Questions tagged [applications]

Applications of mathematics to any field inside or outside mathematics

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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 \...
20 votes
5 answers
2k views

Examples of research on how people perceive mathematical objects

What examples are there on research related to human perception and mathematical objects? For example, the shape of a beer glass influences drinking habits, since people are bad at integrating. ...
77 votes
11 answers
13k views

Applications of mathematics

All of us have probably been exposed to questions such as: "What are the applications of group theory...". This is not the subject of this MO question. Here is a little newspaper article that I found ...
8 votes
2 answers
863 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 \...
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 ...
6 votes
14 answers
5k views

Applications of compactness [closed]

Similar to this question: Applications of connectedness I want to collect applications of compactness. E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
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 ...
27 votes
4 answers
3k views

Algebra and cancer research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research It is well-known that mathematical modeling and computational biology are effective tools in cancer research....
10 votes
3 answers
2k views

Applications of knot theory to biology/pharmacology

What are the applications of knot theory to biology/pharmacology? I guess there should be some, since proteins are quite long and some of their properties are probably related to whether they are ...
9 votes
2 answers
5k views

Applications of algebraic geometry/commutative algebra to biology/pharmacology

Are there applications of algebraic geometry/commutative algebra to biology/pharmacology? It might be that some Gröbner basis technique is used somewhere? I know there are some applications to ...
5 votes
0 answers
343 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 ...
4 votes
2 answers
458 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, ...
10 votes
1 answer
657 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{...
11 votes
1 answer
656 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 ...
14 votes
1 answer
2k views

Which journals publish applied mathematics with mostly pure mathematics content?

In the spirit of Which journals publish expository work? please advise: What consistently high quality journals (1) today publish results that would otherwise go to a pure mathematics journal were ...
5 votes
1 answer
491 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
945 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
522 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 $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ ...
25 votes
5 answers
2k views

Sperner Lemma Applications

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of ...
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
493 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 ...
7 votes
1 answer
3k views

Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$ such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...
8 votes
1 answer
343 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
386 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 ...
28 votes
11 answers
7k views

Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world consequences? [closed]

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
40 votes
4 answers
5k views

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 ...
5 votes
1 answer
308 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 ...
30 votes
4 answers
4k views

Elementary applications of Krein-Milman

This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try. Recall that the Krein-...
7 votes
1 answer
1k views

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 ...
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
665 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 ...
19 votes
4 answers
1k views

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 ...
7 votes
5 answers
4k views

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, ...
5 votes
4 answers
957 views

Applications of Szemeredi's Theorem

Szemeredi's Theorem is a famous theorem in Additive Combinatorics, Ergodic Theory and maybe some other parts of Mathemtatics: (Szemeredi's Theorem) Let $\Lambda \in \mathbb{Z}$ be a subset of ...
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 ...
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
2 answers
906 views

Applications of propositional dynamic logic

Propositional dynamic logic (PDL) is an example of a (multi)modal logic with a structure on the set of modalities. In particular, the set of its modalities is indexed by "programs" and one can use ...
16 votes
3 answers
2k views

Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool. The basis is Mamikon's theorem. The area of a tangent sweep is equal to the area ...
3 votes
0 answers
226 views

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 ...
20 votes
8 answers
11k views

Nice applications of the spectral theorem?

Most books and courses on linear algebra or functional analysis present at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many ...
2 votes
0 answers
154 views

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 ...
0 votes
1 answer
496 views

Mathematical properties of financial prices

Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes. What is known about their mathematical properties ? I know ...
8 votes
2 answers
921 views

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 ...
8 votes
8 answers
6k views

Is Riemannian integration sufficient in physics?

Are there any applications in physics or engineering which require the Lebesgue integral and cannot be treated by Riemannian integration
1 vote
3 answers
528 views

"Graphical models" and "gene finding and diagnosis of diseases" ?

Quote Wikipedia: Applications of graphical models include ... gene finding and diagnosis of diseases... Unfortunately there is no comment what are these applications... Can one comment on this ? ...
1 vote
0 answers
40 views

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 $\...
5 votes
1 answer
430 views

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 ...