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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How to model function with unknown exponents?

I know the Cobb-Douglass function which describes the production quantity: $$Q(K, L) = A \cdot K^\alpha \cdot L^\beta$$ Also I do know multiple assignments of K (...
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68 views

Smooth function to model a well or pit

In order to solve a Physics problem, I have to model an axisymmetric well or pit on a plane. I am looking for a function $f(x)$ that is smooth and monotonic in the domain $(0,1)$, where the origin ...
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129 views

Interperate Jacobian Determinant - Stability of Equilibriums

In my SIR model, I have the following Jacobian Matrix \begin{align*} J =\begin{bmatrix} -\alpha I & -\alpha S & \zeta & 0 \\ \alpha I & \alpha S - \beta - \rho & 0 & 0 \\ 0 &...
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1answer
83 views

Rayleigh-Bénard convection

I have a nondimensionalized linear perturbation system relevant to the appropriate pure conduction solution for Rayleigh-Bénard convection in upper planetary atmospheres under the compressible gas ...
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76 views

Assistance with Bayesian Random Effects and Mixed Effects Models

I am looking to build either a random effects or mixed effects model for a project I am working on, but have had trouble finding good sources and understanding the general structure of the model. ...
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1answer
37 views

Estimating number of slaves imported before 1790

I have a statistics problem I am having a hard time figuring out how to model mathematically. The 1790 US Census counted 697,681 slaves and 59,196 free Africans in the United States. (A) assume ...
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34 views

Establishing a stable state withing a discrete dynamic model

Given the following discrete dynamic model: $P_0=360$ and $R_0=80$ and $t[0,200]$ $P_t=1,18*P_{t-1}-0,002*R_{t-1}*P_{t-1}$and $R_t=0,86*R_{t-1}+0,0004*P_{t-1}*R_{t-1}$ One can rewrite the ...
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485 views

fantasy basketball model

i'm creating a fantasy basketball model (could be used in other games too) where we can project how well a player will do against another team even when the player hasn't played against a certain team ...
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44 views

What does this determinant mean?

I have the following Jacobian matrix for an equilibrium of an SIR model $$J=\left( \begin{array}{cccc} -\text{$\alpha $N} & 0 & \zeta & 0 \\ \text{$\alpha $N} & -\beta -\rho & ...
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1answer
87 views

How to find the basic reproductive number of a discrete SIS epidemic model

I have been following a textbook called Mathematical Models in Population Biology and Epidemiology. The SIS model is given by the system \begin{aligned} S_{n+1} &= \Lambda + S_n e^{-\mu} e^{-\...
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1answer
179 views

Is this probabilistic balls-and-bins problem well-defined and is my solution correct?

Problem definition: There are $n$ bins, labeled with $1, 2, \ldots, n$. Let $X_i$ be a random variable denoting the number of balls contained in the $i$-th bin. The collection of random variables $...
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2answers
71 views

Split cab fare proportional to saving made across the board

Interesting dilemma. I've figured out how to split a cab fare 3 ways (proportional to time, mileage and equally) but I've found that when the cab fare is split proportional to time and mileage, the ...
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42 views

Split a cab fare proportional to the time everyone has spent in taxi

How would you split a cab fare proportional to the time everyone has spent in the taxi? Total journey time = 19 min Fare = $17.25 ...
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0answers
262 views

mathematical model of an assignment/scheduling problem

I am solving a scheduling problem and I am able to abstract it into an assignment problem of assigning 45 machines to 42 jobs. the assignment problem was given as having 14 jobs, each with 3 tasks and ...
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1answer
114 views

Anyone know about definition of weak dictator?

I am trying to prove arrow's impossibility theorem in case which ties are allowed in individual preference lists and so is social preference list. It says that if $p$ is a weak dictator, then we can ...
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190 views

Analyzing/Modeling DynamoDB Usage

I'm seeking the correct mathematical tools to analyze/model a real world process for the purpose of optimizing resource allocation. I want to analyze consumed throughput of a AWS DynamoDB table and ...
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1answer
92 views

How to connect the deterministic and probabilistic descriptions of the SIR model

I am a 17 year old student and I was reading up on epidemic modelling for a math project, specifically the SIR model and I came across this: "This" refers to the assumptions to which the Markov Chain ...
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0answers
42 views

Classifying growth as percent increase.

So I have been thinking about resource consumption a lot after watching the most important video you will ever see. It is pretty long so I will summarize it as follows, the professor makes a strong ...
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2answers
182 views

How to plot a phase potrait of a system of ODEs (in Mathematica)

I am asked to use mathematica to plot the fixed-points of the following system $$ \frac{dN}{dt} = -\gamma N \left( 1 - \left( \beta M + N\right) \right) $$ $$\frac{dM}{dt} = M \left( 1 - \left( \...
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1answer
159 views

How do you scale a set of number such that they sum to 0.5 after scaling

Suppose I have a set of numbers of various different values (>0.0). I want to scale these numbers so that they all sum to 0.5. The scaling is required so that the relative strength of the numbers with ...
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2answers
173 views

Stochastic Calculus For Finance: Conditional Expectation of Binomial Tree Model

I'm reading through Stochastic Calculus For Finance Volume 1, page 32. The conditional expectation of $S_2$ knowing $1=H$ is $p*S_2(HH) + q*S_2(HT) = 0.5*16 + 0.5*4 = 10$ But how do I calculate the ...
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54 views

Establishing a inequality

I need to establish an inequality containing $19$ terms in $7$ variables. My problem arises in the context of proving that the HIV-only Quasi Disease-Free Equilibrium of a HIV-TB co-epidemic model ...
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275 views

Swimming Problem - How can I do this?

I have a problem with the following diagram. The triangle is just a possible path and may not be the shortest. If the boy has to get to the destination in the fastest way possible: What methods/...
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2answers
37 views

Unusual Probability Question

Wheel of fortune: Assume that the probability for an angle φ is P(φ) = λ$φ^2$. The game pays $1000 times the angle. a) What is λ? b) Find the expectation and the variance of the game. Comments: I'...
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60 views

Generalize discrete Lyapunov equation for n-th order linear dynamics system

My specific application is analysis of dynamic textures using linear dynamics systems of the form $$ I(t) = Cz(t) + w(t) \\ z(t + 1) = Az(t) + Bv(t), $$ where $I(t)$ is the original signal, $z(t)$ ...
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43 views

Check if intervals overlap

I have a set of activities $A$, where each activity $i \in A$ has a starting date $s_i$ and an end date $e_i$ (or equivalently a starting date and a duration $d_i$). Therefore, each activity can be ...
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74 views

Predicting profit with price variation

I am currently working on a high school project that aims to predict profit from X amount of items to Y amount of profit based off a deviated sale price. For instance: I sale 10 cookies for 10 ...
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1answer
85 views

Mathematica Question regarding NSolve

I'm trying to solve for 2-cycles for the equation $$x=xe^{r[1-x]}$$ using Mathematica. I've tried using NSolve, FindRoot, and Solve. When I use NSolve I input it as $$\textrm{NSolve}[xe^{r[1 - xe^{r[1 ...
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1answer
283 views

A way to check the accuracy of a Markov chain?

I am not sure whether I should post this question on MSE or SSE. I will post it here 1st to see if I can get some feedback. Say I have a finite discrete Markov chain constructed maybe using some data ...
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1answer
23 views

Regularity of the surface of a crystal

If I want to model the surface of any random crystal, is it safe to assume that it is the graph of a Lipschitz function. Is there a precise result from physicists? How wrong would it be if I assume ...
2
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1answer
91 views

Mathematical Description for Steam Rising from a Cup

I was staring at a cup of coffee I have on the desk just now. The light shines through the water vapor as they rise from the cup. The shape of the steam is not completely random, as it drift from hot ...
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196 views

Efficient free alternative to *Mathematica*

I am searching for a free alternative to Mathematica. By efficient, I mean that it should have every (or at least almost every) function that you can find in Mathematica, including for example Number ...
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52 views

What resources (books, videos, etc) to help develop math thinking skills?

I am a applied mathematics major and have taken basic undergraduate coursework up to multi-variable calculus, ordinary differential equations, and elementary linear algebra. I am currently taking ...
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1answer
107 views

Including non-markovian processes in a birth-death process

Current model I want to model a stochastic system with a birth-death (Markovian) model. I therefore have this kind of $n$ times $n$ (where $n$ is the number of possible states) transition matrix: $$\...
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797 views

Finding a general equation for number of paths through grid

I started with a 4x4 grid (although I want to eventually generalize for an n x n grid). You must move through a grid on the squares, not on the grid lines. The number of paths for path length = 1 is ...
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1answer
100 views

How can you find orbits of period 2 of the tent map

$f(x) = \begin{cases} \mu x &\text{ for } 0\leq x\leq 0.5 \\ \mu(1 - x) &\text{ for } 0.5 < x \leq 1 \end{cases}$ I am to find the orbits of period 2. By using google I found an example ...
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2answers
89 views

Population dynamics calculation

I am trying to solve the question: A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero, the population consists of two members. ...
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3answers
66 views

Linear nonhomogenous ODE

Solve: $$y''(t)=-y(t)-\cos(t),\ \ \ \ \ \ y(0)=y'(0)=0$$ I'm sorry that I have no approach to solve it. In my solution I have the hint to solve it with resonance but I don't know anything about ...
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1answer
36 views

Prove $\sigma_V=x\sigma_S$. (Financial Mathematics)

Prove that the standard deviation of the value $V(T)$ at time of any portfolio $(x,y)$ at time $T$ in a one-step binomial is given by $\sigma_V=x\sigma_S$, where $\sigma_S$ is the standard deviation ...
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0answers
51 views

Describe as a system of differential equations

How is it possible to describe the $\dfrac{du_c}{dt}$ and $\dfrac{di}{dt}$ as a system of differential-equations? $$ u_s = u_r + u_l + u_c\\ i = C\frac{du_c}{dt}\\ u_l = L\frac{di}{dt}\\ u_r = Ri\\ $...
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367 views

How do you go about formalizing a concept?

I am reading Godel Escher Bach. I love it. In the first few chapters, the author shows what a formal system is and gives examples that eventually lead to a typographical formal system of strings that ...
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68 views

Poisson Modeling/ Queue Theory - Reference Material

Can anyone reccomend some practical reference material related to building and implementing queueing theory models. using stochastic (prefferably Poisson) processes? We are looking to build out a few ...
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0answers
24 views

Could the word “continuum” be applied to theposition of the hands of an analog clock?

The word "continuum", in term of measurement, is defined as: models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, ...
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0answers
33 views

What is distinction between Functional Linear Regression and Functional Linear Models?

Thanks in advance for the help. I want to make sure that I understand two concepts correctly. A functional linear model is a particular type of linear model while functional linear regression is the ...
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1answer
324 views

Find the difference equation for {2, 4, 16, 256, …}

Write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence. b. {2,4,16, 256,...} I know that an= 22n but I can't figure out how to ...
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1answer
40 views

Quantifiers and Mathematical Modelling

When we are in the context of pure mathematics, quantifiers are everywhere. When we are in the context of mathematical modelling, quantifiers usually disappear. For instance, in statistics we often ...
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413 views

Game theory: Bidding strategy during an auction in a card game

I'm trying to create a mathematical model for the auction process in a card game called Pitch. The specific question I'm interested in solving is: Let $p_i$ represent the probability of a specific ...
2
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1answer
412 views

Markov Chain with Memory

One of the defining characteristics of a Markov Chain is that it is memoryless: the next state depends only on the current state, and not on the set of preceding states. I'm looking for a ...
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36 views

I need to model a population where every component have a fixed life span.

I have a certain population, which growth is function of a certain factor. I have already modeled the growth. Now I need to impose that every individual in the population have a fixed life length. Can ...