# Questions tagged [applications]

The [application] tag is meant for questions about applications of mathematical concepts and theorems to a more practical use (e.g. real world usage, less-abstract mathematics, etc.)

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8answers
26k views

### Are there real world applications of finite group theory?

I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
7answers
2k views

### Applications of ultrafilters

I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters? I'm ...
20answers
64k views

### Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here. What ...
2answers
260 views

### Any employment for the Varignon parallelogram?

The midpoints of the sides of an arbitrary quadrilateral form a parallelogram, which is called the Varignon parallelogram of the quad. While answering a question about Quadrilateral Interpolation it ...
20answers
83k views

### Real-world applications of prime numbers?

I am going through the problems from Project Euler and I notice a strong insistence on Primes and efficient algorithms to compute large primes efficiently. The problems are interesting per se, but I ...
1answer
5k views

### Physical interpretation of Laplace transforms

One may define the derivative of $f$ at $x$ as $\lim\limits_{h\to0}\cdots\cdots\cdots$ etc., and show that that has certain properties, but it also has a "physical" interpretation: it is an ...
17answers
16k views

### Interesting “real life” applications of serious theorems [closed]

As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: ...
7answers
10k views

### Why does Benford's Law (or Zipf's Law) hold?

Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural ...
17answers
13k views

### Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
8answers
97k views

### What is a simple example of a limit in the real world?

This morning, I read Wikipedia's informal definition of a limit: Informally, a function f assigns an output $f(x)$ to every input $x$. The function has a limit $L$ at an input $p$ if $f(x)$ is "...
19answers
7k views

### What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
26answers
6k views

### Applications of complex numbers to solve non-complex problems

Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating ...
7answers
8k views

### Uses of quadratic reciprocity theorem

I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic ...
4answers
7k views

### Real-world uses of Algebraic Structures

I am a Computer science student, and in discrete mathematics, I am learning about algebraic structures. In that I am having concepts like Group,semi-Groups etc... Previously I studied Graphs. I ...
4answers
9k views

### Does the concept of infinity have any practical applications?

I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world ...
2answers
2k views

### Applications of cardinal numbers

I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$. ...
7answers
7k views

### What is the best base to use?

When I typed this question in google I found this link: http://octomatics.org/ Just from the graphic point of view: this system seems to be easier (when he explains that you can overlap the line). He ...
33answers
25k views

### Can you provide me historical examples of pure mathematics becoming “useful”?

I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...
10answers
2k views

### Surprising applications of topology

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
10answers
9k views

### Real life usage of Benford's Law

I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.
3answers
2k views

### Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
3answers
104 views

### Practical system with the following ODE form

I was wondering if anyone is familiar with an example of a practical / real system with the following ODE form: $\dot{x}_1= a_{11} x_1$ $\dot{x}_2=a_{21} x_1 + a_{22} x_2 + b u$, where $u$ is a ...
9answers
26k views

### How do I sell out with abstract algebra?

My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into ...
12answers
13k views

### What are the applications of functional analysis?

I recently had a course on functional analysis. I was thinking of studying the mathematical applications of functional analysis. I came to know it had some applications on calculus of variations. I am ...
20answers
8k views

### What are some examples of mathematics that had unintended useful applications much later?

I would like to know some examples of interesting (to the layman or young student), easy-to-describe examples of mathematics that has had profound unanticipated useful applications in the real world. ...
4answers
11k views

### How do people apply the Lebesgue integration theory?

This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas. In a lot of calculus textbooks, there is ...
9answers
13k views

### Real world applications of category theory

I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if ...
7answers
2k views

### Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
5answers
52k views

### How does linear algebra help with computer science?

I'm a Computer Science student. I've just completed a linear algebra course. I got 75 points out of 100 points on the final exam. I know linear algebra well. As a programmer, I'm having a difficult ...
5answers
2k views

### Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
12answers
7k views

### Applications of algebraic topology

What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy ...
8answers
15k views

### Real world uses of Quaternions?

I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real ...
6answers
3k views

### Real life uses of prime numbers (in physics/engineering) [closed]

Prime numbers (or coprimes) have few well-known uses but interesting ones. The classical example is that prime numbers are used in asymmetric (or public key) cryptography. Prime numbers and coprimes ...
18answers
41k views

### What is an example of real application of cubic equations?

I didn't yet encounter to a case that need to be solved by cubic equations (degree three) ! May you give me some information about the branches of science or criterion deal with such nature ?
10answers
43k views

### Real world uses of hyperbolic trigonometric functions

I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses ...
4answers
2k views

### Applications of Gröbner bases

I would like to present an application of Gröbner bases. The audience is a class of first year graduate students who are taking first year algebra. Does anyone have suggestions on a specific ...
2answers
6k views

### Applications of Perfect Numbers

I'm preparing a talk on Mersenne primes, Perfect numbers and Fermat primes. In trying to provide motivation for it all I'd like to provide an application of these things. I came up with these: ...
8answers
5k views

### Applications of Probability Theory in pure mathematics

My (maybe wrong) impression is that while probability is widely used in science (for example, in statistical mechanics), it is rarely seen in pure mathematics. Which leads me to the question - Are ...
5answers
3k views

### What is Cramer's rule used for?

Cramer's rule appears in introductory linear algebra courses without comments on its utility. It is a flaw in our system of pedagogy that one learns answers to questions of this kind in courses only ...
9answers
5k views

### Why is the derivative important? [duplicate]

Derivatives, both ordinary and partial, appear often in my mathematics courses. However, my teachers have never really given a good example of why the derivative is useful. My questions: Other than ...
2answers
1k views

### What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...
8answers
2k views

### Practical application of matrices and determinants

I have learned recently about matrices and determinants and also about the geometrical interpretations, i.e, how the matrix is used for linear transformations and how determinants tell us about area/...
3answers
282 views

### Applications of derivatives outside mathematics and physics

I've been teaching calculus for several years and have some doubts about whether derivatives (and integration techniques) of common functions are useful and important outside mathematics and physics. ...
3answers
11k views

### What are the real-world applications of real analysis?

I've read the wikipedia article on mathematical analysis and this, but I can't exactly find an answer. Is real analysis just some pure math, or does it really have something to with physical ...
1answer
342 views

### Closed form of $I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx$ and is it behave similar with error function?

$\newcommand{\erf}{\operatorname{erf}}$ The computation of $\int_{0}^{a}{(e^{-x²})}^{\erf(x)}dx$ for large $a$ gives $0.972106...$ by wolfram alpha, but according to JJacquelin comments which ...
5answers
3k views

### How are complex numbers useful to real number mathematics?

Suppose I have only real number problems, where I need to find solutions. By what means could knowledge about complex numbers be useful? Of course, the obviously applications are: contour ...
3answers
1k views

### Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [closed]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
0answers
186 views

### Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
3answers
3k views

### Differential Equations Skydiver Problem

I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate. The famous differential equation problem models a skydiver jumping out of a ...
24answers
17k views

### Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never ...