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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How to find optimal policy to maximise a desired resource over time?
Each week, we can decide to do a hunt or skip it. Starting at week $1$, the available reward from that hunt is a yellow shard, then a blue shard, then a red shard, repeating in that order (so e.g. ...
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Stick gaming questions
One person would like to go from the origin to point (2000, 2000) using stick with 1 unit length. The stick needs to be place on safe endpoints (e.g., x and y axes or other safe sticks). Please give a ...
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Find a function $\rho:[0,1]\to \mathbb{R}$ such that a total mass of any portion $(a,b]\subset [0,1]$ of the wire is given by $M(a,b)=\int_a^b d\rho$
Consider a wire of length $1$cm with constant density $\mu$(g/cm), containing two point-like masses $m_1$ and $m_2$ grams located at $s_1=\frac{1}{3}$ and $s_2=\frac{2}{3}$, assuming the wire is ...
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Structure to model multiple hierarchies on the same entities
Say I have a set of entities and multiple hierarchies based on them, each hierarchy modeling a different type of relationship. What would be a useful structure to model this scenario?
For instance, ...
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What's a good predictive modeling approach for a very small dataset (50-100 rows) [closed]
I want to model the performance of specific athletes, but instead of taking the big data approach and letting the model generalize (something I've already done), I want to create a high bias model ...
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Mathematical Modelling reference request
I would like to self-study MATHEMATICAL MODELLING based on the following topics:
Monte Carlo simulation modelling: simulating deterministic behavior (area under a curve, volume under a
surface), ...
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How do I find the function and derivative of an unknown curve?
I have $x$ and $y$ values to plot the curve and I need to find a tangent line of slope 1 that intersects the curve (and the point at which it intersects). I was trying to do polynomial and exponential ...
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How to solve $y''(x) + \alpha = 0$ using the Finite volume method for one-dimensional steady state diffusion?
I am sharing something that I am seeing for the first time.
I must solve the following boundary value problem
\begin{align}
\begin{cases}
&y''(x)+ \alpha = 0, \ x \in I = [0,1], \ \alpha:= \text{...
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Are there probability distributions completely characterized by their nth statistical moments for n>2?
The normal/Gaussian distribution is fully characterized by its first and second statistical moments, $\mu$ and $\sigma$, so we can write the distribution only as a function of these two parameters; $F(...
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Linearization of a hydro power plant (Control Theory)
I am simulating the lagoon of a hydro power plant. To begin with I will simulate it as a water tank problem.
The main objective of this project is to control the height of the lagoon by using a radial ...
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Computing gravity on a swinging object
I'm working on a software in which there is a simple model for a swinging object attached to a pivot (e.g. a stick that's nailed to the wall at one of it's extremities).
The object supports one basic ...
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If $f(x)\times n=0$ for every $x$ in some plane then $\text{curl}f(x)\cdot n=0$ ($f$ is a vector field and $n$ a vector orthogonal to the plane) [duplicate]
Let $f$ be a vector field defined in $\mathbb{R}^3$, $\Sigma$ be a plane and $n$ be a vector orthogonal to $\Sigma$. I have to prove the following (question from Mathematical Modeling course):
If $f(x)...
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How to make this mathematical modeling linear?
The problem I'm dealing with has the objective to minimize the amount of people we need to hire. We have 14 contracts with 14 companies, each of which have shared with us the amount of workers that ...
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How to find the basic reproduction number?
I have just begun studying stability of dynamic systems. I came across this model system where I have no information about the coefficients, but I assume that they are strictly positive:
\begin{...
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Why can certain variables be left out of an Objective function in Mathematical Programming?
I have an objective function that has the form:
\begin{equation}
\min \qquad \sum_{i} \omega_{i}x_{i}q_{i}
\end{equation}
Where $\omega_{i}$ is the weight of variable $i$, $x$ is the decision ...
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State Space Representation of a discharge through a gate
I have one equation that describes the discharge through a radial gate placed in a hydro power plant, given by
$$
Q = \frac{dV(t)}{dt} = C_g \cdot D \cdot L \cdot \sqrt{2\cdot g \cdot H_g(t)}
$$
where
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Modelling - Looking for a function that is similar to a combination of a quadratic and exponential
I want to model the following dataset in this plot.plot.
The dataset is posted here.
The left half looks like a parabola and the right half looks like an exponential. If I use a piecewise function to ...
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Help for generic logisitic population growth with variable harvesting problem
I am attempting to work through a common problem entailing variable harvesting of a population growing logistically. I am struggling to determine all the properties for stable harvesting. I will ...
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Find a value $\alpha$ so that $\displaystyle\lim_{n\rightarrow}X_ne^{-\alpha n} = V$
We model a species population on an annual basis, where the number of juveniles in year $n$
is $J_n$ and the number of adults is $A_n$. Our model has the following form
\begin{align*}
&J_{n+1} = ...
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find the max algebraic expression
In one paper, the author concluded the following:
How was the inequality of $C_L$ (highlighted) derived?
The author also mentioned above that
$C_L>{((s - m e_{rm} - b)^2 k \beta)}/{(8 k - \beta m^...
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Did Pasch and Peano use models for their axiomatic systems in geometry?
I'm trying to track down the introduction of models in order to get a better understanding of the modern axiomatic method in general. So far it seems to be Hilbert that first created and made use of a ...
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Understanding a final size distribution histogram
I would appreciate help with understanding how a histogram of a final size distribution for a SIR model is created and how to interpret such.
For example: Consider a SIR epidemic with parameters $\...
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Logistic ODE with negative initial condition
For the ODE with negative initial condition, say for example
\begin{align*}
\dot{x}= rx\left(1-\dfrac{x}{K}\right),x(0)=-K
\end{align*}
From the phase portrait, $x(t)$ should converge to $-\infty$ as ...
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How to find the equilibrium point of this system?
I've been looking for the equilibrium point with the help of SymPy (Python Library). Even one of the equations cannot be found with SymPy because the solution is very long. I'm curious about the ...
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Find the effective reproduction number for a Markovian model
I would appreciate help with what formula I should use to calculate the effective reproduction number for a general stochastic SIR model (Markovian model).
I have a general stochastic SIR model (...
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Solve the graph problem with a specific limitation.
Assume that there are a set of roads that connect the cities between A and B. We model this as a graph of $n$ nodes (cities) connected by $m$ edges (roads). Each edge weight corresponds to the time it ...
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I want to find the most efficient way to build/structure a city in SimCity BuildIt. Where should I even start mathematically to do so?
So for those unaware, Simcity BuildIt is a city-building game where you are the mayor and aim to build and manage the perfect city.
Some of the game mechanics that I wish to model are explained below.....
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How many clients should I trust?
I hope this is the right place to put this problem. Otherwise, I beg you to teach me where to put it
Supposing a laptop company organizes a month-event in which it shows and sells its new 15 laptops (...
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What is the probability that a marble from the urn has been picked up by exactly $n$ people?
An urn starts with $m$ marbles and is then approached by $p$ people, each of which picks up $k$ marbles, discarding one and returning the rest to the urn. The urn now has $m - p$ marbles remaining. ...
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finance mathematics [closed]
A factory contributes £1 million into a small city. It is estimated that 75% of that money is respent back into the community. Economists assume that the money is re-spent again and again at a rate of ...
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The coordinates of vertices in sierpinski triangle [closed]
For a given sierpinski triangle (https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle)
what are the coordinates of vertices of triangles inside the main triangle?
can one have closed formula for ...
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Biological meaning of "eigenvalue" in DDE-based population model
For example, consider a Verhulst model with delay
$$ \boxed{\dot N(t) = r N(t) \left( 1 - \frac{N(t-T)}{K} \right)} $$
where $r$ gives the reproduction rate and $K$ means the carrying capacity of the ...
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Why R-sq Value is Not a Measure for How Good a Model Is
My statistics teacher told me that R-sq measures how your model fits the data, not how good your model is.
I am confused. RMSE is a measure of how good your model is. And R-sq value is
$$1 - \frac{...
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Solution of $R(t)$ in SIR model
Consider the SIR model,
$$
\begin{align}
\frac{dS}{dt}&= -rSI \tag 1\\
\frac{dI}{dt}&= rSI-\gamma I\tag 2\\
\frac{dR}{dt}&= \gamma I\tag 3
\end{align}
$$
from $(1)$ and $(3)$ we get, $\...
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How to solve SDE: $ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$
Problem:
I would like to find a solution to the following SDE:
$$ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$$
Calculations:
by Ito's lemma: $$df(t,X_t) = \Bigg(f_t(t,...
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Is it reasonable to multiply the output of a model by a constant in an attempt to calibrate and better match real life data?
I'm trying to model the volume change of a lake. I've developed a rough mathematical model that somewhat matches the actual volume change of the lake. However, my model doesn't quite match exactly. ...
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How would you calculate the lifetime risk of a specific cause of death for people living today, assuming annual rates are constant?
It's been about seven years since I took differential equations, and I'm trying to wrap my head around something. This is just for my curiosity, and it's absurdly oversimplified, but I just want to ...
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Amplitude and frequency of undamped forced oscillations
I am solving an atypical problem. The problem is of forced undamped oscillations:
$$x''(t) + 9 x(t) = 5\sin(2t) − 6 \cos(4t)$$ with initial conditions $$x(0) = 0$$ $$x'(0) = 4$$
The solution is clear ...
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Fences question verification
For a general system of equations $x’=f(x,y)$ , $y’=g(x,y)$ a $C^1$ curve $γ$ Is called a fence if for some continuous normal $n$ to the curve we have $(f,g) \cdot n \geq 0$ everywhere along the curve....
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Modeling Latency Based Decay
I'm hoping this should be simple. I remember reading something about how this can be done years ago, but I've forgotten exactly how now.
Assume you have some group $A$ which changes in time such that $...
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Is it possible to have the growth rate for a logistic function be fixed relatively to an external variable?
Problem description:
I am currently modelling the behaviour of ants in finding a new nest. With in the model I am changing the attractiveness of a potential nest (scalar value) depending on the ...
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Leaky Buckets: Volume in a system of 2+ buckets that can be empty
What is the volume of water in each bucket in a system of leaky buckets with the following conditions:
There are $N$ buckets with $W_i(t)$ volume of water.
A bucket has infinite capacity, but a ...
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how to calculate the reproduction number $R_0$ in stochastic model
There are several specific ways to calculate the reproduction number $R_0$ for the deterministic model I extracted it properly, but in the stochastic model I found a difference in how to extract it.
...
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What are the odds that De Vries' formula for the fine structure constant $\alpha$ is a numerical coincidence?
The dimensionless fine structure constant $\alpha \approx \frac1{137}$ has intrigued physicists for over a century. Whilst not currently a majority view, there is a school of thought that considers ...
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How do you compute the result of an equation with discrete values
A bit of background - I had a phonecall appointment with the drs and it wasn't until a few hours after the appointment time that I received the call. I decided to try and model how much extra time I ...
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"American Monte-Carlo"
I heard this name in a discussion on market modeling. As I understood from the context, this is an optimization of the classical Monte Carlo method.
Unfortunately, Google does not know such a name. ...
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How to calculate a value based on similarity?
Assume we have a program with different instructions. Due to some limitations in the field, it is not possible to test all the instructions. Instead, assume we have tested 4 instructions and ...
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Mathematically Optimal Way To Pick The Best Team?
I am the president of the Science Bowl Club at my school and am trying to find a mathematically optimal way to pick the best 5 team members for the state competition.
For those who do not know what ...
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Finding a single function $y=f(x)$ (with the minimum possible parameters) satisfying $7$ properties
For one experiment with four different samples, the data points of each sample showing a different shape of the curve:
Sample #1:
$y$ increases as $x$ increases.
the curve has no inflection points.
...
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The conditions of optimization task
Could you please help me to solve the task using optimization method?
The factory produces three types of glue. And four types of chemicals are used for its production: starch, gelatin, alum and chalk....
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