For questions about gaining and achieving motivation.

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2
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2answers
61 views

Do coalgebras arise outside the study of bi/Hopf-algebras?

Hopefully the title is fairly self explanatory. I'm curious as to whether the coalgebra structure (that is, a vector space with a comultiplication and counit) comes up an any area of mathematics not ...
6
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1answer
224 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
3
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1answer
90 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
2
votes
1answer
181 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
1
vote
1answer
78 views

Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
0
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1answer
76 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
4
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0answers
61 views

Why do we care about non-$T_0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
4
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0answers
213 views

What are the advantages of proof-relevant mathematics?

I've read that Theorems in HoTT (homotopy type theory) tend to characterize the space of proofs of a proposition, rather than simply state that the corresponding type is inhabited. So, HoTT ...
3
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0answers
72 views

A good way to explain $\varepsilon$-$\delta$ for chemistry / biology students?

I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry ...
3
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0answers
82 views

What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
3
votes
0answers
118 views

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Let $x$ denote an arbitrary real number. Then $x^n$ makes sense for arbitrary $n \in \mathbb{N},$ via the obvious recursive definition. We can extend this definition by asserting that if $x$ is ...
3
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0answers
173 views

Algebraic Geometry question

Why do we study projective normality of a projective variety ? Does it have anything to do with non-singularity ? Any other purpose to study this ?
2
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0answers
22 views

Motivation for the study of units in cyclotomic fields beyond Washington's book

Right now, I am reading Larry Washington's book "Introduction to Cyclotomic Fields." In Chapter 8 of this book, the unit group of the ring of integers in a cyclotomic field (or its totally real ...
2
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0answers
62 views

Intuition for the Positive Real Number $\epsilon$ in Topology

Although this question might sound a little too simple, it is a problem that I must get addressed. In addition, there is no way for me to formally describe it. If you have something you can add, by ...
2
votes
0answers
93 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
1
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0answers
58 views

Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...
1
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0answers
50 views

How do I visualize algebraic products?

I think I have no or a little problem with analyzing given algebraic products. That is, I know properties of direct, semi-direct and free products. For example, the free product $G\ast H$ is a group ...
1
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0answers
98 views

Big theorems in information geometry?

Working on preparing a talk on information geometry to a young finance/applied math audience. Motivating this area is turning out a little difficult. What are some big theorems or results that I ...
0
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0answers
24 views

What does a functional integral evaluation look like?

I've read the Wikipedia page on functional integration, but it really isn't very easy to understand. There don't seem to be any online videos on the subject either. In addition, when I search online, ...
0
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0answers
55 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...