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18
votes
0answers
467 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
1
vote
1answer
45 views

Do equations that rely on a fractional number of variables exist?

In statistics, data is usually fitted with trend lines. Usually you can get statistics back that say things pertaining to how correlated one variable is with another. For instance if a variable $x$ ...
15
votes
7answers
3k views

Books to learn physics, being a math major

What books would you recommend to learn physics, being a a Math major, from classical mechanics, electricity, etc. to modern physics?
1
vote
2answers
40 views

Representative Pedagogical Examples of Groups, Real Functions, Modules, etc.

In the preface of Munkres's Topology, he writes, Fortunately, one does not need too many counterexamples for a first course; there is a fairly short list that will suffice for most purposes. Let ...
0
votes
0answers
44 views

Collatz algorithm generalization try-out (Collatz k-algorithm)

Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of professor Lagarias about it. Everything was so interesting (and I ...
3
votes
2answers
37 views

Concept of random sample? I have a truly problem understanding it.

I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). Suppose that we have a box with ...
1
vote
2answers
45 views

Numeric system without “zero”, how to explain importance of zero to average person?

As we all knew that Aryabhata (http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero) invented zero ($0$) in our number system. I have few questions about it. How did the numeric system ...
1
vote
1answer
28 views

Question about optimization

I have a question about maximization/minimization problems. I have noticed that for almost all the practice problems that I have had that ask to find the sum of numbers and minimize product or ...
0
votes
2answers
37 views

How should trigonometric expressions be simplified?

I have been learning trigonometric identities and I am having trouble understanding how they should be applied to simplify expressions. For example, the expression $2\sin{x} \cos{x}$ is equal to ...
0
votes
0answers
11 views

Formula to convert currencies

I know this currency exchange rates: 1.000000 USD = 0.943837 EUR 1.000000 USD = 0.683463 GBP What would be the formula to find: 245 EUR = ??? GBP
3
votes
2answers
47 views

Particular case of an Implication

Let's take the following propositions : 1 - "If Bill Gates is poor then Bill Gates is rich". 2 - "If Bill Gates is poor then the moon is made of cheese". Both propositions are inevitably true ...
1
vote
1answer
353 views

T table with infinity degrees of freedom

why is z table the same as a t table with infinity degrees of freedom. For example as df for the T distribution goes to infinity it becomes z ( standard normal) distribution. Is this true and why is ...
22
votes
4answers
905 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
1
vote
0answers
37 views

Is there a companion to the book 'A Synopsis of Elementary Results in Pure and Applied Mathematics' by George S. Carr?

A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr is as most of you probably know a book that was famously used by the great mathematician Ramanujan. It is said he ...
2
votes
0answers
46 views

How does Godel Escher Bach support Artificial Intelligence? [closed]

Typically, Godel's Incompleteness theorems have been used to argue against the possibility that the human mind is essentially equivalent to a formal system. However, in Daniel Dennett's book "Darwin's ...
1
vote
0answers
41 views

How to remember a proof for a long time

A very basic question of mine: Whenever I read a proof I am able to remember it only for a couple of months. But I really want to remember it for at least one year or so if not more. Is it the ...
1
vote
0answers
33 views

Least upper bound and greatest lower bound of the void set.

Let $(L,\ge)$ a partially ordered set. Suppose that for avery $S \subset L$, there exists an element $LUB(S)= a$ such that $x \ge a \iff x \ge u \quad \forall u \in S$ and, with the obvious meaning of ...
45
votes
4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
3
votes
1answer
32 views

Is the closure axiom necessary for algebraic structures defined via a binary operation?

Numerous algebraic structures are often defined as a set $X$ equipped with a binary operation $f:X\times{X}\rightarrow {X}$ that satisfies some set of axioms. Since the image of $f$ is always in $X$ ...
34
votes
3answers
532 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
1
vote
0answers
26 views

What is the value of an Infinitesimal?

In the regular type of math "0.999..." is the same thing as the value 1. In some other different kind of math they say that "0.999..." is not the same thing as the value 1. Where 1 is > "0.999..." ...
4
votes
1answer
99 views

Why is it called the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic is easy enough to understand, saying that every integer greater than 1 is either prime or is the product of a unique combination of prime numbers. What I don't ...
3
votes
1answer
105 views

Mathematics books that tell you what is really happening? [closed]

Many book I've read teach you symblobic manipulations instead of pointing out what's really happening. So my question would be: what mathematics textbooks don't do that? Books that rather than listing ...
10
votes
1answer
955 views

What distinguishes the Measure Theory and Probability Theory?

It is clear that the Theory of Probability works primarily with limited measures on measurable spaces. On the other hand there is a folklore that says that what distinguishes Measure Theory and ...
8
votes
1answer
77 views

Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: ...
2
votes
0answers
52 views

How much math was “Broken” by Russell's Paradox?

As you know, the phrase "the set of all sets that don't contain themselves" caused a paradox that "broke" (made trivial) Naive set theory. How much mathematics had to be redone because of this? Most ...
6
votes
1answer
97 views

Examples of categorical adjunctions in analysis and differential geometry?

In a lot of introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis and ...
3
votes
1answer
117 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
108
votes
33answers
11k views

Can you provide me historical examples of pure mathematics becoming “useful”?

I'm trying to think/know about something but I don't know if my basis premise is plausible, here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...
5
votes
1answer
103 views

Why are fundamental theorems called fundamental?

I am just curious about some theorems that I had been used for some time. We have Fundamental Theorem on Calculus the Fundamental Theorem of Calculus. In number theory we have Fundamental Theorem of ...
3
votes
1answer
52 views

Soft Question about Möbius Transformations

Very soft question and I may be completely wrong about this, but does it make any sense to think about the Möbius transformation matrix as a change of basis for $\mathbb C$?
2
votes
0answers
50 views

What are the current frontiers of mathematics? [closed]

In language suitable for an undergraduate student familiar with the basic objets of interest and classical techniques used in the major subfields of mathematics, what are the ''frontiers'' of modern ...
0
votes
2answers
71 views

High school looking to prepare for university

I am a high student and doing the general math course at my high school, it will cover: Geometry Graphs and Relations Matrices Statistics Next year I want to enroll in a science degree and major ...
2
votes
0answers
55 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
3
votes
1answer
47 views

Generalisation of posets

The notion of (set) monoid can be generalised to that of monoid object in a monoidal category. Can the notion of poset be generalised in a similar fashion? How?
1
vote
3answers
66 views

Are there infinite sequences not reproducible by finite algorithms?

Let me know if this is a repeat question. I was thinking that sequence of integers we deal with (e.g., the digits of $\pi$, the prime numbers, the Fibonacci numbers, pseudorandom numbers) seem to be ...
52
votes
9answers
3k views

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers ...
379
votes
25answers
73k views

How to study math to really understand it and have a healthy lifestyle with free time? [closed]

Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical ...
1
vote
0answers
24 views

proof of Wiener’s criterion

I'm in my first course of PDE and I need to investigate the proof of Wiener's Criterion for Laplace Equation which says, if $\Omega \subset \mathbb{R}^n$$(n>2)$ is a bounded domain and $\partial ...
7
votes
1answer
202 views

Do hom-sets really live in the category Set?

In familiar introductory books on category theory, one of the first examples of a category given is Set. And what category is that? Typically no explanation is given at this stage. But of course ...
6
votes
2answers
165 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
6
votes
1answer
120 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
19
votes
7answers
3k views

Mathematical Career Advice to a young 16 year wannabe mathematician [closed]

I am high school math enthusiast I am very much interested in mathematics. I try to learn stuffs from the internet but I can't seem to find the resources and am also confused about where to start. I ...
36
votes
4answers
2k views

A Young Math Enthusiast's Fear

Having been an avid lover of Mathematics, it is my dream to become a mathematician one day. I have been learning some "Advanced Mathematics" (Real Analysis and some Abstract Algebra mostly, and a ...
4
votes
2answers
65 views

Is there any other constant which satisfy Euler formula?

Every body knows Euler Formula $e^{ix}=\cos x +i\sin x$ Is there any other constant beside $i$ which satisfies the above equation?
8
votes
2answers
144 views

Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
3
votes
2answers
63 views

How can Bayesian and Frequentist approach be different?

Let's say I am trying to add numbers, like say one to ten. I can either add them in order, or I can notice that I can group them into five groups of eleven, so I suppose which method to use depends on ...
0
votes
1answer
69 views

What are eigenvalues and eigenvectors really?

I know how to determine the eigenvalues and eigenvectors of a given matrix $A$, but we were not really explained to what exactly ARE eigenvalues and eigenvectors, what is their purpose and what ...
13
votes
3answers
601 views

Journals of math history?

In a related question to this one, in what journals do math historians publish their article in? Brian M. Scott provided a link to Judy Grabiner's, who is a math historian, home page and it seems that ...
1
vote
0answers
39 views

Going to graduate school in applied mathematics without having taken a topology course?

Due to a critical course conflict, I won't be able to take any topology or geometry courses before I graduate. However, I plan to go to graduate school in [applied] mathematics. Will this hinder me ...