Best mathematical infinity books according to redditors
We found 22 Reddit comments discussing the best mathematical infinity books. We ranked the 13 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 22 Reddit comments discussing the best mathematical infinity books. We ranked the 13 resulting products by number of redditors who mentioned them. Here are the top 20.
If that's what you value in a tofu press I might recommend
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
> Right now I'm looking at Terrence Tao's notes/textbook in the making, but that's only online
Tao's book on measure theory has been out since 2011. See also this book with further topics.
If you want a proper introduction that does things rigorously but is accessible without any advanced math knowledge (except the last section becomes serious analysis pretty quickly), Khinchin's book is probably the best choice (though it is quite old and it translated from Russian): http://www.amazon.com/Continued-Fractions-Dover-Books-Mathematics/dp/0486696308
If you want something more modern, you likely need to find a book on number theory that has a chapter or two on continued fractions since that's mostly where they come up (approximations to rationals etc).
On the other hand, for most continued fraction expressions that come up, Euler's formula is often enough: https://en.wikipedia.org/wiki/Euler's_continued_fraction_formula
I highly recommend this book.
A variety of places, here's some references:
Number Theory: The Mellin Transformation is connected to Dirichlet Series, in particular the Riemann Zeta function, you might try section 5.1 of Montgomery and Vaughan's Multiplicative Number Theory. There is this nice write up Fourier Analysis in Additive Number Theory or the Springer Book Fourier Analysis on Number Fields (a number field is a particular kind of extension of the rationals)
Representation Theory: Fourier Analysis on Finite Groups or Terras' lovely book Fourier Analysis on Finite Groups which has applications in Families of Expander Graphs
I confess that I'm dodging the answer to your question a bit here. Since I don't know of any unified treatment, I don't feel qualified to say "any time you see this phenomena, you should use an X-transformation." (other than what the books might say with regard to Fourier analysis on Groups and the fact that the transforms are (very) roughly equivalent...
I too am fascinated by the Laplace transform (and it's analogues). I'd recommend looking into Control Theory which is expands on the ideas of the Laplace Transform. It's usually treated in an "engineery" manner, but it is very much a mathematical theory.
EDIT: This is by no means the limits of transform methods, this only reflects my interests/knowledge, others will have many more examples.
EDIT2: I will again be teaching the Laplace this semester in DE, every time I do this, I wish I had more time to start an intro to Control Theory as it flows so naturally from the Laplace.
> Does it work only for few 'a' and 'c' ?
Yes, but it depends on the modulus too. For prime modulus, a lot of different combinations will work (i.e. give statistical properties as good as you can expect from an LCG). For power-of-2 modulus, most choices will not work (e.g. think about iterating 2x + c mod 32). In general, the problems arise when the modulus and the multiplier have common divisors, so a highly divisible modulus is already incompatible with most multipliers and increments. In particular, an even modulus rules out a full 50% of possible values.
Unfortunately, even determining the period of a given LCG requires a great deal of thought and a good chunk of elementary number theory: you can see e.g. Knuth, The Art of Computer Programming, vol. 2.
Interesting statistical properties are harder than that. It's possible to give conditions for n-dimensional equidistribution for given multipliers and the increments, but in practice it's easier to do a computational search for the good values. If you're interested in results of the form "these given families of multiplier-increment-modulus combinations will give bad results", I'd suggest looking into the works of Pierre L'Ecuyer.
I think you'll find reading the book of Kuipers-Niederreiter generally interesting. For measuring discrepancy LCGs, see this paper. The Mersenne Twister was created specifically in a way that makes even high-dimensional equidistribution easy to prove: you may read the Mastumoto-Nishimura paper to see how multiple-recursive matrix methods achieve that.
What I immediately thought about was Infinite Sequences and Series by Konrad Knopp; he also wrote a larger book called Theory and Application of Infinite Series, and I will say that the later material in both books is not covered in the calculus sequence.
Actually, sequences and series aren't used much in Calculus III, but moreso in Differential Equations; also, if you ever take Introductory Analysis, it expands on them greatly (which is why major textbooks in that area have been recommended in this thread).
Ahh, awesome. Though I would suggest Richard Courant's "Introduction to Calculus and Analysis I" due to my own bias, that book actually does not give the most complete exposition on infinite series. In fact, it is Richard Courant who suggested in a footnote that Konrad Knopp's "Theory and Application of Infinite Series" is the detailed treaties for infinite series, the book also touches on complex analysis (which is an amazing bonus, of course).
That is the book where I found an extremely straightforward derivation of the exponential generating function for the Bernoulli numbers, that's a very good book in my opinion (derivation on page 183).
You could look into Knopp's book on the subject. He has a section on divergent series. It's just a chapter as opposed to Hardy having an entire book on the topic, but it's more accessible.
Perhaps the Dover book on Continued Fractions by Khinchin. It may be terse but after about a century it remains quite good.
Rudy Rucker's Infinity and the Mind is an excellent book, and spends a decent amount of time discussing the different cardinalities of infinities. Great book for the lay person to explore the concepts.
Infinity and the Mind, by Rudy Rucker
One hell of a book on Mathematics and the concept of infinity. There's a great chapter on Godel and what the Incompleteness Theorems really mean in terms of computer consciousness.
I enjoyed this book a long time ago.
Yes, googling things is not a rigorous way of approaching the topic. But also, the first response agrees with what I'm saying. In fact, if you actually look at the accepted answer of the stack exchange question you found, you will see they also agree with me.
https://math.stackexchange.com/questions/596028/does-cardinality-really-have-something-to-do-with-the-number-of-elements-in-a-in
> What is a number? It is an informal notion of a measurement of size. This size can be discrete, like the integers, or a ratio, or length (like the real numbers) and so on.
> Cardinal numbers, and the notion of cardinality, can be seen as a very good notion for the size of sets.
>
> One can talk about other ways of describing the size of an infinite set. But cardinality is a very good notion because it doesn't require additional structure to be put on the set. For example, it's very easy to see how to define a bijection between ℕ
> and ℤ
>
> , but as ordered sets these are nothing alike. Cardinality allows us to discard that structure.
>
> Once accepting this as a reasonable notion for the size of a set, we can now say that the number of elements a set has is its cardinality.
But none of that matters, here is a excerpt from an actually rigorous book on the topic.
https://i.imgur.com/8IUGcYa.png
Just to note:
>The cardinality of a set X is a way of measuring in precise mathematical terms the number of elements in X.
Go read any math book on these topics and you will see unanimous agreement with this point. This is a mathematical statement that has been proven for well over 100 years.
>I take it you're saying there are other axiomatic systems which also have value where things behave differently?
There are two ways of thinking about it. On the one hand yes, there are weaker axiomatic systems that recover much of our mathematics, the study of which is called reverse mathematics. The big text on this is Subsystems of Second Order Arithmetic. In reverse mathematics, we study which axioms are needed for individual mathematical results. As it turns out, usually very weak systems of arithmetic suffice, but if our systems are weak enough, what we end up with is revisionary mathematics, in which we lose some theories (for instance, without the Weak Konig's Lemma we lose that a continuous real function on any compact separable metric space is bounded). In some of these weak systems of arithmetic, the world of mathematics is finite (or, at least, it appears finite from stronger systems). That is true for instance in Robinson's Q, in which we can't even prove N != N + 1 for all N. Note however that this finitism is only apparent, so that if there is a fact of the matter regarding which system of arithmetic holds for mathematics, and that system is finite, we might still be able to do mathematics involving 'infinite' cardinals, but where such theories are satisfied by intuitively 'finite' models (as you can imagine, this gets philosophically tricky). Parsimonious considerations, along with the physical impossibility of manifested infinities in the real world, have led many to be classical finitists along these lines.
On the other hand, we can think of finitism as a meta-mathematical position, which may or may not be revisionary. A revisionary approach is Sazonov's feasible numbers, and a non-revisionary approach is given by Shaughan Lavine in Understanding the Infinite in which he recovers all our infinitary semantics, systems including large cardinal axioms, anything whatever in set theory, by appeal to the concept of indefinitely large sets as a substitute for infinity.
To piggy back off of danielsmw's answer...
> Fourier analysis is used in pretty much every single branch of physics ever, seriously.
I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.
If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.
A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.
If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.
Random links which may spark your interest:
It won't always be referred to as "Fourier's theorem". Some texts may refer to it as "completeness of fourier series", or they may prove more general versions of the theorem using Sturm–Liouville theory. Note that "completeness" has a technical definition, but in this context roughly means that any square-integrable periodic function can be described as a (possibly infinite) trigonometric series.
I seem to recall that A First Course in Wavelets with Fourier Analysis was readable & yet not long-winded, and had a fairly rigorous proof of Fourier's theorem by the middle of the second chapter. See Amazon.com. A free PDF from an MIT OpenCourseware that might be of use is here. I skimmed over it; it does prove what you are interested in, but I can't vouch for its readability.
sure....ask Barry Simon
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https://www.amazon.com/gp/product/0157850501/ref=dbs_a_def_rwt_hsch_vapi_taft_p1_i1
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https://www.amazon.com/Fourier-Analysis-Self-Adjointness-Methods-Mathematical/dp/0125850026/
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https://www.amazon.com/Scattering-Theory-Methods-Mathematical-Physics/dp/0125850034
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https://www.amazon.com/gp/product/0159850045/ref=dbs_a_def_rwt_hsch_vapi_taft_p2_i0
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