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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Can you modeling complicated dynamics without using differential/difference equations?
Let's imagine there is a phenomenon I want to understand. I have a few multivariate time series about the phenomenon but not a lot. I don't know how the variables are related to each other but from ...
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Rewriting system of second order differential equation as system of first order
Given a charged particle moving in an electromagnetic field. We have $N$ amount of point charges placed in $\mathbb{R}^2$ on the coordinates $p_i$. We also have a free particle moving in $\mathbb{R}^2$...
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Beta binomial regression model: sufficient statistic of the regression coefficients
I have the following beta-binomial with logit link function model:
$$f(y_i|\pi_i,\phi)\sim\operatorname{binomial}(p_i), \text{with }p_i \sim \operatorname{beta}(\frac{\pi_i
}{\phi},\frac{(1-\pi_i)}{\...
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How many vectors do I need to span the vector space for Velocity, Distance, Angles, Acceleration, Position, and Time?
I am working on a Robot Arm simulation project and trying to get those 6 varibles, in my question, coming from the Robot Arm. The robot arm has 6 joints and each joint can run the motor inside to ...
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Mathematical modeling of population using logistic growth
I am currently preparing for my finals and I came across one problem which I couldn't solve. I've been struggling with this for a few days.
The farmer owns $50$ animals and land that can feed $1000$ ...
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Difficulty understanding some inequalities in S,A,I,R disease modelling
Consider this paper (reading the entire pdf is not required) on disease modelling.
There are four real functions of interest, based on constant positive parameters $\mu,\nu,\delta,\beta$ and four ...
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Assume unknown tennis players B & D in three set match. Find Relation in $p$ [Probability B wins First Set] & $q$ [Probability Match ends in two sets]
Problem
If we assume that two completely unknown tennis players B and D are facing each other in a three set match.
Let $p$ be the probability that B wins the first set
Let $q$ be the probability ...
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Modeling body temperature in a continuous framework [closed]
I am reviewing for an exam and was reviewing last year's exam. Since our professor doesn't want to solve it in class, I come here to see if someone is so kind to solve it. The problem has to be ...
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Calculating net value after a series of transactions — family of problems
Is there a generic name for the type of problem which involves transactions between multiple containers with ownership of the containers and ownership of transactions by multiple individuals?
Here is ...
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How can we solve this geometry problem using Lagrange Multipliers? It is must to use the given formula of area.
The problem is belongs to Mathematical Modeling with Excel book.
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Dimensional Analysis - Simple Pendulum
For a simple pendulum consisting of a spherical mass $m$ anchored to a fixed
point $O$ via a mass-less rod of length, perform dimensional analysis
on the problem to find the period of oscillation τ (i....
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Finding nullclines and linearization about the steady state
Here's my system:
$$
\left\{
\begin{aligned}
\frac{du}{dt} &= u(1-u^2)-w \\[5pt]
\frac{dw}{dt} &= u
\end{aligned}
\right.
$$
I'm coming to the conclusion that the only steady state ...
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For the given delay-differential equation, $x'(t)=x(t-T)e^{3-x(t-T)}-x(t)$, how do you find stability for given equilibria?
The delay differential equation given by, $x'(t)=x(t-T)e^{3-x(t-T)}-x(t)$, how to find stability conditions?
So here's my attempt: to find equilibria we have, $x^*e^{3-x^*}-x^*=0$ which leads us to $x^...
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Modeling mosquito movement as a binomial distribution (Shape by Jordan Ellenberg)
I'm reading through Shape by Jordan Ellenberg and came across this claim, modeling the movement of a mosquito as a binomial distribution.
The mosquito is fixed to a straight line. Each day, it can ...
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Difference between mean response at $x_0$, and predicted value at $x_0$
I've read a few accounts on the difference between these two, but none have been very clear. So what exactly is the difference between 'mean response at $x_0$' and 'predicted value at $x_0$'? Also, ...
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For the difference equation $x_{n+1}=ax_n\exp(-x_n)$, find values of $a$ that lead to a period-doubling bifurcation and values that lead to extinction
For the difference equation, $x_{n+1}=ax_n\exp(-x_n)$, find values of $a\in(-1,1)\cup(1,e^2)$ that lead to a period-doubling bifurcation, and values that lead to extinction
So for $x_{n+1}=ax_n\exp(-...
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For given system of ODE's, in a region $\Omega$ in the $uw$-plane, show it's a trapping region for large R
Need help figuring out what this question wants me to do. No need to do entire problem for me, just need a push in the right direction on how to solve
Here's my system:
\begin{gather*}
\frac{du}{...
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how to properly write a sum
What is the correct way to write a sum? I want to write a general formulation for the voltage at the last node of a line of an electrical grid. Here is an example line:
I have written:
\begin{...
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Performing linear stability analysis for nonlinear discrete system by approximating function for large values of the varying bifurcation parameter
Here's my system, \begin{gather*}
N_{t+2}=N_t\exp{[r(1-\frac{N_t}{K})]}\frac{1-e^{-aP_t}}{aP_t} \\
P_{t+1}=N_t[1-\frac{1-e^{-aP_t}}{aP_t}]
\end{gather*}
In the research paper, it states that ...
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The latest best approach to determine the order of the system model
The classical maximum likelihood estimation using Akaike's criteria is defined by
$$\text{AIC}=-2\log^-\text{(maximum likelihood)} + 2 \text{(no. of independently adjusted parameters within the model)}...
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Conditional modelling of a binary variable based on the values of two continous variables
I want to model a binary variable $(b)$ from two continous variables $(x_{in},\:x_{out})$.
These variables are $ 0\leq x_{in} \leq x_{max},\: 0\leq x_{out} \leq x_{max})$. I want the following three ...
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Numerical solution of a Projectile motion problem
Lets consider the situation of a rocket launched from the ground onto some impact function $f(x)$. Assume that the impact function is just the $x$ axis, hence the projectile hit the ground at the same ...
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Is this SEIRD disease model a linear or non-linear least-squares fitting problem?
This is a system of ODEs I have set up to model a disease.
I'm trying to fit the parameters for this model for a given data set using a least squares fitting algorithm. The parameters are the Greek ...
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How to determine equilibrium points numerically for linear stability analysis of the following Predator-Prey system?
Consider the following system,
\begin{gather*}
N_{t+2}=N_t\exp[r(1-\frac{N_t}{K})]\frac{1-e^{-aP_t}}{aP_t} \\
P_{t+1}=N_t[1-\frac{1-e^{-aP_t}}{aP_t}]
\end{gather*}
how should I go about ...
- 365
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Are the replicator dynamics the limit of a discrete model?
The (pure-strategy) replicator dynamics for a symmetric game with $n\times n$ payoff matrix $A$ are given by the system of differential equations
$$ \dot x_i = x_i[e^i \cdot Ax - x\cdot Ax], $$
where $...
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How to determine $R_0$, the basic reproduction number, for following system?
Consider a population of size $N$ with per capita birth rate $b(N)$ and death rate $d(N)$. Assume that it reaches stable steady state $N^*$ in absence of disease, with $b(N^*)=d(N^*)$. Find $R_0$ when ...
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Find one sided limit to show equivalence
I'm working on a problem that looks at growth models and I'm trying to show that the equation,
$\frac{dN}{dt}$ = $aN^{\gamma}$ - $bN^{\gamma}$($\frac{N^{\gamma -1} - 1}{1 - \gamma}$)
is equivalent to ...
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Reference request: multi-armed bandit problems with analytical solutions
Is there a book/survey on (multi-armed) bandit problems that yield analytical solutions? I.e. has an exactly optimal closed-form solution (e.g. derived using dynamic programming).
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When can the boat enter and leave the harbour safely?
A harbour experiences a tidal change in water between $4$ and $8$ meters. The period of successive low tides is every $12$ hours and $24$ minutes. If a boat has a draught of $5$ meters, what ...
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Show that the cash flow of a caplet is equivalent to cash flow of put options
I study Filipovic's book "Term-Structure Models
A Graduate Course". I came across exercise 2.7, which wants me to show that the cash flow of the $i^{th}$ caplet at time $T_i$, i.e $$\delta(F(...
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Romeo & Juliet ODEs
One analytical model of the marriage relationship — Romeo & Juliet — includes the constant term before marriage (uninfluenced state), call it, $x_0$ for one spouse and $y_0$ for the other spouse. ...
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convert between deterministic to stochastic rates?
I have a bit of confusion, if somebody could shed a light that would be appreciated.
Suppose a reaction of the form (reaction 4) in the ref. paper
R4:= S1+S2->> S3 at some rate $k_1.$
S1 and S2 ...
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Equation of motion with non-linear drag
Consider a particle of mass $m$ and electric charge $q$, initially at rest (zero
velocity, $v(0) = 0$), placed in a unidirectional potential $ψ(x) = −E_0qx \hspace{0.05in} \text{sin}(ωt)$. Assume that ...
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Modeling of blood flow using PDEs?
I'm in an intro to PDE class, and so far we've only covered the Diffusion and wave equations. I'm working on a final project, and I'm very interested in how PDEs are used to model blood flow. I've ...
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Applying a Short Force Impulse on my Object During Runge-Kutta Integration
I'm fairly familiar with Runge-Kutta ODE solvers, but recently I have begun to try to add collision to my soft-body and hard-body objects. I believe the safest and most general way to model this would ...
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Extinction condition for Leslie matrix
Good morning, I assume that I have an irreducible Leslie matrix s.t.:
$$
\begin{pmatrix}
0 & \alpha_2 & ... & ... & \alpha_n\\
\beta_1 & 0 & ... & ...& 0\\
0 & \...
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The French Newbie and the Lotka-Voltera Crazy Idea
I am back to talk with you and to have criticisms on ideas crossing my mind!
I am currently working on a disease that is striking french vineyards really hard. I am currently using a model that is ...
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How to remove "root(f(z), z, n)" from solve solution in MatLab
Wondering if it is possible to remove the expression "root(f(z), z, n)" from the solution presented by the solve function. In this case, my code is listed below and the outputs of "...
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Why is the phase space expressed in terms of the cotangent space of the configurational space?
The following is taken nearly verbatim from section 1.1.2.1 Free Energy Computations: A Mathematical Perspective by Mathias Rousset, Gabriel Stoltz and Tony Lelievre.
An excerpt, in which the part I ...
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Derivative pricing and porftofolios
I am struggling a little in understanding a passage in "PDE representations of derivatives with
bilateral counterparty risk and funding cost" by Burgard and Kjaer:
"As in the usual ...
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ODE with time lag
We can build a population growth model based on the following hypothesis:
$$\frac{dN}{dt}=\text{births}-\text{deaths}$$
$$\text{births}=aN(t), a>0$$
$$\text{deaths}=bN(t), b>0$$
We can find its ...
user1007703
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(Generating model of N-Dim data points) Is there any quantitative way to analyze the "probable" relation between an input variable and the output?
Caution: Layman's terms ahead...
Let, prior to building a model of a natural system, we prepare a set of data points consisting of N variables. Thus the dataset became an N-Dim dataset where the ...
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What is $f'(x)$ if $f(x)$ is a cumulative function
Let me preface by saying that I am completely ignorant of the math covered in Calculus 2, which for me includes integration. All I know is that the integral is the area under a curve, which I ...
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Finite Element model of Fourth order PDE in FreeFEM++
I am attempting to finite element model the/a solution to the Kuramoto-Sivashinsky equation
$$u_{t}+\nu u_{xxxx}+u_{xx}+uu_{x}=0 $$
with 1-periodic boundary condition
$$u(\cdot,0)=u_{0} $$
using the ...
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Is vector based distance between a point and a line differentiable?
We calculate the shortest distance between a point and a straight line. Consider a line given by the points $$r=a+b.t$$
The point from which the distance to the line will be calculates is $$x$$
The l1 ...
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Control of flow rate to be 'fair'?
We have a pipe transporting discrete particles from a to c with max flow capacity of $C$ particles/second. It takes $d_{min}$ seconds to get from a to c when the pipe is not fully utilized (details ...
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How to prove that there exist a travelling wave solution?
Im currently studying the SIRS infectious disease model for modelling an epidemic. I have expanded the model to take spatial expansion into account, i.e. I have incorperated a diffusive term. The ...
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Uncertainty propagation in non-samplig methods
I am able to "visualize" how uncertainty propagates from stochastic inputs in sampling methods (Monte Carlo simulation) since it is a consequence of repeated evaluations of the computational ...
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How many different partitions with exactly n parts can be made of a set with k-elements?
Find all partitions of S = {a, b, c, d}. Note first that each partition of S contains either 1, 2, 3, or 4 distinct cells
(1) in 1 distinct cell
Solution: 1 case
[{...
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How to determine degrees and coefficients of a polynomial regression with multiple variables?
I am trying to write my own statistical learning model from scratch and for polynomial regression I can't seem to figure out how to find coefficients/degrees when it involves multiple features. I ...
