talmud gemara - Where can I find examples of R. Yishmael's 13 talmudic rules?

We read Rabbi Yishmael's 13 principles of interpreting torah in the shacharit service and many of the explanations seem fairly straightforward, but I am not a very advanced student and I find I don't really know how some of them work. The only ones I routinely, explicitly encounter with my chevruta seem to be kal v'chomer and gezeirah shava. Presumably I'm encountering the others too but they aren't necessarily called out and I don't recognize them.

What I'm looking for is a straighforward example of each -- an argument that uses only binyan av and one that uses only kelal u'perat and so on. These should be "clean" examples, ones that most people would agree are good examplars of each technique. Where can I find such examples?

(For reference, here is the list and some explanations (not complete) from Wikipedia. Their external links seem to be broken so I can't tell if the two sources they cite would be helpful.)

Answer

The Artscroll siddur has a short example on each one.

Here are the examples given:

1. Kal Vachomer - If a lenient case has a stringency, a stringent case should have a stringency. An example would be if on Yom-Tov one cannot pick an apple (even though one can generally do melachos that involve food-preparations) all the more so on Shabbos.


2. Gzeira Shava - A tradition from Sinai that two ideas are connected through one of two ways:
   
   
   
   • The same word appears two times in two different contexts, thereby connecting the two idea. An example would be in the Gmarah Psachim which says that we learn the law that we bring the Pesach offering on Shabbos from the fact that we bring the daily Tomid even on Shabbos through the Gzierah Shava of the words "Bmoado" (It says "Bmoado" by Tomid and Pesach).
   
   
   
   • Two psukim next to each other (also called Hekesh). An example could be that the law that one could get married through a "Shtar" (contract) is derived from the verse "She left from his house, and became married to another". Through this "Hekesh" we learn that "leaving" is compared to "marrying", and just like a divorce is done through a "shtar", so too could marrying.
   
   
   
   


3. Binyan Av - Also called "Me Matzinu", which means that a detail by law apply to other laws when it logically makes sense. An example would be the Gemarah Yevamos which says that from the Torah's prohibition on marrying one's mother's half-sister and one's mother we derive that marrying one's father's half-sister is also prohibited. (If a mother's half-sister is considered like a mother's sister, than a father's half-sister should be like a father's sister).


4. Klal U'Prat - When a generality is followed by specific cases, only the cases are included in the rule. An example would be the verse: "...A person who brings an offering, you should bring it from the domesticated animals, from the cows and the sheep". Therefore, one can only bring it from cows and sheep.


5. Prat U'Klal - When specific cases are followed by a generalization, the rule applies by any case in the generalization, and is not limited to the cases. An example would be that with regarding lost objects the verse says: "So too you should do to his garment, so too you should do to his donkey, so too you should do to any lost object of your brother". Therefore, one has to return all lost objects, and not just donkeys and garments.


6. Klal Uprat Uklal - When there is a general rule, followed by examples, followed by a general rule. The Halacha is that we follow the general rule, but only on condition it is similar to the specific case. An example would be in Bava Metziya where the Gemara is discussing when a thief must pay back double. The Gemarah brings a verse which says that "On any trespass, on the ox, on the donkey, on the sheep, on the garment, on any lost item...". "On any trespass" is one generalization, "on the ox...garment" are the detailed cases, and "on any lost item" is another generalization. Therefore, one must pay back double on any item similar to the middle items, that is something that has inherent monetary worth and isn't land.


7. Klal Shehu Tzrich Liprat - Sometimes the specific case is necessary to clarify the general rule. An example is the verse which says: "You shall cover it with dirt". The Gemarah says that at first glance, this verse is a classic example of "A general rule followed by a specific case" ("You shall cover it" is the general, while "with dirt" is the specific case). However, "with dirt" is needed to explain HOW to cover it (i.e. that the dirt must be capable of being planted with, not salty earth).


8. Kol davar shehaya bichlal... - An example would be: the verse says: "Whatever remains from any sacrifice on the third day shall be burned in fire" and the next verse says: "If anyone eats from the Shlamim sacrifice on the third day... the one that eats it shall bear his inquity (gets Kares)". This rule teaches that one who eats not only nosar-shlamim sacrifices gets Kares but anyone who eats Nosar gets Kares.


9. Vyatzo Liton...Kinyano... - An example: the verse says that one who kills someone is executed (no differentiation between careless or premeditated murder). When the Torah later on refers to the punishment of exile for a careless murderer, it cannot be adding to execution, but to replace it with exile.



10. Veyetze Liton...Shelo Kinyano... - These verses describe how Jewish male-servants go free. According to logic the laws of the female slave should be the same. However, because it says by a female servant "She should not go out like men-servants", females could go out before six years. However, it is also more stringent, as the master could betroth her against her will.


11. Veyatza Lidon - A Kohen who married an Israelite is not permitted to eat Trumah. If she becomes divorced or widowed she would not be able to eat Trumah unless the Torah returned her back to the family group.


12. 
    

    There are two rules:

    
    
    
    
    • 
      

      Davar Halamed Minyano - It says in the Ten Commandments "Do not steal". The sages say it must refer to kidnapping because the two commandment before (Don't commit adultery and Don't kill) are capital crimes. The only capital form of stealing is kidnapping.

      
      
    
    
    • 
      

      Davar Halamed Misofo - When the Torah refers to "leprosy" on houses, it says that a house with a "leprous" spot must be torn down (it doesn't say which house). From the end of the subject (when it talks about how to clean houses of wood, mortar and stone) we see that leprosy on houses only applies to wood, mortar and stone houses.

      
      
      
    
    
    
    


13. Shnei Ksuvim - In one verse it says that Avraham's progeny will stem from Yitzchak, while later Hashem says to offer Yitzchak as a burned offering. Therefore, Hashem explained that Hashem didn't want Avraham to offer Yitzchak as a burned offering, rather he was supposed to just "bring him up". Therefore, there is no contradiction.

physical chemistry - Entropy - "Wiggle"?

The title is not a reference to a Jason Derulo song.

In any case:

1) How is change in entropy measured, experimentally? I've Googled this for a bit and I've found all sorts of mathematical equations such as delta S = q/T as a way to get a grasp on entropy. However, I want to know how entropy is actually determined in the lab (as opposed to on paper). I understand that the units on entropy are in terms of energy/temperature. So if I had to guess I would suppose that some sort of calorimetry is used.

2) What is the best way to teach entropy (at an introductory level)? I know that textbooks like to say it's a "measure of disorder." On the other hand I have lots of websites telling me that entropy is best NOT described at a measure of disorder. So what is it really, and what do you think about the term "disorder" as a synonym? My professor refers to it as the degrees of freedom in a system; he also notes that longer carbon chains tend to have higher entropy values due to a greater ability to "wiggle" along their carbon chains. What do you think of this?

Discarding the archaic idea of "disorder" in regard to entropy is essential. It just doesn't make scientific sense in the 21st century and its apparent convenience often is flat-out misleading. As of November 2005, fifteen first-year college texts have deleted “entropy is disorder” although a few still retain references to energy “becoming disorderly”. (This latter description is meaningless

Answer

You can't measure entropy directly, any more than you can measure interatomic distances. You measure other quantities -- for instance often you can measure energy gain/loss and temperature, and then you integrate $dS=dE/T$.

How to explain it? One of the best expositions I know is The Second Law by Henry A. Bent. It is full of insightful examples, lays the ground carefully and avoids woolly talk, unlike many other thermodynamics books.

Some elementary but valid comments about the link between information-theoretical entropy and thermodynamical entropy can be found here: http://www.madsci.org/posts/archives/1998-10/909712896.Ph.r.html

Edit: Greg reminds us that energy and enthalpy are not directly measurable either. That's why I was careful to write energy gain/loss, which are somewhat more accessible to measurement; but of course even when doing a calorimetric experiment you're not measuring heat per se but other quantities: how much gas you burned or how intense the current was and how long it was left on. And even some of these quantities again are indirectly measured.

I agree 100% that "entropy is not more abstract than energy" -- they are all abstractions. Ultimately, however, I do think the measurement of thermodynamic entropy is one step more indirect than that of energy, if only because it involves the notion of temperature, which (if you think about it carefully) is subtle indeed.

calendar - What is a ger's birthday in terms of Jewish astrology? conversion date or birth date?

In terms of Jewish astrological signs and influences on people, per Kabbalah, would a ger use their actual 'birth' date or their date of conversion which is considered a rebirth?

particles - Usage of か after a clause?

Here's a sentence I found:

デパートはどこにあるか、知っていますか？
Do you know where the department store is?

What's the purpose of the か particle in 「どこにあるか」? Under what circumstances do I use it?

I'm fairly sure it's not the か that's normally used to form questions, like the second か in that sentence, nor is it the か that means “or.”

tefilla - Explain apparent contradiction regarding "binah" in the bracha "Chonen Hada'at"

The Bracha Chonen Hada'at seems to have an apparent contradiction regarding "binah".

The beginning of the blessing separates "da'at" from "binah" by implying that "G-d gives man graciously da'at and teaches a man 'binah'"

My rav explained this distinction. "Da'at" - wisdom is a gift that G-d gives us. But 'binah' is the ability to distinguish things such as good vs. bad / right vs. wrong. That must be learned and G-d helps man learn this.

OK - I understand this explanation. However, at the end of the bracha, we ask G-d to "Give us graciously ...'binah'". So it seems that we are asking G-d to give us 'binah' as a gift, not to teach it to us.

Is this contradictory, or what is the meaning implied by the request at the end of the bracha?

NOTE: The link uses Nusach Ashkenaz. The phrasing in Nusach Sfard is different, but still implies the same idea.

Answer

In the Siddur Shai LaMora he explains that Bina is earned through toil (as your rav explained) and then goes on to say that in the second half we are asking that despite the fact that it requires this work and effort, we are still asking for it to be given for free (as in Rashi's explination of VesChanan).

So we introduce it - you give Da'at and give us the ability to toil to earn Bina. Please give us both for free.

Perhaps we can understand this according to a story between the Tzemach Tzedek and the Alter Rebbe of Lubavitch. At one point, the Alter Rebbe offered the Tzemach Tzedek to give him a blessing for great insight into Torah, and the Tzemach Tzedek demurred, saying he wanted to earn the Torah through his own efforts.

When he was older, the Tzemach Tzedek said he regretted that decision, since in Torah there are always higher levels to earn, and he could have been so much further.

hashkafah philosophy - Why does belief in Judaism require irrationality?

In conversations with Rabbis, and in watching debates between Rabbis and atheists, I have often found that the response to questions which attack Judaism either on the basis of some of its perceived picayune laws (for example, “Why does God care what food we eat?”) or on the ramifications of its theology (for example, “Over the history of civilizations, Jews have historically been very small in number; why would God want only a minor percentage of his creation to experience ‘the truth’ and serve him properly?”), has been some version of “there are some parts of the Jewish faith which require a person to go beyond rational belief and simply have faith”.

My basic question is, Why would God create Judaism in such a way as to require irrational beliefs? If the goal is to lead a moral life in the face of temptation and to serve God, why not create Judaism in such a way as to be 100% rational? People would still have free will to do bad things, act irrationally, etc.

words - What is a hakira?

In the parlance of Talmud study and lomdus, what does the word "Hakira" mean and how should it be used? Why are Hakiras useful?

suffixes - Is ～がる suffix limited to specific adjectives only?

I noticed that there are some adjectives that have ～がる suffix to make them into verbs. Some examples from WWWJDIC:

ほしがる
こわがる
いたがる
くるしがる
さびしがる
うれしがる

Apparently ～たい form also can take the ～がる suffix to become ～たがる suffix:

いきたがる

Rikaichan popup explains the ～がる as:

to feel (on adj-stem to represent third party's apparent emotion); to behave as if one were

So I thought that I could append this suffix to any adjectives to add those connotations. However, it seems that not all adjectives and ～たい have the suffix. WWWJDIC does not return any exact matches for the following words:

やさしがる (to behave as if one were nice?)
きたがる (to feel that one wants to come? - Since いきたがる exists so logically きたがる should exist too?)
ねむがる (to behave as if one were sleepy?)

My question is, as in the title, is ～がる suffix limited to specific adjectives only? How do we know which adjectives can have the suffix and which adjectives can't?

Answer

As explained by the Rikaichan popup you reference in your question, ～がる is a suffix for representing a third party's apparent emotion. As such, you cannot use it with やさしい or ねむい as these are not adjectives which describe emotions.

As it was explained to me, you cannot generally know another person's emotions or thoughts, and so when speaking of them, you use either ～がる (e.g. うれしがる) or ～そう (e.g. うれしそう). To some extent, this is true even in English -- "John seems sad today" seems more generally applicable than "John is sad today" (without John having said so himself).

To answer your question: you use this for adjectives (including ～たい) which represent emotional or psychological states. It's very difficult to provide an exhaustive list of such adjectives.

chanuka - 9 Days of Chanukah?

All Yomim Tovim we celebrate in Chutz Laretz an extra day because of A safek, why do we not do the same for Chanukah?

Answer

Minchas Chinuch argues that indeed, when the Sanhedrin was functioning and we used an observation-based calendar, Chanukah in outlying places would have had to have been celebrated for nine days. "When the Beis Hamikdash is rebuilt, speedily in our days," he says, "and we go back to sanctifying the months based on observation - then faraway places (for Eretz Yisrael will spread out to include all other lands) will surely have to keep Chanukah for nine days."

history - Known birthdays

In the season of my birthday, I was wondering:

Which figures, from Tanach and Sha"s, do we know the exact birthday of? I am aware of Moshe's birthday being identified as the 7th of Adar. Which other figures have known birthdays?

I am specifically and only interested in knowing which figures have birthdays that are documented in Jewish sources.

I am more interested in those sources which explicitly identify a birthday, but am also interested in sources which figure out an exact date based on analysis of sources.

grammar - What is a subsidiary verb?

What is a subsidiary verb? Could one explain it and give some examples of its application?

Answer

Subsidiary verbs, known as 補助動詞（ほじょどうし） in Japanese grammar, are a small set of verbs which have grammaticalized uses following 〜て. According to Martin^†, these verbs include:

いる・おる・いらっしゃる
くる・まいる
いく

くれる・くださる
しまう
みる
おく
もらう・いただく
ある・ございます
やる・あげる
みせる

In these grammaticalized uses, they have several properties:

1. 
   

   They form a single predicate with the verb they follow.

   
   

   
   

   歩いている is usually a single predicate, not "walks and is".

   
   

   
   


2. 
   

   They no longer have their literal meaning (although their grammaticalized meanings are generally related to their literal meanings). The 〜ている ending, for example, expresses aspectual meaning rather than expressing existence.

   
   

   
   
   

   歩いている is usually the progressive "is walking".

   
   

   
   


3. 
   

   Informally, they often contract with 〜て (see my previous answer for a chart).

   
   

   
   

   歩いている can be contracted informally to 歩いてる.

   
   

   
   


4. 
   
   

   Because they're used grammatically rather than for their literal meaning, they're generally written in kana:

   
   

   
   

   歩いている should not be written 歩いて居る.

   
   

   
   

   Keep in mind, though, that these verbs aren't always used as subsidiary verbs. When they're used as regular verbs, they can in most cases be written with kanji.

   
   

Subsidiary verbs are often referred to as auxiliary verbs, but this can unfortunately cause confusion because the same term is used as a translation of 助動詞（じょどうし）, an unrelated class of inflecting auxiliaries in traditional Japanese grammar including words such as 〜ない and 〜ます.

For this reason, many people choose to avoid confusion and use the unambiguous term subsidiary verbs instead.

---

^† This list from page 512 of Samuel Martin's 1975 Reference Grammar of Japanese. Subsidiary verbs are described in more detail in this book on pages 510-551.

physical chemistry - Why does copper form bubbles in vinegar in this situation?

I watched this video in which the host tries to show the principles behind the voltaic pile.

The host first immersed zinc in vinegar and bubbles are observed forming around zinc. I've learned that this is because hydrogen ions in vinegar get electrons from zinc atoms and become gaseous.

Then the host take a conductor and attach at one end the piece of zinc and at the other a piece of copper. Bubbles start to form around copper too! (fig1)

• Why does this happen?


• Why do some electrons travel through the conductor instead of being attracted immediately by the hydrogen ions in the vinegar?


• What if the conductor instead was very long and the pieces of copper and zinc far apart? (fig2)

Answer

When you put a zinc rod in a acid solution containing $\ce{H^+}$ ions, the zinc piece loses $\ce{Zn^{2+}}$ ions to the solution leaving electrons behind. These electrons attract hydrogen atoms to the zinc rod, and an electron transfer occurs, resulting in bubbles of hydrogen forming on the surface of the zinc. However, the evolution of gaseous hydrogen from zinc surface is rather slow, and that is why pure zinc reacts slowly with dilute acids.

The situation changes when a copper,or any other metal less electropositive than hydrogen is attached to zinc, by a wire. The copper can now be considered as an extension of the zinc rod: a place form where hydrogen can take electrons quickly and become gas. As the rate of evolution of hydrogen from the copper piece is faster, hydrogen bubbles out from the copper piece. This does not mean, however, that the zinc stops reacting with acid; it does, but at a very slow rate.

A common mistake is to think that the copper rod participates in the electrochemical reaction taking place. The copper is merely a pathway to release electrons to H+ ions in the solution. The only reaction taking place is

$\ce{Zn + 2H^+ -> Zn^{2+} +H2\uparrow}$

Copper has no place in the reaction.

So, to answer your questions,

Why does this happen?

This happens because copper can do away with the electrons quicker than zinc, and when electrons get a pathway, they will take the pathway.

Why do some electrons travel through the conductor instead of being attracted immediately by the hydrogen ions in the vinegar?

Well, some electrons do attach to H+ ions from vinegar directly, and hydrogen bubbles should form on the zinc plates, although they are less in amount. And I believe my earlier explanation answers this question.

What if the conductor instead was very long and the pieces of copper and zinc far apart?

It does not have anything to do with distance. If you take a large solution, the reaction will certainly occur. On the other hand, if you separate the poles into two solutions the copper rod will not give off electrons as the circuit is not complete. Think about it this way, if the reaction was to occur, the solution with zinc will become progressively positive and the solution with copper will become negative. This can never happen.

Footnotes:

(1) As per your comment below, I now attempt to answer the question "What makes copper better than zinc in releasing electrons?"

The answer lies in a factor called overpotential, sometimes referred to as overvoltage. There is one type of overpotential which is important in this case— activation overpotential. This is caused due to the fact that when hydrogen atoms are produced on the surfce of a metal ($\ce{H+ +e-> [H]}$), they take some time before they can combine to form diatomic hydrogen and some more time before the hydrogen molecules can be released as gas. This results in the formation of a layer of hydrogen on the surface of metal effectively reducing the rate of reaction and halting it; this effect is known as polarization.

As you can see from the data chart in the Wikipedia page, Zn electrode has -0.77 V overpotential for evolution of hydrogen while there is no overpotential for copper electrode. This means that copper is not polarized easily.

(2) I mentioned in the beginning of my answer that pure zinc reacts with dilute acid slowly. Commercial zinc, on the other hand, reacts quickly with acids, owing of the same effect of polarization. In commercial zinc, impurities of other metals form cathode and the pure zinc forms anode in the acid medium, essentially producing a galvanic cell, where the anode and cathode are connected. Thus the reaction occurs quickly, without the hindrance of polarization.

(3) You should read this paper : http://pubs.acs.org/doi/abs/10.1021/ed078p516. It has some facts which you may find interesting.

fft - DFT as convolution question

I have tried to make this question as readable and consistent as possible. The short of it, is that I am trying to ascertain how one gets from the math equation shown, (which I understand), to the correct implementation. (Which I also understand). However, how/why one gets from one to the other seems unclear to me.

Here is my setup:

We have an $M$ length 'original' signal $x[m]$: (Equation 1)

$$ x[m] , \space m = 0,1,..M-1 $$

We wish to take a 'zoom-DFT' of it, where the DFT length will be, $N$, where $N > M$.

We thus now have an $N$ length zero-padded new signal, lets call it $x_p[n]$, where: (Equation 2) $$ x_p[n], \space n = 0,1,...N-1 $$

Our discrete time variable is now $n = 0,1,...N-1$, and our discrete frequency variable is $k = 0,1,...N-1$.

The $N$-length DFT of a signal is defined as such: (Equation 3)

$$ X[k] = \sum_{n=0}^{N-1} x_p[n]\space e^{-j\frac{2\pi nk}{N}} $$

If the $nk$ in the DFT equation is written as $\frac{-(k-n)^2 + n^2 + k^2}{2}$, then the DFT can now be written as: (Equation 4)

$$ X[k] = e^{-j\frac{\pi k^2}{N}} \space \sum_{n=0}^{N-1} x_p[n]\space e^{-j\frac{\pi n^2}{N}} \space e^{j\frac{\pi (k-n)^2}{N}} $$

This expresses the full DFT, (i.e, the DFT from 0 to nyquist), as a convolution. To evaluate the 'Zoom-DFT', where we want to interrogate only frequencies between $f_1$ and $f_2$, and so the following minor modification is made: If we let $F_{\Delta} = \frac{(f_2 - f_1)}{f_s}$, and $F_1 = \frac{f_1}{f_s}$, then the zoom-DFT, $X_{z}[k]$ becomes: (Equation 5)

$$ X_z[k] = e^{-j\frac{\pi F_{\Delta} k^2}{N}} \space \sum_{n=0}^{N-1} x_p[n]\space e^{-j\frac{\pi n^2}{N}} \space e^{-j 2\pi \space F_1 n}\space e^{j\frac{\pi F_{\Delta} (k-n)^2}{N}} $$

We can then simplify, if we let: (Equation 6)

$$ a[u] = x_p[u]\space e^{-j\frac{\pi u^2}{N}} \space e^{-j 2\pi \space F_1 u} \\ b[u] = e^{j\frac{\pi F_{\Delta} u^2}{N}} $$

, where $u$ is just some general dummy variable, (you will see why I picked it later). Anyway, now finally, we simply have: (Equation 7)

$$ X_z[k] = b^*[k] \space \sum_{n=0}^{N-1} a[n] \space b[k-n] $$

This, as we are told here, and here, page 183, is how the DFT can be re-written as a convolution.

Great! So far so good. The problem comes in interpretation vs implementation.

The problem:

My interpretation of (Equation 7), is that $a[u]$ is of length $N$, and $b[u]$ is also of length $N$, where we evaluate then for $u = 0,1,...N-1$. Then, we do a circular convolution of the series $a[u]$ and $b[u]$, to attain a series also of length $N$, (which is our desired DFT length), before we finally perform an element-by-element post-multiply with another series, also of length $N$. I dont so much care about post-multiply, that is not the point of this question. The point here is that $a[u]$ is length $N$, $b[u]$ is also length $N$, and $u$ for both of them is evaluated for $0 \leq u \leq N-1$, and then we do a circular convolution, modulo-$N$.

If you however now take this interpretation and implement it, you will not get the right answer. In fact, you will only ever get the right answer, if the $u$ variable for evaluation of $a[u]$ is taken from $u = 0,1,...N-1$, (same as above), BUT, the $u$ variable for evaluation of $b[u]$ is taken from $-(M-1) \leq u \leq N-1$. So now the series $b[u]$ is not length $N$ anymore, it is of length $N+M-1$. Then, if we do a modulo-($N+M-1$) circular convolution, and extract the indicies from $M$ to $M+N-1$, we will get the right answer.

...How, from equation-1, does one possibly get this 'correct' interpretation though? This, I cannot seem to explain, however I understand everything else.

FWIW, I have sat down and done the computation 'by hand', so I can 'see' why $b[u]$ MUST be taken from $-(M-1) \leq u \leq N-1$. (It would otherwise be impossible to compute the DFT for $k = 0,1,...N-1$). However I do not understand how one goes from (Equation 7), to this conclusion. Right now, to me, EQ-1 just looks like a garden variety circular convolution, with no hints as to the details for now $a[u]$ and $b[u]$ must be exactly evaluated.

Put another way - AFAIK, when we see equation-1, we simply assume that $a[u]$ and $b[u]$ are evaluated from $0 \leq u \leq N-1$, and then we do a circular convolution, modulo-$N$ - how/why then do we derive the proper evaluations for $a$ and $b$ from this equation??

Thanks!

Answer

You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in Equation 7 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of $a[u]$ and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as $X[N-1]$.

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].

There is an interesting write up of Chirp-Z along with Goertzel's algorithm that I think puts this in proper perspective: http://www.google.com/url?sa=t&rct=j&q=goertzel%20algorithm&source=web&cd=3&sqi=2&ved=0CEIQFjAC&url=http%3A%2F%2Focw.mit.edu%2Fcourses%2Felectrical-engineering-and-computer-science%2F6-341-discrete-time-signal-processing-fall-2005%2Flecture-notes%2Flec20.pdf&ei=rawjUd6sM-LE0QHJ2IGQAQ&usg=AFQjCNEFEo7R7VSmk5jWYon0f4WqZYpvzg&bvm=bv.42553238,d.dmQ

meaning - The の in のに and なのに

I've heard that the の in のに and なのに is the general-noun　の (I don't remember the word for it.). So why, in that light, does the meaning of the two make sense?

Answer

The particle construction　～（な）のに expresses the adversative, i.e. in English (al)though, even though, etc.

The の in ～のに and ～なのに is a suffix that functions as a nominalizer. の turns any inflected expression into a noun, and なの does the some for expressions that cannot be inflected. This happens in order to make the attachment of grammatical markers possible that usually do not directly attach to the expressions, or that do so with a different grammatical meaning. The に is dative/locative case, which only appear with nominals.

 攻撃されたのに、冷静に行動した。
 Even though (he) was attacked, he acted calmly.

攻撃された is an inflected expression (a passivized verb), hence のに.

 重要な発見なのに、無視された。
 Though an important discovery, it was ignored.

発見 cannot be inflected, hence なのに is required.

Some linguists believe that the separation of のに in の and に is not entirely correct because both particles must be present in order to express the adversative. The adversative meaning also does not follow from meaning composition, i.e. combining the nominalizing property of の with dative/locative に does not result in an adversative reading.
That の has a nominalizing property can be seen in other examples, such as

 太郎が行かなかったのを知らなかった。
 (I) didn't know that Taroo didn't go.

The clause 太郎が行かなかった 'Taroo didn't go' is nominalized, and hence the accusative case particle を can attach.

The answer to your question then is: の in ～（な）のに is a nominal particle that produces nouns, or nominal expressions. Therefore, some linguists call it a particle noun (e.g. Jens Rickmeyer). 　　

halacha - May we still say Elohim and other names of God in everyday speech?

My experience (at least in the Orthodox Ashkenazic community) is that it is common knowledge that we may not say Elohim in everyday speech. On the other hand, it seems like in many cases in Tanach or Talmud the name Elohim or similar is used in conversational terms or at least not directly in a blessing.

Is this true Halacha as many Ashkenazim appear to think it is, is it more of a minhag that has the force of Halacha, or is it not true?

And if it is forbidden as many seem to think it is, then why can't we say God's name? Also is there one name that is forbidden while other terms are not? For example many places and names include "el" in them and that seems to be fine.

Also, in practice, if you speak Hebrew and want to teach a child about God, do you actually use a fake word like "Elokim" to discuss it?

How is giyur le’chumra different from a standard conversion in practice?

I am interested to know how giyur le’chumra is different from a standard conversion in practice? Does anyone have any personal experiences?

My understanding of the meaning of giyur le’chumra (from an article I found on Torah Lab) is as follows:

Giyur le’chumra is a term which refers to conversions performed as precautionary measures. They are undertaken when a doubt exists about one’s Jewishness or about the validity of his conversion. Such conversions involve accepting the commandments before a rabbinical court, and immersing in a mikvah.

Answer

This answer is based off of experience of being present at a giyur l'chumra conducted by R' Yosef Berger. The Beis Din does not remind the convert, as is normally done, that this is his last chance to change his mind and not become a Jew, as we are already assuming that he is a Jew, and the conversion is "just in case." In the conversion which I witnessed, they did not inform him about mitzvos kalos v'chamuros (some "light" and some "serious" mitzvos), for what I assume was the same reason - because we don't want to and aren't trying to scare him/her off.

Normally, a convert makes a blessing on going to mikvah, and a shehecheyanu. The giyur l'chumra convert makes neither of these blessings, although I suppose it couldn't hurt for him to bring a new fruit with him and just have it in mind.

The convert was not asked to choose a name for himself, as normally converts do take a name upon conversion. I do not know if this was because it is assumed he wouldn't want to change the name he had been going by as a Jew, or because we aren't interested in changing his name since we are assuming he already was a Jew.

discrete signals - my Butterworth lowpass formulas do not agree with Fisher webpage

I want to implement Butterworth low-pass filter. Thanks to this question, I have found out that the filter coefficients can be generated using Tony Fisher web-site or using his code. But the problem arose when I had tried to verify his formulas myself.

Wikipedia says that the derivation of low-pass formulas is simple: we start with the Butterworth polynomial of order n. I took n=1 $$B_1\left(\frac{s}{\omega_{cutoff}}\right)=1+\frac{s}{\omega_{cutoff}}$$

(note that $\omega_{cutoff}$ is angular frequency, not usual one) then do bilinear transform $$s = 2f_{sampling}\cdot\frac{z-1}{z+1}$$

($f_{sampling}$ is usual frequency, not angular) and rewrite the resulting fraction in the form with non-positive powers of $z$.

To make the story short, my final formula for transfer function is $$H(z) = \frac{Y(z)}{X(z)}$$ where $$Y(z) = \frac{a}{1+a}+\frac{a}{1+a}z^{-1}$$ $$X(z) = 1-\frac{1-a}{1+a}z^{-1}$$ and $$a = \frac{\omega_{cutoff}}{2f_{sampling}}$$ and the resulting formula is $$y[n] = \frac{a}{1+a}x[n]+\frac{a}{1+a}x[n-1]+\frac{1-a}{1+a}y[n-1]$$ For testing I use $f_{cutoff}=1$ (which is $\omega_{cutoff}=2\pi$) and $f_{sampling}=30$.

We should not worry about coefficients in front of $x[]$ since Fisher multiplies them by gain factor anyway, but the coefficient in front of $y[n-1]$ equals

$$\frac{1-a}{1+a} = 0.8104139027$$

while Fisher's web-page for $f_{cutoff}=1$ and $f_{sampling}=30$ gives $0.8097840332$.

If you had a patience to finish this reading, may be you can explain, where I (or Fisher?) am wrong.

Answer

The problem is in the way you apply the bilinear transform. You have to use the appropriate (pre-)warping of the frequencies. Since the bilinear transform warps the frequency axis, you have to make sure that the corner frequency of the discrete-time filter is correct. One way to do that is as follows. The bilinear transform is defined as

$$s=k\frac{z-1}{z+1}\tag{1}$$

with some constant $k$ yet to be defined. If we denote the analog frequency by $\Omega$, and the normalized discrete-time frequency by $\omega$, Eq. (1) becomes for $s=j\Omega$ and $z=e^{j\omega}$

$$j\Omega=k\frac{e^{j\omega}-1}{e^{j\omega}+1}=k\frac{e^{j\omega/2}-e^{-j\omega/2}}{e^{j\omega/2}+e^{-j\omega/2}}=jk\tan(\omega/2)\tag{2}$$

Eq. (2) describes the frequency warping caused by the bilinear transform. If we use an analog lowpass filter with a normalized corner frequency $\Omega_0=1$, we must choose the constant $k$ such that for the desired discrete-time corner frequency $\omega_0$ the term on the right-hand side of Eq. (2) becomes $1$:

$$k=\frac{1}{\tan(\omega_0/2)}\tag{3}$$

where $\omega_0$ is the desired corner frequency of the digital filter. Eq. (3) and Eq. (1) define the appropriately normalized bilinear transform that you must use.

So for your example, the normalized first-order analog Butterworth lowpass transfer function is given by

$$H(s)=\frac{1}{1+s}\tag{4}$$

Applying the bilinear transform gives

$$H(z)=\frac{1}{1+k\frac{z-1}{z+1}}=\frac{z+1}{z(1+k)+1-k}=\frac{1}{1+k}\cdot\frac{1+z^{-1}}{1+\frac{1-k}{1+k}z^{-1}}\tag{5}$$

Ignoring the gain term $1/(1+k)$, the corresponding difference equation is

$$y[n]=x[n]+x[n-1]-\frac{1-k}{1+k}y[n-1]\tag{6}$$

With a desired corner frequency $\omega_0=2\pi/30$ you get from (3) $k=9.5144$ and $(1-k)/(1+k)=-0.80978$, just like on Fisher's website.

verbs - Meaning of "生まれし"

I wanted to ask this question because it is the first time I have honestly been unable to find any information on a given form. I know it involves the verb "to be born," but I've never seen a stem+し form. For some more information, the full line is:

人{ひと｝の世｛よ｝に 生｛う｝まれし頃｛ころ｝より 戦｛いくさ｝道｛みち｝

For those curious, the quote comes from a game called Xenoblade. Could someone explain what construction "生まれし" uses and possibly give a rough translation of the whole thing? Also, I don't know many kanji, so using kana in answers would be appreciated.

Answer

「し」 is the [連体形]{れんたいけい} (attributive form) of the retrospective auxiliary verb 「き」. 連体形 modifies nouns (頃 in this case).

Even though 「き」 is a Classical auxiliary verb, it is listed in any medium-sized dictionary of Modern Japanese because it is still used today in creative writing where the author's aesthetic preference calls for the old-fashioned and/or literary kind of taste.

https://kotobank.jp/word/%E3%81%97-515145#E5.A4.A7.E8.BE.9E.E6.9E.97.20.E7.AC.AC.E4.B8.89.E7.89.88

「生まれし頃」＝「生まれた頃」= "around the time I was born"

「人の世に生まれし頃より戦道」, therefore means:

"I have always walked on the road of war since I was born."

physical chemistry - What exactly is happening when sodium, potassium, or other alkali metal explodes in water?

There are lots of videos on YouTube showing sodium, potassium, etc. exploding when dropped into water (this, for example).

I understand that when an alkali metal is exposed to water, a violent exothermic reaction occurs where a hydroxide and hydrogen gas are produced, but why and how does the sample of metal end up detonating and fragmenting? Physically speaking, how can a block of metal seemingly blast apart from the inside when the reaction occurs on the surface of the sample?

The Wikipedia article on alkali metals explains this, but I still don't seem to understand how this would result in an explosion. Diagrams would be very helpful.

The key of the question is how does the release of hydrogen gas and energy result in an explosion? Is there thermal runaway, and is the metal physically destabilized in some way during the reaction?

Answer

Until recently the answer was unknown, but a short time ago it was discovered that the reaction is in fact a Coulombic explosion. The rapid exchange of electrons between the sodium and the water causes the surface of the sodium droplet to become positively charged, and the ions repel each other. This behaves very like a negative surface tension, and the surface of the droplet increases in area rapidly forming a spiny, porcupine like shape as fingers of the molten metal are shot into the liquid at astonishing speed. The larger the surface area gets the faster the reaction occurs, leading to a runaway effect. The study was published in Nature chem.

http://www.nature.com/nchem/journal/vaop/ncurrent/full/nchem.2161.html

nomenclature - Why is it ACETone?

Acet* indicates Ethyl, but does not offer two Carbon atoms, but three.

Acetaldehyde is Ethanal, Acetic acid is Ethanoic acid, but Aceton is Propanon (yes, I'm aware that there is no Ethanon).

Why is Acetone called Acetone?

Answer

Here's a link to the first page of a book entitled, "The History of Acetone, 1600-1850" by Mel Gorman. The author points out that acetone was known in the Middle Ages and was frequently produced by heating dry lead acetate. I suspect that the "acetate" (or whatever the Latin, French or German term was) root stuck and then it was just modified a bit - to acetone - to make it distinct.

halacha - Opening plastic soda bottles on Yom Tov

In regards to opening plastic soda bottles on Shabbos I'm sure everyone knows there is a maklokes in this. Is the same maklokes applicable on Yom Tov, or is Yom Tov different so that according to everyone it is muter to open the bottles? (My sofek (doubt) is based on there being a heter to do some malachos on Yom Tov l'tzrich ochel nefesh (for the purpose of food for humans).)

theology - Who owns Gehenem (Hell)?

Who owns Gehenem (also known as Hell)?

acid base - Why is HClO4 the strongest oxyacid?

Why is $\ce{HClO4}$ the strongest oxyacid?

My effort: The structure of $\ce{HClO4}$ should be tetrahedral. So when there is a negative charge on oxygen, it should not be in resonance with other oxygen atoms as a condition for resonance is that the atoms taking part in resonance should lie in a plane.

Answer

Assume an acid $\ce{HA}$. An acid is strong if

1. protons are donated easily, i.e., the $\ce{H-A}$ bond is readily broken;


2. the conjugate base $\ce{A-}$ is stable, i.e., $\ce{A-}$ is reluctant to combine back to $\ce{HA}$.

When are protons easily donated?

First parameter to consider is the polarity of the $\ce{H-A}$ bond. More polar bonds mean less electron density on the bond. Thus the dissociation equilibrium

$$\ce{{{HA_{(aq)}}+H2O_{(l)}}<=>H3O_{(aq)}+ + A_{(aq)}-}$$

is tilted strongly towards the right. For this reason, $\ce{HF}$ is a stronger acid than $\ce{NH3}$.

In addition, the size of $\ce{A}$ is important. If $\ce{A}$ is very big, essentially it is harder to hold on to the hydrogen. Thus the $\ce{H-A}$ bond is weaker. This is why $\ce{HI}$ is a better acid than $\ce{HF}$.

Stability of the conjugate base

Charge delocalisation is key here. If all atoms bonded to a central atom are equal, then the hydrogen ion does not have a preferred location at which to bond. A large charge in a small volume would also translate into an unstable ion.

Oxoacids

1. Look at the number of oxygens bonded to the central atoms. More oxygens -> stronger acid, e.g., $\ce{HNO3}$ is stronger than $\ce{HNO2}$.


2. If the number of oxygens is equal, consider the electronegativity of the central atom. Higher electronegativity -> stronger acid. For instance, $\ce{HBrO3}$ is stronger than $\ce{HIO3}$.


3. When electronegativity is similar as well ($|\Delta EN|\le 0.1$), it sometimes helps to look at the size of the central atom. For example, $\ce{H2SeO4}$ is stronger than $\ce{H2SO4}$.

The case at hand

Polarity of the $\ce{O-H}$ bond in $\ce{HClO4}$ is guaranteed via

• the electronegativity of the $\ce{O}$ atom itself;


• the electronegativity of the central atom $\ce{Cl}$;


• the electronegativity of the three additional $\ce{O}$ atoms.

Thus, the electron density is pulled from the $\ce{O-H}$ bond and the proton can be easily donated.

Here is a picture where you can compare the charge delocalisation of various chlorine oxoanions.

Clearly, the perchlorate ion $\ce{ClO_4-}$ is the most stable. Every valence orbital and valence electron of chlorine takes part in the bond formation.

Extra

This reply discussed relative acid strength with respect to Brønsted–Lowry acid–base theory (in water). Do not forget that these are pointers; there will be other parameters to consider and exceptions to take into account.

$\ce{HOF}$ is the single known fluorine oxoaxid. The analog $\ce{HFO4}$ is simply not possible. It would violate the octet rule.

EDIT: Answer to comment

You asked: "When you say charge delocalisation in ClO4- anion is it due to resonance? Is resonance possible?"

From IUPAC Gold Book: "[Resonance] refers to the representation of the electronic structure of a molecular entity in terms of contributing structures. Resonance among contributing structures means that the wavefunction is represented by 'mixing' the wavefunctions of the contributing structures."

Therefore, the answer is yes. The main resonance structures are given in the picture below.

The resulting ion is a resonance hybrid. In classical valence bond theory, charge delocalisation is not due to resonance; rather, resonance is delocalisation.

Furthermore, coplanarity is not a requirement. It is fair to say that there is (almost) always some delocalisation (again, see the definition given). Still, having atoms that lie in the same plane can be advantageous. Qualitative models of aromaticity and conjugation may even give coplanarity as a necessary condition. This is not always the case, however. Quantitative models are used to more rigorously predict aromaticity and antiaromaticity (see NICS, QTAIM models).

P.S. Resonance energy is important. High, positive resonance energy indicates increased stability; slightly positive or even negative values point to unstable particles. For instance, antiaromatic molecules generally have low resonance energies. And even though planarity is a prerequisite of qualitative antiaromaticity, too, it is not actually a rule, instead a rough indicator. Example: 2,3,4,5-tetraphenylcyclopenta-2,4-dienone here and there.

How do I apply a function window to a signal?

How can I apply a window function like Hamming or Lanczos to a signal, using its coefficients?

I mean, which method can I use to do this? FFT? Convolution? Which method has the better performance?

Answer

I wrote this long answer for someone yesterday on stackoverflow.com . .

https://stackoverflow.com/questions/9694297/matlab-fft-xaxis-limits-messing-up-and-fftshift/9699983#9699983

It is a matlab based example showing how to use the FFT for analysis, but it might give you some ideas About half way through the second code block, I apply a window function to a buffered signal. This is effectively a vector multiplication of the window function with each buffered block of time series data. I just use a sneaky diagonal matrix trick to do it efficiently.

isotope - Are stronger bonds always shorter?

This question relates to a few earlier questions I have seen on the site (particularly this one about the bond strength of heavier isotopes). Does a bond necessarily have to be shorter to be stronger?

To my understanding, length and strength are definitely correlated, but I don't think it is necessarily a one to one correspondence. The examples I brought up in the comments of ron's answer in the linked question related to data tables that seemed to show no difference in bond length for deuterium substituted bonds despite significant changes in energy. (HCl NIST, DCl, data table)

To expand on my question, if length and strength are always directly related:

1. How do we explain experimental and theoretical cases where the length doesn't differ with bond energy?


2. If there is some direct relationship between length and strength, what is that relationship?

For the sake of not being too general, I'm looking for an answer focused on comparing bonds of the form $\ce{A-B}$ and $\ce{A-C}$ (similar to the examples I asked about in the linked question). So for a more specific case related to the linked question, does substituting a protium for a deuterium always make the bond shorter in addition to being stronger?

Answer

This is an interesting question and depending on how you define bond strength the answer is different. Let us for simplicity consider only diatomic molecules and let us assume that the electronic potential between the two atoms is well described by a Morse potential

\begin{align} V(r) &=D_e \left( 1 - \mathrm{e}^{-a(r-r_e)} \right)^2,& \text{with } a &=\sqrt{\frac{k_e}{2D_e}}. \end{align}

Here $D_e$ is the depth of the potential (at the minimum position $r=r_e$) and $k_e$ is the (harmonic) force constant. The potential depth is related to the dissociation energy $D_0$ by

\begin{align} D_0 &= D_e - \frac{1}{2}\omega_ehc,& \text{with } \omega_e &=\sqrt{\frac{k_e}{\mu}}, \end{align}

where $\omega_e$ is the harmonic wavenumber, $\mu$ is the reduced mass and $h$ and $c$ have their usual meaning.

We can define the strength of the bond by the magnitude of the dissociation energy $D_0$ or by the "spring constant" $k_e$ of the bond. In your example with deuterium you implicitly assumed the former definition. In the Born-Oppenheimer Approximation the potential (i.e. the Morse potential) does not depend on the mass of the atoms (they are assumed to be infinitely heavy) and the potential is the same for $\ce{H2}$ and $\ce{D2}$. However, because deuterium is heavier than hydrogen, the harmonic frequency is lower and therefore the dissociation energy is larger ($\ce{D2}$ has a smaller zero point energy). Using this definition, the bond length is not easily related to the bond strength (as the reduced mass of the system plays a role as well).

As stated earlier, we can also use $k_e$ as a measure for the bond strength. The larger the value of $k_e$, the steeper the harmonic part of the potential well and the more localized the nuclear wavefunctions will be. In other words, if we take $k_e$ as a measure of bond strength, then stronger bonds are indeed shorter.

As stated by @porphyrin, the separation between $k_e$ and $D_e$ is not very strict (see also the formula's above) and the explanation above implicitly assumed a constant $D_e$, just as we assumed constant $k_e$ for the different isotopologues.

hashkafah philosophy - What if our purpose is to be bad?

How do we know that our purpose is not to be bad? Surely evil men have, by extension, caused some good. And even without that idea, we have no use what our purpose is on this earth -- how do we know that us being bad isn't part of the divine plan?

Or, put another way, how can it be both that God wants us to follow His will, and also evil/bad people be part of the divine plan?

Are there any sources that discuss this idea?

Answer

Devarim 10:12

יב וְעַתָּה, יִשְׂרָאֵל--מָה יְהוָה אֱלֹהֶיךָ, שֹׁאֵל מֵעִמָּךְ: כִּי אִם-לְיִרְאָה אֶת-יְהוָה אֱלֹהֶיךָ לָלֶכֶת בְּכָל-דְּרָכָיו, וּלְאַהֲבָה אֹתוֹ, וְלַעֲבֹד אֶת-יְהוָה אֱלֹהֶיךָ, בְּכָל-לְבָבְךָ וּבְכָל-נַפְשֶׁךָ

What does Hashem want from you? To fear the Lord your G-d, to walk in His ways, to love him, to serve the Lord your G-d with all your heart and all your soul

and many other such verses.

Hashem wants us to follow in His ways and serve Him.

Hashem does not want us to be evil and His plan does not necessitate such. He gives us free will and allows us to be evil. However, in His ultimate control, He weaves all of the choices that we make into His divine master plan, in a way that we cannot comprehend (Ramchal - Da'as Tevunos siman 54). In the time of Moshiach it will be revealed to us how everything fit in exactly to Hashem's plan, but while in this world we cannot comprehend it (ibid).

gentiles - Who do B'nei Noach turn to for psak?

My understanding is that there are a few organized groups of non-Jews who strive to follow the Noahide laws. Just like Jews, B'nei Noach might have situations where the correct action to take is unclear. When Jews have such a problem, they ask a competent halachic authority their question. Presumably, there must be an equivalent type of "posek" who can answer the questions of the non-Jews and determine what their halacha is.

Are there any people who are known as prominent poskim for non-Jews? Are these poskim rabbis or are they non-Jews?

Answer

There are a number of Rabbis, mostly Chabad, involved in teaching Bnei Noach and answering their halachic questions. One of the more prominent ones is Rabbi Yaakov Rogalsky, co-author of Path of the Righteous Gentile. Another is Rabbi Chaim Richman.

heresy - Can an apikores (heretic) count in a minyan?

Suppose there is a person who openly denies the divinity of the Torah, breaks the Sabbath and declares himself an atheist, but goes to synagogue for the sense of community or for cultural reasons. Can he be counted in the minyan, lead the prayers, or read from the Torah? I am asking of course from a traditional or Orthodox perspective.

halacha - Wife's obligation to submit to her husband for relations

Some people say only a husband is obligated to provide relations to his wife but the wife has no obligations. If she wants to she submits, if not she does not (since her husband is not allowed to rape her), and it is only a recommendation for her to submit as much as possible.

Others say she is obligated to submit.

What is the truth?

Please bring sources.

Related: What is the role of a wife towards her husband according to Jewish thought?

electrochemistry - Balancing redox equations in acidic media: to use H+ or H3O+?

Is it preferable to balance redox equations with $\ce{H+}$ or $\ce{H3O+}$? I think the latter, because first, the bare hydrogen proton doesn't exist in water solution. So why write down something we know not to exist?

I ask because in this problem, if we balance using both species, $\ce{H^+}$ and $\ce{H3O+}$, we achieve two different equations. Mass is conserved in both equations as well as charge; all the differs is one coefficient.

So are the two equations equivalent in that they are still describing the same reaction? I think so, because sometimes I see textbook authors write these two equations for $K_\mathrm{w}$:

$$\ce{H2O <=> H+ + OH-}$$

$$\ce{2H2O <=> H3O+ + OH-}$$

However, if we were actually going to run an experiment and needed exact quantities, we would probably prefer the second reaction both above and below because the coefficients in these reactions better represent the chemistry of the system right?

For example, which of the following is "better"?

$\ce{ClO_{3}^- + 6I^- + 6H^+ \leftrightharpoons 3I_2 + Cl^- + 3H_2O}$

$\ce{ClO_{3}^- + 6I^- + 6H_3^+O \leftrightharpoons 3I_2 + Cl^- + 9H_2O}$

Answer

This is a question of "style" so there is no correct answer. In fact, just as $\ce{H^+}$ doesn't really exist in solution, neither does $\ce{H3O+}$ exist in solution. In reality higher hydrates of $\ce{H^+}$ are what exist in solution. See the first paragraph in this Wikipedia article. Again, both $\ce{H^+}$ and $\ce{H3O+}$ are just shorthand notations for reality, choose the style that you like best.

halacha - Can and should I report cheaters?

Sometimes during an exam, the Rabbi or professor leaves the room. This provides an opportunity (which some students take) to openly and publicly cheat. (These students are not wearing kipot nor are Jewish.)

Is there any halachic prohibition or recommendation to report this behavior?

Students aren't officially ranked against one another, and the professor probably won't curve the exam.

What should the halachic onlooker do?

talmud gemara - Proof from Arvos Moav to Har Sinai regarding who was there

גמרא שבת קמו עמוד א

ישראל שעמדו על הר סיני פסקה זוהמתן גוים שלא עמדו על הר סיני לא פסקה זוהמתן אמר ליה רב אחא בריה דרבא לרב אשי גרים מאי אמר ליה אף על גב דאינהו לא הוו מזלייהו הוו דכתיב (דברים כט, יד-טו)ואת אשר ישנו פה עמנו עומד היום לפני ה׳ אלהינו ואת אשר איננו פה וגו׳

Gemara Shabbos 146a

When the Jewish people stood at Mount Sinai, their contamination ceased, whereas gentiles did not stand at Mount Sinai, and their contamination never ceased. Rav Aḥa, the son of Rava, said to Rav Ashi: What about converts? How do you explain the cessation of their moral contamination? Rav Ashi said to him: Even though they themselves were not at Mount Sinai, their Mazal [guardian angels?] were present, as it is written: “It is not with you alone that I make this covenant and this oath, but with he that stands here with us today before the Lord our God, and with he that is not here with us today” (Deuteronomy 29:13–14), and this includes converts.

Question: The passuk which the Gemara cites as proof that the Mazal [guardian angels?] of the geirim, (converts) were at Har Sinai, is not discussing Har Sinai at all, but rather the gathering at Arvos Moav at the end of the 40 years in the desert. So how can the Gemara use this as proof for what took place at Har Sinai?

Answer

Like is explained by the Ba’al HaTurim to Devarim 29:9, Chazal see the same language usage as in Shemot 19:17 and through that make the equation.

אתם נצבים היום גו׳

ויוצא משה את-העם לקראת האלהים מן-המחנה ויתיצבו בתחתית ההר.

There is an amazing discussion of this subject in the introduction to the commentary היכל הברכה by the Komarna Rebbe and properly understanding the nature of why the future generations have any obligation in regard to keeping the mitzvot. It is worth checking out.

translation - What does the expression くそかわ mean?

I read many times the expression くそかわ referred to babies, what does it exactly mean in english?

I know that くそ means "damn" while かわ comes from かわいい, right?

So damn cute?

Thank you!

Answer

Yes, right. くそ・クソ・糞 is used in place here for what usually would be すごく, e.g.

くそかわ = すごく可愛い
くそうめぇ = すごくうまい (=すごくおいしい)
etc.

(As slang is usually more versatile, there are more expressions with クソ, where すごく isn't a valid substition, e.g. 糞美人. Also, note that くそ isn't traditionally a positive interjection/prefix/..., but traditionally used to embellish derogatory terms.)

I'm not 100% on the precise ordering, but クソ is probably the most recent of a series of similar expressions

めっちゃうまい (from むちゃくちゃ)
まじうまい (from 真面目)
超{ちょう}うまい
鬼{おに}うまい
くそうまい

In any case, as you guessed, it is a stronger expression, just as in English:

くそかわ "damn cute"
すごくかわいい "really cute"

As you say, かわ is just a short form of かわいい, as common with all "i-adjectives", うまい => うま(っ), すごい => すご etc.

usage - Why censor this one kanji?

I was watching an old Saturday Night Live sketch from the 1970s, called Night of the Moonies, making fun of the Unification Church by putting them in the context of the movie Night of the Living Dead.

The video I was watching happened to have Japanese subtitles. I noticed that in the subtitles, when they referred to the Unification Church, which is called 統一{とういつ}教会{きょうかい} in Japanese, they replaced 一 with ◯, so it looks like 統◯教会.

My understanding of the use of ◯ is that it is the equivalent of when in English words are blacked out, to censor them so as not to offend. But I'm a little baffled by this instance of censorship, or even if that's what it is.

[統〇]{とういつ}協会{きょうかい}が私{あたし}の家族{かぞく}

親{おや}は嫌{きら}い

"The Unification Church is my family. I don't like my parents."

Why censor the name of the church? Why block out only one of the kanji?

Answer

You are right that ◯ is used here to mask a letter.

There are several related but different reasons as to why one would do this. The comment section already refers to one such use, where certain words are deemed inappropriate (especially on broadcasting), the equivalent of f*ck. But I don't think that explains this one.

In this case, I think the intention is to poke fun at 統一教会 in several ways.

First, by disgracing their name, by implying that it needs masking, as in the same league of f*ck and such. It also implies that the content refers to them in a negative way, which would actually make it more attractive to certain audience (who despise them.)

And finally, because the masking in this case doesn't really mask anything, for those who do not see 統一教会 nagatively, this sends a signal that they should.

(Perhaps you aren't aware of the controversies surrounding 統一教会, and if so, reading upon it a bit might make it easier to understand this.)

image processing - What does zero-mean noise mean?

I don't know where to start from. and this question might be very silly,

I have found such as comment from someone at here,

"I think if you noise is zero-mean it doesn't matter, if not it does (you should de-noise first). I'm not entirely sure though so I won't put it in an answer yet."

Q1. Actually, I want to know what does zero-mean noise mean? and what does it not matter means?

Q2. Why we presume that the noise in Gaussian filter have sigma 1 and mean 0 ?

Answer

It is an easy concept but a sometimes it is hard to express it compactly. The difficulty arises because of additional concepts to be known for any formal description.

As you may know, mathematical analysis of noise variables and signals are handled via the tools of probability theory, in particular by the stochastic (random) processes.

The usual progress is such that first properties of random variables are described and then properties of random signals are derived based on it.

The mean for a random variable, X, is defined to be its Expected Value E(X) and computed based on its Probability Density Function $f_X(x)$ such as $\mu_X = \int {xf_X(x) dx}$ with the limits from minus to plus infinity for a continuous random variable.

So a zero mean random variable is that one for which the above integral is zero. (for a discrete random variable the integral is replaced by a sum and pdf is replaced by a pmf). This is also called as the average value.

Random signals are considered to be made up of an infinite set of random variables each making a single value at a single instance of observation of the instance being created. Hence there is the concept of ensemble and time averages.

Now coming to random signals X(t) (noise) expected value of a random signal is also expressed as E(X(t)) which for a Stationary (or at east Weakly Stationary up to second order) Process is a fixed value.

Your answer is that a zero mean noise is that one for which E(X(t)) is zero for all t.

However above is a theoretical description of Mean. In practice to compute mean and other theoretical parameters, you will refer to statistical computation which deal with practically observed data and computations based on that.

The key which connects statistically made (practical) computations to theoretical computations is the Ergodicity Theorem which states that for an ergodic random process theoretical computations are equal to statistical ones.

experimental chemistry - How can 30 ml of water be heated in less than 10 seconds?

How is it possible to heat a tiny amount (30 ml)[1] of water to a high enough temperature to make a coffee, in less than 10 seconds and possibly instantly?

Most heaters that I know of heat water in no less than 90 seconds (induction heater) or 3 minutes which is way too much for my purposes, and a microwave oven takes me one minute and a half to heat.

[1] A single coffee is between 20 - 30 ml and my coffee machine makes 30 ml for each cup.

Answer

Well, let's do some math:

Assuming 30 mL of water is 30 g, and we want to heat our water from 20 °C to 90 °C, the energy required is: $$\begin{align}E&=C_Pm\Delta K \\ &=\left(4.18 \mathrm{\frac{ J}{gK}}\right)(30\mathrm{\ g})(70\mathrm{\ K})\\ &=8.778\mathrm{\ kJ}\end{align}$$

So how much power do we need to do this in a given time? "Instant" doesn't really mean anything, so let's go with 10 seconds:

$$\begin{align}P&=\frac{E}{t}\\ &=\frac{8778\mathrm{\ J}}{10\mathrm{\ s}}\\ &=877.8 \mathrm{\ W}\end{align}$$

This is not an enormous amount of power, but the trick is that it all has to go into heating the water. A good microwave outputs a fair bit more power than this, but it generally doesn't all get absorbed by such a such a small amount of water in only 10 seconds. Your best bet is probably an electric heating element directly inserted into the liquid, though I don't know if you can get a ~1000 W one small enough to sit in that much water.

As Jon Custer notes, it's not necessary to produce all the heat at once. If you heat some kind of thermal reservoir and flow the liquid past/through it, it reduces the demands on your heat source.

---

Edit: Also, I just tried this with a 1200 W microwave and it only took 15 seconds. How fast do you really need this coffee?

purim - שונא מתנות by Mishloach Manos

Can one rightfully refuse to accept Mishloach manos based on the idea of one should not accept presents, or a person who does so wrong since shalach manos was instituted to promote love? The other reason for shalach manos is that it provides food for the seudah, so maybe the idea of not accepting presents can apply (and he should not accept it)?

Answer

The issue is discussed in the נזר התורה journal of Adar 5767, and in responses thereof.

Among the sources cited by the author are the following:

I. Responsum in Yizchak Burstein's מטעמי יצחק. There, R' Burstein cites Chulin 44b:

Whenever R. Zera was sent a gift he would not accept it but whenever he was invited out to dine he would go, for he used to say, ‘They are honoured by [inviting] me’. (Soncino translation)

R' Burstein explains that when the giver of the gift has some benefit (in the gemara, the sender would be honored if the esteemed Rabbi accepted the gift), the concept of שונא מתנות does not apply. Certainly, R' Burstein continues, by משלוח מנות where the sender is fulfilling a מצוה, the concept of שונא מתנות would not apply. (The discussion in the responsum is more detailed and complicated; what I've written here is just the essence of his argument.)

II. R' Moshe Sternbuch in a footnote in Volume 2, Chapter 186 of his מועדים וזמנים makes a similar argument as that of R' Burstein.

(See also here, here,here and here.)

I would also add that the words of the Maharal (in נתיב העושר:א) are appropriate here:

והכל בודאי הולך אחר הכוונה, אם הוא שונא מתנות רק מכוין להחשיב הנותן שרי, אבל כאשר מכוין להנאת עצמו, בודאי האדם הזה יוצא מגדר מה שראוי אל התלמיד חכם

Everything follows the intent [of the recipient], if he despises gifts but he intends to give importance to the giver it is permitted. But when he intends for his own benefit, certainly that person departs from the bounds of what is befitting a Talmid Chacham.

conversion to judaism - When did Ruth convert?

In the book of Ruth it never openly states that she converts and in fact she is referred to as Ruth the Moabite throughout the text. At what point, chronologically, did she in fact convert?

organic chemistry - Why substitution and elimination reactions are favored respectively at lower and higher temperature?

I read about the competition between nucleophilic substitution and nucleophilic elimination depending on temperature here. Though the webpage clearly says higher temperature favors elimination while lower temperature favors substitution, the explanation on the webpage is unclear.

Need a clear explanation regarding the phenomenon.

Answer

For the nucleophilic substitutions and eliminations, we can draw up these four generalised schemes:

$$\begin{align} &\mathrm{S_N1}{:} & \ce{R-X + Nu- &-> R+ + X- + Nu- -> R-Nu + X-}\\ &\mathrm{S_N2}{:} & \ce{R-X + Nu- &-> R-Nu + X-}\\ &\mathrm{E1}{:} & \ce{RCH2-CR2-X + B- &-> RCH2-CR2+ + X- + B- -> RCH=CR2 + X- + HB}\\ &\mathrm{E2}{:} & \ce{RCH2-CH2-X + B- &-> RCH=CH2 + X- + HB}\end{align}$$

Clearly, each substitution keeps the number of freely diffusing particles the same while each elimination increases it by one. Therefore, the entropic term of the eliminations is likely to be higher than that of the substitutions. The rest is given by the Gibbs free energy equation: $$\Delta G = \Delta H - T\Delta S$$

The entropic factor scales with temperature as Bryce already mentioned so this outweighs any other effects at high temperatures.

translation - Meaning of sentences ending in の

How does putting a の at the end of a sentence change the meaning?

For example, here are some sentences:

明日何時に来るの What time will you come tomorrow
昨日何を食べたの What did you eat yesterday

Answer

Generally speaking a sentence ending in の will be less formal equivalent of the "の／んです" construction. Hence:

Questions ending in の, as in your examples, are less formal equivalents of the same questions ending in んですか.

Likewise

Questions ending in the plain/past form are less formal equivalents of (in your examples) questions ending with 来ます（か？）／食べました（か？）

Notes:
1) I have assumed you are familiar with the use of "の／んです vs です／〜ます" but for reference:

"When んですis used in questions or in the でしょう constructions, it is used to express one's conjectures on the basis of what one has heard or observed" eg 雨が降っているんですか〜"Is it that it is raining" as opposed to "Is it raining" (ref below)

2) I have also assumed your focus is on questions: Sentences that are not questions using "の／んです" (and its variants) are used for explanation, rapport, reproach and "backgrounding". This is covered in the same reference given below. [I would classify "soft sounding affirmation" discussed below as an example of rapport.]
3) Your examples are probably spoken with slightly raised tone at the end of the sentence to make it clear you are asking a question, as you might do in English.
4) If the sentence ends in か then raising your voice at the end of a sentence is less important.
5) I said の is equivalent to のですか rather than のです because I think you are less likely to hear questions asked that way but its not inconceivable (in your sentences it is clear from the the 何時に／何を that they are questions).

Ref: A students' guide to Japanese Grammar, Naomi Hanaoka McGloin, p89/90

translation - Help understanding a JLPT N3 conversation (～くらいだから)

This is from a JLPT N3 listening comprehension test on Youtube. This is the transcript they give in the answer key.

M: すみません、カバンを落{お｝としたみたいなんですが。

F: 電車｛でんしゃ｝の中｛なか｝じゃないですか？よく忘｛わす｝れる人｛ひと｝がいますから。

M: お酒｛さけ｝を飲｛の｝んだのは、電車｛でんしゃ｝を降｛お｝りてからですから。

F: じゃ、お酒を飲んだ所じゃないんですか。

M: 友達｛ともだち｝がカバンを忘｛わす｝れて、注意｛ちゅうい｝したくらいですから。

F: その後どうしました？ ...

I'm not sure I get the last phrase correctly. "It's because my friend forgot (his?) bag and I took care (of it?)." It doesn't seem to answer the question being asked (it's the man's bag that is lost, not his friend's). Yet the police officer seems happy with it and continues with another question.

Answer

The idiomatic construction "Clause A くらいだから Clause B" is used to state that the fact of A serves as a testament to B, a basis for surmising B, or a justification for saying B. The English expressions "Given/Since/seeing as (and the like) A, B", for the most part, will do as a translation though. Examples:

巨大{きょだい}な雪像{せつぞう}が[1週間]{いっしゅうかん}も建{た}っていられるくらいだから、極寒{ごっかん}である。 ≪The fact that those colossal snow sculptures can stand for a full week testifies to the sheer coldness (of the place).≫

“ハルキスト”っていうくらいだから村上春樹{むらかみはるき}のこと大好き{だいすき}なわけじゃん。 ≪Given the label "Harukists," they adore Haruki Murakami, right? ≫

Now to the sentence in question:

M: 友達｛ともだち｝がカバンを忘｛わす｝れて、注意｛ちゅうい｝したくらいですから。

With the above-mentioned pattern in mind, the first thing you notice is, apart from だから changing to ですから for politeness, that there's no Clause B to be seen. But no worries, this is simply because it is ellipted and implied, and luckily for us, recovering the lost clause (or at least what its main import is) from the context is easy enough; It's something that serves to refute the officer's guesswork offering, likewise with M's previous line. So the complemented sentence could be any of these:

友達｛ともだち｝がカバンを忘｛わす｝れて、注意｛ちゅうい｝したくらいですから、｛それはあり得ません/それは違います/そこで忘れたはずはありません/etc.。｝

Another point of note is that "注意｛ちゅうい｝する" means, among other things, to alert someone to, or even scold them for, their mistake or misconduct.

So all in all, M is reasoning that since he is the one who alerted his friend to the fact they'd forgotten their bag in the restaurant/bar, the same watchfulness couldn't have failed to fall upon his bag, at any rate not then and there, could it?

---

It seems there are more interpretations of the sentence than I would have imagined. Mine is the only one that came naturally to me, and, after considering other possibilities, still seems most natural to me but I'm just putting in my two cents' worth.

downsampling - What is anti-alias pre-filter for preventing aliasing after under-sampling?

We know that the under-sampling results in aliasing and frequencies higher than half of the Nyquist rate is not distinguishable. I've a base band signal that I want to use the higher frequencies which are higher than the half of the Nyquist rate (Nyquist frequency) as well as low frequencies(all parts). I've a special process with this path:

$$\textrm{Input}{\longrightarrow}\boxed{\textrm{anti-aliasing pre-filter}}{\longrightarrow}\boxed{\textrm{decimate}}{\longrightarrow}\boxed{\textrm{FFT}}{\longrightarrow}\boxed{\textrm{tune on special part}\\{\textrm{of the signal}}}$$

The low-pass post-filter that people usually use as anti-aliasing filter removes the high frequencies that are of interest for me. What is the digital or analog anti-aliasing pre-filter that I don't lose high frequencies.

particle は - は significance in ゲームは上手いですね

I'm confused by the reaction to は in this exchange

A:「ゲームは上手いですね!」
B:「ゲーム「は」.....」

source context: http://neetsha.jp/inside/up/1/2/12840/24.jpg

Answer

A:「ゲームは上手{うま}いですね!」

B:「ゲーム「は」.....」

This is the contrastive 「は」 or at least that is what B takes it to be.

A's line can be interpreted as "You are good at games (if not anything else)!", which is exactly how B interpreted it. That is why B reacts (jokingly) by emphasizing the contrastive 「は」.

B's line is obviously difficult to translate literally as English has no such particles. In that sense, A's line is already difficult to translate to begin with.

Particles can be so powerful that this shortest exchange can stand as a valid joke in itself. (And this often takes place with the contrastive 「は」 in real life.) The 「は」 in B's line would receive much stress in actual pronunciation.

B could have also said:

「ゲーム『は』って・・・」

「ゲーム『は』かよ！」

「ゲームだけかよ！」

discrete signals - How to include phase in a sinusoidal Kalman Filter

I start with the equation for sinusoidal motion with an offset and differentiate to get the 2nd order ODE describing the motion of the object.

\begin{align} x &= A\sin(\omega t + \phi) + O\\ \dot{x} &= A\omega \cos(\omega t +\phi)\\ \ddot{x} &= -\omega^2 A \sin(\omega t +\phi)\\ \implies \ddot{x} &= -\omega^2(x-O) = -\omega^2x + \omega^2O \end{align}

In matrix form this becomes:

$$ \begin{bmatrix} \dot{x} \\ \ddot{x} \\ \dot{O} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0\\ -\omega^2 & 0 & \omega^2 \\ 0 & 0 & 0 \\ \end{bmatrix} \cdot \begin{bmatrix} x \\ \dot{x} \\ O \end{bmatrix} $$

$\implies$ the system dynamics matrix $\mathbf F$ is:

$$ \begin{bmatrix} 0 & 1 & 0\\ -\omega^2 & 0 & \omega^2 \\ 0 & 0 & 0 \\ \end{bmatrix} $$

then by performing $$ A(t) = \mathscr{\mathbf L}^{-1}\left(s\vec{\mathbf I} - \vec{\mathbf F}\right)^{-1} $$

I get the following discretised $\mathbf A$ matrix:

$$ \begin{bmatrix} \cos(T_s \omega) & \dfrac{\sin(T_s \omega)}{\omega} & -\cos(T_s \omega) + 1\\ -\omega \sin(T_s \omega) & \cos(T_s \omega) & \omega \sin(T_s \omega) \\ 0 & 0 & 1 \\ \end{bmatrix} $$

This works to filter a sine wave of a know frequency with an offset, it predicts the correct phase, amplitude and even if a change the amplitude of the signal being filtered half way in it corrects for this very quickly and fits to the true signal. However if I shift the phase of the true signal part way through it does not correct for it. How might I include phase $\phi$ in my Kalman filter model such that it can still track the signal with sudden changes in phase?

If I try with my current filter I get the following result:

Answer

The answer is that the filter I detailed can correct for the phase, the issue was that the values in my Q matrix were too small and therefore it was overly trusting of the model and was correcting for the change in phase very slowly. By increasing the values in the Q matrix this model can quickly correct for the change in the phase of the oscillator!

The output now looks like the following:

fft - Plotting magnitude and phase for frequency spectrum

I would like to find the frequency spectrum by drawing magnitude and phase for the following signal:

S=sin(2*pi*100*t)+cos(pi*500*t);

X=sinc(2*pi*t);

Is there something particular to this problem that I should be noticing?

We can easily find the spectra giving a time domain like

t=0:0.01:0.1;

And using the fft command for both signals

X=fft(S);

What did they mean by drawing the magnitude and phase?

particles - What's the difference between -ga and -o when they are used to designate a direct object?

During the past month I've been addicted to Japanese. I've listened to about 10 online tutorial video courses and read about as much printed lessons. I am determined to learn Japanese, but I am really a newbie so my question may be very basic, but please bear with me.

If I understand correctly, both -ga and -o particles designate a direct object. For example, I've heard:

Watashi wa ongaku-ga suki desu. = I like music

Watashi wa ongaku-o kiku (or kikimasu, I'm not sure) = I am listening to music

So why is it ga in one case and o in the other? Is it specific to the verb or the object or what?

P.S. I don't know hiragana yet, so I'd appreciate if you could keep your examples, if any, in romaji.

Answer

It depends not only on the verb, but on the form of the verb.

The general rule is that static verbs and adjectives take "ga" and "action verbs" take "o" on the direct object.

piano-o hiku
play the piano

piano-ga hikeru
can play the piano

Here, playing the piano is an action, thus "o" is used. Being able to play the piano is a state, thus "ga" is used.

ringo-ga hoshii
want an apple

ringo-o hoshigaru
act like you want an apple

Again, to want an apple is a state, so use "ga", to act like you want it is an action, so use "o".

grammar - Varying word order for stylistic effect

Sometimes, for stylistic or rhetorical effect, one wants to delay mentioning a word/concept until the end of a sentence. For example, it's often best to save the punchline for the very end:

I was happy to discover that my ex was sentenced to life in prison for arson, murder, and jaywalking.

If we directly translate into a Japanese-style word order—

((My ex)-NOM (Arson, murder, and jaywalking)-for ((life in prison)-to sentenced) was) that I discover happy was.

—the utterance hemorrhages much of "jaywalking"'s comic effect into the bog of unnecessary background and framing information.

Whereas English word order naturally places the word where we want it, if we instead wanted to place it somewhere else, English makes available alternative or periphrastic phrasings, such as "x did y." → "It was y that x did."

So likewise: what techniques or periphrastic constructions are available in Japanese to move words or clauses?

linear systems - In the context of transfer functions, what is the relationship between the terms "proper", "causal", and "realizable"?

I am thinking about these terms in the context of linear control.

A transfer function is proper if the degree of the numerator is not greater than the degree of the denominator. I've read often that improper transfer functions are "not causal". I also often see the word "unrealizable" used often in this context.

If a control transfer function I've designed is improper, does that mean it is "causal" and/or "unrealizable"? What is the difference between these terms? What do they mean in practice?

Answer

Causality is a necessary condition for realizability. Stability (or, at least, marginal stability) is also important for a system to be useful in practice.

For linear time-invariant (LTI) systems, which are fully characterized by their transfer function, we get realizability constraints on the transfer function. For continuous-time LTI systems, if we work at frequencies for which the lumped element model is valid, we require the system's transfer function to be rational for the system to be realizable. Also for discrete-time LTI systems we require rationality of the transfer function, which implies that the system can be realized by adders, multipliers, and delay elements.

For an LTI system to be causal and stable, its poles must lie in the left half-plane (for continuous-time systems), or inside the unit circle (discrete-time systems). From this it follows that the rational transfer function of an LTI system must be proper, otherwise you would get one or more poles at infinity, causing the system to be unstable (or non-causal).