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 does one solve ODE with some domain constraints?

So far I have no clue, how to go about this .. in school we did not go over constraint based ODES. So let's say we have $\ x'' = -ax' $ $\ y'' = -ay' - b $ Note: a,b are constant and x(t), y(t). ...
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Question about noise term in SDEs

Do any properties/assumptions of SDEs prevent the noise term from being extremely large? Using a simple population growth model as an example, $\frac{dNt}{dt} = (r_{t} + W_{t})Nt , N0$ given, ...
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26 views

Finding the correction factor of a model

If I have a model, and data against that model. The model says that it should be linear, and the data begins linear and drops away from the line as x decreases. Is there some function I can multiply ...
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69 views

How to make a cobweb diagram

I am struggling making a cobweb diagram for the function $$x_{t+1}=8x_t/{1+2x_t}$$ So I understand when making the cobweb diagram, that I have to draw the line $y=x$ But where I have trouble ...
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36 views

Discrete Population Model Crashing

Consider the discrete population model $$U_{t+1} = au_t^2/(b^2+u_t^2)$$ Where $a >0$ If $a^2 > 4b^2$ show that the population may be driven to extinction if it becomes less than a critical ...
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52 views

Finding eigenvalues to classify the steady state of a system

I have this system of differential equations which model chemical concentrations in a certain reactions: $$\dot{x}=a-x-\frac{2xy}{1+x^2}\qquad \dot{y}=bx\left(1-\frac{y}{1+x^2}\right)$$ for $a,b ...
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19 views

Graph analysis and phenomenon modeled by the curve $S(t)= 1/(1+e^{-t})$

So the question is: The graph looks like: So where I stand with this question is that I am struggling to find something that it models. Once I find some phenomenon that it looks similar to, I ...
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62 views

Interpreting phase-plane portraits

I have found the phase plane for a "predator-prey model" system of differential equations: $$\dot{x}=x^2-x^3-xy \qquad \dot{y}=yx-ya$$ where $x$ represents the population of prey and $y$ represents ...
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57 views

What probability analysis did the allies use during WWII after they cracked the Enigma, or how can this probability be quantified?

In the recent film, The Imitation Game, after cracking the Enigma they mentioned that the allies didn't simply use every cracked message but instead analysed the probability the Germans would find out ...
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16 views

How define a disturbance figure as input?

I have a 2 dimension figure of disturbance.I have to consider this as input of model predictive control. How can i do this.How can i have a model for this disturbance?
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Interpreting differential equation models

Consider the model $$\dot{x}=x[x(1-x)-y], \qquad \dot{y}=y(x-a)$$ where $x \geq0$ represents the population of prey and $y\geq0$ represents the population of a predator, with $a\geq0$ as a control ...
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51 views

intuition for mixing problems with ODEs

assume we have a mixing problem with a salt solution coming into a water tank. the flow rate in and flow out rate are equal (5 L/hour), the concentration of salt flowing in is 1 g/L and the tank ...
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203 views

Four Color Theorems: Graphs vs. Maps [closed]

This question has changed dramatically from its original form. Please See the improved question. ORIGINAL QUESTION: There are two variants of the four color theorem that are commonly cited: (4CTG): ...
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solving linear ODEs for mixing problems with function notation

consider this mixing problem for differential equations: a tank holds 100 L of water that initially has no sugar in it. Sugar water with 5 grams / L of sugar enters at rate of 2 L per minute. Water is ...
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22 views

Modelling: Use Newton's law to write down an equation for the position $x$ of the mass.

Here is the background for the question: Consider a one-dimensional frictionless spring-mass system, where the forces acting on the mass $m$ at position $x$ are the forces of gravity $F_g =-mg$ with ...
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32 views

Is Principal Component Analysis applicable to this type of situation?

I'm trying to model the response of ant populations to pheromones in this way: The ants are simulated as Self Propelled Particles with internal energy. They undergo acceleration due to their internal ...
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104 views

SEIR model need help solving first order linear ODE

I've recently been working on simulating an SEIR (susceptible, exposed, infected, recovered) project for an endemic disease using matlab solving via Euler's method. I have taken a picture of the ...
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37 views

Parabola properties assumptions

I am trying to model projectile trajectory but I'm having some trouble. I didn't realise parabolas are this complicated... I have some assumptions that I would like to be clarified. If I specify a ...
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53 views

Parabola describing projectile motion.

I am trying to create a function that will generate a parabola that describes projectile motion. Here are my inputs: The starting x-y coordinate of the throw The initial x-y velocity vector. I ...
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35 views

Questions about Disease Model

A non-fatal infectious disease divides a population into two groups: normal or ill. Assume that the average number of contacts that each ill individual has with normal individuals is $a$ multiply the ...
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1answer
46 views

Discrete Dynamical Systems & Credit Card Debt: How to solve for payment

I have the following problem, taken out of Giordano, Fox, and Horton's A First Course in Mathematical Modeling: Your current credit card balance is $\$12,000$ with a current rate of $19.9\%$ per ...
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Lotka-Volterra equations modified

Okay, so I'm just learning about the Lotka-Volterra and the question I have regards the following model: $dx/dt = x(1-y/2)$ $dy/dt = -y(1 - (x/0.8) + (x^2)/4)$ I need to state what term has been ...
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Interpretation of a model

$$ \frac{dx}{dt}=ax(x-b)(1-x)-\frac{xy}{1+cx}$$ $$\frac{dy}{dt}=-ey+\frac{xy}{1+cx}$$ Make a apropriate interpretation of this model. What I thought are: $\frac{dx}{dt}\ \& \ \frac{dy}{dt}$ are ...
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Indicate whether each of the following equations is linear or nonlinear.

Indicate whether each of the following equations is linear or nonlinear. If linear, determine the solution; if nonlinear, find any steady states of the equation. $x_n=(1-\alpha )x_{n-1}+\beta x_n$ ...
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A mathematical abstraction of colours?

Does any formal system exist built upon the mechanics of Additive Colour Theory? Additive Colour Theory is the description of the behaviour of the visible spectrum of light as it combines-the ...
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39 views

Critical points of quadratic forms

Let $A$ be an $n\times n$ symmetric matrix, let $b$ be an $n$-vector, let $c \in \mathbb{R}$ and set $Q(x) = 1/2 x^T Ax-x^T b+c$. Prove that $x_0$, defined as a solution to $Ax_0=b$ is a critical ...
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26 views

find the maximum

I'm stuck in this problem (Instructions and my work is showed on the picture below) I just don't know what to do right after find the values of D, to prove that the maximum of F(D) is attained at the ...
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Modeling Question: Finding a Fit for the form W=kl(g^2)

I have a modeling question on my assignment that I am unsure about. I am given a set of points $W, l,$ and $g$. I have to find some $k$ for the data to optimally fit $W=klg^2$. At first, I tried ...
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36 views

Lotka-Volterra coordinates transformation

I would like to ask the following: Given a Lotka-Volterra predator-prey system, \begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} , with ...
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Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...
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108 views

Differential Equation - Blood Alcohol

Let C(t) be the concentration of alcohol in a person's bloodstream. We propose a mathematical model which states that, once a person stops drinking alcohol, C decreases at a rate that is proportional ...
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33 views

Determining probabilities Markov Chain

If I have a Markov Chain $X_0, X_1, X_2 \dots$ that has a transition probability matrix $ \textbf{P} = \matrix{~ & 0 & 1 & 2 \cr 0 & 0.3 & 0.2 & 0.5 \cr ...
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108 views

Modelling Concentration

I'm currently doing a research project that involves modelling E. Coli growth in a wetland. The data I've been given is the E. Coli mass concentration ($mgC/L$) at various times throughout the two ...
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1answer
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For the three measurements b=0, 3, 12 at times t=0, 1, 2 find the best parabola y=C+Dt+E$t^2$

So I know how to do least squares regression using matrices to solve for Ax=b. I simply do $A^TAx=A^Tb$. However I don't really know how to account for the second power in a typical parabola ...
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25 views

What is the least squares solution given a line passes through original and following points?

So I am looking for the line y=Dt through the origin that fits the data y=4 at t=1, y=5 at t=2 and y=8 at t=3. This is what I have done so far. I know the three equations that are supposed to be ...
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38 views

Write the system of equations describing the populations in this system…

In a closed ecosystem, we have that Frogs,represented by $F(t)$, eat fleas, represented by $f(t)$, and the fleas eat fungus, represented by $g(t)$. Assuming that fungus grows at rate A, fleas eat the ...
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58 views

Model a chemical phenomenom

I have a chemical phenomenon to model for my research. I'll try to be clear while explaining my problem. I created little molecules (let's call them "emitters"), which emit a signal. When the ...
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1answer
84 views

What is the process of nondimensionalizing an equation?

Question: I need to scale time by $\frac{1}{I}$ and species by $P$ for the following equation $\frac{dS}{dt}=I(1-\frac{S}{P})-\frac{ES}{P}$ where P - Size of the source pool of species on the ...
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1answer
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Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
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1answer
18 views

Finding the minimum value of $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$

I am trying to find the minimum value of the following: $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$ I know this is equal to: $1/2 ([x_1 x_2] Id [x_1 x_2]^T )-[x_1 x_2] [0 b_2]^T$ To find minimum we have to ...
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What is the proper name of a model that takes as input the output of another model?

Thanks in advance for the help. I am writing a paper and for the life of me can't remember the proper term for a model that works as follows. rawData -> model1 -> outputModel1 -> model2 -> ...
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40 views

Online resources for Mathematical Modeling

I'm taking mathematical modeling this semester in college and usually there are resources online to read examples and see another point of view but I'm not able to find it for this class. Can anyone ...
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36 views

Calculate migration probability matrix from distances matrix

Imagine the world-wide human populations as a series of interconnecting populations. The distances between any two populations is given by the following kind of matrix $$ \begin{matrix} ...
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How many additional crews should be brought in to minimize the cost of an oil spill?

An oil spill has fouled $200$ miles of Pacific shoreline. The oil company responsible has been given $14$ days to clean up, after which a fine will be $10000$ \$/day. The local cleanup crew cleans $5$ ...
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105 views

Linear algebra of state space representation won't be linear (superposition theorem)…

After answering a question about calculating the state space representation of a circuit with 3 sources in it (the circuit is there), I had a doubt - while checking, it became clear there is something ...
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What is the correct term for the equations which comprise a mathematical model?

I have a mathematical model constructed by myself and my supervisor. In writing my report do I refer to the equations that make up this model as "constitutive equations", or is there another term ...
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104 views

Modeling Forced Oscillations Resonance Given from Second Order Differential Equation (2.13-3)

My answer to this problem does not agree with the text's answer: $y=sin(t)-3cos(t)$ Problem to solve is to find the steady state oscillations of the vibrating system governed by the following ...
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44 views

Saturation Modeling in ODE45

I have a machine with an arm that can move in a linear one dimensional way. There are 3 limits on the arm: The arm has boundary for its location $(x_{min},x_{max})$ The arm has limit on its velocity ...
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103 views

Math theories in Game Theory

What are all the mathematical theories in Game Theory? I have taken Mathematical Modelling, including: application of linear systems, matrix operations, inverse of matrix, leontif input-output model, ...
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Positivity of compartments in epidemiological model

Given the following dynamical model (system of ODEs): \begin{array} $ \frac{dA}{dt}=\Lambda-\mu A-\beta(C+D+E+F)\frac{A}{N}-\tau(B+D)\frac{A}{N} \\ ...