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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Why life expectancy calculate as 1/μ in SIR model?

In the SIR model, when the death rate is μ, life expectancy is calculated as 1/μ. Can anyone explain it intuitively?
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Initial margin covariance matrix [closed]

I am trying to replicate the following example in order to calculate IM : https://www.clarusft.com/isda-simm-in-excel-equity-derivatives/?amp=1 I am stuck at the last step (impossible to find the ...
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What is the physical and mathematical meaning of nonsymmetric mass, damping and stiffness matrix of a linear fluid-solid interaction modeling problem?

Suppose the problem of the subsonic flow over a simply-supported rectangular plate as bellow. According to the assumptions of incompressible, inviscid and irrotational flow, and using perturbation ...
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Need help in quantitatively modeling solubility of carbon dioxide in beer.

I'm trying to model a 3D surface which reflects the solubility of $CO_2$ in beer. An empirically derived chart is available at this link. Solubility is empirically related to the pressure of $CO_2$ in ...
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Find Variants in list<list<int>>

Have a mathematic solution for this: I have some list<list> For example List{ List1 = {1,2,3} List2 = {4,5} List3={6} List(n) {k} } How can I find all ...
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Discover where car is parked when car is using adjacent parking strategy to hide in N garages [duplicate]

The car can be parked in n garages lined up in a row. Each night the car is parked in some garage, and each day it is reparked to garages adjacent to the garage where the car was parked the previous ...
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Reference request: converting discrete models to continuous models

I have been reading a bunch of papers which take discrete models or agent-based models and convert them to continuous PDE models, using Continuum Mechanics, or such methods. Can anyone recommend a ...
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How to develop a continuous function for known pattern? [closed]

I am given a range of x and y and would like to develop a continuous function f(x). Meaning ...
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Positivity of solutions of an ODE system with non-negative initial condition [closed]

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 system is non-negative if the ...
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Laguerre tessellations in real world

Aside from microstructures, where are Laguerre tessellations used in mathematical modelling?
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Geometric mean is to arthithmetic mean as arithmetic mean is to what?

I am interested in a type of "mean" $r$ associated to a set $\{a_1,a_2,\dots,a_n\}$ where $$ e^r=\frac{1}{n}\sum\limits_{i=1}^n e^{a_i}. $$ I will call this the "? mean" for now ...
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Capacity planning and modelling

I have a business case in which I am going to model how many devices are required given the predicted workload in a series of monthly cohorts in the next ten years. The work could come from multiple ...
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Picking numbers from a list with Gaussian distribution (programming implementation)

If we have values: $x \in [0, 100]$ I would like to implement a method, where the bigger the value, the less likely it is that it will be picked. (something like a Gaussian curve, but with maximum at ...
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How do you distinguish euristically using a phase-space diagram if a point is asymptoticaly stable or not? (Using a SI model as example)

In my notes the following SI model is given $\frac{d}{dt}S(t)=(1-S(t))-R_0I(t)S(t)$ $\frac{d}{dt}I(t)=R_0I(t)S(t)$ $S(t)+I(t)=1$ $t \ge 0, R_0 >0$, $ 0 \le S(t)\le 1$, $ 0 \le I(t)\le 1$ where $S(T)...
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Finding the steady state solutions of a model.

Problem: Find the steady states and check for stability of the model \begin{align*} X_{t+1} &= rX_te^{r(1-X/k)-aY_t} \\ Y_{t+1} &= X_t(1-e^{-aY_t}) \end{align*} Attempt: Calculating ...
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Is there a Mathematical Model to check if a paper was folded correctly every time to fit in an envelope? [closed]

I have an A4 piece of paper folded into an envelope. How can i be certain that my folds were done to fit perfectly in the envelope every time from a Mathematical Model?
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Differential Equation Modeling Fish Farm

Question - If the fish population at a given time is 240,000, give an estimate of the number of fish born in one week. Given Constant Harvesting Rate $h = 2100$ fish per week Per Capita Death Rate $α ...
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Can someone help me solve this? Finding revenue function given two different prices and quantities

Have a Math for Econ intro course exam coming up and this question showed up in one of my past papers, I am afraid I do not know exactly what to do. Can anyone help me solve it? The following is the ...
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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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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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1 vote
1 answer
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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 ...
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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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1 vote
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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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