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What Makes a Neural Code Convex?
We provide a complete characterization of local obstructions to convexity of convex neural codes and define max intersection-complete codes. Expand
Polarization of Neural Rings
The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently,Expand
Mapping toric varieties into low dimensional spaces
A smooth $d$-dimensional projective variety $X$ can always be embedded into $2d+1$-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax ourExpand
Separating invariants and local cohomology
The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate theExpand
A singularly perturbed semilinear system
A constructive existence proof is given for solutions of boundarylayer type for the singularly perturbed semilinear system edx/dt = H{t,x,e) subject to either Dirichlet or general Robin boundaryExpand
Algebraic signatures of convex and non-convex codes
A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Expand
Multiplicities of classical varieties
The j-multiplicity plays an important role in the intersec- tion theory of Stuckrad-Vogel cycles, while recent developments confirm the connections between the e-multiplicity and equisingularityExpand
Non-simplicial decompositions of Betti diagrams of complete intersections
We investigate decompositions of Betti diagrams over a polyno- mial ring within the framework of Boij{Soderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. RelaxingExpand
We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissier's volume-theoretic interpretation of theExpand
Quantifying singularities with differential operators.
The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strongExpand