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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Regression analysis model vs mathematical model

Can anyone explain the difference between the equation generated from a regression analysis as opposed to a mathematical model such as $E = m \cdot {c^2}$ type models?
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Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
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Forming a differential equation from game

I was wondering if someone could help me form a differential equation from the following game: A population consists of two types of diets, fish and veg. People play a with every other person and the ...
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48 views

Linear model, Newton's Cooling law

A thermometer is removed from a room where the temperature is 70 and is taken outside, where the air temperature is 10. After half a minute the thermometer reads 50. What is the reading of the ...
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Websites or books on modelling

I have some operational data that I'm looking to model. However I'm a bit lost as to where to start. I'm looking for some websites or books on modelling data. Can anyone provide some advice?
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28 views

Trajectory Code Problem

Problem 2. “Pumpkin chucking” is a competition event to see which team can shoot a pumpkin as far as possible, usually with a pneumatic cannon. In this problem we’re going to write a computer program ...
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50 views

Help needed for a step by step calculation

I hope I can get some help to understand step by step calculations for formulas below. If I have a weighted distance formula like below: $d(O, P) = \sqrt{\frac{\hat{x_1^2}}{\hat{s_{11}}} + ...
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23 views

Question about problem linear programming math modeling

Consider points $A(4.7,−4.1,−1.5)$,$B(−0.4,−2.4,1.9)$,$C(−0.3,−2.1,−6.5)$ and $D(2.7,−3.6,4.0)$. How to discover if segment $AB$ has intersection different of zero with the segment $CD$? Formulate ...
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46 views

I'm having trouble finding the right direction to take in this HIV modeling exercise

Cells that are susceptible to HIV infection are called T (target) cells. Let $T(t)$ be the population of uninfected T-cells, $T*(t)$ that of the infected T-cells, and $V(t)$ the population of the HIV ...
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1answer
99 views

Notions of consistency / heterogeneity in sets of vector values

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
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Modelling a continious-time queue which behaves differently when there are more or less people being served.

For my research I am trying to model a continuous-time queue which behaves differently when there are more or less people being served. The arrival rate in the queue is constant, however the departure ...
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51 views

After how many bounces is the ball subsequently always less than 5cm from the ground?

a). A ball is thrown vertically upwards with a speed of 20 metres per second from a point 3 metres above the ground. Find the speed with which it hits the ground. b). If the ball rebounds with a ...
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32 views

Stability of a function with logaritms

I am trying to proof that the following equation is stable if $|1-ln(\alpha)|<1$; $$f(x)=\alpha x e^{- \beta x}$$ So the equilibrity points would be $x=0$ and $x=\frac{ln\alpha}{\beta}$, the ...
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Learning to Apply Mathematical Concepts ( i.e. function modelling, etc.)

Firsty, I want to state my situation clearly. I am one of those students who are incredibly good at absorbing mathematical concepts but without knowing how to apply them. I get A's but it is growing ...
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43 views

Formulating equation for distances between atoms

I'm trying to formulate notation to describe code that calculates the distances between protein atoms (represented as points in 3D space). Fragments consist of residues (amino acids) and each residue ...
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1answer
28 views

Best method for assessing repeatability between year 1 and year 2

I have a data set of values for a particular measured metric for employees for the year 2013, and a set of values for the same measured metric for the year 2014. I am looking to assess the ...
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42 views

Competition Model - System of differential equations

Two plants are feeding off the same substrate, whose weight at time t is S(t). One of the plants began to feed off the substrate 20 days earlier than the other. Devise a system of differential ...
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54 views

Differential Equation of a Probability Density Function

I have a deterministic rule of my variable (ODE model), $\frac{dY}{dt} = at$, in which $a$ is a parameter in my deterministic model (let's say growth rate) and $t$ is the time. I know that on every ...
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24 views

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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31 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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1answer
86 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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53 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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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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71 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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59 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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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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39 views

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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55 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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209 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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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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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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122 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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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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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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37 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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48 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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1answer
40 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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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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37 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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113 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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1answer
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 ...