We study biased Maker-Breaker games on a graph system $\{G_1,\ldots,G_s\}$, in which Maker's goal is to claim certain rainbow structures, i.e., specified subgraphs consisting of at most one edge from each graph $G_i$. We consider the rainbow-connectivity game, in which Maker wants to claim a rainbow path between every pair of vertices. We analyse this game in detail, essentially determining the threshold bias when played on the system of complete graphs, and observing that whether the random graph intuition holds depends on the size of $s$. The key ingredient of our result is the analysis of a Maker's strategy that combines several randomized strategies with an appropriately designed balancing game. As a byproduct, we find the order of the threshold bias for the Maker-Breaker diameter game, and disprove a conjecture by Balogh, Martin and Pluhár. Another natural and general way to analyse Maker-Breaker games that are played on a colored board is to require Maker to occupy a rainbow winning set of a given positional game. In the case of the connectivity game, Maker's goal is to claim a rainbow spanning tree. For this game played on the system of complete graphs, we establish matching upper and lower bounds on the threshold bias, up to constant factors.

In the finite projective space PG$(2n,q)$ we consider flags of type $(n-1,n)$, that is, pairs $(A,B)$ consisting of an $(n-1)$-space $A$ and an $n$-space $B$ that are incident. Two such flags $(A_1,B_1)$ and $(A_2,B_2)$ are opposite if $A_1\cap B_2=A_2\cap B_1=\emptyset$. Let $Γ_{2n}$ be the graph whose vertices are the flags of type $(n-1,n)$ of PG$(2n,q)$, with two vertices being adjacent if the corresponding flags are opposite. Using the Erdős-Matching theorem for vector spaces shown by Ihringer, we determine, for $q$ large enough, the largest cocliques of $Γ_{2n}$ and obtain a stability result. This EKR-type theorem proves a conjecture of D'haeseleer, Metsch and Werner.

Exact regions between rank correlations describe the set of all pairs of values that two dependence measures can attain simultaneously on the same copula and thus yield sharp inequalities between them. In this paper, we determine the exact region between Chatterjee's rank correlation $ξ$ and Blest's rank correlation $ν$ over the class of all bivariate copulas. Our approach is based on a constrained optimization problem whose solution is characterized by Karush--Kuhn--Tucker conditions. This leads to a novel extremal copula family that uniquely traces the boundary of the region. For this family, we derive closed-form expressions for both $ξ$ and $ν$, which provide an explicit parametrization of the exact attainable region.

This paper is a documentation of author's reseach, focusing on the topic Grassmann Algebra spanning over July, August 2025 under mentorship provided by DRP Turkiye 2025. Grassmann algebra is a fundamental structure in mathematics with wide-ranging applications across multiple areas of mathematics and physics. Most notably, it serves as the foundation for differential geometry, by constituting the natural setting which differential forms reside. This paper begins with presenting the defining properties of Grassmann Algebra, outlining the working principles of the key mechanism of the algebra, wedge product. Following that, we give an exposition of formal construction of Grassmann algebra from free associative algebra with the goal of emphasizing how these properties are imposed in the structure of the algebra. The intrinsic relationship between the exterior product and the determinant is explored in Section 4. Finally, we investigate invariant subalgebras, one of the primary focuses of this paper. Here, we present a novel classification of invariant subalgebras.

We present a mathematical model of the curvature blindness illusion in which sinusoids appear as angular zigzags when drawn with alternating contrast polarity against a gray background. The model identifies two complementary mechanisms, both operating in V1. First, polarity channel separation: simple cells are selective for contrast polarity, and lateral connections link only same polarity neurons; where the line switches from darker than background to lighter than background at each peak and trough, the encoding population changes and the lateral chain is broken, segmenting the contour into half-wavelength pieces. Second, orientation channel fragmentation: at moderate contrast, the active orientation window is narrow, and within each half-wavelength segment no single orientation channel spans the full range of edge normals; the inflection point at the center of each segment anchors a locally straight percept. Together, the two mechanisms produce a zigzag: polarity breaks supply the corners, and fragmentation straightens the segments between them.

We confirm a conjecture of Darmon-Vonk on the antisymmetry of real quadratic singular moduli. The proof relies on a careful analysis of rigid meromorphic cocycles à la Darmon-Gehrmann-Lipnowski for the split orthogonal group on four variables. Moreover, we prove the modularity of a generating series of Kudla-Millson divisors in the spirit of Gross-Kohnen-Zagier.

We compute the capacity of entanglement in two dimensional dilaton gravity in a setting where Hawking radiation, backreaction, and islands can be treated analytically. Our focus is the eternal black hole of the Russo Susskind Thorlacius model coupled to N conformal matter fields. Unlike previous gravitational computations, which were mostly carried out in JT gravity, the RST model forces one to deal with a genuinely dynamical conformal factor and with the global constraints of the replica construction. The main technical step is therefore to solve the replica deformation on the orbifold globally at first order near n=1, including the homogeneous sector fixed by single valuedness and by the requirement of a fixed microcanonical state. For a single interval we obtain a time independent generalized capacity, parallel to the generalized entropy. For two intervals, even in the late time factorization regime, the global solution generates an interaction term between replica fixed points; after Lorentzian continuation this produces a time dependent capacity on the two QES saddle, despite the corresponding entropy plateau. We discuss the regime of validity of the resulting expressions and explain how the large size of the two QES capacity implies a highly non uniform saddle competition near n=1, providing a concrete mechanism for sharp features of the capacity at the Page transition.

Large time-domain sky surveys generate extensive multi-year catalogs of light curves in which scientifically valuable transients, such as supernovae (SNe), are vastly outnumbered by artifacts and routine star variability. While supervised machine learning models can efficiently filter known classes, they struggle with extreme class imbalance and may overlook rare or novel events. Conversely, unsupervised anomaly detection provides broad discovery potential but lacks targeted sensitivity. We present a hybrid strategy that integrates a supervised SN probability score (SN-score) into the PineForest active anomaly detection framework to enhance SN discovery rate in the 23rd data release of the Zwicky Transient Facility. We train a binary classifier using light-curve features of spectroscopically confirmed SNe from the ZTF Bright Transient Survey, achieving a ROC-AUC approximately 0.98. Incorporating the SN-score as an additional feature, together with a small set of labeled priors, significantly accelerates the discovery of SN-like transients across ten extragalactic ZTF fields. This method increases discovery efficiency without compromising the ability to identify diverse astrophysical anomalies. Application of the combined methodology resulted in the discovery of seven previously unreported SN candidates, one AGN candidate, one unusual Galactic variable star SNAD283, as well as two host galaxies exhibiting multiple supernova events. These results demonstrate its value for scalable and expert-guided transient search in current and future surveys, including the Vera C. Rubin Observatory Legacy Survey of Space and Time.

The integration of Large Language Models (LLMs) into scientific discovery is currently hindered by the Implicit Context problem, where governing equations extracted from literature contain invisible thermodynamic assumptions (e.g., undrained conditions) that standard generative models fail to recognize. This leads to Physical Hallucination: the generation of syntactically correct solvers that faithfully execute physically invalid laws. Here, we introduce a Neuro-Symbolic Generative Agent that functions as a cognitive supervisor atop traditional numerical engines. By encapsulating physical laws into modular Constitutive Skills and leveraging latent intrinsic priors, the Agent employs a Chain-of-Thought reasoning workflow to autonomously validate, prune, and complete physical mechanisms. We demonstrate this capability on the challenge of thermal pressurization in low-permeability sandstone. While a standard literature-retrieval baseline erroneously predicts catastrophic material failure by blindly adopting a rigid "undrained" simplification, our Agent autonomously identifies the system as operating in a drained regime (Deborah number De << 1) via dimensionless scaling analysis. Consequently, it inductively completes the missing dissipation mechanism (Darcy flow) required to satisfy boundary constraints, predicting a stable stress path consistent with experimental reality. This work establishes a paradigm where AI agents transcend the role of coding assistants to act as epistemic partners, capable of reasoning about and correcting the theoretical assumptions embedded in scientific data.

Understanding the interaction of charge carriers with lattice vibrations in the quasi-equilibrium regime is crucial for semiconductor functionality. However, the structural signatures of these interactions are often too subtle for conventional ultrafast techniques to detect. We developed a time-resolved spontaneous Raman technique based on time-correlated single-photon counting to track the spectral response following photoexcitation, providing sub-wavenumber spectral resolution and a few-hundred-picosecond temporal resolution. Unlike traditional pump-probe schemes, our method utilizes a modulated continuous-wave probe to maintain high spectral resolution, enabling detection of low-frequency Raman shifts down to 10 cm$^{-1}$. Applied to lightly boron-doped silicon, we resolve intra-valence band and inter-valence band electronic transitions. A coupled-mode analysis of transient phonon asymmetry, resulting from interference with the inter-valence band transitions, reveals electron-phonon coupling parameters that directly relate to carrier recombination. By capturing these subtle dynamical shifts, we demonstrate that this platform offers a powerful probe for investigating electron-phonon interactions in long-lived excited states.

Synthesizing from Corbett and Munneke (2025), who demonstrated that questions originating in human-computer interaction (HCI) and game design can be answered through the theoretical toolkit of cognitive science, this perspective argues that commercial videogames represent a largely underutilized research environment at the intersection of these two fields. Cognitive science has long relied on carefully controlled laboratory paradigms to study perception, attention, and executive functioning, raising persistent questions about ecological validity. HCI, by contrast, has spent decades developing methods for studying behavior in rich, complex, interactive environments, but has been less concerned with what that behavior reveals about underlying cognitive mechanisms. Commercial videogames sit precisely at this intersection. They are cognitively demanding by design, motivating by nature, and consistent enough across players to support systematic behavioral comparison. The affordance structure of a game does the work that experimental manipulations typically require of the researcher, instantiating cognitive demands that are genuine, sustained, and meaningful to the player. We argue that perception, attention, and executive functioning can be meaningfully studied within commercial games using a minimal observational toolkit of screen recording, eye tracking, and behavioral timing. We propose an affordance-cognition mapping framework as a systematic basis for game selection and research design and offer practical methodological recommendations for researchers wishing to work in this space.

To model the dynamics of polymers formed through nucleation, elongated by polymerisation, shortened by depolymerisation and subject to aggregation reactions, we study a nonlinear integro-differential equation. Growth and shrinkage are described by transport terms, nucleation by a positive boundary condition, and aggregation by a Smoluchowski coagulation kernel. Our main result is the existence of steady states for the multiplicative coagulation kernel despite this kernel producing gelation in finite time for the pure coagulation equation. This is made possible by a sufficiently strong decay rate for large polymers. Beyond the existence result, the qualitative properties of the steady states are illustrated through explicit examples and numerical experiments. The analytical results connect the growth behaviour of the transport velocity and of the coagulation kernel to the decay properties of steady states.

We study the problem of simultaneous geometric embedding of two paths without self-intersections on an integer grid. We show that minimizing the length of the longest edge of such an embedding is NP-hard. We also show that we can minimize in $O(n^{3/2})$ time the perimeter of an integer grid containing such an embedding if one path is $x$-monotone and the other is $y$-monotone.

Excitation energy transfer in light-harvesting aggregates is highly efficient, yet whether quantum coherence plays an operational role in transport remains debated. A central challenge is that coherence is usually inferred from spectroscopic signatures, whereas transport performance is assessed through specific observables and depends on both the open system dynamics and the initial state preparation. Here we develop a resource theoretic approach that quantifies the maximum change that initial site-basis coherence can induce in a chosen readout under fixed reduced dynamics. The central quantity is the resource impact functional, which yields state independent, readout specific bounds on coherence-induced changes in signals and transport figures of merit. We apply the framework to two models. For a donor-acceptor dimer, we analyse coherence sensitivity across coupling and bath-timescale regimes and bound trapping efficiency and average transfer time in terms of the impact functional. For a multi-site chain with terminal trapping, we derive rigorous criteria that distinguish population placement from sensitivity to initial state site-basis coherence. These include upper bounds on the largest advantage over incoherent preparations, necessary delocalization requirements for achieving a prescribed improvement, and a simple pairwise sufficient condition that can be checked from local information. For quasi-local reduced dynamics, we further obtain a Lieb-Robinson-type bound that constrains when coherence prepared in a distant region can influence a localized readout at finite times. Together, these results provide operational diagnostics and rigorous bounds for benchmarking coherence effects and for identifying regimes in which they are necessarily negligible or potentially relevant in excitonic transport models.

Machine learning approaches to nuclear mass prediction have achieved remarkable accuracy, but typically rely on existing theoretical baselines or hand-crafted physics features. Here we demonstrate that these prerequisites can be supplanted by structural inductive biases embedded directly in the network architecture. We present the Cooperative Neural Network (CoNN), which predicts binding energies from raw proton and neutron numbers (Z,N) alone by additively combining four structurally constrained modules: a smooth network for bulk liquid-drop trends, discrete scalar embeddings for shell effects, a learnable two-dimensional grid for regional collective correlations, and a parity-aware network for odd--even staggering. On the AME2020 dataset, the CoNN achieves a root-mean-square deviation of 0.269 MeV across all 3558 nuclei, with 0.419 MeV on a held-out interpolation subset and 0.728 MeV on 122 nuclei newly measured since AME2016, placing it among the most accurate baseline-free approaches to direct mass prediction. Notably, the learned embeddings develop pronounced extrema at canonical magic numbers and the pairing module reproduces the expected odd--even staggering along isotopic chains, both emerging from the data without explicit supervision. These results demonstrate that physically motivated architectural constraints can effectively substitute for feature engineering, establishing architecture as physical prior as a promising paradigm for neural-network mass modeling.

We propose and numerically validate an experimentally feasible on-demand protocol for the nucleation and manipulation of stable quantum vortex rings in trapped Bose-Einstein condensates. The method relies on sweeping a laser-sheet barrier that locally constricts the superflow and triggers vortex-ring formation. By tuning the barrier height and width, and by scanning the barrier velocity, we identify the onset of periodic generation of vortex rings above the critical velocity and achieve direct, deterministic control over the ring nucleation position, radius, and hence propagation speed. After its formation, ad-hoc optical potentials are applied to reshape the vortex ring, creating clean Kelvin-wave excitations. Our results provide a foundation for systematic studies of three-dimensional vortices in atomic superfluids and open the door to tailored vortex dynamics and interactions, enabling controlled access to quantum turbulence.

Bridging generative foundation models with non-equilibrium thin-film synthesis remains a central challenge, limiting the practical impact of AI-driven materials discovery on semiconductor dielectrics. Here, we introduce IDEAL (Inverse Design for Experimental Atomic Layers), an inverse-design platform that links generative diffusion models, machine learning interatomic potentials, and graph neural network property predictors with atomic layer deposition (ALD). We demonstrate IDEAL using the Hf-Zr-O system as a stringent benchmark for semiconductor-relevant complex oxides. The platform statistically enumerates thermodynamically plausible structures and constructs a composition-structure-property map. Crucially, it identifies a narrow composition window where low-energy tetragonal and orthorhombic phases cluster, revealing trade-offs between band gap and dielectric response. Experimental validation using atomic layer modulation (ALM) corroborates these predictions, demonstrating predictive guidance under realistic, non-equilibrium thin-film growth. By experimentally closing the loop, IDEAL provides a transferable and generalizable route to the precision synthesis of next-generation semiconductor dielectrics.

$P$-, $f$-, or $h$-wave antialtermagnets yield large non-relativistic spin splitting with out-of-plane spin polarization in momentum space perpendicular to the coplanar non-collinear local magnetic moments. We provide a microscopic explanation of this unconventional spin polarization by linking it to a previously overlooked site-compensated spin density that orders antiparallel when projected onto opposite momenta. We verify this result both by model derivation of the out-of-plane momentum-space spin polarization being proportional to the direct-space cross product of the coplanar non-collinear spin order, as well as by ab initio calculations in the material candidate CeNiAsO. By providing a general classification and analytic expression for the spin polarization of all spinful two-site tight-binding Hamiltonians, we reveal the momentum-resolved spin polarization as a probe of the Bloch-state geometry arising from spin-site coupling. Furthermore, our approach allows for geometric distinction between ferro-, alter-, and antialtermagnets. Our results provide a quantitative guidance for quantized out-of-plane momentum-space spin polarization and large spin splitting, and construction principles for antialtermagnets.

Conditional value-at-risk (CVaR) is a prominent risk measure in financial engineering, energy systems, and supply chain management. In these domains, Markov decision processes (MDPs) with a long-run CVaR criterion effectively mitigate cost variability over a specified horizon. However, implementing MDPs relies on known transition models, which are typically unavailable in practice. This necessitates a model-free approach to risk-sensitive dynamic optimization. To tackle this challenge, we propose a reinforcement learning algorithm that simultaneously conducts policy evaluation and improvement based on a CVaR-specific Bellman local optimality equation. This algorithm employs a nonparametric incremental learning approach for policy improvement, relying on a single sample trajectory to identify the optimal policy. Under appropriate technical conditions, we prove almost sure convergence of the algorithm and derive its convergence rate. Our analysis reveals that the optimal convergence rate, measured by the mean absolute error of policy estimators, is of order O(1/n). Our main algorithm and results are further extended to solving the mean-CVaR optimization problem. Numerical experiments corroborate these results.

The phase-field approach to brittle fracture provides a continuum framework for modeling crack initiation and propagation without explicit representation of discrete crack surfaces, provided the spatial discretization is fine enough to resolve the regularization length scale. However, uncertain local material parameters due to material defects can strongly influence simulation results, such as crack paths and remaining structural strength. At the same time, the ability to continuously monitor structures using sensors allows complementing modeling predictions with, e.g., displacement measurements. In this contribution, we connect these two complementary sources of information and present a Bayesian inference procedure that allows updating the current model state with incoming sensor data. We construct a Bayesian prior for the model state (both displacements and phase-field) and employ an ensemble Kalman filter (EnKF) to perform the update. In the EnKF, the update is computed by performing a Kalman shift on each ensemble member. Since the standard EnKF may produce assimilated states that violate common modeling assumptions, we present a phase field-based regularization technique as a proximal step correction toward model-consistent updates. 1D and 2D numerical examples demonstrate the performance and accuracy of the proposed method and show that the updated state matches the ground truth reasonably well. Unlike traditional Bayesian inversion techniques, which have already been applied to brittle fracture, we infer not the model parameters but the model state, i.e., the displacement field and the phase-field. Although only displacements are observed, the strong correlation between both fields also allows inference of the posterior phase-field.

Given a value computed within a program, an idempotent backward slice with respect to this value is a maximal subprogram that computes it. An informal notion of an idempotent slice has previously been used by Guimaraes et al. to transform eager into strict evaluation in the LLVM intermediate representation. However, that algorithm is insufficient to be correctly applied to general control-flow graphs. This paper addresses these omissions by formalizing the notion of idempotent backward slices and presenting a sound and efficient algorithm for extracting them from programs in Gated Static Single Assignment (GSA) form. As an example of their practical use, the paper describes how identifying and extracting idempotent backward slices enables a sparse code-size reduction optimization; that is, one capable of merging non-contiguous sequences of instructions within the control-flow graph of a single function or across functions. Experiments with the LLVM test suite show that, in specific benchmarks, this new algorithm achieves code-size reductions up to -7.24% on programs highly optimized by the -Os sequence of passes from clang 17.

We present DRES: a 1.5-hour Dutch realistic elicited (semi-spontaneous) speech dataset from 80 speakers recorded in noisy, public indoor environments. DRES was designed as a test set for the evaluation of state-of-the-art (SOTA) automatic speech recognition (ASR) and speech enhancement (SE) models in a real-world scenario: a person speaking in a public indoor space with background talkers and noise. The speech was recorded with a four-channel linear microphone array. In this work we evaluate the speech quality of five well-known single-channel SE algorithms and the recognition performance of eight SOTA off-the-shelf ASR models before and after applying SE on the speech of DRES. We found that five out of the eight ASR models have WERs lower than 22% on DRES, despite the challenging conditions. In contrast to recent work, we did not find a positive effect of modern single-channel SE on ASR performance, emphasizing the importance of evaluating in realistic conditions.

Rankings play a crucial role in decision-making. However, if minor changes to items significantly alter their rankings, the quality of the decisions being made can be compromised. The stability of ranking is a measure used to assess how modifications to the ranking algorithm or data affect results. While previous work has focused on stability of the ranking under changes to the algorithm, we introduce a novel measure we refer to as local stability. Local stability indicates the effect of minor changes to the values of an item in the ranking on its rank. Our proposed definition furthermore takes into account the presence of multiple items with similar qualities in the ranking, called dense regions, permitting minor modifications to swap the positions of items within the region. We show that computing this measure in general is hard, and in turn propose a relaxation of the definition to admit approximation. We present (i) LStability, a sampling-based algorithm for approximating local stability, on which we make probably-approximately-correct-type guarantees through the use of concentration inequalities, and (ii) Detect-Dense-Region, an algorithm based on this approach to detect the dense region an item lies in, if it exists. We introduce a number of optimizations to our algorithms to improve their scalability and efficiency. We validate our proposed framework through an extensive suite of experiments, including case studies highlighting the utility of our definitions.

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

Recent years have witnessed significant progress in the entanglement-based characterization of quantum phases of matter. The primary objects of interest are the reduced density matrix and its associated entanglement Hamiltonian. As intrinsic properties of a quantum state, these quantities theoretically determine all experimentally accessible local observables. In this work, we investigate the response of two entanglement measures to the real-time dynamics driven by the entanglement Hamiltonian--a process known as modular flow. We demonstrate that our results can be unified into a single generating function, $\langleρ_{AB}^α\mathrm{e}^{λ{Q}_{AB}}\mathrm{e}^{μ{Q}_{BC}}ρ_{BC}^β\rangle$. This function is of independent interest as it represents a generalization of the recently proposed Rényi modular commutator. In appropriate limits, this function yields the response of Rényi entropy and its charged version, which we find to be uniquely determined by chiral topological invariants, specifically the chiral central charge and the Hall conductance. Our analytical findings are validated through two independent approaches: (i) free fermion systems using the real-space Chern number formula, and (ii) an effective field theory treatment that regularizes the entanglement cut via chiral conformal field theory. Both methods yield consistent results.

Understanding the mechanism of chemical reaction rate modulation by vibrational strong coupling (VSC) has been the focus of several recent studies. However, a definitive explanation for the mode-specificity of VSC still eludes us. In this study, we highlight the dynamics under VSC by utilizing a model for post-transition state bifurcation (PTSB) reactions coupled to an optical cavity. The minimal two-dimensional PTSB model features a valley-ridge inflection (VRI) point leading to bifurcated energetically asymmetric product wells. Here, we are interested in exploring whether the product selectivity (branching ratios) in such PTSB systems, known to be sensitive to dynamical effects, can be significantly perturbed under VSC conditions. Detailed classical and quantum dynamical calculations, along with systematic variation of the model parameters, reveals that the branching ratio can be enhanced under VSC by nearly a factor of two. Interestingly, for certain parameter regimes we find excellent classical-quantum correspondence. Apart from emphasizing the role of both cavity-system and intramolecular energy transfer in the observed enhancements, our study brings out the complexity of VSC in terms of the choice of the cavity frequency vis--à--vis the various molecular mode frequencies. In addition, our work highlights the potential of cavity quantum electrodynamics as a tool for reshaping dynamical outcomes in reactions with complex potential energy landscapes.

It is well-known that Polish Roelcke precompact groups are the groups that can be represented as automorphism groups of $\aleph_0$-categorical structures in continuous logic and that there is a precise correspondence between properties of the group and properties of the structure. The goal of this paper is to extend this correspondence to the classes of locally Roelcke precompact groups and locally $\aleph_0$-categorical structures, the latter of which we define here. We characterise locally Roelcke precompact groups in terms of their isometric actions. We define locally $\aleph_0$-categorical theories and structures, prove an appropriate version of the Ryll-Nardzewski theorem, and identify the Polish locally Roelcke precompact groups as the automorphism groups of such structures. In all locally $\aleph_0$-categorical structures, there is a definable metric, which we call localising, and which captures the coarse geometric structure of the corresponding automorphism group. We show that two locally $\aleph_0$-categorical structures are bi-interpretable if and only if their automorphism groups are isomorphic. Finally, we show that (the unit ball of) a Banach space is $\aleph_0$-categorical if and only if the corresponding affine space is locally $\aleph_0$-categorical (as a metric space).

In a companion paper (Studia Math., 2023), we proved for every $λ\in(1,2]$ the existence of a $(λ^+)$-injective renorming of $\ell_\infty$ that is not $λ$-injective, thereby establishing a~forgotten theorem of Pełczyński in that range. The complementary range $λ\in(2,\infty)$ was left open. In the present paper, we resolve this remaining case: for every $λ>2$ we construct a Banach space that is $(λ^+)$-injective but not $λ$-injective, completing Pełczyński's theorem for all $λ>1$. The construction uses a single device: the `zero-sum' subspace $Σ_N(Y)\subset Z_\infty^N$, which multiplies the relative projection constant by $μ_N=2-2/N$ while preserving non-attainment. Iterating this operation reduces the problem to the range $(1,2]$ already covered by the companion paper. Since the ambient spaces arising in the iteration are finite $\ell_\infty$-sums of $\ell_\infty$, the resulting examples may be realised as subspaces of~$\ell_\infty$. We also prove that if two Banach spaces are each isometrically isomorphic to their own square and each is isometric to a $1$-complemented subspace of the other, then their Banach--Mazur distance is at most $9+6\sqrt{3}$. Consequently, we obtain the estimate $\operatorname{dist}(L_\infty[0,1],\ell_\infty)\le 9+6\sqrt{3}$, thereby improving a recent result of Korpalski and Plebanek.

A basic problem in the geometry of degenerating period maps is to determine whether their completed images admit an intrinsic algebraic description. For polarized variations of Hodge structure over smooth quasi-projective surfaces, Deng and Robles formulated such a problem in terms of the Kato-Nakayama-Usui completion of the period image and a conjectural Proj description involving the augmented Hodge line bundle and the boundary divisor on a smooth compactification of the base. We show that the essential obstruction to this description is divisor-theoretic: it may be expressed as a Picard-generation statement on the completed mixed period image. We prove this statement when the pure period image is one-dimensional, and consequently obtain the Deng-Robles Proj description in this case.

We use optical-optical double-resonance (OODR) spectroscopy to measure for the first time hot-band transitions of ethylene (C${_2}$H${_4}$) between states in the 3000 cm$^{-1}$ and 9000 cm$^{-1}$ energy ranges. A 3.2 $μ$m continuous wave (CW) pump is used to populate selected states in the $ν$${_9}$ vibrational mode. The sub-Doppler OODR transitions are then probed with two different cavity-enhanced probes tunable around 1.7 $μ$m: a frequency comb probe that allows for broadband measurements and simultaneous detection of many OODR lines, and a CW probe that measures individual lines with higher signal-to-noise ratio and better frequency accuracy. We report center frequencies and relative intensities of 90 ladder-type hot-band transitions from three different states in the $ν$${_9}$ vibrational mode. We exploit combination differences and measurements of polarization-dependent intensity ratios to determine the final state rotational quantum numbers J. Comparison to theoretical predictions from ExoMol allows tentative assignments for 28 transitions. We report improved center frequencies for the three pump transitions in the $ν$${_9}$ band. Furthermore, we observe 18 sub-Doppler V-type transitions from the depleted ground state to the 6000 cm$^{-1}$ region and assign 14 of them to the variational line list of Mraidi et al. [J Quant Spectros Radiat Transfer. 2023;310:108734 doi:10.1016/j.jqsrt.2023.108734].