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0
votes
1answer
42 views

Examples of how abstract algebra is used to find concrete solutions to a mathematical model?

All references I've seen to abstract algebra show how it helps in the representation of mathematical models...are there any examples of using abstract algebra to calculate actual solutions to a ...
4
votes
0answers
49 views

Alternative Proof of Heine-Borel Theorem

This is regarding the proof of Heine Borel Theorem for closed intervals on real line as given in Hardy's Pure Mathematics. Heine-Borel Theorem: Let $[a, b]$ be a closed interval with $a < b$ and ...
5
votes
3answers
123 views

Explain branches of geometry for non-mathematician

Some background - I'm an advanced physics undergrad and lately was motivated to self study basic contemporary geometry to get a better grip on general relativity (maybe there is a more appropriate ...
15
votes
8answers
781 views

How to explain your area of study to non-math people [duplicate]

I'm sorry in advance if here is not the suitable place to ask this question, and people can feel free to vote to close this if that's the case. However, since I'm not sure about this issue, I'll ask ...
5
votes
2answers
149 views

Chess rating calculating algorithm

In competitive chess tournaments, there is a complex rating system that evaluates your rating based on how you do well you do playing games. I am referring to the FIDE system not USCF. Are there FIDE ...
2
votes
3answers
121 views

If $AB\not=BA,$ then do $A$ and $B$ have to be matrices

Is this solvable? Or are there other things that fit the bill for $A$ and $B$?
4
votes
0answers
226 views

Need mathematical explanation for different musical notes sound different on different instruments [migrated]

I am not expert in music. There are number of musical instruments. One (especially a person who knows about music) can blindly recognize which instrument is being played just by listening to it. I ...
0
votes
1answer
62 views

Why is functional analysis so obsessed with sequences?

Beginning functional analysis I thought I would learn about generalized properties of functions and operators but yet I am flipping through pages after pages of texts on cauchy sequence, convergence, ...
1
vote
0answers
18 views

L1 and L2 regularization and L1 and L2 space

I am looking to characterize the difference of the function obtained in the solution process of $L^1$ and $L^2$ regularization. It is known that $L^1$ regularization gives sparse solutions. In $L^2$ ...
1
vote
0answers
55 views

Recommendations for a thorough logic textbook

I'm looking for a (possibly introductory) textbook on logic that covers the motivation behind conventions in logic, like the definition of the implication. Prof. J. Lau has an excellent webpage, ...
1
vote
2answers
119 views

Which should I study first: Logic or set theory?

I'm an undergraduate student in a college of sciences and technics studying maths, physics, computing and some chimestry so we studied elementary materials in logic and set theory. As I am interested ...
-2
votes
2answers
55 views

Function undefined at non-integer values [closed]

Is there a function $f(x)$ which is not defined at integer values? Please do NOT answer $f(x) = \begin{cases} a, & \text{if } x \in \mathbb{Z}, \\ \text{undefined}, & \text{otherwise} ...
10
votes
2answers
171 views

Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda ...
10
votes
0answers
116 views

A “generalized field” with $q$ elements, when $q$ is any number?

It is well-known that if a finite field has $q \in \mathbb{N}$ elements, then $q$ is prime power and $q > 1$. However, various modification of the concept of a "field" have been made in order to ...
6
votes
0answers
185 views

Becoming an Interuniversal Geometer [closed]

Apart from the extensive amount of studying, are there reasons why I should or should not go into Interuniversal Teichmuller theory when I get older? Is there already a second generation of ...
2
votes
0answers
21 views

composition of relations and multiset relations

As is well known, composition of relations is defined as $$R\circ S = \{(a,b):\exists x: (a,x)\in S \land (x,b)\in R\} \tag 1$$ This is formally the very same definition as for function composition. ...
5
votes
2answers
146 views

Characterizations of the cross-ratio

$$ (z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} $$ What are the most prominent or most interesting theorems of the following form? Theorem: The cross-ratio is the only function ...
3
votes
0answers
89 views

Convergence to $\frac{1}{\pi}$

Mathematicians of all times found approximations for the value of $\pi$ using infinite sums. But I was asking to myself: is there any infinite sum that approximates $\frac{1}{\pi}$?
42
votes
14answers
7k views

Can math be subjective?

Often times in math, ever since Kindergarten and before, math has been defined by the fact that there are only one answer for problems. For example: $1+1=2$ and $\frac{d}{dx}x^2=2x$. What I am showing ...
1
vote
1answer
46 views

Database of unsolved problems in mathematics

Is there a good database of unsolved problems in mathematics?
2
votes
1answer
88 views

What shall I write for a reason for applying graduate school for algebraic geometry?

I'm a undergraduate applying a graduate school this year and now I'm writing a letter of self-introduction. To be honest, I don't know what exactly is algebraic geometry and I think 99% of ...
1
vote
1answer
39 views

What is a prime ideal?

I am having some trouble understanding the concept of a prime ideal in ring theory. I have researched what a prime ideal is and the simplest answer I got was this: An ideal $P$ of a commutative ...
7
votes
2answers
243 views

Recommendation for books on topology (light reads)

Are there any books on topology which can be read without having to do any exercises and look up definitions every second line? Something to read while relaxing, and not meant to replace a textbook ...
2
votes
0answers
48 views

Where are multisets used in mathematics? [duplicate]

Outside of factorization (integers into primes, polynomials into irreducibles) where else are multisets naturally useful in mathematics? [edit] Deleting. If anybody wants me to stop, please say so ...
12
votes
0answers
105 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
1
vote
0answers
25 views

Stochastic process, Fourier transform and $L^2$

Consider a time domain signal received at a sensor $x(t)$ over some time $t$, and we have performed Fourier transform on $x(t)$ to obtain $X(w)$. While performing Fourier Transform to find the ...
-1
votes
2answers
80 views

Is this mathematics journal reputable? [closed]

A little background, I have written and had the necessary people revise my first paper that will not be supported directly by an institution (meaning I'm pretty much on my own) and am curious as to ...
0
votes
0answers
42 views

Positive definite function and covariance matrix.

I tend to view positive definite function as a function of elements of positive definite matrix. A reference is: https://en.wikipedia.org/wiki/Positive-definite_function My question in essence: is ...
2
votes
0answers
53 views

How do I make sure that I've learned and mastered a part of the Visual Complex Analysis book?

So I'm reading Visual Complex Analysis by Tristan Needham. It's a beautiful book that's not very hard to understand at all; however, I just don't know if I have sufficiently learned what I'm supposed ...
2
votes
0answers
56 views

Analytical approach of representation theory

I'm doing M.Sc. in mathematics. I want to do my M.Sc. thesis on Representation theory in analytic approach. So I start reading the book Representations of Finite and Compact Groups by Barry Simon. ...
0
votes
0answers
13 views

On statistical analysis and sudden changes in data

Here we see the value of Euro against the United States Dollar, provided by BBC approximately 10.00 GMT on the 6th of June, 2015. On the 5th Greece had a referendum, and it's outcome of "No" to ...
2
votes
1answer
30 views

Example of Non-separable stochastic process.

This question is related to the link: http://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process under ...
5
votes
1answer
120 views

Differential topology versus differential geometry

I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds: Analysis on manifolds, containing: definition of manifold, tangent space (as ...
8
votes
0answers
65 views

Why do people prefer cosine to sine when speaking of harmonic oscillation?

In almost all of the physics textbooks I have ever read, the author will write the oscillating function as $$x(t)=\cos\left(\omega t+\phi\right)$$ My question is that, is there any practical or ...
0
votes
0answers
46 views

How to effectively learn from and use Ramanujan's notebooks? [duplicate]

I will come back and elaborate on the question if necessary (I must be off for a while...). But I'll try being specific. I have all four of Ramanujan's notebooks, with their respective Errata ...
0
votes
0answers
27 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
0
votes
1answer
32 views

Difference between stochastic process and chaotic system [closed]

Can anyone please point out some difference and similarity between stochastic system and chaotic system?
0
votes
0answers
16 views

Question about “non-question” types of exchanges. [migrated]

Sorry if posting this question in the wrong forum, it is more of a meta-question. What if I have something interesting to discus or get feedback on, but am not able to formulate a question about it? ...
1
vote
3answers
65 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
2
votes
0answers
40 views

Calculus on Spaces other than $R$

I was thinking that all that the real numbers (and the complex numbers similarly) need for calculus to work is for them to be a field and a complete metric space, and probably have the usual field ...
7
votes
1answer
66 views

Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
3
votes
1answer
97 views

Difference between proof-based calculus and analysis?

When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere ...
8
votes
1answer
168 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
3
votes
1answer
58 views

A negative third derivative implies a positive first derivative at a point.

Let $f$ be three times differentiable on $\mathbb{R}$ with $f(0)=f(1)$, and for all $x\in[0,1]$, $f'''(x)<0$. Prove that $f'(\frac{1}{2})>0$ I actually have a proof of this question using ...
2
votes
0answers
23 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
35
votes
6answers
2k views

Is there something between summation and integration?

Let's take a general function $f(x)$, we can do a summation like: $$\sum_{k=m}^n f(k)$$ And we can do an integration like: $$\int_a^bf(k)dk$$ The basic difference between the two operation is that ...
0
votes
1answer
43 views

Is there a way to generate groups, rings, fields, etc.? [closed]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
3
votes
1answer
114 views

What motivates the definition of a ring in abstract algebra? [closed]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
0
votes
0answers
21 views

Order of 'Strength' of inequalities

There have been times when we solve an inequality and we get the reverse sign of inequality. The reason is quite simple- we did not choose a strong inequality. So my question is- Is there an order of ...
2
votes
2answers
395 views

Changing from Positive to Negative

I may mess up a little bit...Sorry for that! When we want a summation to go negative in case of odd numbers and positive otherwise , we use: $$\sum\limits_{i=1}^{12} \color{red}{{(-1)}^i} 2x^3 $$ ...