Toolnut

My Proof of one of L'Hôpital's Rules

The proof provided for the ∞/∞ case in L'Hôpital's rule article is lacking. Please review my proof on my user page and comment on it below. Thanks.Toolnut (talk) 10:00, 1 December 2011 (UTC)

Negation of a statement on IVT discussion

Hi, possibly this was addressed later in the text, and I just couldn't find it. It appears that you might have had some mistakes in your negation of the definition of existence of a limit, and this may have contributed to the confusion. First of all, it looks like a quantifier was missing. The correct starting point is:

∀ϵ>0 ∃δ>0 ∀x ( |x|<δ⇒|f(x)-L|<ϵ )

I removed all "0<" as they are superfluous. The next line appears to be an attempt at negation:

∃ϵ>0 ∀δ>0 ( |x|<δ⇒|f(x)-L|≥ϵ ) however in this line, the A⇒B statement hasn't been negated properly. The proper negation ¬(A⇒B) is A∧¬B.

Thus the negation "¬( |x|<δ⇒|f(x)-L|<ϵ )" is "|x|<δ and ¬(|f(x)-L|<ϵ)" which means "|x|<δ and |f(x)-L|≥ϵ" I'll include the whole step by step process of negation:

¬(∀ϵ>0 ∃δ>0 ∀x( |x|<δ⇒|f(x)-L|<ϵ ))

∃ϵ>0 ¬(∃δ>0 ∀x( |x|<δ⇒|f(x)-L|<ϵ ))

∃ϵ>0 ∀δ>0 ¬∀x( |x|<δ⇒|f(x)-L|<ϵ )

∃ϵ>0 ∀δ>0 ∃x ¬( |x|<δ⇒(|f(x)-L|<ϵ) )

∃ϵ>0 ∀δ>0 ∃x ( |x|<δ∧¬(|f(x)-L|<ϵ) )

∃ϵ>0 ∀δ>0 ∃x ( |x|<δ∧|f(x)-L|≥ϵ )

Hopefully I haven't let any typos slip through. Again, if you recognized this later, I apologize for not having the patience to find it. Rschwieb (talk) 02:37, 20 October 2011 (UTC)

Thank you so much: I don't know what you're apologizing for, I'm new to this and haven't noticed the differences b/w my derivation and the one provided to me by User:Sławomir_Biały until you brought it to my attention. Right now, I'm reading the Wikipedia articles on logic to gain sufficient understanding to do this on my own. I never encountered this kind of logic in any of my Boolean algebra learning relevant to digital circuits and probability theory. Thanks again.Toolnut (talk) 03:07, 20 October 2011 (UTC)

Well, sometimes if I make a suggestion without reading all the text, some people will get angry that I didn't see it was already mentioned before, so I just wanted to cover myself on that :) Glad you're having fun learning this basic logic. With a little practice, it's not hard to master, and it will make your thinking clearer. The number one enemy of learning is muddled thinking.

So you have actually learned applying Boolean algebra to circuit design? That's great! I'm an abstract algebraist who is interested in applications of algebra, so I'm kind of going in the other direction: pure math --> applications. I've read about representations of circuits with Boolean algebras, but I don't know how deep the theory goes. Rschwieb (talk) 13:59, 20 October 2011 (UTC)

Functions and their Limits

Hi, I'd like to offer some of what I know related to the last post you put in the Math Project page. You were speculating about three types of functions. Really, functions come in all different sorts. The variety is really too great to describe with only a few classes. I'm going to describe now the "big picture", which hopefully will help give you a view of how big the scope of the matter is.

A function is a relationship between two sets ("a function f from a set X to a set Y"). It seems you are most familiar with functions where X and Y are both the set R, since that is where everyone usually starts. However it's important to realize X and Y can be any other set, including N, or in fact many other strange sets that don't even consist of numbers.

Now, when we want to talk about continuity or differentiability of a function, some ingredients need to be added to our spaces X and Y. We need a sense of when a point is "close" to a set, and the way this is achieved is by specifying topologies on X and Y. Without these topologies, it is impossible to talk about continuity. In this picture, "continuity" of a map means that "if x is close to the subset D of X, then f(x) is close to the subset f(D) of Y". This is very close to the description of continuity in the article we were looking at.

To repeat, "continuity" depends completely upon the topology of the set. If you change topologies, then suddenly a continuous function might become discontinuous in the new topology. Where is the topology in the functions from R into R that we were looking at? That would take a little explaining, but basically the epsilon-delta stuff is the topology in disguise.

Differentiability is even more special. Not only do you need a topology, but you also have to be able to add, subtract and multiply things in X and Y by scalars. "Differentiability" in a nutshell indicates the "smoothness" of a function. Derivatives, as you know, may become discontinuous if you go far enough (e.g., cuberoot(x) is continuous on R, but its first derivative is discontinuous at 0). Some functions though, have every order of derivative, for example, sin(x). This indicates sin(x) is "really smooth". Rschwieb (talk) 17:29, 20 October 2011 (UTC)

Welcome! The point I'm still trying to make regarding the limit of a function article is that, nearly all the examples given in that article belong to the class of functions that are differentiable except at a finite number of points; hence, my insistence that the bounded differentiability (I was wrong about continuity alone being enough) of such functions in a vanishingly small deleted neighborhood of a fixed point is sufficient for a one-sided limit to exist (even about points of discontinuity, so long as the function is defined in said neighborhood). I suggest that my third class of "pathological" functions be expanded to include the kind of functions Slawomir just described, those with infinite numbers of discontinuities and continuities, which class is not at all mentioned in the said article. What's really lacking there are functions in my second class, such as any mention of infinite series (which may be regarded as sequences involving sums) and their evaluation, especially those not derived from known functions, which can be even harder or impossible to evaluate in closed form. They are not discussed even in the companion article Limit of a sequence. By the way, I just thought of another type of limit not discussed in that article: asymptotic limits, such as Stirling's approximation of the factorial for large arguments.Toolnut (talk) 20:23, 20 October 2011 (UTC)

Ok, well this is certainly a potpourri of suggestions. I'll try to address them one at a time succinctly:

1. (the article focuses on functions differentiable except at a finite number of points): Do you wish to increase the variety of the article by pointing to exotic examples of continuous functions which are not differentiable anywhere? We could add a link to the article.
2. (something about "your classes of functions"): What do you hope to achieve by classifying functions this way? Is this related to the previous point I wrote?
3. (infinite series): An series is is indeed a limit of the sequence of partial sums. That could be mentioned after addressing how the limit of a sequence is also the limit of a function.
4. (asymptotic limits): There is an article devoted to this, so a link should be enough. We don't repeat information that has its own nice article.

I also have some advice about how to communicate with your fellow editors more clearly. One: try to succinctly introduce your points. Your above paragraph takes some patience to read. To me it looks like 50% content and 50% fluff. Two: stick to standard terminology (I don't think anybody knows what you mean by "a vanishingly small deleted neighborhood". A "neighborhood" is a term in topology. If you don't understand basic logic well then there is reason to doubt you have license to invoke topology.) Rschwieb (talk) 14:26, 21 October 2011 (UTC)

"Neighborhood" in topology only, really? This word comes up a lot in my textbooks, in the proofs of theorems having nothing to do with topology. Even my calculus book introduces it in its opening chapters. You're right about the "fluff," though: you may have no idea how many times I've edited the message you finally responded to; I was about to make even more changes to it, when you did: I'm in constant flux for improvements in all the work that I do; I'm a perfectionist.

Anyway, I have given this some more thought. It is easy to find sufficient conditions for the existence of a limit, but all of the ones I've come up with so far have been too restrictive to qualify as also necessary. The following is probably a more inclusive sufficiency for the existence of a limit:

• If a function is bounded, continuous, and monotonic in a sufficiently small one-sided deleted neighborhood of a point, its one-sided limit exists at that point.

This leaves out a function that is infinitely oscillatory, though with decaying oscillations, as it approaches the point of the limit, a function such as x*sin(1/x). It also leaves out some "pathological" functions, such as the nowhere-differentiable Weierstrass function.

I'm still in the development stage, and am only offering raw suggestions while, at the same time, trying to convince myself beyond a shadow of a doubt of the necessity for improvements. I know that original research is not welcomed here, so I may only be able to get a small fraction of these suggestions to be taken seriously, but it helps to think and learn stuff I may use in my own projects. Thanks for your input.Toolnut (talk) 15:06, 21 October 2011 (UTC)

There we go, that's a good response! You're interested in some sort of characterization of functions which have a given limit at a point. I think this task is overhard (maybe impossible), even for professionals. Surely being bounded on the half-open interval approaching x is necessary, but f can exhibit all sorts of bad behavior on that interval. Using the squeeze theorem, take any two (different) functions g≤f whose limit is L as we approach x from the left. Every function h such that g≤h≤f also has limit L as you approach x from the left. As you can see, this leaves a LOT of freedom for h to behave poorly.

As for neighborhood, yes, it is a topological term. The reason you see it in your texts is that they have to make use of topology without actually explaining it :)

It's funny, everybody learns about continuity with the epsilon-delta definition first. It's quite complicated, but it helps you learn basic logic. I encountered a paradox while taking topology: the abstraction made things easier! If you were interested, I could try to teach you enough topology to understand continuity. You might find this more valuable than the unforgiving task of characterizing functions with a given limit. Rschwieb (talk) 16:46, 21 October 2011 (UTC)

It seems that the concept of neighborhood makes the concept of derivative and limit more intuitive. The concept of limit is just a “rephrasing of that of neighbourhood”--MagnInd (talk) 00:00, 29 December 2011 (UTC)

I have been reading quite a few of the Wikipedia articles on logic, the quantifiers, and topological space, the last few days. I've been reviewing all this material, blending it with material from my own textbooks and original research into many Word files on my computer, for the past many years, trying to make better and more lasting sense of all kinds of math and science topics. Yeah, that would be cool of you to want to teach me: I guess you mean in this way, having these discussions on Wikipedia. I'll be happy to return the favor with what I know, a lot of engineering stuff. Thanks.Toolnut (talk) 17:28, 21 October 2011 (UTC)

Yes, I was imagining an informal discussion here. I can see you're interested in self-study, so I was thinking of pointing you in a direction. Suppose you have any set X. There are a lot of subsets of X, and a topology is a collection of subsets of X. It has to be "cohesive" in a special sense, and that's where the axioms come in. If you're interested, check out the definition at Topological space#Definition. I can recommend a few exercises if you decide that's something you're interested in. Rschwieb (talk) 18:06, 21 October 2011 (UTC)

Thanks Rschwieb: I shall start a new section, "About Topology", on this page for that purpose. I found out I was wrong about the word "neighborhood" being used at all in my first-year calculus book: it does not even mention it in its definition of continuity for functions to two variables. But my more advanced books do use it. The reason I've recently immersed myself in limits and continuity is that I have had to privately tutor two HS juniors prepping for advanced placement in College math.

Also, please check out my other new section, "Quantification", and my contributions to the WP article by the same name.Toolnut (talk) 05:05, 27 October 2011 (UTC)

Two remarks. First: I feel that the objects that Toolnut strives to classify, describe and use, are not functions but rather exercises about functions. This trend does not fit well into the mathematical tradition, but anyway, can be useful for students. It is not "mathematics", but rather "teaching math to engineers and other non-mathematicians". (And indeed, it is pedagogically inappropriate to require topology as a precondition to calculus.)

Second. Toolnot, if you'll feel that you do not fit into Wikipedia (because WP cannot accept original ideas, not even too original presentation of well-known ideas), you may try other wikis such as [1], [2]. Boris Tsirelson (talk) 16:28, 22 October 2011 (UTC)

Thank you, Boris, for bringing my attention to these new sites: I have checked them out and like them very much. Apparently you had much to do with contributing to Knowino's math topics and administration since its recent birth. I'll try to contribute as soon as I have the time.Toolnut (talk) 05:05, 27 October 2011 (UTC)

Nice. Educational content is quite welcome on Knowino. Originality is not a problem there, due to expert approval system. Boris Tsirelson (talk) 09:14, 27 October 2011 (UTC)

Quantification

I think I have finally grasped the fundamentals of quantification, after much reading of all the Wikipedia material on that and related subjects, and after much reorganizing of this new information in my own head, I have done my best to sum it all up in the subsection titled "Equivalent Expressions" I added to the article Quantification#Logic (without conflict so far, this time); I also added a few helpful points to the succeeding subsection, called "Nesting", such as "[the reason that the order of quantifiers matters] is because the syntax directs that any newly introduced variable cannot be a function of subsequently introduced variables." Let me know how I did.Toolnut (talk) 04:57, 27 October 2011 (UTC)

Looks good, but a word of caution: one can get carried away contributing to topics that they are very new to. I think nothing's gone wrong so far, but beware of using wikipedia as a "class notebook". Some editors fall into this and end up writing down a weird semicorrect understanding of something and it needs to be fixed. :) Rschwieb (talk) 01:02, 28 October 2011 (UTC)

About Topology

Let me know when you've had a look at the topology axioms. Rschwieb (talk) 01:03, 28 October 2011 (UTC)

Happy New Year

A Happy New Year, with many (wikipedian) achievements, fellow wikipedian Toolnut!--MagnInd (talk) 20:35, 5 January 2012 (UTC)

Thanks, and same to you!Toolnut (talk) 20:45, 5 January 2012 (UTC)

Hybrid engineering field - Electrochemical engineering

As an electrical engineer, how do you regard the hybrid field of electrochemical engineering?--MagnInd (talk) 20:19, 9 March 2012 (UTC)

Harmonic mean - arithmetic mean relation

Hi! What is exact relation between the harmonic mean and the arithmetic mean (of some numbers/variables)? Some more details concerning this relation are necessary to be introduced especially regarding the coefficient of variation mentioned.

An equivalent formulation which ignores the number of terms in the mean is the following: given the sum of some numbers/variables is the sum of reciprocal of those numbers fully determined?--MagnInd (talk) 20:25, 23 February 2013 (UTC)

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