We announce an explicit description of the strictly semistable boundary of the GIT moduli space of quintic threefolds. For the natural action of \(\mathrm{SL}(5)\) on \(\mathbb P(\mathrm{Sym}^5\mathbb C^5)\), we classify the 38 boundary components arising from maximal strictly semistable supports and construct closed-orbit normal forms for the general polystable representative in each component. We also determine the singular loci of these general representatives and compute their local and global minimal exponents. The isolated boundary singularities are quasi-homogeneous and fall into eleven analytic types, all with local minimal exponent equal to \(1\). Consequently, the global minimal exponent of a general closed-orbit representative in every boundary component is \(1=(4+1)/5\), the critical value in the stability criterion for quintic hypersurfaces in \(\mathbb P^4\). We further announce the pairwise non-inclusion of the 38 quotient-side boundary families and compute the codimension-one wall-adjacency graph, which has 38 vertices and 184 edges, is connected, and has diameter 4. Detailed proofs and complete case-by-case computations will appear in a forthcoming full-length paper.

We study the role of numerical quadrature in residual-minimisation methods for neural network approximation of partial differential equations. We first present an abstract error framework that separates approximation, quadrature and optimisation errors, and derive a nonlinear Strang-type estimate quantifying how inaccuracies in the discrete loss affect the final approximation. Motivated by this analysis, we propose an anisotropic adaptive composite quadrature strategy that controls the relative quadrature error of the residual loss using richer reference quadratures and bisection-based refinement. We then introduce a refresh-based training methodology that rebuilds the quadrature only when an online error indicator exceeds a prescribed threshold, balancing accuracy and computational cost. Numerical experiments on a range of benchmark problems show that the proposed approach narrows the gap between training and reference losses, uses quadrature points more efficiently and delivers strong approximation accuracy relative to non-adaptive quadrature strategies.

This paper pioneers a novel scheme for artificial-noise (AN)-aided movable-antenna (MA)-enabled physical-layer service integration (PLSI) to harmonize the simultaneous delivery of multicast and confidential messages. By jointly exploiting the spatial reconfiguration capability of MAs and the interference shaping capability of AN, we aim to enhance secrecy performance while guaranteeing multicast reliability. The joint design of MA positions and transmit variables results in a highly coupled and non-convex optimization problem. To address this, we first provide key insights into the role of spatial degrees of freedom in AN design. We then characterize the AN direction under a structured transmission design and derive a closed-form expression for the AN-to-confidential power allocation ratio, which significantly simplifies the overall design. To solve the resulting problem, we further develop a low-complexity block coordinate ascent (BCA)-based scheme that alternates between transmit design and MA position optimization. Numerical results demonstrate that the proposed scheme achieves significant secrecy performance gains with low computational complexity and fast convergence, highlighting its effectiveness for MA-enabled PLSI systems.

In this paper, we prove Aubry's completeness stating conjecture that for a twist map the graph of rotation numbers as a function of the cohomology classes is a purely singularly continuous function (called complete devil's staircase by Aubry) when the set of all minimal configurations is uniformly hyperbolic. Such a phenomenon is crucial for characterizing the chain of atoms being an insulator for the Frenkel-Kontorova model, and can be considered as the analogue of the phase locking phenomenon in critical circle maps as well as the fractional quantum Hall effect. In contrast, in the presence of a positive measure set of KAM tori, we prove that the devil's staircase is incomplete.

Achieving long-term stability in high-capacity lithium-ion battery anodes remains a critical challenge. In this study, we present a materials-intrinsic strategy for extending the cycle life of Ge, a promising next-generation anode material, through trace doping with metal elements. We systematically investigated the effects of small additions of various metals and found that elements with large atomic size, particularly Yb, markedly improved the cycling stability without sacrificing the initial capacity, while appropriate Yb doping enhanced the anode lifetime by approximately a factor of three. Structural and electrochemical analyses revealed that this improvement originates from mechanical softening of the Ge anode, which suppresses lithiation-induced damage such as cracking and delamination. Nanoindentation measurements further showed a strong negative correlation between dopant atomic size and film hardness, establishing anode softening as a new design principle for damage-tolerant electrodes. Although Yb doping reduced the rate capability at high C-rates, the present results demonstrate a clear shift in design strategy from volume-change suppression to mechanical compliance. These findings provide a useful framework for stabilizing high-capacity alloy anodes through atomic-scale mechanical control.

Curtain coating, in which a moving plate is coated by a falling liquid sheet, sustains advancing contact lines at large capillary numbers Ca ~ O(1), based on plate speed. Steady states exist up to a critical capillary number, beyond which wetting failure occurs through air-bubble entrainment. In the steady regime, experiments report that velocity along the fluid-fluid interface accelerates as the contact line is approached, down to tens of micrometres; this has been interpreted as evidence against the Navier slip model. We ask whether this acceleration is compatible with slip models, and show that it is. Although Navier slip implies a vanishing velocity at the contact line, the experimentally accessible microscale region lies outside the slip region. The curtain-coating setup is revealing because the local Reynolds number, based on distance from the contact line r ~ 10 microns, is order unity, so the observable flow is governed by local inertia. Our two-phase Navier-Stokes Volume-of-Fluid simulations with quadtree adaptive mesh refinement resolve the smallest scales and study the flow with a Navier slip boundary condition and fixed contact angle. The simulations reproduce the non-monotonic dependence of the critical capillary number on global Reynolds number, based on feed-flow velocity, and the variation of the macroscopic contact angle at the inflection point, in agreement with Liu et al (2016). The interfacial velocity in the microscale region is well described by an inertially corrected wedge flow solution whose wedge angle is set by the inflection-point value, with agreement improving as slip length is reduced; at larger scales, interface bending follows the Benney solution. These inertial effects, absent from pure Stokes flow, are essential in the experimental region. Thus qualitative microscale observations do not decisively invalidate slip models for advancing contact lines.

A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erdős's seminal 1935 paper. As applications, we prove two 1966 conjectures of Erdős-Sárközy-Szemerédi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erdős Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erdős-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area.

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a Euclidean ball, provided $\frac 1π \arctan((\frac{3A}{4π})^{1/3})$ satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values $A$ satisfying these properties has positive Hausdorff dimension.

DHOL is an extensional, classical logic that equips the well-known higher-order logic (HOL) with dependent types. This allows for concise encodings of important domains like size-bounded data structures, category theory, or proof theory. Automation support is obtained by translating DHOL to HOL, for which powerful modern automated theorem provers are available. However, a critically missing feature of DHOL is polymorphism. We develop the syntax and semantics of polymorphic DHOL and extend the translation accordingly. We implement the translation in the logic-embedding tool and evaluate it on a range of TPTP formalizations. The logic-embedding tool, together with an off-the-shelf HOL theorem prover easily creates a PDHOL theorem prover for experimenting.

Block copolymers often create droplets when placed on a substrate. Such nanostructured droplets can be arranged into regular microstructured arrays, thereby forming hierarchically organized materials that can be used in microelectronics, plasmonics, sensing, photonics, metamaterials production, and even cryptography. However, it is unclear if such materials can be stimuli-responsive, i.e., be able to change their nanostructure on a single droplet level upon applying external stimuli. In this work, we discovered that small (10-100 nm) surface-adsorbed droplets of symmetric diblock copolymers can form a multitude of different externally switchable nanostructures. We obtained a near-equilibrium, comprehensive 4D diagram of droplet morphologies by performing large-scale self-consistent field theory (SCFT) calculations under various wetting and phase separation conditions. The SCFT modeling was augmented with a computational algorithm that established an equilibrium droplet morphology in a given system without assuming potentially equilibrium structures prior to simulation. The discovered droplet nanostructures agreed excellently with previously published experimental data. Crucially, we showed that direct and reversible transitions between different droplet morphologies are possible upon changing the interaction strength between components, which can be tuned externally in experiments by adding surfactants or controlling temperature. We confirmed experimental realizability of such stimuli-responsiveness by modeling surfactant addition that led to a switch between droplet nanostructures. This work demonstrates that even the simplest symmetric diblock copolymers are able to produce versatile and stimuli-responsive structures on a surface when confined to a small nanodroplet. This opens the possibility to produce smart coatings with externally switchable hierarchical micro- and nanostructures.

High-resolution spectroscopic observations of helium emission lines provide a powerful probe of accretion geometry in classical T Tauri stars, revealing regions not well traced by hydrogen lines. Parallel studies in the planetary-mass regime are lacking. In this work, we investigate helium emission from the nearby (47 pc), wide-orbit (~84 au), ~13 $M_{Jup}$ accreting circumbinary companion Delorme 1 (AB)b and use resolved line profiles to constrain their origin. We analyse 33 high-S/N VLT/UVES spectra spanning near-ultraviolet to optical wavelengths at R~50,000. We detect seven He I lines at >5$σ$ confidence - 3890, 4027, 4473, 4923, 5017, 5877, and 6680 Ȧ - with significant epoch-to-epoch variability. The He I 5877, 4923, 4473, and 4027 Ȧ lines are asymmetric, showing a narrow component near 0 km/s and a broad component redshifted by ~15 km/s. The accretion luminosity ($1.3^{+1.6}_{-0.7}\times 10^{-5} L_{\odot}$) and mass accretion rate ($0.7^{+0.9}_{-0.4} \times 10^{-8} M_{Jup} yr^{-1}$) inferred from the median He I line luminosities are broadly consistent with, but slightly higher than, estimates from the ultraviolet excess. We conclude that protoplanet Delorme 1 (AB)b shows asymmetric He I profiles analogous to those of classical T Tauri stars, but with much smaller narrow- and broad-component widths. The triplet-singlet line ratio, a strong correlation with ultraviolet excess and the near-zero, redshifted velocities obtained for the narrow component suggest that it originates within the post-shock region, close to the planet surface. The persistent redshift of the broad component, its line width, and velocity correlation with the narrow component imply an origin within the shock structure, closer to the shock front. Emission seems to be dominated by accretion based on the obtained accretion luminosities, but a contribution from chromospheric activity may be present.

In remote video meetings, visual non-verbal cues, such as facial expressions or head movements, are seen continuously but often only partially. This increases ambiguity compared to in-person settings and can cause misinterpretation or misalignment between intended and perceived meaning. Motivated by communication theories, we designed FaceValue, a technology probe that augments the self-view with private, real-time overlays. These overlays are subtle, suggestive prompts intended to help attendees reflect on how their cues might be interpreted by others. To invite personal interpretation, FaceValue avoids behavioral labeling and instead aims to support meaning-oriented self-awareness: recognizing when visible cues may unintentionally (mis)communicate intent. We deployed FaceValue in the wild with thirteen knowledge workers over multiple weeks, capturing perceived changes in self-awareness and behavior, and impressions on the design concepts, as self-reported by participants through diary entries and exit interviews. Participants felt FaceValue increased their awareness of potentially misaligned cues and motivated in-meeting adjustments, which they believe resulted in improved communication with other attendees. We contribute a conceptual framing that positions visual non-verbal cues as a manipulable communication resource, a technology probe that aims to foster meaning-oriented self-awareness, and empirically-grounded design insights for future meeting systems.

We present a protocol in which sequential weak measurements of a quantum harmonic oscillator enable simultaneous estimation of both quadratures of a displacement channel. Calculations of the quantum Fisher information show that the measurement backaction can increase the information gained for a range of measurement strengths. The protocol distributes information over a $N$-bit string after $N$ weak measurements. Thus we find that post-processing can be used to avoid information loss due to phase wrapping, increasing the effective dynamic range. Finally, the periodic information extraction makes the protocol robust to decoherence. Our results establish mid-sensing measurement as a resource for single- and multi-parameter quantum metrology.

Time-resolved x-ray diffraction (TR-XRD) and ultrafast electron diffraction (TR-UED) are emerging tools for probing ultrafast quantum dynamics. From a theoretical perspective, they are commonly described within different frameworks and modeled using distinct approximations. Here, we present a unified quantum-field-based description of ultrafast diffraction imaging that permits consistent consideration of TR-XRD and TR-UED within a common theoretical formalism. Our approach elucidates the correspondence between TR-XRD and TR-UED and allows their similarities and differences to be systematically disentangled. The developed formalism is sufficiently general to consistently and straightforwardly incorporate additional physical effects of interest, such as relativistic charge-current and current-current couplings. We apply our approach to simulate diffraction measurements of laser-driven electron dynamics in graphene, demonstrating the unique capabilities of diffraction imaging to unravel intricate quantum processes in matter.

We study holomorphic foliations on normal crossings varieties arising as semistable degenerations. We do so by we exploring the notion of foliated d-semistability using the language of logarithmic structures in the sense of Fontaine-Illusie. First, we identify both local and global obstructions to d-semistability. In order to analyze the existence of smoothings, we develop a logarithmic deformation theory of foliations and show that the corresponding moduli functor admits a versal hull.

Conformance checking, one of the main process mining operations, aims to identify discrepancies between a process model and an event log. The model represents the expected behaviour, whereas the event log represents the actual process behaviour as captured in information systems records. Traditionally, the process model and the event log are both accessible to the business analyst performing the conformance checking. However, in some contexts, it is necessary to keep either the model or the log private to protect critical or sensitive information. In this paper, we propose a secure approach to conformance checking based on string processing algorithms and homomorphic encryption, where the process model and event log ar not visible to either the model's or event log's owner. The proposed technique is based on alignments, a well-known formalism used for conformance checking. An evaluation is performed using a synthetic and a real-world event log, showing that conformance checking can be securely computed at the expense of high memory and processing requirements.

Early-stage concept envisioning is a critical juncture in AI design, shaping how designers frame problems and the decisions that follow. Yet values and potential harms are often too abstract or addressed too late to meaningfully shape design. Using a Research-through-Design (RtD) approach, we developed the AI Concept Envisioning Toolkit, comprising an AI Capability Library, 24 Value--Harm Cards, and a Value--Tension Map, to support reasoning by juxtaposing values and harms within AI technical capabilities. Through a survey with 30 designers and in-depth interviews with 12 designers, we find that the toolkit is clear and perceived as valuable, and that it encourages value reflection, helps anticipate potential harms, and makes ethical considerations more transparent in early-stage design. We reflect on our design process and discuss design approaches for tools that promote reflection on values and potential harms, surface and navigate value tensions, and introduce productive friction throughout design workflows.

As AI integrates into design practice, designers increasingly use generative AI tools to envision AI-enabled solutions, positioning AI as both design tool and design material. This dual role creates recursive value tensions distinct from traditional design work. We engaged 18 designers in a concept envisioning activity and interviews to understand how they navigate values and recognize potential harms in this context. Our analysis reveals that (i) designers engage in reciprocal reflection-in-action with AI; (ii) this process surfaces multi-level value tensions across tool, designer, and concept; (iii) designers demonstrate greater attunement to harm recognition as a primary design signal than to articulating positive value fulfillment; and (iv) designers exercise anticipatory judgment through meta-design reasoning about how tool assumptions risk propagating into designed concepts and future use contexts. We extend Schon's reflection-in-action framework and discuss implications for redesigning AI-mediated design tools, supporting harm-centered reasoning, and positioning design as foundational to AI development.

We consider the problem of finding the value of a maximum flow over time in a network with uniform edge lengths where the edge capacities change at specific time instants. To solve this problem, we show how to construct a condensed version of a Time Expanded Network (cTEN) whose standard max flow value is the same as the max flow over time on the original network. In particular, for a graph with $n$ nodes, $m$ edges, and $μ$ {\em critical times} where some edge capacity changes, we obtain a cTEN with $O(n^2μ)$ nodes and $O(μmn)$ edges. This implies that the problem can be solved in $O(μ^2n^3m)$ time using the combinatorial max flow algorithm of Orlin [Orl13], or in $O(μ^{(1+o(1))}(nm)^{1+o(1)}\log (UT))$ time using the algorithm of Chen et al. [CKL+22], where $U$ is the maximum capacity of any edge and $T$ is the time horizon. We focus on graphs that experience many time changes across the period of interest, as in such graphs the $μ$ term dominates the runtime.

Background: Conversational AI chatbots are emerging as scalable mental health tools, but little is known about real world engagement or its relationship to clinical outcomes. Objective: To characterize engagement phenotypes among users of Ash, a purpose-built AI mental health chatbot, and examine associations with clinical change and working alliance. Methods: K-means clustering across eight behavioral features identified engagement phenotypes among 102,684 users. Subsamples completed the PHQ-9 (n=298), GAD-7 (n=298), and MSPSS (social support; n=194) baseline and 3 weeks; 11,437 users completed baseline Working Alliance Inventory (WAI). Results: Five engagement phenotypes emerged: Early Dropouts (52.2%), Power Users (1.6%), Intensive Users (4.1%), Weekly Users (25.3%), and a novel Concentrated User pattern (16.8%); across users, 66.9% had at least one overnight session (9pm-5am). Significant pre-post improvements occurred in depression (d = -0.51), anxiety (d = -0.57), and social support (d = 0.22). An observed dose-response gradient in self-reported depression improvement was replicated in a larger sample with model-predicted PHQ-9 (n = 23,813; Power Users d = -0.54; Early Dropouts d = -0.13). Higher working alliance predicted depression improvement and moderated the engagement-social support relationship. Conclusions: Engagement with AI mental health tools is multidimensional, and different clinical outcomes respond to different dimensions of use. Findings caution against treating session counts as a primary engagement metric and offer naturalistic evidence for the clinical value of purpose-built conversational AI.

Landau theory of superfluidity associates low-temperature flow of the normal component with the phonon wind. This picture does not apply to superfluids in which Galilean invariance is broken either by disorder, porous media, or lattice potential, and the phonon wind is no longer solely responsible for depletion of the superfluid component. Based on Popov's hydrodynamic action with anharmonic terms, we present a general theory for low-temperature ($T$) dependence of the superfluid stiffness, which reproduces Landau result as a special case when several parameters of the hydrodynamic action are fixed by Galilean invariance, and validate it with numerical simulations of interacting lattice bosons. In a broader context, our approach reveals universal low-temperature thermodynamics of superfluids with an intrinsic connection between finite-$T$ and finite-size ($L$) effects implying universal scaling, $T^{d+1}$ and $1/L^{d+1}$, respectively, for a large class of thermodynamic quantities. We discuss the experimental detection of this law, and compare our prediction to the existing literature.

We introduce LNODE, a mechanism-based phenomenological model for amyloid beta (A$β$) dynamics, calibrated using positron emission tomography (PET) imaging. A$β$ is a key biomarker of Alzheimer's disease. LNODE is designed to support the fusion, harmonization, quantitative analysis, and interpretation of Abeta PET scans. We evaluate LNODE on 1461 subjects in the ADNI cohort and 1070 subjects in the A4 Study, using MUSE and DKT anatomical atlases. LNODE is formulated as a regional neural ordinary differential equation (ODE) model that is jointly calibrated on all available scans within a cohort. The model captures the spatial propagation, proliferation, and clearance of A$β$ and incorporates a latent-state representation that modulates A$β$ dynamics. The temporal evolution of these latent states is governed by cohort-shared parameters, enabling LNODE to represent both population-level trajectories and subject-specific deviations. The proposed model demonstrates strong parameter identifiability and stability properties, supported by synthetic experiments and analytical analysis of the Hessian condition number. To mitigate overfitting and reduce spurious correlations, LNODE is intentionally underparameterized, employing approximately five to ten parameters per subject. Despite this parsimonious parameterization, LNODE achieves $R^2 > 0.99$ in both the ADNI and A4 datasets. LNODE exhibits strong predictive performance: in the A4 cohort, it accurately forecasts the A$β$ PET signal in previously unseen follow-up scans, including cases with inter-scan intervals exceeding four years. Clustering in the learned latent-state space reveals distinct subgroups, consistent with the existence of different subtypes of Alzheimer's disease progression.

In much discussed work Artemov has recently shown that, for $\mathrm{PA}$, the consistency schema admits a form of uniform verification via selector proofs, despite the unprovability of the corresponding uniform consistency sentence $\mathrm{Con}(\mathrm{PA})$. In this note, we show that this phenomenon extends to all sufficiently strong arithmetizable theories. For such theories $T$, there exists a primitive recursive selector producing proofs of all instances of the associated consistency schema. This yields a form of computational uniformity, despite the fact that it cannot be internalized as the uniform consistency sentence of Gödel's Second Incompleteness Theorem. We analyze this gap and locate selector proofs within the broader framework of provability and reflection.

The goal of this work is to continue the study the smoothings of 3-dimensional manifolds with singularities obtained as small covers of non simple right-angle Coxeter polyhedral orbifolds. They appear in the study of coaxial intersections of ellipsoids. In particular we introduce the concept of $n$-pyramitoid generalizing the $n$-pyramid.

We have examined the stellar structure of 10 nearby, low stellar mass (10^6 to 6 x 10^7 Msolar) dwarf irregular galaxies by fitting ellipses as a function of surface brightness on ultra-deep V images. These are compared to far ultraviolet images as tracers of the star formation. We find that the often asymmetrical distribution of large patches of star formation activity in dwarfs, even out to low disk surface brightness levels, skews the broad-band optical isophotes in these galaxies. We also looked for evidence of the presence of a stellar halo. Possible hints of such are found in several galaxies from irregularities in the ellipses, but a stack of seven of the galaxies shows a pure exponential out to a V surface brightness of 32.3 mag/arcsec^2 where the stellar surface density is 0.0013 +/- 0.0011 Msolar/pc^2. The extended stellar component, most likely a disk structure, is probably due to internal evolutionary processes rather than external accretion. The UBVI colors of the annuli are consistent with ages of 1-6 Gyr for the far outer stellar disk.

In this paper, we consider remote reconstruction over wireless networks when simultaneous accuracy at the legitimate receiver and confidentiality against eavesdropping are required. These two objectives are often treated separately, even though they arise from the same update process and are marginals of a joint reconstruction event. This paper introduces confidential reconstruction accuracy (CRA), a metric to capture the joint event in which the legitimate receiver reconstructs correctly while the eavesdropper fails. Under randomized stationary policies, we develop a three-dimensional stationary analysis and derive closed-form expressions for the long-term average CRA and the optimal transmission probability. The results show that conventional marginal analysis can misidentify the optimal policy and misestimate the achievable simultaneous accuracy-confidentiality performance. They also reveal nontrivial behaviors: more frequent transmissions or better legitimate channels do not necessarily improve joint accurate and confidential reconstruction, and when the eavesdropping channel is strong, improving the legitimate channel alone may be insufficient. Finally, the framework induces the spatial safety boundary in a geofencing setting for secure remote reconstruction.

Orthogonal frequency-division multiplexing (OFDM) and its static sinusoidal subcarriers have underpinned the 4G and 5G eras, delivering high spectral efficiency and resilience to multipath fading through an efficient multicarrier architecture. However, as future systems move toward doubly dispersive environments driven by high-mobility applications and migration to mmWave/sub-THz bands, the time-invariance assumption underlying OFDM becomes increasingly difficult to maintain, and Doppler-induced degradation becomes prominent. While enhancements such as MIMO, advanced coding, and scheduling provide incremental remedies, they introduce additional overhead, because the sinusoidal subcarrier itself offers no inherent waveform-level robustness to Doppler impairments. Accordingly, two time-frequency spreading philosophies have emerged to improve Doppler resilience by distributing each symbol's energy across both dimensions of the time-frequency plane: (i) 2D isotropic spreading via the delay-Doppler (DD) domain, exemplified by the orthogonal time frequency space (OTFS) family, and (ii) sheared spreading via parameterizable chirps, exemplified by the affine frequency-division multiplexing (AFDM) family. In this article, we examine key considerations for future waveform design across these paradigms and argue that transitioning from the sinusoidal subcarriers of OFDM to the chirp-based subcarriers offers a viable design direction for improving Doppler robustness while retaining much of the mature OFDM infrastructure. This perspective also highlights the suitability of chirp-based waveforms for integrated sensing and communications (ISAC) and their extensibility to emerging physical-layer techniques. Overall, we argue that the transition from sinusoids to chirps is a technically motivated, compelling evolutionary direction for future wireless physical layer design.

Many quantum-network applications require end-to-end Bell pairs whose fidelity exceeds a request-specific threshold, but existing entanglement routing algorithms either optimize only throughput without regard for fidelity or enforce fidelity guarantees using centralized controllers with global link-state knowledge. We present Q-GUARD, an online entanglement routing algorithm that enforces per-request fidelity thresholds within a distributed protocol model in which nodes exchange link-state information only with their $k$-hop neighbors. After link outcomes are realized in each slot, Q-GUARD builds per-link purification cost tables from realized Bell pairs, allocates per-hop fidelity targets using a Werner-state equal-split rule, and selects between candidate path segments using a segment-local expected-goodput (EXG) metric that jointly accounts for swap success, purification overhead, and resource availability. We also introduce Q-GUARD-WS, an extension that exploits per-link hardware quality estimates to allocate purification effort non-uniformly across hops. On synthetic 100-node topologies with heterogeneous link fidelity and stochastic BBPSSW purification, Q-GUARD raises the qualified success rate from under 20\% to over 85\% on 4-hop paths and nearly doubles the qualified service radius in Euclidean distance relative to throughput-only and naive-purification baselines, while Q-GUARD-WS provides additional throughput gains under high hardware heterogeneity.

Interactions between inertia-gravity waves and balanced flows lead to a spectral diffusion of wave action. Prior work has established that this diffusion is weak across constant frequency surfaces in three-dimensional settings, but can be significant in two dimensions with a non-stationary balanced flow. We investigate the two-dimensional setting through numerical simulations that simultaneously evolve a turbulent quasigeostrophic balanced flow and advect rotating shallow water wave packets. In contrast to earlier predictions based on the synthetic flows used by Dong et al. (J. Fluid Mech., 2020, vol. 905, R3), we find that frequency spreading from wave mean-flow interactions is weaker for realistic turbulent flows. We derive a timescale for frequency diffusion and show that frequency spreading with a realistic background flow is an order of magnitude smaller than with the synthetic flow. We narrow the discrepancy between the two- and three-dimensional induced diffusion theories, which suggests other mechanisms are responsible for the broadband frequency spectra seen in the atmosphere and ocean.

This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.