We establish criteria under which stochastic networks in a Markovian environment stabilize, thus confirming Conjecture 7.2 from Levine-Greco [GL23]. The networks evolve on finite connected graphs $G=(V,E)$, and their dynamics are encoded by $V \times V$ toppling matrices $M$, whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter $ρ$ which depends on the largest eigenvalue of the matrix $M+αI$, with $α=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$.

We introduce a theory of uniform K-stability for big line bundles on smooth projective varieties. This extends the existing theory both for varieties with ample line bundles, and for varieties with big anticanonical class. Our main result gives a valuative characterisation of uniform K-stability, through finite collections of divisorial valuations. We further prove that uniform K-stability is preserved under pullbacks and certain pushforwards, which implies that uniform K-stability is well-defined at the level of b-divisors.

Let $A$ be a Dedekind domain of characteristic zero such that for each height one prime ideal $\mathfrak{p}$ in $A$, the local ring $A_{\mathfrak{p}}$ has mixed characteristic with finite residue field. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e., $\operatorname{deg} A=\underline{0}\in \mathbb{N}^n$ and $\operatorname{deg}(X_j)=e_j\in \mathbb{N}^n$. Let $I$ be a $\mathfrak{C}$-monomial ideal of $R$ and let $M:= H^i_I(R)=\bigoplus_{\underline{u}\in \mathbb{Z}^n}M_{\underline{u}}$. Recently, the second author and S. Roy [2025, J. Algebra 681, 1-21] proved that for a fixed $\underline{u}\in\mathbb{Z}^n$, the Bass numbers $μ_i(\mathfrak{p},M_{\underline{u}})$ are finite for each prime ideal $\mathfrak{p}$ in $A$ and for every $i\geq 0$. Let for a subset of $U$ of $\mathcal{S}=\{1, \ldots, n\}$, define a block to be the set $\displaystyle\mathcal{B}(U)=\{\underline{u} \in \mathbb{Z}^n \mid u_i \geq 0 \mbox{ if } i \in U \mbox{ and } u_i \leq -1 \mbox{ if } i \notin U \}$. Note that $\bigcup_{U\subseteq \mathcal{S}}\mathcal{B}(U)=\mathbb{Z}^n$. In this article, the main result we establish is that for a fixed prime ideal $\mathfrak{p}$ in $A$ and $i\geq 0$, the set of Bass numbers $\{μ_i(\mathfrak{p},M_{\underline{u}})\mid \underline{u}\in \mathbb{Z}^n\}$ is constant on $\mathcal{B}(U)$ for each subset $U$ of $\{1, \ldots, n\}$. Our idea is to prove this by carrying out a comprehensive study of the structure theorem for the graded components of $M$ when $A$ is a complete DVR of mixed characteristic with finite residue field.

We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP$_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations in the Jordan-normal basis of the defective matrix at an EP$_n$, we show that the upper-$k$ Hessenberg structure of the perturbation directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto ε^{1/k}$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP$_3$s in P- and C-symmetric systems are restricted to at most $\sim ε^{1/2}$ branch points, while PT-symmetric systems generically support EP$_3$s with the strongest possible singularities (viz. $\sim ε^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band C- and P-symmetric models, establishing the existence of EP$_4$s with $\sim ε^{1/4}$ singularities.

Reactive sputtering is a plasma-based technique to deposit a thin film on a substrate. This contribution presents a novel parameter-interval estimation method for a well-established model that describes the uncertain and nonlinear reactive sputtering process behaviour. Building on a proposed monotonicity-based model classification, the method guarantees that all parameterizations within the parameter interval yield output trajectories and static characteristics consistent with the enclosure induced by the parameter interval. Correctness and practical applicability of the new method are demonstrated by an experimental validation, which also reveals inherent structural limitations of the well-established process model for state-estimation tasks.

The method initiated by Wentzel, Kramers, and Brillouin to find approximate solutions to the Schrödinger equation lies at the origin of the spectacular development of microlocal and semiclassical analysis. When used naively, the approach appears to break down at caustics, but Maslov showed how a simple generalization could overcome this difficulty. In this paper, after a partial historical review, we take advantage of more recent advances in microlocal analysis to present a unified treatment of this generalized Maslov-WKB method, using a microlocal sheaf-theoretic approach. This framework provides a rigorous proof of the Bohr Sommerfeld Einstein Brillouin Keller quantization conditions for the eigenvalues of general semiclassical operators (pseudodifferential and Berezin Toeplitz) in one degree of freedom. We also review some applications and extensions.

Parametric uncertainty in nonlinear dynamical systems can fundamentally alter bifurcation behaviour, leading to qualitative response changes. Predicting operating margins/envelopes under such uncertainties is critical but challenging: conventional uncertainty quantification (UQ) methods struggle to efficiently propagate uncertainties across bifurcation boundaries, where response gradients become singular and solution branches emerge/vanish. We present a general UQ framework for bifurcation analysis of nonlinear dynamical systems with proportional parametric uncertainty, which systematically integrates continuation methods with parametric sensitivities and extremal conditions. The approach uses a two-step scheme: first, the loci of extremal response points are traced as the uncertainty domain is expanded from a deterministic reference point; second, these extremal points are tracked as the bifurcation parameter varies, thus determining the maximum and minimum response margins throughout. The continuation problem scales linearly with the number of uncertain parameters, enabling efficient analysis. The method is demonstrated on a two-degree-of-freedom nonlinear oscillator exhibiting a range of bifurcation phenomena, including multiple solutions, modal interactions, and symmetry breaking. In all cases, the framework efficiently captures uncertainty-induced shifts in bifurcation boundaries and response margins. Notably, the method reveals that parametric uncertainty induces topological changes in the bifurcation structure, including the emergence of an isolated response branch that is absent in the deterministic system with the reference parameters.

We study Generalized Proca theories inspired by the recent regularised Proca theory of four-dimensional Gauss-Bonnet gravity. By abandoning the rigid constraints typically imposed by specific regularization schemes, we treat the coefficients of the terms in the action as free parameters. This approach uncovers a broader solution space that admits static and spherically symmetric black hole solutions characterized by primary hair, where, surprisingly, the cosmological constant arises naturally as a constant of integration even in the absence of a bare cosmological term.

We develop structure-preserving time integration schemes for Gaussian wave packet dynamics associated with the magnetic Schrödinger equation. The variational Dirac--Frenkel formulation yields a finite-dimensional Hamiltonian system for the wave packet parameters, where the presence of a magnetic vector potential leads to a non-separable structure and a modified symplectic geometry. By introducing kinetic momenta through a minimal substitution, we reformulate the averaged dynamics as a Poisson system that closely parallels the classical equations of charged particle motion. This representation enables the construction of Boris-type integrators adapted to the variational setting. In addition, we propose explicit high-order symplectic schemes based on splitting methods and partitioned Runge--Kutta integrators. The proposed methods conserve the quadratic invariants characterizing the Hagedorn parametrization, preserve linear and angular momentum under symmetry assumptions, and exhibit near-conservation of the averaged Hamiltonian over long time intervals. Rigorous error estimates are derived for both the wave packet parameters and observable quantities, with bounds uniform in the semiclassical parameter. Numerical experiments demonstrate the favorable long-time behavior and structure preservation of the integrators.

We investigate a class of interacting dark energy (IDE) models arising from density-driven spontaneous symmetry breaking in a conformally coupled scalar-tensor framework. In this construction, the dark matter-scalar interaction is dynamically activated as the cosmological density evolves, and the redshift dependence of the coupling follows a logistic profile whose steepness is determined by the local curvature of the symmetry-breaking potential. Working in the controlled adiabatic tracking regime, we implement the resulting epoch-dependent interaction in a perturbative background close to $Λ$CDM and confront the model with late-time cosmological data, including Planck 2018 CMB lensing reconstruction, redshift-space distortions, and Pantheon+SH0ES supernova data. We analyze realizations in which the activation index is allowed to vary and compare them with a restricted realization in which it is fixed to the canonical quadratic minimum value, thereby probing the structural role of the activation profile. We find no statistically significant preference for interaction over $Λ$CDM; current observations constrain the model to a hierarchical regime in which the scalar remains heavier than the Hubble scale at activation and background deformations remain perturbatively small. Allowing the activation index to vary preserves an extended degeneracy direction in parameter space, whereas fixing it removes this freedom and leads to a contraction of the allowed posterior region once geometric and growth data are combined. Our results delineate the viable parameter regime of symmetry-breaking IDE and clarify the structural distinction between microphysically motivated scalar-tensor realizations and phenomenological interacting models.

We revisit the double adiabatic evolution equations and extend them to the relativistic and ultrarelativistic regimes. We analytically solve the relativistic, time-dependent drift kinetic equation for a homogeneous, magnetized, collisionless plasma and obtain a solution explicitly dependent on the magnetic field and density variations. In the case of an initial relativistic Maxwellian distribution, a natural extension to an anisotropic Maxwell-Jüttner is obtained. We calculate the moments of this time-dependent solution and obtain analytical expressions for the evolution of the perpendicular and parallel pressures in the ultrarelativistic case. We numerically solve the moment equations in the relativistic case and obtain general expressions for the double-adiabatic equations in this regime. We confirm our results using fully kinetic particle-in-cell simulations of shearing and compressing boxes. Our findings can be readily applied to relativistic species including cosmic-rays and electron-positron pairs, present in astrophysical plasmas like pulsar wind nebulae, astrophysical jets, black hole accretion flows, and Van Allen radiation belts.

Reflecting intelligent surface (RIS) is a promising technology for 6G mobile communications. However, identifying the niche of RIS within the mobile networks is a challenging task. To mitigate the escalating system complexity of mobile networks, we propose the concept of Intelligent Reflection as a Service (IRaaS), and discuss its system architecture, enabling technologies, and deployment strategy, respectively. By leveraging technologies such as resource pooling, service based architecture (SBA), cloud infrastructure, and model-free signal processing, IRaaS empowers telecom operators to deliver on-demand intelligent reflection services without a radical update of current communication protocols. In addition, IRaaS brings a novel deployment strategy that creates new opportunities for the vendors of intelligent reflection service and balances the interests of both telecom operators and property owners. IRaaS is expected to speed up the rollout of RIS from both technical perspective and commercial perspective, fostering an authentic smart radio environment for future mobile communications.

We perform a cluster expansion in the canonical ensemble with periodic boundary conditions, introducing a new choice of polymer activities that differs from the standard ones. This choice leads to an improved bound for the convergence of the cluster expansion, which we compare with the known one. We also recover the irreducible Mayer coefficients for the thermodynamic free energy. The results presented here can also be applied to the case of zero boundary conditions and to the convergence of correlation expansions.

The properties and spatial distribution of large-scale structures of the solar corona determine the observed solar wind structure at 1 au. Coronal holes are a major source of fast solar wind, an important geo-effective component, and appear as large dark patches in extreme ultraviolet images. Solar observatories provide images of the solar corona at different wavelengths, enabling identification of coronal hole morphology and other large-scale structures along a given line of sight. The problem is that although models exist, few work in real time, separate coronal holes from other dark features, or are fully automatic and suitable for comparison with models. The main goal of this work is to develop an automatic threshold-based coronal hole detection tool across solar cycles 23, 24, and 25 using artificial intelligence. The only user input is the date, enabling retrieval of the threshold value used to detect coronal hole contours in line-of-sight extreme ultraviolet images from SDO/AIA and SoHO/EIT. We retrieve data affecting the threshold due to contrast changes from the Heliophysics Events Knowledge database for large-scale features such as active regions, solar flares, coronal mass ejections, and filaments, and engineer them to train the neural network model (POP-CORN). The model input comprises categorical features of large-scale structures in the solar corona, including spatial distribution and properties such as solar flare class by intensity. The neural network model is trained to achieve higher accuracy and determines the threshold needed to detect coronal holes, allowing their boundaries to be identified automatically and consistently. We conclude that properties of large-scale structures affect the determination of coronal hole regions, and incorporating these properties into training improves detection.

We show that several families of asymptotically rigid mapping class groups arise as explicit quotients of the fundamental group of a graph of groups, with mapping class groups as vertex and edge stabilizers. Using this description, and building on the work of Labruère and Paris, we compute explicit presentations for asymptotically rigid mapping class groups of surfaces.

The interaction of ultrafast lasers with plasmas has been studied for many years, primarily with respect to fundamental emission mechanisms. Only in recent years has ionizing radiation emerged as a safety concern in ultrafast laser-based material processing, where high pulse energies, repetition rates, and average powers, combined with continuous material supply, can lead to sustained X-ray emission. These processing-specific findings have informed German radiation protection legislation, which mandates notification or approval for laser systems exceeding irradiances of $1 \times 10^{13}~W/cm^2$. However, this threshold does not distinguish between material processing and other ultrafast laser applications. In this work, we show that the conditions required for X-ray generation are highly specific and are typically only met during material processing. We assess the applicability of existing radiation studies to non-processing environments and present experimental results demonstrating negligible or no dose production under representative laboratory conditions, such as ultrafast laser interactions with underdense gas or stationary solid targets. We conclude that current legislation generalizes a processing-specific hazard to all ultrafast laser applications and does not adequately reflect the relevant physical conditions.

We study a nonlinear multimarginal optimal transport problem arising in risk management, where the objective is to maximize a spectral risk measure of the pushforward of a coupling by a cost function. Although this problem is inherently nonlinear, it is known to have an equivalent linear reformulation as a multimarginal transport problem with an additional marginal. We introduce a Lagrangian particle discretization of this problem, in which admissible couplings are approximated by uniformly weighted point clouds, and marginal constraints are enforced through Wasserstein penalization. We prove quantitative convergence results for this discretization as the number of particles tends to infinity. The convergence rate is shown to be governed by the uniform quantization error of an optimal solution, and can be bounded in terms of the geometric properties of its support, notably its box dimension. In the case of univariate marginals and supermodular cost functions, where optimal couplings are known to be comonotone, we obtain sharper convergence rates expressed in terms of the asymptotic quantization errors of the marginals themselves. We also discuss the particular case of conditional value at risk, for which the problem reduces to a multimarginal partial transport formulation. Finally, we illustrate our approach with numerical experiments in several application domains, including risk management and partial barycenters, as well as some artificial examples with a repulsive cost.

The tame symmetric algebras of period four, TSP4 algebras for short, form an important class of algebras, with interesting links to various branches of modern algebra. The study of this class has been recently developed in two major directions. The first embraces new classes of examples of TSP4 algebras, such as virtual mutations and generalized weighted surface algebras, both extending known class of the weighted surface algebras. The second provides new classifications of TSP4 algebras (based on known results for $2$-regular case), which handle algebras, whose Gabriel quivers satisfy more general properties. An ongoing project shades a new light on the combinatorics of such algebras, introducing a new useful tool for their classification, called periodicity shadows. In this paper, we attack the problem of classification of TSP4 algebras, from another perspective, namely, we give a classification of all TSP4 algebras with not too big Gabriel quivers, i.e. having at most $5$ vertices -- but with no restrictions on their structure, as it was the case for previous classifications. The result is based on the application of the notion of periodicity shadow, which allows to compute all possible Gabriel quivers of such algebras (for small number of vertices), and recent results on interated mutations of algebras with periodic simple modules. The main result show that TSP4 algebras with Gabriel quivers having at most $5$ vertices are generalized weighted surface algebras, confirming a general conjecture in this case.

Particle physics and gravitational waves provide complementary probes of the deep structure of the Universe. Gravitational waves from the mergers of neutron stars and black holes are sensitive to the structure of dense quark matter and to different dark matter scenarios. Measurements of stochastic gravitational waves backgrounds can teach us about possible first order phase transitions in the early Universe, including providing sensitivity to the TeV scale which is of key interest to future particle collider experiments. Gravitational waves measurements will also give new probes of the evolution and expansion of the Universe, complementary to measurements with electromagnetic radiation. This Perspectives article explores the physics synergies between the science opportunities provided by next generation gravitational waves measurements and particle physics experiments. Gravitational waves can also probe deep into the early Universe reaching physics much above possible collider energies if the signals can be detected.

We theoretically study the Josephson effect in ultracold Fermi gases, where the two sides of the Josephson junction are independently tuned to different regions of the Bardeen-Cooper-Schrieffer (BCS)-Bose-Einstein condensation (BEC) crossover. Using the nonequilibrium Green's function approach combined with the tunnel Hamiltonian formalism, we evaluate the DC and AC Josephson currents throughout the entire crossover region. We calculate the DC Josephson current as a function of interaction strength by tuning both sides of the junction synchronously from the BCS to the BEC regimes, and give the asymptotic expression of the current in the deep BCS and BEC limits. We also study the AC Josephson junction through the interaction-asymmetric junction by fixing the interaction in one reservoir and tuning that of the other one. A peak of the tunneling current is found when one side is fixed in the BCS limit and the other side is tuned into the BEC regime, which corresponds to the interaction-biased Riedel peak. Our results indicate the competition between contributions of increasing pair spectral weight and decreasing chemical potential to Josephson tunneling throughout the BCS-BEC crossover, and demonstrate the realization of the Riedel peak in strong-coupling quantum gases.

The number of low Earth orbit (LEO) satellite constellations has grown rapidly in recent years, bringing a major change to global wireless communications. As LEO satellite links take on a growing role in critical services such as emergency communications, navigation, wide-area data collection, and military operations, keeping these links secure has become an important concern. In particular, verifying the identity of a satellite transmitter is now a basic requirement for protecting the services that rely on satellite access. In this article, we propose an active challenge-response authentication framework in which the verifier checks the satellite at randomly chosen times that are not known in advance, removing the fixed measurement window that existing passive methods expose to adversaries. The proposed framework uses the deterministic yet unpredictably sampled nature of orbital observables to establish a physics based root of trust for satellite identity authentication. This approach transforms satellite authentication from static feature matching into a spatiotemporal consistency verification problem inherently constrained by orbital dynamics, providing robust protection even against trajectory-aware spoofing attacks.

We introduce the notion of birth and death cochains as generalized versions of birth and death simplices in persistent cohomology. We show that birth and death cochains (unlike birth and death simplices) are always unique for a given persistent cohomology class. We use birth and death cochains to define birth and death content as generalizations of birth and death times. We then demonstrate the advantages of using that birth and death content as loss functions on a variety of topological optimization tasks with point clouds, time series and scalar fields. We close with a novel application of topological optimization to a dataset of arctic ice images.

This paper proposes a physics-informed learning framework for a class of recurrent neural networks tailored to large-scale and networked systems. The approach aims to learn control-oriented models that preserve the structural and stability properties of the plant. The learning algorithm is formulated as a convex optimisation problem, allowing the inclusion of linear matrix inequality constraints to enforce desired system features. Furthermore, when the plant exhibits structural modularity, the resulting optimisation problem can be parallelised, requiring communication only among neighbouring subsystems. Simulation results show the effectiveness of the proposed approach.

This paper studies a fractional attraction-repulsion system with time-space dependent growth source and nonlinear productions: \begin{equation*} \left\{ \begin{aligned}\label{1.1} &u_t = -(-Δ)^αu - χ_1 \nabla \cdot (u \nabla v_1) + χ_2 \nabla \cdot (u \nabla v_2) + a(x,t)u - b(x,t)u^γ, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = Δv_1 - λ_1 v_1 + μ_1 u^k, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = Δv_2 - λ_2 v_2 + μ_2 u^k, &x \in \mathbb{R}^N, \, t > 0. \end{aligned} \right. \end{equation*} We first establish the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial data in two different cases: $γ\geq k + 1$ and $γ< k + 1$, respectively. For a fixed $γ$, when $k$ exceeds the critical value $γ- 1$, a larger $b$ must be chosen to suppress the blow-up of the solution. Moreover, we show the persistence of the global solutions for both cases $γ= k + 1$ and $γ\neq k + 1$.

A high-flux source of strontium atoms is required for cold atom quantum technology applications. We present a re-entrant oven design that avoids the need for any vacuum feed-throughs and has an inherent temperature gradient to guard against clogging of the nozzle. The nozzle is fabricated by micro-machining of fused silica using selective laser etching; this specialised technique is capable of making many thousands of fine microchannels and is suitable for batch production. Operating with only electrical heating, using <20W of electrical power, a total flux of $8(1)\times 10^{14}$ atoms/s is achieved at an oven temperature of 475°C, of which we estimate $1.8(2)\times 10^{13}$ atoms/s could be captured. A heated in-vacuum sapphire window grants optical access directly opposite the oven, and can be cleared of metallization without breaking vacuum. We used this optical access to modulate the flux of the atomic beam by direct illumination of the nozzle and the strontium metal with high-power laser light. Heating by laser light increased the useful flux by a factor of up to 16(3) on a timescale of 40s, and a factor of 2.5(5) on a timescale of 1s. This flux modulation serves to increase the operating lifetime of the oven. We report experimental measurements of the performance of the oven in long-term operation over many months.

In user-generated content (UGC) transcoding, source videos typically suffer various degradations due to prior compression, editing, or suboptimal capture conditions. Consequently, existing video compression paradigms that solely optimize for fidelity relative to the reference become suboptimal, as they force the codec to replicate the inherent artifacts of the non-pristine source. To address this, we propose a novel perceptually inspired loss function for learning-based UGC video transcoding that redefines the role of the reference video, shifting it from a ground-truth pixel anchor to an informative contextual guide. Specifically, we train a lightweight neural quality model based on a Selective Structured State-Space Model (Mamba) optimized using a weakly-supervised Siamese ranking strategy. The proposed model is then integrated into the rate-distortion optimization (RDO) process of two neural video codecs (DCVC and HiNeRV) as a loss function, aiming to generate reconstructed content with improved perceptual quality. Our experiments demonstrate that this framework achieves substantial coding gains over both autoencoder and implicit neural representation-based baselines, with 8.46% and 12.89% BD-rate savings, respectively.

We compute the murmuration density function for the family of Hecke forms of weight $k$ and prime power level $N=\ell^a$, with $\ell$ a fixed odd prime and $a\to \infty$.

This paper explores an efficient method for entanglement quantification in two-qubit and qubit-qutrit quantum systems based upon the framework of collective measurements in conjunction with machine learning. We introduce an adaptive measurement procedure in which measurement settings are dynamically adjusted based on prior measurement outcomes aiming to optimize the inference precision given a limited number of these measurement settings. The procedure makes use of the Long Short-Term Memory networks to recurrently process collective measurements on two copies of the investigated states. Obtained results demonstrate the tangible benefits of the adaptive measurements in comparison to previously described non-adaptive strategies.

Let $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ be the number of $3$-way contingency tables of size $m \times n \times p$ with two of its three plane-sum margins fixed by $\mathbf{a}=(a_1, \ldots, a_m) \in \mathbb{N}^m$ and $\mathbf{b}=(b_1, \ldots, b_n) \in \mathbb{N}^n$. When $p=1$, this is the number of $m \times n$ non-negative integer matrices whose row and column sums are fixed by $\mathbf{a}$ and $\mathbf{b}$. In this paper, we study the numbers $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ through the lens of quiver invariant theory. Let $\mathcal{Q}^{p}_{m,n}$ be the $p$-complete bipartite quiver with $m$ source vertices, $n$ sink vertices, and $p$ arrows from each source to each sink. Let $\mathbf{1}$ denote the dimension vector of $\mathcal{Q}^{p}_{m,n}$ that takes value $1$ at every vertex of $\mathcal{Q}^{p}_{m,n}$, and let $θ_{\mathbf{a}, \mathbf{b}}$ denote the integral weight that assigns $a_i$ to the $i^{th}$ source vertex and $-b_j$ to the $j^{th}$ sink vertex of $\mathcal{Q}^{p}_{m,n}$. We begin by realizing $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ as the dimension of the space of semi-invariants associated to $(\mathcal{Q}^{p}_{m,n}, \mathbf{1}, θ_{\mathbf{a}, \mathbf{b}})$. Using this connection and methods from quiver invariant theory, we show that $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ is a parabolic Kostka coefficient. In the case $p=1$, this recovers the formula for the number of the $m \times n$ contingency tables with row and column sums fixed by $\mathbf{a}$ and $\mathbf{b}$, which in the classical $2$-way setting can also be obtained via the Robinson-Schensted-Knuth correspondence.

In this paper, we establish local decay estimates for the bi-Laplacian Schrödinger equation with time-dependent (in particular, quasi-periodic) potentials in spatial dimension $n\ge14$. Moreover, under stronger spectral regularity hypotheses, the same result can be extended to dimension $n\ge9$. Our approach, based on asymptotic completeness and the existence of the channel wave operator, departs from standard resolvent-based methods. In addition, global-in-time Strichartz estimates are derived from the local decay estimates.