Is Eliezer's daughter anybody?

Rashi to Breishit 24:39 mentions Eliezer's daughter. Oftentimes unnamed characters have traditions linking them to named characters. Is Eliezer's daughter anybody? (and don't say "oh, she was the daughter of Eliezer")

impulse response - LTI system output

I compute the output of a LTI system, can someone tell me if my answer is right..? and help me with my others questions?

The impulse response is: $h(n) = \left(\frac{1}{2}\right)^nu(n)$ , entry is $x(n)=u(n)-u(n-1)$ in which $u(n)$ is unit sequence.

(1) We know that the outpout of this LTI system is $y(n)=x(n)*h(n)$

(2) If replace we take $y(n)=(u(n)-u(n-1))*h(n)=u(n)h(n)-u(n-1)h(n)$

(3) $u(n)*h(n)=h(n)$ and $u(n-1)*h(n)=h(n-1)$

As a result: $y(n)=h(n)-h(n-1) = \left(\frac{1}{2}\right)^nu(n) - \left(\frac{1}{2}\right)^{n-1}u(n-1)$

My Questions:

1. First of all is this solution right?


2. How we know that the equations (3) stand?


3. Always in these systems in the entry is the unit sequence?

halacha - Why don't most American orthodox Jews send their kids to public school?

Given the benefits of a public school education over private day school education, such as $0 tuition and technology and extra curriculars which many schools do not have, why don't most orthodox American Jews send their kids to public school for their secular education and then hire tutors for Jewish studies? Is this an entirely socio-cultural phenomenon or are there halachic reasons as well?

arayot - Androgynal self-impregnation

Is a naturally self-impregnated androgynus's child a mamzer?

Answer

A mamzer is the result of a relationship prohibited to the point of spiritual excision (Mishna, Yevamos 4:13). There is no prohibition of any severity for relations with one's self. Every act of normal relations that is prohibited has a verse and explanation of what relationship between those two people forbids their relations. Homosexual relations, which can be (hypothetically) violated with one's self, is prohibited without any consideration of who the individuals involved are, and therefore one's self could be (hypothetically) included. However, relations with the female part of an androgynous is not considered homosexual relations (Rambam Isurei Biah 1:15). There is therefore no scriptual basis for such a prohibition - it does not fit into any of the 15 relations for which the punishment is spiritual excision.

The gemara in Sanhedrin 55a is indicative of this -

בעא מיניה רב אחדבוי בר אמי מרב ששת המערה בעצמו מהו אמר ליה קבסתן אמר רב אשי מאי תיבעי לך בקושי לא משכחת לה כי משכחת לה במשמש מת למאן דאמר משמש מת בעריות פטור הכא פטור ולמאן דאמר חייב הכא מיחייב תרתי מיחייב אשוכב ומיחייב אנשכב

Summary: Rav Achdavuy: What is the halacha of one who commits sodomy with themselves? Rav Sheshes: Stop bothering me with ridiculous questions! (See Rashi.) Rav Ashi: This case cannot exist with an erection. You can only ask without an erection. According to the opinion that there is a prohibition without an erection, it violates two prohibitions. According to the opinion that there is no prohibition without an erection, there is no prohibition. (Halacha follows the latter opinion.)

So the Gemara was unable to find a case of someone violating themselves with an erection. If the case of this post's question provided such a case, Rav Sheshes should not have scolded Rav Achdavuy and Rav Ashi should not have needed to resort to an opinion that is not normative halacha. One is forced to conclude that either such a case is impossible, and an androgynous could not impregnate themselves naturally, or such a case is not an erva, as there is no verse in the Torah prohibiting the "relative" of one's self. As opposed to the case the Gemara did ask about, in which the prohibition is sodomy which has nothing to do with the relationship between the individuals.

Therefore, the case of the question either does not exist or is not prohibited by the Torah (as a forbidden act of relations).

fft - What statistic is used to determine presence of a signal in noise?

This is a detector problem I believe:

I am being stumped by what appears to be a simple problem. Basically, I have a band of interest. If signal energies exist within this band of interest, then I perform operation X on my signal.

My problem is that I am not sure exactly how to go about 'deciding' if a signal exists or not. In that, after I perform an FFT, I can look for peaks.

But now what?

• Is the statistic used usually comparing this peak to the surrounding mean of the spectrum? Or is it some other statistic?


• What statistical measure do I use to simply determine if a signal is present, and go from there?


• How do I set this value? Simple thresholding?

EDIT Based on feedback:

For this simple case, I am assuming a tone, in white gaussian noise. What I am trying to get a handle on are:

1. 
   

   How exactly does one generated a ROC curve. Does one have to go and label all the data first, and then get the true-positive and false-positive rates for a multitude of thresholds?

   
   
   


2. 
   

   How does decreasing SNR affect the ROC curve? Move it towards the diagonal?

   
   


3. 
   

   What is adaptive thesholding doing to a given ROC curve that was otherwise generated without an adaptive threshold?

   
   

   3a. What are some common adaptive threshold techniques I can look at that are common?

   
   

Answer

This is one of the oldest signal processing problems, and a simple form is likely to be encountered in an introduction to detection theory. There are theoretical and practical approaches to solving such a problem, which may or may not overlap depending upon the specific application.

A first step toward understanding the approaches to the problem is understanding how you would measure the performance of your signal presence detector. There are two important and related metrics used to quantitatively measure how good a detector is: its probability of detection $P_d$ and its probability of false alarm $P_{fa}$.

$P_d$ is specified as the probability that your detector will indicate the presence of the signal of interest, given that the signal is actually there. Conversely, $P_{fa}$ is the probability that your detector will indicate the presence of the signal of interest, given that the signal is not there. As you might expect, then, in a perfect world, we would design a system that yields $P_d = 1$ and $P_{fa} = 0$ and call it a day. As you might also expect, it's not that easy. There is an inherent tradeoff between the two metrics; typically if you do something that improves one, you will observe some degradation in the other.

A simple example: if you are looking for the presence of a pulse against a background of noise, you might decide to set a threshold somewhere above the "typical" noise level and decide to indicate presence of the signal of interest if your detection statistic breaks above threshold. Want a really low false-alarm probability? Set the threshold high. But then, the probability of detection might decrease significantly if the elevated threshold is at or above the expected signal power level!

To visualize the $P_d$ / $P_{fa}$ relationship, the two quantities are often plotted against one another on a receiver operating characteristic curve. Here's an example from Wikipedia:

An ideal detector would have a ROC curve that hugs the top of the plot; that is, it could provide guaranteed detection for any false alarm rate. In reality, a detector will have a characteristic that looks like those plotted above; increasing the probability of detection will also increase the false alarm rate, and vice versa.

From a theoretical perspective, therefore, these types of problems boil down to selecting some balance between detection performance and false-alarm probability. How that balance is described mathematically depends upon your statistical model for the random process that the detector observes. The model will typically have two states, or hypotheses:

$$ H_0: \text{no signal is present} $$ $$ H_1: \text{signal is present} $$

Typically, the statistic that the detector observes would have one of two distributions, according to which hypothesis is true. The detector then applies some sort of test that is used to determine the true hypothesis and therefore whether the signal is present or not. The distributions of the detection statistic is a function of the signal model that you choose as appropriate for your application.

Common signal models are the detection of a pulse-amplitude-modulated signal against a background of additive white Gaussian noise (AWGN). While that description is somewhat specific to digital communications, many problems can be mapped to that or a similar model. Specifically, if you are looking for a constant-valued tone localized in time against a background of AWGN, and the detector observes the signal magnitude, that statistic will have a Rayleigh distribution if no tone is present and a Rician distribution if one is present.

Once a statistical model has been developed, the detector's decision rule must be specified. This can be as complicated as you desire, based on what makes sense for your application. Ideally, you would want to make a decision that is optimal in some sense, based on your knowledge of the distribution of the detection statistic under both hypotheses, the probability of each hypothesis being true, and the relative cost of being wrong about either hypothesis (which I'll talk more about in a bit). Bayesian decision theory can be used as a framework for approaching this aspect of the problem from a theoretical perspective.

In the simplest practical case, the detector might trigger a detection if the detection statistic exceeds a fixed threshold $T$. In a more complicated and practical case, the detector might have some criteria for setting an adaptive threshold $T(t)$ and trigger a detection at time $t$ if the detection statistic breaks the threshold value at that instant. In your description of your problem, you hit on one common method for setting such an adaptive threshold: calculate the neighborhood mean to estimate the "background level", then set a detection threshold some amount above that mean. This can work for some applications, and there are many other ways to arrive at such a threshold.

Given a statistical model for the detector's input and the decision rule used to map that statistic to detection conclusions, one can then calculate the detector's theoretical performance metrics. In the design phase, you would typically calculate these metrics as functions of the free design parameters that you have (for instance, the threshold $T$ above). You can then evaluate the inherent tradeoffs: "if I set $T=5$, then I get $P_d = 0.9999$, but $P_{fa} = 0.01$. That's too high of a false alarm rate, so I better increase the threshold."

Where you eventually decide to sit on the performance curve is up to you, and is an important design parameter. The right performance point to choose depends upon the relative cost of the two types of possible failures: is it worse for your detector to miss an occurrence of the signal when it happens or to register an occurrence of the signal when it hasn't happened? An example: a fictitious ballistic-missile-detector-with-automatic-strikeback-capability would be best served to have a very false alarm rate; starting a world war because of a spurious detection would be unfortunate. An example of the converse situation would be a communication receiver used for safety-of-life applications; if you want to have maximum confidence that it doesn't fail to receive any distress messages, it should have a very high probability of detection of the transmitted signal.

passive voice - How do I interpret the Japanese construction of verb+noun?

A) Let's take just transitive verbs first:

1. 食べる人



2. 食べられる人

B) Now let's take intransitive verbs:

3. 起きる人


4. 起きられる人


5. 起こす人


6. 起こされる人

OK, this thing has confused me for a very long time now, like really long. Thus I am posting this here. I have asked the question once before but did not get a real answer.

What I am confused with about verb+noun sentence is how it is to be translated. e.g from A) we get

1. "The person that eats (person is subject, object omitted)" and


2. "The person that is eaten (person is object here, subject omitted)" right?

Now let's get to B) I think all 4 sentences are grammatically valid and sound as far as usage is concerned. I would translate them as

3. "The person that wakes up (person is subject and object - instransitive verb)"


4. "The person that is awaken (by himself) (person is subject and object - intransitive)" or in other words "The person that awakes himself (by himself)"

Then we have,

5. "The person that wakes (others) up (subject is person, object omitted)"


6. "The person that is awaken (by someone else) (subject omitted, object is person)"

Now what I want you folks to do is to make sure that I interpreted the Japanese sentences correctly. This will make sure that I understand how to make verb+noun sentences.

Thus 食べる食事 means "The food that eats (someone/something)". To refer to the food I eat I shall say 食べられる食事は… "the food I eat is..." right?

organic chemistry - Do indigo and leuco-indigo have fully conjugated pi systems?

I've just come across a figure indicating the positions of conjugated bonds in indigo and leucoindigo, and I'm quite confused about why both molecules aren't fully conjugated.

My reasoning is as follows:

1. Both carbons above and below the bond are sp² hybridized and have π electrons.



2. The carbon to the left is sp² hybridized and has π electrons, and the nitrogen to the right has a lone pair.


3. Same as 2.


4. Same as 1.

I also want to ask about why leuco-indigo isn't planar. According to Wikipedia:

In indigo white, the conjugation is interrupted because the molecule is nonplanar.