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.
2,419 questions
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Exercise in Acoustic Doppler Effect
I am looking for some guidance on the second part of a geometry type problem which I have given working on and described the next parts below (likely with an error). I have given multiple attempts but ...
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I'm working on finding the constant parameter of the Lotka Volterra differential equations. I am using the LINEST function, but my alpha is negative [closed]
My alpha, which stands for the growth factor of the pray without predation, is negative. Is this a valid value, since it indicates that without predation the population will decrease? If not, how can ...
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Parametric leaf profile?
I am trying to look for a nice way of parametrizing a leaf (botanical) profile. First a regular leaf, but I also would like to do an oak. The trivial answer is to use a B-Spline, but for multiple ...
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Is multicollinearity a problem for optimizing the response with respect to inputs?
I have come to learn that while multicollinearity affects the model "stability" and ability to examine individual affects, but does reduce overall model predictive power.
I am interested in ...
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About Fuzzy Logic [closed]
I am reading fuzzy logic and completed basics like fuzzy sets , fuzzy arithmetic , operations and other things. I want to study advanced topics like Interval type 2 Fuzzy sets, ordered fuzzy numbers ...
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4th Order Tensor multiplication Rules for Sparse Regression analysis
I am working on a problem which involves working with stress and deformation tensors of the order 4. I have a set of data at different time steps for 20 cases and each element stress is 3x3 matrix, so ...
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Neglecting higher order terms in a PDE: why?
This is quite non-rigorous question since I don't think there is a clear-cut theorem answering it. I am deriving a PDE from a system which I know in the limit should give a heat equation. The ...
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Expected relative rate of relapse vs onset in a heterogeneous population (frailty effects)
Model.
Consider the following model reflecting dynamics of major depressive disorder (MDD).
Let $x,y,z$ be the expected proportions never, active, and past MDD, as a function of age beyond 10-years ...
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Waggle-Dance Equation 🐝
Once a foraging bee finds food, it returns to the hive and communicates the location of the food source to the colony using the elegant waggle dance.
Bees interpret this dance by combining their ...
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Dynamics of Snail Balls
It looks like the standard equation of motion for a rigid body rolling without slipping down an incline of angle $\theta$ is
$$
a \;=\; \frac{g\sin\theta}{1 + I/(mR^2)},
$$
where $m$ is the mass, $R$ ...
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Probability distribution of number of successes in n trials without replacement and accounting for order
I've been having some problems on how to model a variant of the sampling without replacement problem.
The context is as follows: Imagine a game with a $N = 40$-card deck, where $K = 8$ cards are of ...
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The value of biologically impossible bifurcation analysis in ODE systems from mathematical biology [closed]
I am studying a 3D system of ODEs derived from a biological model (tumor growth). All parameters are defined to be positive, as negative values would lack biological meaning (e.g., negative loss rates ...
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Compactness with graph theory (gerrymandering)
"To deter gerrymandering, many state constitutions require legislative districts to be 'compact.' Yet, the law offers few precise definitions"
See also: https://en.wikipedia.org/wiki/...
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How to derive a dynamic model for a switched system?
I have been solving following problem which originates in the electrical engineering but from my point of view it is basically a mathematical problem.
I have a dc-dc converter which operates as an ...
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How is a probability-independent formula achieved in the Binomial Options Pricing Model?
How is a probability-independent formula achieved in the Binomial Options Pricing Model?
Intro______________
I am self-studying financial math and I found these lectures on Google written by Karl ...
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Discrete SIR model
I'm studying the time-discrete SIR model with time-dipendent parameter. Assume $\Delta t = 1$, so the model is:
$$S_{t+1} - S_t = - \beta_t \frac{I_t}{N} S_t $$
$$I_{t+1} - I_t = \beta_t \frac{I_t}{N} ...
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Is this a valid ODE? Which kind of ODE it is? (ODEs with finite extinction times)
Intro
On previous questions where I am trying to understand ODEs that have solutions that stop moving in finite time (example 1, example2), due other users answers (@md2perpe, @RollenS.D'Souza), I ...
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How sensitive is volumetric flow rate to changes in radius in Poiseuille’s law?
Poiseuille’s law describes the volumetric flow rate $Q$ of an incompressible, viscous fluid through a cylindrical pipe as:
$$
Q = \frac{\pi r^4 \Delta P}{8 \mu L}
$$
where:
$r$ is the radius of the ...
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Coin tossing with an alternate coin tossing schedule
Suppose I have 𝑁 coins arranged in a line. I can toss a coin up to
𝜂 times, but I’m only allowed to toss coins that are not adjacent to each other in the same round. Each toss is independent and has ...
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Properties of dynamical systems symmetric under permutation
Im trying to model a process for which I expect a fair deal of limit cycles, however, it also has to be a system of the form
\begin{align}
x'&=f_1(x,y), \\
y'&=f_2(x,y),
\end{align}
where $f_1(...
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When can product of cosine signals be written as sum of cosine signals? [duplicate]
In some applications such as amplitude modulation, one encounters product of cosines of the form:
$$x(t) = \cos(\omega_1 t + \phi_1)\cos(\omega_2 t + \phi_2) $$
And for specific values of $\omega_1, \...
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Characterizing the sum of $m$ inverse-gamma random variables
Let $f, g \in \mathbb{R} \times \mathbb{R}^m \to \mathbb{R}$ be defined as:
\begin{equation*}
f(x, \textbf{t}) := \frac{xm}{\sum_{i=1}^m t_i}, \qquad g(x, \textbf{t}) :=
\frac{x}{m} \sum_{i=1}^m \...
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Speed Estimation [closed]
I have some questions regarding speed estimation.
I want to know the speed of another object. I have the following parameters:
at t0 I know
My speed
Estimated distance of another object
at t1 I know
...
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What is the optimal geometry of a waffle?
A natural question is modeling the optimal geometry of a waffle. Specifically, the layout of chambers and walls to optimize for conflicting goals:
Syrup coverage: how quickly and thoroughly syrup, ...
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Dependence of travelling wave speed in Fisher-KPP on asymptotic initial conditions
I'm currently reading J.D. Murray's Mathematical Biology I, Chapter 13.2 discussing travelling wave solutions to the Fisher-KPP equation, and in particular their dependence on initial conditions $u(x, ...
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How to formally show these functions are solutions to these ODEs of finite duration? $g''=-\text{sgn}(g')$
How to formally show these functions are solutions to these ODEs of finite duration? $g''=-\text{sgn}(g')$
Summary: I need to formally demonstrate the following is an equivalence:
$$\def\sgn{\...
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Interpreting integrals with respect to a vector
I'm reading a paper titled A non local model for cell migration in response to mechanical stimuli by Marchello et al., and I'm confused by the meaning of the following integral in equation (6):
\begin{...
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Deriving formula for distribution from recursive formula
I've been working on a physics model for my Extended Essay and I am troubled by it.
The gist is that there is a platform oscillating like a sine wave with very high frequency. If you drop a ball on it,...
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Is it reasonable to assert that "strong duality" is useless in practice because function values themselves are usually meaningless?
I find the concept of strong duality hard to appreciate.
In optimization, our ultimate objective is to find the optimizer $x^*$ that lives in the constraint set of the optimization problem
$$\min f(x) ...
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Framing a linear programming optimization problem
In an interview, I was asked to frame the following optimization problem:
A company produces two products $x,y$ using two machines $a,b$.
To produce each product, both the machines are used together. ...
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Formulation of an optimization problem involving the largest ball inscribed in an uncertain polyhedron: is it a linear program?
I wish to check if it is possible to formulate the inscription of the largest ball $B$ inside of a set of uncertain polyhedron $P$ as a linear program, starting from an initial description of the ...
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Models for aging human populations
I'm coming here for some advice on human populations modelling. I have the book Mathematical models for the growth of human populations, by Pollard, published in 1973.
I like the great amount of ...
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What's the name of this probabilistic model for events with fixed average frequency?
Background
While designing a subsystem for a tabletop game, I'm trying to ensure that a player is subject to a recurring event at intervals that are random, yet guaranteed to be close to a fixed ...
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Model a finite lattice with an ILP
Is there a standard optimal way to enforce the finite lattice (order) definition in an Integer Linear Program, for a lattice with a given number of elements $n$?
I have tried a web search but with no ...
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"Surface of Steepest Descent" Given Two Points
Given two points $O := (0, 0), P := (x, y)$ in $\mathbb{R}^2$ and assume a uniform downward (negative $y$-direction) gravitational field is applied. By considering all the possible curves and their ...
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Is there a clearly preferable choice for how to model games within games?
I am interested in how to model games, which may have some theory established about their solution concepts or other properties, when they become encapsulated within broader games. As a real-world ...
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Visualising ReLU fitted functions
I am currently working on neural networks and am trying to visualise how the non-linearities lend the model flexibility/modelling power. Specifically, I am looking for advice on certain areas of 'what ...
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Explain how this objective function is chosen
Problem: A tele company has 4 customers, and the company wants to plant a network tower as close as possible to all of them and serve them with a priority aswell. The 4 customers consume their network ...
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Finding complex Fourier series coefficients of a perturbed square waveform
I am trying to recreate the analysis from a paper published in 2008. I will try to provide a detailed understanding of what I was able to comprehend from the paper so that the readers don't need to go ...
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Regression for power series data: how to compute PS coefficients given a near perfect, finite, polynomial model
I'm working on a model for a criterion $y$ known to be described (ideally) as the sum of a power series in $x$, with all coefficients non-negative, i.e.:
$y = a_0 + a_1x + a_2x^2 + a_3x^3 + ...+a_ix^...
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Brine Tank with precipitation
I'm trying to solve the following problem:
I have a perfectly mixed brine tank, where solid salt is added at a constant rate $R$ $(kg/min)$, there is also some insoluble solids that is added at a ...
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Is it valid to integrate Carrying Capacity in the rate constant of elementary reactions in Chemical Reaction Network?
What the title says. So instead of using the Lotka-Volterra Competition Model, we decided to use the CRN framework to analyze the competition among 3 companies.
For $ i = 1, 2, 3 $, we capture a ...
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Identifying the Ellipse through vectors
I am reading a paper on Social force model for pedestrians. The avoidance force for two pedestrians calculated through an ellipse whose semi minor axis is $b$ given as
$$2b=\sqrt{(||\vec{BA}||+||\vec{...
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Where does the e come from in the SIR model?
I am learning infectious disease modeling, and I have come across something in my textbook that has me stumped. They take the equation:
$$\frac{dS}{dR} = -R_0S $$
Then, they say "upon integrating ...
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How to get State Space representation from Jacobian of a non-linear system?
I have a non-linear system that is described by these equations:
$$\dot{x_1} = \frac{60}{x_2}*(-k_2*\frac{R*T}{V}*\frac{x_2}{k_1}*x_1 +k_2*\frac{R*T}{V}*k_3*u_1)$$
$$\dot{x_2} = \frac{60}{\frac{J*x_2}{...
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Help for fitting a S shaped curve...
(Warning: I have a few remnants of math, stats, and machine learning, but I'm far from an expert in this field, consider me a noob :)
I am trying to fit with a curve the following dataset which is ...
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Any system for determining if a model is applicable to a problem?
Is there any formal system for determining whether a model is applicable or corresponding to a problem? Take the following example:
Problem:
Peter's age is a third of his father's age, but in 10 years ...
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Functions that model polarising light filters
On a video about negative probability, the guy used an example involving polarising light filters. That automatically got me thinking about how to express them mathematically.
When you apply one ...
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Modelling population growth where only the latest generation divides at a point in time
I'm trying to come up with a continuous function to model the growth of a bacterial population under the following conditions.
Bacteria never die
Each bacterium divides to produce two offspring
...
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Show that an ansatz is solving $\Delta|u|^{\frac12}=0$ in 2 dimensions $(\mathbb{R}^{1+1})$
Show that an ansatz is solving $\Delta|u|^{\frac12}=0$ in 2 dimensions $(\mathbb{R}^{1+1})$
I have added how the ansatz solve the equation by brute force, but I am stuck in defining properly it's ...
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