This short note is motivated by a recently discovered connection between a drift-diffusion process in $n$-dimensional Euclidean space with a divergence-free drift sampled from a stationary and isotropic Gaussian ensemble of critical scaling on the one hand, and a geometric Brownian motion on $\textbf{SL}(n)$ on the other hand. This can be seen as a tensorial form of a stochastic exponential; it thus is naturally intermittent, which transfers to the pair distance of the drift-diffusion process. In this note, we quantify the intermittency of the geometric Brownian motion $\{F_τ\}_{τ\ge0}$ on $\textbf{SL}(n)$ also in dimensions $n>2$. We do so in two (related) ways: 1) by identifying the exponential growth rate for the $2p$-th stochastic moment $\mathbb{E}|F_τ|^{2p}$ with its anomalous dependence on $p$ (and $n$), and 2) by quantifying a non-tightness of $|F_τ|^2/\mathbb{E}|F_τ|^2$ as $τ\uparrow\infty$. It is the second property that transmits to the drift-diffusion process. The arguments rely on stochastic analysis: We write $\{F_τ\}_{τ\geq 0}$ as the solution of $dF=F_τ\circ dB$ with $\{B_τ\}_{τ\geq 0}$ a Brownian motion on the Lie algebra $\mathfrak{sl}(n)$. The arguments leverage isotropy: The diffusion projects onto the spectrum of the Gram matrix $G=F^*F$, as captured by ${\rm tr}G^p$.

Let $f(N)$ denote the least integer $k$ such that, if $G$ is an abelian group of order $N$ and $A \subseteq G$ is a uniformly random $k$-element subset, then with probability at least $\tfrac12$ the subset-sum set $\{ \sum_{x \in S} x : S \subseteq A \}$ equals $G$. In 1965, Erdős and Rényi proved that for all $N$, $$ f(N) \le \log_2 N + \left(\frac{1}{\log 2}+o(1)\right)\log\log N. $$ Erdős later conjectured that this bound cannot be improved to $f(N)\le \log_2 N+o(\log\log N)$. In this paper we confirm this conjecture by showing that, for primes $p$, $$ f(p)\ge \log_2 p+\left(\frac{1}{2\log 2}+o(1)\right)\log\log p. $$

Wi-Fi channel state information (CSI), which is originally acquired for communication purposes, has recently been reused for sensing and radar-like functionalities. However, in practical Wi-Fi systems with independent clocks at the transmitter and receiver, the lack of a common delay and phase reference fundamentally precludes phase-coherent radar-like delay--Doppler analysis. By exploiting the line-of-sight (LoS) path component, i.e., the earliest-arriving direct path, as an over-the-air (OTA) reference for delay and phase, we propose an OTA LoS-path referencing scheme, termed LoSRef, that enables delay calibration and phase alignment under this practical constraint. Unlike conventional Wi-Fi bistatic radar systems that rely on wired reference signals or dedicated reference antennas, the proposed LoSRef-based framework enables phase-coherent bistatic radar-like operation that can be integrated into typically deployed Wi-Fi systems. Through human gait and respiration experiments in indoor environments, we demonstrate that phase-coherent channel impulse responses and corresponding delay--Doppler responses can be obtained using only commodity Wi-Fi devices. This enables physically interpretable human motion sensing, including gait-induced range variation and respiration-induced sub-wavelength displacement, as well as the extraction of target-induced dynamics up to 20 dB weaker than dominant static multipath components.

Stationary entanglement between the motion of macroscopic objects and light is a long-standing goal of quantum optomechanics, with implications for both fundamental tests of quantum physics and emerging quantum technologies. We report the generation of quantum entanglement between the center-of-mass motion of a nanosphere levitated in an optical tweezer inside an optical cavity and the electromagnetic field. By heterodyne detection, we reconstruct the full set of optomechanical correlations and observe a violation of separability bounds between the mechanical motion and the quadratures of a propagating optical mode. This demonstrates the distribution of nonclassical correlations beyond the interaction region. The entanglement is generated at room temperature and remains robust over a broad range of parameters. Our results establish levitated optomechanical systems as a promising platform for continuous-variable quantum communication and for tests of macroscopic quantum physics.

Water drops sliding on hydrophobic surfaces spontaneously separate charges at their rear. It is unclear how this charge separation affects the contact angles of a sliding drop. We slide grounded and insulated drops on hydrophobic surfaces at low capillary numbers (\leq 10^{-4}). We find that drop charge leads to spontaneous electrowetting, which decreases the contact angles. Additionally, the deposited charges lead to a surface charge effect and decrease the contact angle. Both phenomena compensate each other at the receding contact line, resulting in an insignificant change in the receding contact angle of a sliding drop.

We prove the existence of Ulrich sheaves on the Hilbert scheme of two points on a polarized K3 surface or an abelian surface. The construction proceeds by descending Ulrich bundles on the surface to the symmetric square and lifting them to the Hilbert square via the crepant Hilbert--Chow resolution. Finally, we estimate a bound for Ulrich complexity of the Hilbert Square.

Continuous Integration (CI) provides early feedback by automatically building software, but long build durations can hinder developer productivity. CI services use caching to speed up builds by reusing infrequently changing artifacts, yet little is known about how caching is adopted in practice and what challenges it entails. In this paper, we conduct a large-scale empirical study of CI caching in Travis CI, analyzing 513,384 builds from 1,279 GitHub projects. We find that only 30% of projects adopt CI caching, and early adopters are typically more mature, with more dependencies, commits, and longer CI lifespans. To understand non-adoption, we submit pull requests enabling caching in non-adopting projects, and nearly half are accepted or merged. Developer feedback indicates that non- or late adoption mainly results from limited awareness of CI caching support. We further study cache maintenance and identify five common activities, performed by 24% of cache-enabled projects. While one-third of projects see substantial build-time reductions, cache uploads occur in 97% of builds, and 27% of projects contain stale cached artifacts. An analysis of reported caching issues shows developers mainly struggle with corrupted or outdated caches and request broader caching features. Overall, CI caching does not benefit all projects, requires ongoing maintenance, and is more complex in practice than many developers expect.

Magnetic hopfions are three-dimensional topological solitons -- knotted, vortex-like spin configurations. In chiral magnets, hopfions can appear as isolated structures or they can be linked to skyrmion strings. Previous studies employed a sophisticated protocol and a special sample geometry to nucleate such hopfions linked to one or a few skyrmion strings. Here, we introduce an electric-current-assisted nucleation protocol that is simple and independent of the sample shape and size. The resulting hopfions exhibit extraordinary stability in the presence of both positive and negative magnetic fields, in perfect agreement with micromagnetic simulations. We also present a comprehensive framework for classifying hopfions, skyrmions, and merons by deriving the corresponding homotopy group.

The temporal degree-of-freedom of light is incredibly powerful for modern quantum technologies, enabling large-scale quantum computing architectures and record key-rates in quantum key distribution. However, the generalized measurement of large and complex quantum superpositions of the time-of-arrival of a photon remains a unique experimental challenge. Conventional methods based on unbalanced Franson-type interferometers scale poorly with dimension, requiring multiple cascaded devices and active phase stabilization. In addition, these are limited by construction to a restricted set of phase-only superposition measurements. Here we show how the coupling of spatial and temporal information inside a single multi-mode fiber can be harnessed to program completely generalized measurements for high-dimensional superpositions of photonic time-bin. Using the multi-spectral transmission matrix of the fiber, we find special sets of spatial modes that experience distinct dispersive delays through the fiber. By exciting coherent superpositions of these spatial modes, we engineer the equivalent of large, unbalanced multi-mode interferometers inside the fiber and use them to perform high-quality measurements of arbitrary time-bin superpositions in up to dimension 11. The single fiber functions as a scalable, common-path interferometer for time-bin qudits that significantly eases the experimental overheads of standard approaches based on unbalanced Franson-type interferometers, serving as an essential tool for quantum technologies that harness the temporal properties of light.

We study the local evolution of the Universe around galaxy formation sites in the EAGLE50 large-volume reference simulation. Using the reduced inertia tensor (r-TOI), we followed the anisotropic evolution of initially spherical Lagrangian volumes (LVs) centred at galaxy formation sites, both in dark matter (DM) and in cold baryons (CB), from very high redshift $z=15$ onward. We describe LV deformation in terms of the r-TOI eigen-directions, principal axes, their derived shape parameters, and the timescales for the freezing-out of these principal directions and axes. Of particular interest are the age of the Universe, $t_{\rm U}$, when the local Cosmic Web (CW) spine emerges, and that when anisotropic DM mass arrangements (i.e., migrant mass flows) cease. We find that the shapes LVs acquire along their evolution affect the halo and stellar mass of their central galaxy: prolate-shaped LVs show a tendency to host low-mass galaxies at $z=0$, while massive galaxies tend to form within triaxial or oblate LVs. Also, the local CW spine tends to set in earlier on in LVs that are to host massive galaxies than in those harbouring less massive galaxies. In addition, anisotropic DM-mass rearrangements stop late on average, at $t_{\rm U}\sim 10.5\,$Gyr, and even slightly later for CB. Interestingly, $z=0$ LVs with either flattened configurations in CB or those that are highly prolate in DM, are more likely to host rotation-dominated galaxies. This effect increases from $z=1$ to $z=0$. Finally, the CB spine of LVs that are more likely to host rotation-dominated galaxies emerges at later times.

Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(Σ^k):=\int_ΣF(T_xΣ)\,d\mathcal{H}^k.$$ While a monotonicity formula is not available for critical points, when $k=2$ and $n=3$ we show that the Michael-Simon inequality holds if $F$ is convex and close to $1$ (in $C^1$), meaning that $\mathcal{F}$ is close to the usual area. Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem. As another byproduct, we also show Michael-Simon for another class of integrands which includes the $\ell^p$ norms for $p\in(1,\infty)$. For a general $F$ satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds.

Electric vehicles with multiple motors provide a flexibility in meeting the driver torque demand, which calls for minimizing the battery energy consumption through torque allocation. In this paper, we present an approach to this problem based on approximating electric motor losses using higher-order polynomials with specific properties. To ensure a well-behaved optimization landscape, monotonicity and positivity constraints are imposed on the polynomial models using sum of squares programming. This methodology provides robustness against noisy or sparse data, while retaining the computational efficiency of a polynomial function approximation. The torque allocation problem based on such polynomials is formulated as a constrained nonlinear optimization problem and solved efficiently using readily available solvers. In the nominal case, the first-order necessary conditions for optimality can also be used to obtain a global solution. The performance of the proposed method is evaluated on several certification driving cycles against a grid search-based benchmark. Results show a modest influence on electric energy consumption, while enabling real-time optimization and integration with other vehicle control systems.

To detect ultra-high-energy neutrinos, experiments such as the Askaryan Radio Array and the Radio Neutrino Observatory in Greenland target the radio emission induced by these particles as they cascade in the ice. This is done by, amongst others, using deep in-ice antennas at the South Pole or in Greenland. A crucial step toward this goal is the characterization of the in-ice radio emission from cosmic-ray-induced particle showers. These showers form a primary background for neutrino searches, but can also be used to validate the detection principle and provide calibration signals for in-ice radio detectors. In this work, we use the Monte-Carlo framework FAERIE to perform the first characterization of cosmic ray signals with simulations that incorporate both their in-air and in-ice emissions. We investigate cosmic ray signatures such as their radiation energy, timing, polarization and frequency spectrum and quantify how they depend on shower properties. These results provide key guidelines for cosmic-ray identification and cosmic-ray/neutrino discrimination in future in-ice radio experiments.

We investigate the simultaneous estimation of two optical phases in a three-mode interferometer assisted by optical parametric amplification (OPA). By employing the normally ordered characteristic-function formalism, we analytically obtain all photon-number moments of the output quantum state, enabling an explicit evaluation of the quantum Fisher information matrix for multiparameter phase estimation. In the lossless scenario, we show that uniformly applied OPA significantly enhances the attainable precision beyond that of an unamplified interferometer. By analyzing the second-order correlation functions, we demonstrate that this enhancement originates from the amplification of intra-mode photon-number correlations, rather than from inter-mode correlations. We further extend our analysis to realistic interferometers with photon loss using a purification-based variational approach. Although loss degrades the achievable precision, the OPA-assisted scheme retains a clear advantage for moderate loss, indicating a degree of robustness against dissipation. Our results clarify the physical mechanism underlying OPA-enhanced multiparameter quantum metrology and provide guidelines for optimizing phase estimation protocols in realistic noisy environments.

Understanding fluid flow through porous media with complex geometries is essential for improving the design and operation of packed-bed reactors. Most existing studies focus on spherical packings, having as a consequence that accurate models for irregular interstitial geometries are scarce. In this study, we numerically investigated the flow through a set of packed-bed geometries consisting of square bars stacked on top of each other and arranged in disk-shaped modules. Rotation of each module allows the generation of a variety of geometrical configurations at Reynolds numbers of up to 200 (based on the bar size). Simulations were carried out using the open-source solver OpenFOAM. Selected cases (e.g., $α= 30^\circ$, $\mathrm{Re}_\mathrm{p} = 100, 200$) were compared against Particle Image Velocimetry measurements. Results reveal that, based on the relative rotation angle, the realized geometries can be classified as channel-like ($α\leq 10^\circ$) and lattice-like ($α\geq 15^\circ$), fundamentally altering the friction factor. Furthermore, the maximum friction factor obtained in the creeping regime occurred at $α= 25^\circ$, whereas in the inertial regime, this occurred at $α= 60^\circ$. The module-equivalent diameter, based on the angle-dependent wetted surface area, collapses the friction factor onto the Ergun correlation and yields good permeability predictions for the lattice-like geometries.

In this paper, by means of a standard model problem, we devise an approach to computing approximate dual bounds for use in global optimization of coefficient identification in partial differential equations (PDEs) by, e.g., (spatial) branch-and-bound methods. Linearization is achieved by a McCormick relaxation (that is, replacing the bilinear PDE term by a linear one and adding inequality constraints), combined with local averaging to reduce the number of inequalities. Optimization-based bound tightening allows us to tighten the relaxation and thus reduce the induced error. Combining this with a quantification of the discretization error and the propagated noise, we prove that the resulting discretization regularizes the inverse problem, thus leading to an overall convergent scheme. Numerical experiments illustrate the theoretical findings.

We express a family of basic cellular integrals over moduli spaces of curves explicitly in terms of multiple zeta values, answering a question of Brown. Moreover, we study a priori the weights appearing in these integrals and find a relation that expresses the odd-dimensional integrals in terms of the even-dimensional ones. We also sketch an explanation of this relation in the spirit of Grothendieck's Period Conjecture.

Parametrically driven oscillators can emerge as a basis for the next generation of qubits. Classically, these systems exhibit two stable oscillatory states with opposite phases. Upon quantization, these states turn into a pair of closely spaced Floquet states, which can serve as the logical basis for a qubit. However, interaction with the environment induces phase-slip events which set a limit on qubit coherence. Such phase slips persist even at zero temperature due to a mechanism known as quantum activation \cite{QuantumActivation}. In contrast to conventional tunneling, the quantum activation is described by a {\em real-time} instanton trajectory in the complexified phase space of the system. In this work, we show that the phase-slip rate is exponentially sensitive to weak AC perturbations. The spectrum of the system's response -- captured by the so-called logarithmic susceptibility (LS) -- enables a direct observation of characteristic features of real-time instantons. Studying this spectrum suggests new means of efficient qubit control.

We present the results of customised single-pulsar noise analysis of 27 millisecond pulsars from the second data release of the Indian Pulsar Timing Array (InPTA-DR2). We model various stochastic noise sources present in the dataset using stationary Gaussian processes and estimate the noise budget of the InPTA-DR2 using Bayesian inference, involving model selection, Fourier harmonics selection, and parameter estimation for each pulsar. We check the efficacy of our noise characterisation by performing the Anderson-Darling test for Gaussianity on the noise-subtracted residuals. We find that all 11 pulsars with time baseline $\lesssim2.5\,\text{yr}$ show Gaussian residuals and do not have evidence for any red noise process in the optimal model, except for PSR J1944$+$0907, which shows presence of DM noise. PSRs J0437$-$4715, J1909$-$3744 and J1939$+$2134 show preference for the most complicated noise model, having achromatic and chromatic red noise processes. Only 4 out of 15 pulsars with time baseline $\gtrsim2.5\,\text{yr}$ show significant non-Gaussianity in noise-subtracted residuals. We suspect that this may require more advanced methods to model noise processes properly. A comparative study of six pulsars with data removed near solar conjunctions showed deviations from the parameter estimates obtained with the original dataset, indicating potential bias in red noise processes due to unmodeled solar-wind effects. The results presented in this work remain broadly consistent with the InPTA-DR1 noise budget, with better constraints obtained on noise processes for several pulsars and support for achromatic red noise in PSR J1012$+$5307 due to the extended time baseline.

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-induced-minor-free if no induced minor of $G$ is isomorphic to a member of $\mathcal{H}$, We denote by $W_{t\times t}$ the $t$-by-$t$ hexagonal grid, and by $K_{t,t}$ the complete bipartite graph with both sides of the bipartition of size $t$. We show that the class of $\{K_{t,t},W_{t\times t}\}$-induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer $t$ there exist $ε\in (0,1]$ and $c \in \mathbb{N}$ such that every $n$-vertex $\{K_{t,t},W_{t\times t}\}$-induced minor-free graph with no clique of size $t$ has treewidth at most $2^{c\log^{1-ε}n}$.

Quantum Monte Carlo (QMC) methods deliver highly accurate electronic structure calculations but are computationally intensive. The quantum Monte Carlo kernel library (QMCkl) provides a modular, portable collection of high-performance kernels implementing the core building blocks of QMC calculations. It offers a C-compatible API, supports the TREXIO standard for input, and covers essential QMC kernels including atomic and molecular orbitals, cusp corrections, Jastrow factor, and the necessary derivatives also to perform variational and structural optimization. QMCkl separates algorithmic development from hardware-specific tuning by combining human-readable reference implementations with performance-optimized kernels that produce identical numerical results. The library enables consistent, efficient, and reproducible simulations across different QMC codes and architectures, and achieves substantial speedups in the evaluation of the energy and its derivatives. Beyond QMC, QMCkl can accelerate deterministic quantum chemistry workflows and visualization tools, promoting cross-code interoperability and simplifying high-performance scientific software development.

Europium-doped gallium nitride (GaN:Eu) is a promising platform for classical and quantum optoelectronic applications. When grown using organometallic vapor-phase epitaxy, the dominant red emission from Eu exhibits an inhomogeneous photoluminescence (PL) spectrum due to contributions from several non-equivalent incorporation sites that can be distinguished with combined excitation emission spectroscopy. Energy transfer from the GaN bandgap to the majority site is inefficient, limiting the performance of GaN:Eu LEDs and resulting in an inhomogeneous emission spectrum dominated by disproportionate contributions from minority sites. In this work, we use site-selective spectroscopy to characterize the photoluminescence properties of delta-doped structures with alternating doped and undoped layers of varying thicknesses and demonstrate that they selectively enhance emission from the majority site when compared to uniformly-doped samples. Samples with 2-nm and 10-nm doped layers show much greater PL intensity per Eu concentration as well as more efficient energy transfer to the majority site, which are both highly desirable for creating power-efficient LEDs. Meanwhile, a sample with 1-nm doped layers shows emission only from the majority site, resulting in a narrow, homogeneous emission spectrum that is desirable for quantum technologies. This utilization of delta-doping has the potential to be broadly applicable for engineering desirable defect properties in rare-earth doped semiconductors.

We develop a general method for determining the effective interaction potential between two or more droplets suspended within a fluid phase. Our approach is based on classical density functional theory. Here, we apply the method to determine the interaction potential between oil droplets suspended in water and also consider the influence of adding a third species, alcohol. This ternary mixture is that found in the ouzo beverage. The ouzo system exhibits spontaneous emulsification when the neat spirit is mixed with water. The oil emulsion that forms has been observed to be surprisingly long-lived. Here we show that the alcohol in the system does indeed play a role in making the droplets more stable, by decreasing the oil-water interfacial tension and therefore also the strength of the attractive interactions between droplets. Within our theory, the surfactant nature of the alcohol can be enhanced without changing the bulk fluid thermodynamics. In fact, our theory can be used to model surfactant mixtures. In this model, the effective interaction between pairs of oil droplets can become repulsive, with a free-energy barrier to droplets merging, thus making them stable.

We address the numerical solution of second-order Mean Field Game problems through Newton iterations in infinite dimensions, introduced in [14], where quadratic convergence of the method was rigorously established. Building upon this theoretical framework, we develop new numerical discretization techniques, including both a finite difference and a semi-Lagrangian scheme, that enable an effective computational implementation of the infinite-dimensional iterations. The proposed methods are tested on several benchmark problems, and the resulting numerical experiments demonstrate their robustness, accuracy, and efficiency. A comparative analysis between the two schemes and existing approaches from the literature is also presented, highlighting the potential of Newton-based solvers for MFG systems.

Most real-world networks exhibit a significant degree of modularity. Understanding the effects of such topology on dynamical processes is pivotal for advances in social and natural sciences. In this work we consider the dynamics of Kuramoto oscillators on modular networks and propose a simple coarse-graining procedure where modules are replaced by effective single oscillators. The method is inspired by EEG measurements, where very large groups of neurons under each electrode are interpreted as single nodes in a correlation network. We expose the interplay between intra-module and inter-module coupling strengths in keeping the coarse-graining process meaningful. We show that, when modules are well synchronized, the phase transition from asynchronous to synchronous motion in networks with 2 and 3 modules is very well described by their respective reduced systems, regardless of the network structure connecting the modules. Applications of the method to real networks with small modularity coefficients reveals that the approximation is also very accurate if oscillators in each module are identical. The method reproduces global synchronization patterns despite the low synchronizability of some modules, possibly allowing for the inference of the mean synchrony of each module when individual dynamics are not known.

We present new spherically symmetric solutions of the Einstein-Klein-Gordon equations in a quasi-stationary approximation that describe self-gravitating scalar field configurations around a black hole, including angular momentum number $\ell$. An approach analogous to the one which gives rise to $\ell$-boson stars is used here to construct self-gravitating ``gravitational atoms" with $\ell\ge0$. We refer to these new solutions as {\it noble gravitational atoms}, by analogy with noble atoms, which are characterized by closed electron shells. We show that, in the proper limit, noble gravitational atoms approach $\ell$-boson stars globally, displaying noticeable differences only in a region very close to the event horizon. Noble gravitational atoms with $\ell>0$ sometimes present density maxima located at relatively large radii, with small density close to the horizon for $\ell>1$. Furthermore, they do not always present the typical density spike at the event horizon if $\ell > 0$; on the contrary, they sometimes exhibit a small dip there. When $\ell=0$, a spike can appear, but its contribution to the total mass density is always negligible. The size, density, and lifetime of these objects vary significantly depending on the parameters, being in some cases as large as galaxies, as dilute as dark matter, and as long-lived as the Universe itself.

We derived and implemented the calculation of the one-body reduced density matrix for Aufbau suppressed coupled cluster theory, from which excited state natural orbitals and one-body properties, like atomic populations and dipole moments, are obtained. We utilized the natural orbitals to refine the ASCC solution for simple valence and Rydberg systems, exploring the process of repeatedly solving the ASCC equations in successive natural orbital bases to achieve independence from the starting molecular orbitals. For dipole moments in small molecules where high-level comparison data is available, we find that the accuracy of ASCC essentially matches that of linear response and equation-of-motion coupled cluster as long as care is taken to preserve the response's perturbative completeness.

We describe the loop corrections to supercharges in supersymmetric quantum field theories using the holomorphic twist formalism. We begin by reviewing the relation between supercharge corrections and the "twice-generalized" Konishi anomaly, which corrects the semi-chiral ring. In the holomorphic twist, these corrections appear as BRST anomalies and are computed using the higher operations of an underlying $L_\infty$ conformal algebra. We then apply this formalism to obtain the complete one-loop corrections to the supercharge of four-dimensional Lagrangian supersymmetric gauge theories, including $\mathcal{N}=4$ SYM, where it admits a remarkably compact expression in terms of superfields.

An Erdös matrix $E$ is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum of all the $σ$-traces for permutations $σ$'s). We characterize all Erdös $E$ by the patterns of their zero entries; showing that each such skeleton has at most one $E$. We present an algorithm to find all $n\times n$ Erdös matrices, which finds them up to $n\leqslant 5$ quickly and also size $n=6$. We further show some presently known RCDS matrices to be Erdös.

Nucleon short-range correlations (SRCs) and their high-momentum tails (HMTs) encode key short-range dynamics in nuclei and dense matter. This review provides a concise overview of SRC features relevant to the Equation of State (EOS) of isospin-asymmetric nuclear matter. We summarize empirical and theoretical properties of the single-nucleon momentum distribution $n(k)$, emphasizing the role of the neutron--proton tensor force, the dominance of correlated np pairs, and the enhancement of minority-species HMTs. Links to nucleon effective E-masses, quasi-deuteron components, and orbital entanglement are briefly noted. We examine how SRC-induced HMTs modify kinetic and potential contributions to the EOS in both non-relativistic and relativistic frameworks, including the softening of the kinetic symmetry energy and departures from the isospin parabolic approximation of asymmetric nuclear EOS. Sensitivity to high-momentum components and generalizations to arbitrary dimensions are also highlighted. Implications for heavy-ion reactions are summarized, including effects on particle yields, collective flows, deeply sub-threshold particle production and hard photon emission, driven by modified initial nucleon momentum distributions and abundant high relative-momentum np pairs during the reaction. Finally, we outline SRC-HMT consequences for neutron-star matter, covering proton fractions, tidal deformabilities, $Z$-factors, cooling, and the core--crust transition, as well as possible connections to dark-matter interactions in dense environments.