passover seder hagada - Why is Chad Gadya a mix of Hebrew and Aramaic?

Chad Gadya (recited at the end of the Passover seder) is written in a mix of Hebrew and Aramaic. Specifically, most of the verbs are in Hebrew, except for זבין (bought) and אתא (came), and most of the nouns are in Aramaic, except for HaKadosh Baruch Hu, Malach HaMaves (angel of death), and HaShochet (slaughterer). Why?

Answer

The first attestation of Chad Gadya in print is in the Prague Haggadah of 1590 (online source). It is thought by some to have been modelled on a German folk song, and thus composed at such a time as Aramaic was no longer being spoken. This may account for what some have seen as grammatical errors within the text (online source).

An example of a "grammatical error" (taken from that website, which is in Hebrew) is שונרא... דאכלה ("the cat... which ate"). Aside from possible problems with vocalisation, which may be found in other passages in Chad Gadya also, שונרא (cat) is a masculine noun, but אכלה (ate) is a feminine verb. Really, the text should say either שונרתא (which is the feminine word for "cat") or דאכל (which is the masculine verb, "ate").

In any case, an admixture of Hebrew and Aramaic in mediaeval Jewish literature (a time when people are neither speaking Aramaic nor Hebrew outside of liturgical settings) is par for the course, and is the result of both languages being read with roughly equal frequency. Hebrew, the language of scripture and prayer; Aramaic, the language of scholarship.

purim torah in jest - What to do if your husband is muktzeh

My husband is muktzeh (he was born in the eighth month and is now nineteen). Our marriage is going well. However, sometimes on the three-day Yom Tov it becomes difficult that neither I nor anyone else can p̶a̶s̶s̶ ̶h̶i̶m̶ ̶t̶h̶i̶n̶g̶s̶ pass him to things at the table. Is there any solution for this?

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What is the math behind median filter's noise reduction property?

I am interested in understanding the mathematical reason for why does applying a median filter on an image (or signal) result in reduction of noise.

Answer

Intuition: The intuition is this: Your noise is some event or events that are rare, and that when compared to other events, look like outliers that shouldn't really be there.

For example, if you are measuring the speeds of every car on the highway as they pass by you and plot them, you will see that they are usually in the range of say, $50$ mph to $70$ mph. However as you are inspecting your data for your boss, you see that you recorded a speed of $1,000,000$ mph. Not only does this value not make physical sense for the speed of an actual car on a highway, but it also sticks out wildly from the rest of your data. Chalking this event up to some strange measurement error, you remove it, and give the rest of your data to your boss.

However as you continue your measurements day in and day out, you notice that every now and then, you get those wild measurements of speed. For example, over the span of 1 hour, you measure 1000 cars, and their speeds are nicely between $50$ and $70$ mph, however 3 of those have speeds of $23,424$ mph, $12,000,121$ mph, and $192,212,121,329,982,321,912$ mph, breaking not only local state laws, but also those of theoretical physics.

You get tired of continuously having to go in, and remove those errant data points caused by your cheapo-radar by hand. Afterall, your boss is really only interested in the statistics of the speeds, not so much every actual value. He likes to make nice histograms for his bosses.

Those errant and large numbers are a kind of 'noise' you reckon - 'noise' caused by your cheapo-radar that you bought from a shady pawn shop. Is the noise additive white gaussian noise? (AWGN). Yes and no - It's spectrum is wideband and white, but it is temporally rare, sparse, and very localized. It is better referred to as 'salt and pepper' noise, (especially in the image processing domain).

Thus, what you can do, is run your data through a median filter. Your median filter will take a block of say, $5$ speed points, (points 1 to 5), find the median, and spit that value as the 'average' speed. Then it will take the next 5 points, (points 2 to 6), take that median, and spit out this as the average, etc etc.

What happens when you come across one of your faster-than-light speeds?. Let us say that your 5 speeds were [45, 65, 50, 999999, 75]. If you took the normal average, your 'average' speed here will be something quite large. However if you take the median, your 'average' will be 65. Which best approximates the average that you are really trying to measure? The median metric.

Thus, if you filter your data with a median filter, you will be sure to remove those outliers - and you have thus faithfully 'de-noised' your signal. In contrast, if you tried to remove your noise via traditional filtering, (nothing but a moving weighted sum), you will instead 'smear' the error across your data, and not get rid of it.

Math: The math is this: The median measurement is what is referred to as an order statistic. That is, it returns the value of your data, along some point, after it has been ordered. The max and min are also both order statistics - they return the extreme points of your data after it has been ordered. Taking the median also returns the value of your ordered data, but right from the middle.

But why are they different from mean filters? Well, mean-filters compute an average using all the data. If you notice from max, min, and median, you are getting an answer without using all the data. In fact, all the median does is order your data, and pick the value in the middle. It never 'touches' the outliers, like those large speeds that you measured.

This is why median - an order statistic - is able to 'remove' outlier noise for you. Outlier noise segregates itself in front of the median, and the median never comes near it or considers it, while still giving you a nice estimate of central tendency.

halacha - Halachot of Lashon Naki (clean speech)

What is the halachik source, if any, for lashon naki (clean speech)? Is there a portion of Talmud that deals with the specifics? On the surface, it seems to be employed inconsistently, by which I mean there are times where the torah or chazal will go out of their way to use euphemism instead of the proper noun or description (the below example from Pesachim is great). However, other times we find very graphic descriptions for instance, of female genitalia in shir hashirim, (albeit assumedly metaphorical) or tzoah rosachas (gitin 57a) with no compunction for 'lashon naki'.

Answer

The Talmud discusses this issue in Pesachim 3a.

There it brings a number of instances where a verse uses extra letters to avoid saying a negative word. Here's one example that it brings:

The verse by Noach (Genesis 7:8) says to bring into the ark animals that are טהורה (pure) and animals that are אשר איננה טהורה (lit. that are not pure). This is instead of the shorter and more conventional טמאה (impure). I know this doesn't work out in translation so well, but in Hebrew it uses 8 extra letters to avoid saying 'impure'.

See there for more proofs and examples.

tefilla - How do I say vidui in Selichos in a meaningful way?

We say vidui three times in selichos. Many words are uttered in a relatively short time. How can we do this in a meaningful way that is not "vidui peh"?

I have heard tell of one of the gedolei Torah that in the vidui of everyday Tachanun, he said only one word of the “oshamnu, bogadnu” prayer. Is that a derekh for non-gedolim?

Answer

1. Learn the meaning behind the vidui before saying it.


2. Buy a sefer that translates them or an interlinear siddur, and daven from that.


3. Write your own commentary to them whilst learning the meanings.

physical chemistry - Is there any formula that can be used to find loss of mass due to binding forces in atomic and sub atomic particles?

Atomic weight of Br-79 is 79.641 if you add the masses of protons and neutrons. However, in periodic table, it is less than the value given here. How is the difference arrived at for all the elements listed in the periodic table?

molecular orbital theory - Methane T2 SALCs

So I have a question on the form of the $T_2$ SALCs of methane. Below I show the $T_d$ character table and the accompanying reducible representation for the sigma framework (using, in the case of methane, the $4$ $1s$ H orbitals).

It is easy to show, and many people have, that the irreducible representations contained therein are $A_1$ and $T_2$.

If we then perform the projection operation for all of the elements of $T_2$, I get the following:

(The X's are intended to be "Chi's" here, signifying the $4$ $1s$ H orbitals that go into the procedure).

Meaning that, if we take the first line for instance, and perform the sum in the projection operator, we get: $$ 3\chi_1-\chi_2-\chi_3-\chi_4 $$

As our first un-normalized $\phi_t$ SALC. I can easily see by intuition how we can obtain the (again, not normalized) known SALCs for a $\sigma$ framework tetrahedron from this: $$ 3\chi_1-\chi_2-\chi_3-\chi_4 = $$ $$ \chi_1-\chi_2+\chi_3-\chi_4 $$ $$ \chi_1+\chi_2-\chi_3-\chi_4 $$ $$ \chi_1-\chi_2-\chi_3+\chi_4 $$

What I do not understand, is how we can derive this result without intuition. If I perform any number of Schmidt orthoganalizations using the other 3 rows of the projection operator table, I always end up with an exceedingly strange SALC, regardless of normalization and inclusion of overlap integrals $\chi_i{\times}\chi_j = S_{ij}$.

Is there any mathematical tool that can be used to generate these known SALCs from the one above?

Again, let me stress, I can clearly see the intuitive path, but I always rest easier when I know there is a reproducible, rigorous theory that leads there.

Answer

UPDATE: added more detail for those without access to the referenced text.

As noted in the comments, the details of this process can be found in "Group Theory and Chemistry" by David M. Bishop (Courier Corporation, 1993) p. 238-9. Here's a summary:

Following the same process that you started, you can get two other projections as (using your notation):

$-\chi_1+3\chi_2−\chi_3−\chi_4$

and

$-\chi_1−\chi_2+3\chi_3−\chi_4$

All three of these normalize with a factor of $\frac{1}{\sqrt{12}}$.

The challenge with degenerate orbitals is that there is an infinite number of linearly independent combinations that will combine to give the desired $T_2$ symmetry properties. Schmidt orthogonalization can be used to generate an orthogonal set, but there is an infinite number of those as well.

To prove that the canonical set is valid, one need only show that all three are linear combinations of the above projections and that they are orthogonal to each other. But to derive the canonical set without prior knowledge, we use the procedure below.

We want a set that is orthonormal and matches up with the central atomic orbitals. That is, each has symmetry that matches a member of the $p$ orbital set of the central atom. First, we set the $t_2$ central atom orbitals as $p_x$, $p_y$ and $p_z$. Then we consider that we must be able to make the hybrid orbital that coincides with each ligand bond from a linear combination of the $s$ and $p$ atomic orbitals. This is essentially a statement of the principle underlying the concept of hybridization. Furthermore, since the $s$ orbital only adds magnitude, not direction, an unnormalized hyrbid orbital can be made from a linear combination of $p$ orbitals only. Finally, any member of the $t_2$ ligand group orbitals must similarly be able to be constructed from a linear combination of the central atom $p$ orbitals.

Thus, $(3\chi_1-\chi_2−\chi_3−\chi_4)/\sqrt{12} = (a_1^2+b_1^2+c_1^2)^{-1/2}(a_1p_x+b_1p_y+c_1p_z)$ for some $a_1, b_1, c_1$ and likewise for the other two projections (adding a minus sign where necessary for directionality with respect to the positive lobes of the $p$ orbitals).

We can then take advantage of the linearity of the symmetry operators to set up equations and solve for $p_x$, $p_y$, and $p_z$ in terms of $\chi_1$, $\chi_2$, $\chi_3$ and $\chi_4$, and the coefficients of those solutions are the coefficients for the H1s orbitals in each of the three $t_2$ ligand group orbitals.

For example, let $\chi_1$ be the hybrid orbital oriented in the positive $x,y$ and $z$ octant of Cartesian space, and let the $C_{3a}$ axis run through it. Thus,

$O_{C3a}(\chi_1)=\chi_1$

$O_{C3a}(\chi_2)=\chi_3$

$O_{C3a}(\chi_3)=\chi_4$

$O_{C3a}(\chi_4)=\chi_2$

Because the operator is linear, we now have that

$O_{C3a}[(3\chi_1-\chi_2−\chi_3−\chi_4)/\sqrt{12}]=(3\chi_1-\chi_2−\chi_3−\chi_4)/\sqrt{12}$, i.e. this projection is unchanged by that transformation because $\chi_2$, $\chi_3$, and $\chi_4$ are equivalent in the expression.

Therefore, we can further state that

$O_{C3a}[(a_1^2+b_1^2+c_1^2)^{-1/2}(a_1p_x+b_1p_y+c_1p_z)]=(a_1^2+b_1^2+c_1^2)^{-1/2}(a_1p_x+b_1p_y+c_1p_z)$

We can also determine by inspection that $O_{C3a}(p_x)=p_z$,$O_{C3a}(p_y)=p_x$, and $O_{C3a}(p_z)=p_y$. Therefore, we also have that

$O_{C3a}[(a_1^2+b_1^2+c_1^2)^{-1/2}(a_1p_x+b_1p_y+c_1p_z)]=(a_1^2+b_1^2+c_1^2)^{-1/2}(a_1p_z+b_1p_x+c_1p_x)$

and we can conclude that $a_1=b_1=c_1$. Thus,

$(3\chi_1-\chi_2−\chi_3−\chi_4)/\sqrt{12}=(p_x+p_y+p_z)/\sqrt{3}$.

Repeating this for the other symmetry transformations (for example $C_{3b}$ and $C_{3c}$) gives a system of equations that can be solved to yield:

$p_x=\frac12 (\chi_1−\chi_2+\chi_3−\chi_4)$

$p_y=\frac12 (\chi_1−\chi_2-\chi_3+\chi_4)$

$p_z=\frac12 (\chi_1+\chi_2-\chi_3−\chi_4)$

The coefficients of $\chi_1$, $\chi_2$, $\chi_3$ and $\chi_4$ in these three expressions are the coefficients of the H1s orbitals to each canonical $t_2$ ligand group orbital.