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Questions tagged [applications]

Applications of mathematics to any field inside or outside mathematics

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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 \\ ...
TUHOATAI's user avatar
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 ...
Max's user avatar
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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 ...
Bart's user avatar
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1 answer
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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
Luật Trần Văn's user avatar
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} -\...
Paul's user avatar
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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 ...
Lorenzo Del Vecchiopontopolos's user avatar
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,...
Silly Goose's user avatar
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-...
Hao Chen's user avatar
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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 ...
Dora's user avatar
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1 answer
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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 ...
shintuku's user avatar
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6 votes
1 answer
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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 ...
Alan's user avatar
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0 votes
0 answers
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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 ...
Marco Ripà's user avatar
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15 votes
4 answers
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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
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 ...
Sam Hopkins's user avatar
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 ...
H.C Manu's user avatar
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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) ...
tparker's user avatar
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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 ...
Max's user avatar
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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 ...
JustWannaKnow's user avatar
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 “...
mathoverflowUser's user avatar
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 ...
Mark Backhaus's user avatar
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} ( \...
shuhalo's user avatar
  • 4,896
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 \...
Motaka's user avatar
  • 291
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 ...
Jordan Mitchell Barrett's user avatar
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 \...
Maria's user avatar
  • 133
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^...
wlad's user avatar
  • 4,863
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 ...
windsheaf's user avatar
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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 ...
fisura filozofica's user avatar
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 ...
ShBh's user avatar
  • 271
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 ...
Emily's user avatar
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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 ...
student's user avatar
  • 101
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 $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ ...
Yellow Pig's user avatar
  • 2,530
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} ...
Olivier's user avatar
  • 468
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 ...
Turbo's user avatar
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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 ...
Patrick Elliott's user avatar
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 ...
user avatar
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, ...
nobody's user avatar
  • 417
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 ...
Nikita Kalinin's user avatar
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 ...
Matteo Raffaelli's user avatar
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 ...
Serge the Toaster's user avatar
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 ...
Michael's user avatar
  • 2,175
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 ...
Sam OT's user avatar
  • 560
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,...
No One's user avatar
  • 1,553
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'...
user avatar
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 ...
NongAm's user avatar
  • 187
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{...
yoyostein's user avatar
  • 1,219