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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Robby Robby
Center for Mathematics and Society, Faculty of Science, Parahyangan Catholic University, Bandung 40141, Indonesia
Mathematics

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Geometry-Based Differentially Private Synthetic Tabular Data Generation via K-Means Clustering with Bounded and Discrete Feature Constraints Robby Robby; Agus Sukmana; Erwinna Chendra
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 2 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i2.41403

Abstract

Most clustering-based differentially private synthetic data generation methods assume unconstrained continuous feature spaces and offer no mechanism for hard feature bound enforcement or discrete-valued attribute handling, which limits their practical applicability to real-world tabular data where such constraints are common. This paper proposes a geometry-based mechanism that generates synthetic tabular data by application of Laplace noise jointly to K-means cluster centroids and within-cluster radial distances, calibrated via a data-dependent sensitivity approximation. Three components distinguish the approach from prior work: coordinate-wise centroid reflection to enforce hard feature bounds after perturbation, coordinate-wise clipping to enforce bounds on reconstructed synthetic points, and randomized rounding for discrete features as a post-processing step. A utility-driven calibration strategy selects the privacy budget to meet a user-specified target Adjusted Rand Index (ARI), which makes the privacyutility trade-off directly interpretable. Baseline comparisons on a two-dimensional illustrative example show that the proposed mechanism achieves ARI=0.666 at 1.60, which substantially outperforms direct coordinate-wise noise addition at the same budget (ARI=0.199), while it matches the non-private synthesis baseline (ARI=0.624). Across 30 independent runs the mechanism achieves mean ARI=0.6290.108, which confirms that the calibration target is reliably met under stochastic variation.
A Remark on Resonance and Beat in a Homogeneous Linear Delay System Robby Robby
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 10, No 2 (2025): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v10i2.35283

Abstract

We consider a homogeneous system of linear delay differential equations (DDEs) with internal coupling. It is well known that such systems can exhibit oscillatory solutions via Hopf bifurcation when the system parameters, including the delay, satisfy certain critical conditions. In this paper, we show that this oscillatory behavior can lead to resonance-like amplification in one variable, even in the absence of any external forcing. The phenomenon arises when the natural frequency of the internal forcing, induced by delay and coupling, matches the system’s oscillatory mode. Furthermore, when distinct delays are introduced in each equation, a beat phenomenon naturally occurs due to the detuning between internal frequencies.
Co-Authors Agus Sukmana Erwinna Chendra