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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mathematical model for a differential drive robot [closed]
I need to write the mathematical model equations for a two-wheeled differential robot (mini sumo robot), whose center of mass is not on the wheel axis.
Robot tire radius {$R_{right}$ and $R_{left}$}, ...
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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:
$$\dot{x_1} = \frac{60}{x_2}*(-k_2*\frac{R*T}{V}*\frac{x_2}{k_1}*x_1 +k_2*\frac{R*T}{V}*k_3*u_1)$$
$$\dot{x_2} = \frac{60}{\frac{J*x_2}{...
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Problem Set Question: Math 205 (Mathematical Problem Solving) [duplicate]
Without the use of a calculator, determine which is larger:
EDIT: $\log \left(\frac{4.4^{5.5}}{2}\right) \text{ or } \log \left(\frac{5.5^{4.4}}{2}\right)$?
Be sure to explain your answer and justify ...
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Help for fitting a S shaped curve...
(Warning: I have a few remnants of math, stats, and machine learning, but I'm far from an expert in this field, consider me a noob :)
I am trying to fit with a curve the following dataset which is ...
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Econometric Model for Vector-Valued Variables
In classical econometric models, we typically study linear relationships of the form:
$$ Y = C_1 X_1 + C_2 X_2 + \dots + C_n X_n + \epsilon, $$
where $Y$ is a scalar dependent variable, $X_1, X_2, \...
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Traffic Flow Modelling From one Lane Into Two Lanes
I need to derive a model that demonstrates the flow from one lane splitting into two lanes. I am a bit stumped on how to start. I imagine it would be similar to the way light traffic in front of heavy ...
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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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From the perspective of mathematical optimization, why is $B_2$ a better battery than $B_1$ even though $B_1$ has better summary statistics? [closed]
Consider two batteries being offered by BatteCo, LLC. $B_1$ offers 8 hours of use and 10 minutes of charge time until full. $B_2$ offers 8 days of use and 1 day of charge time until full.
From the ...
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Ergodicity breaking in the Ising model
Let's consider the Ising model in dimension $d \geq 2$ with constant coupling $J= 1$ between direct neighbors and without external field.
It is common to read in physics texts that, under the critical ...
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Approximate solutions to a system of differential equations
This system of differential equations corresponds to a compartmental model in epidemiology called SIRD:
\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta IS}{N}}\\[6pt]&{\frac {dI}{dt}}={\frac {\...
user1385231
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On the maximum rate of change of solutions to the wave equation, Could it be limited?
On the maximum rate of change of solutions to the wave equation, Could it be limited?
Intro_________
In the classic 1D Wave Equation $u_{tt}-c^2u_{xx}=0$ with a traveling wave solution $f(x-ct)$, ...
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Sliding frame optimization
I have a challenge regarding optimization of a laser die-cutter. Material is scrolled from left right through the laser chamber and the laser needs to cut out various shapes on the material. The ...
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Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind)
Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind)
In this other question I did about Bump functions, other user told in an answer that these kind of functions "...
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ODE with additional parameter - Comparison of various limiting solutions
I am trying to solve the following ODE, where s is the independent variable, and k is a parameter
$$
\left( {\frac {{\rm d}^{2}}{{\rm d}{s}^{2}}}T \left( s \right)
\right) {s}^{3}+ \left( -2\,k{s}^{...
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Methods of analysis in non linear coupled PDEs
I'm trying to study a system of coupled second order nonlinear PDEs with zero flow boundary conditions. The system is given by
$$\frac{\partial u_{j}(x,y)}{\partial t}=u_{j}f_{j}(\textbf{u})+ D_{j}\...
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How can I improve this model to better fit the data?
I have some data as a function of several variables,
$$k_{ap}=f(k_{ij},\ \alpha,\ D/L).$$
The data is as shown in the three plots below.
Fig 1
Plots a-c show the ratio of $k_{ap}/k_1$ as a function ...
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Exact solution of SIR model [closed]
I am trying to solve the equation: $\mathbf{\frac{dR}{dt}= \gamma(N_0 - R -S_0e^{-R/\rho})}$, where $S_0e^{-R/\rho} = S(R)$ and $N_0=S+I+R$. I want to find the exact solution using separation of ...
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Intuition and Timescale Analysis for a Nonlinear Differential Equation Dependent on Parameter $K$
I am exploring the behavior of the following first-order nonlinear differential equation:
$$
\tau \frac{dy}{dt} = 2Ky^K(a - y^Kb)
$$
where $ y, a, b \in \mathbb{R} $, $ K \in \mathbb{N} $, and $\tau$ ...
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finding the transfer equation of $\frac{H(s)}{Q_d(s)}$ in liquid control system with hydraulic integral controller.
On problem B-4-10
So my attempt is as follows:
from the liquid control system we have
$(q_d+q_i-q_o)dt=Cdh$ since $R=\frac{h}{q_0}$ and $R=0.5$ then $q_0=2h$.
By the equibilirium system on the lever ...
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How was this equation for the stability of a finite difference model Wave Equation derived? Von Neumann analysis?
Wave Equation
In this paper, the following equation is given for a wave equation:
Stability Equation
They then go on to state the stability of such a wave equation in sample based solutions (in terms ...
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Numerical example for a large system with a quadratic Hamiltonian and a state-dependent energy conservation matrix.
I am currently looking for physics motivated examples of the form $$\dot{x} = J(x)\nabla H(x),$$ where $J(x)^T = -J(x)$ is a skew-symmetric state-dependent matrix ($J: \mathbb{R}^n \longrightarrow \...
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Biological model
I have the following model:
$
B_{t+1} = (a + b B_t) B_t - c,
$
where $ B_t \geq 0 $, $ B_0 > 0 $, $ c > 0 $, and $ a < 1 $. I need to determine the conditions under which $ B_t \to 0 $ as $ t ...
user1135300
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What shape is the waterfall start line on a track?
The most common shape of a 400m outdoor running track is called a stadium in mathematics and consists of a rectangle capped off by semicircular ends. The length of the straight and the radius of the ...
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When can you use the $SEID$ subsystem to study the dynamics of the $SEIHDR$ epidemic model?
I am working on the following model:
\begin{equation}\label{eq1}
\begin{split}
\frac{dS}{dt} &= \Pi- \left( \lambda + \nu + \mu \right) S\\
\frac{dE}{dt} &= \lambda S - \left( \...
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Use of smooth bump functions in probabilities: make any sense?
Use of smooth bump functions in probabilities: make any sense?
After reading this answer I started to wonder if it make sense to define a Cumulative distribution function as follows:
$$F(x)=\begin{...
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Open Ended Problem: Estimating Productivity in a Factory
This is a statistical problem I am working on and I would like to request some guidance from the respected community.
Suppose there is a factory that receives food orders.
The food sits in a ...
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Correlation coefficient and coefficient of determination
I meet the concept: Correlation coefficient ($r$) and coefficient of determination ($r^2$). I also know that we can deduce $r$ from $r^2$.
$r^2$ helps us to find that whether the model fits well or ...
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Computing $\sum_{k=1}^{m-1} \frac{m-k}{n-k}$ for some arbitrary integer $n$ and with $m \leq n$
My question was: is there a "nice" expression of the following sum (for some $n\in\mathbb{N}$ and $m\leq n$) :
$\sum_{k=1}^{m-1} \frac{m-k}{n-k}$
For a bit of context, this came up when ...
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Question About Approximation in Formula For cGMP Synthesis
In this paper, it has a formula.
$\alpha \propto \frac{1}{1 + (\frac{C}{K_{Ca}})^m}$,
Where $\alpha$ is the rate of synthesis of the molecule cGMP, $C$ is the calcium concentration, and $K_{Ca}$ is ...
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generalized poisson moment function approach to initalize input output model
I am trying to understand how the Generalized Poisson Moment Functions apporach is a way to initalize the nominator and denominaor of a LTI-Model that describes the connection between Input and Output ...
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Solution of the "reciprocal of the heat equation"?
I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$ ...
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Time difference between peaks of Lotka-Volterra equations [closed]
In the Lotka-Volterra equations:
\begin{align}
\frac{dx}{dt} & = \phantom{-}\alpha x - \beta xy\\
\frac{dy}{dt} & = - \gamma y + \delta xy
\end{align}
The peak of the predator population ...
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Global existence of solution of ODE - Gray-Scott model
I am dealing with the ODE version of Gray-Scott model:
\begin{equation}
\begin{split}
\dot{x} &= -xy^2 + F(1-x)\\
\dot{y} &= xy^2 - (F + k)y,
\end{split}
\end{equation}
where $F>0$ and $k&...
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On the role of Homogeneity constraints
I refer here to the following paper by ECB: "What caused the euro area post-pandemic inflation?" I attach an equation about short term inflation expectations. What is unclear to me is the ...
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2D rigid body dynamics thrust vector-controlled rocket and optimal control
I have to do some calculations about the optimal control of a powered landing of a thrust vector-controlled rocket. As a start I found [1] which is almost what I need, which is the system (1) but have ...
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Model exposure-time vs gain relationship
Cameras can increase exposure either by applying gain or by increasing exposure-time and I would like to model this relationship.
For this, given a target exposure, I measured the necessary exposure-...
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A question related to physical application of differentiable functions in Zorich's Mathematical Analysis
A body that can be regarded as a point mass is sliding down a smooth hill under the influence of gravity. The hill is the graph of a differentiable function $y = f (x)$.
a) Find the horizontal and ...
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Modelling a queueing system with equal processing speed across n workflows
Hey all need some help on a real world queuing system. Apologies if we are a little loose with some terminology.
Let a workflow $w_i$ consist of steps $s_k$. A workflow can have n steps, which are ...
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question regarding modelling motion of particle using a step function
I have a particle which starts off with a velocity of 30 meter per second and reduces its velocity by 5 meter per second instantly every two seconds i thought a good way to model this kind of motion ...
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Question About Comparative Statics and Optimization
Let me try to describe my question first. By solving the following maximization problem
$$
\max_{t_P} U_{B_i}(t_P) = (1-t_P)y_{B_i} + V_{B_i}(g),
$$
where $V_{B_i}$ is a function of $g$ and $g=t_P\...
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How to triagulate multiple sound locations
Other people posed the question of how to triangulate sound from multiple locations.
Approximate (but as accurate as it can) location of sound
Sound Triangulation
My question is how to seperate ...
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Consecutive binary block in MIP modeling with variable length
I'm currently modeling a MIP and face a problem on how to tackle consecutive binarys.
I have a integer variable $A_v$ which marks the start time and a integer processing time $P_v$. I want to model ...
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Lagrangian for catenary problem with non-uniform/external force
It is well known that the Lagrangian for the catenary (hanging wire) with uniform gravitational force is.
$$\mathcal{L}=\mu g y\sqrt{1+y'}$$
Now Suppose that there is a nonuniform, possibly external ...
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Smooth periodic extension of a function on $[0,1)^d$ to $\mathbb R^d$
Here is my goal: Given $n\in\mathbb N_0\uplus\{\infty\}$, I want to construct a function $q:[0,1)^d\to(0,\infty)$ such that $q\circ\iota\in C^n(\mathbb R^d)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;...
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Forming an O.D.E for $y=A\sin(Bx)+B\cos(Bx)$
Forming an O.D.E for $y=A\sin(Bx)+B\cos(Bx)$, Where $A$ and $B$ are the arbitrary constants.
My efforts:
We have $$y'=AB\cos(Bx)-B^2\sin(Bx)$$
Also $$y''=-AB^2\sin(Bx)-B^3\cos(Bx)=-B^2y$$
Now I am ...
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How do generative models generate?
I am trying to understand the difference between discriminative models and generative models in machine learning. One of the helpful answers at Stack Overflow is here: https://stackoverflow.com/...
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Circular Breakout Game: time or collisions needed to reach nth layer
Shower thoughts...
A tiny ball starts inside a unit circle, surrounded by fixed concentric circles of increasing integer radius length. So it starts in is the 0-th "level".
The ball moves in ...
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