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We introduce the notion of the Bruce-Roberts Tjurina number for holomorphic 1-forms relative to a pair $(X,V)$ of complex analytic subvarieties. When the pair $(X,V)$ consists of isolated complex analytic hypersurfaces, we prove that the Bruce-Roberts Tjurina number is related to the Bruce-Roberts number, the Tjurina number of the 1-form with respect to $V$ and the Tjurina number of $X$, among other invariants. As an application, we present a quasihomogeneity result for germs of holomorphic foliations in complex dimension two., Comment: 15 pages

We introduce the notion of the \textit{Bruce-Roberts number} for holomorphic 1-forms relative to complex analytic varieties. Our main result shows that the Bruce-Roberts number of a 1-form $\omega$ with respect to a complex analytic hypersurface $X$ with an isolated singularity can be expressed in terms of the \textit{Ebeling--Gusein-Zade index} of $\omega$ along $X$, the \textit{Milnor number} of $\omega$ and the \textit{Tjurina number} of $X$. This result allows us to recover known formulas for the Bruce-Roberts number of a holomorphic function along $X$ and to establish connections between this number, the radial index, and the local Euler obstruction of $\omega$ along $X$. Moreover, we present applications to both global and local holomorphic foliations in complex dimension two., Comment: 31 pages. arXiv admin note: text overlap with arXiv:2107.01967 by other authors

Black box deep learning models trained on genomic sequences excel at predicting the outcomes of different gene regulatory mechanisms. Therefore, interpreting these models may provide novel insights into the underlying biology, supporting downstream biomedical applications. Due to their complexity, interpretable surrogate models can only be built for local explanations (e.g., a single instance). However, accomplishing this requires generating a dataset in the neighborhood of the input, which must maintain syntactic similarity to the original data while introducing semantic variability in the model's predictions. This task is challenging due to the complex sequence-to-function relationship of DNA. We propose using Genetic Programming to generate datasets by evolving perturbations in sequences that contribute to their semantic diversity. Our custom, domain-guided individual representation effectively constrains syntactic similarity, and we provide two alternative fitness functions that promote diversity with no computational effort. Applied to the RNA splicing domain, our approach quickly achieves good diversity and significantly outperforms a random baseline in exploring the search space, as shown by our proof-of-concept, short RNA sequence. Furthermore, we assess its generalizability and demonstrate scalability to larger sequences, resulting in a ~30% improvement over the baseline.

Healthcare data contains some of the most sensitive information about an individual, yet sharing this data with healthcare practitioners can significantly enhance patient care and support research efforts. However, current systems for sharing health data between patients and caregivers do not fully address the critical security requirements of privacy, confidentiality, and consent management. Furthermore, compliance with regulatory laws such as GDPR and HIPAA is often deficient, largely because patients typically are asked to provide general consent for healthcare entities to access their data. Recognizing the limitations of existing systems, we present S3PHER, a novel approach to sharing health data that provides patients with control over who accesses their data, what data is accessed, and when. Our system ensures end to end privacy by integrating a Proxy ReEncryption Scheme with a Searchable Encryption Scheme, utilizing Homomorphic Encryption to enable healthcare practitioners to privately search and access patients' documents. The practicality and benefits of S3PHER are further validated through end to end deployment and use case analyses, with tests on real datasets demonstrating promising execution times., Comment: 20 pages, 1 figure, 2 tables in the appendix

The use of Neural Networks (NNs) for sensitive data processing is becoming increasingly popular, raising concerns about data privacy and security. Homomorphic Encryption (HE) has the potential to be used as a solution to preserve data privacy in NN. This study provides a comprehensive analysis on the use of HE for NN training and classification, focusing on the techniques and strategies used to enhance data privacy and security. The current state-of-the-art in HE for NNs is analysed, and the challenges and limitations that need to be addressed to make it a reliable and efficient approach for privacy preservation are identified. Also, the different categories of HE schemes and their suitability for NNs are discussed, as well as the techniques used to optimize the accuracy and efficiency of encrypted models. The review reveals that HE has the potential to provide strong data privacy guarantees for NNs, but several challenges need to be addressed, such as limited support for advanced NN operations, scalability issues, and performance trade-offs., Comment: 11 pages, 1 figure, DCAI 2023 Conference Pre-print

Genetic Programming (GP) is an heuristic method that can be applied to many Machine Learning, Optimization and Engineering problems. In particular, it has been widely used in Software Engineering for Test-case generation, Program Synthesis and Improvement of Software (GI). Grammar-Guided Genetic Programming (GGGP) approaches allow the user to refine the domain of valid program solutions. Backus Normal Form is the most popular interface for describing Context-Free Grammars (CFG) for GGGP. BNF and its derivatives have the disadvantage of interleaving the grammar language and the target language of the program. We propose to embed the grammar as an internal Domain-Specific Language in the host language of the framework. This approach has the same expressive power as BNF and EBNF while using the host language type-system to take advantage of all the existing tooling: linters, formatters, type-checkers, autocomplete, and legacy code support. These tools have a practical utility in designing software in general, and GP systems in particular. We also present Meta-Handlers, user-defined overrides of the tree-generation system. This technique extends our object-oriented encoding with more practicability and expressive power than existing CFG approaches, achieving the same expressive power of Attribute Grammars, but without the grammar vs target language duality. Furthermore, we evidence that this approach is feasible, showing an example Python implementation as proof. We also compare our approach against textual BNF-representations w.r.t. expressive power and ergonomics. These advantages do not come at the cost of performance, as shown by our empirical evaluation on 5 benchmarks of our example implementation against PonyGE2. We conclude that our approach has better ergonomics with the same expressive power and performance of textual BNF-based grammar encodings.