Omid Khormali

Nuh Aydin, Mohammad K. Azarian, Omid Khormali et al.

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a technique for fast exponentiation commonly used in modern cryptography and computational number theory. Despite its prominence, the historical origins of the algorithm are not known with certainty. This paper critically examines the origins and formalization of the algorithm through primary source analysis. We focus on Jamshid al-Kashi's fifteenth-century Miftah al-Hisab where the algorithm is articulated explicitly as a general method and claimed by al-Kashi as his own innovation. To contextualize this, we trace earlier instances of successive squaring in the works of al-Uqlidisi and al-Biruni, who applied these techniques for specific calculations, but did not formalize them into a general procedure. The earliest known work on this method of computation is found in Pingala's prosodic studies in ancient India (c. 200 BCE). Even though it was not fully developed as a general technique, Pingala's work seems to contain the conceptual foundation of the algorithm which is to employ the binary representation of a positive integer. By mapping this intellectual progression, the paper illustrates the historical background of an algorithm that is prominent in modern computation.

Omid Khormali, Mia Pesavento

This paper proposes an integrated framework for machine learning-guided quota optimization applied to multi-round sorority recruitment, a small two-sided market where approximately 100 potential new members (PNMs) are matched to three chapters through a structured process governed by the Release Figure Methodology (RFM). Our framework combines a Random Forest classifier trained on historical registration data to generate PNM-chapter compatibility scores, integer linear programs for Round~2 and Round~3 invitation quota optimization balancing fairness, coverage, and efficiency objectives, and a Deferred Acceptance algorithm for final matching. Applied to five years of de-identified recruitment data from a small Midwestern university, and working with only 282 matched training pairs, the compatibility model achieves a cross-validated ROC-AUC of 0.5822, reflecting the inherent difficulty of predicting social compatibility from pre-recruitment registration data in a data-limited setting. Because fairness and coverage constraints dominate quota allocation under noisy scores, the framework is designed to degrade gracefully when ML signal is weak. Optimized quotas closely align with actual coordinator decisions for active chapters, and the Deferred Acceptance algorithm replicates actual 2025 recruitment outcomes with a 96.4% individual-level agreement rate and a 100% match rate across 56 PNMs. An interactive web application implementing the framework is made available to recruitment coordinators. These results support the viability of data-driven approaches to small-market matching with broader applicability to other constrained two-sided markets.

Omid Khormali

We introduce the Overlap-Weighted Hierarchical Normalized Persistence Velocity (OW-HNPV), a novel topological data analysis method for detecting anomalies in time-varying networks. Unlike existing methods that measure cumulative topological presence, we introduce the first velocity-based perspective on persistence diagrams, measuring the rate at which features appear and disappear, automatically downweighting noise through overlap-based weighting. We also prove that OW-HNPV is mathematically stable. It behaves in a controlled, predictable way, even when comparing persistence diagrams from networks with different feature types. Applied to Ethereum transaction networks (May 2017-May 2018), OW-HNPV demonstrates superior performance for cryptocurrency anomaly detection, achieving up to 10.4% AUC gain over baseline models for 7-day price movement predictions. Compared with established methods, including Vector of Averaged Bettis (VAB), persistence landscapes, and persistence images, velocity-based summaries excel at medium- to long-range forecasting (4-7 days), with OW-HNPV providing the most consistent and stable performance across prediction horizons. Our results show that modeling topological velocity is crucial for detecting structural anomalies in dynamic networks.

Umar Islambekov, Hasani Pathirana, Omid Khormali et al.

Large time-varying graphs are increasingly common in financial, social and biological settings. Feature extraction that efficiently encodes the complex structure of sparse, multi-layered, dynamic graphs presents computational and methodological challenges. In the past decade, a persistence diagram (PD) from topological data analysis (TDA) has become a popular descriptor of shape of data with a well-defined distance between points. However, applications of TDA to graphs, where there is no intrinsic concept of distance between the nodes, remain largely unexplored. This paper addresses this gap in the literature by introducing a computationally efficient framework to extract shape information from graph data. Our framework has two main steps: first, we compute a PD using the so-called lower-star filtration which utilizes quantitative node attributes, and then vectorize it by averaging the associated Betti function over successive scale values on a one-dimensional grid. Our approach avoids embedding a graph into a metric space and has stability properties against input noise. In simulation studies, we show that the proposed vector summary leads to improved change point detection rate in time-varying graphs. In a real data application, our approach provides up to 22% gain in anomalous price prediction for the Ethereum cryptocurrency transaction networks.