arXiv subjects

Fatemeh Jalalvand

Publications and source records attributed to Fatemeh Jalalvand.

5 recordsLinked to original sources

arXiv2026

On the Geometry and Shapes of Rank 2 Log Unit Lattices

Every number field canonically gives rise to two lattices: its ring of integers and its log unit lattice. While the shapes of the former have undergone extensive research, far less is known about the shapes of the latter, referred to as unit shapes. This paper presents an in-depth analysis of the unit shapes of number fields with unit rank 2. Our first main result characterizes, in many cases, the location of unit shapes within the fundamental domain of the space of rank 2 lattice shapes in terms of the Galois group of the field's Galois closure, and determines when these unit shapes are transcendental. Next, we establish that the unit shape uniquely determines the field up to isomorphism for totally imaginary $D_6$ non-CM sextic fields; this result fails in the CM case. Finally, for certain subfamilies of $D_6$ non-CM imaginary sextics, we offer a simple sufficient condition for their log unit lattices to be orthogonal and provide lower bounds on the proportion of fields with orthogonal log unit lattice.

math.NT↗

arXiv2026

On the Squarefree Values of Degree-$2q$ Polynomials

In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.

math.NT↗

arXiv2025

Unit lattices of $D_4$-quartic number fields with signature $(2,1)$

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on $D_4$-quartic fields with signature $(2,1)$; such fields have a rank $2$ unit group. Viewing the unit lattice as a point of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Moreover, we produce three explicit (algebraic) points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ which are limit points of the set of (points associated to) unit lattices of $D_4$-quartic fields with signature $(2,1)$.

math.NT↗

arXiv2025

LLMs in the SOC: An Empirical Study of Human-AI Collaboration in Security Operations Centres

The integration of Large Language Models (LLMs) into Security Operations Centres (SOCs) presents a transformative, yet still evolving, opportunity to reduce analyst workload through human-AI collaboration. However, their real-world application in SOCs remains underexplored. To address this gap, we present a longitudinal study of 3,090 analyst queries from 45 SOC analysts over 10 months. Our analysis reveals that analysts use LLMs as on-demand aids for sensemaking and context-building, rather than for making high-stakes determinations, preserving analyst decision authority. The majority of queries are related to interpreting low-level telemetry (e.g., commands) and refining technical communication through short (1-3 turn) interactions. Notably, 93% of queries align with established cybersecurity competencies (NICE Framework), underscoring the relevance of LLM use for SOC-related tasks. Despite variations in tasks and engagement, usage trends indicate a shift from occasional exploration to routine integration, with growing adoption and sustained use among a subset of analysts. We find that LLMs function as flexible, on-demand cognitive aids that augment, rather than replace, SOC expertise. Our study provides actionable guidance for designing context-aware, human-centred AI assistance in security operations, highlighting the need for further in-the-wild research on real-world analyst-LLM collaboration, challenges, and impacts.

cs.CR↗

arXiv2025

Adaptive alert prioritisation in security operations centres via learning to defer with human feedback

Alert prioritisation (AP) is crucial for security operations centres (SOCs) to manage the overwhelming volume of alerts and ensure timely detection and response to genuine threats, while minimising alert fatigue. Although predictive AI can process large alert volumes and identify known patterns, it struggles with novel and evolving scenarios that demand contextual understanding and nuanced judgement. A promising solution is Human-AI teaming (HAT), which combines human expertise with AI's computational capabilities. Learning to Defer (L2D) operationalises HAT by enabling AI to "defer" uncertain or unfamiliar cases to human experts. However, traditional L2D models rely on static deferral policies that do not evolve with experience, limiting their ability to learn from human feedback and adapt over time. To overcome this, we introduce Learning to Defer with Human Feedback (L2DHF), an adaptive deferral framework that leverages Deep Reinforcement Learning from Human Feedback (DRLHF) to optimise deferral decisions. By dynamically incorporating human feedback, L2DHF continuously improves AP accuracy and reduces unnecessary deferrals, enhancing SOC effectiveness and easing analyst workload. Experiments on two widely used benchmark datasets, UNSW-NB15 and CICIDS2017, demonstrate that L2DHF significantly outperforms baseline models. Notably, it achieves 13-16% higher AP accuracy for critical alerts on UNSW-NB15 and 60-67% on CICIDS2017. It also reduces misprioritisations, for example, by 98% for high-category alerts on CICIDS2017. Moreover, L2DHF decreases deferrals, for example, by 37% on UNSW-NB15, directly reducing analyst workload. These gains are achieved with efficient execution, underscoring L2DHF's practicality for real-world SOC deployment.

cs.CR↗