arXiv subjects
Publications and source records attributed to Rafael Diaz.
We study the second law in the context of combinatorial processes, focusing on the mechanisms that give rise to irreversible behavior from an underlying deterministic, invertible, and reversible dynamics.
The rapid growth of smart devices such as phones, wearables, IoT sensors, and connected vehicles has led to an explosion of continuous time series data that offers valuable insights in healthcare, transportation, and more. However, this surge raises significant privacy concerns, as sensitive patterns can reveal personal details. While traditional differential privacy (DP) relies on trusted servers, local differential privacy (LDP) enables users to perturb their own data. However, traditional LDP methods perturb time series data by adding user-specific noise but exhibit vulnerabilities. For instance, noise applied within fixed time windows can be canceled during aggregation (e.g., averaging), enabling adversaries to infer individual statistics over time, thereby eroding privacy guarantees. To address these issues, we introduce a Cooperative Local Differential Privacy (CLDP) mechanism that enhances privacy by distributing noise vectors across multiple users. In our approach, noise is collaboratively generated and assigned so that when all users' perturbed data is aggregated, the noise cancels out preserving overall statistical properties while protecting individual privacy. This cooperative strategy not only counters vulnerabilities inherent in time-window-based methods but also scales effectively for large, real-time datasets, striking a better balance between data utility and privacy in multiuser environments.
In this paper, we explore the use of generative artificial intelligence (GenAI) for ship propeller design. While traditional forward machine learning models predict the performance of mechanical components based on given design parameters, GenAI models aim to generate designs that achieve specified performance targets. In particular, we employ conditional flow matching to establish a bidirectional mapping between design parameters and simulated noise that is conditioned on performance labels. This approach enables the generation of multiple valid designs corresponding to the same performance targets by sampling over the noise vector. To support model training, we generate data using a vortex lattice method for numerical simulation and analyze the trade-off between model accuracy and the amount of available data. We further propose data augmentation using pseudo-labels derived from less data-intensive forward surrogate models, which can often improve overall model performance. Finally, we present examples of distinct propeller geometries that exhibit nearly identical performance characteristics, illustrating the versatility and potential of GenAI in engineering design.
The PWP map was introduced by the second author as a tool for ranking nodes in networks. In this work we extend this technique so that it can be used to rank links as well. Applying the Girvan-Newman algorithm a ranking method on links induces a deconstruction method for networks, therefore we obtain new methods for finding clustering and core-periphery structures on networks.
The second law of thermodynamics states that the entropy of an isolated system is almost always increasing. We propose combinatorial formalizations of the second law and explore their conditions of possibilities.
Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an application we develop a continuous analogue for the binomial distribution.
We introduce a program aimed to studying problems arising from the theory of complex networks with differential geometric means. We study the propagation of influences on manifolds assuming that at each point only a finite number of propagation velocities are allowed. This leads to the computation of the volume of the moduli spaces of directed paths, i.e. paths satisfying the imposed tangential restrictions. The proposed settings provide a fertile ground for research with potential applications in geometry, mathematical physics, differential equations, and combinatorics. We establish the general framework, develop its structural properties, and consider a few basic examples of relevance. The interaction between differential geometry and complex networks is a new and promising field of study.
We study using combinatorial methods the structural coefficients of the formal homogeneous universal enveloping algebra Uh(sl2) of the special linear algebra sl2 over a field of characteristic zero. We provide explicit formulae for the product of generic elements in Uh(sl2), and construct combinatorial objects giving flesh to these formulae.
We give explicit formulae and study the combinatorics of an identity holding in all Rota-Baxter algebras. We describe the specialization of this identity for a couple of examples of Rota-Baxter algebras.
The PWP method was introduced by Diaz in 2009 as a technique for measuring indirect influences in complex networks. It depends on a matrix D, provided by the user, called the matrix of direct influences, and on a positive real parameter which is part of the method itself. We study changes in the method's predictions as D and the parameter vary.
We provide an explicit description of the quantum product of multi-symmetric functions using the elementary multi-symmetric functions introduced by Vaccarino.
We introduce quantum Boolean algebras which are the analogue of the Weyl algebras for Boolean affine spaces. We study quantum Boolean algebras from the logical and set theoretical viewpoints.
We address the problem of gauging the influence exerted by a given country on the global trade market from the viewpoint of complex networks. In particular, we apply the PWP method for computing indirect influences on the world trade network.
We consider four combinatorial interpretations for the algebra of Boolean differential operators. We show that each interpretation yields an explicit matrix representation for Boolean differential operators.
In these notes we study several categorical generalizations of the Möbius function and discuss the relations between the various approaches. We emphasize the topological and geometric meaning of these constructions.
We introduce PWP a method for counting indirect influences and compare it with three well-known methods. We provide combinatorial as well as probabilistic interpretation for the PWP method.
Replacing the usual notion of quotient sets by the notion of orbiquotient sets we obtain a generalization of Pólya theory. The key ingredient of our extended theory is the definition of the orbicycle index polynomial which we compute in several examples. We apply our theory to the study of orbicycles on orbiquotient sets. Keywords: Orbifolds, Pólya Theory, Partition Lattice.
We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Diaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.