Papers

We construct modified wave operators for the nonlinear Schrödinger equation $i\partial_tu+\frac12\Delta u=|u|^pu$ in the full subcritical long-range case $0<p<2/d$, with small, nonvanishing, analytic final data $W(x)$ with bounded logarithmic gradients. Previous results established large-time asymptotics for selected classes of Cauchy data. Moreover, the exact asymptotic expansion for $p<1/d$ remained unknown. When $1/d<p<2/d$, our result gives the approximation $$\frac{1}{(it)^{\frac{d}{2}}} e^{\frac{i|x|^2}{2t}} W\left(\frac{x}{t}\right) \exp\left[ -i\frac{t^{1-\frac{dp}{2}}-1}{1-\frac{dp}{2}} \left|W\left(\frac{x}{t}\right)\right|^p \right]$$ The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When $p\le 1/d$, we construct the profile from a finite truncation of a Fuchsian equation coupled with a transport equation. This profile still leaves a long-range triangular coupling whose terminal integral does not preserve the required fast decay class. The construction yields quantitative $L^q$ asymptotics for $2\le q\le\infty$ and uniqueness in the prescribed analytic asymptotic classes. The central new ingredients are a nonlinear final-state normal form that removes this long-range coupling and a mixed iteration in particular analytic spaces.

Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition

Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on $\mathbb R_+$ whose conditional drift depends on both time and position and is of order $\rho x^\alpha t^{-\beta}$ with $\rho>0$. The case on the critical line $2\beta-\alpha=1$, with $\alpha\in(-1,1)\setminus\{0\}$, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for $-1<\alpha<1$ and $\beta\ge 0$, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive Hölder class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-Hölder continuity of the integrated density of states for the associated quasiperiodic Schrödinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

Quasiperiodic order is characterized by irrational frequencies whose rational approximability known as irrationality measure defines distinct arithmetic classes. We establish that this arithmetic classification has direct physical consequences for long-wavelength charge fluctuations. In translation-covariant quasiperiodic systems, the infrared scaling of charge fluctuation is governed by the interplay between the irrationality exponent of irrational frequency and the large-harmonic decay of the hull charge profile: the latter determines the available charge weight, while the former controls how efficiently that weight is transferred to the infrared. Consequently, all algebraic irrational frequencies share the same arithmetic scaling, whereas exceptionally well-approximable transcendental frequencies can exhibit strongly enhanced infrared fluctuation scaling. We further prove that occupied states separated from the Fermi level by a gap stable throughout the hull contribute only an analytic infrared background, leaving the nontrivial scaling to near-Fermi states. Our results extend to general translation-covariant multi-frequency quasiperiodic systems.

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion “direct observation prediction” (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

We perform a numerical study of uniform meandric systems via their encoding by pairs of simple random walks and their interpretation as a model of critical percolation on random planar maps. We investigate certain natural box-crossing events and provide numerical evidence for a strong negative association between opposite open crossings and formulate conjectures on their asymptotic behavior. Our second contribution is a simple law-preserving dynamics on meandric systems, by local peak/valley flips in the walk encoding. Our simulations suggest that the percolation structure decorrelates much faster than the underlying random planar map.

The AKSZ construction encodes the Batalin-Vilkovisky formulation of a topological field theory in terms of finite-dimensional geometric data. There are at least two ways to extend this approach to nontopological gauge theories. The first, due to Costello, replaces the spacetime exterior algebra with a more general DGCA that is not freely generated. The second replaces the symplectic structure with a degenerate presymplectic structure, which may also be non-regular. In this work, we show that both approaches are special cases of a more general algebraic AKSZ construction, in which the underlying algebras are allowed to be constrained and the presymplectic structure may be degenerate and non-regular. Within this framework, we explicitly relate Costello's formulation of the Chalmers-Siegel model to the presymplectic formulation, the latter admitting a first-principles derivation. We also construct Costello-like formulations of higher-form analogues of the Chalmers-Siegel model and of self-dual higher-spin gauge theories of Yang-Mills type, obtaining in the latter case a remarkably concise description. In so doing we propose a simple algebraic construction for the corresponding source space DGCAs underlying these examples.

We study the particle number distribution of locally interacting bosonic quasi- free states, which arise in various areas of mathematical physics. We show that the particle number decomposes into an infinite sum of independent geometrically distributed random variables, confirming predictions from the physics literature. This representation yields exponential tail bounds for the particle number, a law of small numbers together with a large deviation principle at logarithmic speed when the lattice spacing diverges, and a central limit theorem when it vanishes. As an application, we obtain a detailed description of the statistics of the quantum depletion in Bose–Einstein condensates, with the constants explicit in terms of the scattering length of the interaction potential.

In this paper, the log-conformation reformulation of a Giesekus fluid near a planar stick-slip singularity is investigated analytically. In what is, to our knowledge, the first use of the log-conformation formulation beyond its numerical purpose, we present results that match the asymptotic stress analysis of Evans [JNNFM 222 (2015) 24-33] for the stick, slip, and core regions under the assumption of a given Newtonian velocity field. Furthermore, the logarithmic reformulation allows us to extend the existing asymptotic theory substantially, showing, e.g., that the determinant of the conformation tensor is asymptotically constant, as well as thoroughly characterizing the transition behavior between the boundary layers and the core region. Combining these results, we obtain a composite asymptotic solution of the log-conformation equation that is valid uniformly throughout a close neighborhood of the singularity.

We present a phenomenological framework for fully-heavy tetraquark spectroscopy based on spectral graph theory. The four valence partons are the vertices of \(K_4\), with edge weights fixed by the colour-Casimir factors of the two colour-singlet diquark–antidiquark channels, \(\bar3\otimes3\) and \(6\otimes\bar6\); a physical state is a coherent mixture of the two, controlled by one mixing angle. Unlike an earlier version, which passed the colour-weighted adjacency matrix through an absolute-degree graph Laplacian, here the spectrum is obtained by directly diagonalising the adjacency matrix; we show analytically that the Laplacian step inverts the physical correspondence between colour attraction and mass ordering. Fixing the model's three parameters (mixing angle, energy scale, effective charm mass) against \(X(6900)\), \(X(7100)\), and the more tentative \(X(7200)\) gives \(\alpha=0.740\), \(\gamma=23.2\,\mathrm{MeV}\), \(m_c=1.77\,\mathrm{GeV}\); with three parameters fixed by three inputs this has zero residual degrees of freedom and is not itself a statistical test. The framework's value rests instead on two genuinely predictive, zero-additional-parameter applications: the all-bottom ground state falls at \(18.7\)–\(18.9\,\mathrm{GeV}\) for standard \(m_b\), with no retuning; and for \(T_{cc}^+\) the same mechanism under-binds by \(\approx93\,\mathrm{MeV}\), a quantitative diagnostic of missing long-range (molecular) dynamics. We discuss the sensitivity to the tentative status of \(X(7200)\), the tension between this static colour-configuration picture and the diquark radial-excitation interpretation favoured by recent CMS data, and what further data would make the model falsifiable in a stronger sense.

Anti-de Sitter (AdS) spacetimes are not globally hyperbolic, so boundary conditions are generally needed for well-defined dynamics. Ishibashi and Wald (IW) determined boundary conditions yielding well-posed dynamics with conserved positive energy for scalar, electromagnetic, and linearized gravitational fields, using nonlocal potentials and individual spherical harmonic modes. In this paper, we determine which IW conditions are local in the primary fields and which are AdS-invariant. In four spacetime dimensions, the standard AdS-invariant boundary conditions set the rescaled magnetic field to zero for electromagnetism and the rescaled magnetic Weyl tensor to zero for Einstein gravity on the conformal boundary. We generalize these standard boundary conditions by setting arbitrary linear combinations of the electric and magnetic fields, or the corresponding Weyl tensors, to zero, and we conjecture that these exhaust the local, AdS-invariant, conservative boundary conditions in AdS$_{1,3}$. Boundary conditions with non-AdS-invariance and in different spacetime dimensions are also studied. The AdS-invariant conditions in AdS$_{1,3}$ extend to Yang-Mills theory and nonlinear gravity in a general Fefferman-Graham setting. We construct phase spaces for these general boundary conditions. The covariant phase spaces for general conditions are naturally associated with Lagrangians containing Pontryagin terms and, in gravity, a Gauss-Bonnet term. In gravity, except for the standard boundary condition, no nontrivial asymptotic symmetries exist and all conserved charges such as ADM mass vanish identically, as in a closed universe. Spacetimes with Killing fields including AdS then exhibit linearization instability in the sense of Fischer and Marsden and Moncrief. For nonstandard boundary conditions, the modified symplectic structure yields new expressions for the entropy formula of black holes.

We develop a time–frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

We develop a machine-checked construction of non-Gaussian functional integrals using the Henstock–Kurzweil gauge integral and Chernoff product approximations. The central object is a finite family of bosonic modes with action S(phi) = (1/2) phi^T A phi + lambda * sum_i phi_i^4, where A is positive definite. We prove that the one-mode integral I(omega, j, lambda) is finite, strictly positive, monotone and infinitely differentiable in the coupling lambda on [0, infinity). Its derivatives are given by convergent integrals of phi^{4k} with the same weight, not by the divergent perturbative series. The M-mode influence functional factorises into one-mode integrals and is bounded by its Gaussian value. A Chernoff / Lie–Trotter splitting handles the non-commutativity of the free and non-Gaussian generators. All statements are formalised in Lean 4 with Mathlib; the accompanying file HkNonGaussian.lean is free of sorry and uses only the standard axioms propext, Classical.choice, Quot.sound. Four illustrative applications are worked out at the level of explicit formulas: the Duffing oscillator, local volatility (CEV) in finance, Wilson–Cowan neural fields, and non-Gaussian quantum reservoirs. The construction is completely direct and does not use Wick rotation, Wiener measure, zeta-regularisation or analytic continuation back from imaginary time.

Altermagnetism has recently emerged as a distinct collinear magnetic phase exhibiting momentum-dependent spin-splitting despite vanishing net magnetization, as a consequence of inequivalent non-magnetic environments. Lately, it has been proposed that strong electronic correlations may yield spontaneous altermagnetism due to orbital ordering even for equivalent non-magnetic environments. While previous studies have largely focused on the electronic structure, a unified understanding of the collective excitations associated with these two different microscopic mechanisms stabilizing altermagnetism remains absent. Here, we develop an extended Kugel'-Khomskiĭ spin-orbital model on a decorated square lattice that simultaneously incorporates inequivalent non-magnetic environments and correlation-driven orbital ordering within a common theoretical framework. Employing a self-consistent mean-field spin-wave orbital-wave formalism, we demonstrate the emergence of mutually unhybridized but interdependent magnon and orbiton excitations exhibiting characteristic chiral-splitting. We show that the splitting originates from two distinct microscopic contributions: a lattice-dependent term arising from inequivalent non-magnetic environments and an orbital-order-induced exchange-anisotropy term that survives even for equivalent non-magnetic environments. The proposed framework therefore unifies the collective excitations associated with both types of altermagnetism. We further investigate the finite-temperature evolution of the coupled spin-orbital system, revealing the breakdown of spin-wave and orbital-wave approximations through spurious 1st-order transitions, while complementary classical Monte Carlo simulations recover the expected continuous 2nd-order behaviour. Our work establishes a unified microscopic framework for understanding collective magnon and orbiton excitations in altermagnets.

We show that a formal deformation quantisation of degree-2 differential graded symplectic (QP) manifolds gives rise to the complete classical kinematics and dynamics of NS-NS supergravity, or, equivalently, its duality-covariant formulation as double field theory (DFT). Our construction extends the standard tensor hierarchy, encoding gauge parameters and their redundancies, to a complex encoding the physical fields, field strengths, and Bianchi identities. We present two realisations of such a complex: 1) a differential graded Lie algebra built from the algebra of functions on the QP-manifold; 2) a cochain complex that reproduces the tensor hierarchy representations conjectured in the literature and can be interpreted as a classical BV complex of the linearised theory. The deformation that gives rise to the dilaton comes from a normal-ordered graded Moyal–Weyl star product. This naturally extends the duality structure group from $\mathrm{O}(D,D)$ to $\mathrm{O}(D,D)\times\mathbb{R}^+$. Finally, our framework provides a direct route to constructing the DFT action via local double Lorentz invariance, and offers a transparent algebraic foundation for curvature tensors in generalised Cartan geometry.

We study the representation theory of the vertex operator algebras (VOAs) associated with rank-one 4d $\mathcal{N} = 3$ superconformal field theories. For the $\mathbb{Z}_3$ S-fold theory, whose VOA $\mathcal{W}_{\mathbb{Z}_3}$ has central charge $c_{\mathrm{2d}} = -15$, we use the $\mathcal{N} = 1$ Lagrangian description to obtain the unflavored Schur index in terms of Dedekind eta functions, while Wilson-loop indices yield the unflavored non-vacuum characters. These characters all solve a modular linear differential equation (MLDE) whose solution space also contains a logarithmic character. Combining flavored MLDEs from null states with Zhu's associative algebra and a free-field realization, we study four highest-weight modules of $\mathcal{W}_{\mathbb{Z}_3}$ and their flavored characters in closed form. A parallel analysis applies to the $\mathcal{N} = 3$ theories obtained by gauging a discrete $\mathbb{Z}_n$ flavor subgroup of $\mathcal{N} = 4$ $U(1)$ and $SU(2)$ super-Yang–Mills, for which we also obtain closed-form Schur indices and a new free-field realization of the VOA of the $\mathbb{Z}_4$ quotient.

This study evaluates viscosity-dependent limitations in refined potential flow theory (RPT) applied to incompressible flow over an impulsively started circular cylinder. RPT adapts classical potential flow theory to account for viscous effects, flow fluctuations, and three-dimensional variables. By utilizing a quasi-irrotational viscous stream function, the framework solves the continuity and Navier-Stokes equations while satisfying both free-stream and no-slip boundary conditions.

Consider a one-dimensional branching Brownian motion. Let $W_\infty(\beta)$ denote the limit of the additive martingale in the subcritical regime $\lvert \beta\rvert < \beta_c$ and $Z_\infty$ be the limit of the derivative martingale at criticality. Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470–502) established the following convergence \[ \frac{W_\infty(\beta)}{\beta_c-\beta}\xrightarrow[\beta\nearrow \beta_c]{\mathbb{P}} 2Z_\infty. \] The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving \[ \frac{1}{\beta_c-\beta}\left( \frac{W_\infty(\beta)}{\beta_c-\beta} - 2 Z_\infty +2(\beta_c-\beta)\log(\beta_c-\beta) Z_\infty\right) \xrightarrow[\beta\nearrow \beta_c]{(d)} S, \] where, conditionally on $Z_\infty$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_\infty$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.

We study an inextensible open filament with free ends in a planar Stokes fluid. The system reduces to a third-order nonlocal curvature equation coupled to an elliptic equation for the tension. We prove local well-posedness for nearly critical initial data in supported Sobolev spaces $\widetilde H^s$, $-1/2<s\le0$, satisfying the arc-chord condition. For positive times, we prove improved Sobolev regularity and derive a $d^{3/2}$-type expansion at each free end using Wiener–Hopf factorization, where $d$ denotes the distance to that endpoint. We further prove global existence and exponential convergence to a straight filament for sufficiently small initial data and for finite-energy initial data satisfying $E(0)<\pi^2/4$. The finite-energy result follows from an energy identity and a geometric estimate relating the bending energy to the arc-chord constant. More generally, any finite-time breakdown must be accompanied by loss of the arc-chord condition, while every global solution either converges exponentially to a straight filament or has arc-chord constants tending to zero along a sequence of times tending to infinity.

Reciprocal contacts can favor cooperation by protecting a lineage until its descendants compete as a group. An individual is chosen to reproduce from the whole population, with a probability increasing with its averaged cooperation payoff. Its offspring replaces a neighbor through contacts weighted equally in both directions. We compare a single cooperator taking over defectors with the reverse invasion, using the same distribution of introduction sites. When payoffs weakly affect reproduction, every network with at least six sites and a path visiting them all admits weights favoring cooperation at any benefit-cost ratio $b/c>1$. This holds for mutants placed uniformly or arising in newborn offspring. Unequal contacts sustain aligned domains around two protected sites, converting individual reproduction into collective competition. Cooperation cannot have an advantage when its benefit does not exceed its cost, at any positive selection strength with positive reproduction rates. Where mutants arise changes which contact patterns can use this mechanism. On a five-site path, the weak-selection threshold has the sharp lower bound seven under uniform placement; newborn introduction never favors cooperation at this order. We classify the contact patterns with at most five sites that can favor cooperation under either rule; newborn introduction requires a four- or five-site cycle. Tracing copies backward to their ancestors separates domain competition from the probability of reaching it. Exact polynomial inequalities prove sharp threshold bounds for paths and the four- and five-site cycles. The same ancestry reconstructs evolution without selection on weighted paths and cycles and fixes the slowest relaxation rate on every connected reciprocal network. Cooperation thus depends on both the collective competitors created by contacts and the route by which a new lineage enters them.

We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis–Marchioro–Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.

A compressible, non-isothermal dilute polymer solution is formulated as a Class-II binary mixture with separate solvent and polymer mass and momentum balances and with total energy and entropy balances for the complete mixture. The entropy exploitation is sharpened by a balance-anchoring axiom introduced here: in a fixed balance representative, each selected binary dissipative product contains a constitutive factor from an unclosed balance flux or source. Polymer configuration is resolved first by a connector distribution and then by its conformation tensor. Population kinematics fix polymer transport and, by moments, yield a two-velocity upper-convected rate; objectivity verifies covariance rather than selecting it. Within the total entropy balance, configurational transport and deformation powers cancel their chemical-potential and elastic partial-stress counterparts when deformation and stress-decomposition weights match. A Gordon–Schowalter test independently requires the affine upper-convected choice for the stated dumbbell free energy and Kramers stress unless an additional reversible channel is supplied. Coordinated stress–interaction changes shift the local entropy flux/production pair by a divergence, exposing representation dependence of local mechanism-wise production. An entropy-invariant Class-II-to-Class-I reduction selects a descendant entropy flux preserving the parent production and yields thermo-chemical, configurational-stress and partial-viscous-stress diffusion terms. The omitted quadratic relative-inertia flux is paired with relative kinetic-energy storage and transport and is a reversible truncation, not missing entropy production. The resulting compressible non-isothermal Hookean stress and temperature equations reduce to Oldroyd-B/UCM only after one-velocity, incompressible and isothermal limits.

We study factorizations of the single-particle spectrum of non-interacting quantum systems and their consequences for many-particle statistics. Representing the single-particle partition function by a spectral alphabet, tensor factorizations become multiplicative factorizations of the alphabet, which can be lifted to canonical Bose and Fermi partition functions using standard symmetric-function and $\lambda$-ring identities. We show how product spectra arise from different factorizations, how antisymmetrization can be assigned across an odd number of factors, and which tensor factorizations are compatible with a fixed spectrum. The paper therefore studies spectral factorization and provides a consolidated combinatorial framework for canonical Bose and Fermi partition functions.

We numerically study the bulk correlation functions in the two-dimensional Abelian sandpile model, aiming both to compare with the predictions of logarithmic conformal field theory and to extend the analysis to lattices where analytical methods are difficult to apply. Wilson's algorithm efficiently generates large-scale uniform spanning trees in parallel, which can be mapped to independent recurrent configurations via the Majumdar–Dhar burning bijection, eliminating sample autocorrelations and yielding fast convergence. On the square (single-sublattice) and honeycomb (two-sublattice) lattices our results agree well with the analytical predictions. For the kagome lattice we provide the first systematic numerical study of the bulk correlation functions, and cross-check the bulk height-1 probability against a closed-form analytical expression that we also derive here via the lattice Green function.

We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus $\mathbb T_N^D$, $D\ge2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times \[ t_N(s)=\frac{\log(N^D)+s}{2\gamma_N}, \qquad \gamma_N=2-2\cos\left(\frac{2\pi}{N}\right), \] the Radon–Nikodym density of the process relative to equilibrium converges in $L^2$ to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence $(S_N)$, if the covariance-normalized squared amplitude $\mathfrak q_N^{S_N}(s)$ converges to a scalar $\mathfrak q$ at a fixed $s$, then the distance to stationarity converges to $ 2\Phi\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where $\Phi$ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong–Rayleigh property.

We develop a thermodynamically consistent diffuse-interface model for multicomponent membranes. The proposed model is derived from a coupled free energy functional that incorporates protein-dependent bending elasticity, diffuse surface tension, a volume penalty, and a membrane-associated Ohta–Kawasaki energy. Applying the Onsager variational principle, we derive a coupled $L^2$ gradient flow system and its energy dissipation law. We then construct a stabilized alternating ETD1 scheme and a stabilized alternating ETDRK2 scheme with Strang-type composition (alternating Strang-ETDRK2). To the best of our knowledge, the proposed alternating Strang-ETDRK2 scheme has not previously been developed and analyzed for coupled phase field systems. We further prove the discrete energy dissipation for both schemes under some regularity assumptions on the numerical solutions. Numerical experiments in two and three dimensions validate the discrete energy dissipation law, and present the protein segregation and membrane deformation produced by the proposed model.

A shell finite-element model of a stiffened aluminium wingbox panel, a laboratory structure shared by several experimental studies, is built with its plate-to-stiffener joints at the $78$ physical fastener positions and calibrated against measured modal data, a six-parameter sensitivity update of the substructure moduli within $\pm20\%$ reproducing the first five measured modes to $1.1$–$4.3\%$ with modal-assurance values of $0.88$–$1.00$. Making the fasteners nonlinear, at equal mesh, mass and damping, shows that joint nonlinearity enters the response strongly asymmetrically. Hardening is almost invisible, at most $+1\%$ in frequency, whereas softening or slipping the joints moves the first three resonances by $-2.1$, $-4.9$ and $-9.2\%$, and friction removes up to $80\%$ of the resonant peak before it recovers. Newmark integration of all $18{,}804$ degrees of freedom and a harmonic balance condensed exactly onto the joints and continued in arclength agree to $5.5\times10^{-4}$. A projection-based nonlinear model order reduction then locates the criterion that a jointed structure imposes on a reduced basis. The binding quantity is not the linear response, which eleven vectors reproduce to better than $0.01\%$, but the receptance of the structure between the joints, of which $126$ global eigenvectors carry about $2\%$, and without which the reduced model overpredicts the hardening shift of the fundamental by a factor of thirty and returns a value beyond the rigid-joint limit of the panel.

Two sharp yes/no questions are mutually unbiased if and only if their involutions anticommute. A set of unbiased questions is therefore a Clifford algebra, whose maximal size is odd, meaning four anticommuting questions typically imply a fifth: the two-level systems of real, complex and quaternionic quantum theory allow two, three and five questions, while four is skipped. Without this operator product restriction, a two-level system with four unbiased questions exists as the four-dimensional Bloch ball, the hyperbit. The single-system postulates that pick out the Jordan state spaces allow such a model, and only energy observability rules it out. We show what it lacks, and realise it inside two qubits: its Kirkwood-Dirac imaginary parts are hidden $\mathfrak{so}(4)$ generators, its composites allow entanglement but no interaction, and its 24-cell of states stays Spekkens preparation contextual down to depolarising strength $2/3$. In fermionic terms, the four questions are the Majorana operators of two modes while the fifth is parity: the four-question system is therefore the sector removed by parity superselection.