Computations on a manifold often involve constructing an operator on the tangent space and computing its inverse, which can be time-consuming in many applications. In order to reduce the computational costs and preserve the benign properties of tangent operators, we develop the Riemannian Nyström approximation on manifolds, a low-rank approximation of tangent operators through subspace projections onto the tangent space. The developed approximation is intrinsically constructed and inherits desirable properties from the classical Nyström approximation, e.g., positive semidefiniteness and approximation errors. Instead of the Gaussian sketching, we introduce the Haar--Grassmann sketching condition with a coordinate-free representation, which remains compatible under isometric vector transport across tangent spaces. Moreover, we propose a randomized Newton-type method for optimization on manifolds in which the linear system is constructed via the Riemannian Nyström approximation. Numerical experiments on the SPD and Grassmann manifolds, together with principal geodesic analysis on real data, illustrate that the proposed approximation reduces the computational cost of operators while maintaining comparable accuracy.

Autonomous agents based on large language models (LLMs) are rapidly emerging as a general-purpose technology, with recent systems such as OpenClaw extending their capabilities through broad tool use, third-party skills, and deeper integration into user environments. At the same time, these agentic systems introduce substantial security risks by combining unconstrained capabilities with access to sensitive user data. In this work, we investigate the security of LLM-based agents through the lens of operating systems. We argue that both face strikingly similar challenges in isolating resources, separating privileges, and mediating communication. Guided by this perspective, we survey the current landscape of open-source agents, derive a unified agent architecture, and systematically analyze potential attack vectors. To validate this analysis, we conduct a case study evaluating four widely used OpenClaw-like agents. Even under modest attacker capabilities, we find that several protection mechanisms fail in practice and that secure operation requires detailed system knowledge and careful configuration. However, we also observe that while some agentic capabilities remain insecure by design, many vulnerabilities can be mitigated using well-established techniques from operating system security. We conclude with a set of recommendations for the secure design of agentic systems.

The environment plays a critical role in galaxy evolution, with galaxy clusters and their infall regions offering diverse conditions that shape galaxies before they enter the dense cluster core, a process known as ``pre-processing''. However, identifying environmental substructures, particularly galaxy groups in these transitional zones, remains challenging due to projection effects and ``fingers-of-god'' distortions. In this work, we present a supervised machine learning framework for classifying galaxies into three environmental categories: main cluster, group, and neither, using observable galaxy properties such as positions, line-of-sight velocities, and stellar mass. The model is trained on mock observations derived from cosmological simulations designed to replicate survey conditions and achieves an overall accuracy and class-size-weighted precision of 81%. The neither and main cluster populations are reliably recovered, whereas group galaxies remain the most challenging to identify, achieving 30% completeness and 76% purity. Within $1\times R_{200}$, classification performance is suppressed, but it improves beyond this radius, reaching 40% completeness and 80% purity. Resampling and thresholding strategies allow the model to be tuned toward either higher purity or higher completeness; in this study, we adopt first-past-the-post thresholding to emphasise purity. Model performance is consistent across cluster masses and dynamical states, and it outperforms both Friends-of-Friends and Gaussian Mixture Modelling. This flexibility makes it well suited to upcoming spectroscopic surveys of cluster infall regions, providing a robust statistical tool for disentangling environmental influences on galaxy evolution.

We present a deep DREAMS JWST/NIRSpec MSA medium-grating spectrum of MACS0416-Y1, a galaxy at $z=8.312$ with the highest-redshift ALMA dust continuum detection to date, in order to characterize its properties together with archival IFU and ALMA data. The deep NIRSpec spectrum reveals a broad H$β$ line with a width of $\sim1100$ km s$^{-1}$. We interpret it as a broad-line AGN whose line diagnostics are consistent with AGN activity across its clumpy structure, given the absence of little red dot signatures. MACS0416-Y1 clearly shows [OIII]4363 emission, suggesting a moderately low metallicity of $12+\log(\mathrm{O/H})=7.86^{+0.09}_{-0.08}$ ($0.15~Z_\odot$). The combination of [CII]158$μ$m and dust continuum emission indicates low dust mass ratios of $\log (M_{\rm dust}/M_{\rm gas})=-3.60^{+0.29}_{-0.22}$ and $\log (M_{\rm dust}/M_{\rm metal})=-0.95^{+0.29}_{-0.20}$. Because the metallicity of MACS0416-Y1 is around the critical metallicity of $0.1\textrm{-}0.2~Z_\odot$, the system is expected to undergo dust growth, explaining these low dust mass ratios as well as its small dust mass, $M_{\rm dust}\sim10^6~M_\odot$. The intense UV radiation from the AGN may contribute to a high dust temperature of $T_{\rm dust}\simeq 91^{+62}_{-35}$ K, boosting the dust-continuum emission above the ALMA detection limit despite the small $M_{\rm dust}$ at $z>8$. We find a very high total flux ratio of [OIII]88$μ$m/[OIII]5007 = $0.26 \pm 0.06$ in MACS0416-Y1, above predictions from single ionized nebular models at any electron density. This discrepancy suggests that the [OIII]88$μ$m and [OIII]5007 trace largely distinct regions, with the optical line suppressed in dusty nebulae, and thus requires careful interpretation when combining optical and infrared emission lines in JWST+ALMA studies.

We answer a question of Simental by providing a combinatorial interpretation of a formula which generalizes rational Catalan numbers and which appears in the study of Springer fibers. We provide an interpretation in terms of binary necklaces as well as anchored Dyck paths.

Transmit beamforming for underwater acoustic communication is challenging because it requires perfect knowledge of the channel to the receiver in advance. In practice, channel estimates must be learned through feedback and are often noisy or outdated because of feedback delay and channel variation. In this paper, we investigate angle-based beamforming strategies for a single-user link that reduce dependence on full channel knowledge by exploiting stable components of the geometric structure in the propagation field. In particular, we focus on scenarios in which there exists a dominant path that remains relatively stable over time, making it a suitable candidate for transmit beamforming. Experimental results using the SPACE and MACE data sets demonstrate the effectiveness of the proposed method in terms of data-detection mean-squared error and bit error rate.

The ability of a small set of coordinated actors to manipulate opinions in online social networks poses a serious challenge to the fairness and integrity of public debate. We investigate this problem by studying how targeted stubborn agents can shift the average opinion of a network governed by the Hegselmann-Krause bounded-confidence dynamics. Experiments are conducted on weighted LFR benchmark networks with community structure, using multiple node-selection strategies based on degree, strength, PageRank, betweenness, k-coreness, s-coreness, and salience. We compare static interventions, in which stubborn agents keep a fixed extreme opinion, with dynamic interventions, in which their opinion gradually evolves from moderate to extreme values. Results show that dynamic strategies are substantially more effective than static ones, as they exploit bounded-confidence dynamics to progressively recruit intermediate agents and extend influence across the network. In contrast, static strategies tend to create early opinion separation and therefore have a more limited reach. We also find that while some centrality measures offer advantages in static settings, dynamic interventions can achieve strong performance even with simple or random node selection. Overall, the study clarifies how intervention design and target selection interact in shaping collective opinions, with implications for understanding and countering manipulation in social networks.

Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers (SCS) showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos, establishing a paradigmatic link between random connectivity, dynamical instability, and internally generated fluctuations in neural circuits. Here, we extend this framework to two-population firing-rate networks with segregated excitatory and inhibitory neurons and target-specific inhibitory couplings that break excitation--inhibition balance. Using dynamical mean-field theory, we derive self-consistent equations for the macroscopic mean activities and autocorrelations, together with stability criteria distinguishing mean-driven and fluctuation-driven instabilities. We show that target-specific inhibition organizes the phase diagram into three qualitative classes: inhibition-dominated or strictly balanced networks display only quiescent activity and asynchronous chaos; excitation-dominated networks display persistent activity together with either synchronous chaos with non-vanishing mean activity or coherent oscillations, depending on the stability-matrix eigenvalues. Crucially, coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory; rather, their onset suppresses the chaotic component, reminiscent of input-induced suppression of chaos. These results generalize SCS theory to recurrent networks with explicit excitatory--inhibitory structure and identify target-specific inhibition as a key control parameter for large-scale neural dynamics.

We study numerically the evolution of one-dimensional FKPP fronts initiated from steep initial conditions in the presence of a quenched random growth rate. Compared to both the homogeneous case (with velocity $v_0$) and deterministic disorder, quenched randomness increases the average propagation speed. We show that the velocity shift relative to the homogeneous case scales linearly with the disorder variance $σ^2$, with a universal prefactor -- independent of the specific distribution of the disorder -- such that $v = v_0 + a σ^2$, with $a \approx 0.02432 \pm 0.00002$. Moreover, the front position exhibits diffusive fluctuations across disorder realizations. The corresponding effective diffusion coefficient scales quadratically with $σ$, $D = \frac{b^2 σ^2}{2}$, with $b \approx 0.223 \pm 0.002$. These results suggest a universal statistical response of FKPP fronts to quenched heterogeneity.

Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization of this question, asking whether a fixed loop (or, more generally, a topological sphere) can escape to infinity via a tube formed by random plaquettes. We refer to this phenomenon as tube percolation. We first compare tube percolation with previously studied higher-dimensional percolation phenomena, including face and cycle percolation. For tubes of codimension one, we further relate the critical probability for tube percolation to those for percolation of finite clusters and shielded percolation in the dual bond percolation model. Next, we introduce a tubular analogue of the classical one-arm event, the tubular one-arm event, and prove that it exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. The proof relies on the O'Donnell-Saks-Schramm-Servedio (OSSS) inequality together with an exploration algorithm adapted to the topology of tubes. Finally, we study the tubular box-crossing property. Unlike ordinary path connectedness, "tube connectedness" is not transitive, and thus there is no natural notion of clusters. Nevertheless, we establish an analogue of the uniqueness of the infinite cluster from classical bond percolation. Combining this result with the sharp threshold for the tubular one-arm event, we prove that the existence of a box-crossing tube also exhibits a sharp threshold.

We show that $κ$-solutions to the Ricci flow in dimensions $n\geq 4$ whose asymptotic shrinking Ricci soliton is the round cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.

Circulant graphs are a widely studied family of graphs whose members possess varying amounts of symmetry. Although considerable progress has been made in finding the automorphism groups of circulant graphs under certain restrictions, a complete classification is elusive. In general, the structure of the automorphism group of a graph with twins can be simplified by separating the effect of automorphisms that permute mutually twin vertices and those that operate on the twin quotient graph. Further simplification can be achieved in twin-free, vertex-transitive graphs that have co-twins, which we define to be vertices whose neighborhoods are complementary. In this paper, we demonstrate how the these simplifications can be used provide insight into the automorphism groups and symmetry parameters of vertex-transitive graphs in general and circulant graphs in particular.

The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bounds behind these results have traditionally been so large that the resulting algorithms are regarded as "galactic". Our main result is a general irrelevant-vertex theorem for a common generalisation of $k$-Disjoint Paths and Rooted Minor Checking for graphs of size at most $d,$ commonly called the $(k,d)$-Folio problem. Specifically, we show that for any graph $G$ in which the $k$ terminals are chosen from some set $R,$ if the treewidth of $G$ exceeds $β(k,b,d)\in$ $2^{{\bf poly}(b + d)}$ $\cdot {\bf poly}(k)$ then we can locate an irrelevant vertex for the $(k,d)$-Folio problem. Here, the quantity $b$ is the bidimensionality of $R,$ that is, the largest $b$ for which a $(b\times b)$-grid minor in $G$ can be rooted on $R$. Thus, the exponential component of the irrelevant-vertex threshold is driven by the bound on the bidimensionality, rather than by the number of terminals, and we argue that this dependence is essentially optimal up to polynomial factors. As a consequence, the Linkage Function satisfies $\ell(k) \in 2^{{\bf poly}(k)}$. Beyond its structural significance, our result yields improved parameter dependencies for algorithms for Disjoint Paths and Rooted Minor Checking}, and provides a quantitative improvement for a broad range of graph-minor-based algorithmic frameworks.

We study mean field games for large non--exchangeable populations with moderate local interactions and common noise. The finite--player system is driven by two complementary interaction mechanisms : a graphon--type structure, which encodes heterogeneous large--scale interactions between agents, and a rescaled local kernel, which produces a density-dependent interaction term in the limit. The limiting model is a non--exchangeable mean field game in which the representative player is indexed by a label \(u\in[0,1]\), interacts through a graphon--weighted local density, and is affected by a graphon--induced environment law. We introduce a relaxed formulation of the limiting mean field game, adapted to the presence of common noise, and prove existence under general continuity and non--degeneracy assumptions. Under additional convexity assumptions, relaxed equilibria can be realized in strict form. In the deterministic case without common noise, we obtain deterministic equilibria and provide a probabilistic characterization of strict equilibria through a nonlinear Feynman--Kac representation. We then establish the asymptotic connection with the finite--player game. We prove that every limit point of approximate closed--loop Nash equilibria is a relaxed solution of the limiting mean field game, and that the corresponding averaged equilibrium payoffs converge. Conversely, every relaxed mean field game equilibrium can be approximated by Markovian approximate Nash equilibria of the finite--player systems. These results give a complete asymptotic characterization of equilibrium behavior for non--exchangeable games with moderate interactions and common noise.

$D$ meson decays into neutral kaons involve both Cabibbo-favored and doubly Cabibbo-suppressed amplitudes as well as final-state kaon mixing, providing abundant sources of CP violation. In this work, we analyze CP asymmetries in the $D\to K^0_{S,L}P$ and $D\to K^0_{S,L}V$ decays, where $P$ and $V$ denote pseudoscalar and vector mesons respectively. The formulas of the time-dependent and time-integrated CP asymmetries in these modes are derived, in which the $D^0-\overline D^0$ mixing effects and the $K^0_L$ modes are considered for the first time. The hadronic parameters that determine CP asymmetries are extracted by the global fit of branching fractions within the topological diagram approach. A significant result is that the tension between theoretical predictions and experimental data for the $K_S^0-K_L^0$ asymmetries in $D^0\to K_{S,L}^0ω$ and $D^0\to K_{S,L}^0ϕ$ modes is mitigated. The CP-violating effects arising from the interference between Cabibbo-favored and doubly Cabibbo-suppressed amplitudes with neutral kaon mixing could reach to $\mathcal{O}(10^{-3})$ order in the $D^+\to K^0_Sπ^+$, $D^+_s\to K^0_SK^+$, $D^0\to K^0_Sρ^0$, and $D^0\to K^0_Sϕ$ modes. The difference between the CP asymmetries in the $D^+\to K^0_Sπ^+$ and $D^+_s\to K^0_SK^+$ modes is available on LHCb and Belle II in the near future.

Establishing structure-property linkages in polycrystalline materials requires representative two- (2D) and three- (3D) dimensional microstructural inputs for full-field simulations. A core objective of microstructure characterization and reconstruction is the generative synthesis of 2D and 3D microstructures that reflect a target statistical ensemble using limited 2D data as a reference. This work introduces an orientation-based differentiable microstructure characterization and reconstruction framework, implemented in MCRpy, to perform reconstructions of voxelized images. Unit quaternions in combination with symmetrized hyperspherical harmonics are utilized to derive a continuous, symmetry-invariant representation of crystallographic orientations to overcome the numerical singularities and discontinuities associated with traditional Euler-based methods. The descriptor-based reconstructions are driven by a set combining two-point spatial correlations, a novel hybrid three-point variogram, and a mean variation regularizer to capture both global texture and local interfacial topology. The framework's efficiency is demonstrated by reconstructing 3D realizations from 2D orientation data of an aluminum alloy after thermo-mechanical processing, successfully recovering both morphological features and crystallographic distribution. Systematic benchmarking indicates that second-order gradient-based optimization, utilizing the L-BFGS-B algorithm, effectively navigates the complex loss landscape to generate high-fidelity realizations with minimal residuals. This methodology provides a versatile, open-source framework for the digital synthesis of polycrystalline representative volume elements to facilitate the rapid development of microstructure-informed materials design workflows.

In this paper, we investigate generalized bent functions (GBFs) from $\mathbb{Z}_3^n$ to $\mathbb{Z}_m$. We show that GBFs exist whenever $3$ divides $m$, while several nonexistence results are obtained when $3\nmid m$. In particular, we prove that no GBFs exist for $n=1,2$ when $m$ is odd and not divisible by $3$. For the case $n=3$, we establish the nonexistence of GBFs $f:\mathbb{Z}_3^3 \rightarrow \mathbb{Z}_{5\cdot11^r}$ for all nonnegative integers $r$. Finally, we show that no GBF exists from $\mathbb{Z}_3$ to $\mathbb{Z}_{2m'}$ and $\mathbb{Z}_3^2$ to $\mathbb{Z}_{2m'}$, where $m'$ is odd and not divisible by $3$.

Brown dwarfs, being transitional objects between giant planets and low-mass stars, possess dense, cool interiors that provide optimal conditions to explore non-standard physics. Capture and accumulation of dark-matter particles can alter the thermal, structural and dynamic of these substellar objects. We aim to apply a self-consistent two-fluid framework to model the internal structure of self-gravitating brown dwarfs and to quantify how the presence of a dark-matter component modifies their mass--radius relations and dynamical properties. The brown dwarf is modeled as a composite system of a baryonic fluid, described by a polytropic equation of state, and an independent dark-matter fluid. Both components are coupled through their shared gravitational potential in hydrostatic equilibrium. We solve numerically the coupled Lane-Emden equations for a range of dark-matter mass fractions. We find that dark matter accumulating in the core reshapes the baryonic density profile, modifying both the radius and the second-order Love number. Radius and dynamical anomalies in brown dwarfs can serve as diagnostic tools to constrain dark-matter properties. Future high-precision astrometric missions could identify these structural signatures, establishing brown dwarfs as possible detectors of dark matter in the Galaxy.

The largest matching root of a $k$-graph is the largest real root of its matching polynomial, which is equal to the maximum modulus of all the zeros of the matching polynomial. In this paper, we investigate the perturbation of the largest matching root of $k$-graphs. We determine all $k$-graphs whose largest matching root attains the maximum among all $k$-cacti and linear $k$-cacti with a given number of cycles and edges, where a $k$-cactus is a $k$-graph in which every two distinct cycles have at most one vertex in common. To achieve this, we prove that the celebrated shifting operation of $k$-graphs, introduced by Erdős, Ko and Rado, does not decrease the largest matching root. This result extends a classical result by Csikvári (Electron. J. Combin. {\bf 18} (2011) $\#$P182) stating that the Kelmans transformation does not decrease the largest matching root of graphs.

We present a tool QSeqSim, a Qiskit-integrated symbolic backend that fills the current gap of having no Qiskit-native support for simulating while-loop quantum programs and their induced sequential quantum circuits. QSeqSim takes Qiskit QuantumCircuit objects, translates them into OpenQASM 3 code, and organises the resulting program into a combination of combinational, dynamic, and sequential circuits, thereby assigning while-loops a precise sequential circuit semantics with explicit internal and external qubits. Building on this semantics, QSeqSim adopts a Binary Decision Diagram (BDD)-based symbolic representation and integrates weighted model counting to compute measurement probabilities efficiently by exploiting sharing in structured and sparse BDDs. On top of this Boolean backbone, it introduces dedicated symbolic operators for state composition and state retention, thereby enabling efficient symbolic execution of sequential quantum circuits. Our experiments demonstrate that QSeqSim scales to substantial while-induced sequential circuits; in particular, in the quantum random walk benchmark we successfully simulate circuits with over 1000 qubits for more than 10 loop iterations. QSeqSim is available at https://github.com/Veri-Q/QSeqSim.

Automatic recognition of affective state from wearable physiology has clear societal impact for public health, preventive care, and stress-aware interventions, but real deployments require robustness to non-stationary dynamics, artefacts, and missing sensors. We study this problem on WESAD, using baseline, stress, and amusement conditions, where common fixed-basis spectral features such as FFT bandpower and Welch PSD can oversmooth short-lived discriminative patterns. We propose a non-stationary pipeline that combines Fourier-Bessel Series Expansion (FBSE) with EWT data-driven spectral segmentation to extract mode-wise transient descriptors. For multimodal integration, we adopt decision-level aggregation over per-modality predictors and weight each modality by predictive uncertainty and modality reliability. Results on WESAD, using 15 subjects and ECG, EDA, BVP, EMG, and ACC signals across three classes, indicate that decision-level aggregation is approximately 84 percent of the time at least as good as feature-level aggregation, and approximately 48 percent of the time strictly better, suggesting improved robustness under heterogeneous and partially reliable sensing.

We perform a systematic study of two-meson form factors of the scalar, vector, and anti-symmetric tensor types within the framework of the $U(3)$ resonance chiral theory. The complete perturbative form factors in both the strangeness-conserving and strangeness-changing channels are calculated by incorporating one-loop light-flavor pseudoscalar meson contributions and tree-level resonance exchanges. With these newly calculated chiral results, we construct the corresponding unitarized form factors by incorporating meson-meson final-state interactions. The parameter values obtained in previous meson-meson scattering studies are then exploited to predict the corresponding form factors. Different types of form factors are found to exhibit rather distinct resonance structures across channels.

Attosecond electron pulses enable real-time probing of ultrafast matter dynamics, yet conventional modulation schemes suffer from drastically shortened longitudinal focal lengths when targeting sub-attosecond durations. To address this bottleneck, we propose and demonstrate a compact scheme utilizing two counter-propagating lasers that reveals a previously unidentified stable modulation regime. Contrary to established models of ponderomotive forces and stochastic acceleration in dual-laser fields, we show that a specific parametric resonance condition permits the electron beam to be stably modulated into highly periodic attosecond trains with rapid energy gain. Using a sub-relativistic electron beam, simulations confirm the generation of ~1 as pulses with a Lorentz factor up to 15 and a relative energy spread below 0.02%, extending the focal length by three orders of magnitude compared with conventional approaches. This work identifies the critical transition from ordered modulation to stochastic acceleration, offering a viable route to overcoming the focal-length barrier in attosecond electron-pulse applications.

Motivated by the central role of the Zel'dovich approximation in the description of cosmic structure formation through gravitational collapse, we investigate Burgers-type dynamics in a spherically symmetric gravitational field. In the Newtonian setting, we derive perturbatively the catastrophe time for radial motion by imposing the loss of invertibility of the Lagrangian map. We show that the perturbative expansion is controlled by the dimensionless parameter $ α=μ/{r_0^3 v_0(r_0)'^2}, $ rather than by the local gravitational acceleration alone. Hence, the expansion remain valid even when gravity is strong. We then extend the analysis to radial geodesic motion in Schwarzschild spacetime.

The spectrum and composition of ultra-high-energy cosmic rays (UHECRs) suggest that the population dominating above the ankle releases particles with an unusual hard spectrum at low rigidity, below the EV scale. In self-confinement scenarios, such an apparent hardening arises from transport: escaping UHECRs generate magnetic turbulence that delays their own release from the magnetized environments surrounding their sources. We extend the self-confinement scenario based on the non-resonant streaming instability to a mixed nuclear composition. We describe the confinement region with an effective leaky-box model including escape, photodisintegration, and secondary production. We then compare the resulting spectrum and composition with Auger measurements and compute the associated cosmogenic neutrino and gamma-ray emission. We find that self-generated turbulence can suppress the escaping flux below the EV scale for source luminosities and magnetic-field coherence lengths compatible with UHECR sources hosted in galaxy clusters and propagating through cosmic filaments. During confinement, heavy nuclei efficiently photodisintegrate, producing secondary protons that contribute below the ankle and help account for the observed composition. The predicted neutrino flux remains compatible with current limits, while the diffuse gamma-ray background provides a potentially strong constraint on the most extreme configurations.

This paper constructs derived autoequivalences of Gorenstein orders as twists around spherical functors. More precisely, given a Gorenstein order $A$ and a quotient $p \colon A \to B$, then we specify natural conditions on $B$ under which the twist around the corresponding derived restriction of scalars functor is a derived autoequivalence of $A$. In the process, we show that the associated cotwist is a shift of the Nakayama functor of $B$. These results, together with local-to-global technology, are then used construct new derived autoequivalences for skew group algebras and $G$-Hilbert schemes, and we apply this theory to explicit examples.

The process of engineering and deploying applications in the edge/embedded space is massively complicated by the non-homogeneous nature of the software stack and the complexity of diagnostics & debugging. Often different languages and runtimes are used for different components of the system forcing designers to, irrevocably, make decisions about what components run on the periphery and what components run in the cloud. Further complications arise when handling and diagnosing failures in the system. Multiple stacks and, often, limited support for debugging complicate the already difficult task of analyzing distributed applications. This paper presents a work-in-progress vision for a unified language and runtime system that allows applications to scale seamlessly across the edge and cloud. Using a single language and runtime, applications can be developed and tested in a single environment, and then deployed to any component of the system -- from resource limited controllers to large cloud servers. Further, we outline how this retargetable stack can provide integrated diagnostics and debugging tools that allow developers to record and replay distributed events locally for analysis and debugging.

The commutative subgroup width of a group $G$ is the smallest $k$ such that there are abelian subgroups $A_0,A_1,\ldots,A_{k-1}\leq G$ with $G=A_0A_1\cdots A_{k-1}$. Commutative (inverse) submonoid width is defined analogously. In 2002, Abért showed, rather surprisingly, that the commutative subgroup width of the symmetric group on an infinite set is always finite. It was later shown by Seress that it is always bounded above by $14$. We answer a question of Seress and show that in fact the commutative subgroup width of $\operatorname{Sym}(\mathbb{N})$ is at most $9$. We improve the best known lower bound to $4$. We also study standard monoid analogues of the symmetric group; showing that the commutative submonoid widths of the full transformation monoid $\mathbb{N}^\mathbb{N}$, the partial transformation monoid $P_\mathbb{N}$ and the symmetric inverse monoid $I_\mathbb{N}$ are exactly $3$. We conclude by showing that the commutative inverse submonoid width of any infinite symmetric inverse monoid is always infinite.

We study Lévy-like and truncated Lévy-like flights with step probability distribution of the form $r^{-1+ν}$ for negative, positive, and zero $ν$, focusing on the appearance of fractal geometry characteristics in the generated point sets. Forming ensembles of such point sets with fixed multiplicity, we develop simulation techniques leading to the desired value of correlation dimension in a vast continuous interval of scales. In particular, we demonstrate the possibility to produce ensembles of data sets with a low number of points with the needed properties. Furthermore, we show that the positive $ν$ distributions, apart from a region near the upper scale limit, show fractal behaviour that extends to infinitesimally low scales. As an example, we apply our findings to producing simulations relevant to the search for critical fluctuations, related to QCD critical endpoint, in heavy-ion collision experiments.

Ultrafast continuous-variable quantum states offer new opportunities for advanced quantum technologies, but efficient homodyne detection of these states remains challenging. Here, we present a method for efficient ultrafast homodyne detection by exploiting temporal correlations in detector signals. By optimizing the temporal weight used to extract quadrature outcomes, we achieve a substantial increase in the signal-to-noise ratio of ultrafast homodyne detection, thereby improving the detection efficiency. We analyze the autocorrelations of shot noise and electronic noise and determine the optimal weight by solving a generalized Rayleigh quotient problem. The optimal weight enhances the squeezing and anti-squeezing levels observed experimentally. These results highlight the importance of optimized signal processing for efficient quantum measurements.