(Part 3) Best applied mathematics books according to redditors
We found 2,147 Reddit comments discussing the best applied mathematics books. We ranked the 845 resulting products by number of redditors who mentioned them. Here are the products ranked 41-60. You can also go back to the previous section.
Poor Americans are more liberal than rich Americans in general. But there are distinct patterns of political preferences by income among racial groups and geography. Poor Blacks vote overwhelmingly Democratic, as do poor Whites, except in the South where poor White's preferences are a little murkier. Andrew Gelman does a good job of explaining this in Red State, Blue State, Rich State, Poor State.
The book's paradoxical conclusion is this: Rich states and poor people vote Democratic, while poor states and rich people vote Republican. The way to reconcile the contradiction is that in red states income is a much more robust predictor of your voting habits than in blue states. So, in Connecticut, a rich blue state, income does a less good job of predicting the voting habits of wealthy and middle class voters, many of whom vote Democratic despite their wealth. In Alabama, a poor red state, the votes of people who have above-average incomes are very well predicted by their incomes. And rich, white, Southerners are the most conservative people in the United States. In Mississippi in 2012, Obama only got 10% of whites in the state to vote for him. If you went to a polling station in a rich, white suburb of Atlanta, or Tallahassee, or Jackson, I'd guess that well over 95% of people would be voting Republican. Why? Gelman goes into the nuances in his book, but it has a lot to do with religion and values which are more important to rich Republicans than any other group.
Another interesting finding that comes out of people researching voting habits by demographic characteristics is the existence of two kinds of whites. White people's voting habits basically differ based on what side of the Mason-Dixon they live on. Southern Whites are extremely conservative, whereas Northerners basically split the vote between Democrats and Republicans. That's why Republicans handily win southern states with the country's largest minority populations.
TL;DR The poorest 10% of Americans are more liberal than the richest in general as measured by preference for the Democratic Party. This relationship is less strong outside the South
Source:
Red State, Blue State, Rich State, Poor State
Gelman's research
Crooked Timber
I've wasted too much time trying to find the so-called "right" statistics book. I'm still early in my journey, going through calculus using Prof. Leonards videos while working through a Linear Algebra book all in prep for tackling a stats book. Here's a list of books that I've had a look at so far.
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These seem to be of a similar level with regards to rigour, as they aren't that rigourous. That's not to say you can get by without the calculus prereq and even linear algebra
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The other two I've been looking at which seem to be a lot more complex are
And then there's Casella and Berger's Statistical inference, which I looked at once and decided not to look at again until I can manage at least one of the aforementioned books. I think I'm leaning most to the first book listed. Whichever one you decide to use, good luck with your journey.
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>You are not ruining the economy by shopping there.
Well, yes, you are. Shopping at Wal-Mart is the same as defecting in the Prisoner's Dilemma problem, except with a gazillion players instead of two. By shopping there, you get some small gain for yourself at the expense of a net larger loss to the world at large (including you). It is pretty classic game theory.
Free markets in general are known to fail at that kind of choice: people tend to pick the path that yields personal short-term gain over collective benefit, even if the choice yields long-term ruin. In the case of environmental destruction, the costs are external to the system as a whole, and there are whole branches of economics discussing how to tweak the market to account for external costs of actions.
In the case of economic plundering (like Walmart engages in) the costs are internal to the eeconomy but are deferred and homogenized so that the cost to each individual isn't directly visible at the time of purchase -- one might call them "artificially externalized" costs.
Edit: I seem to be attracting a fair number of downvotes. I'll charitably assume they're not knee-jerk responses. Here are some some nice references: The Bully of Bentonville; Fishman's nice book on the Wal-Mart Effect; a nice documentary DVD; and Davis's fun pop-level introduction to game theory.
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.
> Furthemore, why did the South shift from being a Democratic stronghold to a Republican one?
There have been plenty of threads here about the Southern Strategy and the partisan realignment. The tl;dr is that the Republican Party appealed to racism against blacks and opposition to civil rights among southern white voters. Those voters, previously strongly Democratic voters, switched to supporting the Republican Party, where they remain to this day. (For an academic look, you can see here.)
> Why is it that after '92 the Northeast and West coast became consistently Democratic, and the South and midwest become consistently Republican?
It's largely a function of population demographics. Another tl;dr is that the coasts are far more urbanized than the South and midwest. Highly urban areas tend to be more Democratic-leaning. Essentially, blue states are blue because they're disproportionately urban, while red states are red because they're disproportionately rural. Even in states like California, you see large swaths of Republican counties because they're heavily rural areas.
As for the central thrust of your question, Andrew Gelman would likely argue that, even though rich people tend to vote Republican quite overwhelmingly, 1) there are far more poor people in those blue states, and poor people tend to vote Democratic, and 2) rich people on the coasts care more about social issues that Democrats favor. In general, I think it's just a function of population demographics.
Analyzing Baseball Data with R is basically the textbook when it comes to learning R with baseball data. Second edition just came out with some updates with regards to Statcast data. It's like $50, though, so if that's a bit out of your reach, I might suggest looking at DataCamp's R tutorials or something of that nature.
It is hard to provide a "comprehensive" view, because there's so much disperate material in so many different fields that draw upon probability theory.
Feller is an approachable classic that covers all of the main results in traditional probability theory. It certainly feels a little dated, but it is full of the deep central limit insights that are rarely explained in full in other texts. Feller is rigorous, but keeps applications at the center of the discussion, and doesn't dwell too much on the measure-theoretical / axiomatic side of things. If you are more interested in the modern mathematical theory of probability, try Probability with Martingales.
On the other hand, if you don't care at all about abstract mathematical insights, and just want to be able to use probabilty theory directly for every-day applications, then I would skip both of the above, and look into Bayesian probabilistic modelling. Try Gelman, et. al..
Of course, there's also machine learning. It draws on a lot of probability theory, but often teaches it in a very different way to a traditional probability class. For a start, there is much more emphasis on multivariate models, so linear algebra is much more central. (Bishop is a good text).
Algebra
Trigonometry
Functions and Graphs
These are three books that I would recommend to somebody trying to prepare for calculus. They're all written by the mathematician Gelfand and his colleages, and they're some of the best-written math books I've ever read. You come away from reading them really understanding the subject matter. I'd read them in that order, too.
This is a really popular theoretical differential equations book http://www.amazon.com/Ordinary-Differential-Equations-V-I-Arnold/dp/0262510189
It's Ordinary Differential Equations by V.I. Arnold, it's highly regarded and I see people recommend it over on math.stackexchange all the time.
However I'm not sure if it's the kind of book you're looking for because I don't believe it's an introductory book at all. From what I've heard it's pretty advanced.
Hopefully someone more knowledgeable than I can explain whether this book is appropriate for you or not.
For undergrad probability, Pitman's book or Ross's two books here and here.
For graduate probability, Billingsley (h/t /u/DCI_John_Luther), Williams or Durrett.
Sample size is in regards to the battle count. If you don't think 4,317 battles are enough to determine the probability of what is something with only three possible outcomes to a reasonable certainty, I don't know what to tell you except you may need to do some further reading. I suggest Jordan Ellenberg's "How Not to Be Wrong", which covers a wide variety of problem types and is actually a pretty entertaining read.
Anti-disclaimer: I do have personal experience with all the below books.
I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.
It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.
If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.
I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .
Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.
Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.
I've heard nothing but good things about this book.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
I would start with Cover & Thomas' book, read concurrently with a serious probability book such as Resnick's or Feller's.
I would also take a look at Mackay's book later as it ties notions from Information theory and Inference together.
At this point, you have a grad-student level understanding of the field. I'm not sure what to do to go beyond this level.
For crypto, you should definitely take a look at Goldreich's books:
Foundations Vol 1
Foundations Vol 2
Modern Crypto
Took a numerical statistics course first year of my masters which used C&B as well. I found Hogg, McKean, and Craig - Introduction to Mathematical Statistics to be a really good companion volume. It covers a lot of the same material, but in a more accessible fashion.
You beat me to it! Well, here are the recommendations:
> On advanced Bayesian statistics, Cyan recommends Gelman's Bayesian Data Analysis over Jaynes' Probability Theory: The Logic of Science and Bernardo's Bayesian Theory.
> On basic Bayesian statistics, jsalvatier recommends Skilling & Sivia's Data Analysis: A Bayesian Tutorial over Gelman's Bayesian Data Analysis, Bolstad's Bayesian Statistics, and Robert's The Bayesian Choice.
The 1st Course In Math Analysis by Brannan
Analysis I by Terrence Tao
Yet Another Intro To Real Analysis by Bryant
Understanding Analysis Stephen Abbott
Elementary Analysis: The Theory of Calculus Kenneth A. Ross
Metric Spaces by Robert Reisel
A Problem Text in Advanced Calculus by John Erdman. PDF
Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF
I've been going through this book. Would recommend.
Andrew Gelman is the one of the authors of Bayesian Data Analysis. He generally favors Bayesian approaches to statistics, although I get the impression he sees them as means to getting robust/tractable and partially-pooled estimates from data, rather than as the only coherent way to make any inferences, ever.
You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:
Real Mathematical Analysis by Charles Pugh
Understanding Analysis by Stephen Abbot
A Primer of Real Functions by Ralph Boas
Yet Another Introduction to Analysis
Elementary Analysis: The Theory of Calculus
Real Analysis: A Constructive Approach
Introduction to Topology and Modern Analysis by George F. Simmons
...and tons more.
In my opinion the hardest part of the course is the first 3 weeks, and the last 3-4 weeks.
FIRST THREE WEEKS:
Probability at first was extremely confusing, and in some ways still is a bit confusing for me since I almost never use it and forget stuff over time. You may be like me in this regard, the reason I always would get tricked by probability is there are cases where the wrong answer just seems like pure common sense (until you learn probability better) which will leads you down a very wrong path because you are convincing yourself you're right when you are not. The trick I found for myself was to aggressively do every problem I could get my hands on and understand exactly why I was wrong. I went through 2-3 different textbooks outside of the course and only then finally started to understand how to think in a probabilistic way whereby the tricks that tend to destroy people on exams and such would not catch me off guard.
The textbook for the course (Grinstead and Snell? I may be spelling this wrong) was extremely verbose and I started reading elsewhere out of boredom, in retrospect I regret this decision since it was the closest book to all the topics covered in 247.
The lecturer felt like he threw examples at us (I assume this is your complaint too?) and my biggest mistake in that course was not spending ample amounts of time understanding exactly why they worked. Despite this, what he did explain was good and I liked his teaching a lot, but I had to go to office hours to understand things that were vague in lectures.
As an example, do you know why the permutation formula is defined the way it is? Do you know why n choose k is defined the way it is, or rather, how can you get to the formula for n choose k if you know the permutation formula?
The unfortunate thing is it took me going through books like A First Course in Probability, which is probably insane to go through if you aren't comfortable with math/proofs/some stats already despite the book name... but the massive amount of examples gave me some pretty huge insights. I did this for I think 1-2 other books, and then I read the textbook for the course. It was not easy, you will invest probably 2x the work of any other class if you try what I did, and I didn't even do as much as I'm telling you here and tried half of this after the course.
The best thing you may be able to do if you're like me is just practice more and more problems, make sure you fully understand exactly why you were wrong if so, and double confirm why you were right just to make sure you didn't arrive on the answer via some fluke -- which I actually had happen to me on the midterm and gave me an over-inflated mark because... luck. You must understand every detail of why the formula exists the way that it does. I say this because the amount of dumb tricks on the midterms will not be pleasant if you get caught up on "am I actually right?" like I do and choke.
Also the fact that each question on the midterm was 1-2% of your mark also caused a great deal of stress, and I don't perform too well under it.
LAST n-1 TO n-4 WEEKS:
I rushed moment generating functions because I fucked up my study time when CSC236 came around for midterms and shallowly understood them as a result. This was a mistake, so don't do this. It's quite cool what you can do with it actually so let that inspire you.
Chapter 9 (or the last few weeks minus the very last week) was double integration with stuff, and I was not only extremely rusty at this but unable to find any external practice whatsoever. I went through the entire lectures having no clue why we were doing it, and due to a severe lack of time I memorized a ton of formulas instead of understanding... and paid the price on the final for that very reason (causing me to drop from an A-/A to a B, which pissed me off tremendously and was all my fault).
The very last unit which was Markov Chains for me was common sense and extremely interesting, and the exam questions were very straightforward with no tricks... or so it seemed...
And that is my experience with the course.
That class average was the lowest out of every course I've ever had, also was my lowest mark too, I wish I spent more time understanding. I found the middle weeks (mainly 3 - 6) to be straight forward and number crunchy with a lot more intuition, but you'll likely still have to haul ass for that section too if its your first time looking at that.
Maybe I would have had better luck in STA257 if they go into deeper understand of why with proofs, I don't know...
Try the two:
https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431
https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
introduction to mathematical statistics by craig and statistical inference by george casella.
Then read Analyzing Baseball Data With R
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
EDIT:
Most likely Rice will be the best book for a comprehensive look at prob/stat, and it is sufficiently technical.
Start With "Foundations Of Analysis" By Edmund Landau
http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X
It's a tiny book, but is very good at explaining basic abstract algebra.
Here is the description from Amazon:
"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."
With the above book you should then have enough knowledge to move on to calculus.
I recommend the two volume series called "Calculus" by Tom M. Apostol.
The first volume is single variable calculus and the second is multivariate calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&s=books&qid=1239384587&sr=1-4
http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&s=books&qid=1239384587&sr=1-3
Schaum's probability and statistics was enough for me when I was a physics undergrad.
There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:
First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.
Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.
And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.
After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.
The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).
If you have any other questions about learning math, shoot me a PM. :)
I found Geometrical methods of mathematical physics by Bernard Schutz to be helpful, though it doesn't have many problems and it doesn't go into much depth on covariant differentiation. But it is good about discussing the modern view of tensors.
I would recommend watching the first half of the International Winter School on Gravity & Light (check the YouTube channel as well ) if you are interested in learning tensor calculus for use in differential geometry for GR.
I learned tensor calculus in bits from several different courses and texts, so I'm not sure what the best ones that are actually dedicated to the subject might be. In any case, I think you'll have a lot of fun learning the subject.
Ross is the standard probability book. Its on its 9th edition, so it most likely has few typos (I have the 5th, and that was a solid book). Also, you can probably (get it?) find an older edition for next to nothing.
I really enjoyed Dantzig's, Number. It more explores the development of the numerical system, but I think that is tied into what you are interested in. The book doesnt really get too far into modern notation though.
I learned from Baez & Muniain; Gauge Fields, Knots, and Gravity.
Toward the end of the course, I met Brian Greene at a public talk, and he recommended Schutz; Geometrical Methods of Mathematical Physics.
Khan Academy, free.
If you want problems and answers, I highly recommend the Schaums guides. You’ll need to pick the right one for her level, but basically there are a lot of problems and answers to help understand the issues.
https://www.amazon.ca/Schaums-Outline-Probability-Statistics-4th/dp/007179557X
There's a 2nd edition of this book coming out in December with added chapters on tidyverse and using statcast data
I think learning proofs-based calculus and linear algebra are solid places to start. To complete the trifecta, look into Arnold for a more proofy differential equations course.
After that, my suggestions are Rudin and, to build on your CS background, Sipser. These are very standard references, though Rudin's a slightly controversial suggestion because he's notorious for being terse. I say, go ahead and try it, you might find you like it.
As for names of fields to look into: Real Analysis, Complex Analysis, Abstract Algebra, Topology, and Differential Geometry mostly partition the field of mathematics with corresponding undergraduate courses. As for computer science, look into Algorithmic Analysis and Computational Complexity (sometimes sold as a single course called Theory of Computation).
That's nothing. At least you are comparing different books, so maybe the new, expensive one benefits from something that has changed since 1960.
Look at this: Apostol, "Calculus", Volume 2. A brand new copy of the current edition in hardback is $270. That's the 2nd edition.
That book was about $20 when I bought a hardback copy in 1976 at Caltech. Guess what edition we were using? The 2nd edition, from 1969.
Same story with volume I. The nearly $300 edition they sell new today is the 1967 2nd edition. (Some sites list it as 1991, but it's still just the 1967 2nd edition text).
Feller. I forgot the author. It is a bit more rigorous. I would try to get it from your university or local library first to see if you will need another book to prepare.
http://www.amazon.com/Introduction-Probability-Theory-Applications-Edition/dp/0471257087
Its become a bit more expensive since when I bought it.
http://www.math.uah.edu/stat/index.html
Might not be exactly what you're looking for, but it's free.
You might also want to try Feller's book. I got a used copy of the third edition on ebay for 5 dollars...I wouldn't recommend paying the price they want for a new one. It's also on bookfi if you're into that.
If you want an inexpensive book just to get a taste of graph theory, I'd recommend Richard J. Trudeau's Introduction to Graph Theory.
If you want a more in-depth textbook, try one of these:
I would recommend these, as well as
http://www.math.upenn.edu/~wilf/DownldGF.html
http://www.amazon.com/Walk-Through-Combinatorics-Introduction-Enumeration/dp/9814335231
http://www.amazon.com/Introduction-Graph-Theory-Douglas-West/dp/0130144002/
The book for my undergrad diff eqs class. I highly recommend it if you have an introductory background in ODEs, but even if you don't (I didn't going in), it's a great book.
If you're looking for a thorough and rigorous introduction into probability theory, I'd recommend going with Introduction to Probability Theory and Its Applications Vol.1 and 2 by Feller. Another well recommended book is Probability and Random Processes by Grimmett and Stirzaker (this starts from the get-go with measure theory).
If you're looking for general statistics, then you may want to look at All of Statistics by Wasserman and perhaps Bayesian Data Analysis by Gelman, et al.
Finally, since you're a physicist, you'll probably want to take a look at Monte Carlo methods in particular, such as with Monte Carlo Statistical Methods by Robert and Casella.
Have you had a look at Carroll's general relativity notes? Chapters 2 and 3 are predominantly about developing the mathematics behind GR, and are very good introductions to this. I have a copy of Carroll's book and I can promise you that those chapters are almost unchanged in the book as compared to the lecture notes. This is my main suggestion really, as the notes are freely available, written by an absolute expert and a joy to read. I can't recommend them (and the book really) enough.
Most undergraduate books on general relativity start with a "physics first" type approach, where the underlying material about manifolds and curvature is developed as it is needed. The only problem with this is that it makes seeing the underlying picture for how the material works more difficult. I wouldn't neccessarily say avoid these sort of books (my favourite two of this kind would be Cheng's book and Hartle's.) but be aware that they are probably not what you are looking for if you want a consistent description of the mathematics.
I would also say avoid the harder end of the scale (Wald) till you've at least done your course. Wald is a tough book, and certainly not aimed at people seeing the material for the first time.
Another useful idea would be looking for lecture notes from other universities. As an example, there are some useful notes here from cambridge university. Generally I find doing searches like "general relativity site:.ac.uk filetype:pdf" in google is a good way to get started searching for decent lecture notes from other universities.
If you're willing to dive in a bit more to the mathematics, the riemannian geometry book by DoCarmo is supposed to be excellent, although I've only seen his differential geometry book (which was very good). As a word of warning, this book might assume knowledge of differential geometry from his earlier book. The book you linked by Bishop also looks fine, and there is also the book by Schutz which is supposed to be great and this book by Sternberg which looks pretty good, although quite tough.
Finally, if you would like I have a dropbox folder of collected together material for GR which I could share with you. It's not much, but I've got some decent stuff collected together which could be very helpful. As a qualifier, I had to teach myself GR for my undergrad project, so I know how it feels being on your own with it. Good luck!
Evolution of numbers is the "basic story". This is what Einstein said about this book
"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."
Read Amazon reviews to get more info.
The Cauchy-Schwarz Master Class is a great book on inequalities that will really improve your understanding of how and when to apply specific techniques. Highly recommended, and the paperback version isn't too overpriced.
That being said, competition mathematics also requires that you be able to recognize which technique to use (and then do it) quickly. Improving your speed is really accomplished by doing problem sets, and I don't have a good collection to suggest here.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
(Note: I wrote this elsewhere)
Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:
When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
I don't know of any good online introduction to discrete math, but here are a few book ideas.
Entire books have been written on this subject--here is a good place to start
I know this is a boring suggestion, but nothing beats the old, venerable Schaum's Outlines for their combination of problems, solutions, and inexpensiveness. If you are just starting, perhaps start with this one, and once you get some exposure to the basics you'll have a better idea of what you might want to pursue next- perhaps the next step would be analysis using a computer instead of by hand.
Lots of us have free YouTube videos on the basics that you can reference if/when you need them as you go. Try me or Kahn Academy, there are many others. Let me know if this idea doesn't fit with what you had in mind, and I can try to point you in a different direction.
Did any of your calc classes include multivariate/vector calculus? E.g. things dealing with double and triple integrals.
If not, take another calc class or two; calculus is very important for statistics. It shouldn't be too hard to pick up the rest of the necessary calc since you've already got a good calc background.
If so, start taking probability and statistics courses in your school's math department if you can. The mathematical way (read: the right way) of understanding probability and statistics is based on probability distributions (like the normal distribution), defined by their probability functions. As such, you can use calculus to obtain a myriad of information from them! For instance, among many other things, within the first one or 2 courses, you'd likely be able to answer at least the Spearman's coefficient question, the Bernoulli process question, and the MLE question.
If you don't have room in your schedule to do the stats course, you could get a textbook and try learning on your own. There are tons of excellent resources. Hogg, Tanis, and Zimmerman is pretty good for an introduction, though I'm sure there's better out there.
I quite like Schutz’s book:
Geometrical Methods of Mathematical Physics
https://www.amazon.co.uk/dp/0521298873/
Is this the book you are looking for https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/ref=cm_cr_arp_d_pdt_img_top?ie=UTF8 ?
There are more freely available books from erstwhile USSR published by Mir Publishers https://mirtitles.org/
https://www.amazon.com/Schaums-Outline-Probability-Statistics-4th/dp/007179557X
https://www.amazon.com/Probability-Through-Problems-Problem-Mathematics/dp/038795063X/
https://www.amazon.com/Problems-Probability-Mathematical-Statistics-Functions/dp/0486637174
https://www.amazon.com/Thousand-Exercises-Probability-Geoffrey-Grimmett/dp/0198572212
It's a good idea to abuse the library if still in school.
> Are there key ideas and considerations when working with particular game mechanics? Are there any critical questions I should be asking myself before the game starts? While its underway?
There is a whole field of study based around these questions. If you don't mind a little reading try looking into Game Theory.
If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:
Algebra
Trigonometry
Functions and Graphs
The Method of Coordinates
Not interested, but related to your mention of R, there's a book about analyzing baseball data using R. Haven't checked it out yet but I plan to.
The technically correct cop-out response is that voting itself isn't rational, so I'd say the really interesting question is about whether or not voters vote according to their interests. I think the diversity of answers in this thread is a good reflection of the diversity of thought on this question. There isn't really an easy answer.
Some reading I'd recommend would be Gelman's Red State, Blue State, Rich State, Poor State and any of the work on decision theory by Daniel Kahneman (often accompanied by Amos Tversky).
Calculus Volume 2 - Apostol.
found it! Apparently they've gone through several editions and added a coauthor since I bought my copy.
My father is a statistician and he is the one who recommended Hogg and Craig when I complaining about Casella and Berger. I spent a summer working my way through Hogg and Craig and then reviewed everything from my classes that previous year as my way for studying for the written quals. I passed so it worked. And then I promptly forgot everything.
This might get poo pood, but I really like some of the schaums outline books.
https://www.amazon.com/Schaums-Outline-Probability-Statistics-4th/dp/007179557X
Why? They are packed full of sample problems and answers, and they tend to provide really concise definitions. I think one of the better ways to understand conditional probability is to see it applied to a range of clear examples.
Also, these books are ridiculously cheap. Tiny investment to make on the off chance you don’t love the format.
I still use this book to quickly brush up on specific concepts at least once a year.
My intro/grad class used Bulmer's book. Its an enjoyable read, easy to follow and answers to odd exercises in the back... and a hell of alot cheaper than current textbooks.
Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.
And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/
And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/
And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/
If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/
You might try the aptly named book Introduction to Hilbert Spaces. I haven't spent much time with it myself (I slogged through Conway at the beginning of grad school), but it looks good for your application.
Also, I was reminded of this cool book:
Cauchy-Schwarz masterclass.
I'd recommend doing mathematics, It's much important than learning a language. It helps you grab the logic of solving a problem.
Discrete Mathematics by Rosen is the best book from my experience.
Graph Theory by Bollobas is recommended by many but i prefer Graph Theory by Douglas West
Algorithms by Cormen. No introductions needed this book encompasses most of the problems you'll encounter.
However if you're keen on learning a C/C++/Java i'd recommend the Head First Series from O'Reily .
Goodluck!
Schaum's Outline of Probability and Statistics is a good review with lots of practice problems. Check out the videos on Khan academy too, they really helped me with some of the concepts.
These are, in my opinion, some of the best books for learning high school level math:
These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.
Smooth manifolds play an important role in the theory of dynamical systems, so I would suggest learning that after, or concurrently with smooth manifolds. There's a good book by V. I. Arnol'd.
It's a parametric model. The parameters of the model are simply the parameters of the distributions he assumes (or the "hyperparameters" if there's some sort of multilevel modelling) over the visible data he feeds into the model (previous years' results). He's fitting using the Stan software (which uses No-U-Turn-Sampling, other reference and another). Once he gets all the posterior probability distributions over the parameters, it's pretty trivial to simulate the model a bunch of times to see the distribution over outcomes.
The advantage of MCMC is that you don't HAVE to calculate the normalization constant (which is hard). Look at the formal derivation of Metropolis-Hastings on wiki. The basic idea is that it relies on a fraction of posterior probability distributions for generating samples from a distribution. Since the normalization constant is present in both numerator and denominator, it cancels out. So you don't need to calculate it directly, and you only need to know the posterior up to a constant of proportionality. And this is generally much easier to do.
If you want a book to look through this stuff, the classic reference is Gelman's Bayesian Data Analysis (and he'll be coming out with a third edition pretty soon).
https://amzn.com/052138835X
/u/istvan_magyary have you tried libgen?
That's the name of the book. Yet Another Introduction to Analysis
PDF WARN: Introduction to Math Stat by Hogg
Not to be confused with Probability and Math Stat by Tannis and Hogg which is a "first semester" course.
Good blend of theory and "talky-ness", good exercises that test your understanding, most should be do-able from just applying the basics.
Is it Number: The Language of Science by Dantzig? It covers the early history of math pretty well. I strongly recommend it. So does Albert Einstein.
My math stats textbook is Hogg McKean Craig. I don't think the math would be too much for a computation statistics major, but it would give you a great overview if you're interested in that direction.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431
Here's the one I used for the class I took. It's a bit more of a reference though
https://www.amazon.com/Introduction-Graph-Theory-Douglas-West/dp/0130144002/ref=sr_1_18?ie=UTF8&qid=1510241743&sr=8-18&keywords=graph+theory+textbook
Here are some books I'd recommend.
General Books
These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.
Topical Books
For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
General Advice
One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
But how do you know? How do you know there's not a third root lurking out there somewhere?
To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)
Absolutely. The stress tensor is a (2, 0) tensor (called contravariant in the physicists definition), which means that it takes two vector inputs to produce a real number.
If you input a vector, say e1 (this may be x-hat, the unit vector in the x direction in Cartesian coordinates), it will return a vector which represents the force per unit area in that direction. It actually returns a 1-form (covariant vector), but in the case of the stress tensor, which is a Cartesian tensor, covariant vectors are the same as contravariant vectors, their duals.
This operation is called tensor contraction, where the tensor only acts on one input and returns another tensor of rank (n-1, m-1), or in the case of the stress tensor, it returns a (1, 0) tensor which is just a covariant vector, or in the case of cartesian tensors, it is just a vector (contravariant).
I encourage anybody who is interested in this stuff to read Schutz's Geometrical Methods of Mathematical Physics, as this book describes tensors fully in the newer language (my definition number 2), and does so with applications to physics. Most tensors in physics are taught in the old indices/transformation law language, and can be quite confusing for first timers.
Principles of Statistics (Bulmer) - this is a very nice introduction to probability and statistics. It takes you through the important distributions (binomial, normal, poisson, etc), laws of probability, central limit theorem, etc. And it's like $10 as an eBook or $15 in paperback.
http://www.amazon.com/Principles-Statistics-Dover-Books-Mathematics/dp/0486637603/ref=sr_1_5?ie=UTF8&qid=1463424228&sr=8-5&keywords=statistics
I use [this book]http://www.amazon.com/Statistics-Manual-Edwin-L-Crow/dp/048660599X) as a reference. It's very small and inexpensive (you may have to buy it direct from Dover, though). It won't go through any derivations, but it covers most of the very important, basic, topics. I also have my old textbook on backup as well. Old editions of textbooks are cheaper, and all the information is the same.
Looking through amazon, this one looks pretty good as well, especially for an inexpensive text. A reviewer said it ends with what we just talked about! Any book that lays down the foundations well enough should be fine. See if you can find one that does correlation analysis, since you'll probably use that later.
Getting more advanced than this may depend on your field. If you're in biology, I'd recommend a book on designing and analyzing scientific experiments. I can't recommend a good title, though, because I'm only familiar with computer experiments (which tend to be easier).
I hope this helps!
Game Theory: A Nontechnical Introduction might be a good place to get started.
Number: The Language of Science
Changed my thinking about math/science to be more compatible with my artistic/philosophical leanings.
I would suggest you take a look at the following book, https://www.amazon.com/Analyzing-Baseball-Data-Second-Chapman/dp/0815353510, or the Stanford sports stats class, https://web.stanford.edu/class/stats50/references.html
Not sure what your background is, but I quite liked Steele's book.
What do you mean by advanced Calculus? Multivariate Calculus without proofs?
Anyway,
Mathematical Analysis and Proof by Stirling
A First Course in Mathematical Analysis by Brannan
Yet Another Introduction to Analysis by Bryant
Game Theory: A Nontechnical Introduction
I'm a Statistics student, so naturally, I love the idea of applying numbers and statistics to decision making. I find the book fascinating.
I'd recommend Discrete Mathematics, Elementary and Beyond By Lovász, Pelikán, and Vesztergombi. It's the book I'm using in my undergraduate discrete math course, and I think it's a great introductory book that explores many areas of discrete math, and should allow you to see which field interests you most.
> 1) I was in a school like that. I didn't join. No one hassled me. No one ever said anything to me. The really pro-union people kept to themselves and the vast majority did whatever and could actually care less.
That's good they didn't hassle you. Olof I decided to join a union I would lay back in the shadows and not be adamantly going on tangents why people should join. Glad you weren't harassed. During student teaching there was a teacher without fail that every Friday would wear her union shirt.
>2) Probably not, coming from a perspective of Power. Because it is so large and controls all of CPS, I doubt it would ever want to be split up -- even if those smaller unions are basically CPS lite.
Great point, probably.
>3) I know. Tell me about it. It did all across the state (WI). Most of the old teachers that were stuck in their ways were either asked not to come back by the district; felt like they had to retire or else they would lose all of their benefits (I'm still unclear where this hysteria came from); and, more district flexibility allowed districts to better craft budgets reflective of their priorities. It was a good 5-year window to get hired here.
I remember several years ago it had made news. Is hiring better now? I know you got a lot of flack as a state about the education stuff.
>4) There are many possible answers for this. One answer I've seen is that more conservative-minded people are in professions that typically pay more (accounting, business (management), etc.). Another answer is that that conservative ethos of conserving your wealth (being thrifty) is something harped on if you grow up in a conservative household and it is, therefore, something carried one through one's life. And there are other reasons but you should avoid blanket statements because, actually, if you (taking Republican and Democrat to be proxies for conservative-liberal, respectively) measure it, you'd see that Democrats have slightly, on average, a higher income. Believe it or not, wealth at the top quintile isn't a really good predictor of political ideology. It's actually pretty even split between R and D. In the lower quintile, you'd find a stronger correlation between income and D or R: the poorer one is, the more likely they are to vote D. Yet, a better way to examine that would be racial. There you'd see a clear split between black low income (D) and white low income (R). This whole idea of wealth impacting voting habits and ideology is something political scientists are trying to still better understand. One of the better books, written for the general public, on this subject is (still) (Red State, Blue State, Rich State, Poor State)[https://www.amazon.com/Red-State-Blue-Rich-Poor/dp/0691143935].
Thanks for that in depth answer, I'll be sure to look into that more.
First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.
Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:
Geometry and solutions
Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.
A First Course in Calculus
For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.
Some more very good books that should be used more, by Gelfand:
The Method of Coordinates
Functions and Graphs
Algebra
Trigonometry
Lines and Curves: A Practical Geometry Handbook
This imo is a good book for basic probability and mathematical statistics. Super easy read with a lot of examples. [You also mentioned pdf's for books and someone told you library gensis. I can promise this one is on there :)]
I disagree; I think there is a need to be a dick about it, because you wrote this:
>I feel like there’s a thread of Democrats who just don’t understand the Prisoner’s Dillema at all.
Not only does this comment carry a superior tone, it's also wrong. r/politics is full of amateurs--many of whom are teenagers--who think they're experts in law, politics, and economics. They all congratulate each other for "getting it" when there are people who actually study these things. It is the acme of ignorance.
Game theory is math-intensive, and there is probably no way around that once you get past the oversimplified models. But this book seems to give a reasonable explanation without being too rigorous [1]. I haven't read that one. But Fudenberg and Tirole's text on game theory is often considered the standard.
https://www.amazon.com/Game-Theory-Nontechnical-Introduction-Mathematics/dp/0486296725
How Not to Be Wrong: The Power of Mathematical Thinking is my most recent awesomebook.
My go to book for anything graph theory related is the intro book by West.
Great book for undergrad / first year grad students. Goes into detail on numerous topics and if I can recall, you can find a bit of good application there. I know computer science replies on applications of graph theory quite a bit, so you may be able to delve further into that.
For probability I'd recommend Introduction to Probability Theory by Hoel, Port & Stone. It has the best explanations of any probability book I've seen, great examples, and answers to most of the problems are in the back (making it well-suited for self-study). I think it's still the best introductory book on the subject, despite its age. Amazon has used copies for cheap.
For statistics, you have to be more precise as to what you mean by an "average undergraduate statistics" course. There's a difference between the typical "elementary statistics" course and the typical "mathematical statistics" course. The former requires no calculus, but goes into more detail about various statistical procedures and tests for practical uses, while the latter requires calculus and deals more with theory than practice. Learning both wouldn't be a bad idea. For elementary stats there are lots of badly written books, but there is one jewel: Statistics by Freedman, Pisani & Purves. For mathematical statistics, Introduction to Mathematical Statistics by Hogg & Craig is decent, though a bit dry. I don't think that Statistical Inference by Casella & Berger is really any better. Those are the two most-used textbooks on the subject.
I think it's this: https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431/ref=mt_hardcover?_encoding=UTF8&me=
But really, if I remember right, they "use" it the same way we "used" the textbook in 400. (I do like both books though.)
Ah, this is great! Thank you :)
I didn't manage to find the book I need for MATH 3215 though. Is there any way you could get Probability and Statistical Inference, Ninth Edition by Hogg, Tanis and Zimmerman?
A First Course in Probability Theory by Sheldon Ross is the book that was used in my undergrad class. The book is currently on the 9th edition, but you can pick up a copy of the 7th edition in like new condition for under $15 plus shipping.
This is also one of the books that is suggested by the Society of Actuaries for the Probability (P) exam.
To the nay-sayers, I'll offer a contrary opinion: It is doable. Especially if you do linear algebra and multivariable calculus at the same time, since a lot of the underlying ideas and techniques are the same. It will, however, take focus.
I am by no means a mathematical genius, but with consistent, daily studying, I was able to take calc III and linear algebra in the same 5 weeks, and differential equations in the regular semester following that. By prepared to work hard, do lots of problems, and carefully dissect new ideas as they are presented, but it can be done.
EDIT:
In fact, I'd like to recommend a superb textbook that covers all three of these topics:
http://www.amazon.com/Calculus-Vol-Multi-Variable-Applications-Differential/dp/0471000078
If you're interested in self-study, it's often difficult when different textbook authors use different notation, or different but practically equivalent definitions and methods. Not only does this avoid that problem, but it's an extremely lucid and thorough book, with lots of exercises, and you can keep it for the rest of your career for reference.
If you want theoretical / mathematical I would suggest reading a few math, stats or engineering books.
Dover is a great place to find some cheaper reading material. They republish old scientific and math texts that were popular in their time in a smaller sized paperback. They're a nice size to bring around with you and they don't cost much.
Math and stats findings of today build on this knowledge, and much of it is still used in state-of-the-art applications. Or, that math/stats is used as part of some state-of-the-art algorithm. Lots of the newest ML algorithms are blending math from a variety of areas.
Statistical analysis of experimental data
Principals of Statistics
Information Theory
Statistics Manual
Some theory of sampling
Numerical Methods for Scientists and Engineers (Hamming)
Mathematical Handbook for Scientists Engineers
Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables
==
There is also the Data-Science Humble Bundle for more technical / practical skill building.
Discrete mathematics and any proof based math in general is what college based math should be like- if you continue to take upper level math and CS courses, you will undoubtedly face this style of math again. Plug and chug (which is what a lot of calculus is) will no longer be the norm.
There is often a very large learning curve for students who are not used to seeing this type of math- so don't stress out too much about it. Eventually, you'll break a point where everything will make (sort of) sense. I went through the exact same thing when I took discrete for the first time, and I felt like I was getting destroyed on everything (I still suck at some topics) until I suddenly hit a point of clarity where I could see how most topics were tied in together. Mathematics, and especially an introductory discrete course, is cruel in that way- that every topic you learn is inherently related to each other, so if you already fall behind just a little, the mountain to catch up just becomes incredibly massive incredibly fast- and it's hard to even pinpoint a place to even start to catch up.
You may be lost in learning elementary proof techniques, or number theory, and then the next topic (say it's graph theory) utilizes a bunch concepts and previous proofs from number theory, and then the next topic might use something proved in graph theory and number theory, and so on. All of a sudden, nothing makes sense, and to learn topic ___, you need to know graph theory, but to know graph theory, you need to know number theory, but you don't know number theory that well, and some topics in number theory can perhaps be explained by another topic in graph theory (or any topic for that matter) The chain is all interlinked and it may difficult to even see where to start- but it is for this reason that once you cross this steep barrier, most things will suddenly become clear to you.
So I'd advise you to just continue visiting professor office hours, asking more questions, asking for other students' help, doing more and more practice. It may seem like you're getting nowhere, but you're essentially learning a new language right now, so it'll obviously take sometime until you feel as if you know what you're doing. Figuring out where people get the intuition to suggest seemingly random functions or a set of numbers or some assumption will come to you slowly, and slowly you'll break more and more of this chain.
https://www.amazon.com/Discrete-Mathematics-Laszlo-Lovasz/dp/0387955852 is another book my professor enjoyed using as a supplmenet.
Wow, ambitious! I'd highly recommend V.I. Arnold's book on ODEs: http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 ... not only is it a great book in itself, but it should give you an excellent foundation for differential geometry and more advanced geometric mechanics (e.g., Lagrangian/Hamiltonian mechanics, dynamical systems, etc.).
There is a very interesting book called Number which explores this a bit. It goes through the history of the development of the concept of what a number is, including the shift to writing math with symbols instead of words.
I can't recommend this book highly enough.
I would also recommend Cauchy-Schwarz Master Class. It has a lot of interesting problems without requiring a whole lot of prerequisite equipment. It also does an incredible job explaining how to go about solving problems.
http://www.amazon.com/The-Cauchy-Schwarz-Master-Class-Introduction/dp/052154677X
What about Number: The Language of Science by Tobias Dantzig? I came across it at a Border's closing and rather enjoyed it.
http://www.amazon.com/Number-Language-Science-Tobias-Dantzig/dp/0452288118/ref=sr_1_1?ie=UTF8&qid=1347469608&sr=8-1&keywords=Number
Dude, Feller
http://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257087
I believe those were the books used during the 2016-2017 school year (thats when I took discrete II)
From what I understand now, the newest renditions of the course use
Discrete Mathematics and Its Applications by K. Rosen
and
A First Course in Probability by Ross
But it'll depend entirely on who it is that's offering the course during the summer and what they include on their syllabus so I'd wait until seeing what they say to purchase either of the books.
The first book you listed (Mathematics for Computer science) is available for free for anyone to use here
The second is available for free on the Rutgers libraries website so I'd advise you not waste your money buying either of those two.
Hope this helps
Since you're an applied math PhD, maybe the following are good. They are not applied though.
This is the book for first year statistics grad students at OSU.
http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126/ref=sr_1_1?ie=UTF8&qid=1368662972&sr=8-1&keywords=casella+berger
But, I like Hogg/Craig much more.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431/ref=pd_sim_b_2
I believe each can be found in international editions, and for download on the interwebs.
Sure, there are a few directions you could go:
Algorithms: A basic understanding of how to think about and analyze algorithms is pretty necessary if you were to go into combinatorial optimization and is a generally useful topic to know in general. CLRS is the most famous introductory book on algorithms, and it gets the job done. It's long, but I thought it was decent enough. There are also plenty of video lectures on algorithms online; I liked the MIT OpenCourseWare of this class.
Graph Theory: Many combinatorial optimization problems involve graphs, so you would definitely want to know some graph theory. It's also super interesting, and definitely worth learning regardless! West is a good book with lots of exercises. Bondy and Murty and Diestel also have good books, which are freely available in PDF if you do a google search. Since you're doing a project on traffic optimization, you might find network flows interesting. Networks are directed graphs, where you think about moving "flow" across the edges of the graph, so they are useful for modelling a lot of real-life problems, including traffic. Ahuja is the best book I know on network flows.
Linear and Integer Programming: Many optimization problems can be described as maximizing (or minimizing) some linear function subject to a set of linear constraints. These are linear programs (LPs). If the variables need to take on integer values, then you have an integer program (IP). Most combinatorial optimization problems can be formulated as integer programs. Integer programming is NP-hard, but in practice there are methods that can solve most IPs , even very large ones, relatively quickly. So, if you actually want to optimize things in real-life this is a very useful thing to know. There's also a mathematically rich field of developing methods to solve IPs. It's a bit of a different flavor than the rest of this stuff, but it's definitely a fertile area of research. Bertsimas is good for learning linear programming. Unfortunately, I don't have a good recommendation for learning integer programming from scratch. Perhaps the chapters in Papadimitriou - Combinatorial Optimization would be a good introduction.
Approximation Algorithms: This is about algorithms which quickly (in polynomial time) find provably good but not necessarily optimal solutions to NP-hard problems. Williamson and Shmoys have a great book that is freely available here.
The last book I'd recommend is Schrijver. This is the bible for the field. I put it here at the end because it's more of a reference book rather than something you could read cover to cover, but it's REALLY good.
Lastly, if you like traffic optimization, maybe look up what people are doing in operations research departments. A lot of OR is about modelling real problems with math and analyzing the models, so this would include things like traffic optimization, vehicle routing problems, designing smart electric grids, financial engineering, etc.
Edit: Not sure why my links aren't all formatting correctly... sorry!
this book really helped me in undergrad. Has a lot of really good concepts. It went along with a course but it does a great job on its own explaining some of the most relevant concepts to computer science.
This one?
Read this book and it will make the 1:6 odds clearer, plus ensures you will never be wrong (not even on reddit):
http://www.amazon.com/gp/aw/d/1594205221?pc_redir=1410606086&robot_redir=1
I can't remember, honestly. That's how bad it was. I'll dig out my notes tomorrow.
Prob Theory and Math Stats together was basically this book.
My favorite introductory discrete math textbook is https://www.amzn.com/0387955852. (It also appears to be available for less unreasonable prices.)
no you have just to use common sense when interpreting a poll. I recommend the following book it does discuss the problems with different models and how corrupt math is in the business world. https://www.amazon.ca/How-Not-Be-Wrong-Mathematical/dp/1594205221
http://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/1594205221
You need to read this book. Converting the percentage to the number of dead Americans is absurd.
Specifically, just read this chapter: How Much Is That In Dead Americans?