arXiv subjects

Mohamed Omar

Publications and source records attributed to Mohamed Omar.

At least 19 recordsLinked to original sources

arXiv2026

Multiple Distance Ramsey Bounds For Graphs in Euclidean Spaces

For a finite set $A \subset \mathbb{R}_{>0}$ and a finite graph $H$, let $χ_H(\mathbb{R}^n;A)$ be the minimum number of colors required to color $\mathbb{R}^n$ while avoiding a monochromatic copy of $H$ whose edges have distances in $A$. Extending the graph-copy framework of Axenovich, Liu, and Sagdeev and a multiple distance theorem of Naslund, we prove for any positive integer $m$, \[χ_H(\mathbb{R}^n;m):=\max_{\substack{A \subseteq \mathbb{R}_{>0} \\ |A|=m}} χ_H(\mathbb{R}^n;A) \geq \left(Γ_χ\sqrt{\frac{m+1}{Ξ(H)}}+o(1)\right)^n.\] Here, $Γ_χ$ is a constant and $Ξ(H)$ is an explicit structural parameter that can be substantially smaller than $|V(H)|-1$, thereby recovering Naslund's similar bound for complete graphs and improving the general bound inherited from the corresponding clique for many graph families. Along the way, we construct a weighted strengthening of the semi-diagonal flattening rank theorem of Correia, Sudakov, and Tomon.

math.CO↗

arXiv2026

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

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 its Füredi-Hajnal limit, which is the limit $\displaystyle \lim_{n \to \infty} \text{ex}(n,P)/n$ where $\text{ex}(n,P)$ is the maximum number of ones in an $n\times n$ $0$-$1$ matrix avoiding $P$. Cibulka proved the universal quadratic bound $s_P\leq 2.88\,c_P^2$. In this note we improve the constants in Cibulka's result through a so-called ``block contraction" argument. Defining \[ F(c)=\inf_{t\in\mathbb{N}} \frac{(t!)^{1/t}\,15^{\,c/t}}{c}, \] for $c>0$, this leads us to the revised inequality $s_P\leq F(c_P)\,c_P^2$. In particular, $F(c)=\log 15+o(1) \approx 2.70805\ldots +o(1)$ as $c\to\infty$, and the constant improves $2.88$ once $c_P \geq 17$.

math.CO↗

arXiv2026

Ehrhart Theory over Abelian Group Rings

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 classical counts. We prove that fundamental results of Ehrhart theory extend to this setting through a generalized Brion theorem, including rational generating functions, reciprocity phenomena, connections to volume, and vertex-cone decompositions. We further show how to derive $q$-enumerative and weighted theories from this setting, recasting several major refinements of Ehrhart theory as consequences of a single algebraic mechanism. We also show how our framework combines with equivariant Ehrhart theory.

math.CO↗

arXiv2026

Permutations with a Given X-Descent Set

Building on the work of Grinberg and Stanley, we begin a systematic study of permutations with a prescribed $X$-descent set. In particular, for a set $X \subseteq \mathbb{N}^2$, and $I \subseteq [n-1]$, we study the permutations $π\in \mathfrak{S}_n$ whose $X$-descent set is precisely $I$, meaning $(π_i,π_{i+1}) \in X$ precisely when $i \in I$. The central focus is enumerating these permutations for a fixed $X,I$ and $n$: this count is denoted by $d_X(I;n)$. We derive a recursion which under expected conditions simplifies to a binomial-type recurrence determined entirely by the values $d_X(\emptyset;n)$. This extends the work of Díaz-Lopez et al.\ on descent polynomials. The resulting reduction shows that the general statistic $d_X(I;n)$ is typically governed by the ``descent-free'' quantities $d_X(\emptyset;n)$, motivating a closer analysis of these numbers. We observe that $d_X(\emptyset;n)$ enumerates Hamiltonian paths in a directed graph canonically associated to $X$. We then record several families of sets $X$ for which $d_X(\emptyset;n)$ is explicit or effectively computable. This includes families with periodicity for which transfer matrix methods apply, and families with succession-type relations where inclusion-exclusion applies. We then investigate the typical behavior of $d_X(\emptyset;n)$ from a probabilistic perspective.

math.CO↗

arXiv2026

Real Reliability Roots of Simple Graphs are Dense

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 corresponding density result for multigraphs due to Brown and Colbourn. The crux of the proof is demonstrating that real reliability roots of edge-substitution graphs $G[H]$, where $G$ ranges over connected multigraphs and $H$ ranges over complete graphs missing an edge, are dense.

math.CO↗

arXiv2026

New Perspectives On The Unimodality Of Domination Polynomials

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 Alikhani and Peng asserts that these polynomials have unimodal coefficient sequences. We develop three complementary perspectives that strengthen existing tools for resolving the conjecture. First, we view dominating sets as transversals of the closed neighborhood hypergraph. Motivated by the relationship between the unimodality of a polynomial and its roots, we use this perspective to expand on known root phenomena for domination polynomials. In particular, we obtain a bound on the modulus of domination roots that is linear in the maximum degree of a graph, improving related exponential bounds of Bencs, Csikvári and Regts. The hypergraph viewpoint also yields explicit combinatorial formulas for top coefficients of $D(G,x)$, extending formulas in the literature and offering fruitful ground for combinatorial approaches to the unimodality conjecture. Second, we strengthen the coefficient-ratio method of Beaton and Brown. This includes tightening their inequalities, and combining a union bound for non-dominating $k$-element sets with an overlap correction based on spanning trees. This produces a new parameter $τ_k(G)$ measuring maximal pairwise neighborhood overlap and yields an overlap-corrected sufficient criterion for unimodality. Third, we prove that the domination polynomial of threshold graphs are log-concave, and hence unimodal, by a planar network argument from total positivity. This offers a new tactic for resolving the unimodality of hereditary graph classes.

math.CO↗

arXiv2025

Revisiting The Rédei-Berge Symmetric Functions via Matrix Algebra

We revisit the Rédei-Berge symmetric function $\mathcal{U}_D$ for digraphs $D$, a specialization of Chow's path-cycle symmetric function. Through the lens of matrix algebra, we consolidate and expand on the work of Chow, Grinberg and Stanley, and Lass concerning the resolution of $\mathcal{U}_D$ in the power sum and Schur bases. Along the way we also revisit various results on Hamiltonian paths in digraphs.

math.CO↗

arXiv2025

Growth Rates Of Permutations With Given Descent Or Peak Set

Given a set $I \subseteq \mathbb{N}$, consider the sequences $\{d_n(I)\},\{p_n(I)\}$ where for any $n$, $d_n(I)$ and $p_n(I)$ respectively count the number of permutations in the symmetric group $\mathfrak{S}_n$ whose descent set (respectively peak set) is $I \cap [n-1]$. We investigate the growth rates $\text{gr} \ d_n(I) = \lim_{n \to \infty} \left(d_n(I)/n!\right)^{1/n}$ and $\text{gr} \ p_n(I) = \lim_{n \to \infty} \left(p_n(I)/n!\right)^{1/n}$ over all $I \subseteq \mathbb{N}$. Our main contributions are two-fold. Firstly, we prove that the numbers $\text{gr} \ d_n(I)$ over all $I \subseteq \mathbb{N}$ are exactly the interval $\left[0,2/π\right]$. To do so, we construct an algorithm that explicitly builds $I$ for any desired limit $L$ in the interval. Secondly, we prove that the numbers $\text{gr} \ p_n(I)$ for periodic sets $I \subseteq \mathbb{N}$ form a dense set in $\left[0,1/\sqrt[3]{3}\right]$. We do this by explicitly finding, for any prescribed $L$ in the interval, a set $I$ whose corresponding growth rate is arbitrarily close to $L$.

math.CO↗

arXiv2024

Multiplex Imaging Analysis in Pathology: a Comprehensive Review on Analytical Approaches and Digital Toolkits

Conventional histopathology has long been essential for disease diagnosis, relying on visual inspection of tissue sections. Immunohistochemistry aids in detecting specific biomarkers but is limited by its single-marker approach, restricting its ability to capture the full tissue environment. The advent of multiplexed imaging technologies, like multiplexed immunofluorescence and spatial transcriptomics, allows for simultaneous visualization of multiple biomarkers in a single section, enhancing morphological data with molecular and spatial information. This provides a more comprehensive view of the tissue microenvironment, cellular interactions, and disease mechanisms - crucial for understanding disease progression, prognosis, and treatment response. However, the extensive data from multiplexed imaging necessitates sophisticated computational methods for preprocessing, segmentation, feature extraction, and spatial analysis. These tools are vital for managing large, multidimensional datasets, converting raw imaging data into actionable insights. By automating labor-intensive tasks and enhancing reproducibility and accuracy, computational tools are pivotal in diagnostics and research. This review explores the current landscape of multiplexed imaging in pathology, detailing workflows and key technologies like PathML, an AI-powered platform that streamlines image analysis, making complex dataset interpretation accessible for clinical and research settings.

q-bio.TO↗

arXiv2024

Partition Rank and Partition Lattices

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 by generalizing Naslund's distinctness indicator to what we call a partition indicator. The advantages of partition indicators are two-fold: they diagonalize tensors that are constant when specified sets of variables are equal, and even in more general settings they can often substantially reduce the partition rank as compared to when a distinctness indicator is applied. The key to our discoveries is integrating the partition rank method with Möbius inversion on the lattice of partitions of a finite set. Through this we unify disparate applications of the partition rank method in the literature. We then use our theory to address a finite field analogue of a question of Erdős, thereby generalizing results of Hart and Iosevich and independently Shparlinski. Furthermore we generalize work of Pach, et al. on bounding sizes of sets avoiding right triangles to bounding sizes of sets avoiding right $k$-configurations.

math.CO↗

arXiv2023

Audio-Enhanced Text-to-Video Retrieval using Text-Conditioned Feature Alignment

Text-to-video retrieval systems have recently made significant progress by utilizing pre-trained models trained on large-scale image-text pairs. However, most of the latest methods primarily focus on the video modality while disregarding the audio signal for this task. Nevertheless, a recent advancement by ECLIPSE has improved long-range text-to-video retrieval by developing an audiovisual video representation. Nonetheless, the objective of the text-to-video retrieval task is to capture the complementary audio and video information that is pertinent to the text query rather than simply achieving better audio and video alignment. To address this issue, we introduce TEFAL, a TExt-conditioned Feature ALignment method that produces both audio and video representations conditioned on the text query. Instead of using only an audiovisual attention block, which could suppress the audio information relevant to the text query, our approach employs two independent cross-modal attention blocks that enable the text to attend to the audio and video representations separately. Our proposed method's efficacy is demonstrated on four benchmark datasets that include audio: MSR-VTT, LSMDC, VATEX, and Charades, and achieves better than state-of-the-art performance consistently across the four datasets. This is attributed to the additional text-query-conditioned audio representation and the complementary information it adds to the text-query-conditioned video representation.

cs.CV↗

arXiv2023

Selective Structured State-Spaces for Long-Form Video Understanding

Effective modeling of complex spatiotemporal dependencies in long-form videos remains an open problem. The recently proposed Structured State-Space Sequence (S4) model with its linear complexity offers a promising direction in this space. However, we demonstrate that treating all image-tokens equally as done by S4 model can adversely affect its efficiency and accuracy. To address this limitation, we present a novel Selective S4 (i.e., S5) model that employs a lightweight mask generator to adaptively select informative image tokens resulting in more efficient and accurate modeling of long-term spatiotemporal dependencies in videos. Unlike previous mask-based token reduction methods used in transformers, our S5 model avoids the dense self-attention calculation by making use of the guidance of the momentum-updated S4 model. This enables our model to efficiently discard less informative tokens and adapt to various long-form video understanding tasks more effectively. However, as is the case for most token reduction methods, the informative image tokens could be dropped incorrectly. To improve the robustness and the temporal horizon of our model, we propose a novel long-short masked contrastive learning (LSMCL) approach that enables our model to predict longer temporal context using shorter input videos. We present extensive comparative results using three challenging long-form video understanding datasets (LVU, COIN and Breakfast), demonstrating that our approach consistently outperforms the previous state-of-the-art S4 model by up to 9.6% accuracy while reducing its memory footprint by 23%.

cs.CV↗

arXiv2023

Multiscale Audio Spectrogram Transformer for Efficient Audio Classification

Audio event has a hierarchical architecture in both time and frequency and can be grouped together to construct more abstract semantic audio classes. In this work, we develop a multiscale audio spectrogram Transformer (MAST) that employs hierarchical representation learning for efficient audio classification. Specifically, MAST employs one-dimensional (and two-dimensional) pooling operators along the time (and frequency domains) in different stages, and progressively reduces the number of tokens and increases the feature dimensions. MAST significantly outperforms AST~\cite{gong2021ast} by 22.2\%, 4.4\% and 4.7\% on Kinetics-Sounds, Epic-Kitchens-100 and VGGSound in terms of the top-1 accuracy without external training data. On the downloaded AudioSet dataset, which has over 20\% missing audios, MAST also achieves slightly better accuracy than AST. In addition, MAST is 5x more efficient in terms of multiply-accumulates (MACs) with 42\% reduction in the number of parameters compared to AST. Through clustering metrics and visualizations, we demonstrate that the proposed MAST can learn semantically more separable feature representations from audio signals.

cs.SD↗

arXiv2022

Sets Avoiding Full-Rank Three-Point Patterns in $(\mathbb{F}_q^n)^k$ Are Exponentially Small

We prove that if a subset of $(\mathbb{F}_q^n)^k$ (with $q$ an odd prime power) avoids a full-rank three-point pattern $\vec{x},\vec{x}+M_1\vec{d},\vec{x}+M_2\vec{d}$ then it is exponentially small, having size at most $3 \cdot c_q^{nk}$ where $0.8414 q \leq c_q \leq 0.9184 q$. This generalizes a theorem of Kovauc and complements results of Berger, Sah, Sawhney and Tidor. As a consequence, we prove that if $3$ is a square in $\mathbb{F}_q$ then subsets of $(\mathbb{F}_q^n)^2$ avoiding equilateral triangles are exponentially small.

math.CO↗

arXiv2021

Burning Graph Classes

The Burning Number Conjecture, that a graph on $n$ vertices can be burned in at most $\lceil \sqrt{n} \ \rceil$ rounds, has been of central interest for the past several years. Much of the literature toward its resolution focuses on two directions: tightening a general upper bound for the burning number, and proving the conjecture for specific graph classes. In the latter, most of the developments work within a specific graph class and exploit the intricacies particular to it. In this article, we broaden this approach by developing systematic machinery that can be used as test beds for asserting that graph classes satisfy the conjecture. We show how to use these to resolve the conjecture for several classes of graphs including triangle-free graphs with degree lower bounds, graphs with certain linear lower bounds on $r$-neighborhood sizes, all trees whose non-leaf vertices have degree at least $4$, trees whose non-leaf vertices have degree at least $3$ (on at least $81$ vertices), trees whose non-leaf vertices are less than $\frac{2}{3}$ concentrated in degree $2$, and trees with a low concentration of high degree non-leaf vertices (the last two results holding for sufficiently many non-leaf vertices).

math.CO↗

arXiv2020

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

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence of measures, characteristic functions), and harmonic analysis (Fourier transforms of distributions). We provide instructive examples which demonstrate the power and generality of our techniques. We also highlight unexpected consequences in the theory of homogeneous symmetric functions.

math.CO↗

arXiv2020

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

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.

math.CO↗

arXiv2020

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

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 symmetric polynomials of even degree $2p$, the positivity of which is a classical result by D. B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees $μ$ with $|μ-2p|< 1/2$. We also prove a positivity criterion for linear combinations of the classical complete homogeneous symmetric polynomials and a sufficient criterion for the positivity of linear combinations of products of such polynomials.

math.CO↗