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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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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81 views

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 ...
2
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1answer
790 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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3answers
432 views

How to find the length of a curved path.

We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog ...
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1answer
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
64 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 ...
2
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1answer
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 ...
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64 views

Curvature of surface

So lets say I have a mesh and for each face I have the position of its $3$ vertices and the area of the face. So let's say I have a point $p$ on this face and a vector $v$ that goes from $p$ to the ...
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1answer
221 views

Looking for a Lyapunov function for the next system

I am really stuck looking for a Lypaunov candidate for the next system (which in simulation is stable). $$ \dot{x} = -(A+A^T)x + Ay \\ \dot{y} = K(x-y) $$ where x and y are vectors in R^3, A is a ...
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2answers
363 views

Slide Puzzle logic??

say there is an image made into a slide puzzle in a grid of sections 4 wide and 6 high, the missing piece that is missing so you can slide the other pieces of the puzzle around is always the bottom ...
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3answers
871 views

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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25 views

Is there a trade model which takes in account credulity?

I have to admit I'm not to familiar with mathematical economics but, as a student in mathematics, I was trying to play a bit with a toy trade model I'm try to build over a finite population. My ...
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1answer
107 views

how to estimate a variable's upper limit with 95% confidence level?

Suppose I have a variable, $S(t)$, for stock price. So always have $S(t) > 0$. $S(t)$ is a random variable, with some volatility $\sigma$ and trend. Now the requirement is to estimate $\hat ...
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1answer
78 views

Computing the length of a path

The rectangular grounds at Hillingham University is going to have a new path built which takes the curved shape of $y=2.2x^2$, starting from the south-west corner - (taken as the origin in a graph) ...
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1answer
80 views

Calculating the expected number of items shared by chance in a Venn Diagram

I have a Venn Diagram that looks like this: $$A) \, 213 \quad B) \, 160 \quad A\cap B) \, 100$$ The items from $A$ come from a population where their probability to be selected is ...
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1answer
199 views

How large initial angular velocity is needed for pendulum to go completely around?

If a pendulum is initially at its unstable equilibrium position, how large an initial angular velocity is necessary for it to go completely around?
5
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2answers
132 views

Are there rigorous mathematical definitions for these waves?

My friend linked this .gif to me tonight, and asked me if I knew of any equations that might model these bottom two waves (the blue and green waves). Unfortunately, I am not far enough in my education ...
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1answer
54 views

a line perpendicular to a given line

I am confused now, I have a 2D line. If its equation is $r = x\cos(\theta) + y\sin(\theta)$, then what will be the line which is perpendicular to that line? Where $r, \theta$ is described ...
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2answers
451 views

Bookshelf problem

There is this thought problem I've been trying to solve, it goes as follows Imagine a bookshelf with a finite number of books in it, to which a finite number of people have access. Each person has a ...
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1answer
256 views

Calculating Humidity, Wet Bulb Temp and Dew Point [closed]

I have been trying to simulate a climate system, but I have hit a wall. I am trying to calculate Humidity. But to calculate Humidity you need the Wet Bulb Temperature, but to calculate the Wet Bulb ...
3
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1answer
382 views

exponential population growth models using $e$?

Im trying to understand this write up [1] of cell population growth models and am confused about the use of natural logarithms. If cells double at a constant rate starting from 1 cell, then their cell ...
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2answers
117 views

Linear Models - Regression Analysis

As a student learning Applied Regression Analysis, I come from a background with very little information about this topic. I understand that given $y = \beta_0 + \beta_1x_1 + \epsilon$ $E(y\mid x) = ...
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1answer
70 views

hopf bifurcation for an ode

I understand how to analyse a system of equations like $x'(t) = f(x,y)$ $y'(t) = g(x,y)$ set $x'$ and $y'$ to zero and find the fixed points etc, and find the stability. What Im am not sure of ...
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50 views

How to model a system with multiple probability distributions, each for a part of the system?

I need to build a complex probability model to describe some "real world" scenarios. The system consists of several types of objects, and the contraints upon these objects and their interactions are ...
2
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1answer
502 views

Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?

On this math.stackexchange on url What is Octave Equivalence? in an answer on the related ( octave equivalence ) question is stated: Mathematically, this signifies that the mammalian cochlea ...
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Is there a general solution for the “Spider and the Fly Problem”?

(I would be appreciative if somebody could give a more formal formulation of this problem.) The Spider and the Fly Problem is a problem in which the objective is to minimize the distance the spider ...
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1answer
130 views

Creating a model

Here's a seemingly simple pondering. If one item is more valuable the higher it is (i.e., $a=5$ is worth more than $a=2$) and another item is more valuable the lower it is (i.e., $b=2$ is worth more ...
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55 views

Transport problem wth time management

Are there general methods for solving the transportation problem with additional conditions: Time management (transportation from point A to point B is so much) Transportation of more than one ...
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0answers
196 views

How to parametrize a function such that it approaches $f(0)=0$ and $f(1)=1$ with different speed

I need a function (polynome) that values $0$ at $0$ and $1$ at $1$ and has these values as local maxima and minima. So far so easy the straight solution is: $$f(x) = -x^4+2x^2$$ Now I want to ...
2
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1answer
146 views

Could you help me to find a model for this curve?

I am very bad in mathematics and I'm not able to find by myself the model corresponding to this kind of curve. I wish to have a quick growth at the beginning, then it should increase slowly for a ...
0
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1answer
46 views

How to solve these simultaneous equations:

$1 = 1/2y - bx,$ $1 = 1/2x - ay$ I need x,y it in terms of constants, but I can't figure it out!
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1answer
138 views

Iterative model fitting

I have a sequence of points $\{(x_k,y_k,z_k)\}$ and I need to fit some $2D$ model $P(x,y)$ that approximates $z$ in some sense. The $z_k$$'s$ are noisy samples of some $2D$ function $z_k = f(x,y) + ...
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1answer
161 views

Actuarial / Modelling question - difference between RUN date and VAL date

My understanding is that RUN date is the date you do the RUN (eg in prophet / moses / igloo / whatever, the date on which you performed the calculations) Whereas VALuation date is the initial 'as at' ...
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0answers
147 views

Should I get the absolute value of the result of the inverse discrete fourier transform?

The result of equation 36 can be positive and negative.And if I don't get the absolute value of it,the ocean surface tend to be very regular.But according to the paper,the author never get the ...
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1answer
458 views

Complex results in inverse Fourier transform for simulating ocean water

I don't understand the equation37 in simulate ocean water by Jerry Tessendorf.The result is all complex number, how to be the slope.Even if I compute the magnitude of it,the result is just positive ...
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0answers
25 views

Model to reward adaptative communication

I'm currently working in different simulations focused on the study of artificial evolution of communications in robots, and I have stumbled against the problem of its mathematic formulation. The ...
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3answers
2k views

Cat Dog problem using integration

Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation ...
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1answer
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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0answers
106 views

How to model bending, folding of 2D figures?

2D shape can be folded in various ways. For example, trapezoid can have its sides (with the acute angles) folded, so that it will effectively become rectangle – with the difference, that part of the ...
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0answers
124 views

Best way to statistically compare two polynomial models?

I'm working on modeling and comparing data sets, and I need a good way to see if there is a significant difference between the data from the 2 models. An example of the models are here. I initially ...
2
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0answers
152 views

Considering the predator prey model to find the range of values to be a spiral

I have the following problem: Consider the predator-prey model: $$\frac{du}{dt}=u(1-\alpha(u)-v), \frac{dv}{dt}=\rho(-1-\alpha(v)+u),$$ where $\rho$ and $\alpha$ are positive parameters with ...
188
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16answers
7k views

Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
3
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1answer
75 views

Inverse of itinerary function

Let $S^{-1}:\Sigma \to \Lambda$ be inverse of itinerary function. I showed that $S$ is continuous and bijective. How to show that $S^{-1}$ is continuous?
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1answer
441 views

Time-evolving probability distribution functions with an equation of motion

I came up with this question a while ago and haven't been able to gain any insight on it. You are playing baseball. As a batter with finite vision capabilities, the only information you have about ...
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1answer
283 views

a mathematical model for file system

We are supposed to come up with a mathematical model describing a file system and its operations. The file system stores metadata in a database, and retrieves it when necessary. How does one go about ...
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1answer
104 views

Finding a logical expression (under some constrains) s.t. it is equivalent to another one

In this question, it was made clear, when $\bullet$ some statement $A$ is stronger than another statement $B$, namely if $A\Rightarrow B$ holds; and when the statement $A$ is weaker than another ...
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1answer
77 views

Which variable represents the predator and which the prey.

Just a quick question I have the following system: $$\frac{dx}{dt}=x\left(2-x-\frac{y}{1+x}\right),\qquad \frac{dy}{dt}=3y\left(\frac{x}{1+x}-4y\right)$$ It asks which variable represents the predator ...
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1answer
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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1answer
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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41 views

using a scientific paper to model the temperature of pools in an irrigation system

Quite a general question... I am referring to a scientific paper : http://www.springerlink.com/content/41p817v1584q1850/ I am trying to model water temperature over a year. I have the initial ...