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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Implications of Choosing Incorrect Variables in Buckingham Pi

One form of the Buckingham Pi Theorem says that for $n$ variables with $k$ dimensions, the number of dimensionless quantities (or pi groups) is $n-k$. This theorem is often applied when the ...
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Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
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Physics related initial value problem (horizontal spring mass system)

Consider the horizontal spring-mass system where the spring-force is the only force acting on the mass. Suppose that a mass is initially at $x=x_0$ with an initial velocity $v_0$. Show that the ...
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How to solve the model parameters?

Find a region of the world in which arms race is occurring or has occurred. Find data about arms spending. Estimate the model parameters. I found some data on Zimbabwe's military expenditure and ...
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Apprehension in Feller's solution of a model of breakage of spores

This one is a solved example from Feller. Spores of the Fungus are produced in chains of eight.The chain may break into several parts into projectiles containing 1 to 8 pores. Find the ...
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37 views

How to quantify a relationship so that a small value increases the result and a larger value decrease the result?

The title pretty much says it all but let me break it down. I thoroughly appreciate any help. I'm trying to quantify a relationship between the difference in peaks and troughs of a wave function and ...
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Turning a word equation into a differential equation (heat transfer problem)

Question: Heat is flowing in a wall with constant cross sectional area, inside which heat is generated at a constant rate of Q $ W m^{-3}$. Write a word equation and show that the equilibrium ...
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Differential Equation Model of Motion problem.

I'm having trouble with this question. "A body is released from rest and travels the last half of the total distance fallen in precisely one second. How far did the body fall and how long did it take ...
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52 views

Solving for projection of vector intersection on earth surface

I need advice regarding a geometry problem. I illustrate this geometry in the figure below: Background of question: There are two points (represented by blue and orange stars) with associated ...
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84 views

An introduction to Lane-Riesenfeld algorithms

I am looking for a good introduction to Lane-Riesenfeld algorithms, which are a family of subdivision methods for generating uniform $B$-splines. Any suggestions? (The more "basic" the exposition the ...
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34 views

Proof that any curve in two dimensions has a modelling differential equation.

I was wondering if anyone knew of a proof stating that any smooth, continuous line drawn in two dimensions must have a differential equation that models it. Best, James.
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Matlab - Storing information in vectors and modelling a system of equations

I'm working on solving a system of differential equations and I'm having trouble setting up a vector of parameters. This is the code for one of my model files: function ...
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constrained differential in a chemical reaction model

I have a system of reversible elementary chemical reactions for a catalytic cycle. Each reaction has a forward reaction rate constant $k_{f}$ and a equilibrium reaction constant $k_{eq}$ (we could ...
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Model equation for customer numbers.

I am trying to formulate an equation to determine the number of customers signed up to a subscription as a function of time. It's been 10 years since I studied maths, so bear with me! The assumptions ...
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Problems with Exchange Procedure in Remez Algorithm

So first off: *** This code is not being used in production software. It is a personal project of mine, trying to understand approximation theory and advanced curve fitting. In other words, I'm ...
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Finding tangent to curve fitting sampled data.

Let $(x_i, y_i)$ be a finite sample of points. I want to find the slope of the tangent at $x$ to a curve "best fit" to the given data. In my problem area, (see context below), the simplest of ...
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Is there a rigorous approach to solving this, instead of trial and error?

This is a question from a 4th grade math book. The solution in the teacher's guide suggests that one might not be able to model the question easily and the best approach may be to solve by trial and ...
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149 views

Solving for a 3D point in a 5D graph given 3 pairs of 2D points.

I am attempting to solve the values $C$, $D$, and $S$, given three pairs of $[M,R]$. $$R = \frac {M}{C - MDC + DC\left(MS\right)^2}$$ I have been able to solve for a related equation (or rather, ...
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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. An infinite cylindrical rod (radius $a$) is initially at temperature ...
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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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19 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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53 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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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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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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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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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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35 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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46 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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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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237 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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118 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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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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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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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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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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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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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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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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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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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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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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316 views

exact solution to lotka-volterra equations [closed]

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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142 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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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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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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45 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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49 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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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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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, ...