Noether normalizations, reductions of ideals, and matroids

We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which is a common generalization of a topological space and a matroid.Comment: 13 pages; to appear in Proceedings of the American Mathematical Societ

Alien Registration- Epstein, Joseph (Bath, Sagadahoc County)

https://digitalmaine.com/alien_docs/9520/thumbnail.jp

Random strings and tt-degrees of Turing complete C.E. sets

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truth-table degrees do not meet to the degree~0, even within the c.e. truth-table degrees, but when taking the meet over all such truth-table degrees, the infinite meet is indeed~0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truth-table degrees form a minimal pair.Comment: 25 page

Closed Quasi-Fuchsian Surfaces In Hyperbolic Knot Complements

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.Comment: 69 pages, 27 figures. Made small changes suggested by refere

Statistics and compression of scl

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length $n$ is of order $n/\log{n}$. This establishes quantitative refinements of qualitative results of Bestvina-Fujiwara and others on the infinite dimensionality of 2-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length $n$ in a mapping class group cannot be written as a product of fewer than $O(n/\log{n})$ reducible elements, with probability going to 1 as $n$ goes to infinity. We also show that the translation length on the complex of free factors of a random walk of length $n$ on the outer automorphism group of a free group grows linearly in $n$.Comment: Minor edits arising from referee's comments; 45 page

Common and Unique Representations in pFC for Place Attractiveness

Although previous neuroimaging research has identified overlapping correlates of subjective value across different reward types in the ventromedial pFC (vmPFC), it is not clear whether this “common currency” evaluative signal extends to the aesthetic domain. To examine this issue, we scanned human participants with fMRI while they made attractiveness judgments of faces and places—two stimulus categories that are associated with different underlying rewards, have very different visual properties, and are rarely compared with each other. We found overlapping signals for face and place attractiveness in the vmPFC, consistent with the idea that this region codes a signal for value that applies across disparate reward types and across both economic and aesthetic judgments. However, we also identified a subregion of vmPFC within which activity patterns for face and place attractiveness were distinguishable, suggesting that some category-specific attractiveness information is retained in this region. Finally, we observed two separate functional regions in lateral OFC: one region that exhibited a category-unique response to face attractiveness and another region that responded strongly to faces but was insensitive to their value. Our results suggest that vmPFC supports a common mechanism for reward evaluation while also retaining a degree of category-specific information, whereas lateral OFC may be involved in basic reward processing that is specific to only some stimulus categories

How Good a Deal Was the Tobacco Settlement?: Assessing Payments to Massachusetts

We estimate the increment in Massachusetts Medicaid program costs attributable to smoking from December 20, 1991, to 1998. We describe how our methods improve upon earlier estimates of analogous costs at the national level. Current costs to the Massachusetts Medicaid program approximate the payments to Massachusetts under the tobacco settlement of November 1998. Whether these payments are viewed as appropriate compensation for Medicaid costs over time depends upon the rate of increase in future health care costs, the rate of decline in smoking, the proportion of smoking that should be attributed to the actions of the tobacco companies and the liklihood that state would have prevailed at trial. The costs to the Medicaid program are dwarfed by the internal costs to smokers themselves.