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3
votes
1answer
64 views

Are there any instances of significant progress deriving from mathematical 'silliness'? [closed]

Last night I thought I'd be silly finding the eigenvalues of a $2\times2$ matrix $A$ with real components. Instead of calculating $\det(A-\lambda I)=0$ I tried to compute the determinant by ...
1
vote
2answers
81 views

Earth population growth rate is exponential or logarithmic?

How many points on a monotonically increasing curve is needed to determine if it is exponential or logarithmic? For example can we tell that in the most recent history population is increasing ...
5
votes
1answer
121 views

What is meant in the quotation of Terry Tao?

Terrence Tao commented of internalizing [here: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/] "It is true that some mathematicians can be vastly more ...
1
vote
0answers
33 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
0
votes
0answers
19 views

Condition on tightness of an inequality

How is generally the tightness of an inequality measured? Is the inequality tight when it is held with equality under certain conditions?
2
votes
0answers
58 views

Pocket Calculators- Obsolete? [closed]

I hope this fits into "software that mathematicians use" within the acceptable topics for this site. Are there still any instances where pocket calculators prove useful to professional mathematicians ...
6
votes
1answer
237 views

A book on advanced math for a “novice” mathematician, but “mature” thinker

I enjoy giving interesting problems to my peers who have not seen a lot of mathematics. It is a constant conversation I have with many friends who would interchange "mathematical thinking" with ...
4
votes
1answer
66 views

Studying Graduate Level Mathematics Outside Lecture Hours

I am not exactly sure if this should be posted on math.se or academia.se, but I think the question is more mathematical in nature. I am a first year graduate student in mathematics, and I would like ...
11
votes
5answers
632 views

Defining the Determinant

The concept of determinant is quite unmotivational topic to introduce. Textbooks use such an "strung out" introductions like axiomatic definition, Laplace expansion, Leibniz'a permutation formula or ...
54
votes
8answers
6k views

Is linear algebra laying the foundation for something important?

I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear ...
1
vote
3answers
81 views

Why do inequalities matter and what are they used for?

I am currently studying mathematical course at my college, and I cannot seem to grasp the concept of inequalities. What troubles me is that, like it's said, "triangle inequality matters because many ...
4
votes
2answers
78 views

Why is it so hard to translate some proves into machine-readable form?

I have just read a topic on mathoverflow about man vs. machine in mathematics. The topic was inspired by the recent victory of Alpha Go over the World Go Champion, Lee Sedol. It reminded me of an ...
2
votes
1answer
75 views

Necessity of Category Theory for understanding Algebraic Topology [duplicate]

I am studying Algebraic Topology and increasingly find that when I turn to the internet for help, the explanations and even definitions I need are given in terms of Category Theory (which I know ...
2
votes
1answer
55 views

Much used compass and straightedge constructions

I am a editor of wikipedia and would like to know which compass and straightedge constructions deserve a place in the list ...
2
votes
0answers
51 views

Can one cite REU papers written by students?

I am writing my Master thesis right now. Two arguments in the proofs are based on REU papers written by students. As those are not officially published but made available on the universities' websites ...
1
vote
1answer
62 views

Why should I expect the product of sum of four squares to be a sum of four squares? How did Euler come up with it?

Euler discovered the lovely identity shown here: https://en.wikipedia.org/wiki/Euler%27s_four-square_identity Is there a natural reason to assume a solution can be found? Any intuition? I saw that ...
0
votes
2answers
39 views

Dealing with computations that are tedious and time consuming

In attempting mathematics homework question one of the greatest redundancies I generally face is having to deal with long winded and tedious computations. For instance, in the field of Quantum ...
0
votes
0answers
43 views

What are some interesting results that could be derived if some conjectures were true/false?

I recently came up on Djikstra's idea of a fictional company called Mathematics.Inc which produced proofs as trade secrets which could then be used by customers.. For example, it produced the proof of ...
2
votes
2answers
47 views

What is wrong with my logic and derivatives?

Since the derivative of a function is analogous to a description on how fast the function grows, I thought that $\lim_{x\to\infty}\frac{f'(x)}{f(x)}=0$ for the following reason. Assume $f(x)$ is ...
4
votes
2answers
36 views

Applications of partitions of natural numbers [closed]

This is a question for applications of partitions in science or technology. I know partitions is an interesting field in combinatorics and in modern algebra because it can be related with symmetric ...
0
votes
0answers
53 views

Probability at a point

If X is a continuous random variable then we say that probability at a point is neglected or id zero.but if it is discrete then the probablity of that point will be taken into consideration I know i ...
3
votes
1answer
62 views

What are Stochastic Processes and Combinatorics useful for?

I am a math major and I am trying to figure out what math classes I want to take next semester. My question is what industries are Stochastic Processes and Combinatorics useful for? I haven't ...
1
vote
1answer
51 views

How to become very good in probability and combinatory [closed]

How someone become very good in probability and combinatory? Is there is a set of methods used to solve problems in this field? If you can suggest any resources or tips to improve myself in this ...
3
votes
2answers
157 views

How to become good in Mathematics? [closed]

I don't know if this is the right place to ask this question...but let's try.... I have seen lots of people all around me, interested in mathematics-ranging from teachers to friends.But,I have seen ...
4
votes
0answers
75 views

Functions with “ugly” inverses

Inspired by this post: I was amazed to see that (at least according to wolframalpha) the inverse of such a nice and simple function as $f(x)=x^3+x$ is: $$ f^{-1}(x) = \sqrt[3]{\frac{2}{3( ...
3
votes
1answer
44 views

What is the complex line integral measuring? How do we motivate it?

How do we motivate it? I am not necessarily looking for a geometric answer to my question so much as I am looking for a way to motivate the idea of a complex line integral. For any path $\gamma$ and ...
2
votes
1answer
55 views

Discrete Mathematics Wording Difference Between “Show” and “Prove” [duplicate]

I just took a midterm for a Discrete Mathematics class. On couple of questions, it says "Show why this is true". For example, a sample question might have said "Show that five consecutive numbers is ...
1
vote
2answers
46 views

Can $X \ne \{0\} \implies X^*\ne\{0\}$ be proved without Hahn-Banach theorem?

We know that if $X \ne \{0\}$ is a NLS then $X^*\ne\{0\}$ ; is there any way to prove it without using Hahn-Banach theorem ? Thanks in advance
3
votes
0answers
29 views

Cohomology of affine hypersurface

I'm interested in calculating cohomology of a smooth affine hypersurface (over an arbitrary field). I know I can use (algebraic) de Rham cohomology, which is "easy" in the complex case, but even there ...
2
votes
3answers
56 views

Why is it okay to write…“for $n\in\mathbb{Z}^+,n\ge 3.5$”?

So I've been playing around with some mathematical induction proofs and I usually open with a statement similar to: (example below) Denote the statement involving $n$ for $n\in\mathbb{Z}^+,n\ge 4$ ...
0
votes
1answer
43 views

Simplicial Complexes/homology Reference request

I am taking an introductory Algebrqic Topology course, and we have just finishe talking abour fundamental group/ covering spaces in Munkres' Topology. However, his treatment of simplicial homology in ...
1
vote
1answer
78 views

Cobordism and h-cobordism

Is there a way to simply explain cobordism and h-cobordism? I am not looking for a math based explanation, but rather, just the main ideas behind the concepts.
2
votes
2answers
132 views

complex equation that has the answer of “40” [closed]

My brother is a college Math Professor and he is turning 40. I want to make him a t-shirt with a cool looking giant (long, complicated, etc.) equation that equals the number 40. Only problem is I'm a ...
2
votes
1answer
35 views

How to prove $\left(1+\frac{x}{n+1}\right)^{n+1} > \left(1+\frac{x}{n}\right)^n$ for all $n \in \mathbb Z^+$ and $x>0$

How to prove $\left(1+\frac{x}{n+1}\right)^{n+1} > \left(1+\frac{x}{n}\right)^n$ for all $n \in \mathbb Z^+$ and $x>0$ I tried by using AM-GM inequality. Can anyone give me some help.
2
votes
2answers
75 views

Ideas for a history of math paper (with an emphasis on the mathematics), having to do with 19/20th century logic?

So I'm currently taking a history of math course and I need to write a 15 page paper in place of my final. It's a 400 level course (high undergrad) so the paper needs to have emphasis on the ...
0
votes
2answers
87 views

Don't understand this maths pun $\sin\left(\frac{\pi}{2} - \theta\right)$ [closed]

I bought these cool maths reward stickers. Unfortunately, there's one that I just don't "get" - the blue one in the lower middle. I know that $\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)$, ...
3
votes
4answers
43 views

How to understand operations with negative numbers in a simple, intuitive way?

I have to teach my $29$ year old brother math. He remembers basic arithmetic from school, but he always hated math, which is why I want to start him from the beginning with intuitive explanations for ...
6
votes
1answer
64 views

Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?

This is not a very well-defined question. Are there any standard constructions of metric spaces, parameterized by real-valued $n \ge 1$, such that: When $n$ is an integer, the metric space is ...
6
votes
2answers
106 views

Is it good to use mean value theorem in $\epsilon-\delta$ continuity proofs?

I wanted to prove $f(x) = \cos(x)$ is continuous using $\epsilon-\delta$ proof Couple of posts on MSE appealed to MVT to resolve this problem. Namely: $\exists c \in [x,x_o]$ s.t. ...
0
votes
1answer
37 views

What is a good criteria for naming theorems/conjectures/lemmas?

Wikipedia, proofwiki, planetmath, and many others have various ways of naming theorems (and proofs and conjectures). I'm making a similar project, and am trying to create a searchable database of ...
1
vote
1answer
32 views

multiplication and division first, then addition and subtraction in equations written in prose

This might be a pretty stupid question, but the following was asked in a german quiz show: "How much is two times two plus two divided by two minus two?" My question now is, is there any literature, ...
0
votes
0answers
46 views

Equality on classes

On every set $A$ there is an equality $=_A$ defined. This is necessary to speak about injectivity, surjectivity, the group axioms and so ... In category theory, one uses classes to be collections of ...
3
votes
2answers
39 views

Comparing Coefficients for Partial Fractions

See this simple example : $$\frac{x+1}{(x-1)(x-2)}\equiv \frac{A}{(x-1)}+\frac{B}{(x-2)}$$ Then we can get $x+1 \equiv A(x-2)+B(x-1)$ for $x \neq 1 ,2$ My Question : Is it correct to put $x=1$ ...
-1
votes
1answer
41 views

Can I make it without times table memorization? [closed]

I am 24 years old and I am planing to get into mathematics sense it is essential In science. I have big plans and Math is the very first thing I want to focus on. I have very important question that ...
1
vote
0answers
23 views

what can we say if we just know the global section has a given universal algebra structure?

Given an universal algebra A, how to reconstruct a topological space X with some universal (and natural) property (e.g initial,terminal...)in the category of topological spaces whose global sections ...
3
votes
2answers
67 views

Why do we require that a simple Lie algebra be non-abelian?

We say that a Lie $k$-algebra is simple if it is a simple object in the category of Lie algebras, and also nonabelian. The only simple object which we do not consider to be a simple Lie algebra under ...
4
votes
4answers
105 views

Should I be concerned if I cannot solve most exercises in my textbook?

I'm self studying introductory real analysis. Out of the proofs in each lesson section and the proofs in each exercise section, I can usually work out$\frac{1}{4}$ of them on my own. After trying to ...
4
votes
0answers
52 views

Is there a dictionary for math notation?

As a non-mathematician who loves the elegance of mathematics, I'm often confused about certain syntax I see. For example, $2+2$ is "2 plus 2" obviously. But things get more complicated when, for ...
2
votes
0answers
24 views

Very Basic Numerical Methods Book for Freshman students

To cut a long story short; the nature of this degree (it's not a college degree) is such that numerical methods is treated shortly after Calc I (single-var) and linear algebra, but before multi-var ...
1
vote
1answer
27 views

Group morphism or group homomorphism?

I apologize for my question that might sound stupid, but i noticed that my lecturer in abstract algebra course uses always "group morphism" instead of "group homomorphism". In the books i see it ...