16,475 research outputs found

### Beginner Modeling Exercises

The goal of this paper written as part of the MIT Systems Dynamics in Education Project is to teach the reader how to distinguish between stocks and flows. A stock is an accumulation that is changed over time by inflows and outflows. The reader will gain intuition about stocks and flow through and extensive list of different examples and will practice modeling simple systems with constant flows. STELLA modeling examples include, but are not restricted to, skunks populations, landfills, a bank account and nuclear weapons. Educational levels: High school, Middle school, Undergraduate lower division, Undergraduate upper division

### WINGS-CF Face-to-Face Meeting 2004

This report focuses mainly on workshop discussions and has been written from detailed notes taken by workshop scribes. Where there was overlap in discussion topics, some points have been combined: this is not just a transcript of the workshop discussions. The report starts with a summary of the implications for WINGS-CF from the meeting, and an overview of the workshops. For anyone who wants to delve more deeply into how a topic was explored at the gathering, Section 4 gives details of discussion, drawn from notes taken by each group and the "post-it" thoughts provided by participants before the working groups started the discussions

### Tridiagonal realization of the anti-symmetric Gaussian $\beta$-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal transformation proof for both cases (Method III

### Analogies between random matrix ensembles and the one-component plasma in two-dimensions

The eigenvalue PDF for some well known classes of non-Hermitian random matrices --- the complex Ginibre ensemble for example --- can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices $G$, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix $G_1^{-1} G_2$, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to a exhibit an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature $\beta = 4$, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.Comment: 25 page

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