Mohamed Omar

What Makes a Neural Code Convex

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.

On Volumes of Permutation Polytopes

This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups,dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also briefly discuss the theta body hierarchy of various permutation polytopes.

On the Hardness of Counting and Sampling Center Strings

Given a set S of n strings, each of length \ell , and a nonnegative value d , we define a center string as a string of length \ell that has Hamming distance at most d from each string in S . The \#{\rm CLOSEST STRING} problem aims to determine the number of center strings for a given set of strings S and input parameters n , \ell , and d . We show \#{\rm CLOSEST STRING} is impossible to solve exactly or even approximately in polynomial time, and that restricting \#{\rm CLOSEST STRING} so that any one of the parameters n , \ell , or d is fixed leads to a fully polynomial-time randomized approximation scheme (FPRAS). We show equivalent results for the problem of efficiently sampling center strings uniformly at random (u.a.r.).

On the hardness of counting and sampling center strings

Given a set S of n strings, each of length ℓ, and a nonnegative value d, we define a center string as a string of length ` that has Hamming distance at most d from each string in S. The #CLOSEST STRING problem aims to determine the number of center strings for a given set of strings S and input parameters n, ℓ, and d. We show #CLOSEST STRING is impossible to solve exactly or even approximately in polynomial time, and that restricting #CLOSEST STRING so that any one of the parameters n, ℓ, or d is fixed leads to a fully polynomial-time randomized approximation scheme (FPRAS). We show equivalent results for the problem of efficiently sampling center strings uniformly at random (u.a.r.).

Asymptotics of Largest Components in Combinatorial Structures

Given integers m and n, we study the probability that structures of size n have all components of size at most m. The results are given in term of a generalized Dickman function of n/m.

A proof of the peak polynomial positivity conjecture

We say that a permutation

“I felt like a mathematician”: Problems and assessment to promote creative effort

\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n

Homomorphisms preserving neural ideals

has a peak at index

Recognizing Graph Theoretic Properties with Polynomial Ideals

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Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers

if