Continuous-variable quantum systems are central to quantum technologies, with Gaussian states playing a key role due to their broad applicability and simple description via first and second moments. Distinguishing Gaussian states requires computing their trace distance, but no analytical formula exists for general states, and numerical evaluation is difficult due to the exponential cost of representing infinite-dimensional operators. We introduce an efficient numerical method to compute the trace distance between a pure and a mixed Gaussian state, based on a generalized Lanczos algorithm that avoids explicit matrix representations and uses only moment information. The technique extends to non-Gaussian states expressible as linear combinations of Gaussian states. We also show how it can yield lower bounds on the trace distance between mixed Gaussian states, offering a practical tool for state certification and learning in continuous-variable quantum systems.

Questions at the intersection of the AdS/CFT correspondence and quantum information theory motivate the study of projectors in sequences of subalgebras of finite-dimensional commutative associative semisimple algebras $\mathcal{A}$, obtained by incrementally adjoining one generator at each step to produce a non-linear generating set for $\mathcal{A}$. We define degeneracy graphs, which are finite layered tree graphs whose nodes represent projectors in the successive subalgebras. Using combinatorial properties of the degeneracy graph, we give a simple formula for constructing a linear basis of $\mathcal{A}$ in terms of monomials in the generators. The nodes can be labelled by formal variables corresponding to the eigenvalues of the generators added at each layer. We prove that the construction is compatible with the required counting of projectors in $\mathcal{A}$, and give explicit constructions of the projectors in terms of the monomials, in the cases of one- and two-layer degeneracy graphs with arbitrary numbers of nodes. More generally, we provide extensive computational evidence for the invertibility of the matrix relating the proposed monomial basis to the projector basis, by evaluating its determinant. In the 1-layer case, this is a Vandermonde determinant. A simple formula for the non-vanishing determinant in the general layer case is conjectured and supported by the computational data. The construction is illustrated with examples including centres of symmetric group algebras and maximally commuting subalgebras generated by JucysMurphy elements. We outline applications of the monomial basis to algorithms for constructing matrix units in non-commutative semisimple algebras, with relevance to orthogonal bases of multi-matrix gauge-invariant operators and to quantum information theory.

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros.

We compare the structures and methods in the theory of causal fermion systems with generalized trace dynamics and non-commutative geometry. Although the three theories differ on many aspects, they agree in that the geometric structure to be recovered in the continuum limit is not the bare spacetime but a suitable fiber bundle. Furthermore, the comparison leads us to the conclusion that the key innovation in causal fermion systems lies in the manner in which the relation between different spacetime points is encoded. The role of Synge's classical world function $σ(x,y)$ that encodes the geodesic distance between any two points in the manifold, is taken by a generalized two-point correlator. We show that this idea can be transferred to non-commutative geometry and generalized trace dynamics.

Let $G$ be a finite simple graph with edge ideal $I(G)$. For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by products of $q$ pairwise disjoint edges of $G$. It is the Stanley-Reisner ideal of a simplicial complex $\mathsf{MF}^q(G)$, called the $q$-matching-free complex, whose faces are those subsets $F\subseteq V(G)$ for which the induced subgraph $G[F]$ contains no matching of size $q$. We study $\mathsf{MF}^q(G)$ when $G=W(H)$ is a whisker graph. We first characterize purity. If $H$ is bipartite, then $\mathsf{MF}^q(G)$ is pure for all $q$. Otherwise, let $\ell$ denote the length of the smallest odd cycle of $H$ and set $n=|V(H)|$. Then $\mathsf{MF}^q(G)$ is pure if and only if $q<\lceil \ell/2\rceil$ or $q>n-\lfloor \ell/2\rfloor.$ We next determine the exact range of shellability. Let $m=\operatorname{girth}(H)$, with $m=\infty$ if $H$ is acyclic. Then $\mathsf{MF}^q(G)$ is shellable for \[ 1\le q\le \begin{cases} \lceil m/2\rceil, & \text{if } m<\infty,\\ ν(G), & \text{if } m=\infty. \end{cases} \] Consequently, $I(G)^{[q]}$ is Cohen-Macaulay for $1\le q\le\lfloor m/2\rfloor$ when $m<\infty$, and for all $1\le q\leν(G)$ when $m=\infty$. If $m$ is odd, then $I(G)^{[q]}$ is sequentially Cohen-Macaulay for $q=\lceil m/2\rceil$. We further obtain extremal characterizations: $\mathsf{MF}^{2}(G)$ is Cohen-Macaulay if and only if $H$ has no induced $3$-cycle, and $\mathsf{MF}^{\,n-1}(G)$ is Cohen-Macaulay if and only if $H$ is acyclic. Finally, we compute the depth of $I(G)^{[q]}$ for whisker graphs and verify a conjecture on the depth of squarefree powers of whisker cycles in the relevant range.

Computing excitonic spectra in quasicrystal and super-moiré systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic $10^{18}$-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectral functions for systems exceeding one billion lattice sites, several orders of magnitude beyond the capabilities of conventional approaches. Our method combines a tensor-network encoding of the real-space Bethe-Salpeter Hamiltonian with a Chebyshev tensor network algorithm. This strategy bypasses explicit storage of the Hamiltonian while preserving full real-space resolution across widely different length scales. We demonstrate our methodology for one- and two-dimensional super-moiré systems, achieving the simultaneous resolution of atomistic and mesoscopic structures in the excitonic spectra in billion-size systems, showing exciton miniband formation and moiré-induced spatial confinement. Our results establish a real-space methodology enabling the simulation of excitonic physics in large-scale quasicrystal and super-moiré quantum matter.

Very recently, Thejitha, Sellers, and Fathima defined the function $a_{r,s}(n)$, which enumerates the number of multicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. In this paper, we extend the concept to overpartitions.

Next-generation gravitational-wave detectors will make black-hole ringdown an increasingly sensitive probe of small departures from General Relativity in the strong-field regime. This motivates obtaining high-precision predictions of gravitational effective field theory, as spectral shifts can be quite small. Here we perform a focused stress test of the modified-Teukolsky framework by designing two null diagnostics. First, we consider an action with redundant operators that must produce zero first-order vacuum QNM shifts. Second, we exploit a Ricci-flat identity relating two physical cubic Riemann to test such a relation is satisfied by the ringdown spectra obtained. We compute the shifts using two independent numerical approaches: the eigenvalue-perturbation and generalized continued-fraction (Leaver-type) methods. Both null tests are passed across multiple multipoles and overtones, and the control-operator results agree in magnitude with the benchmark values reported in Ref. [1]. These validations support using the framework for obtaining accurate predictions for robust strong-field tests, with straightforward extensions to rotating backgrounds and coupling with matter fields.

The accretion-ejection activities of black holes play a vital role in shaping the Universe. Bright and recurrent black hole X-ray binaries are ideal objects for studying accretion physics across a wide range of accretion rates, providing insights into the understanding of their supermassive counterparts. This short review summarizes X-ray techniques capable of measuring accretion geometry, our current understanding, and open questions. In particular, X-ray spectroscopic studies indicate that the accretion disk can extend close to the innermost stable circular orbit in the bright hard state. Some hints of disk-corona-jet connections are also discussed.

Earth tidal perturbations affecting laser-ranged satellites are critical for refining satellite orbital dynamics modeling, and their accurate computation represents a prerequisite for high-precision fundamental physical effects and geodetic investigations based on satellite orbit analysis. This study focuses on the tidal perturbations induced by the asymmetric responses of LARES 2 and LAGEOS on their orbital nodes and inclinations. Perturbations induced by a total of 402 (392 2nd and 10 3rd-order) earth tide constituents on the two satellites were calculated, based on Kaula's orbital perturbation theory and Lagrange's planetary equations for satellites, considering the frequency dependence of Love numbers. The asymmetric characteristics of tidal perturbations between the two satellites were quantitatively analyzed. The minimum resolutions of orbital inclinations and nodes, used as screening thresholds for significant constituents, were derived from the RMS of overlapping orbit differences using orbital geometry and error propagation law. With these thresholds, 83 significant constituents were identified from the 402. The cumulative effect of the 319 minor constituents was further evaluated, and it was found that their total impact, from coherent superposition, noticeably exceeds the thresholds, thus becoming non-negligible. The results of this study provide accurate tidal perturbation parameters for LARES 2 and LAGEOS, and offer methodological references for the screening of Earth tide constituents in high-precision satellite orbital dynamics research, laying a foundation for subsequent studies on inverting geophysical parameters from satellite orbits and verifying fundamental physical effects, particularly the relativistic Lense-Thirring effect.

All the finite order entire solutions of \begin{equation*} f^n(z)+q(z)e^{Q(z)}f^{(k)}(z+c)=P(z) \end{equation*} are given, where $ q(z) $, $ Q(z), P(z) $ are polynomials, $ k $ and $ n \geq 2 $ are integers, and $ c \in \mathbb{C} \setminus \{0\} $.This solves an open problem of Heittokangas-Ishizaki-Tohge-Wen [Bull. Lond. Math. Soc. 55, 1-77 (2023)].

Hypothetical light scalar particles trigger the superradiant instability around spinning black holes (BHs), causing clouds of scalars to grow around the BH. In the presence of sufficiently strong particle self-interactions (characterized by the decay constant $f$), scalars are ejected from BH orbits, resulting in coherent, non-relativistic emissions that continuously carry away the BH's angular momentum. Parameters exist for which cloud growth is much faster, and scalar depletion is much slower, than the age of the Galaxy. This defines a distinct class of astrophysical sources of scalars, which we call BH scalar sirens -- BHs that persistently emit scalars effectively forever. We compute the scalar background from the expected population of $N_\text{BH}\sim 10^{8}$ isolated stellar-mass BHs in the Milky Way, which are sirens for scalars in the mass range $10^{-13}$--$10^{-11}\,$eV and $f\lesssim 10^{14}$--$10^{9}\,$GeV. This provides a detection target independent of early-universe scalar production or cosmological initial conditions. The generated observable signals are up to two orders-of-magnitude larger than those expected from a misaligned cosmic scalar in this mass range. The energy spectrum of emitted scalars is distinctly broader and at higher velocities (up to $\sim 10^{-1}c$) than that of virialized dark matter, and encodes the mass and spin distributions of the BH population. While stellar-mass Milky Way BHs are our primary target, our framework extends to supermassive, intermediate-mass and light BHs. Given the difficulty of directly observing populations of isolated BHs, scalar emissions offer a novel probe of these otherwise invisible objects, highlighting the potential for joint discovery between scalars and BHs, and broadly motivating searches for scalars over many orders-of-magnitude in mass.

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ into $\mathbb{S}^2$, restricted to the $k=2$ co-rotational setting, admits arbitrarily large numbers of concentrating concentric $n$ bubble profiles. For any $n\in\mathbb{N}$, we construct an $n$-bubble solution concentrating at scales $λ_1(t)\gg λ_2(t)\gg \ldots\gg λ_n(t)$, where $λ_n(t)=t^{-1}\vert \log t\vert^β$, and $λ_j(t)\gtrsim \exp( \int_t^{t_0} λ_{j+1}(s)ds)$, for any $j<n$. Here $β>\tfrac32$ is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs.

The discovery of novel carbon allotropes with tailored thermal and mechanical properties is critical for advanced thermal management. However, exploring the vast configurational space of carbon using \textit{ab initio} calculations remains computationally prohibitive. Driven by the rich topological landscape of carbon, where the competition between $sp, sp^2,$ and $sp^3$ hybridization states dictates material performance, we establish a closed-loop AI framework to explore this complex configurational space. We introduce a hybridization entropy descriptor to guide the search beyond conventional forms. Here, we establish a closed-loop AI framework that synergizes a Large Language Model (LLM) for structural generation with a Machine Learning Potential (MLP) for accelerated evaluation. Leveraging CrystaLLM to generate candidates and an iteratively refined MLP for high-fidelity validation, we screened thousands of structures to identify several stable allotropes with exotic properties. Specifically, we report ``yne-diamond C$_{12}$'' and ``yne-hex-diamond C$_{8}$'', which exhibit extreme thermal anisotropy and ultralow in-plane shear stiffness arising from their mixed $sp$-$sp^3$ hybridization. Furthermore, we discovered a complex $sp$-$sp^2$-$sp^3$ hybridized C$_{12}$ phase that combines metallic conductivity with an anomalous negative Poisson's ratio. Notably, we identified a superhard phase (C16_3) possessing a calculated Vickers hardness (103.3 GPa) exceeding that of diamond 96 GPa). Microscopic analysis reveals that thermal transport in these materials is governed by the interplay between rigid frameworks and flexible linkers. This work expands the known carbon phase space and demonstrates the efficacy of coupling generative AI with machine learning potentials for the accelerated inverse design of functional materials.

In this note we initiate the study of ``twisted holography'' for the dualities involving the BFSS matrix quantum mechanics and the IKKT matrix model in their $N \rightarrow \infty$ limits. We identify the admissible twists of each model, compute their cohomology in the BV-BRST formalism, and identify them -- in the planar limit and in perturbation theory around the trivial background -- with corresponding twists of IIA and IIB string theories, respectively. The twisted gravitational duals make manifest certain infinite dimensional symmetry algebras. In the BFSS example, the dual IIA supergravity twists are also obtained as certain zero mode truncations of the minimal (1/16-BPS) and maximal (1/4-BPS) twists of eleven-dimensional supergravity.

We test the Transient Time Correlation Function (TTCF) method to compute nonequilibrium transport coefficients, highlighting its conceptual and practical difference from the standard time-average approach. While time averages extract transport properties from long stationary trajectories and discard transient dynamics, TTCF adopts the complementary strategy: it exploits the information contained in short-time transients following the onset of an external perturbation, while discarding the long-time evolution once stationarity is reached. We revisit the theoretical framework of TTCF and assess its numerical performance through representative case studies, the Lorentz gas and a many-body system, namely a chain of oscillators with anharmonic pinning potential. By direct comparison with time averages, we show that for the Lorentz gas TTCF yields consistent transport coefficients in both linear and nonlinear regimes at a reduced computational cost. Moreover, the TTCF displays superior precision in the linear-response regime, and remains reliable in non-ergodic situations, revealing the presence of regions of phase space corresponding to different behaviors, as well as the possibility of phase transitions. For the anharmonic chain, we show that TTCF is a scalable and efficient alternative for the numerical study of nonequilibrium transport.

Planar germanium is currently the only semiconducting platform where high-coherence spin qubits and proximity-induced superconductivity have each been demonstrated. Recent research into spin qubits in Ge/SiGe heterostructures has focused on increasing the thickness of the SiGe capping layer, reporting improvements in the electrostatic noise levels. Meanwhile, heterostructures with thinner capping layers remain rather unexplored, despite the potential advantages for proximity-induced superconductivity. Here, we study a Ge/SiGe heterostructure with a thin SiGe cap $d \approx 4\ \mathrm{nm}$ and investigate its viability to host low-noise quantum dots. To keep the thermal budget compatible with superconducting layers, low-temperature oxide deposition processes were developed and implemented for the gate dielectrics. The charge-noise level of fabricated devices is estimated to be $1.8 \pm 1.0\ μ\mathrm{eV}/\sqrt{\mathrm{Hz}}$, comparable to devices fabricated on shallow heterostructures $\left(d \sim 20\ \mathrm{nm}\right)$ with high-temperature deposited oxides. Low charge-noise levels, together with the straightforward integration of superconductors, make this heterostructure an attractive platform for prototyping hybrid semiconducting-superconducting devices.

In this paper, we establish a two-way equivalence between the incompressible Navier- Stokes equation (INSE) and the principle of minimum pressure gradient (PMPG). We prove that a candidate smooth flow field is a solution of the INSE if and only if its instantaneous evolution minimizes, at every instant, the norm of the pressure force, required to enforce incompressibility. We show that the PMPG is precisely the minimization formulation of the Leray-Helmholtz projection. Any admissible instantaneous evolution (e.g., onset of separation) resulting from the INSE necessarily minimizes the PMPG cost. Conversely, any other kinematically admissible evolution, requiring a strictly larger pressure force to ensure the same constraints, does not satisfy the INSE. Thus, the PMPG offers a variational perspective through which intricate incompressible flow behaviors may be interpreted. In a finite-dimensional setting with divergence-free modes, we show that the PMPG yields the same dynamics as classical Galerkin projection. Moreover, the PMPG provides a natural generalization of classical Galerkin projection beyond linear modal expansions, accommodating nonlinear and non-modal representations. We then examine the relation between instantaneous dynamical minimization and steady variational selection, including its connection to the variational theory of lift. Motivated by these observations, we formulate conjectures concerning necessary conditions for stability and the convergence of Navier-Stokes solutions to Euler's in the vanishing-viscosity limit.

We establish the convexity of the weighted twisted Mabuchi K-energy functional along geodesics in the finite energy space $\mathcal{E}^{1,T}(X,ω)$, covering the case of divisors with mixed cusp and conic singularities. We then prove that coercivity (relative to the complex torus) of this functional is an open condition under cone angle perturbations. This is obtained from a general result of independent interest, which shows the stability of the coercivity under perturbations by certain twist currents. In particular, this yields the openness for the existence for cscK cone metrics and proves that coercivity at the cusp limit implies existence of cscK cone metrics for small cone angles.

Recent works by the second author and Maxwell et al. have shown that the Einstein-scalar field conformal constraint equations are highly complex and generally intractable, even in the vacuum case. In this article, to gain a clearer understanding and offer a new perspective, we study these equations under special assumptions: the manifold $(M,g)$ is harmonic and all data are radial. In this setting, the system reduces to a single nonlinear equation and is completely resolved in the standard cases. In particular, on the sphere, our results reveal phenomena that contrast with the well-known achievements on compact manifolds without conformal Killing vector fields, including nonexistence of solutions in the near-CMC regime and instability when the mean curvature is non-constant. By contrast, on Euclidean or hyperbolic manifolds, the equations are always solvable, with all expected properties of solutions satisfied. These findings support the view that, although the conformal method appears to present some drawbacks on compact manifolds, it remains a promising tool for parametrizing solutions to the constraint equations on asymptotically flat and hyperbolic manifolds in arbitrary mean curvature regimes. In this article, we also investigate the sign of mass, showing that the ADM and asymptotically hyperbolic mass of vacuum constraint solutions can take arbitrary sign when the decay rate of symmetric $(0,2)$-tensor $k$ at infinity is critical. Finally, most solution classes in our framework are explicit, providing a variety of models in general relativity and offering insights into the behavior of initial data, particularly in numerical applications.

I--MR charts commonly estimate the process standard deviation $σ$ via the span-2 average moving range divided by the unbiasing constant $d_2$; unlike the unbiased sample standard deviation ($S/c_4$), this estimator depends on ordering through adjacency, so permuting a fixed sample changes it. We formalize this by introducing an independent uniformly random permutation and applying the law of total variance, yielding an exact decomposition into a values component (variance of the permutation mean) and an adjacency component (expected conditional variance over permutations). The permutation mean is order-invariant and equals $\GMD/d_2$, where $\GMD$ is the sample Gini mean difference. Under i.i.d.\ Normal sampling, both components admit closed forms; the adjacency fraction converges to $0.3813$, and the familiar asymptotic efficiency loss relative to $S/c_4$ is almost entirely an adjacency effect.

Diffraction tomography is a widely used inverse scattering technique for quantitative imaging of weakly scattering media. In its conventional formulation, diffraction tomography assumes monochromatic plane wave illumination. This assumption, however, represents a simplification that often fails to reflect practical imaging systems such as medical ultrasound, where focused beams are used to scan a region of interest of the human body. Such measurement setups, combining focused illumination with scanning, have not yet been incorporated into the diffraction tomography framework. To bridge this gap, we extend diffraction tomography by modeling incident fields as Herglotz waves, thereby incorporating focused beams into the theory. Within this setting, we derive a new Fourier diffraction relation, which forms the basis for quantitative tomographic reconstruction from scanning data. Using this result, we systematically analyze how different scan geometries influence the reconstruction.

Diffraction tomography aims to recover an object's scattering potential from measured wave fields. In the classical setting, the object is illuminated by plane waves from many directions, and the Fourier diffraction theorem provides a direct relation between the Fourier transform of the object's scattering potential and the Fourier transform of the measurements. In many practical imaging systems, however, focused beams are used instead of plane waves. These beams are then translated across the object to bring different regions of interest into focus. This article discusses what information about the scattering potential can be extracted from such measurements. As in the classical case, the analysis is based on a recently derived Fourier diffraction relation that relates the measurements to the Fourier coefficients of the scattering potential. However, this relation does not immediately provide an explicit reconstruction formula, but instead leads to a linear equation system for the Fourier coefficients. We therefore prove in this work that all Fourier coefficients appearing in these relations are in dimensions higher than two generically uniquely determined. In the two-dimensional case, on the other hand, only a particular subset of the Fourier coverage is uniquely recoverable, while on the remaining region distinct coefficients may produce identical data.

Building on work of Briggs, Grifo and Pollitz arXiv:2506.10827, we give an example of two cohomological support varieties of monomial ideals which are not unions of linear subspaces. We provide a procedure for the computation of the cohomological support varieties of certain other monomial ideals - including those with homogeneous generators - with improved computational efficiency, leading to a computer-assisted verification of the existence of a third support variety of a monomial ideal which is not a union of linear subspaces and a computer-assisted proof of a classification of cohomological support varieties of homogeneous monomial ideals over $\mathbb{Q}$ with 6 generators.

We report a direct demonstration of quantum-enhanced sensing in the Fourier domain by comparing single- and two-photon interference in a fiber-based interferometer under strictly identical noise conditions. The simultaneous acquisition of both signals provides a common-mode reference that enables a fair and unambiguous benchmark of quantum advantage. Spectral analysis of the interferometric outputs reveals that quantum correlations do not increase the amplitude of the modulation peak, but instead lower the associated noise floor, resulting in the expected 3 dB improvement in signal-to-noise ratio. This enhancement persists in the sub-shot-noise regime, where the classical signal becomes buried in the spectral background while the two-photon contribution remains resolvable. These observations establish Fourier-domain quantum super-sensitivity as an operational and broadly applicable resource for precision interferometric sensing.

Electron diffraction from nanogratings provides a platform for free-electron interferometry, yet controlled manipulation of electron spin in such geometries remains largely unexplored. In particular, the role of the self-generated magnetic field arising from electron motion and the feasibility of coherent spin control without disrupting diffraction coherence have not been quantitatively investigated. In this article, a self-consistent Maxwell-Pauli framework is developed to study spin-resolved electron diffraction from nanogratings in the presence of magnetic fields. The model incorporates geometric confinement, image-charge interactions, self-generated magnetostatic fields, and externally applied magnetic fields. Numerical simulations show that the intrinsic magnetic self-field produced by the electron probability current is several orders of magnitude too weak to induce measurable spin mixing, demonstrating that nanogratings act as spin-conserving beam splitters under field-free conditions. When a uniform magnetic field is applied upstream of the nanograting, coherent Larmor precession enables controlled spin rotation without modifying the diffraction geometry or degrading coherence. The magnetic field required for a $π$ spin rotation scales inversely with the interaction length and electron de Broglie wavelength $λ_{dB}$. Furthermore, a downstream nonuniform magnetic field applied after the nanograting imparts a spatially varying Zeeman phase, producing opposite transverse momentum shifts for the two spin components. The spin-dependent transverse dynamics is analyzed using Husimi Q-function phase-space maps, which visualize spin-dependent population redistribution and momentum separation. The proposed approach enables tunable spatial separation of spin-resolved free electron beams and establishes an all-magnetic route for coherent spin rotation, control, and interferometry.

This study numerically investigates magnetisation reversal processes driven by an external magnetic field in three-dimensional antiferromagnetic spin models with weak random field disorder. Considering an extremely weak disorder and low temperature, we observe a step-wise hysteresis loop and the appearance of short magnetisation bursts of a characteristic triangular shape; the number of bursts increases with disorder, indicative of Barkhausen-type noise. These phenomena are attributed to the simultaneous reversal at a given external field of segments composed of spins with identical neighbourhoods. A local random field orients one or more spin neighbours, resulting in small, ferromagnetic-like clusters distributed throughout the system. As disorder increases, these clusters may merge to form a labyrinthine structure within the antiferromagnetic background, facilitating brief avalanche propagation. The results demonstrate that, compared with familiar random-field ferromagnets, the observed antiferromagnetic Barkhausen noise and the related avalanche sequence have a profoundly different structure, organised into peaks associated with the transition between magnetisation plateaus. They exhibit prominent cyclical trends and disorder-dependent multifractal fluctuations, with the singularity spectrum quantifying the degree of disorder. The activity avalanches exhibit scale invariance resembling that recently found in experiments with disordered ferr\textit{i}magnets and martensites, as well as in quantum Barkhausen noise, which are associated with active geometric regions rather than individual-spin dynamics. The observed scaling behaviour is interpreted in terms of self-organised critical dynamics.

We describe a simple Monte Carlo method for estimating $π$ by tossing a coin. Although the underlying Catalan-number series identities appear implicitly in the probability theory literature, the interpretation of $\fracπ{4}$ presented here seems to be new.

Spectral and spatial fringes in polarized light are produced by the interference of transmitted and reflected waves at the interface between materials with different indexes of refraction. These instrumental artifacts can affect the accuracy of optical designs conceived for high-sensitivity spectroscopy and polarimetry. We consider the principal sources of these artifacts and the possible design pathways to mitigate them. In order to do so, we have developed an approximate yet agile treatment of the problem of the transmission and reflection of light in birefringent materials, which fundamentally relies on the assumption of small birefringence of the modeled materials for its implementation. The comparison of our results with those from more rigorous treatments, such as Berreman calculus, thus also serves as a test of the limits of the small-birefringence approximation in optical design applications. The treatment presented in this work is limited to isotropic materials and uniaxial crystals, which are the most common types of optics employed in polarimetric instrumentation. An extensive set of modeling examples is provided to illustrate the salient characteristics of polarization fringes and their dependence on optical design parameters.

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetries. Specifically, we consider (1) linear maps on $M_d(\mathbb{C})$ which are covariant under conjugation by unitary symplectic matrices $S$, and (2) $d\otimes d$ bipartite states which are invariant under $S\otimes S$ or $S\otimes \overline{S}$ actions, each parametrized by two real variables. We provide a complete characterization of all $k$-positivity and decomposability conditions for these maps and explicitly compute the Schmidt numbers for the corresponding bipartite states. In particular, our analysis yields a broad class of PPT states with Schmidt number $d/2$ and the first explicit constructions of (optimal) $k$-positive indecomposable linear maps for arbitrary $k=1,\ldots, d/2-1$, achieving the best-known bounds. Overall, our results offer a natural and analytically tractable framework in which both strong forms of positive indecomposability and high degrees of PPT entanglement can be studied systematically. We present two further applications of symplectic group symmetries. First, we show that the PPT-squared conjecture holds within the class of PPT linear maps that are either symplectic-covariant or conjugate-symplectic-covariant. Second, we resolve a conjecture of Pal and Vertesi concerning the optimal lower bound of the Sindici-Piani semidefinite program for PPT entanglement.