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5
votes
2answers
132 views

How did self-similarity come into mathematics?

As far as I understand the interest in self-similarity was born outside of mathematics. The textbooks I came across give a few objects as examples (tree, broccoli, river, etc) yet it's clear that the ...
0
votes
1answer
25 views

The indeterminate form problem

How to solve for $a$ $a\pm k$(k being any real number other $0$)=$\frac{a}{a}=a^2=a$ Find $a$ ( $k$ is any real number other than $0$ The only solution I could think of is $$\frac{0}{0}$$ But does ...
17
votes
2answers
373 views

Results in graph theory proved using other areas of math, and vice versa

I'm curious about learning graph theory, as it seems to pop up in some unexpected places. In order to get a partial feel for the subject, I was wondering if anyone could point me to some survey ...
1
vote
0answers
54 views

Algebraic extension of an abelian group.

Let $H \leqslant G$ be abelian groups. Suppose there were $k \gt 1$, $x \in G \setminus H$, such that $x^k = b \in H$. Then Define $H(x) = \{h x^n : n \in \Bbb{Z}, h \in H\}$ to be a simple ...
3
votes
2answers
779 views

Lebesgue Integral, Riemann Integral and Integrals of all sorts

I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I ...
0
votes
1answer
24 views

List of Bounds of $n$-th composite

I am looking for a list of all the bounds on $c_n$, the $n$-th composite. There is a trivial bound $2n \geq c_n >n$ $\forall n \geq 5$. But I am looking for bounds stronger than this. I have ...
1
vote
1answer
46 views

Classification systems for mathematics.

Mathematics is a very broad topic nowadays, and it seems to be coming more and more obscure. I was wondering as to whether any organisations have implemented a classification system or organisational ...
2
votes
1answer
170 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} \...
0
votes
2answers
86 views

Reflexive and relation

Please give me feedback on my answer to this question. Question: For all $x,y\in R$ define that $x\equiv y$ if $x^{2}=y^{2}$. Then $\equiv$ is an equivalence relation on $R$, there are infinitely ...
1
vote
1answer
212 views

Are there other models for 2 dimensional hyperbolic geometry?

I was a bit browsing the internet for models for (2-dimensional) hyperbolic geometry. and realised that besides the well known Poincare half plane model Poincare disk model Beltrami-Klein disk ...
2
votes
3answers
156 views

Does $f(x)\,dx$ denote multiplication of $f(x)$ by $dx$? [duplicate]

In the integral form $\int \! f(x) \, \mathrm{d}x$ does $f(x)\,\mathrm{d}x$ can be seen as a multiplication of $f(x)$ and $\mathrm{d}x$?
2
votes
1answer
43 views

What is the meaning of $<$ in a preorder?

Let $(P,\le)$ be a preorder, i.e. $P$ is a set and $\le$ is a relation on it that is reflexive and transitive. In this context for myself I can find two interpretations for the symbol $<$ 1) $a<...
1
vote
2answers
239 views

Questions about the field scientific computing

I have heard about the field of Applied and Computational Mathematics, Scientific Computing and want to get some information. Is this a combination of computer science and mathematics? What subjects ...
1
vote
0answers
31 views

Has an order with this property a special name?

If $a$ is an element in a preorder then you can eventually go 'a step back' (and repeat this) in the sense of finding an element with $b\leq a$ and not $a\leq b$. Is there a special name for (pre)...
1
vote
3answers
150 views

Explaining something to the half

I'm a private tutor in my free time, teaching some basic high school mathematics and I've often been asked: ''Why is something to the half equal to the root of that something?''. And I'm having ...
7
votes
4answers
550 views

What is the relationship between “recursive” or “recursively enumerable” sets and the concept of recursion?

I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am ...
2
votes
1answer
95 views

Post Baccalaureate Studying

So I'm currently finishing up my last quarter of undergrad, but unfortunately I have to take a year off before starting grad school. My degree is in physics, but I've been far more interested in ...
1
vote
2answers
48 views

Analysis on using Unconventional underlying fields

I'm curious if people study analysis while using fields that are not $\mathbb{R}$. I remember seeing a post about doing analysis on $\mathbb{Q}$, but $\mathbb{Q}$ is not complete! Mostly I'm ...
1
vote
1answer
136 views

Ebooks in advanced calculus

This semester I'm taking advanced calculus and in the sources list, the lecturer recommends on "An introduction to analysis-by w. wade" and "Advanced calculus by Edwards". Both of them cost around 100 ...
33
votes
13answers
1k views

Interesting but short math papers?

Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new ...
1
vote
1answer
24 views

Notation for Model-Relation of formulae with free variables

Lets assume we have a formula $\mathsf{path}(x,y)$ with free variables $x,y$, and $\mathsf{acyclic}$ with no free variables on the signature $\tau = \{E\}$ (i.e. Graphs). Informally, what the formula $...
2
votes
1answer
126 views

First year A-Level Mathematics student, wanting to learn more while on vacation!

So I've just finished the first year of A-Level Mathematics (AS - Pure Core 1 and Pure Core 2), and I have a big vacation coming up and so I'm wanting to learn more. My question is, what would you ...
3
votes
3answers
123 views

Colloquialisms in Math Terminology

What are some of your favorite colloquial sounding names for mathematical objects, proofs, and so on? For example, manifolds are often described using an atlas and a neighborhood describes a small ...
0
votes
2answers
78 views

Need some help about computer programming.

In mathematics, sometimes we have to do very long and time consuming work related to some functions or algorithm or some other work. For example if we want to check that for which last number $n$, $0$ ...
14
votes
3answers
8k views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ ...
2
votes
1answer
42 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
7
votes
3answers
819 views

Graduate School?

I'm completely clueless on the process, but on track to graduate in two years, so I have a few questions about what I should do. 1) What's the difference between a Master's degree in Mathematics and ...
2
votes
2answers
146 views

Mathematical recommendations [closed]

I was just wondering if anyone with some experience could recommend a book for one still in the early stages of their mathematical studies(first year). Maybe something related to Algebra or history of ...
0
votes
1answer
110 views

Research Proposal help

Soft question, so I hope this is still in the right place. I have applied for a PhD position at the university I have just finished my undergrad degree at. I have to write a research proposal in the ...
6
votes
4answers
168 views

What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?

It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ ...
1
vote
0answers
197 views

Proofs of “Table of Integrals, Series, and Products” formulas

http://www.amazon.com/Integrals-Series-Products-Seventh-Edition/dp/0123736374 This book is amazing. But there are no proofs in it. Is there book or site which contains all of the proofs of this book'...
3
votes
1answer
101 views

Has anyone considered axioms to the effect that: “The axiom of constructibility fails very very badly?”

If I'm not mistaken, the axiom of constructibility basically says that the universe has no (non-trivial) inner models. Has anyone considered axioms of the opposite flavour, basically asserting that ...
1
vote
2answers
81 views

Formalizing the Fallacy of Composition

Is there a well-known formalization of the fallacy of composition? More generally, where in mathematics is it true that if a property holds for all of some elements of a set it holds for the whole ...
3
votes
2answers
118 views

What's the significance of defining group as a group object in category $\mathcal{Set}$?

At first sight, redefining group as a group object in the category of sets $\mathcal{Set}$ seems just like a meaningless restatement, but when we apply this definition to other categories, ...
1
vote
2answers
107 views

Should the theory be studied thoroughly before solving exercises?

Most of the books claim in the preface that the important part of the book is in the exercises, which makes sense considering that solving problems improves in great depth the understanding of the ...
9
votes
2answers
547 views

Automatizing computational skills

"It is a profoundly erroneous truism, repeated by copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise ...
2
votes
2answers
71 views

Deciding which questions to do in a maths exam

I wasn't quite sure if this was the best place for this question. If faced with a maths exam where you can choose the questions you do, how do you approach which ones to pick? Difficulty (any ...
29
votes
6answers
2k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
8
votes
2answers
236 views

I feel the need to prove every result for myself

I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I ...
6
votes
1answer
82 views

Conservativity of $\mathrm{ZFC}+\varphi$, where $\varphi$ contradicts CH.

It is well-known that ZFC with the continuum hypothesis is a $Π^2_1$-conservative extension of ZFC. General question. What is known about the conservativity of $\mathrm{ZFC}+\varphi$ over $\mathrm{...
2
votes
2answers
238 views

Second Course in Number Theory - Self Study

I just finished a first course in number theory using Dudley's Elementary Number Theory. This was by far my favorite math course and I want to learn more number theory this summer. As far as ...
5
votes
2answers
227 views

How to think of zeros of the derivative of a holomorphic funcion?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, then, for example, if $f'(x_0)=0$ and $f'$ is again differentiable at $x_0$ and $f''(x_0)\neq 0$, then $f$ has a maximum or minimum at $x_0$. ...
13
votes
3answers
291 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
20
votes
5answers
1k views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
6
votes
2answers
2k views

Why is abstract algebra so important?

In my studies of physics and mathematics, I have encountered a fair bit of geometry, Lie group and representation theory, and real and complex analysis and I understand why these branches of ...
1
vote
1answer
53 views

What is the remainder useful for when dividing a polynomial?

I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems. I understand the factor theorem and its application: it helps me find roots of a ...
2
votes
4answers
278 views

What is the most fundamental trigonometric function: cosine or sine? [closed]

$$\cos(\theta) = \sin \left(\tfrac{\pi}{2} - \theta\right)$$ $$\sin(\theta) = \cos \left(\tfrac{\pi}{2} - \theta\right)$$ Both are the same entity. But is sine the copy of cosine, or is cosine the ...
13
votes
4answers
885 views

How do mathematicians think about the existence of numbers?

Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept? I know that existence of numbers is a big ...
80
votes
10answers
11k views

Why can't you add apples and oranges, but you can multiply and divide them?

What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided? This looks ...
2
votes
2answers
56 views

A result that follows from Stokes' theorem— Important?

From Stokes theorem, it is easy to prove the folowing proposition: $\int\int_\vec{S}\vec{F}d\vec{S}=0$ if $\vec{F}=curl$ $\vec{G}$ for vector fields $\vec{F}$ and $\vec{G}$ and a closed parametrized ...