someone's anger".

This may make it easier to understand.

• Aは怒った : A got angry


• AはBに怒った : A got angry at B (in this usage, B is usually not a person, but rather an event or state of affairs).


• AはBを怒った : A scolded B (B is a person)


• BはAに怒られた : B got scolded by A.

Molecular kinetic interpretation of pressure

Regarding the molecular kinetic interpretation of pressure, we arrive at a point in the $F=\frac{\Delta P}{\Delta t}=\frac{2mvx}{\Delta t}$, that is to say, the force exerted in the shock on the wall is calculated by dividing the change in the amount of movement of the wall by that time, which I understand should be the time of duration of the shock, but nevertheless the time used is the time between two shocks, the time in which the molecule travels the distance of the edge of the vessel for one side and returns...this does not contradict the definition of impulse that refers to the duration of the shock??

halacha - Do parts of a minor fast start before morning?

I've always understood that minor fasts run from morning (a bit before dawn) until nightfall. An answer to What activities (other than eating and drinking) are forbidden on minor fasts?, however, cites the Mishnah Berurah as saying:

[A] pious person should be stringent on all of them as on Tish'ah Be'Av, but if one of them falls on the evening when [a man's wife is due to] immerse, he should fulfill his obligation.

I wondered how this could even be an issue -- a woman immerses at night, so by the time she's home from the mikvah the fast would surely be over. Joel K there speculated that the MB might be talking about the night before (e.g. the night between 9 Tevet and 10 Tevet). I also note that the MB is talking about a pious person, not necessarily halacha.

This all leaves me confused about pre-fast activities. According to halacha, which if any fast-day changes begin before the start of the fast in the morning?

parashat bereishit - Adam and Eve as parable or literal

Was the story of Adam and Eve meant as a parable for the dawn of civilization after human species evolved for millennia or is Adam and Eve to be literally read as the first humans to ever be created.

avodah zarah - Abram's worship of El Elyon

In Genesis 14:18-20 Abram recognizes the deity El Elyon of the Jebusite monarch Melchizedek.

Some scholars have investigated into the meaning of this name El Elyon and propose that it actually consists of two separate names of two deities. El being the son of Elyon.

According to Marvin Pope in his book "El in the Ugaritic texts".

Philo of Byblos identifies Elyon as being a separate deity who is the grandfather of the Ugaritic god El. He then goes on to compare Elyon to the 13th century BCE Hittite god Alalu.

https://books.google.com/books?id=ns4UAAAAIAAJ&pg=PA55&lpg=PA55&dq=elyon+in+ugarit&source=bl&ots=TNQkXWyGMW&sig=RlsjOctbw4rwdwkBQpsOzSpeQ4g&hl=en&sa=X&ei=goU4VZy0BcPAsAWEsIDADQ&ved=0CCQQ6AEwAw#v=onepage&q=elyon&f=false

Lastly, there is an Aramaic inscription known as the Sefire stele from 750 BCE that also uses the term El Elyon.

According to John Day:

The inscriptions may, under one possible interpretation, record the names of El and Elyon, "God, God Most High" possibly providing prima facie evidence for a distinction between the two deities first worshipped by the Jebusites in Jerusalem, and then elsewhere throughout the ancient Levant.

If this is indeed the case would that mean that Abram actually proclaimed his allegiance to numerous deities by accepting a blessing in the name of El (son of) Elyon?

inorganic chemistry - What is hybridisation of XeF6 in solid state?

According to the following formula:

Type of hybridisation/steric no. = (no. of sigma bonds + no. of lone pairs)

it should be $\ce{sp^3d^3}$, however according to my textbook it is $\ce{sp^3d^2}$ in solid state. That is how it was written(in my textbook), emphasising on the fact that "in solid state it is $\ce{sp^3d^2}$". How does solid state of a compound changes its hybridisation?

Answer

1. 
   

   Forget about applying hybridization outside the second row, especially in 'hypervalent' compounds. I know, that it is common to use and sometimes works, but it is incorrect.

   
   


2. 
   

   The $\ce{XeF6}$ molecule is a hard spot. While, indeed, experimental data suggest that it adopts distorted octahedral geometry in the gas phase, there is evidence that the minimum is very shallow.

   
   


3. 
   

   The octahedral structure of $\ce{XeF6}$ (which is probably a local minimum, or at least found to be one in more than one calculation) can be pretty easily described in terms of three-center four-electron bonds.

   
   
   


4. 
   

   There is no good agreement on the nature of the stereo-active electron pair in $\ce{XeF6}$. It seems, that repulsion with core d-shells is important, but the recent articles I could get my hands on has little to no rationalization on the fact. Still, it must be noted that the minimum is very shallow, meaning that rough qualitative theories like VSEPR and MO LCAO are too rough for a good rationalization anyway.

   
   


5. 
   

   The solid state typically involves a lot more interaction, and, as result, compounds may adopt very different structures. The simplest one example I can think of, I could point to $\ce{SbF5}$ for example. A monomer adopts trigonal-bipyramidal structure. However, in the solid state, a tetramer with octahedrally coordinated antimony and four bridging fluorines is formed. In terms of VSEPR, it is a move from $\mathrm{sp^3d}$ to $\mathrm{sp^3d^2}$. Similarly, $\ce{XeF6}$ has a crystal phase (one of six) that involve bridging atoms.

   
   

halacha - Are We Not Supposed To Eat Fat?

In Leviticus 3-17 it states:

All fat is the lord's. It is a law for you for all time throughout the ages, in all your settlements: you must not eat any fat or any blood.

כָּל-חֵלֶב לַיהוָה חֻקַּת עוֹלָם לְדֹרֹתֵיכֶם, בְּכֹל מוֹשְׁבֹתֵיכֶם כָּל-חֵלֶב וְכָל-דָּם, לֹא תֹאכֵלוּ

http://www.mechon-mamre.org/p/pt/pt0303.htm#17

Does this mean we are supposed to remove all the fat from meat as we do blood, or is it specifically talking about the sacrifices?

Answer

Chelev (the word translated as "fat" in the quoted verse) in Halacha refers to certain fats which in a sacrifice are offered on the altar and in regular meat are forbidden to be eaten, while Shuman refers to other fats which are completely permitted. A list of which fats on which body parts are in which category is something which pretty much can only be taught with a dead cow/sheep in front of you or some really good pictures.

If you buy certified kosher meat then they have already removed the relevant Chelev or are selling you meat from a part of the animal in which no Chelev can be found.

tanach - What are the reasons or rules for the different variations in vowels on the Tetragrammaton?

I'm aware that the vowels of the Tetragrammaton are unknown and i'm not asking what they are.

I see a lot of variation in the vowels placed or not placed, on the Tetragrammaton.

Sometimes a hataph patach on the first letter, sometimes a shva, sometimes a hataph segol.

Sometimes a holam on the second letter, sometimes no vowel.

Sometimes a kamatz on the third letter, sometimes a hiriq on the third letter.

I'm interested to know what reasons there are for when it's written with some vowels and not other vowels.. I see a pattern where it is spelt with a yud under the vav, every time it follows aleph,dalet,nun,yud. So it seems maybe there are some rules or conventions.

Examples below

The Mechon Mamre site for that verse has the word with just the two vowels, and the yud with the shva. http://www.mechon-mamre.org/p/pt/pt26e4.htm

I see in bibleworks the cholam

I see it seems maybe every case of aleph,daled,nun,yud followed by yud,heh,vav,heh there is a chirik under the third letter of yud,heh,vav,heh

e.g.

and in that Kings example a vav with a cholam

Answer

There are kabolistic ideas in different nekudot of the Name see this.
(See second line from the bottom (and footnote 25) here)

---

But the way I was tought all the examples you bring are just ways to hint to us how to read it

Most of them are hinting (having very similar nekudot) to Adonoy
in the the last 3 examples they are hinting (having very similar nekudot) to Elokim

Not always are they exact:

The shva is a hint to the hataph patach and to hataph segol, depending on the rest of the nekudot.
The holam is sometimes skipped

Since the nekuda on the 3rd letter is enough for the hint

---

There is mishnah Berurah that when you imagine the name before you at all times it should be with the nekudot of the word fear
Source mishna Berurah 1.4

---

Edit
another reason of the 2 versions when it is pronounced Adonoy

shulchan aruch harav 5.2

When, however, [the name] is written as י‑ה‑ו‑ה, there are those who pronounce the alef with a shva, for this is the vowel of [the corresponding initial letter which is] the yud of the name י‑ה‑ו‑ה. Others pronounce the alef with a chataf patach, using the same vowel as the alef of the name A-donai.

Why are edges in spatial images represented as edges in their Fourier transform image?

Here is a well-known image and its Fourier Transform (magnitude).

If I understand correctly the theory behind the FFT, each pixel in the FFT image represents a certain 2D sine wave with frequency depending on the distance from the center of the image, and orientation depending on angle with the horizontal. Intensity of these pixels indicates the coefficients with which each sine wave is added, which, combined with (hidden here) phase information, gives the original image back if we do an inverse Fourier transform.

While I fail to grasp is that when treating real images such as the ones presented, why are edges in the images visible as edges in the frequential domain ?
In this example, there is a diagonal line in the FFT image (let's forget about the vertical and horizontal line which I think are artefacts based on the way FFT is computed, needs a periodical image, etc). This diagonal line is probably caused the girl's hat. But as I understand it, the line in FFT domain means a sum of sines oriented in the same way but with different frequencies. How does that result in an edge when we convert back ? Since edges are high frequency information, wouldn't an edge be represented by one very bright point in the FFT ? Does it have to do with adding different sines so that they cancel each other out on some portions of the image ? Does the phase image has anything to do with it ?

A more compelling example might be the following image set :

Answer

Why are edges in spatial images represented as edges in their Fourier transform image?

They are not edges composed of the same "thing", to the spatial image, and they do not correspond to the same orientation. The image you are using in your example is a bit misleading.

An edge is basically a square pulse whose Fourier Transform is a sinc.

The bandwidth of the sinc is inversely proportional to the width of the pulse. Short pulses have wide bandwidths and long pulses have short bandwidths.

What you see in a magnitude plot, which is what is depicted in the example you show, is the magnitude of each spatial frequency's complex coefficient. This means that this sinc function, when plotted, gets rectified but more importantly, it is at a 90 degree angle to the line that causes it.

Here is an example in Octave but easily adaptable to other platforms:

I = zeros(64,64); % A simple 64x64 image
I(:,30:34)=1; % A vertical line of 4 pixels running vertically in the middle of our image
F = abs(fftshift(fft2(I))); % The FFT Magnitude plot
imshow(F, [min(min(F)), max(max(F))]);

This image and its transform looks like:

Notice that "flutter" in the brightness (and therefore magnitude) of the line in the spatial frequency domain (second image)? These are the sinc coefficients. What you see there is a sinc but it is actually the DC values of each time instance of the sinc.

The two dimensional Discrete Fourier Transform (DFT) is obtained as two successive applications of the one dimensional DFT. The first time around we apply one DFT to each row of the image matrix (vertically to our white line) and the second time we apply one DFT to each column of the previous step DFTs (parallel to our white line).

Each row of the image is basically a pulse and this gives us one sinc for every line. Here is the magnitude of "half" a 2D DFT, as a surface plot to demonstrate this one-sinc-per-pulse better:

In this image, the main lobe of the sinc is still aligned with the white line. To conclude the 2D DFT, we run DFTs to the columns of this array, which is basically a stable signal at some DC value and this is how the spatial frequency "line" forms and appears to be vertical to the spatial domain line.

If you rotate the spatial line, you also rotate the spatial frequency "line".

The image that you show in your example is an unfortunate choice, in terms of orientation in the spatial frequency domain, because what appears to be a spatial frequency line corresponds to the sides of the triangular roof. But, the spatial frequency "line" that runs bottom left to top right corresponds to the image edge of the roof that runs top left to bottom right and vice versa.

For more information, please see this and this link.

Hope this helps.

organic chemistry - Hybridisation of terminal nitrogen in diazomethane

I have a few questions about the terminal nitrogen (highlighted in red) in diazomethane, $\ce{CH2N2}$.

1. 
   

   Is that nitrogen $\mathrm{sp}$ or $\mathrm{sp^2}$ hybridised?

   
   


2. 
   
   

   What type of orbitals do the lone pairs on that nitrogen reside in?

   
   


3. 
   

   Does the negative charge on nitrogen undergo resonance?

   
   

Answer

In da wikipedia you can see that the C=N=N fragment is linear AND another resonance structure involves triple N-N bond. So, the answers

1. it is sp in one resonance structure and probably sp2 in another resonance structure.



2. depends on the resonance structure considered


3. yep.

Diazomethane typically reacts with its negatively charged carbon, so this is of little consequence though.

Confusion regarding writing a word in Kanji and Katakana

In the One Piece manga, it's quite common to see the names of character's attacks written in both Kanji and Katakana. Take as examples:

Gekko Moriah's Doppelman [影法師]{ドッペルマン} Dopperuman, literally meaning "Silhouette"
Daz Bones' Spider [斬人]{スパイダー} Supaidā, literally meaning "Beheading/Cutting Man"

My question is: can those kanji really work together to be pronounced like that, and the katakana is just there so people won't think it's pronounced some other way? Or is it that the kana actually DEFINES how the word is supposed to be pronounced, and I could in theory take any pair of kanji and katakana and say "you write it just like that other word, but this is a different word and here's how it's supposed to be pronounced"?

I used to think it was just clever wordplay, but there's some things that just sound too good to be true.

Answer

In most cases, the kanji can't be pronounced that way, as in that pronunciation does not match the standard on-yomi or kun-yomi for the characters at all. Basically, the kanji provide the meaning and the katakana show how the author wants it to be pronounced. It's a stylistic choice.

This can be seen in song lyrics, too, where a word will have kanji but it will be pronounced with its English meaning, for example 道{ロード}, or with the pronunciation of another Japanese word that is usually written differently, like 永遠{とわ}, 仲間{きみ}, or 現実{いま}.

What are advantages of having higher sampling rate of a signal?

Being a non signal processing science student I have limited understanding of the concepts.

I have a continuous periodic bearing faulty signal (with time amplitudes) which are sampled at $12\textrm{ kHz}$ and $48\textrm{ kHz}$ frequencies. I have utilized some machine learning techniques (Convolutional Neural Network) to classify faulty signals to the non faulty signals.

When I am using $12\textrm{ kHz}$ I am able to achieve a classification accuracy $97 \pm 1.2 \%$ accuracy. Similarly I am able to achieve accuracy of $95\%$ when I applied the same technique on the same signal but sampled at $48\textrm{ kHz}$ despite the recording made at same RPM, load, and recording angle with the sensor.

• What could be the reason for this increased rate of misclassification?


• Are there any techniques to spot differences in the signal?


• Are higher resolution signals prone to higher noise?

---

Details of the signal can be seen here, in chapter 3.

Answer

Sampling at a higher frequency will give you more effective number of bits (ENOB), up to the limits of the spurious free dynamic range of the Analog to Digital Converter (ADC) you are using (as well as other factors such as the analog input bandwidth of the ADC). However there are some important aspects to understand when doing this that I will detail further.

This is due to the general nature of quantization noise, which under conditions of sampling a signal that is uncorrelated to the sampling clock is well approximated as a white (in frequency) uniform (in magnitude) noise distribution. Further, the Signal to Noise Ratio (SNR) of a full scale real sine-wave will be well approximated as:

$$SNR = 6.02 \text{ dB/bit} + 1.76 \text{dB}$$

For example, a perfect 12 bit ADC samping a full scale sine wave will have an SNR of $6.02\times 12+1.76 = 74$ dB.

By using a full scale sine wave, we establish a consistent reference line from which we can determine the total noise power due to quantization. Within reason, that noise power remains the same even as the sine wave amplitude is reduced, or when we use signals that are composites of multiple sine waves (meaning via the Fourier Series Expansion, any general signal).

This classic formula is derived from the uniform distribution of the quantization noise, as for any uniform distribution the variance is $\frac{A^2}{12}$, where A is the width of the distribution. This relationship and how we arrive at the formula above is detailed in the figure below, comparing the histogram and variance for a full-scale sine wave ($\sigma_s^2$), to the histogram and variance for the quantization noise ($\sigma_N^2$), where $\Delta$ is a quantization level and b is the number of bits. Therefore the sinewave has a peak to peak amplitude of $2^b\Delta$. You will see that taking the square root of the equation shown below for the variance of the sine wave $\frac{(2^b\Delta)^2}{8}$ is the familiar $\frac{V_p}{\sqrt{2}}$ as the standard deviation of a sine wave at peak amplitude $V_p$. Thus we have the variance of the signal divided by the variance of the noise as the SNR.

Further as mentioned earlier, this noise level due to quantization is well approximated as a white noise process when the sampling rate is uncorrelated to the input (which occurs with incommensurate sampling with a sufficient number of bits and the input signal is fast enough that it is spanning multiple quantization levels from sample to sample, and incommensurate sampling means sampling with a clock that is not an integer multiple relationship in frequency with the input). As a white noise process in our digital sampled spectrum, the quantization noise power will be spread evenly from a frequency of 0 (DC) to half the sampling rate ($f_s/2$) for a real signal, or $-f_s/2$ to $+f_s/2$ for a complex signal. In a perfect ADC, the total variance due to quantization remains the same independent of the sampling rate (it is proportional to the magnitude of the quantization level, which is independent of sampling rate). To see this clearly, consider the standard deviation of a sine wave which we reminded ourselves earlier is $\frac{V_p}{\sqrt{2}}$; no matter how fast we sample it as long as we sample it sufficiently to meet Nyquist's criteria, the same standard deviation will result. Notice that it has nothing to do with the sampling rate itself. Similarly the standard deviation and variance of the quantization noise is independent of frequency, but as long as each sample of quantization noise is independent and uncorrelated from each previous sample, then the noise is a white noise process meaning that it is spread evenly across our digital frequency range. If we raise the sampling rate, the noise density goes down. If we subsequently filter since our bandwidth of interest is lower, the total noise will go down. Specifically if you filter away half the spectrum, the noise will go down by 2 (3 dB). Filter 1/4 of the spectrum and the noise goes down by 6 dB which is equivalent to gaining 1 more bit of precision! Thus the formula for SNR that accounts for oversampling is given as:

Actual ADC's in practice will have limitations including non-linearities, analog input bandwidth, aperture uncertainly etc that will limit how much we can oversample, and how many effective bits can be achieved. The analog input bandwidth will limit the maximum input frequency we can effectively sample. The non-linearities will lead to "spurs" which are correlated frequency tones that will not be spread out and therefore will not benefit from the same noise processing gain we saw earlier with the white quantization noise model. These spurs are quantified on ADC datasheets as the spurious-free dynamic range (SFDR). In practice I refer to the SFDR and usually take advantage of oversampling until the predicted quantization noise is on level with the SFDR, at which point if the strongest spur happens to be in band, there will be no further increase in SNR. To detail further I would need to refer to the specific design in more detail.

All noise contributions are captured nicely in the effective number of bits (ENOB) specification also given on ADC data sheets. Basically the actual total ADC noise expected is quantified by reversing the SNR equation that I first gave to come up with the equivalent number of bits a perfect ADC would provide. It will always be less than the actual number of bits due to these degradation sources. Importantly, it will also go down as the sampling rate goes up so there will be a diminishing point of return from oversampling.

For example, consider an actual ADC which has a specified ENOB of 11.3 bits and SFDR of 83 dB at 100 MSPS sampling rate. 11.3 ENOB is an SNR of 69.8 dB (70 dB) for a full scale sine wave. The actual signal sampled will likely be at a lower input level so as not to clip, but by knowing the absolute power level of a full scale sinewave, we now know the absolute power level of the total ADC noise. If for example the full scale sine wave that results in the maximum SFDR and ENOB is +9 dBm (also note that this level with best performance is typically 1-3 dB lower than the actual full scale where a sine wave would start to clip!), then the total ADC noise power will be +9dBm-70 dB = -61 dBm. Since the SFDR is 83 dB, then we can easily expect to gain up to that limit by oversampling (but not more if the spur is in our final band of interest). In order to achieve this 22 dB gain, the oversampling ratio N would need to be at least $N= 10^{\frac{83-61}{10}} = 158.5$ Therefore if our actual real signal bandwidth of interest was 50MHz/158.5 = 315.5 KHz, we could sample at 100 MHz and gain 22 dB or 3.7 additional bits from the oversampling, for a total ENOB of 11.3+ 3.7 = 15 bits.

As a final note, know that Sigma Delta ADC architectures use feedback and noise shaping to achieve a much better increase in number of bits from oversampling than what I described here of what can be achieved with traditional ADC's. We saw an increase of 3dB/octave (every time we doubled the frequency we gained 3 dB in SNR). A simple first order Sigma Delta ADC has a gain of 9dB/octave, while a 3rd order Sigma Delta has a gain of 21 dB/octave! (Fifth order Sigma Delta's are not uncommmon!).

Also see related responses at

How do you simultaneously undersample and oversample?

Oversampling while maintaining noise PSD

How to choose FFT depth for ADC performance analysis (SINAD, ENOB)

How increasing the Signal to Quantization noise increases the resolution of ADC

halacha - Boruch Shem Kevod Malchuso L'Olam Voed - why do we whisper it?

Why do we say Boruch Shem Kevod Malchuso L'Olam Voed in Kriyas Shema in a whisper?

Answer

1. This was a declaration of loyalty by the Shevatim on Yaakov's deathbed, so it's a worthwhile prayer. But since it's not a verse found in the Torah, we say it quietly (Pesachim 56a)


2. This was a prayer of the angels so it's not appropriate for sinful mortals to say it aloud, except for on Yom Kippur

From the Artscroll Yom Kippur Machzor, pp.69-70

grammar - The different usages of ことがある

If I understand correctly, the main usages of ことがある seem to be the following:

• Verb(plain)+ことがある: There are times when (I)


• Verb(past)+ことがある: (I) have experienced (something) before

But after this question, I'm starting to wonder when ことがある means "there is a thing/there are things" instead of "there are times when".

When looking through Space ALC and Google searches, I'm thinking ～たいことがある might mean "there's a thing I want to (do)" and ～たくなることがある "there are times when I want to (do)" based on the way they're used, but I can't be sure.

書きたくなることがあります
"there are times I want to write" (?)

書きたいことがあります
"there are things I want to write" (?)

What are the different usages of ことがある? When does ことがある mean "there are things" rather than "there are times when" and how can they be told apart?

Answer

The clause used in ...ことがある is structurally ambiguous between an appositive clause and a relative clause.

書きたくなることがあります

1. As appositive clause
　書きたくなる 'I become tempted to write something' is the content of the formal noun こと 'occasion'
　'There are occasions that I become tempted to write something.'
2. As relative clause
　The こと '(factual) thing' is the missing object of the relative clause 書きたくなる 'I become tempted to write'
　'There are things that I become tempted to write.'

It is just as the same in English. Depending on whether you interpret the English write in:

　There are occasions that I become tempted to write

as intransitive or transitive, you can interpret the clause as appositive or relative, and will get the two meanings.

sources mekorot - Omnipotence and Omniscience

Where was it first claimed that God is omnipotent or omniscient?

Did God or an authorized prophet ever actually make this claim?

hashkafah philosophy - Why was an imperfect world instructed in capital punishment?

So, according to wikipedia (I know... but still it was the best source I could find)

Leading rabbis in Reform Judaism, Conservative Judaism, and Orthodox Judaism tend to hold that the death penalty is a correct and just punishment in theory, but they hold that it should not generally be used (or not used at all) in practice. In practice the application of such a punishment can only be carried out by humans whose system of justice is nearly perfect, a situation which has not existed for some time or never existed at all.

Why then did G-d instruct Jews in this manner? Do I have to interpret this as meaning that the old Jewish society is considered nearly perfect (which I have a hard time believing in the light of the mistakes that were made as described in the Torah)?

Answer

I'd like to answer along two dimensions, one about capital punishment and one more broad.

First, it is possible for the conditions to be met under which capital punishment can apply. Tractate Sanhedrin in the talmud discusses in great detail the relevant laws. We know that sentences of capital punishment were carried out in the past. They were rare, with Rabbi Eleazar famously saying in the talmud (Makkot 1:10) that a court that executes once in 70 years is bloodthirsty (others say once in seven years).

Because these laws can be applied -- not today, but in the past and maybe in the future -- the torah needs to give them and we need to learn them.

Second, the broader point: the torah sometimes gives us laws that we cannot carry out. According to tractate Sanhedrin (h/t user6591, DoubleAA), the laws of the stubborn and rebellious son and of the idolatrous city were never carried out. (However, some disagree.) So why were these laws given? I was taught that, according to those who say it never happened, it's so that we will have the merit of torah study in learning it. Even if we do not apply a law, God had some reason for wanting us to learn it -- some principle we can derive from it, perhaps. God, being perfect, gave us a perfect torah, even if we imperfect people cannot understand why in some cases.

particle は - Which is correct: こんばんわ or こんばんは?

I've seen both こんばんわ and こんばんは used; which is correct here? If we interpret the は as the topic particle, は would seem correct, but it seems that わ is used quite frequently anyway...

Answer

こんばんは is correct, according to that page in Japanese.

My gut feeling is the same - 今晩は ->　こんばんは.

That said, a cursory Google of こんばんは yielded 13M hits, whereas こんばんわ yielded 26M.

halacha - "Yotzer Or" procedure

Right after borochu in shacharit there is a bracha “yotzer or uvoreh choshech, oseh shalom uvoreh et ha kol”. Should this bracha be said out loud by the shaliach tzibur? Regardless of whether it should or should not, if it is said out loud should amen be answered at the end?

Wavelet transform: How to compute the initial coefficients when only samples are available?

In standard MRA we have that the space of functions at scale J can be expressed as
$$V_j = V_0\oplus \left(\bigoplus_{j=0}^{J} W_j\right)$$ where $V_0$ is spanned by the orthonormal system of the translates of the scaling function $\phi(x)$, i.e. $V_0 = \overline{\text{span}}\{\phi(x-k)\}_{k\in\mathbb{Z}}$ and the detail spaces $W_j$ are spanned by the wavelets ,i.e. $W_j = \overline{\text{span}}\{\psi_{j,k}\}_{k\in\mathbb{Z}}$, being $\psi_{j,k} = 2^{j/2}\psi(2^j x - k)$ and $\psi(x)$ the mother Wavelet. From the two scales equation and from the wavelet equation we obtain the iterative algorithm to compute the coefficients at the coarser scale from those of the finer one. That is, given a function $f(x)$ in $V_J$ (which can be for instance an approximation of another function up to such scale) then since by definition of MRA we have that $\{\phi_{j,k}\}_{k\in\mathbb{Z}}=\{2^{j/2}\phi(2^jx-k)\}_{k\in\mathbb{Z}}$ spans $V_j$, we can compute the coefficients $a_{J,k} = \langle f,\phi_{J,k} \rangle$ and then I can compute the coefficients at each scale $0\le j1) Use the low of large numbers and approximate $$\hat{a}_{J,k} = \dfrac{1}{N}\sum_{i=1}^N f(x_i)\phi_{J,k}(x_i)$$

2) Solve the system of N equations $$ f(x_i) = \sum_{k} \hat{a}_{J,k}\phi_{J,k}(x_i)$$
The first one is the simplest, and has the advantage that it can be computed very fast as a new sample arrives, however it has the problem that many coefficients will be undefined and many will be defined with very few terms. The second asks to solve a possibily big system of equations any time a new sample arrives. Are there any smart way to compute the tarting coefficients $a_{J,k}$?

Answer

I remember that using the discrete samples was term, somewhere, "the wavelet sin" by Gilbert Strang. On prefiltering discrete signals to obtain "better" wavelet coefficients, here are some references I have in store:

• Wavelet coefficient computation with optimal prefiltering, IEEE Trans. Signal Processing, 1994, Xia, X.-G. and Kuo, C.-C.J. and Zhang, Z.


• On the initialization of the discrete wavelet transform algorithm, IEEE Signal Processing Letters, 1994, Abry and Flandrin


• From the wavelet series to the discrete wavelet transform - the initialization, IEEE Trans. Signal Processing, 1996, Xiao-Ping Zhang and Li-Sheng Tian and Ying-Ning Peng


• Sard-optimal prefilters for the fast wavelet transform, Numer. Algorithms, 1997, S. Ehrich


• A new prefilter design for discrete multiwavelet transforms, IEEE Trans. Signal Processing, 1998, Xia, X.-G.



• Quadrature prefilters for the discrete wavelet transform, IEEE Trans. Signal Processing, 2000, Johnson, B. R. and Kinsey, J. L.


• Meaningful MRA initialization for discrete time series, Signal Processing, 2000, D. Veitch and M. S. Taqqu and P. Abry


• Efficient Wavelet Prefilters with Optimal Time-Shifts, IEEE Trans. Signal Processing, 2003, Ericsson, S. and Grip, N.

It seems however that no prefiltering does not induce too much errors for compression, denoising, deconvolution. But for detection, or using multiple different wavelets at once (like with dual-tree wavelets, or unions of wavelet bases), or with multiwavelets, it is quite mandatory.

halacha - Having dessert "in mind"

This has always confused me. One is not supposed to make unnecessary Berachoth, so one should "have in mind" to eat all of the foods that will be served in the meal when one begins the meal and makes his first Berachah/oth. Yet, one is supposed to make a separate Berachah on dessert, because it is clearly (really?) not included in the actual meal that one has in mind.

Perhaps I am mistaken as to the underlying reason, so first, is the above correct?

Second, if I do specifically have in mind to exempt dessert with my Motzi, is that effective?

halacha - Could a Cohen burn a body to escape the impurity?

Excuse the extreme hypothetical, but:

If a Cohen were locked in a room with no windows and a dead body, and just so happened to have lighter fluid and a pack of matches (and there were small holes in the ceiling for the smoke to escape), should he/could he burn the body in order to escape the prohibition of being in the same enclosure of a corpse (Rama YD 372:1 - the prohibition is not just to enter a place of impurity, but also to remain there)?

In more general terms, would alleviating his prohibition of being with a corpse override the seeming denigration of the body through burning it¹?

(I've heard before that in such a situation, the Cohen would be required to actually eat the body in order to get rid of it, but even if true it seems rather impractical, as he would have to eat practically all of the bone mass as well.)

^{1 Yes I am aware that the issue of burning a body does not have such a clear source}

hybridization - Determine the bond angle in a compound

How can we find the bond angle between 3 atoms in a compound? Please specify an equation which can be used for all compounds.

Answer

In carbon compounds Coulson's Theorem can be used to relate bond angles to the hybridization indices of the bonds involved.

$$1+\lambda_{i} \lambda_{j} \cos(\theta_{ij})=0$$

where $\ce{\lambda_{i}}$ represents the hybridization index of the $\ce{C-i}$ bond (the hybridization index is the square root of the bond hybridization) and $\ce{\theta_{ij}}$ represents the $\ce{i-C-j}$ bond angle.

Let's look at some examples.

• In the case of methane, each $\ce{C-H}$ bond is $\ce{sp^3}$ hybridized and the hybridization index of each $\ce{C-H}$ bond is $\sqrt3$. Using Coulson's theorem we find that the $\ce{H-C-H}$ bond angle is 109.5 degrees.



• In the case of the methyl carbocation ($\ce{CH3^{+}}$) the $\ce{C-H}$ bonds are $\ce{sp^2}$ hybridized and the hybridization index of each $\ce{C-H}$ bond is $\sqrt2$. In this case, using Coulson's theorem we find that the $\ce{H-C-H}$ bond angle is 120 degrees.


• Finally, in the case of acetylene the $\ce{C-H}$ bond is $\ce{sp}$ hybridized and the hybridization index of each $\ce{C-H}$ bond is $\sqrt1$. In this case, using Coulson's theorem we find that the $\ce{C-C-H}$ bond angle is 180 degrees.

Conversely, if the bond angle is known, the hybridization can be determined.

halacha - Virtual Avodah Zara

My nephew showed me a new video game of his, Zelda: Breadth of the Wild. It's a massive action adventure game.

The relevance of this game to MY is as follows. Part of the game involves collecting objects, and once you collect enough you have to bring them to a Temple. Once there, there's a statue of a god and you have to interact with it. You have the option to pray to it, and by doing so you give it the objects and get some upgrade to your stamina.

Is there any halachic or hashkafic issue with virtual Avodah Zara? One couldn't do this in real life, so I'm wondering if doing it in a video game is just as bad or no. You're actively choosing to pray to what's classified as an idol. My understanding of Avodah Zara is the actions are prohibited even if one doesn't believe in what their doing. Here you're not doing any actions, but you're thinking about doing them. If I recall correctly from the gemarra in Kiddushin (30b?) The thoughts of Avodah Zara are just as bad as the action (although this might refer exclusively to belief).

This question assumes there's no inherent issue in playing video games in the first place

physical chemistry - Which one, Mulliken charge distribution and NBO, is more reliable?

Sometimes the Mulliken and NBO turn out to be so different that I can't decide which one I can trust. I've heard that Mulliken is inaccurate, but is NBO always accurate? And should I use Gaussian or other software to calculate charge distribution? Are there methods more accurate than Mulliken or NBO?

And charge distribution doesn't represent charge density, right?

Answer

As I do not want to drag this out in the comments, I decided to give a brief answer. Let me first cite the very helpful comments to keep the for (relative) eternity and add my thoughts at the end.

1. 
   

   There is no proper, strict and agreed definition of atom border, that's the real problem. Gaussian includes option to fit charges on atoms (assumed for the sake of the procedure as points) so they produced electrostatic field as accurately fitting once calculated from the electronic density distribution as possible, and maybe (I haven't used Gaussian for a long time, I'm not sure) dipoles too. – permeakra

   
   


2. 
   

   NBO is more accurate than Mulliken in very many ways. However, NBO is not always accurate. Generally speaking, you would never want to use Mulliken charges for any sort of production level work unless its been calibrated. – LordStryker

   
   

Mulliken is of course the cheapest and fastest way to compute charges. However, this method tends to give qualitative results at best. The reason for this is obvious. It will divide the canonical orbitals equally amongst the participating atoms. There is no polarisation whatsoever. Another problem is, that they are very basis set dependent. And that being said, the worst part is, that the description becomes worse by increasing the basis set.

NBO charges are much more reliable, since the operate on the electron density instead. Localised natural atomic orbitals can be used to describe the computed electron density. Polarisation of bonds is therefore considered. This method obviously only works very well for systems, that can actually be (quite well) described by a Lewis structure. However, they usually give a nice qualitative picture and are quite robust when it comes to increasing the basis set.

There are a couple of more population analysis tools, that can be used and should be considered when it is necessary to derive some conclusions from charges.

Quite common is also the Hirshfeld charges approach, which divides the electron density into spherical basins around the nucleus. Maybe interesting to read is "Are the Hirshfeld and Mulliken population analysis schemes consistent with chemical intuition?" (I can't tell, I do not have access.)

There are also several schemes to obtain partial charges from the electrostatic potential.

Probably the most rigorous approach to atomic partial charges is the Quantum Theory of Atoms in Molecules (QTAIM). Here the Laplacian of the electron density is used to describe the partitioning in electronic basins. The best feature of this analysis is, that it can be also used for experimentally obtained electron densities. However, as everything, this has also major drawbacks. One of the most important is the computational effort. Unfortunately the integration scheme is not always completely robust when it comes to some particular molecules. However, it is almost completely basis set independent and results could be cross referenced with experimental data.

Since permeakra's initial statement holds true, especially for the agreed part, the best way to deal with partial charges is to be careful and maybe check with different methods. In gaussian there are several schemes implemented, most commonly, Mulliken, Hirshfeld and NBO (3.1) and some others, see the population keyword for more detail.