# Questions tagged [applications]

Applications of mathematics to any field inside or outside mathematics

156
questions

1
vote

0
answers

149
views

### Laplace transform

\begin{equation}
\begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...

0
votes

2
answers

95
views

### Points based partial ranking

I want to rank a population $P=\{P_1,\ldots,P_n\}$. I am given a set $R=\{R_1,\ldots,R_k\}$ of partial rankings. The partial rankings may have varying sizes (e.g. the first ranking ranks only 8 ...

9
votes

2
answers

512
views

### What are applications of asymptotic freeness of random matrices?

In around 1990 Voiculescu showed asymptotic freeness of certain random matrices,
i.e., free independence when the matrix size goes to infinity.
Since then this link between free probability and random ...

-5
votes

1
answer

72
views

### Application of Resultant in Computer Algebra [closed]

Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much

1
vote

1
answer

266
views

### Application of Yamabe and Liouville type equation

Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
The Yamabe Type Equation (for $n>2$):
\begin{equation}
-\...

6
votes

1
answer

401
views

### Do the exceptional root systems arise in the real world?

I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...

20
votes

6
answers

3k
views

### What are some nice uses of ultraproducts/ultrapowers?

Motivated by a recent post (Non-definability of graph 3-colorability in first-order logic), I was wondering: what are some nice arguments based on ultraproducts? I don't mind definability results, but ...

2
votes

1
answer

384
views

### Does there exist a Python package that samples random special unitary matrices such that the matrices are parameterized

For reference, the linked paper is Composite parameterization and Haar measure for all unitary and special unitary groups by Christoph Spengler, Marcus Huber and Beatrix C. Hiesmayr (J. Math. Phys. 53,...

3
votes

1
answer

134
views

### Applications of maximal surfaces in Lorentz spaces

I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces.
I can clearly see the mathematical motivations. But I wonder if zero-...

10
votes

4
answers

1k
views

### Applications of the Dold-Kan correspondence

The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...

0
votes

1
answer

138
views

### Journals of applied mathematics with an economics bent?

I'm asking here instead of the economics stackexchange because I'm interested more in the applied mathematics part, instead of just the economics; I'm interested in seeing what new research is being ...

6
votes

1
answer

341
views

### Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrodynamics?

Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics?
Tried my luck with Google's search engine, didn't show much info.
I guess you can try to use these features ...

0
votes

0
answers

129
views

### Which consequences can be deduced from this peculiar property of tetration?

Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...

15
votes

4
answers

6k
views

### Mathematicians learning from applications to other fields

Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers ...

15
votes

5
answers

1k
views

### Longest increasing subsequence as measure of randomness

Although I am by no means an applied mathematician, I like to occasionally explain applications of the math I teach to real world problems. Right now I am teaching some students about longest ...

9
votes

6
answers

2k
views

### Surprising applications of the theory of games?

I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are ...

13
votes

1
answer

648
views

### Would efficient factoring have any *other* useful applications?

This question is certainly somewhat opinion-based, but hopefully not hopelessly so.
The granddaddy of all applications for an efficient period finding or factoring capability (e.g. Shor's algorithm) ...

19
votes

4
answers

2k
views

### Applications of complex exponential

In calculus we learn about many applications of real exponentials like $e^x$ for bacteria growth, radioactive decay, compound interest, etc. These are very simple and direct applications. My question ...

38
votes

4
answers

4k
views

### Interesting and surprising applications of the Ising Model

One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in ...

37
votes

17
answers

11k
views

### Listing applications of the SVD

The SVD (singular value decomposition) is taught in many linear algebra courses. It's taken for granted that it's important. I have helped teach a linear algebra course before, and I feel like I need ...

0
votes

0
answers

234
views

### How to measure perceived note similarity in music / simplicity of ratios?

I have discovered a method to measure the similarity of two successive musical notes which I wanted to share with a question:
It is known in music theory that two successive pitches $a,b$ which sound “...

5
votes

0
answers

125
views

### Applications of $FP_\infty$ groups preserving direct systems

In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...

3
votes

1
answer

195
views

### Are there applications for the PDE $ - \operatorname{grad} ( \operatorname{div} \vec u ) = \vec f$?

As in the title: given a vector field $\vec f$, are there any interesting applications (in physics, biology, or economy, or ...) of the partial differential equation
$ - \operatorname{grad} ( \...

3
votes

1
answer

94
views

### Applications of coupled Volterra-Hammerstein in Banach space

I'm looking to study the existence solutions of the following coupled equation:
\begin{equation}
\left\{\begin{matrix}
x(t)&=&\int_{0}^{t} K\big(t, s\big) f\big(s, x(s),y(s)\big) d s, \quad t \...

8
votes

2
answers

866
views

### Non-set-theoretic consequences of forcing axioms

This article by Quanta Magazine states:
... forcing axioms ... are workhorses that regular mathematicians “can actually go out and use in the field, so to speak,” ...
What are some examples of uses ...

2
votes

0
answers

112
views

### What is the definition of this function?

I'm reading a paper and I didn't understand this notation used by the author:
Let E be a vector space and F be a subspace of E.
Let $S(E/F)$ be the symmetric algebra of $E/F$. For every element $P \...

5
votes

1
answer

221
views

### Are there any applications of the algebraic polar decomposition?

One of the decompositions mentioned in the Wikipedia page on matrix decompositions is the algebraic polar decomposition. This factors a square complex matrix $M$ into $M = SQ$ where $S = S^T$ and $QQ^...

2
votes

0
answers

77
views

### What are possible applications of 'fast arithmetic' in the Jacobian (degree zero Picard group) of projective curves over fields?

It is well known that there are plenty possible applications of 'fast arithmetic' (that is, 1. having an algorithm at hand that actually computes in..., and 2. the running time of that algorithm is ...

15
votes

9
answers

2k
views

### Tools from other disciplines useful to mathematics research?

Obviously, mathematics provides essential tools for physicists, biologists, economists, engineers and many others to use in their research. Equally obviously, physics, biology, economy and engineering ...

1
vote

0
answers

82
views

### What are some interesting applications of the Archimedean Property?

So a wile back I managed to prove the The Remainder Theorem starting from the Archimidean property and since then I've thought what could be other results which can be proved using it. But I haven't ...

5
votes

0
answers

345
views

### Applications of Tits' alternative in algebraic number theory

I have recently studying Tits' alternative. The theorem statement goes like the following:
Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...

11
votes

1
answer

698
views

### Abstract mathematical concepts/tools appeared in machine learning research

I am interested in knowing about abstract mathematical concepts, tools or methods that have come up in theoretical machine learning. By "abstract" I mean something that is not immediately related to ...

5
votes

1
answer

494
views

### Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc

Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these ...

2
votes

0
answers

983
views

### Applications of linear algebra in the design of aircraft [closed]

David Lay mentioned one application of linear algebra in the design of aircraft in the introductory part of chapter 2 of his book:
[...] A computer creates a model of the surface by first ...

6
votes

0
answers

526
views

### Status of the Salmon Conjecture

The set-theoretic version of the Salmon Conjecture (that is, finding the equations that cut out the fourth secant variety of the Segre embedding of $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ ...

30
votes

7
answers

6k
views

### Applications of mathematics in clinical setting

What are some examples of successful mathematical attempts in clinical setting, specifically at the patient-disease-drug level?
To clarify, by patient-disease-drug level, I mean the mathematical work ...

3
votes

2
answers

279
views

### Reference request on a bijection on trees related to Narayana numbers

The following bijection on rooted plane trees arises in the following context : the counting sequence of (rooted plane) trees with $n$ edges ($n+1$ vertices) and $k$ leaves is given by:
$$\frac{1}{n} ...

1
vote

0
answers

99
views

### Plethora of variant neural networks?

Since a decade ago when new life was breathed in to neural networks in the form of deep learning a plethora of different architectures have come about. Is there a reference that gives compendium of ...

3
votes

0
answers

499
views

### Is there such a field as applied $\infty$-category theory?

It seems that applied category theory has exploded in popularity in recent years.
My question is simple: had there been any work using $\infty$-category theory in applications?
Edit: By ...

25
votes

0
answers

1k
views

### Caramello's theory: applications

In this text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation):
In any case, contemporary mathematics provides an example of ...

4
votes

2
answers

468
views

### Applications of the PBW theorem on enveloping algebras

What are some nice corollaries or applications of the Poincaré Birkhoff Witt theorem? There's this immediate corollary that a Lie algebra sits inside the universal enveloping algebra so in particular, ...

8
votes

1
answer

344
views

### Constants of motion for Droop equation

There is an important ODE system in biochemistry, Droop's equations:
$$s'=1-s-\frac{sx}{a_1+s}$$
$$x'=a_2\big(1-\frac{1}{q}\big)x-x$$
$$q'=\frac{a_3s}{a_1+s}-a_2(q-1)$$
Relatively easy one finds a ...

4
votes

2
answers

387
views

### Applications of flat submanifolds to other fields of mathematics

Developable surfaces in $\mathbb{R}^{3}$ have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).
I am a curious about potential or actual applications to other ...

5
votes

1
answer

310
views

### Applications of De-Bruijn Sequences in "Pure Mathematics"

I know of a few applications of De-Bruijn Sequences and De Bruijn Graphs in combinatorics, applied mathematics, Engineering and computer science. But I have only found one application of De Bruijn ...

3
votes

2
answers

100
views

### Maximizing minimal distance between consecutive brushstrokes when painting a checkerboard torus

Suppose you have a 2-torus and you want to paint an $m\times n$ checkerboard pattern on it.
Every brushstroke could paint a single square.
How does one maximize the minimal distance between ...

13
votes

2
answers

669
views

### Reference Request: Theoretical Mixing Times Research in Machine Learning / Artificial Intelligence (AI)

I'm doing a PhD in probability theory, focusing mostly on mixing times. It's a pure maths PhD, considering precise models and showing rigorous mixing results. I'm also interested in stuff like machine ...

24
votes

6
answers

4k
views

### Research in applied algebraic geometry that essentially needs a background of modern algebraic geometry at Hartshorne's level

By applied algebraic geometry, I don't mean applications of algebraic geometry to pure mathematics or super-pure theoretical physics. Not number theory, representation theory, algebraic topology,...

1
vote

0
answers

258
views

### Industrial research projects on "mathematical modeling and PDEs" [closed]

Apparently there are several companies in a great variety of fields (medical, biological, engineering, etc.) that need "consulting on mathematical modeling and PDEs" from applied mathematicians.
I'...

5
votes

3
answers

1k
views

### Application of simple Lie algebras over finite fields

I am now interested in simple Lie algebras over finite fields. In Lie algebras over the complex numbers, there are several applications and some related topics.
Is there any potential application for ...

10
votes

1
answer

671
views

### Persistent homology over the integers

Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?
Currently, persistent homology is usually done over a field (like $\mathbb{...