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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is this the right approach for investigating models for population growth models represented by a differential equation?
we are investigating different models for population growth in class.
Model 1 was:
A.Find the constant rate of change for the deer population which gives a solution that fits the two given ...
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The Mean time of Moran Model
I am currently reading an article on Wikipedia on how to derive the mean time before fixation $k_i$ of the Moran Model.
But the derivation of this result rests on this equation:
$$ k_i^j = \delta_{ij} ...
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How do I build a state space model of a given phenomenon?
I am a control engineering student and I would like to create a state space model of a given real world problem. The idea is to consider something that can be experienced in the real world, for ...
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Fish tank population modelling using differential equations
I have been given the basic differential equation for population modelling, which is
$$p'(t)=rp(t)\left(1-\frac{p(t)}k\right)$$. The coefficient $r$ is $10$ and constant $k$ is $500$, so the carrying ...
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SIR infection model rate of mortality and recovery
In a simple SIR model if the average infection duration of the disease and the infection fertility ratio are known what are the rates of mortality and recovery for the infected (I) compartment? I ...
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Finding the design matrix of a multiple linear regression
I'm looking for the design matrix $X$ for the following problem:
Suppose the relationship between the variables $x$ and $y$ is linear:
$y = f(x) =\beta_0f(x_0) + \beta_1f(x_1) + ... + \beta_nf(x_n)$....
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Solving a system of differential equations by substitution
I am being asked to solve a system of differential equations. We are provided with a hint to use substitution to simplify the process, but I don't follow how to use this process to move forward.
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How to calculate the variance-covariance matrix of AX in a linear mixed effects model?
For a linear mixed effect model,
$$\mathbf{Y}_i=\mathbf{BW}_i+\mathbf{A}_i\mathbf{X}_i+\mathbf{E}_i$$
where $\mathbf{Y}_i$ is a $d\times 1$ vector, $\mathbf{X}_i$ is a $p\times 1$ vector, and $\mathbf{...
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Phase plane using Quiver plot [closed]
I have set of NonlinearODE equations with state variables (s,y,ys,yt,fs,z,h).
I need to find 1)Equilibrium points 2)stability 3)draw the quiver plot in z-h plane.
I am not understanding if I need to ...
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Efficient way to model this “peaking filter” function.
CONTEXT
I'm working on an image processing algorithm, which uses pixel lightness values to determine how much the pixel should be darkened.
A small range of lightness values should be "selected&...
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Modelling of virus growth [closed]
We have performed an experiment in the lab on 4 different viruses to check their replication density (population density) over time. Suppose I have 4 viruses (V1, V2, V3, and V4). Let be 100 ...
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How to derive a model for a recursive situation
An animal use 50% of its fat every night, but is able to add $5$ grams to its fat reserve every day. This same animal begin with 3 grams of fat on the first night.
Derive a model for its fat reserve ...
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“All models are wrong, but some are useful”. George E. P. Box Question [closed]
“All models are wrong, but some are useful”. George E. P. Box
What is the meaning and context of this statement? Is it math, physics, the universe, our limited capacity as humans, probability, the ...
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1answer
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Expected Value of MSR in General Linear Model
The setting of my problem is as follows:
Consider the general linear model $Y = X\boldsymbol{\beta} + \epsilon$, where $\beta = \begin{bmatrix} \beta_0 \\ \boldsymbol{\beta_1} \end{bmatrix}$ and $X = \...
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relate one coordinate system reference with another coordinate system
There is one point (P) which has been localized inside a cylindrical shape and this point is represented in cartesian co-ordinate system. So the position of point(P) has been expressed as (x,y,z).I ...
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How to account for Natural variation(statistics)
I have a population and they were all exposed to a specific stimulus after a period of time. There wasn't a control group, I'm using the same population for before(I'm using this as control) and after....
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Trying to model how to space/how many transverse yellow lines to slow down cars on roads
I'm trying to create a model that progressively places those yellow road lines closer together. They give cars the illusion that they are travelling faster than they are when they are going at a ...
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Can you use the logistic equation for modelling population decline?
I was wondering if one could use Verhulst's logistic growth model to simulate decline rather than growth of a population (r < 0). Or should I then use the exponential decay formula instead?
The ...
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Moving Average Processes and Autoregressive Processes in Time Series
Foremost, I'd like to apologise if this is considered a rudimentary question, but I've been struggling to grasp this topic for a while now.
I'm new to Time Series, and I understand that these two ...
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Do repeated roots (and Real Jordan form) for ODE's come up in real world applications of ODE's
An equation like $y^{\prime \prime} + 2 y^{\prime} + y = 0$ has repeated roots: The characteristic polynomial is $r^2 + 2r + 1$ which has repeated roots $(-1,-1)$. Two basic solutions of the ODE are ...
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Improving description of vector systems and aggregation rules for social sciences
I am working on a simple individual based model that aggregates information. I am not a mathematician, but I would like to be as precise as possible with the terminology used to describe the system ...
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Creating a score that fits within the value boundaries of an existing model
First let me say that I'm sorry if I don't explain this in the clearest way. I'm a researcher, and not trained in mathematics. I'm looking at an existing algebraic model and trying to create an ...
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Recommendation for resources about forecasting storm surge levels [closed]
I have a number of concerns related to storm surge forecasting:
A large number of different models have been developed in order to forecast storm surge levels. I've read a lot of papers mentioning ...
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29 views
How to reduce steepness of logistic curve?
I am simulating an assay with the following equation:
$$y = xd(1 - \frac{x}{k})$$
where $d$ is the duplication time and $k$ is the asimptote. However, the simaluted curve (in red) is too steep ...
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2answers
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Production Scheduling over Time
A company has a permit to operate for five seasons. It can manufacture only during the first four seasons and in the fifth period it is only allowed to sell any leftover products. It can manufacture ...
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3answers
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Smoothly connecting PDEs with finite differences
A PDE with non-smooth inhomogeneity
Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$.
I'm numerically solving the inhomogeneous PDE
\begin{...
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Finding transfer function of a temperature model, complexity in algebra.
Here is the equations, I'm trying to find a transfer function relating Ti (internal temperature, output) with Ta (outside temperature, input), I know the differential equation will equate to a first ...
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2answers
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what is the circle with the smallest radius that contains the three points $(0,0)$ , $(1,1)$, and $(2,3)$?
How to write a model that gives us a circle with the smallest radius that contain three points $(0,0)$ , $(1,1)$, and $(2,3)$?
I've tried to model this as following:
if $x, y$ be the location and $r$ ...
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1answer
42 views
How to calculate a pdf of a random variable
I'm reading a paper about kernel estimation and struggling with this part, which constructs the pdf of logZ^2 (Z has Standard normal distribution)
I don't ...
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Damped oscillation; Induction of a sinusoidal solution in 2nd order ODE through intentional creation of an imaginary number
Sorry for the bad title. I didn't know how to call it.
1. Motivation.
To show the motivation of the question more explicitly, I will use an example of damped oscillation which occurs very commonly in ...
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How to constructing a binary regular expression for 8 bits, where “1 1 1” occurs twice in any configuration and the other two bits can be whatever?
I'm trying to make a regex for my language with alphabet $\Sigma = \{"0","1"\}$ that captures any 8 bits where the sequence "1 1 1" occurs twice without overlap. These two sequences can be ...
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2answers
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Symmetric Matrices of Multinormal Distributions
I'm working on some computational mathematics for development in three.js and have been working with out the first parameters of multinormal distributions in ...
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1answer
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Basin of attraction Matlab [closed]
I have set of NonlinearODE equations with state variables (s,y,ys,yt,fs,z,h).
I have used ode45 to solve the equation. I have studied the equilibrium point say E1 by checking at the last point where ...
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1answer
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What is the mathematical equivalent of an axle?
I am talking about axles in machines, on which we can put gears, belts and other devices that help us convert the motion.
So this structure has to have (at least) the following properties:
distance - ...
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2answers
55 views
What maths concepts should you know to model infectious diseases?
I'm a high school student and for my math research paper, I've to model and predict spread of an infectious disease (e.g. covid) in a country of my choice. Which math concepts should I know to do this ...
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Bond cashflow grouping and matching
120 monthly liability cashflows need to be met by asset cashflows: 100, 98, 97, 95, ..., 3, 2 (a fairly random diminishing series). I hope to use an optimiser to select 80~100 bonds from a universe of ...
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Using generalized logistic curve to create a mathematical model from data.
The first row is time and the second row is height of a plant. We need to use generalized logistic curve to model the behavior of the plant. The equation of the logistic curve is :
$$N = \frac{N_*}{1+(...
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Confusion about parameterizing birth rate?
So I'm starting out learning how to parameterize ODE models with demographic data. I'm having a slight problem with parameterizing birth rates due to some various ways I have seen?
For simplicity sake,...
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1answer
40 views
The pressure of a fluid as the change of internal energy.
I never do fluid mechanics or physics so go easy on me!
I am trying to understand the description of pressure given by Villani in his book Optimal Transport Old and New. This is the discussion right ...
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1answer
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When do I decide a logistic model with no roots equals 0?
I have a logistic model function:
$$ y=\frac{424.92}{1+0.37027e^{0.000715x}}$$
This exponential model has an asymptote at $y=0$, and hence doesn't ever meet the $x$-axis. However, I need to discern ...
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How to Solve Non-Autonomous System of Non-Linear Ordinary Differential Equations?
Given the Non-Autonomous System of Non-Linear Ordinary Differential Equations,
$$\frac{dx}{dt} = -\alpha {t}{ e^{-t^2}}-{k_1}{x(t)}+\gamma {k_2}{y(t)}$$
$$\frac{dy}{dt} = \beta {t}{ e^{-t^2}}-{(\mu+...
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How can one decide between structure or coincidence regarding the fine structure constant?
This question originates the following observations about the fine structure constant, $\alpha$.
Measurements of $\alpha^{-1}$ yield values slightly smaller than $137.036$, and a good rational fit for ...
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2answers
232 views
Impose PDE itself as Boundary Condition?
Consider, for example, the elliptic PDE $u_{x}+u_y+u_{xx}+u_{yy}=0$ for $(x,y)\in[0,\infty)^2$.
Solution methods often require me to impose boundary conditions. Often, these arise naturally from ...
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Is there a continuous transition between deterministic and stochastic?
This is perhaps a more philosophical question. Hence, I'm looking for philosophical answers but more concrete ones or maybe directions to books or articles about it would also be of help.
During my ...
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2answers
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Probabilistic modelling
This question is about how to make sense of a probabilistic model I'm reading about:
You have three random variable $A$, $B$ and $C$ that are all real-valued. You want to model an input-output ...
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The function field model of the integers and “spillover”
This question was originally posed on MathOverflow. Even with a bounty, it got only a couple of comments and no proper answers, so I thought I'd try it here.
In a classic blog post, Tao discusses the ...
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Cubic spline for a spiral function
I would like to plot the underneath spiral graph with a cubic spline using the following points depicted within the graph.
(4,8), (7,10), (10,6), (10,10), (10, 16), (11, 2),(11, 14), (12, 8), (16, 8), ...
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1answer
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Mathematical modelling of a function
I wonder if how you represent this scenario.
It has been determined that 2400 clocks would be sold if the wholesale price is 3.20 dollars. However, if the price were reduced to 2.4 dollars, there ...
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Is there a function that is both exponential or linear at one end and then logarithmic at the other (joining two types of functions generally)
This is more a general big picture conceptual question. I observe in life some phenomena display exponential growth. Wealth accumulation or growth of a new business or success as an artist can be ...
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How to derive a Boltzmann-like equation for an opinion formation model?
I'm studying an opinion formation model as follows. Opinions are given by $w \in \mathcal{I} = [-1,1]$. If an individual with opinion $w$ meets an individual with opinion $v$, the interaction is ...
