My way of doing this is to first assume that the limit does exist and the value is c.
Then when n0 is big enough, Sum[n^p, {n,1,n0}]/n0^{p+1} = c is a very good approximation. Thus,
Sum[n^p,{n,1,n0}]=c n0^{p+1} (*)
For the same reason,
Sum[n^p{n,1,n0+1}=c (n0+1)^{p+1} (**)
From equation (*) and (**), you can solve that c=1/(p+1).//Done
It is actually more difficult than what you think.
One naive algorithm is:
(1) Think of the sphere as the earth and give it a fixed north pole and south pole and a fixed arch as zero longitude.
(2) Choose a uniformly distributed angle between [0, 2pi] as the longitude of the new point and a uniformly distributed angle between [0, pi] as its latitude.
You might think, through this method, the chosen point is randomly distributed on the spherical surface. However, it turns out this method is wrong! It would give more probability on the poles than on the equator.
This is the talk I gave for journal club in physics dept in November, 2005. An interesting topic: “How E.coli find its middle”. The quick answer is that E.coli uses Min-proteins oscillations to position its middle. For more detail, I hope How E.coli find its middle helps.
This is the slide for my talk during APS Marching Meeting, 2005. The meeting was held in downtown LA. The name of my talk is “Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac:Insights From Physical Modeling”. The slides can be downloaded: Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac:Insights From Physical Modeling.
BacSim
Resources
* Homepage
* Jan-Ulrich Kreft’s homepage
* Local directory: BacSim
Version
No version info.
Distribution method
Free. Open source.
Documentation
I am not able to find. But since the source is provided, at least we can read the source.
What is it?
BacSim
* is an extension of Gecko.
* written by Jan-Ulrich Kreft (University of bonn, Germany)
* I am not able to find any document files. The comments in the source file seems not complete
Gecko
* is an ecological flight simulator written by Ginger Booth (a programmer with the Center for Computational Ecology, Yale Institute for Biospheric Studies) .
* started as a prototype based on John Holland’s Echo models. It then became one of the first programs to be based on the Swarm toolkit for multi-agent based simulations written in Objective-C, originally developed at the Santa Fe Institute.
* Now is pure java, using Ginger’s own CourseWare, a Java Toolkit for online simulators for the teaching of ecology.
Jan-Ulrich Kreft
* Wissenschaftlicher Assistent (~ Assistant Professor)
* His CV jan_kreft_cv
Individual-Based Models
* Online resource
* has gained a lot of popularity in the 1990’s
* are simulations based on the global consequences of local interactions of members of a population. These individuals might represent pretty much everything
* typically consist of an environment or framework in which the interactions occur and some number of individuals defined in terms of teir behaviors and characteristic parameters
* the characteristics of each individual are tracked through time.
* are also known as entity or agent based models, and as individual/entity/agent-based simulations.
* are a subset of multi-agent systems which includes any computational system whose design is fundamentally composed of a collection of interacting parts.
* differences with CA: there is an overlap between ibm and ca. Perhaps the significant difference is whether the simulation’s inner loop proceeds cell by cell, or individual by individual. (Although that distinction is muddied by parallel-processing hardware.)) The philosophical issue is whether the simulation is based on a dense and uniform dissection of the space (as in a CA), or based on specific individuals distributed within the space.
The structure of BasSim
* Currently using diffusion-reaction algorithm. This was made possible by decoupling growth and diffusion on the basis of the different timescales of these processes.
* was extended to multi-species, multi-substrate systems with biofilm geometry (inert substratum at the bottom and bulk liquid at the top of the vertical dimension, periodic boundaries in the other two dimensions), and the bacteria are now living in a 3D space.
* the ability of bacteria to produce and excrete EPS, where the EPS is essentially treated as inert biomass (no metabolism or growth), subject to the same rules of biomass spreading.
Some screen shots
* Colonies
* Biofilms
* Slimy biofilms
* Spoiled eggs
* BacSim: The program behind the IbM scenes
* Quantification of biofilm structure
* How biofilms promote altruism
Some concepts:
ibm: Individual based System
EPS: extracellular polymeric substance
Aquasim: Computer program for the identification and Simulation of Aquatic Systems
Resources
* Homepage
* manual
Purpose
To support environmental scientists in finding an «adequate» model of the system they are investigating. It allows its users to perform simulations using different models, to assess the identifiability and to estimate the values of model parameters (using measured data), and to estimate prediction uncertainty.
Versions
The latest version is 2.1a released in March 2001.
Distribution
* AQUASIM was developed mainly for internal use, but it is also available to commercial companies at a price of CHF 2000 (ca. US$ 1200) and to university institutes at a price of CHF 1000 (ca. US$ 600). This price includes binary versions of the program for one platform, a user manual and additional documentation. Technical support for program usage is not provided.
* Demo version for MS Windows is available.
Documentation
* Complete manual is available for download
* Mailing list
* AQUASIM Training Courses
About
* Developed by the computer and systems sciences department of Swiss Federal Institute for Envronmental Science and Technology (EAWAG).
* Written in C++.
Program Intro
* Allow users to define the spatial configuration of the system to be investigated as a set of compartments which can be connected to each other by links. The available compartment types include mixed reactors, biofilm reactors (consisting of a biofilm and a bulk fluid phase), advective-diffusive reactors (plug flow reactors with or without dispersion), saturated soil columns (with sorption and pore volume exchange), river sections (describing water flow and substance transport and transformation in open channels) and lakes (describing stratification and substance transport and transformation in the water column of the lake and in adjacent sediment layers).
* Different links: (1) Advective links represent water flow and advective substance transport between compartments, including bifurcations and junctions. (2) Diffusive links represent boundary layers or membranes, which can be penetrated selectively by certain substances.
* User can specify any set of state variables and transformation processes to be active within the compartments.
* For the model as defined by the user, the program is able to perform simulations, sensitivity analyses and parameter estimations using measured data.
* Calculated results can be plotted to the screen (window interface version only), written to a Post-Script file for transfer to a printer, or written to a text file for external post-processing
IBIS
Resources
* Home page
* Manual
* Local directory: IBIS
Version
* No version info.
* The last modify date for manual is 06/19/1998, so it seems that this package is an old one.
* The webpage states that it is in the alpha development stage.
Distribution method
* Disclaimer: This software is made available “as is” without any assurance that it will work for your purposes. The software may in fact have defects, so use the software at your own risk.
* Open source, free.
About
* Written in Fortran
* Developed by David J. Eyre and Aaron L. Fogelson (Department of Mathematics,University of Utah)
* Designed to run on SGI workstations that have OPenGL
* A software system for immersed boundary and interface simulations
* Depends on C-shell, TCl/Tk, gnuplot, Net PBM library, MPeg library
How to use
* A user is required to write a short IBIS program that defines the initial conditions, properties of the fluid and properties of the immersed boundary
* The user is also required to write a short FORTRAN subroutine to specify the motion of the immersed boundaries
* Post-processing the data by using ibisiew
Program Architecture: (3 parts)
* IBIS Programming language: is a tool that was designed to allow users to easily set the parameters that are needed for an immersed boundary simulation.
* IBIS Numerical Software: it takes the parameters set by the user and calculates the evolution of the system. An interface to a user defined FORTRAN subroutine has been provided. In this interface, the immersed boundary and fluid data is passed to the user in a managable format, and can then be manipulated as needed.
* IBISVIEW Visualization Interface: The interface provides the user with point and click access to the data computed by IBIS. The data has two basic modes, color plots and line plots.
TUDelft Framework
Resources
* Homepage
* Tutorial 1: Start using the program with Eclipse
* Tutorial 2: Implementation of the case study using the framework
* Electronic poster
* Local directory: TuDelft
Version
No version info
Distribution method
Free. I am not sure if it is open source because I could not find source file.
About
* Developed by Department of Biotechnology, Delft University of Technology, The Netherlands
* Written in java
* Nice and easy to use
* The framework also has functions for output of biofilm structure representations using POVRAY. The POVRAY program is available for download at www.povray.org.
Program Architecture
Class diagram (UML) for biomass species, particulate species, solute species and reactions, which constitutes the basic structure for the mass balances in the biofilm-modelling framework. The generality of the framework allows for arbitrary numbers of species and reactions to be defined, providing the infrastructure for the creation of 2D/3D biofilm models with any type of microbe/chemicals interactions.
Class diagram (UML) for all classes used by the multigrid solver implementing relative diffusivity values (RelativeDiffusivity), boundary layer scenarios (classes inheriting from BoundaryLayer) and the concentration fields of particulates (ParticulateSpecies) and solutes (SoluteSpecies). The hierarchy used for implementation of diverse bulk concentration behaviours is also shown.
Documentation
No manual. The document for each java class is provided with little comment in it.
Several sample codes are provided.
