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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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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162 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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140 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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153 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 ...
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1answer
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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265 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 ...
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36 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: ...
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58 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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70 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
83 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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237 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
22 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 ...
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1answer
87 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 ...
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1answer
191 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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51 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
102 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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750 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
97 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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88 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 ...
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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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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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362 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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1answer
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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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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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
286 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
39 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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399 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 ...
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1answer
351 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 ...
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1answer
213 views

How to represent R function dexp mathematically?

I need help representing this line of code mathematically. exp(sum(dexp(x,prop_theta,log=TRUE)) - sum(dexp(x,cur_theta,log=TRUE))) Thanks for your help in ...
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111 views

How to model a real-world graphical structure into a mathematical formulation?

I am trying to learn how to model programming problems in a mathematical way. I am a software engineer, but have recently been running into road blocks where I can't solve some problems very ...
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2answers
402 views

How to determine new dimensionless variables when non-dimensionsionalizing a system of ODEs?

I am following this a research paper called "Evolution of Within-Host Antibiotic Resistance in Gonorrhea", which can be found here. I have one question regarding the method of non-dimensionalization. ...
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68 views

Find the expression of a function close to this drawing

I am trying to model mechanical response under constraint in materials, and I know that I need this kind of function if I want my model to be accurate. The point is, I have absolutely no idea about ...
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1answer
94 views

A question in application of derivatives and vectors

Consider a skier who is sliding without friction on the hill ${y = h(x)}$ in a two dimensional world. The skier is subject to two forces. One is gravity. The other acts perpendicularly to the hill. ...
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1answer
68 views

Computational compass-and-straightedge constructions

I recently came across Ancient Greek Geometry, a web toy by Nico Disseldorp, and it got me wondering: is there a way to exactly model the points generated by geometric constructions, starting from two ...
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1answer
130 views

Mathematical Modelling Homework

Considering a situation in which two motorists, person A and B, share the same driving route but own different sized vehicles. Person A fills up the vehicle’s tank at a station along normal route for ...
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1answer
396 views

Expected utility and St. Petersburg paradox

Can someone explain to me how they get the $10.94$ at the Expected utility theory section of the solutions to the St. Petersburg paradox? My problem is that they use a formula to calculate the ...
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18 views

Number of significant linear predictors if predictors are not independent?

I would like to determine which of a set of candidate predictors $\{x_1, x_2,\ldots, x_n\}$ are significantly relevant to the linear prediction of $y$. Typically, one can compare a full model ...
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48 views

Form of a function I can fit like a polynomial that has an asymptote on x-axis and is always positive

So I have a small list of pairs of the form $(\mathbb{R}, \mathbb{R}_{\geq 0})$ and I want to fit a function to this data. Additionally I know that as $x$ grows large in either direction $y$ tends to ...
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1answer
195 views

Start studying mathematical biology from basics

I am really passionate about theoretical and quantitative biology and I would like to build my future career around this topic. I've just got my bachelor's degree in biology (ecology) but scince ...
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2answers
47 views

Rounding a real number w.r.t. a given amount of steps

Let $x$ be a real number, $x \in [0,1]$. Suppose a system can only provide a noisy signal about the value of $x$, given the granularity allowed by the system, $N \in \mathbb{N}^*$. I'm looking for an ...
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1answer
130 views

Definition of “epidemic” when using SIR models

I haven't studied differential equations for a long time, but I have just started looking at material on the SIR model of epidemics. My problem is that the resources that I've looked at haven't given ...
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492 views

There are 5 servers. Each server has 1% downtime. What's the probability that at at least three servers are down?

There are 5 servers. Each server has $1$% downtime. What's the probability that at at least three servers are down? My reasoning is the following: A) There is $(1-0.01)^5$ probability that $5$ from ...
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60 views

Finding the largest eigenvalue of a sparse matrix

I would like to find the largest eigenvalue of a sparse matrix by hand- this is part of analyzing a mathematical model for infectious diseases. The nonzero entries are very complicated - hence Maple ...
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1answer
66 views

Finding a value that makes an expression negative

Background: I am working on the mathematical modeling of infectious diseases, namely HIV and TB. In the process of proving global asymptotic stability of the Disease-Free Equilibrium, I must ...
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2answers
111 views

Units of time in simple differential equation

Very simple question: There is a well-known model in epidemiology called SIR model. It describes the changes in the number of susceptible, infectious and recovered individuals in a population. It is ...