We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid hat tilings, and conversely, in an appropriate sense that is made precise in the paper, that every valid hat tiling can be obtained in this way.

We use the 3D fully kinetic simulation to study different turbulence modes and turbulence anisotropy of relativistic turbulence in magnetically dominated collisionless plasmas. We extend the method developed by Cho & Lazarian (2002) for decomposing non-relativistic magnetohydrodynamic (MHD) turbulence into Alfvén, fast, and slow modes to the regime of collisionless plasmas. We find that Alfvén and slow modes are anisotropic, following the Goldreich & Sridhar (1995) scaling, while fast modes are isotropic. We observe a larger kinetic energy fraction of fast modes compared to that in the non-relativistic MHD turbulence, suggesting a stronger coupling of Alfvén and fast modes in relativistic magnetized turbulence in collisionless plasmas. We further examine the dynamic alignment and find a weaker scale dependence of the alignment angle than previously proposed. The dominant thermal fluctuations in the kinetic range can cause flattening of the turbulent velocity structure function and weakening of the turbulence anisotropy and dynamic alignment near the kinetic scales.

Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k, k) \geq 4k-O(k^\frac{2}{3})}$, and conjectured that $n(k,k)=4k-4$. We prove this conjecture, and also establish the optimal lower bound in the more general case where $k_1$ and $k_2$ are arbitrary. We further consider the generalisation of the problem to $r$-edge-coloured complete graphs in which every vertex is contained in a size-$k$ monochromatic clique of each colour, and obtain upper and lower bounds on the size of such graphs.

We explore the application of the Variational Quantum Eigensolver (VQE) to investigate the ground state properties, particularly the entanglement entropy, of the Transverse Field Ising Model (TFIM) in one, two, and three dimensions, considering systems of up to 27 spins. By benchmarking VQE results against exact diagonalization and analyzing the entanglement properties across different system sizes and geometries, we assess the algorithm's effectiveness in capturing critical phenomena. Using results of TFIM, we also investigate how VQE's expressivity and optimization influence the simulation of highly entangled quantum states. We employ different ansätze: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational ansätz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization. We further discuss the implications for scaling these methods to larger quantum systems.

We present a covariant and gauge-invariant formulation of the theory of radial adiabatic linear perturbations of self-gravitating, non-dissipative imperfect fluids within the theory of general relativity. By codifying the thermodynamical properties of the source into an equation of state and an ansatz on anisotropic pressure that involves both matter and kinematic variables, we obtain a set of equations that is directly applicable to a wide variety of thermodynamic theories for matter fields. As examples, we evaluate and compare the predictions of the Eckart theory, the Bemfica-Disconzi-Noronha-Kovtun theory, and the Truncated Israel-Stewart theory on the properties and evolution of radial adiabatic perturbations of stellar compact objects modeled by classical equilibrium solutions. Introducing a new solution of the Einstein field equations, and imposing causality, we propose an upper bound for the maximum compactness of dynamically stable stars with non-trivial radial and tangential pressures.

GD-1 is among the longest, coldest stellar streams in the Milky Way, making it an ideal target for probing dark matter substructure through dynamical heating. We present a catalog of 608 spectroscopically confirmed GD-1 members from the first three years of Dark Energy Spectroscopic Instrument (DESI) observations. This constitutes the largest homogeneous spectroscopic sample of GD-1, doubling the number of members previously available only through heterogeneous compilations combining multiple surveys with different systematics. Using these data, we derive updated stream tracks in sky position, proper motion, and radial velocity that extend over $100^\circ$ of the stream. We apply a Gaussian mixture model to decompose the stream into a dynamically cold thin component ($σ_V = 2.49\pm 0.28$ km s$^{-1}$, width $= 0.23\pm0.01^\circ$) and a kinematically hot cocoon ($σ_V = 6.13\pm0.75$ km s$^{-1}$, width $= 2.18\pm0.17^\circ$). The cocoon contains $\sim30\%$ of members and its velocity dispersion is consistent with $\sim11$ Gyr of heating by cold dark matter subhalos. We also detect a large proper motion dispersion ($41.36\pm4.98$ km s$^{-1}$) along the stream direction in the cocoon component. This feature indicates a significant line-of-sight distance spread in the cocoon, and its origin will be further explored in a forthcoming paper. These measurements demonstrate the power of DESI spectroscopy for characterizing the multi-component phase-space structure of stellar streams and constraining small-scale dark matter substructure.

This paper presents a comprehensive study of the reconstruction of sub-GeV neutrino charged-current interactions within a Liquid Argon Time Projection Chamber (LArTPC). We demonstrate that traditional charge-based calorimetry is fundamentally limited at sub-GeV scales by significant recombination fluctuations and missing hadronic energy. We show that energy reconstruction using energy deposited as scintillation light (L) partially benefits from the previously reported self-compensating light effect. At neutrino energies above 400 MeV, the light-only reconstruction still outperforms charge-only methods that can separate EM and hadronic objects. The performance of the two remains comparable below 300 MeV. Using the energy-deposit information from both detector signals, we demonstrate a 70% efficiency in separating electron neutrinos and antineutrinos. By using a proximity-based algorithm coupled with a geometric lepton-exclusion cone, we also demonstrate the ability to isolate neutron-induced energy depositions from background. This enables an improvement of sub-GeV direction reconstruction by about 20 degrees for antineutrinos. This study provides new insights into how to enhance the physics reach of future LArTPC atmospheric neutrino analyses.

The determination of leptonic CP violation is a primary goal of future long-baseline neutrino experiments such as DUNE and T2HK. The extraction of $δ_{\mathrm{CP}}$ relies on the near-to-far extrapolation and on the assumed knowledge of the cross-section ratios $σ_{ν_e}/σ_{ν_μ}$ and $σ_{\barν_e}/σ_{\barν_μ}$, which are typically inferred under theoretical assumptions such as lepton universality and depend on nuclear modeling. In this work, we quantify how much of the sensitivity of DUNE and T2HK arises from these assumptions by performing a model-agnostic, data-driven estimation of systematic uncertainties in $ν_e$ and $\barν_e$ cross sections. We find that adopting such an agnostic approach can substantially degrade the CP-violation sensitivity, reducing it by nearly $3σ$ at maximal CP violation for DUNE, and $4σ$ for T2HK. We then assess the impact of the proposed $ν$SCOPE experiment, which, through a combination of neutrino tagging and the Narrow-Band Off-Axis technique, can provide percent-level measurements of $σ_{ν_μ}$ and $σ_{\barν_μ}$ and constrain the ratio $σ_{ν_e}/σ_{ν_μ}$ at the $\sim 2\%$ level. We show that including prospective $ν$SCOPE measurements largely restores the lost sensitivity, highlighting that precise external cross-section measurements may be essential for a fully robust determination of $δ_{\mathrm{CP}}$ and for breaking its degeneracy with nuclear mis-modeling or possible new physics affecting neutrino detection.

We present SubTropica, a Mathematica package that performs symbolic integration of multi-polylogarithmic integrals using recent advances in tropical geometry. It focuses on the class of linearly-reducible Euler integrals, such as Feynman integrals, and expands them using a tropical subtraction scheme. The engine behind it is HyperIntica, a native Mathematica package for hyperlogarithm integration that can be used independently. This paper documents both packages and illustrates their usage on examples from across different physics applications. Additionally, we introduce an AI-driven library of Feynman integrals, which catalogs diagrams discussed in the literature and serves as a database for computed results. Its online version is available at: https://subtropi.ca and features a graphical user interface for diagram input and retrieval of records.

In this work, we show that Berry phase estimation admits a natural and universal adiabatic error-cancellation mechanism, making it a promising candidate for practical quantum computing before full fault tolerance. Combining finite-runtime evolutions under $\pm H$ along the loop cancels the leading $O(T^{-1})$ phase error exactly, and Richardson extrapolation further reduces the residual error to an oscillatory term with endpoint-controlled coefficient $O(\|\dot H(0)\|^2Δ(0)^{-4}T^{-2})$. Beyond this deterministic cancellation, we establish that, for suitable smooth runtime distributions, runtime randomization suppresses the remaining oscillatory contribution to $O(T^{-M})$ for any fixed $M$, leading to a randomized Hadamard-test algorithm for Berry phase estimation over the full range $[0,2π)$ with improved runtime scaling under standard sample complexity.

Upcoming long-baseline neutrino oscillation experiments such as DUNE aim to achieve unprecedented precision, but their physics reach is ultimately constrained by systematic uncertainties in neutrino flux predictions and neutrino-nucleus cross sections. These limitations are especially critical for new-physics searches in neutrino oscillations at the near detector, including non-unitarity and sterile neutrinos, where the signal manifests as small distortions in the energy spectrum and is therefore highly sensitive to spectral uncertainties. The PRISM (Precision Reaction Independent Spectrum Measurement) technique offers a robust strategy to mitigate these effects by exploiting measurements at multiple off-axis angles, effectively providing a data-driven handle to reduce systematics. In this work, we demonstrate that PRISM can significantly reduce the impact of large systematic uncertainties, restoring sensitivity to non-unitarity and sterile neutrino scenarios in the electron and muon sectors to a level comparable to that obtained with small spectral uncertainties. We also include the results for the $τ$ sector with PRISM; however, in this case, since the majority of the flux measured at off-axis angles lies below the $τ$ production threshold, we find the improvement to be marginal. As part of this work, we have obtained neutrino and antineutrino fluxes for different off-axis angles with higher statistics than those provided by the DUNE collaboration. We make available these fluxes as auxiliary material to this manuscript.

Radio continuum observations are widely used to study cosmic ray (CR) electron populations and transport processes in star-forming galaxies, but their interpretation relies on several simplifying assumptions. Here, we revisit three common assumptions: that some vertical radio profiles can be explained by CR advection alone, that radio spectra directly trace the galaxy-wide CR electron spectrum, and that bremsstrahlung and Coulomb losses are negligible for radio-emitting electrons. We model radio emission using time-dependent CR electron evolution in a magnetohydrodynamical simulation of an isolated Milky Way-mass galaxy. CR electron spectra are evolved self-consistently along Lagrangian tracer particles with the CREST framework, including injection at supernova remnants, advection with the gas, and spatially and temporally varying radiative losses. We compare these results to commonly adopted steady-state models. We find that advection-only transport in self-consistently driven galactic winds fails to reproduce the extended vertical radio intensity profiles observed in edge-on galaxies, despite reproducing the observed steepening of spectral indices with height. This is because slowly accelerating winds keep electrons in strong cooling environments for too long. Matching observed radio haloes with advection alone requires unrealistically high midplane wind velocities, implying that additional transport or re-acceleration processes are required. Although galaxy-integrated CR electron spectra at radio-emitting energies are similar across models, the resulting synchrotron spectra differ systematically because radio emission is biased toward young electrons in dense, strongly magnetised regions. Finally, we show that bremsstrahlung and Coulomb losses significantly shape radio spectra even when their loss rate is subdominant and therefore cannot be neglected.

Mental health challenges are rising globally, while traditional support services face limited availability and high costs. Large language models offer potential for conversational support, but often lack personalization, empathy, and factual grounding. A virtual agent framework is introduced to provide empathetic, personalized, and reliable wellbeing support through retrieval-augmented architecture, structured memory, and multimodal interaction. Objective benchmarks demonstrate improved retrieval and response quality, particularly for smaller models. A cross-cultural study with university students from Vietnam and Australia shows the system outperforms LLM-only baselines in coherence, perceived accuracy, and empathy, with most participants clearly preferring the proposed approach.

For some important families of complete infinite lattices, we study some generalizations of two fundamental notions which are mostly treated for finite lattices. Specifically, for well-separated $κ$-lattices, and also for weakly atomic completely semidistributive lattices, we generalize the notions of left modularity and extremality. These two families of lattices coincide if restricted to finite lattices, but are distinct when infinite lattices are also included. For both families, we prove that extremality and left modularity imply each other. Furthermore, for weakly atomic completely semidistributive lattices, we give several conceptual characterizations of left modular elements, and show that the set of left modular elements form a complete distributive sublattice. Our results, combined with some recent work on finite lattices, imply that the weakly atomic completely semidistributive lattices that are left modular (or extremal) generalize the semidistributive trim lattices; from finite to infinite lattices. We then apply our results to the lattice of torsion classes of finite dimensional algebras, which are known to fall in the intersection of the two families treated in our work. For an algebra $A$, we obtain that the lattice of torsion classes is left modular (equivalently, extremal) if and only if $A$ is brick-directed. This leads to an abundance of concrete examples and non-examples.

A comprehensive analysis of viral mutations is essential for understanding viral evolution, disease epidemiology, diagnosis, drug resistance, etc. However, challenges remain in capturing complex mutation patterns and supporting diverse viral families with varying genome architectures. To address these needs, we present VARIANT, an web server for mutational analysis of RNA viral genomes and associated viral products across both single- and multi-segment virus genomes. The server takes as input a viral reference genome, a reference protein sequence, and/or multiple sequence alignment, and automatically provides full annotation of mutation types, including standard categories such as point mutations (missense, silent, and nonsense), insertions, deletions, or frameshift events in both coding and non-coding regions. In addition, VARIANT detects three biologically significant mutation patterns that are overlooked by conventional software/packages: ``row mutations'' (consecutive substitutions within a window of 3 nts), ``hot mutations'' (two non-consecutive substitutions within a window of 3 nts), and potential programmed ribosomal frameshifting (PRF) regions. The server currently contains automatic analysis of major viral pathogens, including SARS-CoV-2, HIV-1, Influenza H3N2, Ebola virus, and Chikungunya virus. It also allows users to analyze customized viruses. Users can track VARIANT analysis progress in real time, visualize mutation distributions, and download structured results in ZIP format. VARIANT also incorporates dual graph topology analysis to classify frameshifting element structures from dot-bracket notation input. This feature enables systematic comparison of RNA secondary structure motifs across viral families by mapping structures to a comprehensive library of dual graph topologies. The web server is freely available at https://variant.up.railway.app.

Recent LIGO-Virgo-KAGRA (LVK) analyses have revealed complex structure in the binary black hole (BBH) population, including distinct features in the primary mass spectrum and nontrivial spin-mass correlations. However, the phenomenological models used to capture these features often lack analytic transparency, making it difficult to isolate robust physical laws from modeling artifacts. To address this, symbolic regression is applied to the posterior inference products of the GWTC-4 catalog, discovering compact, closed-form analytic expressions for four key population relationships: (i) the merger-rate evolution with redshift; (ii) the mass-ratio dependence of the effective-spin distribution; (iii) the redshift evolution of the effective-spin distribution; and (iv) the conditional mass-ratio distributions associated with the 10 solar mass and 35 solar mass primary mass peaks. This framework successfully compresses both rigid and highly flexible models into differentiable phenomenological laws, dynamically recovering a consistent low-redshift merger-rate slope without assuming an a priori power-law form. The exact analytic derivatives provided by symbolic regression show that the mass ratio--effective spin and redshift--effective spin correlations are robustly driven by broadening of the posterior widths rather than shifts in the mean. Furthermore, qualitatively distinct functional forms for the mass-ratio distributions conditioned on the 10 solar mass and 35 solar mass primary mass peaks are identified. These closed-form expressions enable exact analytic gradient diagnostics and compact surrogate summaries, particularly for flexible numerical posteriors that are not otherwise available in low-dimensional analytic form. They also facilitate rapid downstream calculations for rate forecasting, formation channel comparison, and stochastic background estimation.

Traditional X-ray computed tomography (CT) scanning strategies typically select projection angles uniformly and allocate dose equally. In practice, however, CT scans often need to be fast, radiation-efficient, and adaptive. Sparse-view tomography addresses these requirements by reducing both the number of angles and the total dose budget. Under such constraints, angle selection and dose allocation should be information-driven, with more dose assigned to informative directions. To this end, we propose a dose-aware acquisition and reconstruction framework that combines a PWLS-PnP reconstruction backbone with an RL-based strategy for adaptive angle selection, explicitly accounting for angle-dependent photon statistics. Numerical experiments show that the proposed approach improves overall reconstruction quality and enhances defect detectability compared with conventional strategies, particularly when only a small number of projections or a constrained dose budget is available.

This is the third and last of three papers introducing generalised Cesaro convergence and is split into two parts. In part 1 we introduce the notion of a "Cesaro-adapted scale" and use it to prove the key generalised Cesaro summation/convergence theorems developed in the first paper in this series. We also use it to trivially extend these results to the case of remainder Cesaro summation/convergence relative to arbitrary $z_{0}\in\mathbb{C}$ (not just $z_{0}=0$). In the course of the working we introduce the concepts of "formal symbols" and "formal function elements", which allow us to express many results in extremely compact form and simplify our arguments considerably. Part 2 is self-contained and devoted to further exploring this "formal" world. We express a number of additional results in surprisingly compact form using formal symbols and function elements, and use them to give simple proofs of several non-trivial results. We also investigate their fascinating properties. These include the need to avoid evaluating too early; the consequent need to retain stand-alone zeros (both "to the left" and "to the right") lest they be brought back to life before evaluation; and the need to use continuous limits to resolve singular ratios in final evaluation when required. Finally, we consider in detail the formal extension we have introduced of our Cesaro-adapted scale to a 1-parameter continuum of period-1 functions $\overset{\lor}{q}_ρ(α)$, $ρ\in\mathbb{C}$. We analyse their distributional aspects when $ρ\in\mathbb{Z}_{<0}$ and derive their Fourier-series coefficients in general. We conclude with a miscellany of further observations, including a formal re-casting of the general Euler-McLaurin sum formula in very compact form, and a number of additional analytical and combinatorial characteristics of the $\overset{\lor}{q}_ρ(α)$ and associated operators.

This paper investigates the equivalence reduction for several classes of multivariate polynomial matrices and their Smith forms, establishing some criteria for such reduction. In particular, we employ algebra isomorphisms as a key tool to study this equivalence problem. We then leverage the Quillen-Suslin and Lin-Bose theorems to extend these results to non-square and rank-deficient matrices. Moreover, the verification of our criteria is achievable algorithmically via existing Gröbner basis methods.

Building on recent work introducing the idea of Quasilocal Probability in curved spacetime, we develop its observational implications for black hole ringdown in detail. We show that horizon-induced probability flux leads to an effective non-Hermitian dynamics producing three distinctive signatures, which are correlated multi-mode deviations, weak amplitude dependence and a mismatch between waveform damping and energy accounting. These effects arise from a single boundary-flux mechanism and therefore exhibit a constrained, low-dimensional structure not expected in generic modified gravity scenarios. We demonstrate that while individual deviations may be mimicked, their combined pattern provides a robust discriminator of quasilocal probability. We further argue that upcoming gravitational wave observations can probe these signatures at meaningful precision. We also establish that black hole ringdown is a novel arena to test whether quantum mechanical Hermiticity is really a fundamental property or an emergent symmetry in quantum gravity.

Speculative decoding has emerged as a promising technique for large language model (LLM) inference by accelerating autoregressive decoding via draft-then-verify. This paper studies a new edge scenario with multi-user inference, where draft tokens are generated locally on devices and subsequently offloaded to a centralized edge server for batch verification. The key challenge is to sustain high throughput under coupled decisions of (i) batching and pipeline scheduling and (ii) per user draft token length. We propose DiP-SD, which exploits two complementary parallelism dimensions: device-level distributed drafting and phase-level draft-verify pipelining. We formulate a throughput-maximization objective, defined as the expected number of accepted tokens per unit time, and jointly optimize the number of batches, user-to-batch assignment, and integer draft lengths. To solve the resulting fractional mixed-integer program, DiP-SD scans the batch number and iteratively alternates between an association subproblem and a draft-length subproblem. Numerical results under a Qwen3-1.7B/Qwen3-32B device-edge deployment show that DiP-SD achieves up to 17.89x throughput over autoregressive decoding (AD) and 1.93x over AD with greedy batching.

Design automation has the potential to substantially improve the efficiency of analog integrated circuit (IC) design. However, existing algorithms and tools typically focus on individual stages, such as device sizing, placement, or routing, and still require significant manual intervention to complete the full design flow. While large language models (LLMs) have recently demonstrated remarkable success in automating digital IC design workflows, these advances cannot be directly transferred to analog IC design. Key challenges include strongly coupled performance metrics, the predominance of unstructured circuit schematic images, and the fact that most prior approaches address only isolated stages of the analog design process, limiting their ability to capture end-to-end performance impact. To address these challenges, we propose AnalogMaster, an extensible, LLM-based framework that enables end-to-end automation of analog IC design through a unified pipeline spanning circuit image-to-netlist generation, parameter optimization, placement, and routing. AnalogMaster integrates a joint reasoning mechanism that leverages in-context learning and intent reasoning to achieve accurate and robust image-to-netlist conversion. A parameter search agent integrating self-enhanced prompt engineering and context truncation is developed for effective device sizing and downstream physical design. Experimental evaluations on 15 representative circuits with varying levels of complexity demonstrate strong and consistent performance across multiple models. In particular, GPT-5 achieves success rates of 92.9% and 99.9% on Pass@1 and Pass@5, respectively. These results validate the effectiveness and robustness of the proposed framework and establish a practical paradigm for applying LLMs to full-stack analog IC design automation.

Ringdown signals from perturbed black holes (BHs) offer a clean window into BH spacetime, strong-field gravity, and fundamental physics. Presently the quasi-normal modes of stellar-mass BH ringdowns have been successfully extracted in the ground-based gravitational wave (GW) observations. Looking ahead, the future space-borne observatories will listen to the ringdowns from massive BH binary coalescences more loudly and resolve multiple modes to unprecedented precision, which calls for efficient approaches to mitigate the sharply increasing computational burden. We develop a practical ringdown analysis pipeline for space-borne detectors by implementing FIREFLY, a novel acceleration algorithm validated in ground-based detectors, and for the first time demonstrate its compatibility and effectiveness with the time-delay interferometry (TDI) observables. With high fidelity, we achieve a $\sim 200$-fold speedup for a simulated ringdown signal including six modes, providing a viable and scalable route for multi-mode ringdown analysis in the space context. This new approach has sound statistical interpretation and is extensible to other GW sources in band.

Drawing on the Data and Predictions strand of the Indicazioni Nazionali per il curricolo 2012, this study proposes a problem based instructional approach to the teaching of probability. More specifically, the study adopts a design based research methodology structured in a single cycle consisting of two teaching interventions in the same class, carried out in two consecutive years. Within this framework, a set of carefully selected problems is employed to foster students engagement. These problems are designed not only to introduce probabilistic concepts, but also to stimulate students' communicative and argumentative skills. The selected tasks provide opportunities to promote key process goals (such as reasoning and proving, communicating, representing, and making connections) which are often overshadowed by a predominant focus on content goals. This approach aims to support teachers in addressing the difficulties they frequently encounter in guiding students conceptualization processes, particularly in bridging the gap between students intuitive reasoning and formal abstraction. At the same time, it seeks to help students develop more robust and flexible forms of thinking, enabling them to better navigate situations involving uncertainty in everyday life.

We propose Sensitivity-Uncertainty Alignment (SUA), a framework for analyzing failures of large language models under adversarial and ambiguous inputs. We argue that adversarial sensitivity and ambiguity reflect a common issue: misalignment between prediction instability and model uncertainty. A reliable model should express higher uncertainty when its predictions are unstable; failure to do so leads to miscalibration. We define a scalar score, SUA_theta(x), capturing the difference between distributional sensitivity and predictive entropy. We show that minimizing its positive part bounds worst-case perturbed risk and relates to calibration error. We also formalize ambiguity collapse, where models produce overconfident outputs despite multiple valid interpretations. We introduce SUA-TR, a training method combining consistency regularization and entropy alignment, along with an abstention rule for safer inference. Across tasks including question answering and classification, SUA better identifies model failures than entropy or self-consistency alone. The framework is model-agnostic and provides a basis for improving reliability in evolving language models.

Nitrogen-vacancy (NV) centers in diamond are of great interest for nano- and macro-scale magnetic field sensing. Most sensing protocols rely on conventional optical readout, which is limited by photon shot noise. The recently developed photoelectrical (PE) readout of the NV center electron spin state promises to overcome these limitations. However, the noise of the PE readout and its influence on readout efficiency have not been thoroughly studied. In this work, we perform magnetic field sensing and estimate the sensitivity using optical and PE readout with a single and an ensemble of NV centers in diamond. We investigate the electronic noise associated with the photoelectric detection and estimate the readout efficiency, using Gaussian statistics. Our quantitative analysis shows that the Johnson-Nyquist noise-limited photoelectric magnetic field sensitivity could outperform optical measurements by an order of magnitude. This work is an essential step towards the development of on-chip magnetometers using photoelectrical detection in diamond.

The primary objective of this paper is to investigate the notions of geometric and sequential convexity within a graph-theoretic framework, with the aim of examining various structural properties and exploring the connection between these two branches of mathematics. A simple connected vertex-weighted graph $G(V,E)$ with a non-empty set of leaf vertices is said to be star-convex if there exists at least one node $u\in V(G)$ such that, for every chosen leaf vertex $v$, there is a monotone path (either increasing or decreasing) connecting $v$ to $u$. One of the main results states that a graph $G$ is star-convex if and only if there exists a tree $T\subseteq G$ that contains all leaf vertices and is itself star-convex. On the other hand, a sequence $\big(u_n\big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_{i}\leq u_{i-1}+u_{i+1}\qquad \mbox{for all}\quad i\in \mathbb{N}. $$ We demonstrate that, under minimal assumptions, a class of convex sequences can be embedded into a spider graph so as to make it star-convex.

In our previous work: Adv. Math. 455 (2024), no. 109880, solubility of solutions was introduced as an extension of solubility of skew braces in the classification context of non-degenerate solutions of the Yang-Baxter equation. One of our main results (Theorem C) proved that a skew brace is soluble if, and only if, its associated solution is soluble. A minor step depending on the definition of homomorphism of solutions was overlooked. In this work, proof of Theorem C is repaired by means of a new class of homomorphisms of solutions: i-homomorphisms of solutions. The importance of this new class is twofold: indecomposable solutions are characterised by means of i-simplicity of solutions, and i-kernels of i-homomorphisms generate ideals in structure skew braces of solutions. Hence, solubility of solutions is redefined as an opposite class of indecomposable solutions. The results obtained with this definition improve our previous outcomes: every soluble solution is proved to have a soluble structure skew brace, and consequently, Theorem C still holds. Several results stemming from this new analysis are outlined.

In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's Lemma. In this paper, we extend Rousseau's approach to $\mathbb{F}_{q}[t]$, providing a new, elementary proof of the reciprocity law for the $d$th power residue symbol, where $d$ is any divisor of $q-1$.

This note provides an introduction to selected topics in algebraic graph theory, including strongly regular graphs, Steiner systems, and automorphism groups. We describe constructions and properties of notable graphs such as the Petersen graph, Paley graphs, Hamming graphs, and the Hoffman-Singleton graph, with emphasis on their symmetry and combinatorial structure. Connections with permutation groups are also discussed. Computational examples using SageMath are included to illustrate key concepts and to compute automorphism groups and related invariants.