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0
votes
0answers
30 views

trigonometric-like function that do not care about the cartesian coordinate system

Is there a common name for the function that maps from an angle inside a circle - to a relation between the length of the radius and the length of the edge (AB) between the two points A and B where ...
5
votes
1answer
74 views

Exact rank of Elkies curve

A naïve question. We definitely know an elliptic curve of rank $28$ or more exists by Elkies but no one knows exactly what the rank is for this curve (and for similar examples given previously). ...
2
votes
1answer
219 views

Structured Self-Learning Program for Calculus I & II

I'm interested in a organised program which comprehensively covers the topics of Calculus I and Calculus II. I've recently finished taking my secondary school's university-level Calculus I course, ...
3
votes
2answers
103 views

Research Area Choice: PDE vs Optimization [closed]

I am on track to starting in applied math PhD this coming Fall. My area of interest is PDE where I have a strong background and have even published a paper. It seems the particular area of PDE I am ...
41
votes
5answers
2k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
1
vote
1answer
76 views

I'm planning on taking an Introduction to Geometry class next term. What should I expect?

For reference, I'm heading in to my second year of my bachelor of Computer Science. I'm required to take a second year level math course and I was thinking about taking Introduction to Geometry. It ...
0
votes
3answers
102 views

Inner Product Spaces, suggestion for book.

Can you suggest me name of some books which would help me visualize IPS better? Like, books having diagrams and stuff?
0
votes
1answer
38 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 ? ...
0
votes
2answers
46 views

Method to Multiply many Matrices simultaneously

Imagine you have three random matrices $$ A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right] ; \quad B = \left[ \begin{array}{cc} 3 & 4 \\ 5 & 6 \end{array} \right]; ...
4
votes
2answers
148 views

Examples of useful Category theory results?

I'm layman to Category theory. Trying to understand it, I just read a bit briefing on it and wiki pages of Category, Functor, Morphism. However I still could not see the merits of it. Category ...
4
votes
2answers
114 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
21 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 ...
21
votes
8answers
2k views

Statements with rare counter-examples [duplicate]

This is a soft question. I'm searching for examples of mathmatical statements (preferably in number theory, but other topics are also fine), that seem to be true, but are actually not. Statements ...
1
vote
0answers
53 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 ...
1
vote
1answer
41 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 ...
17
votes
4answers
2k views

An equation that generates a beautiful or unique shape for motivating students in mathematics

Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet: $$ \large\color{blue}{ ...
31
votes
6answers
2k views

Why are integrals called integrals?

What is the historical background for this term? I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems a strange choice of ...
2
votes
3answers
122 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
38 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) ...
8
votes
2answers
407 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 ...
0
votes
1answer
19 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 ...
2
votes
1answer
166 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} ...
1
vote
0answers
28 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 ...
0
votes
5answers
94 views

Proof of basic properties

Can anyone provide proof of properties such as: $$a(b+c) = ab+ac$$ $$(a+b)^2 = a^2+2ab+b^2$$ And exponent rules: $$a^n \cdot a^m = a^{n+m}$$ $$(a^n)^m = a^{n \cdot m}$$ For $a, b, c \in ...
1
vote
3answers
149 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 ...
1
vote
0answers
40 views

Set and Function

Please provide feedback for my answer to this question. Question: If we define for set S,T that |S|-|T|=|S-T| , then this is well defined in the sense that for all sets S,T,S',T', if |S|=|S'| and ...
0
votes
2answers
78 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 ...
-4
votes
1answer
78 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 ...
1
vote
2answers
562 views

Multivariable Calculus for GRE

This is going to sound strange, but I am a third year math major who never took multivariable calculus (despite having taken courses on Galois and Lebesgue theory, etc). I plan to take the GRE next ...
2
votes
1answer
82 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
40 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
209 views

Why do different countries/regions have different methods of counting large numbers?

When we start counting large quantities of $10's$, the number system varies by country/region: Europe/US: $10^3$ (thousand, million, billion are all multiples of $10^3$) Japan/China/Korea: $10^4$ ...
3
votes
1answer
190 views

Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
22
votes
4answers
556 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
23
votes
9answers
2k views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
2
votes
1answer
104 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 ...
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 ...
0
votes
2answers
73 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 $n$, $0$ ...
0
votes
1answer
102 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 ...
7
votes
3answers
486 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 ...
1
vote
0answers
116 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 ...
3
votes
1answer
93 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 ...
39
votes
7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
2
votes
1answer
38 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 ...
3
votes
2answers
106 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
81 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 ...
1
vote
2answers
69 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 ...
6
votes
2answers
300 views

How should one go about obtaining “mathematical maturity”?

tl;dr: Is mathematical maturity better obtained by doing hard subjects slightly out of your reach, or by doing more simple subjects to gain experience? The end of the semester is close, and I ...
2
votes
2answers
67 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 ...
6
votes
1answer
80 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 ...