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3
votes
0answers
127 views

Any comments on Lax's “Calculus with Applications, 2e”

There's a new calculus book titled Calculus with Applications by Peter Lax (2nd edition of an old one). I really liked his linear algebra and functional analysis books, and I was wondering if this ...
6
votes
0answers
134 views

Mathematicians who were not intelligent [closed]

Were all the mathematicians born intelligent or some of them worked harder to achieve their goals? This question has been haunting me for years. I would love to be a mathematician but I am afraid I am ...
21
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
1
vote
1answer
55 views

Video/audio lectures on differential topology?

Do there exist decent online video lectures, or even audio lectures, covering differential topology? I'm aware of Milnor's talk, but it is more like exposition and doesn't go very far.
-3
votes
1answer
64 views

Feedback on question about integer solutions to $X^n + Y^n = Z^n$, $n \geq 3$

Please give me feedback on my answer to this question. Q: The equation $X^{n}+Y^{n}=Z^{n}$,where $n\geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are integers. A: False. Let ...
3
votes
1answer
90 views

(translated)Russian mathematics books?

Most russian mathematician(generally) are known to do and teach mathematics in a very original manner,they do in a very intuitive yet rigorous way, with/through wonderful connection to physics. ...
1
vote
2answers
114 views

Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox ...
10
votes
6answers
6k views

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to ...
2
votes
2answers
144 views

How to prevent doing the same mistakes over and over again?

At the moment I am preparing for the GMAT. However, a phenomenon that occured and has occured in the past is that sometimes I always make the same mistakes at similar problems, especially when doing ...
1
vote
1answer
54 views

Is there anything we can add to the present Euclidean Geometry?

I did not study the detail history of Euclid. As far as I know, the geometry starts from a few postulates and keeps on expanding (through theorems developing) to the present state. Even with these ...
6
votes
3answers
131 views

Algorithm for multiplying numbers

Background Today I had to explain to some kid how to multiply numbers with multiple digits in them. Then I recalled, that some other day I answered this question describing one of the numerous ...
140
votes
78answers
12k views

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
8
votes
8answers
3k views

How to become proficient in Calculus?

It has been a while since I wanted to ask this question, but couldn't find a right forum. My question might come across as trivial, but its important to me to find an answer to that. Let me give you ...
1
vote
3answers
296 views

Have any one studied this binomial like coefficients before?

Consider the following identities. $\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n}{r}\dbinom{n-r-1}{r-1}$ ...
4
votes
1answer
80 views

Who are some blind or otherwise disabled mathematicians who have made important contributions to mathematics?

Two prominent mathematicians who were disabled in ways which would have made it difficult to work were Lev Pontryagin and Solomon Lefschetz. Pontryagin was blind as a result of a stove explosion at ...
0
votes
1answer
32 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
0
votes
2answers
106 views

Explain cosmic distances to a child

I seem to have lost all my sense for simple calculations. I'd like to explain to my son how long cosmic distances are. As an example: Our sun has a diameter of 1392684 kilometers. The distance to our ...
4
votes
2answers
96 views

How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
0
votes
0answers
34 views

Reference: Fields of characteristic p

I am interested in learning more about fields of characteristic $p\neq 0$. Does anyone know of a good reference that covers the basics of this topic and possibly galois theory over fields of prime ...
9
votes
3answers
447 views

Proof for '$AB = I$ then $BA = I$' without Motivation?

I have read this question page (If $AB = I$ then $BA = I$) by Dilawar and saw that most of proofs are using the fact that the algebra of matrices and linear operators are isomorphic. But from a ...
270
votes
35answers
27k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
1
vote
1answer
76 views

How to make a good definiton

The reason why I come up this idea may due to Banach–Tarski paradox. The process we make a definition may consist of several steps. First step is that we observe a phenomenon. Second is to make a ...
2
votes
1answer
57 views

Starting with ring theory

Can anyone suggest a book on rings explaining concepts using visual diagrams, similar to the one visual group theory book by Nathan Carter for groups.The problem with me is that after reading that ...
2
votes
2answers
66 views

square root solutions

Is there a specific rule to get square root of any non-negative number?. The main reason why I'm asking this is that my maths teacher told me there is only one solution can be contained for any ...
3
votes
3answers
159 views

Provocations on the existence of mathematical objects

The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not ...
2
votes
2answers
115 views

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
0
votes
1answer
32 views

How should this definition of a family with corresponding index set be interpreted?

Does the following definition of a family imply that there exist a surjection from $I$ onto the family ? Or should it be interpreted as a injective partial function from $I$ to the family ? ...
66
votes
16answers
10k views

How do you explain the concept of logarithm to a five year old?

Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
3
votes
0answers
58 views

Attemping Qualifying Exam Problems — and failing

My question is concerning learning strategy. I can solve the majority of the exercises in a typical graduate mathematics textbook like, say, Dummit/Foote's Abstract Algebra. To supplement my education ...
1
vote
0answers
40 views

Most important results from pure math in applied probability?

I'm taking a course next semester at my university on applied probability (with relevance to signal and information theory). Although the nature of probability is mostly problem solving and applying ...
10
votes
3answers
319 views

A question from an engineering undergraduate

My question primarily concerns the necessary transition from an undergraduate program in electrical engineering to graduate program in applied mathematics or pure mathematics. I'm an electrical ...
7
votes
3answers
292 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? I'd like the lectures to cover main topics of the books ...
2
votes
1answer
80 views

Literature on Chern-Weil Theory and the Chern-Gauß-Bonnet Theorem

At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and ...
10
votes
2answers
158 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of mine has only a bachelors degree in pure math and got hired by ...
5
votes
1answer
144 views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
7
votes
2answers
264 views

Graduate level elementary logic books

I've done two courses on Logic during my Bachelor course, but they were very basic. Now I'm going to start by PhD, and I'm interested in learning "real Logic". Could you please provide some references ...
8
votes
3answers
535 views

Mathematical Discoveries that were made or supported by savants

I just read something about Rüdiger Gamm, who recited $81^{100}$ (191 digits), which took approximately 2 minutes and 30 seconds. So I asked myself: Are there any kind of mathematical ...
12
votes
5answers
748 views

In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
30
votes
5answers
2k views

Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
8
votes
6answers
1k views

Should I do all the exercises in a textbook?

The problem sets that you usually get in a university course is a small fraction of the exercises in your textbook. Which raises a question: do you need to solve all the exercises from your textbook? ...
46
votes
49answers
4k views

What was the book that opened your mind to the beauty of mathematics?

Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But ...
2
votes
0answers
42 views

Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$ \int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
10
votes
2answers
214 views

Articles on ideas in the history of mathematics notation?

I'm teaching a course this term on the history of scripts (writing systems) and rather than talking interminably about Semitic and Chinese and their spawn, I'd like to give students a more varied ...
0
votes
3answers
63 views

Infinitesimal Unit of Measurement

This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even ...
3
votes
2answers
99 views

Applications of powerful theorems in Bruns -Herzog's book “Cohen-Macaulay Rings”

It seems that theorem 1.4.13 and it's corollary of Bruns and Herzog's book Cohen-Macaulay Rings, are powerful tools but I don't see any example that shows the power of it. My original question was an ...
9
votes
3answers
214 views

Applications of advanced number theory to other areas of math

In a recent conversation with a friend, I was discussing the fact that out of all of the fields of math, number theory seems to be among those that apply ideas from a large number of different fields. ...
4
votes
6answers
717 views

Book suggestion for linear algebra “2”

I am almost finishing Gilbert Strang's book "An introduction to linear algebra" (plus video lectures at MIT OCW). First and foremost, I would like to suggest this course for everyone. It has been ...
1
vote
1answer
268 views

Callahan's Advanced Calculus: A Geometric View vs Hubbard's Vector C, L A, and Di Forms vs Ad- Calculus: A Differential Forms Approach by Edwards

My friend has given me the chance to get one of the books mentioned in the title of this question for free. I've learned single variable calculus. Which one of these books do you think would best ...
3
votes
1answer
91 views

What are the problems that you tried to find their solutions and you did not know that it is impossible?

Tell us your story about Mathematics. Have you dream one day to do a big contribution in Mathematics because you are curious and love challenges. What are things that you tried to prove which then ...
1
vote
1answer
120 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. ...