Exemplar Dynamics Models of the Stability of Phonological Categories

We develop a model for the stability and maintenance of phonological categories. Examples of phonological categories are vowel sounds such as "i" and "e". We model such categories as consisting of collections of labeled exemplars that language users store in their memory. Each exemplar is a detailed memory of an instance of the linguistic entity in question. Starting from an exemplar-level model we derive integro-differential equations for the long-term evolution of the density of exemplars in different portions of phonetic space. Using these latter equations we investigate under what conditions two phonological categories merge or not. Our main conclusion is that for the preservation of distinct phonological categories, it is necessary that anomalous speech tokens of a given category are discarded, and not merely stored in memory as an exemplar of another category.Comment: 6 pages, COGS201

Diversities and the Geometry of Hypergraphs

The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into $\ell_1$. Here we show that this theory can be generalized considerably to encompass Steiner tree packing problems in both graphs and hypergraphs. Instead of the theory of $\ell_1$ metrics and minimal distortion embeddings, the parallel is the theory of diversities recently introduced by Bryant and Tupper, and the corresponding theory of $\ell_1$ diversities and embeddings which we develop here.Comment: 19 pages, no figures. This version: further small correction

Phase Field Crystals as a Coarse-Graining in Time of Molecular Dynamics

Phase field crystals (PFC) are a tool for simulating materials at the atomic level. They combine the small length-scale resolution of molecular dynamics (MD) with the ability to simulate dynamics on mesoscopic time scales. We show how PFC can be interpreted as the result of applying coarse-graining in time to the microscopic density field of molecular dynamics simulations. We take the form of the free energy for the phase field from the classical density functional theory of inhomogeneous liquids and then choose coefficients to match the structure factor of the time coarse-grained microscopic density field. As an example, we show how to construct a PFC free energy for Weber and Stillinger's two-dimensional square crystal potential which models a system of proteins suspended in a membrane.Comment: 5 pages, 4 figures, typos corrected, more explanation in parts, equilib vs non-equilib clarifie

Constant distortion embeddings of Symmetric Diversities

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of $L_1$ spaces. In the metric case, it is well known that an $n$-point metric space can be embedded into $L_1$ with $\mathcal{O}(\log n)$ distortion. For diversities, the optimal distortion is unknown. Here, we establish the surprising result that symmetric diversities, those in which the diversity (value) assigned to a set depends only on its cardinality, can be embedded in $L_1$ with constant distortion.Comment: 14 pages, 3 figure

Calcium content and distribution as a function of growth and transformation in the mouse 3T3 cell

Total Ca content and that fraction of Ca sensitive to removal by the chelator ethylene glycol-bis(β-aminoethyl ether)N,N,N',N'-tetraacetate (EGTA) have been investigated in the mouse 3T3 cell as a function of growth stage, transformation with SV40 virus, and serum levels of the media. Cells were allowed to grow through several doublings in media containing (45)Ca. The cellular content of (45)Ca was used to access total cell Ca. That fraction of (45)Ca removed by EGTA was presumed to represent primarily surface-localized Ca. The data are expressed on a per cell volume basis to compensate for size differences as a function of growth stage and transformation. During exponential growth phase, the 3T3 cell contains 525pmol Ca/μl cell volume. Of this, approx. 457 pmol/μl is not removable by EGTA and, presumably, is cytoplasmically located. This value is in close agreement with previous studies on the HeLa cell (470 pmol Ca/μl cell water after the removal of the surface Ca). The low level of EGTA- removable Ca present in the 3T3 cell during early exponential growth (68 pmol Ca/μl cell volume) increases progressively with increasing cell density, and upon quiescence it is sevenfold greater. In contrast, SV40- transformed 3T3 cells growing exponentially possess total levels of Ca which are approximately two-thirds the levels of the normal 3T3 cell. However, their EGTA-sensitive Ca is not significantly different from that of exponentially growing, normal 3T3 cells. As the transformed cells continue to grow at high density, their total ca and their sensitivity to EGTA do not change, in contrast to the normal 3T3 cell. Thus, an increase in Ca associated with the cell surface appears to be correlated with growth inhibition. This has been investigated further by regulating growth of the normal and transformed cell with alterations in the serum level of the media. In 4 percent calf serum the normal cell is stopped from continued proliferation. Growth stoppage under these conditions is characterized by a nearly fourfold increase in EGTA-removable Ca, similar to the increase observed upon quiescence in depleted 10 percent serum. Similar treatment of the transformed cell does not reduce its growth rate, nor does it significantly alter Ca distribution. However, at 0.5 percent medium serum levels, the SV40 3T3 growth rate is substantially reduced and, under these conditions, EGTA-removable Ca increases twofold