A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modelling.
162
votes
16answers
6k views
Optimizing response times of an ambulance corp: short-term versus average
Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
9
votes
3answers
469 views
Is there a general solution for the “Spider and the Fly Problem”?
(I would be appreciative if somebody could give a more formal formulation of this problem.)
The Spider and the Fly Problem is a problem in which the objective is to minimize the distance the spider ...
9
votes
2answers
387 views
Geometrical construction for Snell's law?
Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the ...
8
votes
0answers
83 views
Undergraduate thesis advice
I am presently in my senior year and I am considering fluid mechanics for my thesis. What area of research of fluid mechanics or models of transport mechanism in Biological systems which is purely ...
7
votes
1answer
683 views
Probability in single-lane traffic flow: What are the odds of “choke points” being encountered?
Let's say you have a single-lane road (single in this case meaning single lane in each direction, or what you could also call two-lane). Let's say you have a random number of vehicles on a given ...
6
votes
3answers
256 views
What sort of mathematical methods and models are used to model the brain
What sorts of mathematical tools, models and methods and theoretical frameworks do people use to simulate the function of the brain's neural networks? What mathematical properties do different brains ...
6
votes
1answer
137 views
Sunk cost auction modeling.
Consider the following auction concept. I call it a "SUNK COST AUCTION" Each person bids, but you pay all of the money that you bid for every bid you make. So, if you bid \$1 that \$1 is gone, even if ...
5
votes
2answers
92 views
Are there rigorous mathematical definitions for these waves?
My friend linked this .gif to me tonight, and asked me if I knew of any equations that might model these bottom two waves (the blue and green waves). Unfortunately, I am not far enough in my education ...
5
votes
0answers
88 views
Looking for a Lyapunov function for the next system
I am really stuck looking for a Lypaunov candidate for the next system (which in simulation is stable).
$$ \dot{x} = -(A+A^T)x + Ay \\
\dot{y} = K(x-y)
$$
where x and y are vectors in R^3, A is a ...
5
votes
0answers
537 views
Mathematical Modeling of Real-World Situations
I was considering how to model the following problem mathematically and was wondering what you thought the best way to do it is.
"A takes 1 minute to cross a bridge, B takes 2 minutes, C takes 5 ...
4
votes
2answers
1k views
What is a “free parameter” in a computational model?
In many articles regarding computational models of some particular phenomenon, there seems to be a consensus: "the smaller the number of 'free parameters' in the model, the better". So, what is meant ...
4
votes
2answers
189 views
Stochastic predator-prey
My system is a simple $P$ vs $I$ foxes- vs rabbits model given by:
$$
\begin{cases}
\frac{\mathrm{d}I}{\mathrm{d}t}=& \alpha_I+\lambda_IP- \gamma_II -\delta_IPI;\\
...
4
votes
4answers
242 views
Preparing for the Mathematical Contest in Modeling
The Mathematical Contest in Modeling (MCM) is coming up in a couple of weeks. With the idea that the competition is ultimately about teaching students to become better at mathematical modeling, I was ...
4
votes
1answer
325 views
Automation of 3D Paper Modeling
I recently saw this creative paper contraption online this prior weekend and wanted to see if I could automate the process of creating all ~35 layers of an equation. Essentially what I would want the ...
4
votes
1answer
546 views
How to formulate constrained optimization problems
For a constrained optimization problem, there can be different formulations.
For example, consider the problem with the following formulation:
$$\min_{x \in X \subseteq \mathbb{R}^n} \, f(x), $$
...
3
votes
3answers
87 views
Is there a formal name for a mathematical model that has an overly sensitive parameter
I'm a software engineer with a bit of mathematics expertise working with an NGO that has developed mathematical models for ecosystem services. It turns out that one of these models is overly ...
3
votes
3answers
447 views
Cat Dog problem using integration
Consider this equation :
$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$
where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$.
Now how to proceed ?
This equation ...
3
votes
3answers
192 views
Mathematical (or physical) formulation of life
I recently started to have interest in the characteristics of life. Especially, I wonder if there are some kind of mathematical description of the true meaning of life and how it is emerged. Why does ...
3
votes
2answers
1k views
How to start creating Mathematical models?
I know this is a quite general question , but I've always heard about creating mathematical models in economics, social sciences, engineering,... And I would want to know what are the starting points ...
3
votes
1answer
38 views
Inverse of itinerary function
Let $S^{-1}:\Sigma \to \Lambda$ be inverse of itinerary function. I showed that $S$ is continuous and bijective. How to show that $S^{-1}$ is continuous?
3
votes
3answers
271 views
Predicting the next vector given a known sequence
I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a ...
3
votes
2answers
221 views
Practical physics problem about distance between lines.
I am a software engineer and I’m developing a soccer game. I have a solution for this problem based on Newton law and I’m using Newton method to solve the equation I got. I’m here because I think ...
3
votes
1answer
90 views
Help with recursion/difference equation manipulation
I'm trying to reproduce the analysis of a simple biological model from an old paper. The model describes the frequency of two organism types in a population as:
$p_{t+1} = \frac{p_t ( s + (1 - ...
3
votes
1answer
55 views
Iterative model fitting
I have a sequence of points $\{(x_k,y_k,z_k)\}$ and I need to fit some $2D$ model $P(x,y)$ that approximates $z$ in some sense.
The $z_k$$'s$ are noisy samples of some $2D$ function $z_k = f(x,y) + ...
3
votes
2answers
119 views
How to build a function that gives recent years higher weight?
I want to have a value for users between 0-1 , that shows how much they like movies of specific type depending on how many movies they have watched and movie type ( comedy , etc ). I have a data of ...
3
votes
1answer
321 views
Mathematics From Futurama
Dear Professor Farnsworth,
We at D.O.O.P are trying to mathematically model a rocket ship fueled by your employee Leela's pet Nibbler's pooped Black matter. Obviously this rocket ship is fueled by ...
3
votes
1answer
513 views
Understanding the Pareto distribution as applied to wealth
The Pareto distribution is used to say, given a particular person X, what is the pdf of his wealth.
I would like to explore the reciprocal question: Given the total amount of wealth in a population, ...
3
votes
0answers
26 views
Regression model for a shearing process
30 Widgets are randomly assigned to a shearing process.
There are 3 such processes, each getting 10 widgets.
The lengths of each widget are recorded before undergoing the shearing.
The amount that ...
3
votes
0answers
60 views
How to model bending, folding of 2D figures?
2D shape can be folded in various ways. For example, trapezoid can have its sides (with the acute angles) folded, so that it will effectively become rectangle – with the difference, that part of the ...
3
votes
0answers
316 views
How did Target figure out a teen girl was pregnant before her father did?
First of all I do not have a mathematics degree only a B.S. in finance so please take that into account when writing an answer. Generally what type of mathematics is involved here? And specifically ...
3
votes
1answer
141 views
Conditional Expectation and the floor function
I have a piece of code the produces random integers.
...
2
votes
2answers
133 views
why does $\frac{b\partial a}{a \partial b} = \frac{\partial \log a}{\partial \log b}$?
I am doing some background research on sensitivity and elasticity analysis, and I came across the following definitions of elasticity:
$$e_{ij}=\frac{a_{ij}\partial \lambda}{\lambda \partial ...
2
votes
3answers
181 views
Biological processes described by the exponential function
What are biological or biochemical processes that are described (stochastically or deterministically) by exponential function $f(x) = e^{ax}$?
Edit: What about examples that are not related to the ...
2
votes
1answer
53 views
64 possible values - continuous or discrete?
I have a response variable in a model that is calculated from binary values of an 8x8 grid of pixels, giving a proportion. Technically, there are only 64 possible values the result can take, so would ...
2
votes
1answer
77 views
How large initial angular velocity is needed for pendulum to go completely around?
If a pendulum is initially at its unstable equilibrium position, how large an initial angular velocity is necessary for it to go completely around?
2
votes
1answer
101 views
Finding a logical expression (under some constrains) s.t. it is equivalent to another one
In this question, it was made clear, when
$\bullet$ some statement $A$ is stronger than another statement $B$, namely if $A\Rightarrow B$ holds; and when the statement $A$ is weaker than another ...
2
votes
1answer
32 views
Amount of information a hidden state can convey (HMM)
In this paper (Products of Hidden Markov Models, http://www.cs.toronto.edu/~hinton/absps/aistats_2001.pdf), the authors say that:
The hidden state of a single HMM can only convey log K bits of ...
2
votes
1answer
33 views
hopf bifurcation for an ode
I understand how to analyse a system of equations like
$x'(t) = f(x,y)$
$y'(t) = g(x,y)$
set $x'$ and $y'$ to zero and find the fixed points etc, and find the stability.
What Im am not sure of ...
2
votes
2answers
206 views
Recommended book on modeling/differential equations
I am soon attending a undergrad course named differential equations and modeling. I have dealt with differential equations before, but in that course just learned a bunch of methods for solving them. ...
2
votes
2answers
126 views
How to determine units in a partial differential equation
How do we determine the units used in a differential equation? Yes, in theory a PDE has nothing to do with units, but I'm interested in this question from a modeling point of view. By units, I mean ...
2
votes
1answer
120 views
Problem with ratios, dimensionless quantities
I've been trying to understand how a ratio of two of the same units gives a dimensionless quantity. It makes no sense to me. For example, density in terms of mass and volume:
$$d = \frac{m}{V}$$
...
2
votes
1answer
151 views
Population Dynamics model
I am currently searching for population dynamics models.
Concerning animal population growth, I have found the following so far :
Growth models for fish
Predator-prey : Lotka-Voltera and ...
2
votes
1answer
353 views
How to calculate perceived value?
Is there a known formula to calculate perceived value? If I am auctioning some popular product that is worth $100 (shelf price), how much would people bid if they:
Have a 1 in ...
2
votes
2answers
488 views
How to 'minimize' correlation between series
Hi fellow mathemagicians,
let's say that I have 3 series of numerical results (they represent 'drawdowns') :
...
2
votes
1answer
380 views
How do I calculate the instantaneous angular speed of a spindle given a spooled radius and required feed speed?
In an attempt to model the cassette tape spooling action, I'm hoping to solve this problem.
Required feed speed 5cm/second.
Two Spindles with starting radius of
S1: 0.8cm (empty spool)
S2: 1.8cm ...
2
votes
1answer
58 views
How to classify equilibrium points
I have the two differential equations:
$$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$
$$\frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$
I worked out the equilibrium points to be at $N_1 = 0, \frac{4}{5}$ and ...
2
votes
1answer
33 views
Shape of a Kite String
A kite is flown at some angle and elevation. Without wind, the string takes the shape of a catenary curve. But the wind is crucial.
Is there a name for the shape of a kite string -- is it some ...
2
votes
1answer
35 views
Trouble understanding a part of the book Cities and Complexity
I am writing my bachelor thesis or whatever it is called about modelling of city growth, using a book called Cities and Complexity by Michael Batty. It is not that mathematical, to be honest, it has ...
2
votes
1answer
53 views
Could you help me to find a model for this curve?
I am very bad in mathematics and I'm not able to find by myself the model corresponding to this kind of curve.
I wish to have a quick growth at the beginning, then it should increase slowly for a ...
2
votes
2answers
244 views
Sound Triangulation
Hello guys I would appreciate it if someone could help me out here. I have been asked to create a simple visualization for an assignment. I am required to triangulate the source of a artillery cannon ...
