Mohamed Omar

activity

20102026

most citedSparse Neural Codes and Convexity

10 citations · 27 across the 20 of their papers we have counts for

collaborators

25 papers

math.CO2026

Multiple Distance Ramsey Bounds For Graphs in Euclidean Spaces

Ayşegül Kula, Mohamed Omar, Jonah Stockwell +1

For a finite set $A \subset R_{> 0}$ and a finite graph $H$ , let $χ_{H} (R^{n}; A)$ be the minimum number of colors required to color $R^{n}$ while avoiding a mon…

math.CO2026

Real Reliability Roots of Simple Graphs are Dense

Mohamed Omar

We prove that the closure of the real roots of all-terminal reliability polynomials is exactly $[- 1, 0] \cup {1}$ , resolving a conjecture of Brown and McMullin and refining the co…

math.CO2026

On a Quadratic Relation Between Stanley-Wilf Limits and Füredi-Hajnal Limits

Mohamed Omar

For a permutation matrix $P$ , let $s_{P}$ denote its Stanley-Wilf limit, the exponential growth rate of the number of $n \times n$ permutation matrices avoiding $P$ . Let $c_{P}$ denote…

math.CO2026

New Perspectives On The Unimodality Of Domination Polynomials

Mohamed Omar

The domination polynomial of a graph $G$ is given by $D (G, x) = \sum_{k = 0}^{n} d_{k} (G) x^{k}$ where $d_{k} (G)$ records the number of $k$ -element dominating sets in $G$ . A conjecture of Alik…

math.CO2025

Ehrhart Theory over Abelian Group Rings

Robert Davis, Jesús A. De Loera, Alexey Garber +3

We introduce a unified framework for Ehrhart theory in which lattice point enumerators take coefficients in an Abelian group ring, encoding substantially richer algebraic data than…

math.CO2022

Sets Avoiding Full-Rank Three-Point Patterns in $(F_{q}^{n})^{k}$ Are Exponentially Small

Mohamed Omar

We prove that if a subset of $(F_{q}^{n})^{k}$ (with $q$ an odd prime power) avoids a full-rank three-point pattern $x, x + M_{1} d, x + M_{2} d$ then it is ex…

math.CO2022

Partition Rank and Partition Lattices

Mohamed Omar

We introduce a universal approach for applying the partition rank method, an extension of Tao's slice rank polynomial method, to tensors that are not diagonal. This is accomplished…

math.CO2021

Burning Graph Classes

Mohamed Omar, Vibha Rohilla

The Burning Number Conjecture, that a graph on $n$ vertices can be burned in at most $⌈ n ⌉$ rounds, has been of central interest for the past several years. Muc…

math.CO2020

Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators

Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrar…

math.CO2020

Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials

Albrecht Boettcher, Stephan Ramon Garcia, Mohamed Omar +1

We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symm…

math.CO2019

Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators

Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational r…

math.CO2018

On Distinct Distances Between a Variety and a Point Set

Bryce McLaughlin, Mohamed Omar

We consider the problem of determining the number of distinct distances between two point sets in $R^{2}$ where one point set $P_{1}$ of size $m$ lies on a real alge…

math.NT2017★ 2 cited

Lattice point visibility on power functions

Pamela E. Harris, Mohamed Omar

It is classically known that the proportion of lattice points visible from the origin via functions of the form $f (x) = n x$ with $n \in Q$ is $\frac{1}{ζ ( 2 )}$ where $ζ (s)$ is…

math.CO2017

Descent polynomials

Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko +2

Let $n$ be a nonnegative integer and $I$ be a finite set of positive integers. In 1915, MacMahon proved that the number of permutations in the symmetric group $S_{n}$ with…

math.CO2017

Peaks on Graphs

Alexander Diaz-Lopez, Lucas Everham, Pamela E. Harris +3

Given a graph $G$ with $n$ vertices and a bijective labeling of the vertices using the integers $1, 2, \dots, n$ , we say $G$ has a peak at vertex $v$ if the degree of $v$ is greater…

math.AC2016★ 8 cited

Neural Ideal Preserving Homomorphisms

R. Amzi Jeffs, Mohamed Omar, Nora Youngs

The neural ideal of a binary code $C \subseteq F_{2}^{n}$ is an ideal in $F_{2} [x_{1}, \dots, x_{n}]$ closely related to the vanishing ideal of $C$ . The n…

math.CO2016

A proof of the peak polynomial positivity conjecture

Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko +1

We say that a permutation $π = π_{1} π_{2} \dots π_{n} \in S_{n}$ has a peak at index $i$ if $π_{i - 1} < π_{i} > π_{i + 1}$ . Let $P (π)$ denote the set of indices where $π$ has…

math.CO2015★ 10 cited

Sparse Neural Codes and Convexity

R. Amzi Jeffs, Mohamed Omar, Natchanon Suaysom +2

Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by…

math.CO2015★ 3 cited

The $q$ -analog of Kostant's partition function and the highest root of the classical Lie algebras

Pamela E. Harris, Erik Insko, Mohamed Omar

Kostant's partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra. For a given weigh…

q-bio.NC2015

What makes a neural code convex?

Carina Curto, Elizabeth Gross, Jack Jeffries +5

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…

math.CO2015

Low Degree Nullstellensatz Certificates for 3-Colorability

Bo Li, Benjamin Lowenstein, Mohamed Omar

In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable…

math.CO2013

Chromatic Bounds On Orbital Chromatic Roots

Dae Hyun Kim, Alexander H. Mun, Mohamed Omar

Given a group $G$ of automorphisms of a graph $Γ$ , the orbital chromatic polynomial $O P_{Γ, G} (x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of…

math.CO2011★ 1 cited

On Volumes of Permutation Polytopes

Katherine Burggraf, Jesús A. De Loera, Mohamed Omar

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. W…

math.AG2011★ 1 cited

Strong nonnegativity and sums of squares on real varieties

Mohamed Omar, Brian Osserman

Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic p…

Multiple Distance Ramsey Bounds For Graphs in Euclidean Spaces

Real Reliability Roots of Simple Graphs are Dense

On a Quadratic Relation Between Stanley-Wilf Limits and Füredi-Hajnal Limits

New Perspectives On The Unimodality Of Domination Polynomials

Ehrhart Theory over Abelian Group Rings

Sets Avoiding Full-Rank Three-Point Patterns in (Fqn​)k Are Exponentially Small

Partition Rank and Partition Lattices

Burning Graph Classes

Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators

Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials

Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators

On Distinct Distances Between a Variety and a Point Set

Lattice point visibility on power functions

Descent polynomials

Peaks on Graphs

Neural Ideal Preserving Homomorphisms

A proof of the peak polynomial positivity conjecture

Sparse Neural Codes and Convexity

The q-analog of Kostant's partition function and the highest root of the classical Lie algebras

What makes a neural code convex?

Low Degree Nullstellensatz Certificates for 3-Colorability

Chromatic Bounds On Orbital Chromatic Roots

On Volumes of Permutation Polytopes

Strong nonnegativity and sums of squares on real varieties

Sets Avoiding Full-Rank Three-Point Patterns in $(F_{q}^{n})^{k}$ Are Exponentially Small

The $q$ -analog of Kostant's partition function and the highest root of the classical Lie algebras