We study the joint control of arrival and service rates in queueing systems with the objective of minimizing long-run expected cost minus revenue. Although the objective function is non-convex, first-order methods have been empirically observed to converge to globally optimal solutions. This paper provides a theoretical foundation for this empirical phenomenon by characterizing the optimization landscape and identifying a hidden convexity: the problem admits a convex reformulation after an appropriate change of variables. Leveraging this hidden convexity, we establish the Polyak-Lojasiewicz-Kurdyka (PLK) condition for the original control problem, which excludes spurious local minima and supports global convergence guarantees for first-order methods. Our analysis applies to a broad class of $GI/GI/1$ queueing models, including those with Gamma-distributed interarrival and service times, as well as $GI/M/1$ queues with log-concave interarrival times. As a key ingredient in the proof, we establish a new convexity property of the expected queue length under a square-root transformation of the traffic intensity.

We present a fully covariant and gauge-invariant analysis of linear cosmological perturbations in Energy-Momentum Squared Gravity. Working within the 1+3 formalism, we derive the exact propagation equations for scalar, vector, and tensor modes on FLRW backgrounds, in the case of radiation and dust. Two representative subclasses are examined in detail, in which non-linearity enters through $\mathcal{O}(ηρ^2)$ corrections or modifications in the equation-of-state parameter and the sound speed. For scalar perturbations, the density contrast can be enhanced or reduced relative to General Relativity, depending on the coupling parameter and the wavelength regime. A similar behaviour occurs for vector modes, allowing for a non-trivial vorticity at early-times. Tensor modes, described by the magnetic part of the Weyl tensor and the shear tensor propagate as damped waves with slowly varying effective masses. All sectors reduce continuously to their GR limits as $η\!\to\!0$. The framework isolates robust signatures - early-time scalar tilts, tensor damping shifts, and altered vorticity decay - that can be confronted with CMB and large-scale-structure observations to constrain these theories of gravity.

Resting-state brain functional connectivity quantifies the synchrony between activity patterns of different brain regions. In functional magnetic resonance imaging, each region comprises a set of spatially contiguous voxels at which blood-oxygen-level-dependent signals are acquired. The ubiquitous Correlation of Averages (CA) estimator, and other similar metrics, are computed from spatially aggregated signals within each region, and remain the quantifications of inter-regional connectivity most used by neuroscientists. Their popularity is primarily due to computational simplicity despite their demonstrable bias and lack of statistically principled justification. By leveraging linear mixed-effects models, both inter-regional and intra-regional correlation and measurement error can be explicitly modeled as signal variability sources. A novel computational pipeline, focused on subject-level inter-regional correlation parameters of interest, is developed to address the challenges of applying maximum likelihood estimation to such structured, high-dimensional spatiotemporal data. Simulation results confirm the superiority of the proposed estimator relative to CA in terms of both decreased bias and accurate confidence interval coverage across simulation settings. The proposed method is also applied to construct individual human brain networks for subjects from a Human Connectome Project test-retest database. Concordances between inter-regional correlation estimates demonstrate the potentially substantial scientific benefits of the proposed approach that reliably produces more consistent results than CA for test-retest scans of the same subject.

We study abelian anyons at the mean-field/almost-bosonic level, whose dynamics are governed by the Chern-Simons-Schrödinger system. We consider the dimensional reduction of this 2D model by introducing an anisotropic trapping potential, and derive an effective 1D model after tracing out the tight confinement direction. The resulting effective dynamics in the loose confinement direction is captured by a quintic defocusing nonlinear Schrödinger equation. We rigorously establish this dimensional reduction process in the sense of ground state energies and time-dependent solutions, under an $H^2$ well-posedness assumption.

Coherent ferrons, the quanta of polarization waves, can potentially be hybridized with many other quasiparticles for achieving novel control modalities in quantum communication, computing, and sensing. Here, we theoretically demonstrate a new hybridized state resulting from the strong coupling between fundamental-mode (wavenumber is zero) coherent ferrons and cavity bulk acoustic phonons. Using a van der Waals ferroelectric CuInP2S6 membrane as an example, we predict an ultra-strong ferron-phonon coupling at room temperature, where the coupling strength g_c reaches over 10% of the resonant frequency ω_0. We also predict an in-situ bistable electric-field control of mode-specific ferron-phonon hybridization via ferroelectric switching. We further show that CuInP2S6 allows for reaching the fundamentally intriguing but challenging deep strong coupling regime (i.e., g_c/ω_0>1) near the ferroelectric-to-paraelectric phase transition. Our findings establish the theoretical basis for exploiting coherent ferron as a new contender for hybrid quantum system with strong and highly tunable coherent coupling

In many applications, system identification experiments must be performed in closed loop to ensure safety or to maintain system operation. In this paper, we consider the recursive design of informative experiments for ARMAX models by adding a bounded probing signal to the input generated by a fixed output feedback controller. The resulting output perturbations should be kept within user-specified limits. We analyze the identifiability and feasibility conditions of this setting and then proceed to derive a probing signal that can be efficiently computed in closed form. We demonstrate the effectiveness and properties of the design in numerical experiments.

Intermediate Macdonald polynomials for an affine root system $S$ with fixed origin and finite Weyl group $W_0$ are orthogonal polynomials invariant under a parabolic subgroup $W_J\le W_0$. The extreme cases of $W_J=1$ and $W_J=W_0$ correspond to the non-symmetric and symmetric Macdonald polynomials, respectively. In this paper, we use double-affine Hecke algebras to study their basic properties, including that they form an orthogonal basis and that they diagonalise a commutative algebra of difference-reflection operators, and calculate their norms. Finally, we provide two interpretations of intermediate Macdonald polynomials as vector-valued polynomials and connect them to the literature.

Cohesive active assemblies are often regulated by spatially heterogeneous nonequilibrium driving, such as gradients in motility, biochemical turnover, or mechanical activity. Such heterogeneous driving can influence where condensates or cell collectives accumulate, how stable they are, and how they exchange material with their surroundings. However, the minimal physical mechanisms by which activity gradients control the positioning and turnover of cohesive active matter remain unclear. Here, we address this question using a model of attractive active Brownian particles (ABPs) in a spatially varying activity field. Using Brownian dynamics simulations, we show that these particles undergo liquid-gas phase separation, and spatially varying activity fields induce striking emergent dynamics. Attractive active droplets migrate up activity gradients, and at sufficiently high activity, they can fragment or evaporate into a dilute phase. For finite clusters, evaporated ABPs can redistribute through the simulation box, reassemble into new clusters in lower-activity regions, and migrate again toward higher activity, giving rise to cyclic positioning through repeated nucleation, migration, evaporation, and reassembly.

We prove an algebraic decomposition theorem for the unit group $\mathrm{GL}(R)$ of an arbitrary non-discrete irreducible, continuous ring $R$ (in von Neumann's sense), which entails that every element of $\mathrm{GL}(R)$ is both a product of $7$ commutators and a product of $16$ involutions. Combining this with further insights into the geometry of involutions, we deduce that $\mathrm{GL}(R)$ has the so-called Steinhaus property with respect to the natural rank topology, thus every homomorphism from $\mathrm{GL}(R)$ to a separable topological group is necessarily continuous. Due to earlier work, this has further dynamical ramifications: for instance, for every action of $\mathrm{GL}(R)$ by homeomorphisms on a non-void metrizable compact space, every element of $\mathrm{GL}(R)$ admits a fixed point in the latter. In particular, our results answer two questions by Carderi and Thom, even in generalized form.

The vision of a smart wireless factory (SWF) demands highly flexible, low-latency, and reliable connectivity that goes beyond conventional wireless solutions. Reconfigurable intelligent surface (RIS)-empowered communications, when integrated with the open radio access network (O-RAN) architectures, have emerged as a promising enabler to meet these challenging requirements. This article introduces the methodology for the orchestration of RIS with xApps (ORIX), bringing the RIS technology into the O-RAN ecosystem through xApp-based control for SWF environments. ORIX features three key components: an O-RAN-compliant RIS service model for dynamic configuration, an RIS channel simulator that supports 3GPP indoor factory models with multiple industrial scenarios, and practical RIS optimization strategies with finite-resolution control. Together, these elements provide a realistic end-to-end emulation platform for evaluating RIS placement, control, and performance in SWF environments prior to deployment. The presented case study demonstrates how ORIX enables the evaluation of achievable performance gains, exploration of trade-offs among key RIS design parameters, and identification of deployment strategies that balance system performance with practical implementation constraints. By bridging theoretical advances with industrial feasibility, ORIX lays the groundwork for RIS-assisted O-RAN networks to power next-generation wireless communication in industrial scenarios.

We present a generalized, data-driven collisional operator for one-component plasmas, learned from molecular dynamics simulations, to extend the collisional kinetic model beyond the weakly coupled regime. The proposed operator features an anisotropic, non-stationary collision kernel that accounts for particle correlations typically neglected in classical Landau formulations. To enable efficient numerical evaluation, we develop a fast spectral separation method that represents the kernel as a low-rank tensor product of univariate basis functions. This formulation admits an $O(N \log N)$ algorithm via fast Fourier transforms and preserves key physical properties, including discrete conservation laws and the H-theorem, through a structure-preserving central difference discretization. Numerical experiments demonstrate that the proposed model accurately captures plasma dynamics in the moderately coupled regime beyond the standard Landau model while maintaining high computational efficiency and structure-preserving properties.

Quantum Fisher information (QFI) is a measure of multipartite entanglement accessible via inelastic neutron scattering. Here we demonstrate that QFI reveals thermal and dynamical properties of quantum spin ice (QSI), a three-dimensional quantum spin liquid with fractionalized excitations. By developing a multi-directed loop update quantum Monte Carlo algorithm, along with exact diagonalization and gauge mean-field theory, we compute the QFI for the pyrochlore lattice. The temperature and momentum dependence of QFI maps the phase diagram, distinguishing the ferromagnetic ordered phase, its critical region, the zero-flux QSI, and the $π$-flux QSI. QFI also captures two crossover scales: from trivial paramagnet to classical spin ice, then to QSI. We discuss the $π$-flux QSI in light of experiments on cerium-based pyrochlores. Our results suggest that QFI not only detects entanglement but also serves as a sensitive thermal and dynamical probe for frustrated quantum magnets.

We extend the Galois-theoretic Borovoi-Springer interpretation of algebraic bands to a class of étale-locally represented bands on the fppf site of an arbitrary field $k$, which we call separable bands. Next, a band represented étale-locally over $k$ by a pseudo-reductive group is shown to be globally representable when $[k : k^p] = p$, with counterexamples in general. When $k$ is a global or local field, we deduce a generalization of Borovoi's abelianization theory to separable bands represented by smooth connected algebraic groups. As an application, we prove that the Brauer-Manin obstruction is the only obstruction to both the Hasse principle and weak approximation on a homogeneous space of a pseudo-reductive group (more generally, of a smooth connected affine algebraic group with split unipotent radical) having a smooth connected geometric stabilizer.

We present a catalogue of spectroscopically-selected [OIII]+Hb emitters at 6.75<z<9.05 and the resulting [OIII] 5008 ÅLuminosity Function (LF) in the COSMOS field. We leverage the 0.33 deg$^{2}$ covered by COSMOS-3D using NIRCam/WFSS F444W to perform the largest spectroscopic search for [OIII] emitters at 6.75<z<9.05. We present our catalogue of 249 [OIII] emitters and their associated completeness function. The inferred constraints on the [OIII] LF enable us to characterise the knee of the [OIII] LF, resulting in improved [OIII] LF constraints at z~7,8. Notably, we find evidence for a decline of the [OIII] luminosity density between z~7 and z~8, which could be expected if the metallicity of [OIII] emitters, as well as the cosmic star-formation rate density, is declining at these redshifts. We find that theoretical models that reproduce the z~7,8 [OIII] LF do not reproduce well the [OIII] equivalent width distribution, pointing to potential challenges in the modelling of[OIII] and other nebular lines in the early Universe. Finally, we provide the first constraints on the cosmic variance of [OIII] emitters, estimating at 15% the fractional uncertainty for the z~7,8 [OIII] LF in the 0.33 deg$^2$ field. This estimate is in good agreement with that inferred from clustering, and shows that the [OIII] LF derived from smaller extragalactic legacy fields is strongly affected by cosmic variance. Our results highlight the fundamental role that wide-area JWST slitless surveys play to map the galaxy large-scale structure down into the reionisation era, serving as a springboard for a variety of science cases.

Post-quantum cryptography (PQC) is moving from evaluation to deployment as NIST finalizes standards for ML-KEM, ML-DSA, and SLH-DSA. This survey maps the space from foundations to practice. We first develop a taxonomy across lattice-, code-, hash-, multivariate-, isogeny-, and MPC-in-the-Head families, summarizing security assumptions, cryptanalysis, and standardization status. We then compare performance and communication costs using representative, implementation-grounded measurements, and review hardware acceleration (AVX2, FPGA/ASIC) and implementation security with a focus on side-channel resistance. Building upward, we examine protocol integration (TLS, DNSSEC), PKI and certificate hygiene, and deployment in constrained and high-assurance environments (IoT, cloud, finance, blockchain). We also discuss complementarity with quantum technologies (QKD, QRNGs) and the limits of near-term quantum computing. Throughout, we emphasize crypto-agility, hybrid migration, and evidence-based guidance for operators. We conclude with open problems spanning parameter agility, leakage-resilient implementations, and domain-specific rollout playbooks. This survey aims to be a practical reference for researchers and practitioners planning quantum-safe systems, bridging standards, engineering, and operations.

Numerical relativity simulations provide a full description of the dynamics of binary systems, including gravitational radiation. The waveforms produced by these simulations have a number of applications in gravitational-wave detection and inference. In this work, we revisit the accuracy of the waveforms produced by the Spectral Einstein Code. Motivated by the wide range of waveform applications, we propose and explore three accuracy metrics between simulation resolutions: (i) the generalized frequency-weighted mismatch, (ii) the relative amplitude difference, and (iii) the phase difference at different times. We confirm that numerical errors accumulate over the binary evolution, but the error is not intrinsically larger during the latest, more dynamical stages. Studying errors across the parameter space, we identify a positive correlation between both the mismatch and the phase difference with precessing spin, but little correlation with aligned spin or eccentricity. Lastly, amplitude and phases differences are symmetric upon exchanging resolutions across the catalog, suggesting that the dominant source of error is random, rather than something systematic that affects all waveforms similarly.

Let $G$ be a split, simply connected, almost simple algebraic group, and let $P$ be a maximal parabolic subgroup of $G$. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on $X_P$ using the Fourier theory. We also establish the Poisson summation formulae in these cases.

We prove recurrence of the vertex-reinforced jump process on the hierarchical lattice for spectral dimension \(d<2\) for every value of the conductance parameter \(\overline{W}\), and at the critical spectral dimension \(d=2\) for sufficiently strong reinforcement, i.e., sufficiently small \(\overline{W}\). The key estimate is a fractional-moment bound for the Green's function of the associated random Schrödinger operator, expressed as geometric decay across hierarchical scales for the effective \(H^{2|2}\) field. The proof combines the fractional-moment method with an exact hierarchical coarse-graining identity, which turns the path expansion into a recursion over block scales and controls the combinatorial growth created by long-range edges. Together with existing long-range-order results in the regime \(d>2\), these estimates identify the recurrent side of the hierarchical VRJP phase diagram, leaving only the weak-reinforcement critical regime open.

We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields $\mathbb{F}_p$. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by $p$. This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod $p$.

In previous works, we showed that both time and space can emerge from entanglement within a globally constrained quantum Universe, with no background coordinates. By extending the Page and Wootters quantum time formalism to include both quantum clocks and rods, and imposing global constraints on total energy and momentum, we constructed a fully relational model of quantum spacetime. Here we take a further step: working in 1+1 dimensions, we show that the standard wave equations governing quantum particles (the Schrödinger, Klein-Gordon and Dirac equations) emerge naturally from this framework. The solutions of the equations are derived directly from the constraints, without assuming any external spacetime structure. The second quantization formalism is also implemented and discussed. Our results provide further support for the idea that quantum dynamics in spacetime may emerge from entanglement and constraints.

Adapting the definition of ``extended Sobolev scale" on compact manifolds by Mikhailets and Murach to the setting of a (generally non-compact) manifold of bounded geometry $X$, we define the ``extended Sobolev scale" $H^φ(X)$, where $φ$ is a function which is $RO$-varying at infinity. With the help of the scale $H^φ(X)$, we obtain a description of all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of Sobolev spaces $[H^{(s_0)}(X), H^{(s_1)}(X)]$, with $s_0<s_1$. We use this interpolation property to establish a mapping property of proper uniform pseudo-differential operators (PUPDOs) in the context of the scale $H^φ(X)$. Additionally, using a first-order positive-definite PUPDO $A$ of elliptic type we define the ``extended $A$-scale" $H^φ_{A}(X)$ and show that it coincides, up to norm equivalence, with the scale $H^φ(X)$. Besides the mentioned results, we show that further properties of the $H^φ$-scale, originally established by Mikhailets and Murach on $\mathbb{R}^n$ and on compact manifolds, carry over to manifolds of bounded geometry.

We introduce the framework of quadratic-form optimal transport (QOT), whose transport cost has the form $\iint c\,\mathrm{d}π\otimes\mathrm{d}π$ for some coupling $π$ between two marginals. Interesting examples of quadratic-form transport cost and their optimization include inequality measurement, the variance of a bivariate function, covariance, Kendall's tau, the Gromov--Wasserstein distance, quadratic assignment problems, and quadratic regularization of classic optimal transport. QOT leads to substantially different mathematical structures compared to classic transport problems and many technical challenges. We illustrate the fundamental properties of QOT and provide several cases where explicit solutions are obtained. For a wide class of cost functions, including the rectangular cost functions, the QOT problem is solved by a new coupling called the diamond transport, whose copula is supported on a diamond in the unit square.

Koopman spectral analysis turns nonlinear stochastic dynamics into a linear evolution of observables and gives access to decay rates, oscillatory modes, and metastable behavior. In practice, however, EDMD, SDMD, and related estimators depend strongly on where the trajectory data are collected. If most trajectories start in regions that carry little spectral information, the leading eigenvalues and eigenfunctions can be poorly estimated even with a rich dictionary. We propose \emph{Reinforced SDMD}, a data-acquisition method that couples Stochastic Dynamic Mode Decomposition with reinforcement learning. The RL agent chooses trajectory-initialization regions, SDMD updates the Koopman approximation, and a spectral-consistency reward evaluates the estimated eigenpairs on the newly generated data. An exploration bonus is added to avoid repeatedly sampling only a small part of the state space. We test multi-armed bandits, DQN, and PPO on stochastic double-well, Duffing, and FitzHugh--Nagumo systems. The learned policies place more samples in regions that are useful for estimating the leading Koopman eigenpairs. We also give an error-propagation analysis showing how SDMD operator error enters the corresponding bandit, approximate value-iteration, and approximate policy-iteration bounds.

Early results by Borel and Cantelli and Chung (Rendiconti del Circolo Matematico di Palermo, 1909) and Erdős (Transactions of the AMS, 1952) have provided bounds for the measure of a limsup set in terms of measures of its constituent sets and their intersections. Recent work by Beresnevich and Velani (Journal of Mathematical Analysis and its Applications, 2023) states that for sequences of balls the measure of the corresponding limsup set being positive is equivalent to a condition on the relationship between the measures of these balls and their pairwise intersections. In this paper, we show that the condition that the sets are balls is strictly necessary in the result of Beresnevich and Velani. Moreover, let $d \in \mathbb{N}$ and let $[0,1]^d$ be equipped with Lebesgue measure $μ$. Fix $m \in \mathbb{N}$. When we drop the condition that the sets are balls, we can find two sequences of sets $(A_i)_{i \in \mathbb{N}}$ and $(B_i)_{i \in \mathbb{N}}$ in $[0,1]^d$ such that $μ(A_i)=μ(B_i)$ for all $i \in \mathbb{N}$ and for any sequence $(i_1,i_2,...,i_l)$ where for all $l \leq m$ we have $μ(A_{i_1}\cap A_{i_2} \cap... \cap A_{i_l})=μ(B_{i_1}\cap B_{i_2} \cap... \cap B_{i_l})$ but $μ(\limsup_{i \rightarrow \infty} A_i)=1$ and $μ(\limsup_{i \rightarrow \infty} B_i)=0$.

Automated Code Review (ACR) is crucial for software quality, yet existing benchmarks often fail to reflect real-world complexities, hindering the evaluation of modern Large Language Models (LLMs). Current benchmarks frequently focus on fine-grained code units, lack complete project context, and use inadequate evaluation metrics. To address these limitations, we introduce SWRBench , a new benchmark comprising 1000 manually verified Pull Requests (PRs) from GitHub, offering PR-centric review with full project context. SWRBench employs an objective LLM-based evaluation method that aligns strongly with human judgment (~90 agreement) by verifying if issues from a structured ground truth are covered in generated reviews. Our systematic evaluation of mainstream ACR tools and LLMs on SWRBench reveals that current systems underperform, and ACR tools are more adept at detecting functional errors. Subsequently, we propose and validate a simple multi-review aggregation strategy that significantly boosts ACR performance, increasing F1 scores by up to 43.67%. Our contributions include the SWRBench benchmark, its objective evaluation method, a comprehensive study of current ACR capabilities, and an effective enhancement approach, offering valuable insights for advancing ACR research.

PDF parsers have recently improved on page-level layout understanding. However, recovering a document-global section hierarchy with reliable boundaries remains brittle for deeply structured books: many systems expose only page-local heading roles, assume shallow depth, or rely on high-quality PDF tags or Table of Contents (TOC) metadata, and public gold-standard data for deep book hierarchies is scarce. We present HiPS for hierarchical PDF segmentation of doctrinal legal books and make two main contributions. First, we release a gold-standard benchmark of 49 open-access law books with 9,812 manually curated headings, hierarchy levels, and page anchors, enabling evaluation of title detection, hierarchy reconstruction, and section boundary assignment. Second, we introduce complementary segmentation pipelines: a TOC-based parser for books with reliable outline metadata and a TOC-free LLM-refined pipeline that combines OCR whitespace cues, XML typography, and local context. Across a broad comparison against open-source parsers and multimodal/LLM baselines, the TOC-based pipeline is strongest when metadata is complete, while the LLM-refined pipeline improves heading precision, deep-level recovery, and boundary quality when metadata is missing or noisy.

In recent years, charge-channel order in strongly correlated metals has attracted much attention. Representative examples include electronic nematic order in cuprates and iron-based superconductors, and Star-of-David order in kagome metals. Critical phenomena and unconventional superconductivity arising from fluctuations of such charge-channel orders are central issues today; however, the essential role is played by many-body effects (vertex corrections) beyond the mean-field approximation, and their origin and computational methods have not been established. In this study, we propose the Bethe-Salpeter equation method to evaluate electron-electron interactions in two-dimensional Hubbard models beyond the mean-field approximation. Based on the Baym-Kadanoff conserving approximation, we find that an attractive interaction in the charge channel emerges from the Aslamazov-Larkin vertex corrections that describe the interference processes among spin fluctuations. Applying this method to the square-lattice Hubbard model shows that the cooperation of attractive charge fluctuations and repulsive spin fluctuations yields high-$T_c$ $d$-wave superconductivity together with enhanced effective mass. These results provide a natural explanation for the phase diagram of cuprate superconductors, in which $d$-wave superconductivity is strongly enhanced near the charge-order critical point.The theory can also be applied to iron-based and nickelate superconductors, suggesting broad potential for future applications.

We investigate an interesting interplay of destructive interference due to lattice geometry and band folding due to enlargement of the Brillouin zone in generating and subsequently modifying the band topology in a twisted bilayer $α$-$T_3$ system. The pronounced degeneracy of the emergent flat band in the dice limit of the $α$-$T_3$ lattice is removed on alignment with h-BN layers, resulting in the formation of sub-bands with varying topological characteristics. Remarkably, while the sub-band near charge neutrality exhibits a trivial behavior, a topologically non-degenerate singular sub-band emerges away from charge neutrality. The topological band remains isolated from the rest of the bands for a substantial area of the $α- θ$ plane (where $α$ and $θ$ correspond to the hopping ratio and twist angle respectively) while exhibiting multiple phase transitions as a function of the aforementioned parameters via hybridization with its nearest bands. We study the evolution of the hybrid Wannier charge center and the Chern number to characterize the different emergent topological phases. Finally, the degree of flatness of the topological band is studied as a function of both $α$ and $θ$ to explicitly show the influence of quantum interference and band folding on the width of the topological band.

Network reduction simplifies complex electrical networks to address computational challenges of large-scale transmission and distribution grids. Traditional network reduction methods are often based on a predefined set of nodes or lines to remain in the reduced network. This paper builds upon previous work on optimal Kron-based reduction of networks, which was formulated as a mixed-integer linear program, to enhance the framework in three aspects. First, the scalability is improved via a cutting plane restriction, tightened Big M bounds, and a zero-injection node reduction stage. Next, we introduce a radiality-preservation step to identify and recover nodes whose restoration ensures radiality. A linearized voltage magnitude error constraint is incorporated to explicitly bound the difference between full and reduced networks. The model is validated through its application to the 533-bus distribution test system and a 3499-bus utility feeder for a set of representative loading scenarios. In the 533-bus system, an 85% reduction was achieved with a maximum voltage error below 0.0025 per unit, while in the 3499-bus feeder, over 94% reduction was obtained with maximum voltage errors below 0.002 per unit. Additionally, we show that the radialization step accelerates the runtime of optimal voltage control problems when applied to Kron-reduced networks.

Two geometric interpretations for complex multivectors and determinants are presented: a little known one in terms of square roots of volumes, and a new one using fractions of volumes. The fraction is determined by a new holomorphy index, which measures the lack of holomorphy of real subspaces of $\mathbb{C}^n$ via generalized Kähler angles or a disjointness angle, and relates real and complex exterior products. The interpretations are completed with a natural but uncommon concept of complex orientation, related to elementary complex transformations. We also propose graphical representations for complex blades (decomposable multivectors) as fractions of parallelotopes, and discuss how Clifford algebras relate (or not) to Hermitian geometry.