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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Damped Driven Oscillator Dimensional Analysis

So, I am studying the damped driven oscillator with the following drive force: \begin{equation} M\ddot{x}+\gamma \dot{x}+kx=F_0 \cos({\omega t}) \end{equation} where $M$ is the mass, $\gamma$ is the ...
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Is there any tool that could turn a Petri Net into system of math equations?

I have a colored, descrete Petri Net. I have a picture. How to get a math (ODE\DE\system of equations) representation of it?
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AHP (Analytical Hierarchical Process) for individual selection decision with equally important objectives

How can I apply AHP for individual selection decision when all the objectives are important? For example, I want to select a car. The car needs to have all the characteristics (all the components). In ...
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32 views

Adding two sinusoidal functions together (i.e $10\sin x+\cos x$ or $5\sin x+\sin x$)

I'm doing my math exploration on modeling tide levels and wanted to know if it was possible to find a function by just looking at a graph, and knowing their points that looks like (see image on ...
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Trying to model an equation for this problem:

A drug is administered once every four hours. Let Dt be the amount the drug in blood system at t-th interval. The body eliminates a certain fraction p of the drug during each time interval. If the ...
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How to iterate a function 8 times about a given interval of x in a Discrete Dynamical System

This is Dynamical Systems, specifically a discrete system. We are using L and R as in Left and Right such as: L=[0,0.5] R=(0.5,1] and LL=[0,0.25] LR=(0.25,0.5] and so on like that. We keep ...
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Prove multiple OLS t-test follows t-distribution

I'm trying to prove the multiple regression test has a t distribution, i.e.: $\frac{\hat B_j - Bj}{se(\hat B_j)} \sim t (df=n-k-1)$ I was able to prove $\frac{\hat B_j - B_j}{sd(\hat B_j)} \sim ...
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Stability type of logistic population equation's equilibrium points

I was working on this question: Here is what I have so far: $ (a) \ \ \text{We have the algebraic expression to find equilibrium points:} \\ \frac{dy}{dt} = ry \big[1 - \left(\frac{y}{K}\right) ...
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52 views

How To Evaluate How Well a Mathematical Model Works… Modelled VS Measured Values?

Let's say I am working with a mathematical model. The output of the model is always a value between 0 and 1, and is dependent on a number of input variables. I am also measuring the same phenomenon ...
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How can I solve this ordinary differential equation? What are the Initial/Boundary Conditions used?

I am having trouble following the solution to this problem, I can understand how to obtain the general solution but cannot figure out the conditions used to obtain the particular solution with the ...
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63 views

Disadvantages of Non-orthogonal basis

What are some disadvantages of using a basis whose elements are not orthogonal? (The set of vectors in a basis are linearly independent by definition.) One disadvantage is that for some vector $\vec ...
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Writing explicit scheme - numerical methods for PDEs

I've got a problem writing a code for the explicit scheme. In the end i'd like to plot a temperature destribution for a given case. Let's say i have a material of a lenght L with given boundary and ...
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25 views

A property of Piece-wise continuous simple linear regression model

My fellow members I attempted to model the growth of capital of a small business person who does business with the aim of just raising his or her capital, as follows: Assumptions A fixed capital ...
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33 views

Equilibrium of an ordinary differential equation [closed]

I have been trying to show that the following ODE has either one or three non-zero equilibria, depending on the value of $r$, but with no real progress. $$\frac ...
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Calculating remaining balance [Monthly compounding question with rate= 0.01]

Suzanne opens a line of credit at a local bank, and immediately borrows 1870 dollars. 6 months later, she repays 1060 dollars. 5 months after the repayment, she borrows 570 more dollars. 6 months ...
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48 views

interarrival time

I am trying to model the arrival rate (delay between the arrivals) of the cars in my city. I have some real data with the resolution of one minute. For example, Number of counted cars at position x at ...
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22 views

What do we call the harmonics in a discrete Fourier series representation?

In harmonic analysis using discrete Fourier series, if I'm using the 0f, 1f, 2f, 3f and 4f for representation where f = frequency, what is the correct way to say how many harmonics I'm using for ...
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How to prove that the solution of a given ODE is bounded?

Let's have an equation $$ y''(t) + \left(\frac{3}{16 t^{2}} + \frac{A}{t} -\frac{B}{t^{\frac{5}{4}}}cos(2t)\right)y = 0, \quad A > 0, \quad B > 0,\quad A < B, \quad t\in (1, \infty) $$ ...
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Implications of Choosing Incorrect Variables in Buckingham Pi

One form of the Buckingham Pi Theorem says that for $n$ variables with $k$ dimensions, the number of dimensionless quantities (or pi groups) is $n-k$. This theorem is often applied when the ...
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Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
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79 views

Physics related initial value problem (horizontal spring mass system)

Consider the horizontal spring-mass system where the spring-force is the only force acting on the mass. Suppose that a mass is initially at $x=x_0$ with an initial velocity $v_0$. Show that the ...
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How to solve the model parameters?

Find a region of the world in which arms race is occurring or has occurred. Find data about arms spending. Estimate the model parameters. I found some data on Zimbabwe's military expenditure and ...
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Apprehension in Feller's solution of a model of breakage of spores

This one is a solved example from Feller. Spores of the Fungus are produced in chains of eight.The chain may break into several parts into projectiles containing 1 to 8 pores. Find the ...
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28 views

How to quantify a relationship so that a small value increases the result and a larger value decrease the result?

The title pretty much says it all but let me break it down. I thoroughly appreciate any help. I'm trying to quantify a relationship between the difference in peaks and troughs of a wave function and ...
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Turning a word equation into a differential equation (heat transfer problem)

Question: Heat is flowing in a wall with constant cross sectional area, inside which heat is generated at a constant rate of Q $ W m^{-3}$. Write a word equation and show that the equilibrium ...
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Differential Equation Model of Motion problem.

I'm having trouble with this question. "A body is released from rest and travels the last half of the total distance fallen in precisely one second. How far did the body fall and how long did it take ...
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41 views

Solving for projection of vector intersection on earth surface

I need advice regarding a geometry problem. I illustrate this geometry in the figure below: Background of question: There are two points (represented by blue and orange stars) with associated ...
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55 views

An introduction to Lane-Riesenfeld algorithms

I am looking for a good introduction to Lane-Riesenfeld algorithms, which are a family of subdivision methods for generating uniform $B$-splines. Any suggestions? (The more "basic" the exposition the ...
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Proof that any curve in two dimensions has a modelling differential equation.

I was wondering if anyone knew of a proof stating that any smooth, continuous line drawn in two dimensions must have a differential equation that models it. Best, James.
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Matlab - Storing information in vectors and modelling a system of equations

I'm working on solving a system of differential equations and I'm having trouble setting up a vector of parameters. This is the code for one of my model files: function ...
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24 views

constrained differential in a chemical reaction model

I have a system of reversible elementary chemical reactions for a catalytic cycle. Each reaction has a forward reaction rate constant $k_{f}$ and a equilibrium reaction constant $k_{eq}$ (we could ...
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Model equation for customer numbers.

I am trying to formulate an equation to determine the number of customers signed up to a subscription as a function of time. It's been 10 years since I studied maths, so bear with me! The assumptions ...
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Problems with Exchange Procedure in Remez Algorithm

So first off: *** This code is not being used in production software. It is a personal project of mine, trying to understand approximation theory and advanced curve fitting. In other words, I'm ...
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Finding tangent to curve fitting sampled data.

Let $(x_i, y_i)$ be a finite sample of points. I want to find the slope of the tangent at $x$ to a curve "best fit" to the given data. In my problem area, (see context below), the simplest of ...
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Is there a rigorous approach to solving this, instead of trial and error?

This is a question from a 4th grade math book. The solution in the teacher's guide suggests that one might not be able to model the question easily and the best approach may be to solve by trial and ...
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Solving for a 3D point in a 5D graph given 3 pairs of 2D points.

I am attempting to solve the values $C$, $D$, and $S$, given three pairs of $[M,R]$. $$R = \frac {M}{C - MDC + DC\left(MS\right)^2}$$ I have been able to solve for a related equation (or rather, ...
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Rescaling of a problem

I'm having a tough time trying to figure this question out mainly because I haven't any formal training in "scaling" of a problem. An infinite cylindrical rod (radius $a$) is initially at temperature ...
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29 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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Modelling using Shapley Value

I have a Shapley Value related problem that I am unable to solve. Instead of using integers, I have used a percentage (a conversion rate). I have attached a spreadsheet for you to have a look at. ...
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Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
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Definition of a function whose codomain is set of probability measure over cartesian product with dependency between sets in the product

I am thinking about the following function: $$ p : A \to \Delta \big( F(x, y(t) ) \times T \big) ,$$ where $t \in T$ denotes continuous time, and $\Delta (X)$ denotes the set of all probability ...
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A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
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Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
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What is the notation for 'asymptoticly approximate mapping'? If any…

I've learned about the notation 'maps to': $\mapsto$ And also asymptotic approximation: $\simeq$ Is it valid to suggest the notion of 'asymptoticly approximate mapping'? If so, what is ...
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Solving a pair of ODEs

I'm trying to solve a pair of ODEs for which I've obtained a solution. However, my problem is that my answer is slightly different from mathematica's answer. $$ \frac{dA}{dt} = \theta - (\mu + ...
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Population balance model

I have some experimental data and I need to make a population balance model to compare the experimental results with. The experimental results are from the bubble size distribution in a bioreactor. I ...
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Statistical/ML models when observations have different amounts of input

Let's say we're predicting an employee's performance review score for the following year based on his/her performance review scores from each previous year of their employment. We might have these ...
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231 views

The theory in probability

Consider a real-life experiment (perhaps written as a problem in a textbook): A coin is continually tossed until two consecutive heads are observed. Assume that the results of the tosses are mutually ...
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Assignment of initial probability values

Suppose a coin is tossed until a head is observed for the first time. It is given that the coin lands heads with probability $p$ and tails with probability $1-p$. Based on only this information, can ...
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How do you formulate a reaction diffusion model with 3D volume and surface compartments simultaneously?

Suppose we have a two compartment reaction diffusion model, for chemical species $\psi$ and $\phi$. Suppose $\psi(\vec{x},t)$ and $\phi(\vec{x},t)$ exist in two 3D compartments $\Omega_a$ and ...