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-5
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1answer
109 views

How to practically make use of Mathematics? [closed]

How to practically make use of Mathematics ? I have a basic question.How to use Mathematics in our modern day lives? Are there any ways by which we can make Mathematics come out of our classrooms and ...
1
vote
1answer
52 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
1
vote
1answer
34 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is ...
3
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0answers
71 views

How to understand if one is eligible for research?

What are the eligibility criteria for one to undergo research in Mathematics? Or should I place the question as what virtues of a student are given importance when one is interviewed for a PhD ...
1
vote
5answers
134 views

Why do counits go that way?

Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors $\mathcal{C} \begin{array}{c} \stackrel{\large ...
1
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0answers
36 views

Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
1
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1answer
57 views

How is problem solving ability on an olympiad level relevant to mathematical graduate study/research?

I am starting on math later than many of my peers and hence have little to no experience in competitive problem solving. Is this a disadvantage during the study of the more abstract mathematics that ...
0
votes
0answers
39 views

Suggestions(Anything) regarding GRE Math Subject test [duplicate]

I am to appear for GRE Math Subject test probably this year or next year .I have basic knowledge of calculus 1,2,3 ,group theory ,Linear Algebra , Small part of real analysis . I haven't yet studied ...
0
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0answers
17 views

I hear that some operators don't have analytical properties. What does that mean?

The floor, ceiling, and mod functions are very useful operators, but in general discussion I've heard their usefulness called into question because of their lack of analytical properties. For instance ...
1
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1answer
49 views

Building a training program… for mathematics

I want to ask for your advice building my one-year training program for mathematics. Objectives: Keep 'mathematically fit' Improve for the pleasure Get competent at high-level economics and ...
1
vote
1answer
22 views

Expressing a line as a linear combination of two points on the line.

I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and y is the set of linear combinations $sx + ty$ where $s + t = 1$. I'm satisfied that this is true, ...
4
votes
3answers
97 views

Difficulty faced in solving maths problems

I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often ...
1
vote
0answers
42 views

Ideas for math problem solving class for undergraduate students in university

In our university there is a huge gap between two group of students. a group of them came from Math Olympiad competitions and have a very strong background from high school but others, they have just ...
5
votes
4answers
840 views

Albert, Bernard and Cheryl popular question (Please comment on my theory)

Here is the problem, I think that there is one point that makes the question ambiguous, I think they should explicitly say the reason why Albert knows that Bernard does not know the date. Case 1: ...
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 $x$ ...
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 ...
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 ...
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 ...
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 ...
4
votes
2answers
97 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
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
34
votes
10answers
2k views

How do mathematicians find formulas? [on hold]

How do mathematicians find formulas? For instance, the area of a triangle is $$\mathrm{area}=\frac{\mathrm{base}\times \mathrm{height}}{2}.\tag{1}$$ When I study maths, the book I am using tells ...
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 ...
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
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1answer
74 views

Learning mathematical concepts

Our teacher loves to test us on pure concept based questions and test if we really know what we are doing when learning a particular lesson. For example, when we first started learning about ...
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 ...
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 $X$ ...
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..." ...
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, ...
4
votes
1answer
100 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 ...
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 ...
1
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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 ...
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 ...
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 ...
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?
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 ...
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 ...
34
votes
3answers
533 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 = ...
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 ...
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 ...
18
votes
0answers
468 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 ...
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 ...
5
votes
2answers
66 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?
11
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4answers
2k views

Is it necessary for one to understand analysis?

Is it necessary for one to understand analysis in order to pursue a career in mathematics? Basically, I am very weak at analysis. But the problem is that most of the topics listed in the syllabus ...
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 ...
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 ...