A generalization of weight polynomials to matroids

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.Comment: 21 page

Neural Coding of Green Flash in Retinal Bipolar Pathways

What visual information do the graded potentials among retinal bipolar pathways actually transmit from photoreceptors to ganglion cells? The answer does not exist. Even the graded electric signals have not been understood completely. Here, this paper tries to analyze the encoding mechanisms of graded signals among the parallel bipolar pathways in response to brief green flash. The typical ON, OFF and ON-OFF bipolar cells simultaneously abstracted vectors from green flash stimulus with sine-like functions on their dendritic plane. Atypical bipolar cell had the synchronously monopolar response in contrast to the bipolar responses of typical bipolar cells, they also annotated green flash with facilitated stochastic (asynchronous and rate-coded) responses. Some complex ON-OFF bipolar cells with large voltage-gated Na+ current could generate high-frequent asynchronous responses, others had synchronous ON-OFF responses to green flash. The green flash was synchronously and asynchronously represented by the multiple-dimension signaling space among the parallel bipolar pathways. These results unraveled the multiple-dimension neural codes for brief green flash, demonstrated the superior encoding capability of parallel bipolar pathways, and suggested the electrophysiological mechanisms of vision such as color space

Water as a Part of the Public Trust: A Review of Select State Codes

The purpose of this paper is to survey state codes primarily in the Western States for the purpose of determining the extent to which water resources have been explicitly identified as a part of the state's public trust. Our primary interest was in states wherein water law is based on the prior appropriation doctrine. A strict limitation was found to be impractical, given that some states have water rights based on prior appropriation and the riparian doctrine.Each code is arbitrarily graded A, B, or C, to indicate the strength of public trust statements found in the state codes. We recognize that the State's Supreme Court, not the legislature, decides what resources are included in a public trust, as well as any limits on the legislature's issuance of usufructuary rights to use the resource. Therefore, this was taken into account when grading the state's code.The method used to access data reported here was using a keyword search in Lexis-Nexis. The reader should keep in mind that the authors are not lawyers nor were the state codes reviewed in their entirety. The key words used in the search were: "public trust", "public interest", and "water". Working Paper #2002-00

Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids

To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid

Stanley-Reisner resolution of constant weight linear codes

Given a constant weight linear code, we investigate its weight hierarchy and the Stanley-Reisner resolution of its associated matroid regarded as a simplicial complex. We also exhibit conditions on the higher weights sufficient to conclude that the code is of constant weigh