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# A recruiter calls me to talk about a Data Science position in Pune…

A recruiter calls me this morning, from Hyderabad, all unexpectedly. No emails beforehand, no recruiter messages at a jobs-site, no SMSs, nothing. Just a direct call. They are considering me for a Data Science position, in Pune. She says it’s a position about Data Science and Python.

Asks about my total and relevant experience. I tell: 23 years in all, ~12 years in s/w development. She asks about my Python experience. I tell: Familiarity for, may be, 10 years if not more; actual use for, may be, 5–6 years. (Turns out to be since 2006, and since at least 2013–14 times, in connection with scripting while using the open-source FEM libraries, respectively.)

I tell that I’ve been into it for about a year by now, but no professional, paid experience as such. Also add that I do understand kernels from the Kaggle competitions. (In fact, I can think of bringing about meaningful variations in them too.)

She asks about my last job. I tell: Academia, recently, after PhD. (She sounds a bit concerned, may be confused. She must be looking at my resume.) But before that, I was in the software field, I say. And now, am now looking for a Data Science position. I then add: In the software development field, my last job was as a Systems Architect, reporting directly to the CEO. … By this time, she must have spotted this software experience listing in my resume. She says “OK,” with just a shade of a sense of satisfaction audible in the way she sounds.

She then again asks me about my Data Science experience. I now tell her directly: Paid experience, 0 (zero) years.

Hearing it, she keeps the phone down. Just like that. Without any concluding remarks. Not even just a veneer of a courtesey like a hurried “OK, if you are found suitable, we will get back to you” etc. Nothing. Not even that. No thanks, nothing.

She. Just. Keeps. The. Phone. Down.

It must be a project for one of those companies from America, especially from California, especially from the San Francisco Bay Area. Only they can be as dumbidiots* as that. And, they could very well be one of those “Capitalist”s, esp. Indians—there and here. “You are just as good as your performance on your last job!” Said sternly. And, the quote taken literally. In the current context, it is obviously taken to mean that I am as good as zero, when it comes to Data Science positions.

Dumbidiots*. Zeno’s descendents. They don’t deserve to hire me.

But these stupididiots* do amass a lot of money for themselves. Help build the nation. Etc.

Rich idiocy.

*By the rules of the Sanskrit grammar, this “sandhi” is correct. English is an Indo-European language. So, such a “sandhi” should be allowed. The jointed word means something like “k’mt’om” [^] “moorkha”. (You look up “moorkha”.)

A song I like:
(Hindi) “hum the, woh thee, aur, samaa rangeen…”
Singer: Kishore Kumar
Lyrics: Majrooh Sultanpuri
Music: S. D. Burman

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Oakay… My bookmarks library has grown too big. Time to move at least a few of them to a blog-post. Here they are. … The last one is not on Data Science, but it happens to be the most important one of them all!

On Bayes’ theorem:

Oscar Bonilla. “Visualizing Bayes’ theorem” [^].

Jayesh Thukarul. “Bayes’ Theorem explained” [^].

Victor Powell. “Conditional probability” [^].

Explanations with visualizations:

Victor Powell. “Explained Visually.” [^]

Christopher Olah. Many topics [^]. For instance, see “Calculus on computational graphs: backpropagation” [^].

Fooling the neural network:

Julia Evans. “How to trick a neural network into thinking a panda is a vulture” [^].

Andrej Karpathy. “Breaking linear classifiers on ImageNet” [^].

A. Nguyen, J. Yosinski, and J. Clune. “Deep neural networks are easily fooled: High confidence predictions for unrecognizable images” [^]

Melanie Mitchell. “Artificial Intelligence hits the barrier of meaning” [^]

Ijad Madisch. “Why I hire scientists, and why you should, too” [^]

A song I like:

(Western, pop) “Billie Jean”
Artist: Michael Jackson

[Back in the ’80s, this song used to get played in the restaurants from the Pune camp area, and also in the cinema halls like West-End, Rahul, Alka, etc. The camp area was so beautiful, back then—also uncrowded, and quiet.

This song would also come floating on the air, while sitting in the evening at the Quark cafe, situated in the middle of all the IITM hostels (next to skating rink). Some or the other guy would be playing it in a nearby hostel room on one of those stereo systems which would come with those 1 or 2 feet tall “hi-fi” speaker-boxes. Each box typically had three stacked speakers. A combination of a separately sitting sub-woofer with a few small other boxes or a soundbar, so ubiquitous today, had not been invented yet… Back then, Quark was a completely open-air cafe—a small patch of ground surrounded by small trees, and a tiny hexagonal hut, built in RCC, for serving snacks. There were no benches, even, at Quark. People would sit on those small concrete blocks (brought from the civil department where they would come for testing). Deer would be roaming very nearby around. A daring one or two could venture to come forward and eat pizza out of your (fully) extended hand!…

…Anyway, coming back to the song itself, I had completely forgotten it, but got reminded when @curiouswavefn mentioned it in one of his tweets recently. … When I read the tweet, I couldn’t make out that it was this song (apart from Bach’s variations) that he was referring to. I just idly checked out both of them, and then, while listening to it, I suddenly recognized this song. … You see, unlike so many other guys of e-schools of our times, I wouldn’t listen to a lot of Western pop-songs those days (and still don’t). Beatles, ABBA and a few other groups/singers, may be, also the Western instrumentals (a lot) and the Western classical music (some, but definitely). But somehow, I was never too much into the Western pop songs. … Another thing. The way these Western singers sing, it used to be very, very hard for me to figure out the lyrics back then—and the situation continues mostly the same way even today! So, recognizing a song by its name was simply out of the question….

… Anyway, do check out the links (even if some of them appear to be out of your reach on the first reading), and enjoy the song. … Take care, and bye for now…]

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# Non-Interview Questions on Data Science—Part 1

This entry is the first in a series of posts which will note some of the questions that no one will ever ask you during any interview for any position in the Data Science industry.

Naturally, if you ask for my opinion, you should not consider modifying these questions a bit and posting them as a part of your own post on Medium.com, AnalyticsVidhya, KDNuggets, TowardsDataScience, ComingFromDataScience, etc.

No, really! There would be no point in lifting these questions and posting them as if they were yours, because no one in the industry is ever going to get impressed by you because you raised them. … I am posting them here simply because… because “I am like that only.”

OK, so here is the first installment in this practically useless series. (I should know. I go jobless.)

(Part 1 mostly covers linear and logistic regression, and just a bit of probability.)

Q.1: Consider the probability theory. How are the following ideas related to each other?: random phenomenon, random experiment, trial, result, outcome, outcome space, sample space, event, random variable, and probability distribution. In particular, state precisely the difference between a result and an outcome, and between an outcome and an event.

Give a few examples of finite and countably infinite sample spaces. Give one example of a random variable whose domain is not the real number line. (Hint: See the Advise at the end of this post concerning which books to consult.)

Q.2: In the set theory, when a set is defined through enumeration, repeated instances are not included in the definition. In the light of this fact, answer the following question: Is an event a set? or is it just a primitive instance subsumed in a set? What precisely is the difference between a trial, a result of a trial, and an event? (Hint: See the Advise at the end of this post concerning which books to consult.)

Q.3: Select the best alternative: In regression for making predictions with a continuous target data, if a model is constructed in reference to the equation $y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \beta_3 x_i^3$, then:
(a) It is a sub-type of the linear regression model.
(b) It is a polynomial regression model.
(c) It is a nonlinear regression model because powers $> 1$ of the independent variable $x_i$ are involved.
(d) It is a nonlinear regression model because more than two $\beta_m$ terms are involved.
(e) Both (a) and (b)
(g) Both (b) and (c)
(f) Both (c) and (d)
(g) All of (b), (c), and (d)
(h) None of the above.
(Hint: Don’t rely too much on the textbooks being used by the BE (CS) students in the leading engineering colleges in Pune and Mumbai.)

Q.4: Consider a data-set consisting of performance of students on a class test. It has three columns: student ID, hours studied, and marks obtained. Suppose you decide to use the simple linear regression technique to make predictions.

Let’s say that you assume that the hours studied are the independent variable (predictor), and the marks obtained are the dependent variable (response). Making this assumption, you make a scatter plot, carry out the regression, and plot the regression line predicted by the model too.

The question now is: If you interchange the designations of the dependent and independent variables (i.e., if you take the marks obtained as predictors and the hours studied as responses), build a second linear model on this basis, and plot the regression line thus predicted, will it coincide with the line plotted earlier or not. Why or why not?

Repeat the question for the polynomial regression. Repeat the question if you include the simplest interaction term in the linear model.

Q.5: Draw a schematic diagram showing circles for nodes and straight-lines for connections (as in the ANN diagrams) for a binary logistic regression machine that operates on just one feature. Wonder why your text-book didn’t draw it in the chapter on the logistic regression.

Q.6: Suppose that the training input for a classification task consists of $r$ number of distinct data-points and $c$ number of features. If logistic regression is to be used for classification of this data, state the number of the unknown parameters there would be. Make suitable assumptions as necessary, and state them.

Q.7: Obtain (or write) some simple Python code for implementing from the scratch a single-feature binary logistic regression machine that uses the simple (non-stochastic) gradient descent method that computes the gradient for each row (batch-size of 1).

Modify the code to show a real-time animation of how the model goes on changing as the gradient descent algorithm progresses. The animation should depict a scatter plot of the sample data ($y$ vs. $x$) and not the parameters space ($\beta_0$ vs. $\beta_1$). The animation should highlight the data-point currently being processed in a separate color. It should also show a plot of the logistic function on the same graph.

Can you imagine, right before running (or even building) the animation, what kind of visual changes is the animation going to depict? how?

Q.8: What are the important advantage of the stochastic gradient descent method over the simple (non-stochastic) gradient descent?

Q.9: State true or false: (i) The output of the logistic function is continuous. (ii) The minimization of the cost function in logistic regression involves a continuous dependence on the undetermined parameters.

In the light of your answers, explain the reason why the logistic regression can at all be used as a classification mechanism (i.e. for targets that are “discrete”, not continuous). State only those axioms of the probability theory which are directly relevant here.

Q.10: Draw diagrams in the parameters-space for the Lasso regression and the Ridge regression. The question now is to explain precisely what lies inside the square or circular region. In each case, draw an example path that might get traced during the gradient descent, and clearly explain why the progress occurs the way it does.

Q.11: Briefly explain how the idea of the logistic regression gets applied in the artificial neural networks (ANNs). Suppose that a training data-set has $c$ number of features, $r$ number of data-rows, and $M$ number of output bins (i.e. classification types). Assuming that the neural network does not carry any hidden layers, calculate the number of logistic regressions that would be performed in a single batch. Make suitable assumptions as necessary.

Q.12: State the most prominent limitation of the gradient descent methods. State the name of any one technique which can overcome this limitation.

Advise: To answer the first two questions, don’t refer to the programming books. In fact, don’t even rely too much on the usual textbooks. Even Wasserman skips over the topic and Stirzaker is inadquate. Kreyszig is barely OK. A recommended text (more rigorous but UG-level, and brief) for this topic is: “An Introduction to Probability and Statistics” (2015) Rohatgi and Saleh, Wiley.

Awww… Still with me?

If you read this far, chances are very bright that you are really^{really} desperately looking for a job in the data science field. And, as it so happens, I am also a very, very kind hearted person. I don’t like to disappoint nice, ambitious… err… “aspiring” people. So, let me offer you some real help before you decide to close this page (and this blog) forever.

Here is one question they might actually ask you during an interview—especially if the interviewer is an MBA:

A question they might actually ask you in an interview: What are the three V’s of big data? four? five?

(Yes, MBA’s do know arithmetic. At least, it was there on their CAT / GMAT entrance exams. Yes, you can use this question for your posts on Medium.com, AnalyticsVidhya, KDNuggets, TowardsDataScience, ComingFromDataScience, etc.)

A couple of notes:

1. I might come back and revise the questions to make them less ambiguous or more precise.
2. Also, please do drop a line if any of the questions is not valid, or shows a poor understanding on my part—this is easily possible.

A song I like:

[Credits listed in a random order. Good!]

(Hindi) “mausam kee sargam ko sun…”
Music: Jatin-Lalit
Singer: Kavita Krishnamoorthy
Lyrics: Majrooh Sultanpuri

History:

First written: Friday 14 June 2019 11:50:25 AM IST.
Published online: 2019.06.16 12:45 IST.
The songs section added: 2019.06.16 22:18 IST.

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# The Machine Learning as an Expert System

To cut a somewhat long story short, I think that I can “see” that Machine Learning (including Deep Learning) can actually be regarded as a rules-based expert system, albeit of a special kind.

I am sure that people must have written articles expressing this view. However, simple googling didn’t get me to any useful material.

I would deeply appreciate it if someone could please point out references in this direction. Thanks in advance.

BTW, here is a very neat infographic on AI: [^]; h/t [^]. … Once you finish reading it, re-read this post, please! Exactly once again, and only the first part—i.e., without recursion!. …

A song I like:

(Marathi) “visar preet, visar geet, visar bheT aapuli”
Music: Yashwant Dev
Lyrics: Shantaram Nandgaonkar

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# Learnability of machine learning is provably an undecidable?—part 3: closure

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.

Gosh! I gotta get this series out of my hand—and also head! ASAP, really!! … So, I am going to scrap the bits and pieces I had written for it earlier; they would have turned this series into a 4- or 5-part one. Instead, I am going to start entirely afresh, and I am going to approach this topic from an entirely different angle—a somewhat indirect but a faster route, sort of like a short-cut. Let’s get going.

Statements:

Open any article, research paper, book or a post, and what do you find? Basically, all these consist of sentences after sentences. That is, a series of statements, in a way. That’s all. So, let’s get going at the level of statements, from a “logical” (i.e. logic-thoretical) point of view.

Statements are made to propose or to identify (or at least to assert) some (or the other) fact(s) of reality. That’s what their purpose is.

The conceptual-level consciousness as being prone to making errors:

Coming to the consciousness of man, there are broadly two levels of cognition at which it operates: the sensory-perceptual, and the conceptual.

Examples of the sensory-perceptual level consciousness would consist of reaching a mental grasp of such facts of reality as: “This object exists, here and now;” “this object has this property, to this much degree, in reality,” etc. Notice that what we have done here is to take items of perception, and put them into the form of propositions.

Propositions can be true or false. However, at the perceptual level, a consciousness has no choice in regard to the truth-status. If the item is perceived, that’s it! It’s “true” anyway. Rather, perceptions are not subject to a test of truth- or false-hoods; they are at the very base standards of deciding truth- or false-hoods.

A consciousness—better still, an organism—does have some choice, even at the perceptual level. The choice which it has exists in regard to such things as: what aspect of reality to focus on, with what degree of focus, with what end (or purpose), etc. But we are not talking about such things here. What matters to us here is just the truth-status, that’s all. Thus, keeping only the truth-status in mind, we can say that this very idea itself (of a truth-status) is inapplicable at the purely perceptual level. However, it is very much relevant at the conceptual level. The reason is that at the conceptual level, the consciousness is prone to err.

The conceptual level of consciousness may be said to involve two different abilities:

• First, the ability to conceive of (i.e. create) the mental units that are the concepts.
• Second, the ability to connect together the various existing concepts to create propositions which express different aspects of the truths pertaining to them.

It is possible for a consciousness to go wrong in either of the two respects. However, mistakes are much more easier to make when it comes to the second respect.

Homework 1: Supply an example of going wrong in the first way, i.e., right at the stage of forming concepts. (Hint: Take a concept that is at least somewhat higher-level so that mistakes are easier in forming it; consider its valid definition; then modify its definition by dropping one of its defining characteristics and substituting a non-essential in it.)

Homework 2: Supply a few examples of going wrong in the second way, i.e., in forming propositions. (Hint: I guess almost any logical fallacy can be taken as a starting point for generating examples here.)

Truth-hood operator for statements:

As seen above, statements (i.e. complete sentences that formally can be treated as propositions) made at the conceptual level can, and do, go wrong.

We therefore define a truth-hood operator which, when it operates on a statement, yields the result as to whether the given statement is true or non-true. (Aside: Without getting into further epistemological complexities, let me note here that I reject the idea of the arbitrary, and thus regard non-true as nothing but a sub-category of the false. Thus, in my view, a proposition is either true or it is false. There is no middle (as Aristotle said), or even an “outside” (like the arbitrary) to its truth-status.)

Here are a few examples of applying the truth-status (or truth-hood) operator to a statement:

• Truth-hood[ California is not a state in the USA ] = false
• Truth-hood[ Texas is a state in the USA ] = true
• Truth-hood[ All reasonable people are leftists ] = false
• Truth-hood[ All reasonable people are rightists ] = false
• Truth-hood[ Indians have significantly contributed to mankind’s culture ] = true
• etc.

For ease in writing and manipulation, we propose to give names to statements. Thus, first declaring

A: California is not a state in the USA

and then applying the Truth-hood operator to “A”, is fully equivalent to applying this operator to the entire sentence appearing after the colon (:) symbol. Thus,

Truth-hood[ A ] <==> Truth-hood[ California is not a state in the USA ] = false

Just a bit of the computer languages theory: terminals and non-terminals:

To take a short-cut through this entire theory, we would like to approach the idea of statements from a little abstract perspective. Accordingly, borrowing some terminology from the area of computer languages, we define and use two types of symbols: terminals and non-terminals. The overall idea is this. We regard any program (i.e. a “write-up”) written in any computer-language as consisting of a sequence of statements. A statement, in turn, consists of certain well-defined arrangement of words or symbols. Now, we observe that symbols (or words) can be  either terminals or non-terminals.

You can think of a non-terminal symbol in different ways: as higher-level or more abstract words, as “potent” symbols. The non-terminal symbols have a “definition”—i.e., an expansion rule. (In CS, it is customary to call an expansion rule a “production” rule.) Here is a simple example of a non-terminal and its expansion:

• P => S1 S2

where the symbol “=>” is taken to mean things like: “is the same as” or “is fully equivalent to” or “expands to.” What we have here is an example of an abstract statement. We interpret this statement as the following. Wherever you see the symbol “P,” you may substitute it using the train of the two symbols, S1 and S2, written in that order (and without anything else coming in between them).

Now consider the following non-terminals, and their expansion rules:

• P1 => P2 P S1
• P2 => S3

The question is: Given the expansion rules for P, P1, and P2, what exactly does P1 mean? what precisely does it stand for?

• P1 => (P2) P S1 => S3 (P) S1 => S3 S1 S2 S1

In the above, we first take the expansion rule for P1. Then, we expand the P2 symbol in it. Finally, we expand the P symbol. When no non-terminal symbol is left to expand, we arrive at our answer that “P1” means the same as “S3 S1 S2 S1.” We could have said the same fact using the colon symbol, because the colon (:) and the “expands to” symbol “=>” mean one and the same thing. Thus, we can say:

• P1: S3 S1 S2 S1

The left hand-side and the right hand-side are fully equivalent ways of saying the same thing. If you want, you may regard the expression on the right hand-side as a “meaning” of the symbol on the left hand-side.

It is at this point that we are able to understand the terms: terminals and non-terminals.

The symbols which do not have any further expansion for them are called, for obvious reasons, the terminal symbols. In contrast, non-terminal symbols are those which can be expanded in terms of an ordered sequence of non-terminals and/or terminals.

We can now connect our present discussion (which is in terms of computer languages) to our prior discussion of statements (which is in terms of symbolic logic), and arrive at the following correspondence:

The name of every named statement is a non-terminal; and the statement body itself is an expansion rule.

This correspondence works also in the reverse direction.

You can always think of a non-terminal (from a computer language) as the name of a named proposition or statement, and you can think of an expansion rule as the body of the statement.

Easy enough, right? … I think that we are now all set to consider the next topic, which is: liar’s paradox.

The liar paradox is a topic from the theory of logic [^]. It has been resolved by many people in different ways. We would like to treat it from the viewpoint of the elementary computer languages theory (as covered above).

The simplest example of the liar paradox is , using the terminology of the computer languages theory, the following named statement or expansion rule:

• A: A is false.

Notice, it wouldn’t be a paradox if the same non-terminal symbol, viz. “A” were not to appear on both sides of the expansion rule.

To understand why the above expansion rule (or “definition”) involves a paradox, let’s get into the game.

Our task will be to evaluate the truth-status of the named statement that is “A”. This is the “A” which comes on the left hand-side, i.e., before the colon.

In symbolic logic, a statement is nothing but its expansion; the two are exactly and fully identical, i.e., they are one and the same. Accordingly, to evaluate the truth-status of “A” (the one which comes before the colon), we consider its expansion (which comes after the colon), and get the following:

• Truth-hood[ A ] = Truth-hood[ A is false ] = false           (equation 1)

Alright. From this point onward, I will drop explicitly writing down the Truth-hood operator. It is still there; it’s just that to simplify typing out the ensuing discussion, I am not going to note it explicitly every time.

Anyway, coming back to the game, what we have got thus far is the truth-hood status of the given statement in this form:

• A: “A is false”

Now, realizing that the “A” appearing on the right hand-side itself also is a non-terminal, we can substitute for its expansion within the aforementioned expansion. We thus get to the following:

• A: “(A is false) is false”

We can apply the Truth-hood operator to this expansion, and thereby get the following: The statement which appears within the parentheses, viz., the “A is false” part, itself is false. Accordingly, the Truth-hood operator must now evaluate thus:

• Truth-hood[ A ] = Truth-hood[ A is false] = Truth-hood[ (A is false) is false ] = Truth-hood[ A is true ] = true            (equation 2)

Fun, isn’t it? Initially, via equation 1, we got the result that A is false. Now, via equation 2, we get the result that A is true. That is the paradox.

But the fun doesn’t stop there. It can continue. In fact, it can continue indefinitely. Let’s see how.

If only we were not to halt the expansions, i.e., if only we continue a bit further with the game, we could have just as well made one more expansion, and got to the following:

• A: ((A is false) is false) is false.

The Truth-hood status of the immediately preceding expansion now is: false. Convince yourself that it is so. Hint: Always expand the inner-most parentheses first.

Homework 3: Convince yourself that what we get here is an indefinitely long alternating sequence of the Truth-hood statuses that: A is false, A is true, A is false, A is true

What can we say by way of a conclusion?

Conclusion: The truth-status of “A” is not uniquely decidable.

The emphasis is on the word “uniquely.”

We have used all the seemingly simple rules of logic, and yet have stumbled on to the result that, apparently, logic does not allow us to decide something uniquely or meaningfully.

Liar’s paradox and the set theory:

The importance of the liar paradox to our present concerns is this:

Godel himself believed, correctly, that the liar paradox was a semantic analogue to his Incompleteness Theorem [^].

Go read the Wiki article (or anything else on the topic) to understand why. For our purposes here, I will simply point out what the connection of the liar paradox is to the set theory, and then (more or less) call it a day. The key observation I want to make is the following:

You can think of every named statement as an instance of an ordered set.

What the above key observation does is to tie the symbolic logic of proposition with the set theory. We thus have three equivalent ways of describing the same idea: symbolic logic (name of a statement and its body), computer languages theory (non-terminals and their expansions to terminals), and set theory (the label of an ordered set and its enumeration).

As an aside, the set in question may have further properties, or further mathematical or logical structures and attributes embedded in itself. But at its minimal, we can say that the name of a named statement can be seen as a non-terminal, and the “body” of the statement (or the expansion rule) can be seen as an ordered set of some symbols—an arbitrarily specified sequence of some (zero or more) terminals and (zero or more) non-terminals.

Two clarifications:

• Yes, in case there is no sequence in a production at all, it can be called the empty set.
• When you have the same non-terminal on both sides of an expansion rule, it is said to form a recursion relation.

An aside: It might be fun to convince yourself that the liar paradox cannot be posed or discussed in terms of Venn’s diagram. The property of the “sheet” on which Venn’ diagram is drawn is, by some simple intuitive notions we all bring to bear on Venn’s diagram, cannot have a “recursion” relation.

Yes, the set theory itself was always “powerful” enough to allow for recursions. People like Godel merely made this feature explicit, and took full “advantage” of it.

Recursion, the continuum, and epistemological (and metaphysical) validity:

In our discussion above, I had merely asserted, without giving even a hint of a proof, that the three ways (viz., the symbolic logic of statements or  propositions, the computer languages theory, and the set theory) were all equivalent ways of expressing the same basic idea (i.e. the one which we are concerned about, here).

I will now once again make a few more observations, but without explaining them in detail or supplying even an indication of their proofs. The factoids I must point out are the following:

• You can start with the natural numbers, and by using simple operations such as addition and its inverse, and multiplication and its inverse, you can reach the real number system. The generalization goes as: Natural to Whole to Integers to Rationals to Reals. Another name for the real number system is: the continuum.
• You can use the computer languages theory to generate a machine representation for the natural numbers. You can also mechanize the addition etc. operations. Thus, you can “in principle” (i.e. with infinite time and infinite memory) represent the continuum in the CS terms.
• Generating a machine representation for natural numbers requires the use of recursion.

Finally, a few words about epistemological (and metaphysical) validity.

• The concepts of numbers (whether natural or real) have a logical precedence, i.e., they come first. The entire arithmetic and the calculus must come before does the computer-representation of some of their concepts.
• A machine-representation (or, equivalently, a set-theoretic representation) is merely a representation. That is to say, it captures only some aspects or attributes of the actual concepts from maths (whether arithmetic or the continuum hypothesis). This issue is exactly like what we saw in the first and second posts in this series: a set is a concrete collection, unlike a concept which involves a consciously cast unit perspective.
• If you try to translate the idea of recursion into the usual cognitive terms, you get absurdities such as: You can be your child, literally speaking. Not in the sense that using scientific advances in biology, you can create a clone of yourself and regard that clone to be both yourself and your child. No, not that way. Actually, such a clone is always your twin, not child, but still, the idea here is even worse. The idea here is you can literally father your own self.
• Aristotle got it right. Look up the distinction between completed processes and the uncompleted ones. Metaphysically, only those objects or attributes can exist which correspond to completed mathematical processes. (Yes, as an extension, you can throw in the finite limiting values, too, provided they otherwise do mean something.)
• Recursion by very definition involves not just absence of completion but the essence of the very inability to do so.

Closure on the “learnability issue”:

Homework 4: Go through the last two posts in this series as well as this one, and figure out that the only reason that the set theory allows a “recursive” relation is because a set is, by the design of the set theory, a concrete object whose definition does not have to involve an epistemologically valid process—a unit perspective as in a properly formed concept—and so, its name does not have to stand for an abstract mentally held unit. Call this happenstance “The Glaring Epistemological Flaw of the Set Theory” (or TGEFST for short).

Homework 5: Convince yourself that any lemma or theorem that makes use of Godel’s Incompleteness Theorem is necessarily based on TGEFST, and for the same reason, its truth-status is: it is not true. (In other words, any lemma or theorem based on Godel’s theorem is an invalid or untenable idea, i.e., essentially, a falsehood.)

Homework 6: Realize that the learnability issue, as discussed in Prof. Lev Reyzin’s news article (discussed in the first part of this series [^]), must be one that makes use of Godel’s Incompleteness Theorem. Then convince yourself that for precisely the same reason, it too must be untenable.

[Yes, Betteridge’s law [^] holds.]

Other remarks:

Remark 1:

As “asymptotical” pointed out at the relevant Reddit thread [^], the authors themselves say, in another paper posted at arXiv [^] that

While this case may not arise in practical ML applications, it does serve to show that the fundamental definitions of PAC learnability (in this case, their generalization to the EMX setting) is vulnerable in the sense of not being robust to changing the underlying set theoretical model.

What I now remark here is stronger. I am saying that it can be shown, on rigorously theoretical (epistemological) grounds, that the “learnability as undecidable” thesis by itself is, logically speaking, entirely and in principle untenable.

Remark 2:

Another point. My preceding conclusion does not mean that the work reported in the paper itself is, in all its aspects, completely worthless. For instance, it might perhaps come in handy while characterizing some tricky issues related to learnability. I certainly do admit of this possibility. (To give a vague analogy, this issue is something like running into a mathematically somewhat novel way into a known type of mathematical singularity, or so.) Of course, I am not competent enough to judge how valuable the work of the paper(s) might turn out to be, in the narrow technical contexts like that.

However, what I can, and will say is this: the result does not—and cannot—bring the very learnability of ANNs itself into doubt.

Phew! First, Panpsychiasm, and immediately then, Learnability and Godel. … I’ve had to deal with two untenable claims back to back here on this blog!

… Code! I have to write some code! Or write some neat notes on ML in LaTeX. Only then will, I guess, my head stop aching so much…

Honestly, I just downloaded TensorFlow yesterday, and configured an environment for it in Anaconda. I am excited, and look forward to trying out some tutorials on it…

BTW, I also honestly hope that I don’t run into anything untenable, at least for a few weeks or so…

…BTW, I also feel like taking a break… May be I should go visit IIT Bombay or some place in konkan. … But there are money constraints… Anyway, bye, really, for now…

A song I like:


text:
Article Text

RheumaTutor. A case-based rheumatological training program. Stefan Schewe. ($50, DM 98, students$34.00, DM 68). Wiesbaden: Ullstein Medical Verlag, 1999. ISBN 3-86126-934-1 .
1. HEIN J BERNELOT MOENS
1. Medisch Spectrum Twente, PO Box 50 000, 7500 KA Enschede, the Netherlands

## Statistics from Altmetric.com

Two interactive rheumatology tutors on CD-ROM

The widespread availability of information and communication technology offers new possibilities for education of professionals and of patients. One of these possibilities is computer assisted learning.1 Ten years ago, research focused on the development of computer assisted diagnosis and text based help to rheumatologists and others.2 Despite the initial enthusiasm about artificial intelligence (AI), resulting in several diagnostic expert systems, in routine care a precise diagnosis is rarely an important issue and as a consequence these systems are hardly used. The experience with expert systems, however, evolved into systems intended for educational purposes. “AI/Learn”, based on the “AI/Rheum” expert system and exploiting a videodisk with an image library, proved to be an effective means of teaching rheumatology.3 The same image library was used to teach occupational and physical therapy students with the “HP-Rheum” system (HP: Health Professionals).4 In recent years the introduction of sound, images, and movies on PCs, and the replacement of videodisks by CD-ROM facilitated production and access to these methods. The products resulting from the AI/Rheum project are not yet available on this medium, but two European rheumatologists have now completed educational systems on CD-ROM. They are to be honoured for their pioneering efforts. The main characteristics of the systems are summarised in table 1, details are discussed below.

Table 1

Main characteristics of the two CD-ROM tutors

The widespread availability of information and communication technology offers new possibilities for education of professionals and of patients. One of these possibilities is computer assisted learning.1 Ten years ago, research focused on the development of computer assisted diagnosis and text based help to rheumatologists and others.2 Despite the initial enthusiasm about artificial intelligence (AI), resulting in several diagnostic expert systems, in routine care a precise diagnosis is rarely an important issue and as a consequence these systems are hardly used. The experience with expert systems, however, evolved into systems intended for educational purposes. “AI/Learn”, based on the “AI/Rheum” expert system and exploiting a videodisk with an image library, proved to be an effective means of teaching rheumatology.3 The same image library was used to teach occupational and physical therapy students with the “HP-Rheum” system (HP: Health Professionals).4 In recent years the introduction of sound, images, and movies on PCs, and the replacement of videodisks by CD-ROM facilitated production and access to these methods. The products resulting from the AI/Rheum project are not yet available on this medium, but two European rheumatologists have now completed educational systems on CD-ROM. They are to be honoured for their pioneering efforts. The main characteristics of the systems are summarised in table 1, details are discussed below.

This CD is dedicated to education and has won several awards. It uses text, images, sound, and video. The first time user receives clear instructions, and is invited to browse a wide variety of materials and subjects. The interface is straightforward and easy to use.

The materials can be accessed using keywords and text search, or by structured menus. The program provides excellent instructions on history, physical examination, and local injections. Although information on specific diseases is included, methodological topics have more emphasis. This property makes the system particularly useful as an educational tool for medical students, general practitioners, or for paramedical personnel working in rheumatology units. For rheumatologists the CD-ROM is fun to look around, but it is not intended to and cannot replace a textbook at a specialist level.

The quality of the photographs and radiographs is excellent. The video captures are presented in a small window, but do suffice to illustrate physical examination or local injections. There are 10 amusing quizzes, of adequate difficulty for final year medical students or specialist trainees. The CD-ROM is also equipped with a glossary, a set of patient problems presented in a hospital clinic or in a general practitioners' office, and an “Arthritis Expert”. The latter is an elementary expert system that can generate diagnostic hypotheses.

In our hospital final year medical students are now using theRheumatology Tutor during the five day rheumatology course. They are given an instructions sheet to get them started on the first day. The first group enjoyed the program, and agreed that it was a more efficient introduction to rheumatology than a textbook or than instructions by a busy specialist during an outpatient clinic.

A drawback of the system is that it aims at a wide view of rheumatology and therefore is vulnerable for inconsistencies or details about which different opinions may exist. In an earlier review, for example, the different names of drugs between the US and the UK were mentioned as a minor disadvantage.5

This CD-ROM is centred on a set of 25 cases with various rheumatic diseases. The user-interface has a common format, but it takes a couple of cases to learn to navigate through the program.

Cases may be selected from a list, or be chosen randomly by the system. A patient is presented with a chief complaint and data on age, sex and profession. The student is asked to select further questions from a standard list. Subsequently, physical examination is mimicked by photographs of hands, feet, or other parts of the body. The user marks interpretations in standard lists of findings or on a figure, a process that is critiqued and scored by the system. Both correct and incorrect answers are explained and illustrated. The same process is repeated for radiological interpretation, laboratory testing, differential diagnosis, and therapeutic advice. The system's critiquing is supported by texts, based on the Quality Guidelines of the German Rheumatological Association. The quality of photographs and radiographs is adequate for the purpose of the program.

The degree of difficulty in the cases makes it suitable only for students who have at least basic knowledge of rheumatology. Even for an experienced rheumatologist it is hard to obtain the maximum scores. This is because of the degree of detail in which the system requires answers, which also slows down the process of going through a case.

This program is limited to case-based teaching. It lacks direct access to instructions on physical examination or interpretation, although the critiquing by the system substitutes for this. As a consequence of its architecture, originally based on an expert system, it uses long lists of possible signs and symptoms, which are sometimes cumbersome to work with. The author solved this by clustering findings into groups, which can be unfolded by clicking on them.

On the positive side, the system forces the student to observe accurately whatever can be seen, and provides adequate corrections for misinterpretations and funny rewards for the user's efforts. The scoring method, including rewards for cost effectiveness, is interesting, although it is not easy to see how a particular score was reached. A majority of German students evaluated the system positively, as long as it was part of a clinical course.6 It is sad that an official version of the system is only available in German, which limits its use to speakers of that language (an unofficial English translation is available from the author).

## Conclusion

These CD-ROM tutors illustrate both the vast potential and the wide range of possibilities offered by multimedia computers. From a practical point of view, Armstrong's Rheumatology Tutor is the most useful tool for medical students. As an adjunct to an elective week it can help to fill the gaps that generally exist in training based on outpatient clinics or ward rounds. As a case-based tutor, the technology of Schewe's program is more interesting. However, I would prefer the latter to focus less on detail and more on common symptoms. Also, the inclusion of instructions in the form of video-clips would make it even better suited to the needs of medical students.

These CD-ROMs mark the beginning of a new era of education in rheumatology. They will complement clinical teaching, but—despite my enthusiasm—I do not believe that they will replace it!

Two interactive rheumatology tutors on CD-ROM

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Constraining high-redshift X-ray sources with next generation 21-cm power spectrum measurements

@article{EwallWice2016ConstrainingHX,
title={Constraining high-redshift X-ray sources with next generation 21-cm power spectrum measurements},
author={Aaron M. Ewall-Wice and Jacqueline N. Hewitt and Andrei Mesinger and Joshua S. Dillon and Adrian Liu and Jonathan C. Pober},
journal={Monthly Notices of the Royal Astronomical Society},
year={2016},
volume={458},
pages={2710-2724}
}
• Published 12 November 2015
• Physics
• Monthly Notices of the Royal Astronomical Society
We use the Fisher matrix formalism and semi-numerical simulations to derive quantitative predictions of the constraints that power spectrum measurements on next-generation interferometers, such as the Hydrogen Epoch of Reionization Array (HERA) and the Square Kilometre Array (SKA), will place on the characteristics of the X-ray sources that heated the high redshift intergalactic medium. Incorporating observations between $z=5$ and $z=25$, we find that the proposed 331 element HERA and SKA phase…

Figures and Tables from this paper

Constraining the redshifted 21-cm signal with the unresolved soft X-ray background
• Physics
• 2017
We use the observed unresolved cosmic X-ray background (CXRB) in the 0.5-2 keV band and existing upper limits on the 21-cm power spectrum to constrain the high-redshift population of X-ray sources,
Quantifying density-ionization correlations with the 21-cm power spectrum
• Physics
• 2020
The Epoch of Reionization (EoR)---when neutral hydrogen in the intergalactic medium was systematically ionized---is a period in our Universe's history that is currently poorly understood. However, a
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First Results from HERA Phase I: Upper Limits on the Epoch of Reionization 21 cm Power Spectrum
• Physics
The Astrophysical Journal
• 2021
We report upper limits on the Epoch of Reionization 21 cm power spectrum at redshifts 7.9 and 10.4 with 18 nights of data (∼36 hr of integration) from Phase I of the Hydrogen Epoch of Reionization
The impact of non-Gaussianity on the Epoch of Reionization parameter forecast using 21-cm power-spectrum measurements
• Physics
• 2020
Measurements of the Epoch of Reionization (EoR) 21-cm signal hold the potential to constrain models of reionization. In this paper we consider a reionization model with three astrophysical parameters
Emulating Simulations of Cosmic Dawn for 21 cm Power Spectrum Constraints on Cosmology, Reionization, and X-Ray Heating
• Physics, Computer Science
• 2017
This work uses machine learning algorithms to build a fast emulator that can accurately mimic an expensive simulation of the 21 cm signal across a wide parameter space, and embeds it within a Markov Chain Monte Carlo framework in order to perform Bayesian parameter constraints over a large number of model parameters.
Predictions for measuring the 21-cm multifrequency angular power spectrum using SKA-Low
• Physics
• 2020
The light-cone (LC) effect causes the mean as well as the statistical properties of the redshifted 21-cm signal $T_{\rm b}(\hat{\bf n},\nu)$ to change with frequency $\nu$ (or cosmic time).
Robust velocity-induced acoustic oscillations at cosmic dawn
The redshifted 21-cm line of hydrogen holds great potential for the study of cosmology, as it can probe otherwise unobservable cosmic epochs. In particular, measurements of the 21-cm power spectrum
Dark Cosmology: Investigating Dark Matter & Exotic Physics in the Dark Ages using the Redshifted 21-cm Global Spectrum
• Physics
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The Dark Ages, probed by the redshifted 21-cm signal, is the ideal epoch for a new rigorous test of the standard LCDM cosmological model. Divergences from that model would indicate new physics, such
The Evolution Of 21 cm Structure (EOS): public, large-scale simulations of Cosmic Dawn and reionization
• Physics
• 2016
We introduce the Evolution of 21-cm Structure (EOS) project: providing periodic, public releases of the latest cosmological 21-cm simulations. 21-cm interferometry is set to revolutionize studies of

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The epoch of the reionization (EoR) power spectrum is expected to evolve strongly with redshift, and it is this variation with cosmic history that will allow us to begin to place constraints on the
WHAT NEXT-GENERATION 21 cm POWER SPECTRUM MEASUREMENTS CAN TEACH US ABOUT THE EPOCH OF REIONIZATION
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A number of experiments are currently working toward a measurement of the 21 cm signal from the epoch of reionization (EoR). Whether or not these experiments deliver a detection of cosmological
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We consider the evolution of the sky-averaged 21-cm background during the early phases of structure formation. Using simple analytic models, we calculate the thermal and ionization histories,
FOREGROUNDS IN WIDE-FIELD REDSHIFTED 21 cm POWER SPECTRA
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Detection of 21 cm emission of H i from the epoch of reionization, at redshifts z > 6 ?> , is limited primarily by foreground emission. We investigate the signatures of wide-field measurements and an
THE IMPORTANCE OF WIDE-FIELD FOREGROUND REMOVAL FOR 21 cm COSMOLOGY: A DEMONSTRATION WITH EARLY MWA EPOCH OF REIONIZATION OBSERVATIONS
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In this paper we present observations, simulations, and analysis demonstrating the direct connection between the location of foreground emission on the sky and its location in cosmological power
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text:
# American Institute of Mathematical Sciences

## Global bounded and unbounded solutions to a chemotaxis system with indirect signal production

 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France

Received  October 2018 Revised  February 2019 Published  July 2019

The well-posedness of a chemotaxis system with indirect signal production in a two-dimensional domain is shown, all solutions being global unlike for the classical Keller-Segel chemotaxis system. Nevertheless, there is a threshold value $M_c$ of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below $M_c$ while there are unbounded solutions starting from initial conditions having a mass exceeding $M_c$. This result extends to arbitrary two-dimensional domains a previous result of Tao & Winkler (2017) obtained for radially symmetric solutions to a simplified version of the model in a ball and relies on a different approach involving a Liapunov functional.

Citation: Philippe Laurençot. Global bounded and unbounded solutions to a chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019145
##### References:

show all references


text:
## Tuesday, 26 March 2013

### Bayes 250

Of course, a Bayes 500 would be even better and more powerful (although more fuel-consuming...). But: the best we can afford (strictly for time constraints $-$ time since publication, that is) is a Bayes 250.

This is what promises to be a very, very good conference to celebrate the anniversary of the publication of the "Essay towards solving a Problem in the Doctrine of Chances" by the Reverend Thomas Bayes, in the Philosophical Transactions of the Royal Society of London.

The latest number of Significance actually published a paper arguing for a legitimate recognition of Richard Price's contribution to the paper (I think there's possibly more than something true about this, although I thought that the article was trying too hard to make the point).

Anyway, the programme of the two-day conference (to be held on 19-20 June at the Royal Statistical Society in London) is quite impressive! I'm planning to go, although the fee is a bit high $-$ perhaps I'll try to see if I can get some money from random savers across Europe...


text:
Competitions

# Permutations

Given a positive integer n, print all permutations of the integers from 1 to n in lexicographical order.

#### Input

One positive integer n (1n8).

#### Output

Print all permutations of the integers from 1 to n in lexicographical order. Print each permutation on a separate line.

Time limit 1 second
Memory limit 128 MiB
Input example #1
3
Output example #1
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1


text:
# Problems from Operations and limits with sequences

Calculate the limit of the following sequences;

a) $$a_n=\dfrac{(-1)^n\cdot(2n+3)}{9n^3-11n+3}$$

b) $$a_n=\dfrac{(-1)^nn^2+7n^3+21}{4n^3-8}$$

See development and solution

### Development:

a) The sequence is given by the product of the sequences $$a_n=(-1)^n$$ and $$b_n=\dfrac{(2n+3)}{9n^3-11n+3}$$.

The first one is bounded and the second one has limit $$0$$. Applying the presented result we obtain that the product has limit $$0$$.

b) In this case we cannot apply the result used to the previous paragraph. We can say that the limit of the sum is the sum of limits. Namely, $$\lim_{n \to \infty}{a_n}=\lim_{n \to \infty}{\dfrac{(-1)^nn^2}{4n^3-8}}+\lim_{n \to \infty}{\dfrac{7n^3+21}{4n^3-8}}$$$Now we can apply the result commented upon regarding the first limit. Since it corresponds to the product of the sequences $$\{(-1)^n\}_{n\in\mathbb{N}}$$ and $$\Big\{\dfrac{7n^3+21}{4n^3-8} \Big\}_{n\in\mathbb{N}}$$. Then the first limit of the sum is $$0$$. The second limit is $$\dfrac{7}{4}$$. This way we obtain the desired result $$\lim_{n \to \infty}{a_n}= \dfrac{7}{4}$$. ### Solution: a) The limit is $$0$$. b) The limit is $$\dfrac{7}{4}$$. Hide solution and development Calculate the limit of the sum, subtraction, multiplication and division, when possible, of the following pairs of sequences; a) $$a_n=\dfrac{n^3-3}{n-6}$$ and $$b_n=\dfrac{3n^4-7}{3n^2+n-1}$$ b) $$a_n=2+3^n$$ and $$b_n=\dfrac{1}{5^n}$$ See development and solution ### Development: a) The limit of the two sequences is infinite and for the defined arithmetical rules we see that the limit of the sum and that of the product are infinite. To calculate the limit of the subtraction and of the quotient we must solve the indeterminate form. For the limit of the subtraction; $$\lim_{n \to \infty}{\dfrac{n^3-3}{n-6}-\dfrac{3n^4-7}{3n^2+n-1}}=\lim_{n \to \infty}{\Big(\dfrac{(n^3-3)(3n^2+n-1)-(3n^4-7)(n-6)}{(n-6)(3n^2+n-1)}\Big)}=$$$ $$=\lim_{n \to \infty}{\Big(\dfrac{19n^4-n^3-9n^2+4n-39}{3n^3-5n^2-7n+6}\Big)}=\infty$$$For the limit of the quotient; $$\lim_{n \to \infty}{\dfrac{n^3-3}{n-6}/\dfrac{3n^4-7}{3n^2+n-1}}=\lim_{n \to \infty}{\Big(\dfrac{(n^3-3)(3n^2+n-1)}{(n-6)(3n^4-7)}\Big)}=$$$ $$=\lim_{n \to \infty}{\Big(\dfrac{3n^5+n^4-n^3-9n^2-3n+3}{3n^5-18n^4-7n+42}\Big)}=1$$$b) The $$a_n=2+3^n$$ has limit infinity and the $$b_n=\dfrac{1}{5^n}$$ has limit $$0$$. Following the arithmetical rules, the limit of the sum and subtraction are infinity. To calculate the limit of the product; $$\lim_{n \to \infty}{(2+3^n)\cdot\dfrac{1}{5^n}}=\lim_{n \to \infty}{\dfrac{2}{5^n}}+\lim_{n \to \infty}{\dfrac{3^n}{5^n}}=0+0=0$$$

To calculate the limit of the quotient; $$\lim_{n \to \infty}{(2+3^n)/\dfrac{1}{5^n}}=\lim_{n \to \infty}{2\cdot5^n}+\lim_{n \to \infty}{3^n\cdot5^n}=\infty+\infty=\infty$$\$

### Solution:

a) The limit of the sum, subtraction and multiplication is infinite. The limit of the quotient is $$1$$.

b) The limit of the sum, subtraction and quotient is infinite. The limit of the product is $$0$$.

Hide solution and development


text:
lilypond-devel
[Top][All Lists]

## Re: Slanted Beams thickness

 From: Luca Fascione Subject: Re: Slanted Beams thickness Date: Fri, 25 Mar 2022 09:45:07 +0100

Sorry, forgot to say: instead of correcting with 1/cos(\theta) I wonder if
correcting with 1/cos(\theta/2) would be an idea?

sl2 = sl / (1+sqrt(1+sl*sl)) // tan(\theta/2)
th *= sqrt(1+sl2*sl2)

HTH
L

On Fri, Mar 25, 2022 at 9:35 AM Luca Fascione <l.fascione@gmail.com> wrote:

> This video shows Hans Kuehner at work
>
>
> at 4m36 shows beams being engraved, he appears to keep the instrument
> orthogonal to the line direction,
> which makes Valentin's formula appropriate to capture this process.
>
> (I love it when it goes "What happens when you make mistakes?" -> "I
> _don't_ make mistakes!" (7m59 or so) )
>
> As Werner said, I'd have expected something more of a halfwayhouse,
> because in my mind I was expecting more or a nib
> pen feel to this, but even looking at photography based processes
> there seems to be no evidence that any of that technique
> would influence this.
>
> I feel that for more organic looking fonts (such as lilyjazz) this might
> want to change, but I guess that's a somewhat different
> topic.
>
> L
>
> On Fri, Mar 25, 2022 at 8:10 AM Jean Abou Samra <jean@abou-samra.fr>
> wrote:
>
>> Le 25/03/2022 à 01:44, Valentin Petzel a écrit :
>> > Hello,
>> >
>> > Lilypond handles slanted Beams in a geometrically weird way, that is,
>> the
>> > thickness is not measured as the shortest distance between the opposing
>> sides
>> > of the boundary, but as vertical distance. This results in Beams getting
>> > optically thinner and closer the higher the slope is. But we can very
>> easily
>> > factor this out by adjusting the thickness to the slope. In fact if we
>> want to
>> > achieve a real thickness theta the adjusted thickness would need to be
>> > theta·sqrt(1 + slope²). See attached an experimental example.
>>
>>
>>
>> Did you look into engraving literature to back this up?
>> Given the amount of effort put by Han-Wen & Jan in beam
>> formatting, I have trouble imagining this being just
>> an oversight.
>>
>> Jean
>>
>>
>>


text:
Question

# The function f(x) is the total amount spent at a store, when purchasing x items that are $5 each and the items are not taxable. What is the practical domain for the function f(x)? all positive, real, whole nuvbers Factors and multiples ANSWERED asked 2021-08-06 The function f(x) is the total amount spent at a store, when purchasing x items that are$5 each and the items are not taxable.
What is the practical domain for the function f(x)?
1) all positive integers
2) all real numbers
3) all whole numbers
4) all positive integers that are multiples of 5.

Therefore the practical domain is $$\left\{0,1,2,3,- - -\right\}$$, which is set of whole numbers.