# Tagged Questions

A vast area which includes generating results from computer models.

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### Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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### Calculate Density of Values in Cellular Automata

I am working with a special cellular automata that uses hexagonal cells rather than square cells, a hexagonal grid, rather than a square grid, and the set of complex numbers, rather than a finite set, ...
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### Simulation of a diffusion on $[0,1]$

I have a diffusion process $X=(X_t)_{t \ge 0}$ with the generator $$Af(x)=\frac{1}{2}(a(1-x)-bx)f'(x)+\frac{1}{4}x(1-x)f''(x),$$ where $a,b >0$ are constants. I want to simulate $X$ to a ...
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### Simulate a non-homogeneous poisson process

I was trying to simulate a non-homogeneous poisson process with hazard rate function $$\lambda(t)=3+\sin(2\pi t)$$ I tried to use the property that given $N=n$, arrivals in $[0,T]$ are distributed ...
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### Why does this cellular automaton generate circular patterns?

I made a kind of cellular automaton game with the following rules. Each cell in a rectangular grid has a "water level" (a 32-bit floating-point number). In the next generation, water "flows" from each ...
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### Quantile lines of stationary process

Quantile lines of any stationary process are parallel and constant. But for different procesess I've obtained different behavior of quantile lines. First case was process after Lamperti transormation ...
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### Self learning complex systems modeling

I am currently a senior studying mathematics. My program is heavily focused on pure maths, however, my interests lie in the area of applied mathematics. Specifically, I am interested in the study of ...
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### interarrival time

I am trying to model the arrival rate (delay between the arrivals) of the cars in my city. I have some real data with the resolution of one minute. For example, Number of counted cars at position x at ...
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### Determanistically skipping through time of a time homogeneous Markov chain

Suppose I have an infinite number of time steps $X_0,\ldots,X_i,\ldots$, where each $X_i$ is an infinite dimensional random vector consisting of 0's and 1's. I now specify $P(X_i|X_{(i-1)})$ and an ...
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### How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...
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### Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
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### Distribution of phone calls during 24h

I would like to model the amount of phone calls at each time of the day. The phone calls should follow a poisson distribution and at 12:00 there should be the peak. So, semantically what I would like ...
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### Forest fire simulation; analytically constructing a function for tree residual after fire

Consider a Cellular Automaton with an $n \times n$ grid, where each cell corresponds either to a tree or dirt. We assign a tree to cell $(i,j)$ by probability $p$. Next, we initiate a fire in some ...
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### Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
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### Maximizing noisy unknown function

I'm interested in maximizing a function $f(\mathbf \theta)$, where $\theta \in \mathbb R^p$. The problem is that I don't know the analytic form of the function, or of its derivatives. The only thing ...
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### Expected value involving a conditional multinomial distribution

$X|z$ has a multinomial distribution $MN(m, \mathbf{q}(z))$ where $z$ is either 0 or 1 with probability $1/2$. I need to find: $E_X[\max\{\Pr(z=1|X), \Pr(z=0|X\}]$. Is there an analytical form to this?...
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### Semi implicit integration - stability issues

I am trying to decide whether to use semi-implicit integration vs. explicit integration (particularly Position Verlet over Semi implicit Euler). Although the Verlet approach is widely used and is ...
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### Simulation Model for Estimation of Flood qauntile at Ungauged Site

I have proposed a new model to estimate flood quantile at ungauged site. The model is wavelet GMDH. In order to test the strength of my proposed model i want to design a simulation for ungauged ...