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5
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1answer
40 views

Why “singular” in “singular homology/cohomology”?

As the title suggests, I'm curious to know whether there is any reason why the word "singular" appears in "singular homology/cohomology".
2
votes
4answers
79 views

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different ...
8
votes
4answers
374 views

How To Develop A Higher Mathematical Aptitude? [closed]

First off I must say I'm pretty blown away by the vast majority of the people in this forum. I do aspire to reach the knowledge of mathematics as shown on the site, but honestly it's a little daunting ...
0
votes
1answer
50 views

Books on multivariable calculus

I'm looking for a book that covers the following subjects: multivariable functions, extremes of multivariable functions, integration, implicit function theorems, functions defined by integrals, vector ...
5
votes
3answers
685 views

Which background is more suitable to study “Cryptography”

I am a student of Pure Mathematics and also interested in programming .I have learnt C++,SAGE . Recently I have started learning "Cryptography" .But there are many definitions involved here like ...
1
vote
0answers
21 views

General Topology on Complex field?

While Real Analysis is based on spaces of real numbers so we have also Complex Analysis uses complex numbers in its spaces also. General Topology has commons with Real Analysis and whatever I see in ...
0
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0answers
47 views

Chaos theory in stock market

I am doing an IB Extended Essay on chaos theory and fractals in the consumer stock market. It is a high school level essay (4000 words) and should be understandable for a calculus student. I'm having ...
2
votes
0answers
14 views

What kind of subject/book(s) should I get if I'm interested in random motions hitting a barrier and being added to the barrier?

I'll explain in more detail. I'm interested in learning more about diffusion limited aggregation and the fractals that result from that process, however it'd probably be best to know more about ...
6
votes
1answer
164 views

Reading mathematics at the graduate level - does every single proof matter?

I would be interested in hearing from PhD students (and upwards) specializing in pure maths (in particular, the more algebraic aspects). My question is this: When reading to learn mathematics at ...
2
votes
2answers
73 views

Any “DIY Analysis” books?

Are there any good "analysis through problems" type books? I've tried reading analysis books but I literally get bored to death, and, until I manage to concoct a way of transforming a normal textbook ...
1
vote
0answers
40 views

Vector calculus vs Vector analysis?

I was just wondering, is vector analysis the same as vector calculus? What about multivariable calculus? Because my multivariable calculus book (which I assume is the same as vector calculus?) covers ...
54
votes
11answers
7k views

Why there is no sign of logic symbols in mathematical texts?

Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, ...
0
votes
2answers
65 views

Meaning of symbols like $\inf\limits_{\epsilon>0}$

I am very confused at the precise definition of the following symbols. A reference or explanation would be great. $$\Large\inf\limits_{\epsilon>0}\qquad \sup\limits_{\epsilon>0}$$
2
votes
1answer
21 views

what is a spectral function?

My knowledge in spectral theory is very limited, but lately I heard talking about the spectral function of an operator and how it's important. By curiosity I tried to look for a definition and a ...
3
votes
2answers
128 views

Interesting facts and problems to motivate high school combinatorics students

I will give some classes in combinatorics to high school students and I would like to know some facts (and proof) I can show to my students to motivate them to study this beautiful subject. I'm ...
8
votes
0answers
99 views

What was the original motivation for matrix multiplication? [duplicate]

When I took linear algebra class in my freshman year, the multiplication operation for matrices was defined without any apparent motivation. Given an $m$-times-$n$ matrix $A$ and an $n$-times-$p$ ...
0
votes
2answers
28 views

Interpreting a Quotient Group ($D_8/\langle r^2\rangle$) in 2 Distinct Ways

I seem to have a misunderstanding about quotient groups. Let $D_{2n}$ denote the group of symmetries of an $n$-gon and let $V_4$ denote the Klein-4 group. On one hand, if we identify $r^2$ with $1$, ...
2
votes
0answers
66 views

Generalizing beyond proper classes

I noted that issues such as Russell's Paradox involving the set of all sets that don't contain themselves can be resolved by stating that the object that is all set that don't contain themselves is a ...
1
vote
0answers
32 views

Dihedral group and symmetry group of icosahedron

I am studying abstract algebra at the moment, but I have several troubles picturing the dihedral group $D_n$ and the symmetric group of the icosahedron. For instance, I find it really hard to solve ...
3
votes
4answers
77 views

Are there any books with lots of questions of “Fill in The holes” type

Does anyone knows books which have lots of questions ,whose format are like fill in the holes type . . Same goes for theorems and exercises . I am looking on pure math especially Real analysis ...
6
votes
2answers
203 views

Explaining Mathematical Modelling to a nonmathematician

Due to the interdisciplinary nature of my project, I find myself collaborating a lot with nonmathematicians especially biologists, medical doctors, etc. I work mostly on mathematical models as applied ...
1
vote
1answer
49 views

Help in understanding Bochner's theorem

This relates to the Bochner's theorem stated in the link: https://en.wikipedia.org/wiki/Bochner%27s_theorem My question is related to the unique probability measure μ on G. I want to express the ...
1
vote
0answers
26 views

Is there a treatment/development of the Stokes' Theorem using differential forms and the Henstock-Kurzweil integral i.e. the gauge integral?

I'm working through an analysis text independently to prepare for grad school, and the author has discussed the limitations of both the Riemann and Lebesgue integrals and only hinted at the power of ...
2
votes
0answers
41 views

Riemannian Geometry notational tricks or alternatives

I am interested in learning tricks that people have developed to speed up / clean up calculations in Riemannian Geometry. I am hopeful about this question because there is often a lot of symmetry in ...
2
votes
4answers
48 views

Probability versus intuition problem: coins

You are playing with some friends this summer. The game is simple: a fair coin is picked and tossed. Before each toss, you and your friends bet on either heads or tails coming up. Say you begin ...
1
vote
1answer
91 views

What does this infinite product come out to?

$$1\cdot \frac{1}{2}\cdot 3\cdot \frac{1}{4}\cdot 5\cdot \frac{1}{6}\cdots$$ What does this product come out to? It does diverge, but products like this tend to have values $\lt \infty$. Here is what ...
4
votes
1answer
42 views

Is there anything special about a transforming a random variable according to its density/mass function?

Lets say that $X\sim p$, where $p:x\mapsto p(x)$ is either a pmf or a pdf. Does the following random variable possess any unique properties: $$Y:=p(X)$$ It seems like $E[Y]=\int f^2(x)dx$ is similar ...
2
votes
0answers
61 views

Where does the term “Ring” come from in Algebra? [duplicate]

Group and Field make some sense to me, but I can't see why the structures that are closed under two binary operations would indicate "ring".
1
vote
0answers
35 views

Consequences of the Carathéodory conjecture

This is a very stupid question. What are consequences and applications of the Carathéodory conjecture? It seems to me interesting, but completely useless.
2
votes
1answer
60 views

What are the Results of the First and Second Axioms of Countability?

What are the consequences of a space being first or second countable? What was the motivation for these axioms in the first place?
1
vote
1answer
27 views

Ternary equivalence relations that are not equivalent to some binary equivalance

1.Is there such a thing as ternary equivalence that is not equivalent or cant not be expressed as binary equivalence? 2.If there is such a thing as expressed in 1, are there any practical uses for ...
5
votes
1answer
99 views

Is it common to decide not to go to graduate school after extensive undergraduate research experience? [closed]

At the beginning of my undergraduate career, I was almost sure I would want to apply to graduate programs in pure mathematics. However, after participating in two major research experiences and taking ...
6
votes
3answers
189 views

Why is the Gamma function off by 1 from the factorial? [duplicate]

Why didn't they define it as $$ \tilde \Gamma(x) = \int_0^\infty t^x e^{-t} \, dt ?$$ Then the definition would have two less characters than the standard definition of $\Gamma(x)$, and we would have ...
3
votes
1answer
58 views

In what way does replacing maximal ideals with prime ideals yield, intuitively, the theory of schemes?

Background: I've only so far seen the most basic aspects of algebraic geometry from Milne's notes on algebraic varieties, and thus my view of the subject as a whole is quite limited. Thus, when ...
1
vote
1answer
41 views

research in special function with Lie algebra

First of all, I don't know if this is the right place to ask about this. If not, please direct me somewhere I can get more help. I have to research in the field of special functions with a lie ...
1
vote
1answer
21 views

Would a class in Linear Optimization/Programming be useful for a CS degree?

Would a class in Linear Optimization/Programming be useful for a CS degree? if it is useful, how useful it is or what is it used for. Can someone please help me decide. Thanks in advance
7
votes
4answers
334 views

Why do we say “radius” of convergence?

In an intuitive sense, I have never understood why a power series centered on $c$ cannot converge for some interval like $(c-3,c+2]$. Also, I have had a few professors casually mention that a series ...
1
vote
1answer
33 views

Finding conferences and summer/winter schools

I have some funding for travel next term, my colleagues always seem to know of a conference here or there or a workshop in their area. Apart from personal contacts, is there any good web resource for ...
1
vote
2answers
39 views

What is the characteristic of a field in lay terms?

And why do we usually assume a field has characteristic not equal to 2?
0
votes
0answers
31 views

In a chain of equalities/inequalities, is each line referring to the previous one?

In a chain of equalities/inequalities, or equalities/containments: $$\begin{align*} A & = B\\ & \subsetneq C\\ & = D \end{align*}$$ to which element is the last part refering to? In ...
1
vote
2answers
150 views

Why is math fun? [closed]

Let's be honest. More complicated mathematics are often found as more fun than the basic mathematics. By a recent poll I took, the earliest levels of Algebra are the least fun and group theory is the ...
0
votes
0answers
22 views

Prerequisites for Stillwell's Elements of Algebra

I recently posted a thread asking about Cox's Galois Theory text but apparently it is too advancd for me at the moment so I did some searching and found this book by Stillwell. It seems to take a ...
0
votes
1answer
41 views

What is the best way to motivate yourself to review what you have already learned? [closed]

I went through a course in complex variables recently with great zest and enthusiasm and thought that I had mastered the material. Two month after school I turned my interest to one of the chapter ...
2
votes
1answer
56 views

Confusion with Centers, Conjugacy Classes, and Normal Subgroups

Ok this is a bit of a soft question, but I am in my first semester of algebra and getting continually confused by (what seems to me) some competing sets. Let $G$ be a group The center of $G$ is ...
3
votes
4answers
253 views

On the value of proofs vs counterexamples

If a conjecture doesn't hold we usually provide a counterexample. While re-proving theorems is valuable and mathematicians do it usually, I think proving that some statement is wrong without giving ...
1
vote
1answer
53 views

Is this a correct perspective?

Consider I have a sensor which is measuring some disturbance and I am converting the disturbance into complex numbers at a regular rate. I have done it for some time and therefore I get a series of ...
0
votes
1answer
45 views

Unknowledgable of single variabled integral

Browsing Stack-Exchange and other sites, I have noticed this come up quite a few times, an integral with only one letter or number! This would be a great example: $$\int_{a}$$ What in the world does ...
1
vote
1answer
33 views

Introduction to Lebesgue Integration for Statistical Use

I am studying statistics at the graduate level and have a moderate background in real analysis however I unfortunately have no experience with Lebesgue integration. Does anyone have some recommended ...
2
votes
1answer
85 views

Where does the “CW” in CW-complex come from?

I've heard people say that the "CW" in CW-complex comes from the "CW" in JHC Whitehead, though nobody has ever given me a reference for this. Does anyone know where the "CW" in CW-complex comes from?
0
votes
1answer
40 views

Prerequisites for Cox's Galois Theory

I am planning to read Cox's book 'Galois theory' but I don't know any abstract algebra. In the preface which you can read here there is no indication of the assumed knowledge. Could someone please ...