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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Differential Equations & Sinusoidal Functions

I am developing a model for sales forecasting, the basic premise of which is that the rate of change of sales is proportional to the number of possible buyers. In developing this model, I came up with ...
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91 views

I'm trying to understand the solution to this simple mathematical modelling question

This is an example from Gary Chartrand's "Introductory Graph Theory" (Page 8, Example 1.5). He provides an answer without actually showing the steps or working, and I am unable to figure it out. The ...
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Eigenvalue/eigenvector calculation for linearly changed matrices

I'm looking for a way to speed up my computation that involves solving a moderate size generalised eigenvalue problem (matrices 400x400) large number of times depending on the parameter $m$. The ...
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170 views

Summation notation problem

Any help is greatly appreciated! Outline: Hermione has been thinking about the imminent return of the Dark Lord, so she has been busy packing her bag with all the items required for her survival. ...
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252 views

Operations research - linear programming help appreciated.

George Weasley, the owner of Weasleys' Wizard Wheezes, recently found that his Skiving Snackboxes have become extremely popular amongst the students at Hogwarts School of Witchcraft and Wizardry, and ...
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14k views

How can I find the time constant of a first order system transfer function?

How can I obtain the time constant of the transfer function of a first order system, such as the example below? $$ \frac{C(s)}{R(s)} = \frac{2}{s + 3}$$ Where $C(s)$ is the output of the system and ...
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271 views

Interpretation of Phase Portrait

I have the following system $x'=f(X)$ of ODES: \begin{align} x_1'=& -4x_1^3(x_2-2)^2 \\ x_2'=& 2x_1^4(2-x_2) \end{align} Solving for equilibria: I got $1$ at $(0, 2)$. I plotted this and I am ...
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163 views

How to define a surface $z = f(x,y)$ with flat region at centre and sigmoidally tapering towards the edges?

How do we define a continuos function $f(x,y)$ within the bounded domain $x \in [a,b]$ and $y \in [c,d]$ so that $z=f(x,y)$ has a flat surface at the centre (flat means $f(x,y)= C$, $C$ being ...
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352 views

Classify the fixed points at the origin

Consider the linear system $$\frac{dx}{dt}= -3x+2y, \frac{dy}{dt}= ax+6y, a \neq -9$$ classify the fixed point at the origin? Is the correct approach to investigate the steady states and how these ...
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84 views

unit conversion doubt

Im a developer that likes maths, but dont have much experience, I had constructed a graphic that animates a bar that grows on the X axis according to a value. So, I have the values for the x ...
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196 views

Invert this function $y=(1-x)e^{-x}$

Consider the fucntion $f:\mathcal{R}\rightarrow \mathcal{R}$ given by the rule $ f(x)=(1-x)e^{-x} $ Now I want to invert this function(not just for fun but I have a data that seems to fit this ...
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166 views

Given average network diameter, how many nodes are three hops away.

The University of Milan found in 2011 that everyone on the Internet was, on average, 4.74 steps away from anyone else.. is that information sufficient to answer this question: What proportion of ...
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84 views

Closed-form solution

I wrote down this equation which is the mathematical model of a system. Is there any way to get $V_c(t)$ in a closed-form expression? $V_c(t+d) = V_s \cdot U\left(V_r ...
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139 views

How to solve the advection equation with spiral motion

The advection equation is : $$\frac{\partial f(x,y,t)}{\partial t} + \nabla_{(x,y)} \cdot (A f)= 0$$ With initial condition $f(x,y,0) = f_0(x,y)$. If the vector $A$ is constant, ie. $A = ...
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1answer
254 views

Euler angle in ellipsoid rotation

I am modeling an ellipsoid tumbling in a flow field. I have derived an expression for the Euler angle $\phi(t)$ of the rotation in the $x$-$y$ plane as a function of time, but its range is only $\pi$, ...
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47 views

How to visualize a function to aid in optimization?

I am working on minimization of a function with more than two parameters. I want to see variations, local extremes, saddle points, etc. of this function to get some idea of how is this function in ...
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38 views

Predicting time t given time t-n

..Hi, Everyone! I'm having a bit of trouble with a statistics/forecasting problem and I could use some help. I have temperature measurements for each hour of each day for the past 10 years. Given the ...
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simplifying equations from model

I'm using the community land model (CLM) to do some university work and I'm trying to understand some of the equations that go into the model, one example is: $$ ...
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41 views

Linear Fit Issue

Consider a quantity $Q$ that changes as a function of time. The function $Q(t)$ is not explicitly known. We know that $Q(t_{0})=Q_{0}$. Assume for $t-t_{0}\le\epsilon$, we have a method to estimate ...
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159 views

Statistics on three variables/factors

I have what is probably a simple problem. I'm trying to say something about sells of candy bars. I got data of sells for a population of children. There are three parameters (factors) to this data: ...
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110 views

Math model - constrain GDP given different growth rates of industries

ideas needed to model national GDP given different sector growth rates subject to some contraints Given: GDP equations for $n$ industries depend on growth rates and time i.e. $g(r_1,t), g(r_2, t), ...
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40 views

Regression model for a shearing process

30 Widgets are randomly assigned to a shearing process. There are 3 such processes, each getting 10 widgets. The lengths of each widget are recorded before undergoing the shearing. The amount that ...
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1answer
806 views

Python numerical solution for a nonlinear second order ODE with two boundary conditions

I want to solve numerical the next equation, in Python $$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$ it is a nonlinear second order $ODE$ with two $B.C$. ...
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is possible to deduce a formula for vaccination coverage from Whittle threshold theorem?

I am reading the article Application of Whittle’s stochastic threshold theorem to a chickenpox outbreak and I can't understand the meaning of the term "intensity" in the Whittle threshold theorem. In ...
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2k views

True Velocity and Heading

An airplane flies at $670$ MPH directly northwest. Wind blows at $70$ MPH from the west (i.e the wind is blowing towards the east). Determine the true velocity and heading of the plane. Steps: ...
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275 views

Pursuit curves and arc length question

I am studying pursuit curves where a fast pirate ship which pursues a heavily laden treasure ship which tracks along a straight line. The ratio of the speeds of the ships is r > 1 (which is fixed) and ...
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70 views

Acceleration of series convergence

everyone! I am currently struggling with following problem: compute the series $$ \sum\limits_{m=1}^{+\infty}\frac{1}{m}\sin(m\alpha)(\cos(m\beta_1) - \cos(m\beta_2)) $$ and $$ ...
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203 views

Sunk cost auction modeling.

Consider the following auction concept. I call it a "SUNK COST AUCTION" Each person bids, but you pay all of the money that you bid for every bid you make. So, if you bid \$1 that \$1 is gone, even if ...
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How to define an objective function that conveys the concept of selecting the best elements in a set

Consider a set of tasks $\mathcal{T} = \{t_1, \ldots, t_I\}$. Consider also a set of workers $\mathcal{W} = \{w^1, \ldots, w^J\}$, where each worker $w^j \in \mathcal{W}$ is associated with a value ...
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232 views

Water Systems: When can I use buckets of water to simulate an ODE.

It is quite common to use physical systems to perform calculations (see here and here). This is for a number of reasons: sometimes the physical system is efficient, sometimes it helps us understand ...
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618 views

Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf a Turing bifurcation occurs when for an ODE and related PDE $u' = f(u,v), v' = g(u,v)$ $u_t = d_1 \nabla ...
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How do you set up a system of ODE's for this problem?

The problem is as follows: Black and White balls are being created inside an arbitrary volume at rates of $Q_{B}$ and $Q_{W}$. They also disappear from the volume at rates $\lambda_{B}$ and ...
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Calculating time of flight of object (Mechanics - horizontal elevated launch)

I need help with the following question: A smooth spherical object is projected horizontally from a point of vertical height H = 25.78 metres above horizontal ground with a launch speed of u = ...
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55 views

Approximation of beam

Assume that there is a simply supported beam subjected to concentrated moments $M_0$ at each end. The governing equation is $$EI\frac{d^2y}{dx^2}-M(x)=0$$ with the boundary conditions $y(0)=0$ and ...
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How to calculate probability with sigmoid output in feedforward neural network?

first of all I'm sorry for my not very skilled English, but I will do my best to explain my problem. I'm trying to create a feedforward neural network with one hidden layer (with probably arctan ...
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49 views

What method or subject deals with the following?

Can you point me to areas and methods that deals with modeling output from input data? Say we think our output depends on certain parameters, and say we have samples of this output versus time and ...
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117 views

How can I model a rotating system which does not have constant acceleration?

I've sampled a rotating system to come up with a list of positions and velocities at certain times. I want to be able to predict how long it will take to reach a certain speed and how far it will have ...
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36 views

Model selection with different error distributions

I have three different models for a given set of data. Each of the models has a different error distribution. These error distributions are known. How can I decide what is the best model?
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65 views

matrix question - help needed

I am doing revision on matrices and came across this question. The solution (the matrix provided below the question) is there. I am not sure how or why 180 is in the position (1,4) (row and column ...
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2answers
254 views

Stochastic predator-prey

My system is a simple $P$ vs $I$ foxes- vs rabbits model given by: $$ \begin{cases} \frac{\mathrm{d}I}{\mathrm{d}t}=& \alpha_I+\lambda_IP- \gamma_II -\delta_IPI;\\ ...
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154 views

special matrix in terms of its covariance matrix

How can we find a matrix $S\in \mathcal{M}_{n,n}$ and $Z\in \mathcal{M}_{n,m}$ whose $n$ entries of the $i^{th}$ column $Z_i$ are correlated $Z_i \sim \mathcal{N}(0,S)$ where $S \in \mathcal{M}_{n,n}$ ...
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1answer
347 views

How to classify equilibrium points

I have the two differential equations: $$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$ $$\frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$ I worked out the equilibrium points to be at $N_1 = 0, \frac{4}{5}$ and ...
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49 views

Solving a form of the logistic equation to arrive a given solution

I am writing my bachelor thesis on modelling of city growth and using the book Cities and Complexity by Michael Batty. On page 394, while modelling the growth as spatial epidemic, he writes: ...
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134 views

Shape of a Kite String

A kite is flown at some angle and elevation. Without wind, the string takes the shape of a catenary curve. But the wind is crucial. Is there a name for the shape of a kite string -- is it some ...
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Boltzmann machines - motivation for the energy function

I've been studying Boltzmann machines lately and was wondering if anyone could give me a "high-level" explanation or motivation for the energy function used: $$E = -\sum_{i<j} w_{ij} \, s_i \, s_j ...
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2k views

How to make simple iteration in Mathematica

How to make simple iteration in Mathematica for this three equations and save $q$ for each step. If we define $q'=dq/dx$, $q''=dq'/dx$ we have two equations (1) ...
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I need a differentiable function whose plot is a plateau and the steepness and width can be varied arbitrarily and easily

I need to model the solar radiation incident on a solar panel. I tried using $$\tanh(b*(x-a))-\tanh(b*x)$$ but it does not give me a lot of flexibility with the characteristics of the curve, namely ...
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2k views

Calculating equilibrium point of non-linear ODE with free parameter

I have two ordinary differential equations equations: $$ \dot{x}=1+x^{2}y-(1+A)x $$ $$ \dot{y}=Ax-yx^{2} $$ I need to find the single equilibrium point in terms of $A$. So set $\dot{x}$ and ...
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1answer
65 views

Amount of information a hidden state can convey (HMM)

In this paper (Products of Hidden Markov Models, http://www.cs.toronto.edu/~hinton/absps/aistats_2001.pdf), the authors say that: The hidden state of a single HMM can only convey log K bits of ...
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43 views

Trouble understanding a part of the book Cities and Complexity

I am writing my bachelor thesis or whatever it is called about modelling of city growth, using a book called Cities and Complexity by Michael Batty. It is not that mathematical, to be honest, it has ...