It was recently shown by Huang that the category of $C_1$-cofinite modules for any vertex operator algebra $V$ admits a natural braided monoidal structure. Here, we show that this structure extends uniquely to a vertex algebraically natural braided monoidal structure on the completion of the category of $C_1$-cofinite $V$-modules under filtered colimits, within the ambient category of all generalized $V$-modules. A critical point here is that we do not assume the category of $C_1$-cofinite $V$-modules is abelian or that $C_1$-cofinite modules are compact in the cocompletion, since these properties are not known to hold in general. Our results have many applications in the representation theory of vertex operator algebra extensions, since many vertex operator algebras can be realized as objects in the filtered colimit completion of the category of $C_1$-cofinite modules for a vertex operator subalgebra. Generalizing from the specific vertex algebraic setting, we also establish existence and uniqueness for extensions of monoidal structures along a dense inclusion $\mathscr{C}_0 \to \mathscr{C}$ from an abstract, essentially small monoidal category into a well-structured cocomplete target.

Gaussian graphical models in the spectral domain offer a principled approach for recovering conditional dependence structures in stationary high-dimensional time series. Inference on the spectral precision matrix at a fixed frequency enables tests of frequency-specific conditional associations among time series components. The problem is challenging because finite-sample discrete Fourier transforms induce truncation and smoothing biases, while the complex-valued nature of the spectral precision matrix complicates high-dimensional variance estimation, rendering methods for i.i.d. samples not directly applicable. Existing approaches do not provide full likelihood-based inference for the discrete Fourier transforms. We propose a high-dimensional inference framework for sparse spectral precision matrices using the full likelihood of neighboring discrete Fourier transforms. We construct a debiased complex graphical lasso estimator at any fixed frequency. Using asymptotic theory for quadratic forms of multivariate time series, we establish its asymptotic normality and construct entry-wise consistent covariance estimators by aggregating information across neighboring frequencies. The key theoretical contribution is the simultaneous control of regularization, finite-sample truncation, and smoothing biases, enabling valid inference. Simulation studies show reliable coverage away from zero frequency and improved detection power over the benchmark, with false discovery rates near the desired level.

This study investigates a statistical property of Lagrange multipliers in constrained Maximum Likelihood Estimation (MLE) and Least Squares (LS) problems from the perspective of numerical optimization. Building on large-sample theory, we show that the associated Lagrange multipliers converge to zero as the sample size increases, provided the distribution is correctly specified in MLE or the residuals are normally distributed in LS. Although this asymptotic behavior has long been recognized in statistics, it has received little explicit attention in numerical optimization and has rarely been exploited in algorithmic design. Importantly, the insight extends beyond classical low-dimensional settings: even in modern high-dimensional applications, such as deep learning, where the number of parameters may exceed the sample size, the same reasoning applies provided the generalization performance is good. This observation has two main implications. First, many constrained optimization algorithms, including the Augmented Lagrangian Method, Sequential Quadratic Programming, and Interior Point methods, require initial values for the multipliers, and choosing zero is statistically justified. Numerical experiments for constrained regressions and dynamic discrete choice model estimations support this implication by showing that initializing multipliers at zero usually lead to stable and efficient performance. Second, penalty-based approaches that convert constrained problems into unconstrained ones can perform well when the true multipliers are small. This helps explain why penalty-based methods often perform well in practice.

Altermagnets (AMs) offer a compelling platform for exploring novel spin-dependent phenomena in materials with zero net macroscopic magnetization. In this work, we theoretically investigate the emergence of equilibrium spin currents (ESCs) in two-dimensional AM heterostructures using a tight-binding lattice model. We first study an AM-normal metal-AM (AM-NM-AM) junction and demonstrate that the $σ_y$-polarized ESC exhibits a characteristic Josephson-like behavior, fundamentally governed by the relative angle ($θ$) between the Néel vectors of the two AMs pointing in $xz$-plane. Crucially, we show that replacing the central normal metal with a $p$-wave magnet (PM) induces an anomalous ESC. Analogous to the anomalous Josephson effect, the breaking of spatial inversion symmetry by the PM allows a finite, dissipationless spin current to flow even when the Néel vectors are perfectly aligned ($θ=0$). We establish that this anomalous transport is driven by an asymmetry in the quantum phases accumulated by right- and left-moving electrons undergoing spin-flip reflections. Finally, we show that the critical ESC exhibits pronounced fluctuations as a function of band filling, which we attribute to mesoscopic quantum size effects, including transverse subband quantization and longitudinal Fabry-Pérot resonances. Our findings highlight the potential of altermagnet junctions for designing dissipationless, phase-tunable spintronic devices.

While Bayesian inference has become increasingly popular with advances in computational resources, its algorithms can be computationally prohibitive and may not scale with large datasets. This has led to growing interest in alternative algorithms, such as approximation methods and variants of Markov chain Monte Carlo. Among these approaches, prior proposal-recursive Bayesian (PP-RB) inference facilitates scalable Bayesian computation by recursively updating the posterior distribution across stages and utilizing parallel computing resources. While the well-known ``degeneracy'' issue in PP-RB has been studied, another limitation that PP-RB can yield incorrect inferences when posterior distributions shift substantially between stages has remained unsolved. To address this, we propose parallel-tempered prior proposal-recursive Bayesian (PPP-RB) inference, which extends PP-RB by leveraging the key idea underlying Metropolis-coupled Markov chain Monte Carlo. We show both theoretically and empirically that PPP-RB targets the true posterior distribution. We illustrate PPP-RB through numerical studies and real data analysis in application to earthquake count data and sea surface salinity in the North Atlantic region. In these applications, we compare PPP-RB with PP-RB and a standard MCMC, demonstrating that PPP-RB is more efficient in terms of effective sample size per elapsed time.

Transformer-based CTR models face a growing bottleneck at the residual connection: under Pre-Norm, early user-interest signals are diluted layer by layer; the identity skip cannot forget stale interests; and each layer sees only its immediate predecessor, losing long-range cross-layer dependencies. Recent attention-based residual variants (AttnRes) address parts of this in language models, but drop the protective identity skip and have not been tried in recommendation. Drawing on Dual Path Networks (DPN) and the HORNN view of residuals, we present DeRes, which routes each layer through two parallel paths -- an Identity residual path that preserves first-order feature reuse and gradient flow, and a Block Attention Residual path that attends over compressed outputs of all earlier blocks for high-order recall. A vector-wise gate decides, per hidden dimension, the weight given to each path. We further propose Pointwise AttnRes, replacing the Softmax in the cross-layer attention with SiLU so that multiple past blocks can be activated simultaneously and irrelevant ones receive negative (forgetting) weights -- better aligned with CTR's parallel multi-interest patterns. On a large-scale industrial dataset (331M interactions from a major social-media platform), Criteo (45M), and Avazu (40M), DeRes outperforms twelve baselines including OneTrans, TokenMixer-Large, UniMixer, mHC, and AttnRes, achieving up to +0.32% AUC at under 5% extra FLOPs. Beyond a single operating point, DeRes fits a markedly steeper compute-AUC scaling law (gamma=0.118 vs. 0.071 for OneTrans, a 1.66x gap), so an 8-layer DeRes matches a 16-layer OneTrans -- about 2x compute saving at equivalent AUC. Ablations confirm that the dual-path design outperforms either single path, Identity beats learnable residuals, and SiLU beats Softmax.

This paper makes several improvements to existing level set based approaches to computing shape optimizers for the Dirichlet eigenvalues subject to a volume constraint. The most notable changes in formulation include an overhaul of the classical level set construction and root-finding procedures as well the use of a regularized approximation to the standard objective function. Our resulting computational minimizers are either comparable to or improvements on the best known minimizers from the literature. We conclude with a survey of subproblems within the field that may benefit from numerical experiments; these include the existence of cusps on the boundary, the end-behavior of eigenfunction weights in the p-parameterized problem, and the nature of Weyl asymptotics as they relate to the Pólya conjecture.

Master equations in mean field game theory characterize feedback value functions that depend on time, state (space), and the population distribution. Their numerical approximation is challenging because the unknown is defined on a space of probability measures and the equation involves intrinsic measure derivatives and nonlocal population terms. This paper proposes a measure-consistent operator learning method (MCOL) for infinite-dimensional master equations. The population distribution is represented by an empirical measure and encoded through a symmetric pooling structure, so that the network input is built directly from the particles representing the measure. The same particles are used in the empirical quadrature of the nonlocal residual terms, avoiding additional quadrature grids or auxiliary integration points. A key feature is that the intrinsic derivative appearing in the residual is induced by the same measure-dependent representation that defines the approximation of the value function. Consequently, the value function, its measure derivative, and the empirical residual are tied to a common measure representation, leading to a structurally coupled value-derivative approximation. We also introduce an error decomposition separating neural approximation error from empirical discretization error. Numerical experiments on several master equations show that MCOL accurately approximates the value function, intrinsic measure derivatives, and feedback quantities, and remains robust under changes in the input measures.

We study the {\it relation type} of ideals of finite reduced sets of points in the projective plane; for a given ideal, this is the maximal $T$-degree of a minimal generator of the defining ideal of the Rees algebra. Our main focus is on point configurations whose defining ideals are not necessarily linearly presented, with an emphasis on almost collinear configurations. We prove that $\rt(X)\in\{1,3\}$ whenever $X\subseteq \PP^2_k$ is a finite set of at most ten points; and we characterize the configurations of relation type $3$ in this range. We then show that a configuration of eleven points in generic position has relation type $5$, thereby yielding the first occurrence of relation type larger than $3$. Finally, we exhibit a configuration of $17$ points with relation type $4$ and we formulate some questions regarding the spectrum of admissible relation types of point configurations.

This study demonstrates a wafer-scale growth of a 2-inch Si-doped $β$-Ga2O3 (100) epitaxial wafer and the realization of uniform, high-voltage lateral $β$-Ga2O3 MOSFET arrays. The 2-inch homoepitaxial $β$-Ga2O3 (100) film grown by MOCVD exhibit excellent crystalline uniformity with an average rocking curve FWHM of ~27.0 arcsec and a low surface roughness less than 1 nm, alongside a uniform net doping concentration on the value of 4.60 $\times$ 1E17 cm-3. The fabricated MOSFETs deliver a threshold voltage of -31.75 V, a drain-current on/off ratio over 1E9, a specific on-resistance of 126.52 mohm$\cdot$cm2 and breakdown voltage exceeding 3 kV. Statistical analysis across the entire wafer presents good device uniformity, with threshold voltages ranging from -28 V to -36 V, output current densities of 60-75 mA/mm, and a breakdown voltage over 3 kV. These results provide the demonstration using the 2-inch $β$-Ga2O3 epitaxial wafer to realize high-voltage $β$-Ga2O3 MOSFETs with wafer-scale performance uniformity for next-generation power device application.

This paper presents a novel frequency-robust perfectly matched layer (PML) boundary integral equation (BIE) method for solving two-dimensional electromagnetic scattering problems involving periodic arrays of obstacles. In periodic scattering problems, standard BIE formulations based on the quasi-periodic Green's function require the evaluation of lattice sums or challenging Sommerfeld-type integrals, which diverge at Rayleigh--Wood (RW) anomalies. An alternative is to use BIE formulations based on the Helmholtz free-space Green's function, but these are defined on unbounded unit-cell boundaries and therefore require suitable truncation strategies, such as the Windowed Green Function (WGF) method. Although such approaches avoid the use of expensive quasi-periodic Green's functions, they also suffer from breakdowns at RW anomalies unless an appropriate mode correction is incorporated. Similarly, the direct application of PML-BIE techniques to periodic structures experiences comparable difficulties near RW anomalies due to the destruction of exponential convergence near RW anomalies for fixed PML parameters. To overcome this challenge, we propose a modified PML-BIE method that combines the PML technique with a finite-mode correction, ensuring both high accuracy and robustness at and around RW-anomalies. Convergence of the PML-truncated boundary integral operators is proved and several numerical examples are presented to validate the efficiency and performance of the proposed method.

Time-series analysis plays a central role in understanding oscillatory and wave phenomena in solar and stellar atmospheres. However, astrophysical observations are inherently affected by instrumental noise, non-stationary dynamics, and uneven sampling. This review provides a comprehensive overview and comparative analysis of principal methods for detecting and characterizing periodicities in solar and stellar signals. We cover Fourier-transform-based transforms, nonlinear-fitting-based methods (Lomb--Scargle periodogram), time-frequency methods (wavelet and synchrosqueezed transforms), and adaptive decomposition techniques (Empirical Mode Decomposition). Advanced statistical significance tests, including false-alarm probability, autoregressive models, and Bayesian Markov Chain Monte Carlo (MCMC) approaches, are discussed their practical limitations and misuse risks. Through comparative analysis using synthetic benchmarks, we provide guidelines for selecting methods based on signal stationarity, sampling regularity, and noise characteristics. Finally, we outline future directions that integrate Bayesian inference with time-frequency analysis to achieve both statistical rigor and temporal localization in studying non-stationary solar and stellar oscillations.

In an islanded setting, grid-forming inverters must regulate their terminal voltage without support from an external grid, even though the load current depends directly on that voltage. The usual approach is a cascaded proportional--integral (PI) controller, built on a fast inner current loop and a slower outer voltage loop, with feedforward terms used to compensate dq rotational coupling. However, this compensation is only exact at the operating point where the controller is tuned. This tutorial presents an alternative based on full-state feedback linearization. It is shown that the islanded inverter model has full relative degree, which allows exact state-space linearization with no internal or zero dynamics. A single feedback law cancels the main nonlinear effects; rotational coupling, resistive drops, and load conductance, so that the closed-loop system behaves like two independent double integrators. A standard pole-placement design is then used to shape the response. The controller is tested in MATLAB against a cascaded PI baseline under identical conditions at a 20 MW operating point, including reference tracking, load step disturbances, and parameter mismatch. The feedback-linearizing controller settles a reference step in 0.76 ms, while the PI controller does not reach the 2 % band within 50 ms. The cascaded PI controller shows better robustness to filter parameter mismatch due to its inner-loop integral action, which reduces steady-state errors under modeling uncertainty. Overall, the performance improvement and the robustness trade-off both come directly from the controller structures, rather than from tuning choices.

In this paper, we study the B-model categorical enumerative invariants (CEI) associated with derived categories of coherent sheaves on smooth projective Calabi-Yau $3$-folds. We first prove the analogs of the dilaton, string, and divisor equations of CEI in a general context. Then we use these equations and the Givental quantization formula to prove that the B-model CEI for any miniversal family of smooth projective Calabi-Yau $3$-folds satisfies the holomorphic anomaly equations introduced by Bershadsky-Cecotti-Ooguri-Vafa. This provides strong evidence that CEI may be taken as a rigorous mathematical definition of the B-model topological string partition function.

For integers $a,b\ge 2$, let \[ g_n:=\gcd(a^n-1,b^n-1)\qquad(n\ge 1). \] We study the sequence $(g_n)$ from the perspective of divisibility sequences and the Ailon--Rudnick problem. We prove that $(g_n)$ satisfies a constant-coefficient linear recurrence if and only if $a$ and $b$ are multiplicatively dependent. More generally, if $a$ and $b$ are multiplicatively independent, then every integer linear divisibility sequence $(W_n)$ satisfying \[ W_n\mid a^n-1 \qquad\text{and}\qquad W_n\mid b^n-1 \qquad(n\ge 1) \] is periodic. We also determine the local structure of $(g_n)$ through an exact support formula and an exact odd-prime valuation formula. In the normalized setting $\gcd(a-1,b-1)=1$, these formulas identify the bad set $\{n\ge 1:g_n>1\}$ as an explicit union of arithmetic progressions. Finally, we obtain several structural reductions toward the integer Ailon--Rudnick conjecture, including primitive-support, prime-power-ray, prime-index, and resultant formulations.

An effective Bayesian evidence ranking is performed for the monomial potentials \(V_p(ϕ)=λ_pϕ^p/p\), with \(p=2,3,4\), in low temperature warm inflation with the dissipative coefficient fixed as \(Υ=C_ϕT^3/ϕ^2\). In cold single field slow roll inflation, these branches are strongly constrained by the observational upper bound on the tensor to scalar ratio \(r=\mathcal P_T/\mathcal P_{\mathcal R}\), whereas warm inflation can reduce this tension by enhancing the scalar spectrum. The relevant question is therefore which monomial power is favored once \(A_s\), \(n_s\), \(r_{0.05}\), and the viable parameter volume are considered simultaneously. For each branch, the warm background equations including radiation backreaction are solved, and a broadened compressed likelihood for \((A_s,n_s,r_{0.05})\) is integrated over the prior volume to obtain \(Z_{\rm eff}^{(A_s,n_s,r)}\). For \(N_*=55\), \(σ_r=0.005\), and structure conditioned priors covering viable warm branches, the quadratic and cubic potentials are disfavored relative to the quartic branch: $Δ\ln Z_{\rm eff}(p=2)=-32.18,~ Δ\ln Z_{\rm eff}(p=3)=-6.99.$ This hierarchy is stable under changes in \(N_*\), prior ranges, random seeds, and the $r$ bound treatment. A representative quartic trajectory gives \(n_s=0.96420\), \(r_{0.05}=0.02663\), \(Q_*=4.68\times10^{-3}\), and \(T_*/H_*=10.67\), corresponding to a weakly dissipative but thermally occupied CMB window. Decomposing the primordial spectrum shows that the quartic preference is driven mainly by Bose Einstein occupation enhancement for \(T_*/H_*>1\), not by strong dissipative friction. Within the low temperature dissipative effective class and compressed likelihood adopted here, the evidence hierarchy is \(p=4>p=3\gg p=2.\)

Cloud Security Posture Management (CSPM) systems detect known vulnerabilities by maintaining a rule set, distributing it to customers, and evaluating it against periodically-collected asset inventories. To our knowledge, in publicly documented architectures the rule set is environment-agnostic and curated centrally by the vendor; updates are batched into release cycles and shipped on a cadence ranging from hours to days depending on detection severity. The disclosure-to-protection window -- from a CVE being published to the customer's system being capable of detecting affected assets -- is therefore bounded by the vendor's release cadence for version-match detections, and by additional human authoring time for richer detections incorporating configuration predicates beyond the affected-software string. We propose an architecture in which the rule set is not vendor-distributed but continuously derived, within the customer's tenant, from the intersection of public catalogue feeds and the live asset graph. A rule comes into existence when a catalogue entry and an applicable asset are simultaneously present, and goes out of existence when either input ceases to support it. Derivation is bidirectional: new catalogue entries and new assets both trigger it. It incorporates the full structured-field content of catalogue entries, not only the affected-software predicate. The live rule set is bounded by environment diversity rather than catalogue breadth. Prior systems incrementally evaluate a static rule set; we incrementally derive the rule set itself. We present the threat model, the architecture, formal semantics with an equivalence theorem, complexity analysis, a worked example, and an evaluation methodology. The contribution is the architectural shift and its latency and resource consequences; rule correctness and alert prioritization are out of scope.

We study the Minkowski shape of pure number fields \[ K_a=\mathbb Q(θ),\qquad θ^n=a. \] For admissible parameters satisfying an explicit local hypothesis at the primes dividing $n$, we prove a discrete--archimedean factorization \[ \mathrm{sh}(K_a)=\bigl[C(a)^{\mathsf T}\mathrm{diag}\bigl(s_1(a),\dots,s_{n-1}(a)\bigr)C(a)\bigr], \] where the $s_m(a)$ arise from normalized monomials and $C(a)\in\mathrm{GL}_{n-1}(\mathbb Q)$ comes from a normalized integral basis. This yields a uniform odd/even rigidity dichotomy: for every odd $n\geq 3$, the Minkowski shape is a complete invariant among admissible pure degree-$n$ fields, whereas for $n=2r$ it determines the core field $\mathbb Q(|a|^{1/r})$; on the squarefree admissible subfamily it is complete up to sign, although infinitely many non-isomorphic pairs $K_a$ and $K_{-a}$ have the same shape. We also derive explicit formulas for $|\mathrm{disc}(K_a)|$, including exponent-vector and divisor-lattice factorizations. Finally, we show that the pure-field shape locus is supported on rational diagonal leaves in shape space: unconditionally it lies in a countable union of closed leaves, while under the same local hypothesis only finitely many leaves occur in each fixed degree. On a fixed normalized stratum, the shape depends only on ratio variables, whereas discriminant growth is governed by independent product variables.

Building on a recent derivation of spin-charge separation in $(1+1)$d paired Dirac fermions~\cite{Haddad2024}, we develop a self-consistent dressing $ψ(x) = U(x)χ(x)$ for the full Fierz-complete four-fermion model, extending that result and providing a more detailed resolution of the chiral-difermion phase structure. A key feature of this approach is that the composite connection $A_μ^{\rm dress} = i(\partial_μU)U^{-1}$ encodes obstructions to local trivialization of the Dirac operator, i.e., the degree to which the background can be absorbed into the dressing. Using this fact, we prove a trivialization theorem under which three nonperturbative constructions are unified: spin-charge separation in correlated fermion systems, half-infinite Wilson-line dressing in gauge theory, and the holonomy of flat connections. Our approach shows that the three regimes of our model (chiral, difermion, intermediate), are then tied together by an emergent $\mathfrak{sl}(2,\mathbb{R})$ gauge field that binds the spin and charge degrees of freedom. In particular, the chiral-difermion transition is a deconfinement transition for these degrees of freedom, diagnosed by closed boost-sector Wilson loops that develop an area law in the chiral phase for which we compute the associated string tension. This provides a concrete realization of the conjectured Faddeev--Niemi link between spin-charge separation and confinement. We close with a unifying geometric picture in which the order-parameter manifold takes hyperbolic form $ρ^2 - |Δ|^2 = σ^2$, promoted to $\mathrm{AdS}_3 \cong \mathrm{SL}(2,\mathbb{R})$ on inclusion of the charge and difermion phases. The structural matching to the kinematic stage of $\mathrm{AdS}_3/\mathrm{CFT}_2$ is identified explicitly, with the conjecture that the dressed model realizes the inverse Pohlmeyer reduction of the $\mathrm{AdS}_3$ sigma model.

Modulated symmetries are global symmetries with a spatially dependent unit of charge, such as the dipole symmetry and the exponential symmetry. We give the generic condition for a modulated symmetry to be compatible with translationally symmetric Hamiltonians, which we define as a translationally covariant modulated symmetry (TCMS). For Abelian TCMSs, we prove that their units of charge can only contain multipole, exponential and harmonic components. Particularly, we classify all the one-dimensional TCMSs by real Jordan normal form blocks. We further derive the generic Goldstone action for SSB phases of continuous TCMSs, by which we show that a broken multipole symmetry gives higher-order gapless Goldstone modes, a broken harmonic symmetry gives gapless Goldstone modes at finite momenta, and a broken exponential symmetry gives no gapless Goldstone modes, modifying the conventional Goldstone theorem.

Understanding how educational social dynamics evolve is critical for informing effective educational policies and counterfactual interventions. However, traditional methods face a fundamental dilemma: observational studies often lack causal power, while controlled experiments are frequently constrained by ethical concerns. Although LLM-based multi-agent simulations offer a scalable in silico alternative, existing approaches remain limited by weak psychological grounding and insufficient measurement of latent psychological states. To address this, we introduce EduMirror, a multi-agent simulator for the scientific study of educational social dynamics. We provide configurable education-oriented agent forms, including value-driven agents grounded in psychological needs and social value orientation, together with a dual-track measurement protocol for quantifying observable behaviors and latent psychological states. We validate the realism and usability of EduMirror through case studies on school bullying and group cooperation, as well as broader evaluations across diverse educational scenarios. The results show that EduMirror generates educational social dynamics that are realistic, theory-consistent, and measurable by empirical criteria. These properties enable structured in silico educational research, providing a computational tool for hypothesis testing and counterfactual intervention analysis in educational science. Project page: https://edumirror.net.

Fréchet regression provides a versatile framework for modeling responses in metric spaces with Euclidean predictors, yet current methodologies rely almost exclusively on frequentist approaches. We propose a Bayesian framework for Fréchet regression that offers a principled way of incorporating prior information into nonlinear global Fréchet regression. By targeting a novel Fréchet Bayes rule, we reduce the object-valued regression problem to a collection of tractable scalar regression tasks. Our approach allows for a controlled interpolation between the prior and the data-driven frequentist estimate, facilitating effective shrinkage toward informed values. While initially derived under Gaussian assumptions, we demonstrate that our framework is robust to model misspecification by establishing its validity under moment conditions via weak conditional expectations. The numerical properties of the proposed methodology are demonstrated in simulation studies and an application to microbiome compositional data, where we show that leveraging an auxiliary cohort to inform the prior significantly enhances predictive performance in a targeted, small-scale study

For $m \in \mathbb{N}$, let $q_m := \frac{\log(2m+1)}{2\log(m+1)}$. The max-convolution inequality \begin{align*} \sum_{k=0}^{2m}\left(\max_{i+j=k} x_i y_j \right)^{q_m} &\ge \left(\sum_{i=0}^{m} x_i\right)^{q_m} \left(\sum_{j=0}^{m} y_j\right)^{q_m} \end{align*}for arbitrary sequences $x_0 \ge x_1 \ge ... \ge x_m \ge 0, y_0 \ge y_1 \ge ... \ge y_m \ge 0$ implies an affirmative answer to a question of Bourgain, Dilworth, Ford, Konyagin, and Kutzarova \cite{BDFKK} on the sizes of sumsets in product sets. This inequality was proven for $m = 2$ by Becker, Ivanisvili, Krachun, and Madrid \cite{BIKM} by reducing the general case to the geometric block case via a max-tie analysis. We prove the geometric block case $x = (1, t, ..., t^{r}, 0, ..., 0)$ and $y = (1, t, ..., t^s, 0, ..., 0)$, $t \in [0, 1]$, for all $m \in \mathbb{N}$ via a comparison of Stolarsky means. Some perturbations are also verified. Finally, we prove the above inequality when one sequence has only two non-zero terms.

It is known that lower powerdomains preserve and reflect both continuity and quasicontinuity, while the preservation of quasicontinuity by upper and convex powerdomains had long been open. Directed spaces provide a topological extension framework for dcpos. Powerdomains of dcpos can be characterized as the $D$-completions of the corresponding directed powerspaces. Using this observation, the authors proved in 2024 that upper powerdomains do not preserve quasicontinuity. In this paper, we prove that directed convex powerspaces and convex powerdomains do not preserve quasicontinuity, but they do reflect both continuity and quasicontinuity. We also prove that upper powerspaces and upper powerdomains reflect quasicontinuity. Together with the known results for lower and upper powerdomains, these arguments give the complete preservation and reflection profile of the lower, upper and convex powerdomains for continuity and quasicontinuity.

Spectrum aggregation has been a key enabler of LTE and 5G capacity growth, but it will become even more fundamental in 6G as networks expand across low bands, existing mid bands, new upper-mid/centimetric bands, and millimeter wave bands. This article examines how 5G carrier aggregation (CA) and dual connectivity (DC) inform the design of 6G spectrum aggregation. We argue that, while DC was instrumental in accelerating early non-standalone 5G deployment, it also introduced architectural fragmentation and long-term migration complexity. In contrast, CA provides a cleaner and more scalable foundation for multi-band operation in standalone 6G. Building on lessons from 5G, we advocate enhanced CA as the preferred 6G aggregation framework and point out the corresponding key enhancement directions.

We address the inverse problem of reconstructing the state at a final time $T_0$ of a parabolic equation with dynamic boundary conditions of Wentzell type, from a distributed interior observation on a subdomain $ω\subset Ω$. Our approach combines (i) a Carleman-type observability inequality for the bulk-surface system, with (ii) a Tikhonov-regularized optimal-control reformulation of the data assimilation problem, and (iii) a fully implementable spatio-temporal discretization based on a Lie splitting that decouples the bulk and surface dynamics. We prove an exact reconstruction theorem for the continuous problem, derive a stability inequality, and analyze a penalized discrete scheme. We then propose an adaptive post-processing step that optimally combines the regularized reconstruction with the observation itself; under noisy data, the resulting estimator automatically selects between full post-processing and the raw reconstruction without prior knowledge of the noise level. Extensive numerical experiments on a two-dimensional Wentzell heat equation validate the method. A semilinear extension to the Allen-Cahn nonlinearity is treated via a Picard outer loop with Schauder fixed-point convergence; two complementary experiments reveal an intrinsic observability-nonlinearity trade-off in the bulk-surface setting.

Hallucinations in large language models (LLMs) create heightened risks in multi-agent settings, where recursive agent interactions can propagate, reinforce, and amplify unsupported claims. This paper models hallucination as a system-level, time-evolving process across a network of interacting LLM agents, where nodes represent agents and edges encode information exchange. The proposed formulation captures how hallucinated claims diffuse through communication topologies, intensify under adversarial perturbations, and affect collective reliability across reasoning rounds. To suppress error propagation, we introduce an interaction-aware control method that combines confidence-weighted aggregation, adaptive impact regulation, external claim verification, and selective isolation of unreliable agents. Experiments on TruthfulQA and TriviaQA show that the proposed method reduces hallucination by up to 39.0% relative to undefended multi-agent reasoning, improves factual accuracy from 0.79 to 0.87, and increases semantic consistency from 0.75 to 0.84. Under adversarial conditions, the method limits hallucination amplification to 1.08, compared with 1.45 without adaptive control, maintaining stable collective behavior across recursive interaction rounds. These results indicate that hallucination in multi-agent LLM systems is governed by both individual model reliability and system-level interaction dynamics, including communication topology, confidence coupling, and recursive information flow.

Multi-agent large language model (LLM) systems offer strong capabilities for complex reasoning and decision-making, yet coordination across agents introduces error propagation, security risks, and inefficient use of resources. Existing methods often rely on heuristic, static strategies and lack a principled mechanism for balancing performance, security, and computational cost. This paper formulates multi-agent LLM coordination as a constrained optimization problem and proposes a security-aware method for adaptive agent selection. The method integrates trust modeling, risk-aware evaluation, and collective intelligence within a unified optimization objective. To solve the problem efficiently, we use a swarm-intelligence strategy inspired by Gorilla Troops Optimization (GTO), enabling adaptive coordination under varying threat conditions. Controlled experiments across 500 independent runs demonstrate the effectiveness of the proposed method. The system achieves a stable average performance score of 0.5281, with high consensus (0.8764), controlled risk (0.3000), and compact agent subsets averaging 4.04 selected agents. The optimization process converges efficiently, with an average runtime of 24.09 seconds per run and low score variability (standard deviation = 0.0173). Robustness analysis indicates graceful degradation under perturbations, with performance drops limited to 2.5% under agent removal and 5.3% under consensus disruption. These results show that effective multi-agent coordination can be achieved through structured optimization that jointly manages performance, security, and efficiency. The proposed method provides a practical security-aware solution for coordinating multi-agent LLM systems in complex adversarial settings.

Empirical research on the labor-market impact of artificial intelligence has converged, since Frey and Osborne (2017), on a continuous-gradient representation in which each occupation is assigned a real-valued exposure score on [0,1] obtained by linear aggregation across capability dimensions. This continuity is rarely articulated as an assumption and has not been tested at the micro-action level where substitution actually occurs. We decompose 1,961 O*NET Detailed Work Activities into 15,817 micro-actions using a multi-agent LLM pipeline with 31-expert HITL calibration, then project the DWA-level Occupational Automation Index from our prior work onto a 7-macro semantic typology. The result is a bipolar structure. Tool-Mediated Physical (M2, mean OAI = 0.054) and Planning & Design (M7, mean OAI = 0.499) form two extremes separated by Cohen's d = 2.41 (H = 172.88, p = 6.21e-34). The geometry is robust under three independent stress tests: resolution (K=7 to K=15, polar gap widens from 0.45 to 0.57), encoder swap to BGE (LLM-class OAI lead replicates at 3.37x), and Eloundou's GPT-4 task ratings (DWA-level rho = 0.635). The six middle macros form a low-contrast band between the poles (TOST at d=0.2 admits only 1/15 pairs as equivalent), not a flat plain. The geometry's stability does not, however, extend to its content. Across a decade, the polarity has inverted. Frey-Osborne (2013) placed Tool-Mediated Physical near the highest computerisation risk and Planning & Design near the lowest; our LLM-era OAI reverses that order, with macro-level FO-Eloundou Spearman rho = -0.750, p = 0.020, against the original Oxford Martin appendix. Which pole is high is therefore contingent on the era's dominant capability frontier, while the stable geometry itself is the structurally robust object.

Large Language Models (LLMs) generate fluent text but remain vulnerable to hallucinations, producing unsupported, inconsistent, and factually incorrect claims. Most prior work treats hallucination as a static property of isolated outputs. In multi-agent LLM systems, however, responses are exchanged across agents, revised through sequential stages, and reused as context for later reasoning. Hallucination, therefore, becomes a dynamic process shaped by interaction history, cascade depth, and model heterogeneity. This paper analyzes hallucination dynamics in multi-agent LLM cascades by tracking claim-level factual inconsistencies across sequential agent interactions. We conduct 500 cascade experiments across 10 knowledge domains using GPT-5.3, DeepSeek-V3, and LLaMA-3-70B-Instruct, yielding 1,250 evaluated responses. Results show that deeper cascades reduce the normalized hallucination score from 0.422 at the first agent to 0.272 at the final agent in 3-agent chains, with an amplification factor of 0.644, indicating net attenuation. This reduction is accompanied by a decline in factual accuracy from 0.789 to 0.769, revealing a trade-off between hallucination suppression and factual preservation. Transition-level analysis shows that each agent-to-agent refinement reduces hallucination by an average of 0.072, with small but consistent losses in factual consistency and response quality. Model-level results reveal reliability-efficiency trade-offs: LLaMA-3-70B-Instruct achieves the lowest hallucination score, whereas GPT-5.3 provides faster generation with a higher hallucination rate. Domain-level analysis shows that hallucination varies with topic complexity, with lower scores in well-grounded scientific domains and higher scores in more abstract domains.