As LLM-driven agents advance in cybersecurity, Jeopardy CTF benchmarks are approaching saturation and cyber ranges, the natural next evaluation frontier, offer diminishing resistance under their current static design. We validate this observation by deploying an LLM-driven Advanced Persistent Threat (APT) agent across three tiers of increasingly realistic infrastructure (PRO Labs, MHBench, military-grade CYBER RANGES). To counteract this trend, we propose Dynamic Cyber Ranges: cyber range environments augmented with LLM-driven Defender agents that harden infrastructure, monitor for intrusions, and respond in real time. Across evaluated scenarios, Defender agents reduce attacker success to 0-55%, achieving complete prevention on multiple configurations. Since attacker and defender agents draw from the same underlying model capabilities, Dynamic Cyber Ranges preserve evaluation headroom as models improve. Notably, a smaller, specialized on-premise model (alias2-mini) matched the frontier model's defensive outcomes on multiple scenarios under identical, untuned prompts, and detected the attacker 10x faster on a complex enterprise scenario, suggesting that privacy-preserving on-premise
We study spectral constants for convex domains $Ω$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter $γ$ and relating these bounds to geometric properties of $Ω$ and the numerical range $W(A)$. We generalise the proof that the numerical range is a $1+\sqrt{2}$-spectral set to scaled $q$-numerical ranges. We also propose a generalisation of Crouzeix's Conjecture in the context of $q$-numerical ranges.
The paper offers the first systematic study of ordinary and essential numerical ranges of operators on $\ell_p$, $1<p<\infty$, as an atomic picture within a broader $L^p$ project. The paper begins with Banach-space foundations, including the finite-codimensional description of the essential numerical range and a Banach-space convex-hull inclusion for the essential spectrum. It then turns to finite-dimensional $\ell_p$ geometry, where one finds both positive star-shapedness phenomena and explicit $2\times2$ counterexamples. On $\ell_p$, we prove that the essential numerical range is compact and convex, identify it with the algebraic numerical range of the Calkin image, obtain a compact-perturbation formula, and show that, moreover, the closure of the numerical range is star-shaped, while points in the interior of the essential numerical range are exact star-centres of the numerical range. The paper illustrates the developed theory with sequence-space examples, covering tridiagonal Toeplitz operators and the discrete Hilbert transform, and, after relating our study to a variant of the Crouzeix inequality, closes with a brief discussion of extensions to spaces of class $(P)$ and
Let $\mathbf{x}$ be a (non-empty) sequence of positive real numbers. Its achievement set $\mathcal{\mathbf{x}}$ is the set of all the possible sums of the elements of $\mathbf{x}$. The cardinal function of $\mathbf{x}$ is the function $f:\mathcal{A}(\mathbf{x}) \to \mathbb{N}\cup\{ω,\mathfrak{c}\}$ that for every $x\in\mathbb{A}(\mathbf{x})$ the value $f(x)$ is equal to the number of ways $x$ is represented as a sum of elements of $\mathbf{x}$. In this paper we consider possible ranges of cardinal functions of sequences $\mathbf{x}$. We present some general constructions and several criteria that a set has to satisfy in order to be a range of a cardinal function. We put special attention to the case of sets with maximal element equal to $6$. In this case, in particular, we obtained a full characterisation of sets that are ranges of cardinal functions of interval-filling sequences.
Symbolic execution is a software verification technique symbolically running programs and thereby checking for bugs. Ranged symbolic execution performs symbolic execution on program parts, so called path ranges, in parallel. Due to the parallelism, verification is accelerated and hence scales to larger programs. In this paper, we discuss a generalization of ranged symbolic execution to arbitrary program analyses. More specifically, we present a verification approach that splits programs into path ranges and then runs arbitrary analyses on the ranges in parallel. Our approach in particular allows to run different analyses on different program parts. We have implemented this generalization on top of the tool CPAchecker and evaluated it on programs from the SV-COMP benchmark. Our evaluation shows that verification can benefit from the parallelisation of the verification task, but also needs a form of work stealing (between analyses) as to become efficient
We characterize the d x d matrices whose numerical ranges are invariant by rotations of angle 2$π$/d.
In parametric design, the geometric model is edited by changing relevant parameters in the parametric model, which is commonly done sequentially on multiple parameters. Without guidance on allowable parameter ranges that can guarantee the solvability of the geometric constraint system, the user could assign improper parameter values to the model's parameters, which would further lead to a failure in model updating. However, current commercial CAD systems provide little support for the proper parameter assignments. Although the existing methods can compute allowable ranges for individual parameters, they face difficulties in handling multi-parameter situations. In particular, these methods could miss some feasible parameter values and provide incomplete allowable parameter ranges. To solve this problem, an automatic approach is proposed in this paper to compute complete parameter ranges in multi-parameter editing. In the approach, a set of variable parameters are first selected to be sequentially edited by the user; before each editing operation, the one-dimensional ranges of the variable parameters are presented as guidance. To compute the one-dimensional ranges, each variable para
We introduce the new concepts of pseu\-do numerical range for operator functions and families of sesquilinear forms as well as the pseu\-do block numerical range for $n \times n$ operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order $0$ and not even $<1$. As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping.
This paper describes an exploratory empirical study of the effect of named ranges on spreadsheet debugging performance. Named ranges are advocated in both academia and industry, yet no experimental evidence has been cited to back up these recommendations. This paper describes an exploratory experiment involving 21 participants that assesses the performance of novices debugging a spreadsheet containing named ranges. The results are compared with the performance of a different set of novices debugging the same spreadsheet without named ranges. The findings suggest that novice users debug on average significantly fewer errors if the spreadsheet contains named ranges. The purpose of the investigative study is to derive a detailed and coherent set of research questions regarding the impact of range names on the debugging performance and behaviour of spreadsheet users. These will be answered through future controlled experiments.
We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We apply these to get explicit stable ranges for diagonal coinvariant algebras, and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.
Leveraging human grasping skills to teach a robot to perform a manipulation task is appealing, but there are several limitations to this approach: time-inefficient data capture procedures, limited generalization of the data to other grasps and objects, and inability to use that data to learn more about how humans perform and evaluate grasps. This paper presents a data capture protocol that partially addresses these deficiencies by asking participants to specify ranges over which a grasp is valid. The protocol is verified both qualitatively through online survey questions (where 95.38% of within-range grasps are identified correctly with the nearest extreme grasp) and quantitatively by showing that there is small variation in grasps ranges from different participants as measured by joint angles, contact points, and position. We demonstrate that these grasp ranges are valid through testing on a physical robot (93.75% of grasps interpolated from grasp ranges are successful).
A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can, and cannot, be realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.
Accidents are a leading cause of deaths in armed forces. The Aim of this paper is to minimize the accidents caused using weapons in the armed forces. Developing artificial intelligence technologies aim to increase efficiency more and more wherever people exist. Giving guns to inexperienced, untrained, or unpredictable mentally unhealthy people in shooting ranges used for gun training can be risky and fatal. With the use of image processing technologies in these shooting ranges, it is aimed to minimize the risk of life-threatening accidents that may be caused by this people. Artificial intelligence is trained for the targets to be used in shooting ranges. When the camera of weapon sees these targets, it switches from safe mode to firing mode. When a risky situation occurs in shooting range, the gun turns itself into safe mode with various additional security measures.
Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the $σ$-field into an Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two-dimensional vector measures and negative for higher dimensions. We use the existence of maximal ranges to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the case of two measures.
We consider two-variable model spaces associated to rational inner functions $Θ$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_Θ:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and study the geometry of the associated numerical ranges, find formulas for the boundaries, answer the zero inclusion question, and determine whether the numerical ranges are ever circular.
Inspired by Benjamini et al (Ann. Inst. H. Poincaré Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the ranges. Some equivalent conditions for the vanishing of the average entropy of the ranges are given. Particularly, the average entropy of the ranges vanishes if and only if the random walk is recurrent or escaping to negative infinity without left jump. In order to characterize the recurrence further, we study the average entropy of the weighted digraphs formed by the random walk. We show that the random walk is recurrent if and only if the average entropy of the weighted digraphs vanishes.
Range-Based Set Reconciliation (RBSR) synchronizes ordered sets by recursively comparing summaries of contiguous ranges and refining only the mismatching parts. While its communication complexity is well understood, its local computational cost fundamentally depends on the storage backend that must answer repeated range-summary, rank, and enumeration queries during refinement. We argue that a natural storage abstraction for RBSR implementations based on composable range aggregates is a \emph{range-summarizable order-statistics store} (RSOS): a dynamic ordered-set structure supporting composable summaries of contiguous ranges together with rank/select navigation. This identifies and formalizes the backend contract needed for efficient recursive refinement, combining range-summary support with order-statistics navigation for balanced partitioning. We then show that a specific augmentation of B\textsuperscript{+}-trees with subtree counts and composable summaries realizes a RSOS, and we derive corresponding bounds on local reconciliation work in this abstract storage model. Finally, we introduce AELMDB, an extension of LMDB that realizes this design inside a persistent memory-mapped e
We investigate the numerical ranges of weighted composition operators on weighted Dirichlet spaces, focusing on the properties of the inducing functions. We identify conditions on these functions under which the origin lies in the interior of the numerical range. The geometric structure of the numerical range is also analyzed, determining when it contains a circular or elliptical disc and computing the corresponding radius. Next, we introduce a class of Weyl-type weighted composition operators and obtain their Berezin range and Berezin number. Finally, we characterize the convexity of the Berezin range for weighted composition operators on these spaces.
Critical infrastructures increasingly rely on interconnected and software-driven Cyber-Physical Systems (CPS), exposing operational processes to both accidental failures and sophisticated adversarial behavior. While Byzantine Fault Tolerant (BFT) protocols offer robustness against arbitrary faults, evaluating their behavior under realistic cyber-physical conditions remains challenging: traditional cyber ranges lack timing fidelity, and testing in production environments is unsafe. This paper introduces ByzTwin-Range, a dual-layer architecture that integrates a production-grade BFT deployment with a Digital Twin (DT) to enable controlled experimentation, stress testing, and Byzantine fault injection using live operational data. The DT mirrors real system state, executes "What-if" analyses through co-simulation and emulation, and identifies synchrony vulnerabilities, i.e., misconfigured timeouts, timing-sensitive false suspicions, and adversarial delay exploits, configuration weaknesses, and adversarial behaviors that may undermine BFT guarantees. Insights from the twin are fed back into the operational deployment through a secure advisory channel, supporting continuous validation an
We show that the Carathéodory number of the joint numerical range of $d$ many bounded self-adjoint operators is at most $d-1$, and even at most $d-2$ if the underlying Hilbert space has dimension at least $3$. This extension of the classical convexity results for numerical ranges shows that also joint numerical ranges are significantly less non-convex than general sets.