The answer to this question might solve a multiple choice in my previous chemistry exam, which caused an argument with my teacher.
We know that fatty acids form fats and oil due to their carboxylic groups by forming triesters with glycerol. But is it possible to have a fatty acid with two or more $\ce{-COOH}$ groups to exist naturally? If so, is the property "fats and oils are neutral" just a general property, or does it have an exception?
What is the hybridisation of $\ce{F}$ in $\ce{BF3}$ or $\ce{CH3F}$ or $\ce{PF5}$ or $\ce{SF6}$? My book states that the p orbitals of fluorine overlaps, but I feel that sp3 hybridisation occurs in $\ce{F}$, so its sp3 hybrid orbital overlaps with the empty p orbital of boron.
According to Ron's answer, all atoms with p or d orbitals can hybridise.
Answer
TL;DR: As a rule of thumb, terminal (heavy) atoms are almost always best described as having sp hybrid orbitals (at the most).
For more on this, I refer you to my answer on What is the hybridization of chlorine in vinyl chloride?
Here are the relevant visualisations for $\ce{BF3}$ at the DF=BP86/def2-SVP level of theory. First I'll start with the canonical orbitals obtained from a molecular orbital point of view. The molecule has $D_\mathrm{3h}$ symmetry, therefore some orbitals are degenerated.
With appropriate transformations, we can interpret these orbitals in a localised fashion, which will give us hybrid orbitals. In this view the σ bond orbitals, and the lone pairs are triply degenerated. Note that there is a significant amount of delocalisation of the fluorine lone pairs into the empty boron p orbital.
Here is the representative section from the natural bond orbital analysis:
(Occupancy) Bond orbital / Coefficients / Hybrids
------------------ Lewis ------------------------------------------------------
(core and symmetry equivalent orbitals skipped)
5. (1.99345) LP ( 1) F 2 s( 63.79%)p 0.57( 36.19%)d 0.00( 0.01%)
6. (1.94124) LP ( 2) F 2 s( 0.00%)p 1.00( 99.95%)d 0.00( 0.05%)
7. (1.86896) LP ( 3) F 2 s( 0.00%)p 1.00( 99.94%)d 0.00( 0.06%)
14. (1.99778) BD ( 1) B 1- F 2
( 17.29%) 0.4158* B 1 s( 33.25%)p 1.98( 65.98%)d 0.02( 0.76%)
( 82.71%) 0.9094* F 2 s( 36.24%)p 1.76( 63.62%)d 0.00( 0.14%)
---------------- non-Lewis ----------------------------------------------------
17. (0.38551) LV ( 1) B 1 s( 0.00%)p 1.00(100.00%)d 0.00( 0.00%)
18. (0.05374) BD*( 1) B 1- F 2
( 82.71%) 0.9094* B 1 s( 33.25%)p 1.98( 65.98%)d 0.02( 0.76%)
( 17.29%) -0.4158* F 2 s( 36.24%)p 1.76( 63.62%)d 0.00( 0.14%)
Here you can clearly see the approximate two sp orbitals and the remaining p lone pair orbitals.
The above remains true for (almost) all terminal atoms, because of the local linear symmetry (only one bonding partner, negligible external field), although with lesser extend for heavier atoms (like bromine). This is because hybridisation in general becomes a less reliable description due to the increase in the s-p gap.
I noticed that while 「…、電話が切れた」 sounds just like a neutral phrase ("... and the call got cut off."), 「…、電話は切れた」 sounds more like you were negatively affected / that the person on the other side intentionally cut off the call.
This is the sentence which first made me think this:
「そうですか。わかりました。失礼します！」
一方的に三浦は吐き捨てて、電話は切れた。
池井戸潤『空飛ぶタイヤ（上）』、講談社文庫、p. 353
I felt that は flowed marginally better here -- my initial thought was that it's because it was an adversarial situation, but perhaps it's for another reason.
After some Google searching, I found a document which actually includes both phrases, and I think it's interesting:
大分市内在住 82 歳男性宅に、「7 月分の高額療養費の支 給が 35,000 円ある。手続きは今日まで。お客様番号を伝 えるので、保健所に電話するように。お客様番号は 998-573」との電話があった。不審に思い「市役所に出 向けばよいのか？」と聞くと電話は切れた。
vs
平成２４年１０月１１日午後、大分市に住む７８歳（女 性）の被保険者宅に、支所の職員を名乗る男から電話がか かり「保険料の還付が３万円程度ある。８月に緑色の封筒 で通知を送っているが見たか？」と尋ねられた。また番号 （９９８７４５）を告げられ、これをメモに控えようとし た被保険者が何回か聞き返しているうちに電話が切れた。
To me, the first passage seems to feel like the call was cut in response to the question, while the second passage is more like the call just cuts at that point in time (as opposed to being initiated by a human in response to something).
It says in Shulchan Aruch that one may not lean on anything while reciting sh'mona esre. (OC 94:8) The Mishna B'rura defines "leaning" as placing one's weight on another object such that if the object were removed the person would fall over. (ibid. 22) How is this definition to be understood? If a person places almost any amount of weight on an object without exerting any other force on it, he has necessarily shifted his center of mass to a point closer to the object than it was when he was free-standing. Should the object be removed and the person not compensate by shifting his center of mass back, he will fall over. Could it be that even the slightest of "leanings" fits the Mishna B'rura's definition?
Alternatively, if we allow for shifting of mass following the object's removal, then how do we understand the basics of the definition? I could be leaning to an extreme angle and still catch myself by proper placement of an arm, a leg or a torso.
Is there a middle ground? I have always had trouble with this concept and have never received a satisfying explanation.
It seems that the 3rd aliyah in Bamidbar needlessly repeats the information given in the 1st and second aliyah
First Aliyah: Lists the leaders of each tribe
Second Aliyah: gives us the census information for each tribe
Third Aliyah: Describes the order the Tribes camped in.
However, the third aliyah also retells us the leader of each tribe as well as the population of each tribe. Don't we already know that information from the first two aliyahs?
Answer
The repetition of the population figures:
Ibn Ezra (to 2:32) and Ramban (to 2:4) say that it underscores the fact that from the census (on the 1st of Iyar, Num. 1:1) until the next time they traveled (on the 20th of that month, ibid. 10:11), no one in such a vast multitude died (against all natural expectations) - so that the figures remained the same.
[Malbim (to 2:2) says pretty much the same thing, though he takes the view that the figures in ch. 2 were the result of a separate census taken for the purpose of setting up the camp, not just reusing the earlier results.]
Abarbanel writes that this was done to tell us why each tribe was assigned to its particular location. Basically, he says, an army's lead unit ("what people call in the vernacular the 'avant-garde'") and rear unit have to be large in size to repel attackers, while the flanks can be smaller. In keeping with this, Yehudah's camp had the greatest number of people, Dan's the second largest, and the other two had fewer people.
The repetition of the names of the leaders:
Malbim (to 2:5) says that the second counting of the three tribes of Yissachar, Zevulun and Reuven was overseen by their respective nesi'im, and therefore each of these tribes' population is described with the word ופקדיו ("and his enumeration") - by contrast with the others, which are described with ופקודיהם ("and their enumeration"). So those three leaders had to be mentioned again. He doesn't say anything about the other nine (just that those tribes handled the re-counting themselves), but presumably once the Torah had to repeat the names of some, it continues the same throughout.
It's also possible (though I don't know any source for this) that with the information in ch. 1 we might have thought that they were designated only to oversee the census, but not to lead their respective tribes when traveling and encamping. Repeating their names in ch. 2 makes it clear that they are in charge for all purposes. (This might also explain the repetition of the census figures - to clarify the scope of each one's responsibility.)
I am working on some waveform generation software and I am trying to figure out what the best type of filter would be to use a an Anti Aliasing filter for my algorithm. I am generating my waves in a "Raw" mathematical way, meaning that i am creating a ramp for a saw wave and my square wave consists of pure 1's and 0's. The issue is that this technique is inherently not band limited. I would like to implement low pass filter into the algorithm in order to filter of the harmonic being generated that are higher than the nyquist limit.
What would be the ideal filter design for this? What kind of roll of would i need?
I am doing this in c++ but i don't neccesarily need an answer in c++. just an equation or some resources would be good as well.
Thanks
edit: I am not changing the sampling rate at any point in the chain. But i am still experiencing some aliasing somehow.
here is my code:
double _DSP::Saw_Wave(double* _frequency,_DSP::Saw_Data* _data){
double _val = _data->_phasor;
_data->_phasor += 2.0*(1.0/(_DSP::Sample_Rate::Samples_Per_Second()/ *_frequency));
if (_data->_phasor>_data->_tolerance) {
_data->_phasor-=2.0;
}
return _val ;
}
typedef struct Saw_Data{
double _phasor = 0.0;
double _tolerance = 1.0;
}Saw_Data;
where _DSP::SampleRate::Samples_Per_Second() is the sampling rate.
the code generates the proper wave and frequency but has unwanted low frequency information when played at higher frequencies.
Answer
I am generating my waves in a "Raw" mathematical way, meaning that i am creating a ramp for a saw wave and my square wave consists of pure 1's and 0's. The issue is that this technique is inherently not band limited. I would like to implement low pass filter into the algorithm in order to filter of the harmonic being generated that are higher than the nyquist limit.
You can't do this. You've already aliased the frequencies when you generated the waves in that way. You can't remove them with a filter.
You're essentially sampling the pure mathematical signal without using an antialiasing filter first. It's not possible to add the antialiasing filter after sampling. You will just filter the already-aliased signal. Bandlimiting has to occur before sampling. You need to change the way you generate the waveforms if you want them to be bandlimited. You could generate them using additive synthesis, as I've demonstrated here, or you could generate them the simple way at a much higher sampling rate to minimize the aliasing, then antialias filter, then decimate. Or you could use BLIT or BLEP, etc. (I think those are both "reduced-aliasing" not 100% eliminated, but I'm not sure.)
בס"ד
In both Mishpatim and Shoftim there is a verse (Exodus 21:23 and Devarim 19:21) saying "a soul for a soul".
I know that we don't take these verses literally and restitution is the punishment.
But:
What source do we have in the Chumash textually that they aren't taken literally? Or is this purely something that came along through the Oral Tradition? I have heard that there is a gezirat sheva with the word תחת in נפש תחת נפש, but then what do we say about the נפש בנפש verse where תחת isn't mentioned?
Why is this phrase written differently in different places? In one verse (Exodus) it's written נפש תחת נפש and then in Devarim its written נפש בנפש replacing תחת with a ב?
Answer
The Gemara in Bava Kama starting at 83b discusses th issue at great length, bringing numerous sources for the non-literal understanding of the Pasuk. Besides those mentioned in the Gemara, later authorities brought many more hints from the text of the Torah that the pesukim refer to monetary payment. It would be too lengthy of a discussion to quote them all here.
Importantly, the Rambam in his introduction to his explanation of the Mishna deals with this subject. There he writes that there was always a known and undisputed direct tradition that the verses were non-literal. The purpose of the exegesis of the Gemara is not to learn the otherwise-unknown meaning of the pesukim. Rather, it is to demonstrate that all the details of the Oral Torah are hinted to in the text of the Written Torah.
The Rambam compares this to the Gemara in Succah (daf?), which seems to be unsure about what fruit the Torah refers to in the commandment of the Four Species taken on Sukkos. Like in Bava Kama, different Sages offer drashos and pesukim to show that the Pasuk refers to the esrog. Now, says the Rambam, it was undoubtably known throughout Jewish history that the fruit taken on Sukkos is the esrog. Did the Sages not see there parents and communities taking esrogim every year? Thus, writes the Rambam, these pesukim must not be seen as the source for the law, but as the hint to the law that can be found in the Written Torah.
