An edge-coloring of a hypergraph is {\em spanning} if every vertex sees every color used in the coloring. In this paper, we prove that for $k \geq 2r \geq 6$, in any spanning $k$-coloring of the edges of a complete $r$-partite $r$-uniform hypergraph $H$, the vertices of $H$ can be covered by a set of at most $k-r+1$ monochromatic connected components. This proves a conjecture of Gyárfás and Király which is related to a special case of Ryser's conjecture. We also prove that for $k \in \{2,3\}$, every spanning $k$-edge-coloring of a complete bipartite graph admits a covering of its vertices using at most $k$ monochromatic components.

Quadrupole ordering with the ordering wavevector at the L points in PrIr$_2$Zn$_{20}$ under magnetic fields is analyzed using classical Monte Carlo simulations based on an effective $Γ_3$ quadrupole model on the diamond structure. We demonstrate that competition between the magnetic field and quadrupole anisotropy leads to a rich phase diagram for magnetic fields applied parallel to [001], which includes switching between a single-q state and a double-q state. We also show that a symmetry-allowed biquadratic intersite interaction, corresponding to a hexadecapole interaction, is crucial for reproducing the weak-field topology observed in experiments.

Social media platforms typically obtain user consent through Terms of Service (ToS) presented at account creation, rather than through dedicated consent forms. This study investigates whether consent-related information is clearly communicated within these ToS documents. We propose and apply a three-dimensional consent evaluation framework encompassing Textual Accessibility, Semantic Transparency, and Interface Design as declared in ToS documents. Using a combination of computational and qualitative analyses, we assess ToS from 13 major social media platforms. Our findings reveal important shortcomings across platforms, including high linguistic complexity, widespread use of non-committal language, limited disclosure of data retention and sharing practices, and the absence of explicit interface-level commitments to granular or revocable consent. These results indicate that while consent is formally embedded in ToS, it is often presented in ways that constrain clarity and meaningful choice. Rather than treating ToS documents as informed consent instruments, this study positions them as consent-bearing documents whose design and content shape the conditions under which users are asked to agree to data practices. The proposed framework offers a systematic method for evaluating consent information within ToS in the absence of explicit consent forms and informs the design of clearer, more ethically robust consent mechanisms for data-intensive digital platforms.

Nonlinearity in coherent fiber links is fundamentally driven by the temporal statistics and spectral structure of signal intensity. This paper develops a unified framework that links block-level energy statistics of shaped constellations to the low-frequency features of the intensity-fluctuation power spectral density (PSD), thereby enabling spectral-temporal co-design for nonlinear mitigation. A semi-analytical PSD model is derived for finitely block-shaped symbols (including Constant Composition Distribution Matching (CCDM) and Enumerative Sphere Shaping (ESS)), explicitly exposing contributions from self-beating dependent on symbol energy variance, inter-symbol beating dependent on mean symbol energy, and block-induced energy variance terms. A compact expression for the spectral-dip width is obtained that captures the block length, symbol rate, pulse roll-off, and chromatic dispersion. This yields design rules for lowering the low-frequency content. The low-frequency content most strongly drives the induced XPM. Resulting optimal symbol-rate laws are provided for shaped and unshaped systems, and are validated by Monte-Carlo simulations, which also confirm the distinct low-frequency behaviour of CCDM (suppressed DC) versus ESS (finite DC pedestal at moderate block lengths). The framework consolidates prior time- and frequency-domain views and supplies actionable guidance for choosing block length, symbol rate, and shaping method to reduce nonlinear interference in high-capacity WDM systems.

Numerical models are widely used to simulate the earth system, but they are computationally expensive and often depend on many uncertain input parameters. Their effective use requires calibration and uncertainty quantification, which typically involve running the model across many input configurations and therefore incur substantial computational cost. Statistical emulation provides a practical alternative for efficiently exploring model behavior. We are motivated by the Arctic sea ice component of the Energy Exascale Earth System Model (MPAS-Seaice), which generates large spatiotemporal outputs at multiple spatial resolutions, with high-resolution (or high-fidelity, HF) simulations being more accurate but computationally more expensive than lower-resolution (low-fidelity, LF) simulations. Multi-fidelity (MF) emulation integrates information across resolutions to construct efficient and accurate surrogate models, yet existing approaches struggle to scale to large spatiotemporal data. We develop an MF emulator that combines tensor decomposition for dimensionality reduction, Gaussian process priors for flexible function approximation, and an additive discrepancy model to capture systematic differences between LF and HF data. The proposed framework enables scalable emulation while maintaining accurate predictions and well-calibrated uncertainty for complex spatiotemporal fields, and consistently achieves lower prediction error and reduced uncertainty than LF-only and HF-only models in both simulation studies and MPAS-Seaice analysis. By leveraging the complementary strengths of LF and HF data and using an efficient tensor decomposition approach, our emulator greatly reduces computational expense, making it well suited for large-scale simulation tasks involving complex physical models.

Robust invisible watermarking systems aim to embed imperceptible payloads that remain decodable after common post-processing such as JPEG compression, cropping, and additive noise. In parallel, diffusion-based image editing has rapidly matured into a default transformation layer for modern content pipelines, enabling instruction-based editing, object insertion and composition, and interactive geometric manipulation. This paper studies a subtle but increasingly consequential interaction between these trends: diffusion-based editing procedures may unintentionally compromise, and in extreme cases practically bypass, robust watermarking mechanisms that were explicitly engineered to survive conventional distortions. We develop a unified view of diffusion editors that (i) inject substantial Gaussian noise in a latent space and (ii) project back to the natural image manifold via learned denoising dynamics. Under this view, watermark payloads behave as low-energy, high-frequency signals that are systematically attenuated by the forward diffusion step and then treated as nuisance variation by the reverse generative process. We formalize this degradation using information-theoretic tools, proving that for broad classes of pixel-level watermark encoders/decoders the mutual information between the watermark payload and the edited output decays toward zero as the editing strength increases, yielding decoding error close to random guessing. We complement the theory with a realistic hypothetical experimental protocol and tables spanning representative watermarking methods and representative diffusion editors. Finally, we discuss ethical implications, responsible disclosure norms, and concrete design guidelines for watermarking schemes that remain meaningful in the era of generative transformations.

Natural filaments, such as proteins, plant tendrils, octopus tentacles, and elephant trunks, can transform into arbitrary three-dimensional shapes that carry out vital functions. Their shape-morphing behavior arises from intricate patterning of active and passive regions, which are difficult to replicate in synthetic matter. Here, we introduce a filament-centric strategy for programmable shape morphing in which intrinsic curvature and twist are directly encoded within multimaterial elastomeric filaments during fabrication. By harnessing rotational multimaterial 3D printing (RM-3DP), we directly prescribe the filament's natural curvature--twist field $\mathbf{k}(s)$ through controlled material distribution and helical liquid crystal mesogen alignment. When heated above their nematic-to-isotropic transition temperature ($T_\mathrm{NI}$), the helically aligned LCE regions contract along their local director field, while passive regions remain essentially unchanged. This approach enables independent control of bending and torsion at every cross-section along the filament centerline: the principal natural curvatures of the filament along two orthogonal axes as well as the local twist. Next, we printed architected lattices composed of unit cells formed by sinusoidal filaments that either reversibly contract, expand, or exhibit out-of-plane deformations. Discrete elastic rod simulations of Janus filaments with different natural curvatures and twist, which are interconnected within the printed lattices, allow accurate prediction of their observed shape-morphing behavior. By integrating active-passive elastomers, additive manufacturing, and computational modeling, we have created shape-morphing matter with complex programmable responses for applications that rely on adaptive, robotic, or deployable architectures.

This paper considers the combinatorics of continuous and Borel rectangular partitions of free actions of $\mathbb{Z}^n$ on $0$-dimensional Polish spaces, specifically the free part $F(2^{\mathbb{Z}^n})$ of the shift action of $\mathbb{Z}^n$ on the space $2^{\mathbb{Z}^n}$. This is done through the study of a corresponding notion of regulated partitions of $\mathbb{R}^n$. The main concepts studied are the continuous and Borel {\em regulation} numbers of the partition. This is defined as the maximum number of rectangles in the corresponding regulated partition that can intersect in a point. The continuous and Borel regulation numbers $γ_c$, $γ_B$ are the minimum possible values of these numbers as we range over continuous (respectively Borel) rectangular partitions of $F(2^{\mathbb{Z}^n})$. It is shown that for $n=2$ that $γ_c=γ_B=3$, and for $n \geq 3$ that $n+2\leq γ_B \leq γ_c \leq 3\cdot 2^{n-2}$. For $n=3$ we improve this to $γ_c=γ_B=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$.

In this study, we focus on a generalized nonparametric scalar-on-function regression model for heterogeneously distributed and strongly mixing data. We provide almost complete convergence rates for the local linear estimator of the regression function. We show that, under our conditions, the pointwise and uniform convergence rates are the same on a compact set. On the other hand, when the data is dependent, it is proved that the convergence rate can be slower than those obtained for independent data. A simulation study shows the good performance and finite sample properties of the functional local linear estimator (FLL) in comparison to the local constant estimator (FLC). In addition, a one step ahead energy consumption forecasting exercise illustrates that the forecasts of the FLL estimator are significantly more accurate than those of the FLC.

We have combined Gaia astrometry with Gaia-ESO Survey radial velocities to measure the 3D rotation of the open cluster NGC 2516. We compiled a sample of 430 members with astrometry and spectroscopy and use these to determine a distance to the cluster of 406.3 +/- 0.8 pc, which we then use to infer the 3D positions and velocities of all stars in the cluster using a Bayesian model. We identify the axis of maximum cluster rotation and measure a median rotational velocity of 0.12 +/- 0.02 km/s. We find the axis of maximum cluster rotation to be 74 +/- 17 degrees to the plane of our galaxy. We compare this rotation rate to measurements of cluster rotation in other open clusters and find that it is inconsistent with the expected dependences on cluster age and mass.

We introduce an augmented form of the van Trees inequality, that yields uniformly tighter lower bounds on the minimax squared Bayes risk of estimators compared with the classical van Trees inequality. Our augmented inequality also accommodates prior distributions whose densities need not vanish at the boundaries of their supports. We demonstrate how this refinement can be utilised for elementary proofs of a number of minimax lower bounds for nonparametric estimands, that also often attain sharper constants than those obtained by the alternative Le Cam convergence of experiments theory and the classical van Trees inequality, and in some cases obtain exact constants. As an example, our augmented van Trees inequality can be used to obtain the asymptotic minimax pointwise mean squared error when estimating the regression function in the model with normal errors: when the regression function is univariate and differentiable with Lipschitz derivative we obtain this quantity up to a constant factor of $1.37$; and in the high dimensional regime with a Hölder smooth regression function of smoothness $β\in(0,2]$ we obtain exact constants. Both these results do not follow from an application of the classical van Trees inequality. The flexibility of our augmented van Trees inequality accommodates lower bounds for models beyond Gaussianity, loss functions beyond the squared error loss, and we are also able to incorporate this augmentation into generalised versions of the van Trees inequality for irregular models.

We study sequential multiple testing with independent data streams, where the goal is to identify an unknown subset of signals while controlling commonly used error metrics, including generalized familywise rates and false discovery and non-discovery rates. For these problems, procedures that are first-order optimal are known, in the sense that the ratio of their expected sample size (ESS) to the minimal achievable ESS converges to one as the error tolerance levels vanish. In this work, we develop a unified theory of second-order asymptotic optimality. We establish general sufficient conditions under which second-order Bayesian optimality implies second-order frequentist optimality for broad classes of sequential testing procedures. As a consequence, several procedures previously known to be first-order optimal are shown to be second-order optimal: for every signal configuration, the difference between their ESS and the minimal achievable ESS remains uniformly bounded as the error tolerance levels tend to zero. In addition, we derive a second-order asymptotic expansion of the minimal achievable ESS, which refines the classical first-order approximation by identifying the second-order correction term arising from a boundary-crossing problem for a multidimensional random walk. We apply this result to several commonly used error metrics.

A segmented waveguide-enabled pinching-antenna system (SWAN)-based tri-hybrid beamforming architecture is proposed for uplink multi-user MIMO communications, which jointly optimizes digital, analog, and pinching beamforming. Both fully-connected (FC) and partially-connected (PC) structures between RF chains and segment feed points are considered. For the FC architecture, tri-hybrid beamforming is optimized using the weighted minimum mean-square error (WMMSE) and zero-forcing (ZF) approaches. Specifically, the digital, analog, and pinching beamforming components are optimized via a closed-form solution, Riemannian manifold optimization, and a Gauss-Seidel search, respectively. For the PC architecture, an interleaved topology tailored to the SWAN receiver is proposed, in which segments assigned to each RF chain (sub-array) are interleaved with those from other sub-arrays. Based on this structure, a WMMSE-based tri-hybrid design is developed, in which the Riemannian-manifold update used for the FC structure is replaced by element-wise phase calibration to exploit sparsity in analog beamforming. To gain insight into the performance of the proposed system, the rate-scaling laws with respect to the number of segments are derived for both the FC and PC structures. Our results demonstrate that: i)~SWAN with the proposed tri-hybrid beamforming consistently outperforms conventional hybrid beamforming and conventional pinching-antenna systems with pinching beamforming for both the FC and PC structures; and ii)~the PC structure can strike a good balance between sum rate and energy consumption when the number of segments is large; and iii) the achievable rate does not necessarily increase with the number of segments.

This work presents, the first time, analytic solutions for the nearly equatorial, plunging motion of a spinning test-particle in Kerr spacetime. The equations of motion are solved at first-order in the small-body spin for all classes of plunging orbits with energy $E < 1$. The solutions incorporate the small precession of the orbital plane caused by the precession of the particle's spin. Additionally, we present the correction to the radius of the innermost bound circular orbit in closed form, and introduce a novel, Keplerian-like parametrization for generic plunging orbits. Our solutions will be useful in the modelling of inspiral-merger-ringdown waveforms with self-force methods and black hole perturbation theory.

We provide uniform convergence rates for kernel averages on $[0,1]$ under equally-spaced fixed design points of the form $x_{t,T}=t/T,\ t\in\{1,\dotsc, T\},\ T\in\mathbb{N}$. The rates of weak and strong uniform consistency are derived under strong mixing and moment conditions and do not require stationarity. The analysis exploits the grid structure and thus complements existing random-design results such as those of Hansen (2008) and Kristensen (2009), which rely on density-based conditioning arguments. The framework accommodates dependent triangular arrays and is particularly relevant for nonparametric methods applied to time series observed on deterministic grids. As an application, we derive uniform convergence rates for the local linear estimator in a nonparametric regression model with time-varying autoregressive errors. The theoretical results are illustrated through Monte Carlo experiments and an empirical application.

We study the high-pressure melting behavior of titanium using laser-driven shock compression with in situ femtosecond x-ray diffraction and molecular-dynamics simulations based on a machine-learned interatomic potential. The MD simulations predict the solid-liquid coexistence on the Hugoniot in the $\sim$$111-124$ GPa range. Experimentally, we observe the first evidence of liquid at 86 GPa. We also observe pronounced microstructural changes with pressure with strong grain refinement associated with the emergence of liquid, within the solid-liquid coexistence ($\sim$$110-126$ GPa). Above 126 GPa, we observe the persistence of residual levels of highly textured crystalline Ti to $\sim$$180$ GPa, well above the expected melt completion pressure. We discuss the accuracy that current laser-shock experimental platforms have at determining the melt onset and completion pressures.

We present the X-ray Doppler tomography of the Fe K$α$ (6.4 keV) fluorescence line of the low-mass X-ray binary 4U 1822-371 obtained with XRISM. Eleven orbits of this short period (5.57 hr) binary were covered. The Doppler shift of the line shows clear modulation with the orbital period, motivating us to apply the Doppler tomography in the X-ray band for the first time. The resulting velocity map reveals a compact feature at ($v_{\mathrm{x}}$, $v_{\mathrm{y}}$) $\sim$ ($-$550, $+$125) km s$^{-1}$. This is inconsistent with the emission originating from a symmetric accretion disk, an extended corona around the neutron star, or the surface of the neutron or companion star. Instead, it suggests that the emission originates from the accretion stream-disk overflow. Remarkably, the Fe K$α$ velocity map closely resembles that of the O VI 3811 Å, indicating that both X-ray and optical lines arise from the same site irradiated by the central X-ray source. These results provide the first velocity-resolved X-ray map of the fluorescent line, directly localizing the major reflector in an X-ray binary and establishing X-ray Doppler tomography as a new probe of the structures of accreting systems.

We investigate the upper bounds of nodal sets for solutions of bi-Laplace equations without using frequency functions which play an essential role in the study of nodal sets in the celebrated work by Logunov \cite{Lo18}. We obtain some delicate monotonicity and propagation of smallness results by Carleman estimates. A polynomial upper bound for the nodal sets of solutions is obtained.

By the family index theory, we generalize some well-known $SL(2,Z)$ modular forms to the family case and obtain some new anomaly cancellation formulas for the determinant line bundle and index gerbes, and certain results about eta invariants. Moreover, for the higher degree case, we give some anomaly cancellation formulas of residue Chern forms.

The quark-gluon plasma (QGP) is a strongly coupled medium in which both open and hidden charm particles experience substantial nonperturbative interactions. This poses a major challenge for a quantitative description of charmonium transport in ultra-relativistic heavy-ion collisions, as it requires a mutually consistent treatment of pertinent transport coefficients. In this work, we present a coupled charm-charmonium transport framework for a strongly coupled QGP based on thermodynamic $T$-matrix interactions with recent constraints from Wilson-line correlators (WLCs) computed in lattice QCD. For the first time, the same underlying heavy-light interactions and in-medium spectral functions are used to self-consistently evaluate charm-quark diffusion and charmonium kinetics. In particular, the charmonium equilibrium limit, a critical transport parameter for regeneration, is evaluated in the presence of broad spectral functions. Charm-quark diffusion is simulated via Langevin dynamics and coupled to a Boltzmann equation for charmonium dissociation and regeneration. The equilibrium limit of the statistical model is recovered once charm quarks thermalize, and its extension to describe off-equilibrium is constructed. Preliminary applications to charmonium observables in Pb-Pb collisions at the LHC capture the measured centrality and momentum dependence fairly well.

In this paper we consider a population process evolving on a dynamic random graph. The dynamic random graph is an Erdős--Rényi graph that is resampled every time unit, independently of the previous ones, with `edge existence probability' $p$. The population process consists of $M$ individuals which reside at the vertices of the dynamic graph. At each point in time any of the $M$ individuals, supposing it resides at a vertex with $k$ neighbors, jumps to an adjacent vertex with probability $k/(k+1)$ (where this adjacent vertex is picked uniformly at random), and with probability $1/(k+1)$ it stays where it is. We suppose we observe the numbers of individuals at each of the vertices, but not the evolving random graph itself. We propose two estimators for $p$, and establish their consistency and asymptotic normality.

Let $\mathfrak{g}$ be a simple, simply-laced Lie algebra and $f \in \mathfrak{g}$ nilpotent. The Kazhdan-Lusztig category of the W-algebra $W^κ(\mathfrak{g},f)$ associated with $(\mathfrak{g},f)$ at level $κ\in \mathbb{C}$ is obtained from the Kazhdan-Lusztig category of the affine vertex algebra $V^κ(\mathfrak{g})$ via the quantum Hamiltonian reduction associated with $f$. We show that this is a braided tensor equivalence for any $f$ and any irrational level $κ\in \mathbb{C} \backslash \mathbb{Q}$.

Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given.

We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges, and no plane matching of size more than $2d-4$. On the other hand, we prove that every rectilinear drawing of $Q_d$ with vertices in convex position contains a plane path of length $d$ (if $d$ is odd) or $d-1$ (if $d$ is even). We also prove that if a graph $G$ is a plane subgraph of every drawing of $Q_d$ for a sufficiently large $d$, then $G$ is necessarily a forest of caterpillars. Lastly, we give a short proof of a generalization of a result by Alpert et al. [Cong. Numerantium, 2009] on the maximum rectilinear crossing number of $Q_d$.

The development in the early seventies of supersymmetry, in the mid-seventies of gauge supersymmetry and supergravity, and in the early eighties of gravity mediated breaking of supersymmetry and of supergravity grand unification have led to remarkable progress in the pursuit of unification of fundamental interactions of particle physics. They have also led to the intertwining of particle physics, cosmology, and strings. Since supersymmetry and supergravity are manifest in the low energy limit of superstring below the Planck scale, experimental test of them are of interest regarding the validity of the superstring itself. For that reason, over the past decades, after the advent of supersymmetry and SUGRA models, there have been sustained experimental searches for supersymmetry at colliders, in precision experiments, and in astrophysical and cosmological data. The SUSY and SUGRA models have also had deep impact on theories related to inflation, dark matter, and dark energy. The purpose of this article is to provide a view from the bridge of these developments over the past fifty years circa 1974-2024.

The work focuses on Lagrange stability of the origin for the three-state Moore-Greitzer compressor model in closed loop with a nonlinear PI controller, tuned only to stabilize a lower-dimensional invariant surge-dynamics subsystem.The linearization of the system is not stabilizable but the static nonlinearity satisfies a sector condition, and together with a structural property of the stall-dynamics subsystem, this plays an essential role in the analysis. The main contribution provides explicit conditions on the controller parameters together with analytical arguments that guarantee boundedness of all solutions of the closed-loop system. The analysis employs a non-standard application of circle-criterion-based arguments. Together with the additional arguments developed in the work, this stability test also shows that the closed-loop system is robust to certain perturbations and model uncertainties.

Waveguide quantum electrodynamics (WQED) offers a powerful framework for controlling light-matter interactions and realizing collective phenomena such as super- and subradiance. In general waveguide settings, the quantum dynamics spans the full Hilbert space, rendering exact theoretical treatments exponentially difficult and currently out of reach, and only a few models have exact, analytical solutions. Motivated by recent experiments, we treat the thermodynamic limit of the number of atoms, $N \rightarrow \infty$, while the homogeneous atom-waveguide coupling $β\rightarrow 0$ keeping the optical depth $4Nβ$ fixed. In this limit, a second order mean field method is exact, giving analytical solutions for the statistics of the photons emitted in the waveguide both for chiral and symmetric configurations starting from full inversion. As $N \rightarrow \infty$, the emission in freespace approaches that of an independent ensemble. However, until a special time, $\approx 1.59 \times$ the lifetime of a single-atom, we show an exponentially enhanced superradiance in the waveguide as the optical depth increases. After the special time, the emission into the waveguide exhibits subradiance. We also show that the initial shot-to-shot fluctuations in the rate of emission into the waveguide diminish in a chiral system and vanish in a symmetric system as $N$ approaches $\infty$. Additionally, the equal-time second-order correlation becomes trivial, showing that finite-size effects are essential to observe the emergence of second-order coherence. Finally, going beyond the thermodynamic limit requires higher order mean field methods. Our results illustrate finite- and infinite-body collective effects in symmetric and symmetry-lacking systems.

Physical instantiations of a bit of information are subject to thermal noise that can trigger unintended bit-flip errors. Bits implemented with CMOS technology typically operate in regimes that reliably suppress these errors with a large bias voltage, but miniaturization and circuit design for implantable biomedical devices motivate error suppression via alternative low-voltage strategies. We present and analyze an error-suppression technique that involves coupling multiple CMOS units into chains, introducing a natural error correction arising from inter-unit correlations. Using tensor networks to numerically solve a stochastic master equation for the CMOS chain, we quantify the reliability-dissipation tradeoff across system sizes that would be intractable with conventional sparse-matrix methods. The calculations show that the typical time for bit-flip errors scales exponentially with the bias voltage but subexponentially with the chain length. While a CMOS chain adds stability compared to a single CMOS unit for a fixed low bias voltage, increasing the bias voltage is a lower-dissipation route to equivalent stability.

Let $A$ be a complex square matrix, and write its polar decomposition as $A=U|A|$. For $0<λ<1$, the $λ$-Aluthge transform of $A$ is defined by $$ Δ_λ(A)=|A|^λU|A|^{1-λ}. $$ In 2007, Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under $Δ_λ$: for every $0<λ<1$, $$ \|A^*A-AA^*\|_{F} \ \ge\ \|Δ_λ(A)^*Δ_λ(A)-Δ_λ(A)Δ_λ(A)^*\|_{F}. $$ If this inequality held, then the iterated self-commutator norms $$ \Bigl\{\bigl\|Δ_λ^{\,m}(A)^*Δ_λ^{\,m}(A) -Δ_λ^{\,m}(A)Δ_λ^{\,m}(A)^*\bigr\|_F\Bigr\}_{m\in\mathbb N} $$ would form a nonincreasing sequence and necessarily converge to $0$. In this paper we provide a counterexample, thereby disproving the conjecture. We also obtain the quantitative bounds $$ \sqrt{\frac32}\ \le\ \sup_{\substack{A\in\mathbb{M}_n(\mathbb{C}),\ A^*A\neq AA^*\\ 0<λ<1}} \frac{\|Δ_λ(A)^*Δ_λ(A)-Δ_λ(A)Δ_λ(A)^*\|_F}{\|A^*A-AA^*\|_F} \ \le\ 2. $$

In a uniform, source-free, and unbounded medium, Maxwell's equations require electromagnetic waves to be purely transverse. However, when a beam of light is tightly focused or strongly confined, a longitudinal field component can emerge. Strong longitudinal fields enable many novel phenomena and applications, including single molecule detection, near-field imaging, and high-resolution photolithography. Although the behavior of the longitudinal electric (LE) field component of the electromagnetic field in ordinary waveguides is well established, judicious nanostructuring offers unprecedented control over its strength as well as spatial and spectral distribution. Here, we demonstrate a full-vectorial theory and experimental results showing that for specially designed waveguides, such as one-dimensional antislot photonic crystal (PhC) waveguides, the LE field can hybridize with the transverse electric (TE) field in the waveguide and can be subsequently decomposed into independent polarizations through far field imaging. When the in-plane mirror symmetry of a PhC unit cell is broken, coupling between LE and TE modes produces two hybrid LE-TE modes and opens a new photonic bandgap. The LE-TE composition of the hybrid modes and the width of the resulting bandgap can be tuned by changing the rotation angle of the antislot within the unit cell. We show that a 45-degree antislot orientation with respect to the propagation direction yields hybrid modes with the largest LE field contribution and the widest geometry-induced bandgap. Such engineered PhC waveguides enable new on-chip photonic functionalities, including in-plane angle-invariant dipole coupling in quantum systems, higher-order polarization-division multiplexing, and enhanced control of light flow.