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.

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Rescaling of a problem

I'm having a tough time trying to figure this question out mainly because I haven't any formal training in "scaling" of a problem. Question: An infinite cylindrical rod (radius a) is initially at ...
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Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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10 views

Modelling using Shapley Value

I have a Shapley Value related problem that I am unable to solve. Instead of using integers, I have used a percentage (a conversion rate). I have attached a spreadsheet for you to have a look at. ...
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35 views

Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
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1answer
12 views

Definition of a function whose codomain is set of probability measure over cartesian product with dependency between sets in the product

I am thinking about the following function: $$ p : A \to \Delta \big( F(x, y(t) ) \times T \big) ,$$ where $t \in T$ denotes continuous time, and $\Delta (X)$ denotes the set of all probability ...
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27 views

A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
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19 views

Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
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11 views

What is the notation for 'asymptoticly approximate mapping'? If any…

I've learned about the notation 'maps to': $\mapsto$ And also asymptotic approximation: $\simeq$ Is it valid to suggest the notion of 'asymptoticly approximate mapping'? If so, what is ...
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1answer
21 views

Solving a pair of ODEs

I'm trying to solve a pair of ODEs for which I've obtained a solution. However, my problem is that my answer is slightly different from mathematica's answer. $$ \frac{dA}{dt} = \theta - (\mu + ...
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24 views

Population balance model

I have some experimental data and I need to make a population balance model to compare the experimental results with. The experimental results are from the bubble size distribution in a bioreactor. I ...
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10 views

Statistical/ML models when observations have different amounts of input

Let's say we're predicting an employee's performance review score for the following year based on his/her performance review scores from each previous year of their employment. We might have these ...
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202 views

The theory in probability

Consider a real-life experiment (perhaps written as a problem in a textbook): A coin is continually tossed until two consecutive heads are observed. Assume that the results of the tosses are mutually ...
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111 views

Assignment of initial probability values

Suppose a coin is tossed until a head is observed for the first time. It is given that the coin lands heads with probability $p$ and tails with probability $1-p$. Based on only this information, can ...
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23 views

How do you formulate a reaction diffusion model with 3D volume and surface compartments simultaneously?

Suppose we have a two compartment reaction diffusion model, for chemical species $\psi$ and $\phi$. Suppose $\psi(\vec{x},t)$ and $\phi(\vec{x},t)$ exist in two 3D compartments $\Omega_a$ and ...
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16 views

Modelling a solute in reactor by concentration or total amount?

I'm trying to obtain a model of the concentration of a solute, S (g/L) in a reactor with the variable volume V (L). A solution of S is continuously added to the reactor with the flow F (L/min), since ...
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30 views

How to analytically describe the number of peaks on a roughness surface?

I have a question about how to model the number of peaks over a length $l$ on a surface, which is defined as: Let $f$ be the profile of the roughness of a surface. Let $c_1$ and $c_2$ be constant ...
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2answers
17 views

Approximation of a negative exponential model?

I am trying to get an approximation of this model, it is a negative exponential model introduced by Olson in 1963 "Energy storage and the balance of producers and decomposers in ecological systems". ...
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16 views

How to make sense of this contour graph?

Recently, I have been studying the paper "Estimating the basic reproductive ratio for the Ebola outbreak in Liberia and Sierra Leone", published in Infectious Diseases of Poverty. I came across the ...
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29 views

Modeling of Multivariate Function of Dependent Variables

In multi-variable calculus, if I write $f(x,y,z)$, it is assumed that $x,y,z$ are independent. I'd like to model a quantity $F$, that is a function of 3 related quantities, $x,y,z$. In fact, $xy=z$. ...
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32 views

Building a dynamical system

Suppose I have a 4 dimensional system with 4 fixed points: $Q_1 = \left(p_1,0,0,0 \right)$, $Q_2 = \left(0,p_2,0,0 \right)$, $Q_3= \left(0,0,p_3,0 \right)$, and $Q_4 = \left(0,0,0,p_4 \right)$. ...
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16 views

Modeling smoke cloud as expanding Gaussian / ellipse

I am making a simplified model of smoke coming from a train's smokestack. You can imagine that if you want an accurate model you have to think in 3D and use computational fluid dynamics and stochastic ...
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10 views

What are the mathematical descriptions for a spatially dependant concentration to be 'well mixed'?

Suppose we have a concentration $c(x)$, suppose for instance it is a chemical species, in the region $[0,L]\subset\mathbb{R}$. What are the different ways you could mathematically describe the 'well ...
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1answer
21 views

linear programming : Absolute value in constraint in mathematical model

I have a model have an constraint with evaluation of absolute value , a example can be: function objective : $\max \sum(x_i)$ statement: $x_i\geq |(y_i-t_i)|$ for all $i$ but value absolute ...
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27 views

Population models - 13 months in a year

I'm currently studying mathematics at university and have come across a question about finding the population of rabbits in a colony, where the growth rate is given in terms of per month. However ...
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46 views

exact solution to lotka-volterra equations

I am looking for exact or perturbative solution realistic lotka-volterra (the one with logistic term in one of the equations) equations in population dynamics. Any reference where they have done it ...
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1answer
68 views

Heat equation — Modelling a real-life situation

I have read through a lot of books and lecture notes that cover the heat equation and I am still not sure how I would model the easiest real world situations. For example, take a rod at constant ...
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203 views

Explaining Mathematical Modelling to a nonmathematician

Due to the interdisciplinary nature of my project, I find myself collaborating a lot with nonmathematicians especially biologists, medical doctors, etc. I work mostly on mathematical models as applied ...
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17 views

Markov-Chain Monte-Carlo: Are transformations on the inputs valid?

The problem: I am trying to solve a high dimensional (up to ~50) class of data fitting & modelling problems. The user specifies the problem, so I would like to make the configuration as easy as ...
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3answers
29 views

How can mathematical models be applied to image analysis

I'm quite interested in how mathematical models can be used in analysing images. For example, I'm aware that mixed effect models can be using in image analysis but I was just wondering if there are ...
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33 views

How to model compartment chemical concentration with permeation through a membrane

Suppose we have one chemical species $V$ and two compartments in which $V$ can be, $A$ and $B$, with volumes $\Omega_A$ and $\Omega_B$ respectively, where $\Omega_A < \Omega_B$. Compartments $A$ ...
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24 views

Given a set of arbitrary data, is it possible to model this data using differential functions.

Problem At the moment, I have a problem with seven variables: $S, A_1, A_2, R_1, R_2, P_0, P_1 $ and $P_2$. Each of these variables draws a smooth line through time. My question is, is there any ...
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83 views

Math behind No Man's Sky, or Math of Minecraft in Space [closed]

I recently received a question from one of my students which is a bit outside my life experience. However, I expect this may be of interest to many: I was reading up on a new video game that's ...
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Understanding compound distribution and plotting the mixed distribution graph

If $N$ takes the values $0, 1$ and $2$ with probabilities $½, ¼ $ and $¼ $ respectively, and the $X_i$ ’s have a $U(0,10)$ distribution, draw a sketch of the frequency distribution of $S$. $N$ is the ...
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How to generate a function with multiple variables to fit experimental data?

I am researching methods to increase the accuracy of an algorithm that is currently used to analyse radiation patterns as they hit our sensor. (For the non-physicists, we will mainly see Alphas, ...
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3answers
38 views

How to re-write the following equation?

I am a molecular biologist and reading a book on mathematical modeling. In this book I encountered the following algebraic conversion and could not figure out how the conversion is performed. Please ...
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29 views

Survival analysis with a parametric model for recurrent events and time-dependent covariates

My goal is to model waiting times between recurrent events with time-dependent covariats with parametric models (Poisson, Weibull, log-normal etc.). This is not an issue time-dependent covaraits as ...
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65 views

Integrating a First Order Differential Equation (The West Equation)

I am currently doing a project about Growth and have found this really interesting Math Model by Dr. Geoffrey West et al in 2001 while researching. The paper can be found at this link. I was ...
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1answer
34 views

Spring-mass System Phase Plane

If a spring-mass system has a friction force proportional to the cube of velocity, then $$m\frac{d^2x}{dt^2} + a\left(\frac{dx}{dt}\right)^3 + kx = 0$$ (a) Derive a first-order differential ...
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39 views

How to transform one polynomial curve into another

I have a problem: I have two polynomial splines: For red curve I have used Rational fraction function with p=3/q=2 and for blue curve I have also used rational fraction function with p=1/q=3 It ...
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58 views

Differential equations, chemical reactions

A chemical substance A changes into substance B at rate $\alpha$ times the amount of A present. Substance B changes into C at rate $\beta$ times the amount of B present. If initially only substance A ...
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2answers
24 views

ordinary differential equations-morphogen gradient

I am reading a paper by Merkin and Sleeman (2005) Find the approximation solution of $(u')^2=\frac{2}{k}(u-\frac{1}{k}\ln(1+ku)); ~~u(0)=1$ for $k$ sufficiently small. they gave the following ...
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64 views

Can car traffic be managed by mathematical formula? [closed]

How car traffic is managed? Is it managed by mathematical algorithm? Or by human(operator)? If it's by operator, can it be managed mathematically? Or is it by physics? By what theories/formula? ...
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1answer
19 views

System of separable diff. eqns, explicit solution and curves, Lotka-Volterra model

In the book on p.68 is a system of differential equations for a Predator-Prey model (Lotka-Volterra) given as: $$ \dot x=x(\alpha-c\gamma) \\ \dot y=y(\gamma x -\delta) $$ On the next page, it is ...
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1answer
76 views

Modeling, Measuring, and Maximizing “Mixedness”

Possible key-terms: combinatorial optimization techniques; simulated annealing; genetic algorithms; Kirkman's schoolgirl problem; Steiner triple systems; orthogonal regrouping. Background: My class ...
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29 views

Choose Scaling for t

My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my ...
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1answer
36 views

linear or bilinear interpolation

I want to know how to use linear and bilinear interpolation in 2D. Specifically the pairs $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ are given in a quadrilateral. In this case how to ...
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1answer
32 views

Rain forecast model and least squares

Good evening everyone, I'm trying to create a rain forecast model, I have about 720 data, which correspond to monthly rainfall in 60 years during the 12 months of the year. I have a matrix ...
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1answer
28 views

Probability of agreeing to do some work depending on the payment

I am looking for several options of modeling the probability of people agreeing to do some work depending on the price/payment. The payment can only range between $p_1$ and $p_2$, $(p_1 < p_2)$. I ...
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16 views

Modeling reasons for minimizing the distance between parameters

In system identification, when new parameters are calculated, is there any reason for the parameters to be close to the original parameters ?
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19 views

reformulating an integer quadratic problem into a linear integer problem

I am trying to solve an optimization problem, in order to find an optimal runtime schedule for a machine. It involves one boolean variable $x_{t} \in \mathbb{\{0,1\}}$, that describes whetever the ...