# Questions tagged [applications]

Applications of mathematics to any field inside or outside mathematics

144
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### 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

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### 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 ...

8
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6
answers

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### 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 ...

12
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1
answer

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### 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
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### 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
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4
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### 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 ...

35
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17
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### 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

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### 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

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### 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

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### 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

78
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### 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

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### 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

104
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### 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 \...

4
votes

0
answers

96
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### 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

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### 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 ...

14
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9
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### 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
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### 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
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### 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 ...

5
votes

1
answer

337
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### 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

445
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### 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
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### 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 ...

5
votes

0
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### 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
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7
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### 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 ...

9
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2
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488
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### Character theory and Quantum Chemistry

Who (presumably a chemist) realized first the efficiency of character theory in calculations of orbitals of atoms? In which year?

3
votes

2
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### 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

84
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### 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

399
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### 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 ...

22
votes

0
answers

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### 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
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356
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### 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

330
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### 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

354
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### 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

271
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### 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

88
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### 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 ...

12
votes

2
answers

597
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### 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 ...

23
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6
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3k
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### 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

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### 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
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### 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 ...

9
votes

1
answer

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### 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{...

3
votes

0
answers

197
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### Applications of logic in theoretical and practical Computer Science [closed]

Can anyone suggest theoretical and/or practical applications of logic (modal, dynamic, Lukasiewici etc.) in Computer Science (like Markov Chains for linear algebra), as well as some open-source books ...

2
votes

0
answers

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### References on computational PDE (in fluid dynamics, solid mechanics, etc) that emphasize both rigorous analysis and coding

I'm interested in learning about computational aspects of PDE and integro partial differential equations.
In particular, I'd like to know some reference monographs that cover PDE and IPDE from in ...

7
votes

5
answers

3k
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### Applications of Perfect Matching

I'm exploring some applications of perfect matching and I would like some input. I have found many applications in chemistry (storing information, estimating bond lengths, estimating resonance energy, ...

40
votes

4
answers

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### Is algebraic geometry constructive?

Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an ...

1
vote

0
answers

38
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### Envelope of a parametrized family of convolutions

For a certain application I need to compute a pointwise supremum of this family of gaussian convolutions:
$$\sup_s f(x)\otimes e^{-\frac{x^2}{s^2}}$$
where $f(x),x\in \mathbb{R}^2$ is known and $\...

3
votes

1
answer

354
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### Using High Level Probability Theory (eg Markov Chain Mixing) in Cryptography/Cryptanalysis

I'm currently doing a PhD in probability theory, specifically (discrete space) Markov chains and their mixing properties. As well as my current main project, I'm looking to have a side project, eg to ...

11
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3
answers

909
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### What are some applications of Sperner style theorems?

I'm currently working through "Combinatorics of Finite Sets" by Ian Anderson, mostly to improve at a style of mathematics that I've historically been quite bad at, and I find myself wondering why this ...

19
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4
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### Applications of linear programming duality in combinatorics

So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...

3
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2
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### Application for Differential Equation of higher order [closed]

We found some interesting insights in differential equations of the form
$y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$,
i.e. for ordinary differential equations of $n$-th order with $n\geq2$....

7
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1
answer

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### Easy Applications of Model Theory

I've also posted this question on MathSE. I'm posting it here in hopes of a more comprehensive answer. The question is inspired by the following:
Model theoretic applications to algebra and number ...

7
votes

2
answers

802
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### Applications of Topological Complexity of configuration space

I'm starting to work on topological complexity of configuration spaces.
Articles say that this field has applications in robotic and control theory. One of the important articles belongs to Michael ...

1
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0
answers

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### Formalization of adaptive sampling [closed]

The notion on adaptive sampling or adaptive plotting is fairly popular, but I have not found a formal definition.
I have developed an algorithm for plotting implicit algebraic curves in the plane. ...