This paper presents a backstepping approach for the boundary control of first-order hyperbolic equations with spatially varying coefficients posed on domains of arbitrary dimension. The method is based on a change of variables induced by the characteristic flow of the time-invariant transport operator, transforming the original multidimensional system into a continuum of decoupled one-dimensional hyperbolic equations evolving along individual characteristic curves. A backstepping controller is then designed for each equation in the decomposition, and the resulting control laws are reassembled in the original coordinates to achieve finite-time stabilization of the full system. The framework relies on the existence of characteristic curves foliating the spatial domain, with uniformly bounded transit times (non-trapping).

I study the welfare-maximizing allocation of heterogeneous goods when monetary transfers are prohibited. Agents have private values, and the designer chooses a mechanism subject to incentive compatibility and aggregate supply constraints. I characterize the optimal mechanism for two kinds of goods, and show that it either offers one pure option per good or adds a bundle that delivers a larger total quantity. Including the bundle is optimal when narrow preference margins between goods are sufficiently predictive of greater need, allowing the designer to target high-value agents through their willingness to accept mixing. I then consider the case with N types of goods and characterize when the optimal mechanism takes the form of a simple menu, where each option offers some amount of one kind of good and none of the others. When this is the case, it can be implemented as a competitive equilibrium with equal incomes or a choice-based lottery.

The aim of this paper is to apply Hirota's bilinear method to the integrable discrete Manakov system in the focusing dispersion regime in order to construct and analyze soliton and breather solutions. After deriving the general bilinear form of the system, we show how to obtain fundamental solitons, as well as fundamental and composite breathers. We then obtain solutions exhibiting 2 solitons and 2 breathers and combinations of a soliton and a breather, and discuss all ``two-body'' interactions properties, with particular emphasis on explicit formulas, visualization, and long-time asymptotic behavior, thus rigorously confirming the highly nontrivial interaction properties of these coherent structures.

Chomp was introduced by Gale in 1974 \cite{Gale1974}. In the same paper, Gale reported that the $3\times n$ games had been completely analyzed for $n\le 100$, with a unique winning first move in every case, and asked whether winning first moves are unique in general. Although the general uniqueness statement is false \cite[Section~7.1]{BrouwerEtAl2005}, we prove that the three-row uniqueness phenomenon suggested by Gale's computations holds for all $n$: every $3\times n$ Chomp rectangle has exactly one winning opening move. This settles the three-row case of Gale's 52-year-old first-move uniqueness question. The proof is carried out in the two-variable recurrence introduced by Brouwer, Horváth, Molnár-Sáska, and Szabó \cite{BrouwerEtAl2005} for the function $f(q,r)$ whose values encode the $P$-positions. The main local ingredient is a rightmost-hole principle: if a value $p$ is absent from the set $C(q,r)$ but belongs to all corresponding sets $C(t,r)$ for $q<t<p$, then all intermediate values $q+1,\ldots,p-1$ are forced to belong to $C(q,r)$. This separates the diagonal values from the starts of constant rows, and yields a partition of the positive integers into the two possible types of winning opening moves. It also identifies the row of the unique opening move: no first-row opening move is winning; the second-row and third-row cases are precisely the two complementary Chomp sequences A029900 and A029901.

For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$.

The transition toward Industry 5.0 is reshaping industrial work environments with an emphasis on human-centricity, enabling close collaboration between humans and machines to enhance productivity and flexibility. However, such systems typically require monitoring of human workers and operators, often involving sensitive data, raising significant privacy concerns. As a result, affected workers and unions frequently reject human-machine collaboration features due to a lack of transparency regarding privacy threats and implemented mitigation strategies. To enable early stakeholder involvement, establish trust, and support informed decision-making, privacy implications must be communicated in a way understandable to non-technical stakeholders. Yet, current Requirements Engineering (RE) practices provide limited methodological support for making privacy threats and mitigations accessible to non-technical stakeholders (e.g., individual workers or their representative unions). In this paper, we propose a conceptual framework that guides software design from human monitoring-related use cases and requirements to informed decision-making guidance focusing on non-technical stakeholders. Building on principles such as Privacy by Design, the framework leverages Large Language Models (LLMs) to transform technical artifacts into accessible privacy reports. We share initial insights from two industry use cases, evaluate the quality of the generated reports, and outline future research directions toward integrating privacy transparency into RE processes for human-centric industrial systems.

Extremely low-mass white dwarfs (ELM WDs) are helium (He) WDs with masses below $\sim 0.3\ M_{\odot}$, mainly formed through binary interaction. ELM WD binaries typically are formed from two channels, namely the stable Roche lobe overflow (RLOF) channel and the common envelope ejection channel. For ELM WD binaries produced from RLOF channel, the ELM WD mass has a strong correlation with the orbital period, i.e., the so-called WD mass-orbital period relation. However, the observations in the ELM Survey show that the orbital periods of ELM WD binaries from the RLOF channel are typically shorter than the theoretically predicted values. Extra angular momentum loss (AML) may be needed to explain such a phenomenon. In this work, we assumed that part of the transferred mass from the donor is lost at the outer Lagrangian point and simulated the formation of ELM WD binaries. Enhanced AML enables more mass to be lost during thermal-timescale mass transfer, thereby affecting nuclear burning in the transfer phase and producing ELM WDs with distinct internal structures. These structural differences alter the (pre-)He WD mass-radius relation at the end of mass transfer, which in turn shifts the WD mass-orbital period relation downward. These adjustments enable our model to successfully reproduce the majority of observed systems from the relevant survey projects.

We perform a comprehensive analysis of lunar impact flash (LIF) light curve shapes and their dependence on the lunar terrain, using the large sample of LIFs detected by NELIOTA over the last 9 years. We classified 124 multi-frame light curves into mare, highland and `border' regions. Subsequently, we derived analytical expressions for single-size and dual-size ejecta cooling models, which were fitted to the observational data to estimate their physical properties. While impacts on both terrains yield similar peak magnitude distributions, their decay behaviour differs significantly; highland LIFs exhibit a shallower and longer-lasting decay compared to mare flashes, which are faster and steeper. The dual-size model suggests this extended duration is primarily driven by the fine droplets of the ejecta. The profile and duration of the LIF light curves represent the initial stages of the impact cratering process. The observed dichotomy between highland and mare LIFs demonstrates that the initial stages of the impact cratering process are fundamentally dependent on lunar lithology.

Quantum spin Hall (QSH) insulators and Mott insulators are conventionally regarded as distinct insulating phases, arising from band topology and strong Coulomb interactions, respectively. Here, we report the observation of QSH edge transport in a magnetic-field-stabilized Mott insulating state at half filling of the second moire band in a 2.29 degree twisted WSe2 device. This state exhibits a resistance plateau identical to that of the single-particle QSH state at full filling of the first moire valence band, indicating the same number of helical edge channels. Electrical transport measurements reveal nearly quantized resistance that is insensitive to vertical electric field, out-of-plane magnetic field, and temperature below 5 K. Pronounced nonlocal transport and strong negative in-plane magnetoconductance further support helical edge conduction, establishing robust edge transport in the strongly correlated regime. Temperature-dependent Hall measurements reveal a characteristic temperature scale of approximately 10 K, corresponding to an energy scale of about 1 meV. Our results demonstrate that spin-conserved QSH edge states can persist in a half filled, strongly correlated insulating phase and under external magnetic field, opening a route toward interaction-resilient topological transport in moire quantum materials.

Clinical prediction models provide predictions for individuals, typically expressed as point estimates derived from a deterministic function, such as a logistic regression equation. Such 'plug-in' predictions hide inherent uncertainty. In contrast, Bayesian methods offer a coherent mechanism for uncertainty propagation, and allow the computation of the posterior mean as the measure of centrality of choice for clinical decision-making. However, Bayesian methods are not widely utilised in predictive analytics for healthcare. We investigated the feasibility and performance of a Bayesian adaptation of the commonly used frequentist framework for risk prediction modelling. We assessed (i) the use of shrinkage priors with complementary features (simplicity, user input, and automatic shrinkage) that enable Laplace/normal approximation of the posterior, and (ii) exact and approximate methods for efficient computation of the posterior mean. Using examples and simulations, we demonstrate that this Bayesian approach is feasible and improves predictive performance, while enabling uncertainty quantification with suitable coverage. In small-to-medium sample sizes, the gain in clinical utility by using the posterior mean over plug-in predictions was equivalent to the gain from using a noticeably larger sample size. Adapting the widely used parametric regression methods to an approximate Bayesian framework for prediction modelling is both pragmatic and clinically advantageous.

We construct solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present examples arising from optimal transport where the potential of optimal transport maps satisfy Monge-Ampère equations similar to the ones we study.

We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel $K_q(x,y) = -|x-y|^q$, $q \in (0,2)$. The resulting discrepancy is the classical \emph{energy distance} $$\mathcal E_q^2(μ, ω) = -\frac{1}{2}\iint_{\mathbb{R}^d \times \mathbb{R}^d} |x-y|^q \, d(μ- ω)(x)\, d(μ- ω)(y),$$ and we ask how fast the best $N$-point empirical approximation $\inf_{μ_N \in \mathcal{P}^N}\mathcal{E}_q(μ_N,ω)$ decays as $N \to \infty$. Given a probability measure $ω$ on $\mathbb{R}^d$ with compact support satisfying an Ahlfors regularity condition of exponent $β\in (0,d]$, we prove that the sharp two-sided bound $$\mathcal E_q(μ_N, ω) \asymp N^{-\frac{1}{2}\left(1 + \frac{q}β\right)}$$ holds both for the worst-case empirical measure $μ_N$ (lower bound, holding for every configuration of $N$ points) and for an optimally chosen empirical measure $μ_N$ (upper bound). This complements the qualitative consistency result of Fornasier and Hütter \cite{fornasier2014consistency}, who proved narrow convergence of the minimizers of $\mathcal E_q^2(\cdot, ω)$ over empirical measures without quantitative rates.

The elasticity of PDMS has played a central role in advancing important microfluidic technologies, ranging from early valves to sophisticated organ-on-a-chip systems. However, most deformable microfluidic devices are based on geometries that require complex multi-layer PDMS architectures and include thin membranes, leading to difficult microfabrication and poor stability. Recently, Jain, Belkadi et al. (Biofabrication 16.3 (2024): 035010) introduced a single-layer PDMS device in which a wide and long microfluidic channel was deformed by pressurizing two adjacent air chambers. While they demonstrated how the channel ceiling deformation can be leveraged to compress biological materials, it remains unknown how the device geometry influences this deformation. Here, a systematic numerical study is performed on 14,336 variants of this device, through which the height of the PDMS layer is identified as the main feature that determines the ceiling deformation. Three modes of channel deformation are identified as the geometry are varied: a U shape with a central minimum, a W shape with two minima and a central maximum, or an inverse U shape with an upward-bulging single maximum. The numerical results are validated in experiments that reproduce the three modes for the predicted geometries and demonstrate vertical ceiling deformations ranging from a few microns to the millimeter scale. The generality of this approach is demonstrated for two example applications: A fully closing single-layer microfluidic valve and an optical lens of controllable anisotropic magnification. This work leverages the rapid prototyping enabled by 3D printing or micro-milling to open new perspectives in microfluidic actuation.

We develop a class of data-adaptive shrinkage estimators for high-dimensional covariance estimation in which the shrinkage target is a Reynolds projection of the sample covariance under a finite symmetry group selected from a candidate library by held-out predictive performance. The class generalizes the convex shrinkage estimator of Ledoit and Wolf by replacing the scalar-identity target with a structured target derived from a symmetry group when one is available, and generalizes the group-symmetric maximum-likelihood estimator of Shah and Chandrasekaran by combining structural targeting with adaptive convex shrinkage and by selecting the group from data rather than treating it as prespecified. A two-tier procedure performs the group selection: a universal per-candidate evaluation based on held-out negative log-likelihood, optionally preceded by a domain-specific step that constructs the candidate library from structural priors. We establish a finite-sample regret bound for the held-out calibration of the convex combination weight, an oracle inequality for the data-driven group selection, and a quantitative sufficient-match condition under which the proposed estimator dominates Ledoit-Wolf shrinkage in Frobenius mean-squared error. The procedure is illustrated on six real-data problems spanning finance (S&P~500 daily returns), climate (NOAA OISST sea-surface temperature anomalies), genomics (TCGA-BRCA gene expression), radio signal processing (RadioML 2018.A), astronomical imaging (Galaxy10 DECaLS), and natural image patches (CIFAR-10 with a CIFAR-10.1 distribution-shift companion). An empirical comparison is also made against the Bayesian permutation-symmetry estimator of Chojecki and colleagues. Outside the few-shot regime, where structural priors carry the most information per observation, Ledoit-Wolf shrinkage remains the appropriate baseline.

Frequency combs enable precision measurements across timekeeping, spectroscopy, ranging and astronomy, and are now extending to integrated and field-deployable platforms. Realizing their full performance demands a comprehensive account of the quantum noise that arises when broadband optical fields are converted into finite-bandwidth electrical signals. Here we present a rigorous first-principles quantum-mechanical framework for optical frequency-comb metrology based on continuous-mode field quantization and a comb-line-resolved description of quantum fluctuations. The theory describes how quantum fluctuations of the comb field are transduced into electrical measurement noise. Formulated at the level of the comb field, the framework unifies the standard quantum limits of optical frequency division (OFD) and dual-comb spectroscopy (DCS) within a single treatment, and provides a general recipe for other comb-based measurements. On this footing, we identify practical, resource-efficient routes to quantum enhancement through engineered comb states, laying a foundation for the design of next-generation frequency combs operating at and beyond standard quantum limits.

It has been shown that active learning methods are more effective than traditional lecturing at improving student conceptual understanding and reducing failure rates in undergraduate physics courses. Researchers have developed distinct, active learning methods that are now widely implemented in introductory physics. However, the relative benefits of these methods remain unknown. Here we present a multi-institutional comparison of the impacts of four well-established active learning methods -- Peer Instruction, Investigative Science Learning Environment (ISLE), Tutorials and Student-Centered Active Learning Environment with Upside-Down Pedagogies (SCALE-UP) -- on conceptual learning. We find measurable increases in student conceptual learning in all four active learning methods, and significantly larger gains in SCALE-UP than in either Peer Instruction or ISLE. Student development of peer networks is similar across the four methods, but classroom activities differ. In many of the observed Peer Instruction and ISLE courses, instructors lecture for a large fraction of class time. In Tutorials and SCALE-UP courses, instructors dedicate most in-class time to student-centred activities such as worksheets and laboratory work. These results prompt future work to identify causal mechanisms between specific classroom activities and conceptual learning and to examine additional factors related to variation in student learning across different methods.

We derive the exact quantum maximum over all finite-dimensional quantum realizations of the $n$-cycle overlap inequalities, $S_n^{\max}=n\cos^2(π/(2n))-1$, valid for arbitrary cycle length $n\ge3$. The bound is saturated by an explicit family of coplanar qubit states equally spaced along a Fubini--Study geodesic of length $(n-1)π/(2n)$, establishing dimensional saturation of the overlap-cycle hierarchy. Thus, the global optimum over all finite-dimensional quantum realizations is already achieved in dimension two. For three paths, coherence-free models satisfy $\mathcal{V}_{12}^2+\mathcal{V}_{23}^2-\mathcal{V}_{13}^2\le1$, whereas quantum theory allows the larger value $5/4$. Within generalized noncontextuality frameworks, violations witness preparation contextuality. We further derive explicit visibility thresholds for arbitrary cycle length, identifying interference visibility as an operational probe of overlap-based nonclassicality, and discuss feasible photonic implementations.

It is demonstrated how a set of particle representations, familiar from the Standard Model, collectively form a superalgebra. Those representations mirroring the internal behaviour of the Standard Model's gauge bosons, and three generations of fermions, are each included in this algebra, with exception only to those irreps involving the top quark. This superalgebra is isomorphic to the Euclidean Jordan algebra of $16\times 16$ hermitian matrices, $H_{16}(\mathbb{C}),$ and is generated by division algebras. The division algebraic substructure enables a natural factorization between internal and spacetime symmetries. It also allows for the definition of a $\mathbb{Z}_2^5$ grading on the algebra. Those internal symmetries respecting this substructure are found to be $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y,$ in addition to four iterations of $\mathfrak{u}(1)$. For spatial symmetries, one finds multiple copies of $\mathfrak{so}(3)$. Given its Jordan algebraic foundation, and its apparent non-relativistic character, the model may supply a bridge between particle physics and quantum computing. We close by describing current research directions. These include (1) detailing how this construction fits into the larger picture of Bott Periodic Particle Physics, first introduced in [1], [2], [3], (2) investigating how the origin of these particle representations might be grounded in the unsung algebra $\mathbb{O} \oplus \mathbb{H} \oplus \mathbb{C} \oplus \mathbb{R}$, and (3) proposing how these two directions may merge by reframing the model in terms of the 16$\hspace{.5mm}\mathbb{R}$ dimensional sedenion algebra, $\mathbb{S}.$

A tight correlation between nuclear millimeter and X-ray emission has recently been found in nearby ($z < 0.01$) and low-Eddington ratio ($\rm λ_{Edd} < 0.1$) radio-quiet Active Galactic Nuclei (AGN), suggesting a common origin in the hot X-ray corona. We test this relation in nine more distant RQ AGN ($z \sim 0.06-0.16$) with higher bolometric luminosities ($\log(L_{\rm bol}/\mathrm{erg\,s^{-1}})=45.3-46.3$), Eddington ratios ($\rm λ_{Edd} = 0.19-0.85$), and X-ray bolometric corrections ($κ_{2-10}=29-194$), selected from the Burst Alert Telescope (BAT) survey. We obtained quasi-simultaneous observations with Swift at 2-10 keV and the Atacama Large Millimeter/submillimeter Array (ALMA) at 100 GHz and with high angular resolution ($<0.14$"). We find that these high-luminosity AGN lie above the millimeter/X-ray correlation defined by lower-luminosity sources. A joint fit to both samples yields a second-degree polynomial with an intrinsic scatter of 0.32 dex. Furthermore, the millimeter emission correlates linearly with both the UV disk luminosity and $L_{\rm bol}$, with intrinsic scatters of 0.45 and 0.35 dex, respectively. We propose that the deviation from the linear millimeter/X-ray relation arises from a two-component coronal electron population: thermal electrons that produce X-rays, but become less efficient at higher luminosities, and nonthermal electrons that produce millimeter emission and remain tied to $L_{\rm bol}$. Additional millimeter emission from outflow-driven shocks may also contribute, though spectral energy distribution modeling and spectral index studies favor a coronal origin.

We hydrodynamically simulate a core-collapse supernova (CCSN) explosion by launching three pairs of jets in the framework of the jittering-jets explosion mechanism (JJEM), and reproduce a morphology of two opposite circum-jet rings and a bar of dense gas perpendicular to the rings' axis, resembling these morphological features in the CCSN remnant SNR G11.2-0.3. The first pair of wide jets is very energetic; it triggers the explosion and inflates two bubbles that compress the material in an expanding shell. The bubbles also compress material in a plane perpendicular to the jet axis. The second pair of wide jets removes material from this plane, beside along a bar that is on an axis perpendicular to the two pairs' axes. The jets of the third pair, now of narrow jets, penetrate the expanding shell and compress material to their sides to form two opposite rings. These morphological features are qualitatively similar to those observed in the point-symmetric CCSN remnant G11.2-0.3. As competing theoretical CCSN explosion mechanisms cannot explain point-symmetric CCSN remnants, our study provides some support for the claim that the JJEM is the primary explosion mechanism of CCSNe.

Monitoring humans, for example, their movement or location, is essential for safe and efficient human-machine collaboration in Cyber-Physical Systems (CPS). This information allows CPS to ensure safety properties, adapt their behaviour dynamically, and coordinate with humans. To ensure that the design of a CPS respects ethical principles and the privacy of its stakeholders, system requirements, particularly those related to human monitoring, must reflect the human values of all involved stakeholders. However, human values are often underrepresented in Software Engineering -- particularly during requirements elicitation and system design, crucial phases when introducing ethically critical functionality. Stakeholder values are often implicit and conflicting, yet rarely systematically captured. Furthermore, unstructured natural language requirements introduce ambiguity and vagueness, complicating conflict resolution. To address these problems, we propose HM-Req, a requirements elicitation framework including a Controlled Natural Language (CNL) for defining human monitoring requirements. These requirements are then augmented with human values from relevant stakeholders and integrated into a Value Dashboard to detect potential conflicts that require further discussion and resolution. Validation results, applying the CNL to different datasets and conducting a survey and expert interview, provide evidence of the CNL's ability to capture diverse human monitoring requirements and demonstrate HM-Req's usefulness for requirements elicitation activities.

We prove that the theory of abelian groups and R-modules even in infinitary logic is stable and understood to some extent.

Regression to the Mean (RTM) and Regression Dilution are traditionally treated as unrelated issues in the clinical and ecological literatures. In this work, we demonstrate that within a linear errors-in-variables framework where baseline variables are subject to transient temporal or measurement noise, these two phenomena share an identical underlying mathematical signature. We unify these disparate traditions by comparing specialized clinical tools, such as the Berry shrinkage correction, with standard sign-agnostic structural estimators like Major Axis (MA) and Reduced Major Axis (RMA) regression. Using an analytical framework, we evaluate the closed-form population limits and finite-sample performance of these methods across various noise-to-signal ratios and sample sizes. Our results show that the Berry method is a specialized tool designed for clinical scenarios where a 1:1 relationship is expected. However, applying it to ecological trade-offs with negative slopes can lead to severe errors. We provide maps of optimality to identify which estimator most accurately recovers the true biological signal under different conditions. By reconciling these disparate methods, we offer a principled guide for researchers to choose the correct tool based on their data's noise profile rather than their disciplinary tradition.

We explore how the integrated use of quantum spatial distribution (QSD), or more specifically, a superposition of both spin and position states of particles, and gauge symmetry (GS) within Poulin's stabilizer formalism contributes to quantum error correction. The exploration employs $3+2$ particles on nested squares, where three of them encode Shor's nine-qubit code and the remaining two detect errors in this code through their spin state measurements. The first result is that the GS offers resilience against three types of noise acting on a particle: arbitrary decoherence of its spin or position state, and dephasing of both states, which partly or completely destroys its QSD. To show that, we formulate a noise model unifying the above noise sources and prove the correctability of this unified model under our error-correcting scheme. The second result is that the QSD provides architectural flexibility, allowing us to stack the error-correcting systems both vertically and horizontally. Indeed, we show implementations of the error detection (stabilizer measurement), logical Hadamard and Toffoli gates, and a quantum adder with the required interactions only between nearest-neighbor and next-nearest-neighbor particles. Here, our treatment of the dynamics of particles, each having spin and position degrees of freedom, under nontrivial noise and gate operations suggests that the stabilizer formalism would be useful for describing quantum many-body dynamics.

Brzeziński's trusses are ``ring-like'' algebraic structures in which the addition is replaced with an abelian heap operation and the binary product satisfies a natural distributivity rule of the ternary product. The question of how to define ($\mathbb{Z}_2$-graded) super-versions of trusses is addressed in this note. Taking our cue from the theory of algebraic supergroups, we define an affine supertruss as a representable functor from the category of unital associative supercommutative superalgebras to the category of trusses. The representing superalgebras are equipped with a `cotruss' structure--a new concept in itself. We show that from an affine supertruss one can construct an affine superbrace, and so generalise Rump's braces to supermathematics. As an application of these constructions, we propose a generalisation of the set-theoretic Yang--Baxter equation to the setting of affine superschemes.

While periodic $\mathcal{PT}$-symmetric structures offer a versatile platform for wave tailoring, their scattering responses are typically analyzed using stationary methods that presume dynamical stability. This assumption fails when time-growing bound states emerge, signaling a dynamical instability. Here, we analytically derive the instability threshold for a $\mathcal{PT}$-symmetric chain of $N$ unit cells with gain/loss strength $γ$. Our $S$-matrix analysis yields a closed-form threshold, $γ_c = 2\sin[π/(4N)]$, which scales as $\mathcal{O}(1/N)$ and vanishes in the thermodynamic limit. Consequently, enlarging such structures to access richer stationary band phenomena paradoxically triggers instability at weaker gain/loss. As confirmed by time-domain simulations, exceeding $γ_c$ causes exponentially growing bound states to overwhelm the system, rendering standard Bloch-wave descriptions physically irrelevant. Evaluated against this size-dependent threshold, many hallmarks of large $\mathcal{PT}$-symmetric structures, including gain-loss-induced localization, reflectionless transport, and coherent perfect absorbers and lasers, are found to lie within the dynamically unstable regime. Our findings thus establish that physical transport in non-Hermitian periodic systems is governed by a fundamental interplay between stationary band theory and finite-size stability limits.

We introduce the subsum polynomial of a partition $λ=(λ_1, λ_2, \ldots, λ_k)$ defined by $\mathrm{sp}(λ, x)=\prod_{i=1}^k(1+x^{λ_i})$. We study the sum of reciprocals of $\mathrm{sp}(λ, x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.

Realizing controlled, single-clock inflation in string theory is fundamentally obstructed by the backreaction of heavy moduli. We show that in the \emph{penumbra} -- the near-boundary regime of complex-structure moduli space where asymptotic symmetries are partially broken -- this obstruction can be exactly quantified. We derive a covariant control theorem demonstrating that local branch data dictate whether a monodromic valley supports a controlled inflationary plateau, thereby isolating the first controlled penumbral inflationary window. The result turns the penumbra from a geometric regime into a dynamical filter. In the axion-saxion effective theory, a branch-displacing odd term generates a plateau when $Δ\equiv p+2ν-q>0$, while covariant single-clock control further requires $p<2$, or $p=2$ with $12A_pm^2/(dV_0)\gg1$ over the observational window. This splits penumbral valleys into no plateau, uncontrolled plateau, and controlled plateau before global completion is attempted. We identify a minimal analytic family with a closed-form valley and an invariant attractor equation for the full two-field dynamics, providing the first exactly solvable penumbral realization that remains predictive under the next penumbral order. The controlled corridor targets $r\sim10^{-3}$ with the correlated running $α_s\simeq-r/2$ for the $d=q=1$ benchmark, providing a falsifiable target for LiteBIRD/CMB-S4.

Designing spacecraft trajectories remains challenging in the presence of stochastic effects such as maneuver execution errors and observation uncertainties. Although covariance control and belief-space planning provide useful tools for designing robust control policies and information-aware trajectories under uncertainty, practical methods remain limited for partially observable trajectory optimization problems in which trajectory design, orbit determination, and correction maneuver planning are tightly coupled. This paper presents a stochastic differential dynamic programming algorithm for such coupled problems. The proposed method optimizes the nominal control sequence and feedback gains subject to a belief-state transition model and general mission constraints, explicitly accounting for the dependence of covariance propagation on the nominal trajectory without relying on the separation principle. Numerical examples demonstrate that the proposed algorithm produces navigation-aware and uncertainty-robust solutions across a range of dynamical systems, observation models, and uncertainty levels.

To develop a full abstract denotational model of a process language based on prebisimulation preorder, its behavioural semantics has two problems: (1) Two processes related by a standard denotational interpretation afford the same finite observations. (2) Prebisimulation can make distinctions between the behaviours of two processes based on infinite observations. So, finitary part of prebisimulation is needed to obtain full abstract results. There existed two main results on finitary bisimulation: the logical form and the behavioural form. Following the latter one, we give the definitions of truly concurrent prebisimulations and their finitary ones.