The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

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The Riemann zeta function $latex {zeta(s)}&fg=000000$ is defined in the region $latex {hbox{Re}(s)>1}&fg=000000$ by the absolutely convergent series

$latex displaystyle zeta(s) = sum_{n=1}^infty frac{1}{n^s} = 1 + frac{1}{2^s} + frac{1}{3^s} + ldots. (1)&fg=000000$

Thus, for instance, it is known that $latex {zeta(2)=pi^2/6}&fg=000000$, and thus

$latex displaystyle sum_{n=1}^infty frac{1}{n^2} = 1 + frac{1}{4} + frac{1}{9} + ldots = frac{pi^2}{6}. (2)&fg=000000$

For $latex {hbox{Re}(s) leq 1}&fg=000000$, the series on the right-hand side of (1) is no longer absolutely convergent, or even conditionally convergent. Nevertheless, the $latex {zeta}&fg=000000$ function can be extended to this region (with a pole at $latex {s=1}&fg=000000$) by analytic continuation. For instance, it can be shown that after analytic continuation, one has $latex {zeta(0) = -1/2}&fg=000000$, $latex {zeta(-1) = -1/12}&fg=000000$, and $latex {zeta(-2)=0}&fg=000000$, and more generally

$latex displaystyle zeta(-s) = – frac{B_{s+1}}{s+1} (3)&fg=000000$

for $latex {s=1,2,ldots}&fg=000000$, where $latex {B_n}&fg=000000$ are the Bernoulli numbers. If one formally applies

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NON–RESEARCH DISCUSSIONS TOPICS AND READINGS

Becoming a Scientist

Giddings, Morgan C. “On the Process of Becoming a Great Scientist.” PLOS Computational Biology 4, no. 2 (2008): e33.

Erren, Thomas C., et al. “Ten Simple Rules for Doing Your Best Research, According to Hamming.” PLOS Computational Biology 3, no. 10 (2007): e213.

Writing a Fellowship Proposal

Guidelines & Advice on Applying to Graduate Fellowships (PDF)
(Courtesy of Diana Chien and John Casey, Biological Engineering Communication Lab. Used with permission.)

Choosing a Research Problem

Alon, Uri. “How to Choose a Good Scientific Problem.”Molecular Cell 35, no. 6 (2009): 726–8

Writing a Paper

Whitesides, George M. “Whitesides’ Group: Writing a Paper.” Advanced Materials 16, no. 15 (2004): 1375–7.

Doerr, Allison. “How to Write a Cover Letter.” Nature Methods (2013).

Reading Effectively

This resource may not render correctly in a screen reader.How to Navigate a Scientific Paper with Time Constraints: A Graphic Approach (PDF)

Refereeing

Drubin, David G. “Any Jackass can Trash a Manuscript, but it Takes Good Scholarship to Create One (How MBoC Promotes Civil and Constructive Peer Review).” Molecular Biology of the Cell 22, no. 5 (2011): 525–7.

Making Figures

Wong, Bang. “Points of View: Gestalt Principles (Part 1).” Nature Methods 7, no. 11 (2010): 863.

———. “Points of View: Gestalt Principles (Part 2).”Nature Methods 7, no. 12 (2010): 941.

Giving a Talk

McConnell, Susan. “Designing Effective Scientific Presentations.” iBiology.org.

———. “The Importance of Giving a Good Talk.” iBiology.org.

Scientific Ethics

Glass, Bentley. “The Ethical Basis of Science.” Science150, no. 3701 (1965): 1254–61.

Buy at Amazon Engineering, and Public Policy Committee on Science, Institute of Medicine, et al. On Being a Scientist: A Guide to Responsible Conduct in Research. National Academies Press, 2009. ISBN: 9780309119702. [Preview with Google Books]

Curriculum Vitae

Resumes, CVs, Cover Letters, and LinkedIn

Life after the Ph.D.

Austin, Jim, and Bruce Alberts. “Planning Career Paths for Ph.D.s.” Science 337, no. 6099 (2012): 1149.

English Communication for Scientists

  1. http://www.nature.com/scitable/ebooks/english-communication-for-scientists-14053993/communicating-as-a-scientist-14238273
  2. http://www.americanscientist.org/issues/pub/the-science-of-scientific-writing
  3. http://www.cmu.edu/student-org/pcr/media-files/pcr-sessions/swan_handout.pdf
  4. https://www.brown.edu/academics/science-center/sites/brown.edu.academics.science-center/files/uploads/Quick_Guide_to_Science_Communication_0.pdf

Reaction Diffusion Systems/Phenomena

Taeuber U.C.-Critical Dynamics_ A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior-CUP (2014)

Generic reaction-diffusion models are in fact utilized to describe a multitude of phenomena in various disciplines, ranging from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early Universe
in cosmology, equity trading on the stock market in economy, opinion exchange
in sociology, etc. More concrete physical applications systems encompass excitons kinetics in organic semconductors, domain wall interactions in magnets, and
interface dynamics in growth models. Yet most of our current knowledge in this
area stems from extensive computer simulations, and actual experimental realizations allowing accurate quantitative analysis are still deplorably rare

Brownian Dynamics Simulation

LAMMPS and GROMACS are popular simulation packages for molecular dynamics (MD) simulations. Both can be used to simulate Brownian Dynamics by using Langevin dynamics. See “fix langevin” in LAMMPS or “bd integrator” in GROMACS.

LAMMPS: lammps.sandia.gov

GROMACS: http://www.gromacs.org

GROMACS Tutorial: http://www.gromacs.org/WIKI-import/Main_Page/Tutorials

SMOG: Structure-based Models for Biomolecules: http://smog-server.org/


 

BDpack is a Brownian dynamics package developed recently. It is an open source code and is written in parallel with high computational efficiency. It can be found at: http://amir-saadat.github.io/BDpack/


 

BROWNIAN DYNAMICS SIMULATIONS OF POLYMERS AND SOFT MATTER HANDBOOK by P.S. DOYLE