Questions tagged [mathematical-modeling]

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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Modeling scoring functions

I'm looking for general direction on topics to explore for this problem. I think I'm not searching for the right statistical concepts and therefore coming up empty handed. Also, I'm a computer ...
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How to determine number of turns needed to fill spool with known length of wire?

I've been scratching my head on this problem for a while, even bounced it off of some friends who are way better at math than myself with no luck, so I'm hoping someone here can help. I'm trying to ...
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Proof for positivity of solutions of an ODE system

I have an ODE system that takes a mathematical model describing the dynamics between HCV and the immune system. My question is about the proof that the solution of the ODE system is positive if the ...
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What does it mean to use mathematical models to understand qualitative vs quantitative behavior? [closed]

I have been running into this recently, and I want to confirm I have the right idea. For example, Neff et al. (2023) state, "Although we study a mathematically very simple box model in order to ...
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prove existence, uniqueness and non-negativity of the system of ODE

I have an epidemic model with 9 compartments, where all epidemic parameters are non-negative. \begin{equation} \label{model} \begin{split} &\frac{dS}{dt} = \beta N -\delta S-\phi \frac {SI}{...
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For an epidemic model, I got a negative 'basic reproduction number' obtained by next-generation matrix method. How this can be corrected?

I have a system of 6 ODE representing infected subpopulations of a model of epidemic $$ \begin{cases} \dfrac{de}{dt} =\dfrac{dy_2}{dt} = \phi y_1(y_4+y_5)-p_2 y_2\\ \dfrac{dq^*}{dt} =\dfrac{dy_3}{dt} ...
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Latent Change Score Modelling

I am currently undertaking a research project that aims to assess the effectiveness of an intervention program. However, I am encountering difficulties in locating suitable resources for my study. ...
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Non-Dimensionalizing a Traffic Flow PDE for Physics Informed Neural Network Issue

I'm working on analyzing a traffic flow model described by the following partial differential equation (PDE): $V_{\max} \left(1 - \frac{2\rho}{\rho_{\max}}\right) \frac{\partial \rho}{\partial x} + \...
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Math-Bio: Reference/Example needed for a reaction-diffusion system coupled to a conservation law

DISCLAIMER: I also made a post about it in biology stack exchange but I'm not sure whether that forum is the most appropriate. Hence I thought to share my question here as well. While looking at this ...
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Find the best embeddding for this gene/lipid graph

I want to find a 'nice' drawing of the lipids and genes in my database. Lipids belong to one one of several classes, while genes belong to one of several regions. Each gene/lipid pair has an ...
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Permutations of pairs with no pair ending up together.

I'm working on a general version of the problem linked here. To quote the original: Suppose 𝑛=10 married couples are invited to a bridge party. Bridge partners are chosen at random, without regard ...
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1 answer
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How to solve an ODE where the rate is directly proportional to two amounts?

Two chemicals in solution react together to form a compound: one unit of compound is formed from $a$ units of chemical $A$ and $b$ units of chemical $B$, with $a + b = 1$. Assume the concentration ...
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ANFIS Model, tips about improving performance [closed]

I have a question regarding about improving the performance of an ANFIS (adaptive neuro Fuzzy inference system) model. In MATLAB, I have been training a model with 5 inputs, with 816 data point for ...
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Cylindrical Water Reservoir Model and System Behavior [closed]

System Description We're looking for the mathematical model of a cylindrical water reservoir. The reservoir is characterized by three variables: The inlet flow, $Q_e(t)$, at time t The outlet flow, ...
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Find the DE for C

A motorboat starts at the origin and moves in the direction of the positive x-axis, pulling a waterskier along a curve C called a tractrix. See the figure below. The waterskier, initially located on ...
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When fitting a polynomial to data points, how to determine the reasonable degree to use?

I have wondered the following: Suppose that there is a set of data points $(x_i,y_i)$. Then I would like to know if it is more reasonable to assume if there is a polynomial relation of degree $m$ ...
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In search of a model to describe worm behaviour

For my bachelors thesis I am working with tubifex worms, and trying to develop a graph theoretical model that can help explain some mechanical and dynamical properties of the worms once they have ...
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What is happening in this 'multi'-sinusoidal function?

Edit: Thanks to those commenting, I fixed my issue and now have reached the below function, as I had aimed. My remaining question would be how I might describe this function. I have taken the function ...
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1 answer
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Stable coordinates in Indoor localization

I am implementing an Indoor localization demo for a project. In that i am using ultra wide band technology to calculate the distances from three different anchors to the moving object, then I use ...
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1 answer
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Derivation of an epitrochoid

I was working on an assignment and had a good idea to have a water jet model an epitrochoid to create a fountain. While the overall idea was approved by my teachers, I was told that I need to show a ...
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Flow Rate based on Pipe Combination without considering Hydraulic Effects

I would really appreciate help on the following: Let us assume that we have n pumps pumping water through n pipes. The pumps are running at a constant power. It doesn't change. The flow rates in the ...
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Navier Stokes Eqns: Boundary Conditions and Pressure Coupling

I have made it a personal long-term goal of mine to numerically solve the 1D transient Navier-Stokes equations for simple incompressible and compressible flow scenarios. I want to do it for a few ...
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Model for population growth and finding the equilibrium solutions

A model for population growth is given by: $$\frac{dN}{dt} = f(N) = r N \left( 1-\frac{N}{K} \right) \left( \frac{N}{U}-1 \right) $$ where $r,\ U,$ and $K$ are positive parameters and $U < K$. (a)...
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Linear programming constraint involving maximum

I am trying to formulate a much bigger problem as an integer linear program, but I am stuck with one particular constraint and am not sure how to formulate it. To put it shortly, the problem deals ...
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Closed-form solution for polynomial recurrence relation.

Is there a closed-form solution for the recurrence relation $x_{t+1} = x_{t}^\alpha + 1$, where $\alpha \in (0,1)$? If it were $x_{t+1} = x_{t}^\alpha$, then taking the Log of both sides would make it ...
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Trying to find a solution to a problem related to the recursive sequence $S_{n}^{k}(a)$

I'm working on a problem related to a recursive sequence, and I'm having some trouble. The sequence $S_{n}^{k}(a)$ is defined as follows: $$S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = ...
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Exploring Markov Models for Determining Car Insurance Costs

Question: I'm currently delving into the realm of Markov models to understand how they can be applied to determine the cost of car insurance. Thus far, I've come across the bonus-malus system and ...
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General state space form for ARIMA(p,d,q)(P,D,Q)

I cannot for the life of me find a resource which gives the general state space form for an ARIMA(p,d,q)(P,D,Q) model. I am reading Time Series Analysis by State Space Methods, by Durbin & Koopman,...
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1 answer
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Mathematical model building with dependent and independent variables

I have been working with data and building models on data. I have developed models in regression using cubic and power series. It works fine for variables with one dependent and one independent ...
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3d fractal helix modeling

I'm trying to build a 3d visual to illustrate a concept. Imagine a circular helix. We could define a cylinder that contains that helix. But now imagine this cylinder takes the helicoïd shape too ! We ...
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Conversion of Abstract Units to Time in Polynomial Modeling of Snowboarder's Motion

I am currently working on a project where I model the motion of a snowboarder using a 4th-degree polynomial. The polynomial is currently defined in abstract units along the x-axis and y-axis. I have ...
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Why does $I + S \xrightarrow{\lambda} 2S$ correspond to $\frac{di}{dt} = -\lambda \langle k \rangle i(t) s(t)$ in this model?

Chapter 10 of Dynamical Processes on Complex Networks contains a model of information spreading. This model is strongly based on a SIR model of epidemics. In this model, there were three kinds of ...
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How to solve a 2D ODE system of the form $\frac{\text{d}\vec{x}}{\text{d}t}=(M-\Delta e^{-\lambda t})\vec{x}+\vec{x}_0$

Let $x=x(t)$ and $y=y(t)$ for $t>0$, with $x(0)=y(0)=1$. \begin{align} \frac{\text{d}x}{\text{d}t} %%% &= %%% -\left( A + \alpha e^{-\lambda t}\right)x + By + x_0\\ %%% \frac{\text{d}y}{\text{d}...
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Problem in performing sensitivity analysis in prey-predator model

I tried to create a general prey predator model to simulate sensitivity analysis using PRCC and Latin hypercube sampling. But I can a bar plots which are empty. What is wrong in my approach? The code: ...
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1 answer
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Statistics on three variables/factors

I have what is probably a simple problem. I'm trying to say something about sells of candy bars. I got data of sells for a population of children. There are three parameters (factors) to this data: ...
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1 answer
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How to use Euler-Lagrange equation to model a linear particle?

Suppose I have a particle with mass $m$ situated on a frictionless line which we can model using $\mathbb{R}$. Suppose that we supply/push the particle $m$ with a force $f(t)$. Let the position of ...
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How can I get the inverse of the transfer function $\Phi$? a question from Page 56 of Lemma 4.5 in the book "ESSENTIALS OF ROBUST CONTROL" [closed]

Let $\Phi(s) = = \gamma^2 I + B^*\left( (sI-A)(C^*C)^{-1}(sI+A^*) \right)^{-1}B + B^*(sI+A^*)^{-1}C^*D - D^*C(sI-A)^{-1}B - D^*D.$ How to compute the inverse of $\Phi$, in Page 56 of Zhou's book "...
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Did R. Buckminster Fuller reference a corollary of a mathematical theorem in his book _Critical Path_?

From Critical Path (1981) , Part 2, Chapter 6 "The World Game" , Richard Buckminster Fuller & Kiyoshi Kuromiya make the Claim : It has been found that within a 100 mile radius a wind is ...
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Reparametrization of rational Bézier curve

I am trying to solve the following task Using rational Bézier curve find the control points and weights of one sixth of a circle $c_1$, such that $$c_1(0)=\{3,0\},c_1(1)=\{\frac{3}{2},\frac{3\sqrt{3}}{...
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Logistic growth decreasing

In a city, the initial population is $$P(0)=70281$$ and 10 years after, $$P(10)=66277$$ I have to find $P(t)$ using the logistic model with max value equal to 80000, which gives the following equation ...
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Is it possible to integrate real numbers into my custom vocabulary within the framework of model theory?

So, I want to use model theory to approach an applied task. Developing the corresponding vocabulary, I realized I needed real numbers within my formulas and, thus, real numbers within the language. I ...
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Degrees of Freedom in PCA

Suppose we are doing PCA over a historic time series of temperatures. The feature for the PCA to be explained is the time when the temperature was observed every hour. Let’s say for the sake of ...
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1 answer
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Pseudo if statements in LP programming

When designing LPs for exams I often run across problems where I would like to input an "if-statement". For example: $5\leq x_b$ if $p_a\geq 10$ I've tried dividing by itself and using floor ...
8 votes
2 answers
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Linearization of a product of two decision variables

I am trying to solve a problem that involves constraints in which products of two decision variables appear. So far, I read that such products can be reformulated to a difference of two quadratic ...
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How do I describe a Quarter Car Suspension with differential equations?

So I've been struggling with describing this Quarter Car Suspension with a differential equation: The main point of difficulty for me is the fact that there are two x variables (x1 and x2) and I'm ...
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What does a negative value for fuel efficiency mean?

I have found an expression that models the fuel efficiency of a truck and the expression is $(7-0.1(x-110)) km/L$. Where $x$ is the speed of the truck. If fuel efficiency, in this case, means the ...
4 votes
1 answer
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Examples of PDEs that found application after discovery

I am a 2nd year mathematics undergraduate in the UK and recently took an introductory elective in partial differential equations. The focus was on solving some classical examples, all arising from an ...
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Partial differential equations that involve an infinite "continuum" of variables: "Each point in space contributes additional degrees of freedom"?

Page 11, Nonlinear Dynamics and Chaos, by Strogatz, says the following: This is the domain of classical applied mathematics and mathematical physics where the linear partial differential equations ...
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Optimization Models and the chicken-and-egg problem

I have started working with optimization models to minimize the total generation costs of electricity in a system. The basic idea in this model is to minimize the overall system cost given a certain ...
2 votes
1 answer
116 views

How to force my differential equations give a bounded solution?

I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation): $$S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...

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