Name: Real Analysis
ISBN: 9780135113554

What Hacker News says

graycatMay 2022

Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers!

For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals.

But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.

For linear algebra, sure, Nering (a student of E. Artin at Princeton) and then Halmos (a finite dimensional introduction to Hilbert space and written when Halmos was an assistant to von Neumann, office not far from Einstein), etc. For still more, R. Bellman has a good book -- he went all over active and put in everything including the kitchen sink. Then for numerical work, old but a start, Forsythe and Moler and/or the documentation of LINPAK.

For multivariate calculus through the divergence theorem and as needed, e.g., for Maxwell's equations, start with the most elementary treatments of vector analysis and f'get about the lack of proofs. Actually notice, can see via a Google search, that the divergence theorem on a box is just a dirt simple application of the fundamental theorem of freshman calculus. Then the difficulty is proving the thing on other shapes. Then do multivariate calculus again via, say, Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus. That he uses the Jordan curve theorem is a bit much but just go with the flow. Fleming's Functions of Several Variables does it with both measure theory and exterior algebra. Or W. Rudin's, as I recall, third edition, Principles of Mathematical Analysis. For calculus done via Lebesgue's measure theory, Royden's Real Analysis and/or the first half of Rudin's Real and Complex Analysis.

My favorite author on ordinary differential equations is Coddington, and his elementary An Introduction to Ordinary Differential Equations is nicely done and likely enough for nearly any applications might encounter. Will want some of this if do A/C circuit theory or deterministic optimal control.

For basic abstract algebra Herstein, Topics in Algebra should usually be enough and is well written. With that will be well prepared for, say, algebraic coding theory which actually has some important applications.

Once have a background in measure theory, for probability, of course, Loeve or either of his students Breiman or Neveu. Breiman is really nice to read. Neveu is a good candidate for the most polished and elegantly done math book or writing of any kind in all of civilization. Be sure to get to the good stuff on the Radon-Nikodym theorem, conditional expectation, martingales, and ergodic processes.

For relatively good results, better than hardly any students get just from courses, study the theorems and proofs until can do them yourself and can also explain them to any common man in the street and also explain where they fit in more generally. Also be able to do the exercises. E.g., Rudin's Principles has the inverse and implicit function theorems as exercises! Fleming applies them to Lagrange multipliers. These two theorems are the main prerequisites of differential geometry -- some parts of physics, now popular, touch on this math!

In my Ph.D. program I led the class in 4 of the five qualifying exams, and for the analysis and linear algebra exams, the books above were the secret. After I did that, the department tried to have the students do better by offering a course from Rudin's Principles; the course didn't work; that book is too hard for a course and needs weeks of quiet time where a student can chew on the text slowly line by line. Rudin has some of the most precise presentations on Fourier theory. So when physics uses Fourier theory or engineers use the fast Fourier transform, you will likely know Fourier theory quite a lot better than they do. Right, the uncertainty principle in quantum mechanics is just a basic result in Fourier theory. With Fourier theory and linearity, Schroedinger's equation starts to make sense. A good exercise in early measure theory is differentiation under the integral sign, Leibniz's rule -- be sure to cover that with a good proof.

With the background from the books above will be well prepared for a wide range of more narrow topics in pure and applied math.

Nearly all these books are available used in nearly new condition for low prices. In a move I lost some of my library and rebuilt with used copies.

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graycatApr 2022

> All ears

Start with at least a good course in calculus. Then, for the divergence theorem, in one dimension that is essentially just the fundamental theorem of calculus on a closed interval on the real line. Then with multiple integrals can get a proof on any box, rectangular parallelepeped with edges parallel to the three orthogonal coordinate axes. If draw a picture and use the Jacobian for change of variable, can get a somewhat more general proof.

That's enough for most courses in E&M, Maxwell's equations, and fluid flow.

An okay source in PDF from Taiwan is in

https://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/dive...

Apostol does Jordan curves, and that's a bit much; it's a strange physics course that needs Jordan curves.

The last edition of Rudin, Principles of Mathematical Analysis is a good source. With some good lectures to explain to the students just what is going on with compact sets, Rudin does not have to be too difficult or severe.

Fleming in Functions of Several Variables does both the Lebesgue integral and exterior algebra -- at times it has been taught to ugrads.

For the Lebesgue theory, I prefer either of Rudin, Real and Complex Analysis (just the first, real, half) and Royden, Real Analysis to Fleming. The Lebesgue integral does not have to be more difficult to teach or learn than the Riemann integral -- maybe the Lebesgue approach is easier than the Riemann approach where usually dig into compact sets.

Here is the fishy part of improper integrals: Rudin's Principles discusses conditionally convergent series. Then it's just an exercise that a rearrangement can be made to converge to negative infinity, positive infinity, or anything between. Well, can generalize this to improper integrals: Partition the real line into intervals of length 1, (0, 1], (1, 2], .... Then start with a function defined from a conditionally convergent infinite series, say, each interval with integral the same as one of the terms of the series. Then in the limit in the improper integral, integrate over the intervals in whatever order want, not necessarily from -a to a and, thus, converge to anything might want. So, the value of an improper integral depends all on just what order in which add up the pieces -- so that's not a very good definition and stands to resist proofs of good properties.

The Lebesgue theory gets around this issue. First, that theory just takes the given function and takes it apart into the positive part and the negative part. Second, it is easy, then, to define the integral of each part separately. If both parts have integral with absolute value infinity, then the integral of the function is not defined because it would require subtracting infinity from infinity which is not defined, even in the extended real number system. And that approach right away rules out the conditionally convergent issue. Otherwise the Lebesgue integral is defined. It also has good convergence properties, e.g., can prove a good version of Leibniz's rule.

In grad school, the department suggested Courant and Hilbert. So, I got a copy: The content didn't seem to have much to do with the math I was studying or the physics I was interested in (I lost my copy in a move).

For the Feynman Lectures, yes, I like most of it. I didn't like his treatment of the bra-ket notation, but then I don't like that notation anyway. The whole Lectures are available in PDF at https://feynmanlectures.caltech.edu/

The move also cost me my copy of Feynman's Lectures.

There is something, maybe, US STEM field students should understand: In the US, there was The Bomb, the atomic bomb. That made a lot of people in Congress awake and afraid, and they decided that the US research universities should be the world leaders in research in the STEM fields. So, Congress appropriated a lot of money (NSF, NIH, DARPA, etc.) for US research universities and STEM field researchers. The universities like the money, e.g., commonly take 60% for "overhead" and, thus, fund the theater group, the string quartet, the English department, the art department, the new fountain in the quadrangle, the limo for the president, etc. But the STEM field profs and students also get funded. Since the members of Congress tend to be old, they also like funding medical research. For something like the Webb telescope, who's to say, that might lead to some fantastic discovery crucial for US national security (it's happened before).

Well, the Congress is smart enough to want some evaluation, grades, measures, of the work funded. In simple terms, the evaluation is to count prizes and papers in peer-reviewed journals.

Net, the big bucks say that the emphasis is on the research. It's not on the teaching. It's not on writing books that clean up rough edges in old research. E.g., no one questions the basic correctness of Maxwell's equations, Schrödinger's equation, the divergence theorem, and polishing that material will NOT get published in the best peer-reviewed journals and is NOT the results desired from the funding. If some student wants to polish the material, the library is just off the quadrangle, and they can go for it -- or now just use the Internet.

But the emphasis is on the research. What Congress is paying for is the research that might be crucial for US national security (or medical research that might make them live another 10 years). What's wanted is the research, to be the best in the world in research, and polishing and teaching are not much of a part of that.

Actually, Congress has zero curiosity about the research and couldn't care less about the research. All Congress wants is to have the US the world leader in anything that might yield another surprise for US national security such as The Bomb. If there were no competition from Russia, China, etc., the US Congress would cut back the research funding to pennies or less.

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graycatJan 2022

It appears that there are three camps of stochastic processes:

(1) Old. Analyze bumps on railroad tracks, sound noise, the weather over time, ocean waves, .... So, the expectation is fixed, that is, does not vary. The sample paths are bounded. Might assume that the variance is fixed. If increments are independent and identically distributed, then can assume the distributions are Gaussian (have a Gaussian process).

In that case can do interpolation, extrapolation, smoothing, find power spectra, see what happens to sample paths as you run them through a time-invariant linear filter.

(2) Newer. Can get into Markov processes and maybe also martingales. There is a good text by Cinlar, long at Princeton. His chapter on the Poisson process is especially good. Cinlar does not mention measure theory, but other texts do or really are all about measure theory, that is probability based on measure theory. Can get the measure theory out of Royden, Real Analysis and Rudin, Real and Complex Analysis. Should have a measure theory background in probability, and for that try any or all of Breiman, Neveu, Loeve, Chung. Can get into potential theory. Then stochastic optimal control. May end up with some expensive books from Springer written by Russians. Otherwise I gave some names above.

(3) Queuing Theory. Get to do a lot of work with the Poisson processes with the hope of making applications, e.g., in the sense of operations research.

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graycatNov 2021

My view, from math, computing, and computer science is (1) that the author is being too critical and often throwing out a small but good baby with some not very dirty bathwater and (2) that he is not mentioning some larger problems, worse "sins".

Here would be my list of the worst sins or how to avoid them:

(1) A term is a word used with other than a standard dictionary definition. Technical fields are awash in terms. Make sure to try hard to define all the terms you use; with some judgment applied, can omit definitions of some terms certainly well understood by any reader with enough background to get anything from the paper. Poorly or undefined terms can be one of the best ways to lose readers. Here, hoping to get more readers, maybe inviting them into the field, bend over backwards in defining terms, that is, maybe give some term definitions for readers who, really, have little hope of getting much from the paper -- at least, when such a reader gives up they won't blame the author!

In addition, for any term new or relatively new to the field, with some judgment, might also include for the term motivation of its importance and examples of its usage -- i.e., make clear the importance, relevance, value, usage, etc. of the term.

(2) For nearly all acronyms, standard in the field or not, for the first use of the acronym include what it abbreviates. Here to be sure are doing well on this issue, strain and bend over backwards and include, say, even TCP (transmission control protocol), SMTP (simple mail transfer protocol), CPU (central processing unit) -- sure, I'm suggesting bending over backwards. Several slaps on the wrist, an hour in the corner with wearing a dunce cap, and a coating of tar and feathers for each acronym used but not defined with what it abbreviates.

(3) For how to write math, in two parts, (A) and (B):

(A) Take some theorem proving courses from some of the most precisely written texts, say, P. Halmos, Finite Dimensional Vector Spaces, W. Rudin, Principles of Mathematical Analysis, Neveu, Mathematical Foundations of the Calculus of Probability, E. Nering, Linear Algebra and Matrix Theory, Royden, Real Analysis where do a lot of the exercises as homework and where the professor DOES read and remark on your writing. Neveu was a Loeve student at Berkeley. Nering was an Artin student at Princeton. As I recall, Rudin's background was in Austria, although I don't know who his professors were. Of course, Royden was long at Stanford. For more, if have time and insist on some really good examples, read some of Bourbaki.

By the way, on use of we, that is standard. So we might have,

"Given topological spaces X and Y and a function f: X --> Y, we say that function f is continuous provided for each set B open in Y f^(-1)(B) is open in X.

So, that's a vanilla example of using we in mathematical writing.

(B) Do some things none of those texts do: Include some intuitive views, some helpful pictures, motivation via applications in math and also outside math, and outlines of research directions.

There is a lot of question about how appropriate is suggestion (B); in math my guess is that there is no question about the relevance, wisdom, importance, value, etc. of suggestion (A).

For the goal of writing math, suggestion (A) is important: Tough to expect good success without the texts I listed or other texts written with similar care. That level of care is extreme, tough to find and learn, and much tougher to do. Writing math with the care of those texts seems to have been understood and practiced significantly often only after about 1950 or 1940. And for at least one course from at least one such text, DO have the good homework grading of a good math professor.

Personal experience and lesson: While I'm not much interested in being a professor, I have published some papers, and from getting those papers reviewed my guess would be that about half of the reason the papers passed review, and they always did with no significant revision or difficulty, is that I wrote the math with nearly the care and precision of, say, W. Rudin. Point: To critical readers, any lapse in that level of care and precision can be like a worm in a baked apple, perhaps otherwise terrific from brown sugar, butter, etc. Or, might guess that, in nearly any field, the good work is less than 10% of the total, maybe less than 1%, with the rest flawed, maybe as bad as that apple with a worm. In writing math, it is way too easy to be in the bottom 99% or 90% just from the care and precision of the writing, and suggestion (A) is IMHO (in my humble opinion) a good way to have at least the writing quality keep you in the top 90+%.

For math used in computer science, for how to write that math carefully, there are of course examples from D. Knuth.

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graycatJul 2020

Linear algebra, and more than one such book.

IMHO long and still the best linear algebra book is

Halmos, Finite Dimensional Vector Spaces (FDVS).

It was written in 1942 when Halmos was an "assistant" to John von Neumann at the Institute for Advanced Study. It is intended to be finite dimensional vector spaces but done with the techniques of Hilbert space. The central result in the book, according to Halmos, is the spectral decomposition. One result at a time, the quality of von Neumann comes through. Commonly physicists have been given that book as their introduction to Hilbert space for quantum mechanics.

But FDVS is a little too much for a first book on linear algebra, or maybe even a second book, should be maybe a third one.

Also high quality is Nering, Linear Algebra and Matrix Theory. Again, the quality comes through: Nering was a student of Artin at Princeton. There Nering does most of linear algebra on just finite fields, not just the real and complex fields; finite fields in linear algebra are important in error correcting codes. So, that finite field work is a good introduction to abstract algebra.

For a first book on linear algebra, I'd recommend something easy. The one I used was

Murdoch, Linear Algebra for Undergraduates.

It's still okay if can find it.

For a first book, likely the one by Strang at MIT is good. Just use it as a first book and don't take it too seriously since are going to cover all of it and more again later.

I can recommend the beginning sections on vector spaces, convexity, and the inverse and implicit function theorems in

Fleming, Functions of Several Variables

Fleming was long at the Brown University Division of Applied Math. The later chapters are on measure theory, the Lebesgue integral, and the exterior algebra of differential forms, and there are better treatments.

Also there is now

Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

http://vmls-book.stanford.edu/vmls.pdf

Since the book is new, I've only looked through it -- it looks like a good selection and arrangement of topics. And Boyd is good, wrote a terrific book, maybe, IMHO likely, the best in the world, on convexity, which is in a sense is half of linearity.

Some course slides are available at

http://vmls-book.stanford.edu/

For reference for more, have a copy of

Richard Bellman, Introduction to Matrix Analysis: Second Edition.

Bellman was famous for dynamic programming.

For computations in linear algebra, consider

George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems

although now the Linpack materials might be a better starting point for numerical linear algebra. Numerical linear algebra is now a well developed specialized field, and the Linpack materials might be a good start on the best of the field. Such linear algebra is apparently the main yardstick in evaluating the highly parallel supercomputers.

After linear algebra go through

Rudin, Principles of Mathematical Analysis, Third Edition.

He does the Riemann integral very carefully, Fourier series, vector analysis via exterior algebra, and has the inverse and implicit function theorems (key to differential geometry, e.g., for relativity theory) as exercises.

All of this material is to get to the main goals of measure theory, the Lebesgue integral, Fourier theory, Hilbert space and Banach space as in, say, the first, real (not complex) half of

Rudin, Real and Complex Analysis

But for that I would start with

Royden, Real Analysis

sweetheart writing on that math.

Depending on the math department, those books might be enough to pass the Ph.D. qualifying exam in Analysis. It was for me: From those books I did the best in the class on that exam.

Moreover, from independent study of Halmos, Nering, Fleming, Forsythe, linearity in statistics, and some more, I totally blew away all the students in a challenging second (maybe intentionally flunk out), advanced course in linear algebra and, then, did the best in the class on the corresponding qualifying exam, that is, where that second course was my first formal course in linear algebra.

Lesson: Just self study of those books can give a really good background in linear algebra and its role in the rest of pure and applied math.

No joke, linear algebra, and the associated vector spaces, is one of the most important courses for more work in pure and applied math, engineering, and likely the future of computing.

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graycatSep 2019

To get very far in stochastic processes, usually need a good background in measure theory. Here is an overview of about two semesters in a fast, often rare, grad course plus much more:

The integral of freshman calculus is the Riemann integral. Too soon/easily it was seen to be clumsy to inadequate in theoretical developments. Near 1900 in France, E. Borel student H. Lebesgue improved the situation; he did his Lebesgue integral. Don't worry: For every function integrated in freshman calculus, Lebesgue's integral gives the same answer. Otherwise Lebesgue's integral is much more general and, it turns out, is the solid foundation of probability and stochastic processes.

Lebesgue's work starts with measure which is just a notion of area. So, yes, on the real line, the Lebesgue measure of the closed interval [0,1] is 1, just the same as we learned in grade school. But with Lebesgue's work, it's tough to think of a subset of the real line that is not measurable (has a measure or area in Lebesgue's work) -- the usual example of a subset of R that is not measurable needs the axiom of choice. With measure defined on R, Lebesgue can define the Lebesgue integral of functions

f: R --> R

For this we need a meager assumption (which essentially always holds), but lets charge on:

For the generalizations, crucial for probability and stochastic processes, start with a non-empty set M. R is an example of a such a set M. Define a measure on M, say, m. Then for some subsets A of M, we have the measure of set A as m(A). That is, we have the area (volume, whatever) of A. But to do much, we need more assumptions: We want the set of all subsets A of M to which we assign measure m(A) to be a sigma algebra, say, F. As at

https://en.wikipedia.org/wiki/Sigma-algebra

we want M to be an element of F, for each A in F we want the complement, say, the set M - A (obvious but sloppy notation) to be an element of F, and for A_i in F for i = 1, 2, ..., we want the union of the A_i to be an element of F. So, we want F closed under complements and countable unions. No, we don't permit uncountable unions, a point that keeps popping up in stochastic processes. Permitting uncountable unions results in, in one word, a mess.

The sigma part of sigma algebra is supposed to refer to the countable unions as in the capital letter sigma used in calculus, etc., for countable sums.

So, we have the set M, the sigma algebra of subset F, and the measure m. Then the pair (M, F) is a measurable space and the triple (M, F, m) is a measure space. In probability theory and stochastic processes, the measure space used is usually denoted by space, capital omega, the sigma algebra, script F, and the measure, P.

Right -- I just let the cat out of the bag: Probability P, as we saw in elementary treatments of probability, is a measure on the measure space with Omega, script F, and P.

So for A a subset of the space Omega and an element of sigma algebra script F, P(A) is the probability of set A. Set A is called an event. Now we are back to what you saw in elementary treatments of probability.

So, it went too fast -- we mentioned measure theory with (M, F, m) and then took a special case Omega, script F, P, a probability space, and got back to P(A) we saw in elementary treatments.

This jump from a measure space to a probability space was made by A. Kolmogorov in 1933 and has been the foundation of 99 44/100% (as in Ivory soap!) of advanced work in probability and stochastic processes since then.

So, back to the case of measure space (M, F, m). Suppose we have a function

f: M --> R

We want to integrate f as in freshman calculus. Well, for each subset A of M and an element of sigma algebra F, we have the measure (area) of A, that is, m(A). Well with one more assumption, and what Lebesgue did, that's enough! And it's assuming less than Riemann did and, thus, getting more generality in some quite good ways.

Then for the case of Lebesgue integrating some

f: R --> R

compared with the Riemann integral, Lebesgue partitions on the Y axis and Riemann partitions on the X axis. So, Riemann partitions on the domain of the function and Lebesgue partitions on the range. In both cases we are working with lots of little rectangles that get smaller as we converge. So, Lebesgue lets the domain be much more general, e.g., enough for probability theory.

So, back to a probability space and a function

f: Omega --> R

With the meager assumption (that f is measurable -- to be brief I'm skipping over the details here is only because nearly all functions are measurable -- tough to construct one that isn't, so general is Lebesgue's work), then f is a random variable. It is more common to denote random variables by X, Y, Z, etc. Now, finally, maybe after getting lost in a swamp for some years, you have a solid definition of a random variable. Then each point in Omega is one probability or statistical trial.

Then, sit down for this amazing fact: The expectation of real valued random variable X, denoted by E[X], is just the Lebesgue integral of the function

X: Omega --> R

So, in this way, Kolmogorov borrowed from Lebesgue and got a solid foundation for probability and stochastic processes.

Now you know why Lebesgue's measure theory is key to advanced work in probability and stochastic processes -- E[X] is just Lebesgue's integral generalization of the calculus Riemann integral.

Long standard very careful, polished, treatments of measure theory are in Royden, Real Analysis and the first half of Rudin, Real and Complex Analysis. Continuing with probability is Breiman, Probability, Neveu, Mathematical Foundations of the Calculus of Probability, Loeve, Probability, etc. Uh, Loeve was long at Berkeley, and both Breiman and Neveu were Loeve students.

Can use this work to make good sense out of Brownian motion, and there are plenty of books there.

And can do more on stochastic processes, e.g.,

Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus.

R. S. Lipster and A. N. Shiryayev, Statistics of Random Processes.

Ronald K. Getoor, Markov Processes and Potential Theory.

This material is now decades old. With this material it is possible to give high end treatments of some topics in Wall Street type investing, e.g., exotic options and in particular the Black-Scholes formula.

This material was called by Breiman "graduate probability" as he left the field and did, e.g., CART -- classification and regression trees, maybe the most important seed for the current work in machine learning. Breiman was interested in making sense out of some complicated medical data.

The potential of such graduate probability for careers in academics and/or applications seems now to be open to question. But, with a larger view, probability and stochastic processes just will NOT go away and will remain some of the most powerful tools for making sense out of reality -- we can be sure that over the horizon we will be glad to have such a solid foundation and WILL encounter important, new applications, ours or those of others.

There is an old point: Long it was common for the pure mathematicians, yes, to pay close attention to measure theory but essentially to ignore the role in probability theory, saying, that probability theory was just a trivial special case of measure theory for positive measures (can also have measures that have negative or complex values) where the measure of the whole space was 1.

But: Probability theory does a LOT with the assumption of independence tough to find in texts in just measure theory. Also the Radon-Nikodym theorem (an amazing generalization of the fundamental theorem of calculus) IS treated in measure theory but the applications crucial in probability and stochastic processes to conditioning and sufficient statistics usually are not.

So, with this overview, you might find Royden, Rudin, Breiman, Neveu (especially elegant) not so hard reading and a lot of fun.

You will learn about the cases of convergence of sequences of random variables and the classic results of the central limit theorem, the laws, weak and strong, of large numbers, ergodic theory (pour cream into coffee and stir until the cream returns to the distribution it had in the coffee just after pour it in -- Poincare recurrence), and martingales. Margingales are amazing and provide the easiest proof of the strong law of large numbers and some of the strongest inequalities in math.

So, amazing stuff: Go into a lab, measure something, walk out, and have the value of a random variable -- okay, that's good enough for an elementary course. But typically in math, we can't calculate just what we want in just one step so have to iterate and approximate and hopefully converge in some appropriate sense to just what we want. Well, in those classic results, we have notions of convergence. The most amazing case is of strong convergence where we have a sequence of random variables that converge to some random variable. From the elementary treatments, having a sequence of randomness converge to more randomness seems absurd. But with Kolmogorov's treatment, a random variable is just a function, and from math such as Rudin, Principles of Mathematical Analysis we know quite a lot about convergence, and in measure theory we learn more. Then the case of a sequence of random variables converging to another random variable is just a special case of one case of convergence of sequences of functions. So, we can approximate a random variable X by having a sequence of random variables converge to random variable X. How 'bout that! Thank you Borel, Lebesgue, Kolmogorov, von Neumann, etc.

Have fun! Get rich???

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