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1

Sure. Why not? I studied PMA right after I studied set theory. IMO, there's no preliminary for PMA if you skip last few chapters. (Such are Lebesgues measure and multivariable calculus. AND I recommend you to skip those chapters. It's not just me, but many people think those chapters are not that much good as the first half chapters, till chapter 8) It's ...


1

One is that the category of smooth manifolds does not have pullbacks; in particular, the intersection of two smooth submanifolds of a smooth manifold is not necessarily a smooth manifold, and even when it is it doesn't necessarily have the correct behavior (e.g. additivity of codimension). In general you can only take intersections like this under ...


3

Nonlinear dynamical systems can be quite roughly divided in two huge realms: the realm of order (continuous time two dimensional systems) and the realm of chaos (dimension three and above for continuous time systems). The former is mostly about 16'th Hilbert problem about the number of limit cycles on the plane and still is a huge area of research. The ...


1

As with any serious field, there are those in the math community who might take it seriously, and others who wouldn’t. There is also a huge swath of the community for whom it would make little or no difference, and those in that swath might not even ever come across it. If it's significant — and that can probably best be measured by either having a large ...


0

Kaplansky's Linear algebra and geometry does a nice readable overview of the connection with linear algebra. Artin's Geometric algebra goes further in depth! but is a little less fun to read. Then there are Kaplansky's references to Coxeter's books which I imagine are great, but I haven't had the time to read them yet.


0

I will say convex optimization. Yes, convexity has been studied for a long time, but convex optimization as a separate branch is recent. there is a lot of development in last thirty years, especially in the methods related to numerical solutions.


0

Looking at the questions you've asked, it doesn't appear that this site is discouraging your questions. (But maybe you've deleted some.) But if you're wanting to "bounce ideas around" then there's probably a (good) question hidden in there that's worth asking. Also, if you're performing at a research level, I don't think MathOverflow would shun your ...


0

You should definitely not study all by your own. But it is never a good idea to pay someone for tutoring. A few personal suggestions: 1.Attend or audit classes and seek help from the TAs and professors. 2.Watch online lectures if you do not have access to MATH courses locally. But Im afraid there aren't many advanced math video lectures out there. ...


1

Here's a few off the top of my head. Experiments in Topology, Stephen Barr On Knots, Louis Kauffman Gödel, Escher, Bach: an Eternal Golden Braid, Douglas Hofstadter 1,2,3, infinity, George Gamow I've made this community wiki, so other people can feel free to edit this post and add to the list.


0

I would recommend one particular book Rudy Rucker - Infinity and the Mind: The Science and Philosophy of the Infinite. It is a friendly introduction to the concept of infinity, transfinite numbers, and related stuff including discussions of Gödel's Incompleteness theorem and more.


2

Linear algebra applications were theoretically doable, but often completely impractical before computers could be used to speed up computation. There are probably other topics that had a good theoretical base, but the "breakthrough" of fast computation made them finally a lot more actionable.


2

I suggest the rapid development in the early 20th century of game theory, largely through the work of von Neumann. Game theory now has important applications in various fields, especially biology and economics.


4

A lot of theory of computation (started with Turing, Von Neumann and others) brought areas of study for mathematics. An important one is Complexity Theory, which made possible to classify problems in terms of their computational difficulty. This kind of thinking wasn't too much important before the computers arise, but today is very important and has a ...


2

Far away from my expertise but I heard that Homotopy Type Theory may lay a new foundation to the way we analyze proofs. (Google "The HoTT Book")


5

Two things that happened after Galois and created new branches of mathematics: Cantor's foundations of set theory. The creation of functional analysis by the Polish school around Banach.


4

Despite knowing little about it myself, I would say the use of computer applications to develop and/or verify mathematical proofs would count.


4

I suggest category theory, not because of the investigations of more or less exotic categories, but because of how different theories are coupled together through functors and become comparable.


0

More likely your calculator is using table lookup with some simple interpolation (possibly even just linear) between values in the table. Calculators (and computer math co-processor chips) can do this because you just need the answer to some known (10-15 digit most likely) precision. So as long as the error from the interpolation is below that threshold, ...


0

The problem with Taylor polynomials that, while they are very accurate near the center, they get worse and worse the farther away you get from it. What you want for a calculator or computer chip is to be uniformly accurate over the entire interval in question, otherwise you waste computing power with unnecessarily accurate estimates in some areas at the ...


0

$0.01m = 1cm$ and $0.001m = 1mm$ therefore $3.625612m = 3625.612mm$ which is conventionally rounded to $3626mm$.


0

The vertices of a tesseract may be thought of as consisting of the $16$ points of $\{0,1\}^4=\{(a,b,c,d) : a,b,c,d\in \{0,1\}\}$. Two vertices are adjacent (connected by an edge) if and only if their coordinates disagree in exactly one place. For instance, $(0,1,1,0)$ is adjacent to each of the four vertices $(1,1,1,0), (0,0,1,0), (0,1,0,0), (0,1,1,1)$. ...


4

It is all based on our imagination. As you say, we are $3$D organisms, and as such, we have visual memories and experiences from a $3$D world. This $2$D drawing of a cube is not similar at all to a real cube, but the image created in our brains while watching it resembles the one we get when watching an actual $3$D cube. Thus, as far as we can tell (with ...


0

I think this is just a matter of conceptualisation. But Calculus is better because it gives us the concept and the language to do the calculations. Another way is instead of thinking that a 2D area has zero height, I would like to think that height is undefined in the 3rd dimension. Then when you multiply by the height, you give the physical object a 3rd ...


0

A non trivial measure taking only the values $0$ and $\infty$ is non $\sigma$-finite .


0

Counting measure on an uncountable set is not $\sigma$-finite.


2

The book Algèbre linéaire et Géométrie élémentaire by Jean Dieudonné is a highly formal presentation of elementary geometry using linear algebra. It's the way the Bourbaki school might have taught geometry in high school. Two books called Linear Algebra and Geometry, one by Kostrikin and Manin, the other by Shafarevich and Remizov, apply linear algebra to ...


1

Math is a delightfully introspective subject. I know more about my innermost thoughts and the deeper workings of my mind through hours staring at a whiteboard than a psychologist could gather in a hundred years. From this, I know that I cannot take notes. Ever. If I have any form of paper in front of me during a lecture, then I have a blank slate for my ...


2

My opinion on notes has steadily changed over time. Here is what I think now. Warning up front: if you go into class tired or distracted, my suggestion is to write everything down that is written on the board, or even more if your presenter has a more oral style. The reason is that you (a) probably aren't very good at discerning what is important in your ...


1

Banach-Tarkis paradox shows that it is impossible to have a Banach measure in 3 dimensions. This is a bit disappointing, because it would allow us to "measure" all sets, in some sense, if it existed. Also, this is surprising, because Banach measures exist in dimension $1$ and $2$. A Banach measure is a much like a "normal" measure, except: It is only ...


0

It's not a book, but I found Timothy Gowers' address "The Importance of Mathematics" quite inspiring as a graduate student. I wish I had watched it earlier in my career.


0

Real analysis will provide tools to solve many problems, mundane or not. One need only analyse a few problems, and soon you'll need real analysis. The same would happen with computer science: what problem does it solve which cannot be solved by the calculating mind? I don't get if the doubt is why need infinite objects? or abstract? Or only, real analysis? ...


0

See Spectrogram and some of the links there. Basically the signal is going to be a complicated mixture of tones of different frequencies with varying amplitudes. The phases don't really matter (your ears can't detect them except when two signals of close frequencies form beats).


0

What about Maxima from MIT? It's under GNU-license.


1

I have been thinking about this a lot, because i wanted to make something similar. In my opinion the best method for scoring is based on the answers of the other participants in the quiz. You get 100 points if the answer is exactly the right answer, otherwise you get 100 points minus the percentage of participants whose answer is better than yours. An ...


0

If you plot the gradient of the tangent for each point on the original curve, you get a second plot. The second plot gives the rate of change of the first plot. The second derivative is the tangent at any point on your second plot's curve, this value is the rate of change of the rate of change in your first curve. Now repeat the process with your second ...


2

Usually, in the practice, the second derivative is called "curvature" and it is related with the change in direction of the tangent. Think the function describing the curve as a path. Each point is moving along that curve. Then, in a quite simplistic way, the first derivative represents the velocity while the second is the "change in direction" or the ...


0

A "physical" approach to a possible proof of the Riemann Hypothesis: The Spectrum of Riemannium. The idea: the zeros of $\zeta$ are "like" the energy levels of an atomic nucleus.


1

Ther is no way to omit the $\exists$ quantifier in every context. If we use the convention of omitting initial universal quantifiers, we assume that the sentence : $\forall x(x > 0)$ (which is false in the domain $\mathbb N$ of the natural numbers) can be abbreviated as : $x > 0$. Consider now the negation of $\forall x(x > 0)$, i.e. ...


2

If you really want to avoid using quantifiers, from a logical point of view (I mean logically, not in the sense of logic theory, I don't know much about that) the statements $$ \forall x \in A : p(x) \qquad \text{ and } \qquad x \in A \,\, \Rightarrow \,\, p(x) $$ are equivalent just as $$ \exists x \in A : p(x) \qquad \text{ and } \qquad x \in A \,\, ...


4

$15+35+10+80+15+12+30+20+20+40-20-60-10-15-45-30-35-10-5=47?$


1

If you begin a proof with $x \in A$ and demonstrate that $p(x)$ is valid, then its really the same thing as saying $\forall x ( x \in A \Rightarrow p(x))$ ( Because if $x$ is not a free variable of $\phi \in \Gamma$ and $\Gamma \vdash \phi$ then $\Gamma \vdash \forall x \phi$, so its logical unobjectionable) If your question is only about how ...


4

The fact that the universal quantifier on $x$ is omitted doesn't mean it's not supposed to be there. The correct parsing of $\forall x\in A: p(x)$ is $\forall x(x\in A\rightarrow p(x))$. For existential quantifier this would be $\exists x(x\in A\land p(x))$. You may feel it is okay to omit the outer quantifier, but you have to remember it's still there.


1

I presume you must have already taken a basic course into calculus and analysis, so the book Analysis In Vector Spaces-A course in advanced Calculus can be a good place for you to start with. The chapters $3$ to $5$ which I have read were very well-explained ...


-2

Yes, it is possible to use physics to solve a math problem. As a matter of fact, this is exactly what a calculator does. When you enter 2 + 2 and press the equals button, energy and electrons move within the circuitry in such a way that 4 appears on the screen. As another example, consider the harmonic series and the cup of water. With the cup full, ...


3

Here is another bunch of texts. Like the ones suggested by Mathemagician1234, they are not general texts. The level of formality is variable. Classical mechanics F. Scheck, Mechanics, Springer, 2010. Although not specifically geared toward mathematicians, it makes use of mathematically advanced tools. I consider it the best book on classical mechanics ...


0

That will probably be a matter of opinion. The importance also will be a matter of what school you belong. If your're in logicism, than it will matter a lot; if in intuitionism, not very much. Other people already wrote on this subject, you should see what Bertrand Russell says of Wittgenstein and Ramsey in "My philosophical development". And also "The ...


0

Mathematicians seem to be persons that: study mathematics; pick up new ideas and formulate consistent theories that capture the essence of these ideas; formulate and examine statements and try to prove them or find counterexamples; publish their discoveries and results to be reviewed and used by other mathematicians; teach mathematics to others. I ...


2

At Oxford we used http://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710 for the first year of the Physics Course.


6

I think the nicest way to 'picture' this intuitively without having started your course is the 'amplitwist' concept coined by Tristan Needham (I'd recommend his book, Visual Complex Analysis, if this idea interests you). Essentially, to quote from his exposition: Analytic mappings are precisely those whose local effect is an amplitwist: all the ...



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