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1

For full disclosure, I am hardly an expert in this area (indeed, I'm learning from Fulton, just like you), but I thought I'd throw in my two cents. First, a recommendation for a "really beginner book:" Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea. This is one of the friendliest math books I've ever read, and the emphasis on computation and ...


0

All known factorization algorithms (with the exception of very few, like trial division) rely on the fact that GCD is efficiently computable. When trying to factor $n$, the idea is to come up with some other integer $m$ such that $(n,m)$ is a non-trivial factor of $n$ (different from $1$ and $n$).


5

1) "Maybe because this area is very recent?" No, algebraic geometry goes back at least to Descartes and Fermat and is essentially as old as calculus. 2) "Because there aren't many buyers to buy algebraic geometry books?" No, that's irrelevant. Authors don't care, since they know very well that any book beyond the level of linear algebra or calculus won't ...


0

Ronald Brown's text Topology and Groupoids is probably what you want from a topology text. He gives an introduction to general topology and the fundamental groupoid using the language of category theory throughout. It's an excellent textbook.


1

It's about not taking theorems for granted. I'm not a mathematician, but it was quite a realization for me when it hit me that a lot of the things we "found" in high school were... complete magic at that point. For example, the Rational Root Theorem and Descartes' Rule of Signs... did you ever wonder why those work? Or did you--like me--just think "Cool!" ...


5

I'm going to start with your example and work towards a more abstract notion of structure throughout this writing. So let's see, the bijection you give is a function $f:A\rightarrow B$. But all we have are the sets $A,B$. No other information is given. So what does the bijection encode? Well, both sets have $3$ elements. Perhaps that is what we should look ...


5

Talking from my own experience (about 3 years of mathematics at university). I'd like to make few separate points, which somehow connect together, but the connection might be different for different people. High school I was (sometimes am) the kind of person, to whom proofs were something boring (at the high school). Somehow, math has that kind of ...


-3

Math is an art,first and foremost.And as every art its mostly an end by itself.It's also about solving puzzles and having fun while you are at it.It so happens that most of these puzzles are a bit too complex for most people,not so much in the difficulty associated with solving them,but in taking the time to understand the complex rules that characterise ...


0

Most of the time, they try to formulate precise statements about some idea they have, and spend some time proving, or refining, or abandoning them.


12

Mathematics is about understanding things. Proofs are one part of this. The bulk of the time is spent asking questions, working out examples or doing calculations, figuring out what others have already done, positing conjectures/hypotheses, and thinking about ideas for why something should or should not be true. There are many famous unsolved problems ...


21

Yes, mathematicians do a lot more besides writing proofs! While some already pointed out that “they also have lunch” I assume that you are more interested into the intellectual processes involved in the mathematical activity. When I¹ examine my intellectual processes, I can see that some exercise my intuition and some exercise my rationality. My intuition ...


-4

Einstein, Newton, Euler & Hawkins to name but a few of the greats all solved problems, found solutions and created new way's of seeing the world, looking at problems and adding an explanation to all things. Not just maths, but the universe and many other wonders. Most of it was born in their head, an idea, a theory. The theory's all have to be proved, ...


16

Literally all math you've ever done are "proofs". You just get more rigorous about it. "Solve the equation $2x+5=0$" really means "prove that there exists a solution to $2x+5=0$ and give an expression for it". Doing math is just doing logical reasoning with certain rules. In higher math, it is more explicitly logical and rigorous but honestly IMO the ...


1

We don't know if space is infinitely divisible, but if it is, then it has uncountably many points, because if $(x_n)$ is any infinite sequence of points, and if a sequence of regions of space $(S_n)$ is constructed recursively so that $S_{n+1} \subset S_n \setminus \{x_n\}$ ($n = 0, 1, 2, \ldots$), then this nested sequence contains at least one point that ...


0

I guess in a sense it depends if you believe the universe is inherently digital or analog. If analog, there is infinite detail, so the surface area of any actual physical object would be infinite.


-3

Set of stars is infinite in reality. Infact it is countably infinite. Another example of infinite set is "set of all points in a line segment is an infinite set" See this it says the that set of integers is countably infinite whereas set of real numbers is uncountably infinite. In countably infinite sets you cannot count the total number of elements but ...


1

If you accept that one can form infinite sequences from say the set of symbols $\{a,b\}$, then I would say that the set of all such sequences gives a fairly nice example of an uncountable set; as shown by Cantor's diagonal argument. This may be simpler than the uncountability of the reals, or of the interval $[0,1]$ as you don't have to worry about some ...


8

That depends. If by "realistic" you mean something that has to do with physical reality, then I defy you to come up with a set which has exactly $200^{200^{200}}$ elements. If by "realistic" you mean something which comes up naturally in mathematics, then $\Bbb R$ is an uncountable set. As for explaining the difference between them? That's not very easy, ...


6

Always show enough work on your exam so that it is clear, when you make an error, what sort of error it is. Writing clearly makes it easier to grade, but it also makes it easier for you to check later. At heart, if you are making a lot of stupid mistakes, either (1) you don't really understand the steps, and are just applying rote thinking, or (2) you ...


12

A calculus exam is designed to test whether you know calculus, not its prerequisites. Your professor probably assumes you are well-versed enough in arithmetic and algebra that you will not make substantial "simple" errors. If you know the calculus,[*] then, but you're consistently making errors in arithmetic and algebra on the exam, then I'd suggest you need ...


0

I like the Dirichlet function which is periodic but doesn't have a smallest period. Or the always vanishing function which is odd and even.


1

Very intuitive proof that: $$\sum_{n=1}^\infty {(-1)^{n-1}}\frac{1}{3^{n-1}} = \frac{3}{4}$$ (from this web page)


1

Not only are the answers from the Mathematics StackExchange community "more intuitive and instructive," they're much more valid than anything you will find on Wikipedia. Although it's true that a lot of the "community" here are also active on Wikipedia, their talents and insights are mostly wasted over there. There is much tighter control here than there. ...


1

Well I'm not a programmer, but I'm pretty sure that you need math to program like the air you breathe. Especially if you want to become very, very good at it; I mean if you want to understand algorithm running time, do analysis on the complexity of a specified algorithm or trying to find a new one who performs better, you really need to be confident in ...


3

In university departments, mathematics is usually grouped with the physical or natural sciences. The physical or natural sciences usually include physics, chemistry, mathematics, statistics, computer science, etc.


7

(The following is not meant to be serious mathematics.) In decimal there are $1000$ three place numbers. The probability that at the $N^{\rm th}$ decimal place of $\pi$ the last three figures enounce exactly the number $N$ therefore is ${1\over1000}$, and the probability that this does not happen is ${999\over1000}$. Assuming independence of the involved ...


0

A very basic picture of an algebraic structure is that of just a set with "interconnections" between the elements. Imagine a simple set, an alphabet, say, of two letters, "A" and "B". Now suppose that you can combine the letters to get a bigger set by putting the letters together in any combination (I will talk about a monoid for simplicity). So, something ...


2

For Group theory, I can highly recommend "Visual Group Theory" by Nathan Carter. He does a great job delivering a good intuition not only of the concept of a group, but also of different group-related definitions, different groups... (Like the title suggests; in a visual way) http://web.bentley.edu/empl/c/ncarter/vgt/gallery.html For Ring theory, what ...


0

What you have described for $X$ is just the matrix of predictors. A random matrix is quite different. Lets start with a perhaps more familiar example, a bivariate gaussian random variable. This "variable" is a vector valued random variable $X=(X_1,X_2)$, where the two components of $X$ can co-vary with each other (i.e., to get an "oblong" shaped gaussian ...


6

I must disagree with claims that "Algebraic Number Theory" is an algebraic study of anything-whatsoever, possibly including number theory, or, possibly "numbers", whatever the reference may be. That is, in genuine practice, it is "the theory of algebraic numbers", including "algebraic integers", including $p$-adic methods, including complex variables ...


5

It is the study of number theory from an algebraic viewpoint. The methods of algebraic number theory are used to solve many problems in number theory. For example, the study of Gaussian integers sheds light on problem of which prime numbers are the sum of two squares.


8

It's mostly the latter: the study of number theory from an algebraic viewpoint, just as analytic number theory is the study of number theory from the viewpoint of analysis. With algebraic number theory, it is often easier to solve equations that would be more difficult if not impossible with elementary methods. Algebraic number theory often deals with these ...


1

For example in data analysis one could use topological approach. And they actually do. Look the cloud of data (very big data) may have some topological characteristics: cycles, wholes, components, and so on. Traditional statistical tools are not robust enough to deal with certain high dimensional and noisy data. We need additional methods that we can use ...


3

Consider the Wikipedia page for Algebraic Number Theory in other languages: Théorie algébrique des nombres Algebraische Zahlentheorie Teoria algebrica dei numeri Teoria algébrica dos números The exception that proves the rule is Spanish: Teoría de números algebraicos which starts by acknowledging the other form: "La teoría de números algebraicos o ...


0

I can only tell from my personal experience and I go about it like this: I will first try to get a rough overview of what I am supposed to know, with a coverage of maybe 50%-70% . As I go and find things I am supposed to know that I do not know I will make notes. I will then again look at the notes and repeat them. This can hopefully be achieved in 1/3 or ...


0

Ok, I would recommend that you first have a complete intuition of each topic, and, to do so, you may look at all the statements or the propositions of a certain topic and ensure you understand every single thing about them to the most fundamental level. And, to do so, you may also want to understand how the propositions or statements of the topic are ...


0

We often require that both the magnitude, $|z|$, and the denominator of a fraction be positive and real. For $z = x + i y$, $z^2 = x^2 + 2 i x y - y^2$ whereas $\bar{z}z = x^2 + y^2$ which is positive and real, so we define $|z| = \sqrt{\bar{z}z}$. If $z$ is pure real, $|z|$ returns the same result as $\sqrt{r^2}$, so the extension leaves the original ...


1

My professor always called them the shoe-sock theorems or the likes, the idea basically being that you put on your socks first then the shoes, but to do the reverse you must get off the shoes first and then the socks.


1

Considering the dimensions of the various matrices shows that reversing the order is necessary. If A is $m \times p$ and B is $p \times n$, AB is $m \times n$, (AB)$^T$ is $n \times m$ A$^T$ is $p \times m$ and B$^T$ is $n \times p$ Thus B$^T$A$^T$ has the same dimension as(AB)$^T$


3

If $f(x)$ is a polynomial with real coefficients, and $z \in \mathbb C$ is a root of $f$, then $\overline{z}$ is also a root of $f$; in other words complex conjugation acts on the roots of $f$, and we can separate the roots of $f$ into orbits according to this action. An orbit is either a root with $z = \overline{z}$, i.e. a real root, or a pair $\{z, ...


4

The reason is also to acquire inverses and do division $$\frac{z}{q}=\frac{z\bar{q}}{q\bar{q}}=\frac{z\bar{q}}{|q|^2}$$ Where you get $(a+bi)(a-bi)=a^2-(bi)^2=a^2+b^2$


17

One motivation, if you can call it that, is that $i^2=-1$ does not define $i$, because $-i$ also satisfies that equation. So, there are two elements that could be $i$ and there is no algebraic reason for choosing one over the other. In other words, $\pm i$ are interchangeable, hence conjugation. Technically, interchangeable means that there is an $\mathbb ...


2

Numerical Optimization by Nocedal and Wright is quite a classic in the field. It also contains a few appendices introducing an overview of the necessary background.


1

I learned it from Mathematical Modeling by M. Meerschaert. The problems allow for interesting questions that go beyond his suggested exercises, so it's a great source of problems. Also, he writes problems that give you an excuse to learn things like Maple or R Regarding what Calculus to review for this text, you should learn about Newton's Method, the ...


1

A very good Introductory book with detailed explanation http://www.amazon.co.uk/Linear-Nonlinear-Optimization-Igor-Griva/dp/0898716616 The tip I would give is just like all the other maths please practice and write detailed solution. In Optimization solutions can be very very long but i highly recommend solving them. Also one often forgets the algorithm ...


4

The form of the function of course comes first; just as Kevin has said, it is an approximation problem. People do this by least square approximation most of the time, that means you pick a form of a function, for example, $$\frac{a}{b x^2 + c}$$ and then pick as many points as you want, said 1000 points, and calculate the values $a$, $b$, $c$ such that ...


3

The first observation, which is close to something you said, is that you can't approximate the entirety of the curve until you have a rather detailed model-but to get a computer model, for instance, of the curve is almost the exact same problem as approximating it by a function! Instead, notice that when you're tracing a curve you're not really making ...


2

Yes, there are past examples, and it is likely still possible. Here are some recent examples that I've heard of: Newton's polygons (which Newton described in letters he wrote in the seventeenth century) have evolved into Newton's polytopes, which have interesting applications in algebraic geometry. The Bernstein–Kushnirenko theorem (a result from the ...


7

You shouldn't think of a group as a thing in the same way as a vector space is a thing. Groups are not things, groups act on things. If $V$ is a vector space, then the collection of invertible linear transformations $V\to V$ is a group. If $X$ is a set, then the collection of all permutations of $X$ is a group. If $A$ is any object, in any category, then ...


3

It is true there is a lot of high level math in physics, some of which is done by physicists and some by mathematicians. For example the professor I am working with works with (and writes papers with) physicists very often, but the work he does is really math because it comes from a field called algebraic geometry which depends highly on pure math fields ...



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