Adviser: Carina Curto Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How d...

Let C be a clutter with a perfect matching e1, . . . , eg of König type and let ∆C be the Stanley-Reisner complex of the edge ideal of C. If all c-minors of C have a free vertex and C is unmixed, we show that ∆C is pure shellable. We are able to describe in combinatorial terms when ∆C is pure. If C has no cycles of length 3 or 4, then it is shown that ∆C is pure if and only if ∆C is pure shella...

Let ∆ be a simplicial complex and χ be an s-coloring of ∆. Biermann and Van Tuyl have introduced the simplicial complex ∆χ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of ∆χ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of ∆χ over a field is set-theoretic complete interse...

This paper was adopted by the CJLS on December 12, 1990 by a vote of eleven in favor, two opposed, and five abstaining ( 11-2-5). Members voting in favor: Rabbis Kassel Abelson, Ben Zion Bergman, Elliot N. Dorff, Amy Eilberg, Dov Peretz Elkins, Howard Handler, Reuven Kimelman, Lionel E. Moses, Mayer E. Rabinowitz, Joel Rembaum, Morris M. Shapiro. Members voting against: Rabbis Avram I. Reisner ...

This paper was adopted by the CJLS on December 12, 1990 by a vote of thirteen infavor, one opposed, andfour abstaining (13-1-4). Members voting in favor: Rabbis Ben Zion Bergman, Stanley Bramnick, Jerome M. Epstein, David M. Feldman, Sam Fraint, Howard Handler, Reuven Kimelman, Herbert Mandl, Lionel E. Moses, Mayer E. Rabinowitz, Avram I. Reisner, Joel Rembaum, Joel Roth. Member voting against:...

A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible hvectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize...

Perlman, M., and Reisner, S. H. (1973). Archives of Disease in Childhood, 48, 627. Asymmetric crying facies and congenital anomalies. The frequency of hypoplasia of the depressor anguli oris muscle in a newborn population was 41 in 6360 (1 in 155). No adverse factors were noted in the obstetric background of affected infants and the pathogenesis of the lesion is not clear. The incidence of asso...

The notion of toric face rings generalizes both Stanley-Reisner rings and affine semigroup rings, and has been studied by Bruns, Römer, et.al. Here, we will show that, for a toric face ring R, the “graded” Matlis dual of a Cěch complex gives a dualizing complex. In the most general setting, R is not a graded ring in the usual sense. Hence technical argument is required.

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley–Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley–Reisnerand affine monoid algebras. We consider (nonpure) shellable fan’s and the Cohen–Macaulay property. More...