Papers

We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.

In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. We show that, as $N\to\infty$, the boundary of the complex eigenvalue distribution of $\mathbf{A}\mathbf{B}$ is governed by simple equations involving the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$.

We consider the Schrödinger equation for a charged particle in a randomly fluctuating external magnetic field and establish a singular perturbation limit in which the solution converges to a deterministic Schrödinger equation with Gaussian fluctuations. We propose a stochastic Schrödinger equation incorporating the behavior of both the deterministic average and the fluctuations and show that it has the same limiting behavior as the original model, while providing a simpler way to study the distribution of relevant observables.

The nonlinear evolution and mutual interaction of two-dimensional electron acoustic (EA) nonlinear structures in superthermal plasma environment are studied. The plasma model consists of inertial cold electrons, superthermal hot electrons described by kappa ($\kappa$) distribution, and stationary ions providing overall charge neutrality. Using the extended Poincaré-Lighthill-Kuo (PLK) reductive perturbation technique, a pair of two-sided Kadomtsev-Petviashvili (KP) equations governing right- and left-propagating EA solitary waves (EASWs) is derived. Exact analytical solutions of the KP equations, including single soliton, multisoliton, breather, and lump structures, are obtained via the Hirota bilinear method. The effects of key plasma parameters such as hot electron concentration, temperature ratio, and superthermality index on the characteristics of these nonlinear excitations are examined. Particular attention is devoted to the head-on collision dynamics between solitons, breather-soliton, and breather-breather interactions. The results reveal quasi-elastic collisions accompanied by phase shifts, transient amplitude modulation, and localized energy concentration, with clear distinctions between oscillatory and non-oscillatory mode interactions. The present study provides new insights into multidimensional electron acoustic wave (EAW) dynamics and energy redistribution mechanisms in superthermal space plasmas, with direct relevance to planetary magnetospheric environments such as Saturn's ring region.

The dynamic acoustic scattering system is governed by the wave equation $u_{tt}-\Delta u+qu=0$ in $\Bbb R^3,\,\,\,-\infty<t<\infty$, with a compactly supported potential $q$ and infinitely distant sources (controls) $f$, which initiate incoming spherical waves $u=u^f(x,t)$ provided $u^f\big|_{|x|<-t,\,\,\,t<0}=0$. These waves are focused at $x=0$ and fill up the whole space at the moment $t=0$. The system is controllable if the set of waves $u^f(\cdot,0)$ produced by all finite energy controls $f$, covers the space $L_2(\Bbb R^3)$. As we show, if the Hamiltonian $H=-\Delta+q$ has the bound states, then in the space the points $a$ appear such that the system, being refocused at $x=a$, loses controllability. The latter leads to a physical effect: the waves $u^f$ of finite energy appear, which vanish simultaneously in the past and future cones $|x|<\pm\, t$ and leave the region of inhomogeneity of $q$ without reverberation. This effect has some similarities with the wavefront reversal (Time Reversing Mirror), but is more meaningful from a mathematical point of view.

For every $\beta>0$, we establish multi-time soft-edge moment convergence for the Jack–Plancherel process, a discrete $\beta$-analogue of Dyson Brownian motion. The limiting moments admit an absolutely convergent expansion in terms of nonnegative Brownian bridges decorated by pairs of equal and opposite jumps. At equal times, we identify this limit with the joint Laplace statistics of the Airy$_\beta$ process for $\beta\geq1$. The same Brownian expansion yields an asymptotic $\beta$-topological expansion of the marginal $b$-conjecture and, at $\beta=2$, an explicit nonnegative formula for the Witten–Kontsevich intersection numbers.

The macroscopic collective motion of active continuum on curved manifolds is conventionally addressed through perturbative dynamic renormalization or finite-element simulations, often obscuring the underlying geometric mechanisms. Here, an exact algebraic framework is established to reformulate the active phase transition on hyperbolic spaces $\mathbb{H}^2$. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold $\alpha_c = \frac{5}{4}D\kappa^2$ for macroscopic polarization, which is dictated by the geometric mass gap of the Hodge-de Rham Laplacian. We strictly define the parameter subspace $\alpha = 2D\kappa^2$ where the topological free energy reaches the Bogomolny-Prasad-Sommerfield (BPS) limit. This enables the reduction of the complex velocity field to Blaschke products via Möbius gauge symmetry. The flat-space limit ($\kappa \to 0$) exactly degenerates to the topological phase of the classical O(2) model, demonstrating that the constant negative curvature acts as an un-perturbative infrared regularization for non-linear amplitude saturation. Furthermore, mapping the non-variational active convective modes onto the defect translational zero-modes yields an intrinsically non-reciprocal interaction matrix. By analytically extending the singular integral operator of the dynamically condensed defect ring, we identify a macroscopic second-order exceptional point (EP2) characterized by a strictly algebraic dynamic scaling law $\tau \sim |\Delta \nu|^{-1/2}$. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.

A new condition involving the complexity factor, a structure scalar arising from the orthogonal splitting of the Riemann tensor, is generated at the boundary of a spherically symmetric anisotropic fluid. This condition leads to the generation of a model, without invoking any further constraints on geometry or matter variables. The model generated is physically reasonable and suggests that there is a non-zero, positive minimum constraint on the complexity scalar value evaluated at the surface. This suggests that vanishing complexity is a globalized constraint which does not apply at the boundary, specifically for compact objects in the strong-gravity regime.

Let $D=2^n$ and $M=3n+9\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\to0$ and $MQ_D^{3/4}/D^2\to0$, then, for every fixed $0<x<\pi/\sqrt3$, \[ \mu_D\bigl(B_{Q_D}([I],x\sqrt{Q_D})\bigr)\le e^{-c_xD^2}. \] Here $\mu_D$ is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to $(\pi/\sqrt3)\sqrt{Q_D}$ throughout the window $M\ll Q_D\ll D^{8/3}M^{-4/3}$. Choosing $Q_D=\kappa D^{8/3}M^{-4/3}$ with sufficiently small fixed $\kappa>0$ yields a Haar-typical lower bound of order $D^{4/3}M^{-2/3}$ for the corresponding infinite-cliff Carnot–Carathéodory distance, outside an $e^{-\Omega(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4/3$, disproving Brown's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.

In this work, we extend the scattering quantum walk (SQW) framework to a lattice of energy-dependent point interactions. This yields, within the coined quantum walk (CQW) formalism, a coin operator that is directly related to the scattering matrix of zero-range potentials. The model thus provides a discrete-time quantum-walk (DTQW) analog of a periodic array of point interactions of the Kronig-Penney type, where the walker's wavenumber serves as a continuous, physically transparent control parameter for the coin operation. We analyze the spectra and the dynamics of position probability and entanglement, yielding distinct results for specific energies and point interactions. We relate the spectral structure to the spatial probability distribution and explicitly characterize the long-time entanglement behavior for each point interaction. The transmission modulus determines the quasienergy gap, bandwidth, and maximum group velocity, while also controlling the long-time coin-position entanglement for the initial state considered. The four families of one-dimensional point interactions ($\delta$, $\delta'$, crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.

Quantum Schrödinger bridge constructions establish the existence of a conditioned process under a positivity hypothesis on the endpoint data, and three independent ones adopt it. We show that hypothesis fails on a definite locus once the conditioned object is the two-state amplitude $\phi^*\psi$ of a pre- and post-selected pair on configuration space, and that the failure is testable on recorded data. Whenever $\phi^*\psi$ has a moving zero of order $k$ across which probability flows, no diffusion of constant diffusion coefficient can both carry the conditioned density and avoid the nodal curve: the current velocity it would need grows as the inverse $2k$-th power of the distance to that curve and points toward it on one side, placing the curve at finite scale distance, so it is reached and not merely approached. Identifying the weak momentum with a bridge drift fails independently: for a free particle post-selected in position the current velocity is exactly minus the drift of the bridge to the target. The positive Doob $h$-transform and the Bernstein conditioning built on it exist wherever $\phi^*\psi$ is nodeless and fail on the nodal curve. There the osmotic part of the weak momentum diverges as the inverse distance to the zero and changes sign across it with a coefficient set by $k$, while the current part stays bounded; that boundedness rests on the signed factorization of the real-envelope class and hides the obstruction from any picture built on the current velocity alone. A measured fringe visibility fixes a single length, which puts a Lorentzian of order $10^{-3}$ into the current channel reconstructed in the two-slit trajectory experiment if the contrast is limited by an off-axis zero, and nothing there if by incoherence. The results are one-dimensional, covering the separable transverse field of a two-slit geometry, not general two-dimensional vortices.

For positive exponents $\alpha_1,\ldots,\alpha_4$ we evaluate the cyclic four-electron Slater integral, the topology recorded as case V in Bonham's 1965 classification, in a finite real expression of weight at most three. The general answer is an explicit sum of eighteen blocks involving only logarithms, dilogarithms and trilogarithms of rational functions of the exponents; all logarithm arguments are positive and all nonconstant polylogarithm arguments may be placed in $(-1,1)$, so no radical, generalized polylogarithm, contour prescription or unevaluated limit is required. At equal exponents the value collapses to $I(\lambda,\lambda,\lambda,\lambda)=(4\pi)^4\lambda^{-4}[\mathrm{Li}_3(1/4)/2-7\zeta(3)/24+\pi^2\log 2/12-2\log^3 2/3]$, a weight-three period over $\mathbb{Q}$ ramified only at 2 and 3. Three independently implemented numerical routes agree at four exponent tuples to at least 32 decimal places, an exact-dyadic evaluation of the classical formula gives eight 100-place certificates, and three published four-electron calibration values are reproduced to every printed digit. We also prove the small- and large-exponent limit laws with closed-form coefficients, dihedral symmetry, and a strict rearrangement law for the three inequivalent cyclic placements of four exponents. The formula applies to the four-cycle and does not evaluate the two chorded topologies.

In this paper, we consider one-dimensional discrete Schrödinger operators \begin{align} Hu(n)=(\Delta+V)u(n)\nonumber \end{align} on $\ell^2(\mathbb{N})$ with a self-adjoint boundary condition at $n=0$, where $\Delta$ denotes the discrete Laplacian and $V(n)$ is a real-valued perturbation satisfying $$V(n)=\frac{O(1)}{1+n}.$$ We determine the sharp transition for the asymptotic coefficient of \(V\) governing the existence and nonexistence of embedded eigenvalues.

Symmetry reductions of partial differential equations (PDEs) inherit more geometric structures than other types of reductions: invariant conservation laws, variational structures, and, under suitable conditions, Hamiltonian-type structures of the original model descend to the reduced model through the mechanism of invariant reduction. This paper develops computational algorithms that carry out such reductions explicitly. We use the interpretation of variational $p$-forms as conservation laws of an enlarged system — the (degree-shifted) tangent system — consisting of the original equations together with their linearizations, in which the perturbation variables are treated as anticommuting. This allows us to formulate the algorithms using well-known concepts from the theory of conservation laws, naturally adapted to the graded-commutative setting. The reduction of conservation laws, variational $1$-forms, and presymplectic structures thereby becomes a single algorithmic procedure. We present (i) a homotopy-based reduction algorithm for systems of evolution equations, implemented in Maple; (ii) a descent reduction algorithm applicable to general $\ell$-normal systems for $p>0$; and (iii) a simple reduction algorithm available for point symmetries under suitable conditions, including the description of reductions in terms of systems involving fewer independent variables. Examples include nonlinear evolution equations in one and two spatial dimensions, the Laplace equation, the incompressible Euler equations, and the cotangent system of Pavlov's equation.

Motivated by the conjectural existence of M-theory, we investigate a web of correspondences between enumerative invariants of a Calabi-Yau fivefold $Z$ with a torus action. This includes a correspondence between the fivefold Gromov-Witten invariants and K-theoretic Pandharipande-Thomas invariants of a threefold $X \subset Z$, mediated by so-called membrane indices, which generalise Gopakumar-Vafa invariants. We establish the correspondence for strip geometries for restricted torus actions and test it numerically for general torus actions. Further numerical evidence is provided for the closed vertex, local surfaces and local projective spaces. For the latter we present conjectural formulae for their low-degree membrane indices. When the fivefold is the product of the affine plane with a suitable toric variety, we prove geometric engineering and equate the generating series of Pandharipande-Thomas invariants with the corresponding instanton partition function while equality with the Gromov-Witten series is probed numerically.

We prove a Berry-Tabor theorem for the repulsive three-particle trigonometric Sutherland Hamiltonian. For fixed total momentum, its relative spectrum is an affine translate of the positive-definite $A_2$-form. For a Diophantine coupling, the desymmetrized spectrum has Poissonian pair correlation. The proof reduces the problem to a finite-congruence and Weyl-sector version of Marklof's theorem on inhomogeneous quadratic forms.

We establish the existence of standing waves for a nonlinear Schrodinger equation with potentials belonging to the Kato class and an $L^\infty$-bounded nonlinearity, whose Lipschitz constant is smaller than the distance from zero to the essential spectrum of the linear part. We consider both the nonresonant and resonant cases. Our approach is based on the Conley index theory applied to study invariant sets of the associated parabolic semiflow. Using properties of the Schrodinger semigroup with Kato-class potentials, which follow from its Feynman-Kac representation, we derive a priori estimates for bounded solutions in the $L^\infty$ and $L^2$ norms, as well as regularity bounds in Sobolev spaces. As a consequence, we obtain conditions ensuring the existence of connecting orbits between stationary solutions, which in turn yield the existence of nontrivial standing waves.

It was recently shown by Yeliussizov \cite{Yeliussizov} that integer partitions in dimensions $d \geq 7$ asymptotically grow strictly faster than MacMahon numbers. As MacMahon numbers match with integer partitions in dimensions $d = 1,2$, the comparison of asymptotics of integer partitions with MacMahon numbers in intermediate dimensions ($d = 3,4,5,6$) is an open question. In this work, we perform Markov chain Monte Carlo (MCMC) simulations till $N=15000$ by using adaptive weight learning followed by conventional MCMC steps to numerically estimate the asymptotics of integer partitions in these intermediate dimensions. We numerically establish that in these intermediate dimensions, partitions asymptotically grow faster than MacMahon numbers. More specifically, assuming that the limits exist, we show: $\lim_{n\to\infty}n^{-3/4}\log p_3(n) = 1.8196 \pm 0.0019$, $\lim_{n\to\infty}n^{-4/5}\log p_4(n) = 1.7215 \pm 0.0045$, $\lim_{n\to\infty}n^{-5/6}\log p_5(n) = 1.6521 \pm 0.0059$, and $\lim_{n\to\infty}\log n^{-6/7}p_6(n) = 1.652 \pm 0.021$ for partitions in dimensions $d=3,4,5,$ and $6$ respectively. These numbers are all larger than MacMahon leading order asymptotic coefficients of $1.7898, 1.6614, 1.5737,$ and $1.509$ respectively. Additionally, we also find estimates for some of the sub-leading asymptotic terms in $\log p_d(n)$ in each of the dimensions.

We study general holographic fluctuation equations with apparent singularities, and the associated spectral functions. We consider the regime where an apparent singularity approaches the boundary, for which the near-boundary behavior of solutions can be determined via an inner/outer matching type of calculation. Depending on properties of the equation, we find that the merging of an apparent singularity with the boundary may generate either a zero or a pole for the corresponding spectral function. Our results also indicate that the recently introduced holographic product formula still applies in the presence of apparent singularities.

We derive an exact transport equation for the quantum Calogero model, i.e. inverse-square interacting bosons on a line, from microscopic first principles.

Physical decoherence can preserve the microscopic strategic neutrality (i.e. zero effective field and equal potential of the two homogeneous strategic orientations) of a quantum game while changing the thermodynamic regime of the corresponding interacting population. We demonstrate this for an Eisert–Wilkens–Lewenstein (EWL) Stag Hunt embedded as independent nearest neighbor encounters on a square lattice. For the restricted strategies $\mathsf{Q}=i\mathbb{Z}$ and $\mathsf{D}=i\mathbb{Y}$, noisy two-player payoff matrices are determined for phase damping, depolarization, and amplitude damping and mapped exactly to channel dependent Ising parameters $\mathfrak{J}(\Gamma,p)$ and $\mathfrak{H}(\Gamma,p)$. Phase damping and depolarization show the clearest contrast: they share the same microscopic neutrality branch $\mathfrak{H}=0$, while only depolarization suppresses the interaction as $(1-p)^2$. At $\beta=1$, this produces an exact depolarization-driven square-lattice critical point at $p_{*}\approx 0.233460\ldots$, whereas phase damping remains in the ordered coexistence regime along the same neutrality branch. Monte Carlo finite size scaling is consistent with two-dimensional Ising criticality and distinguishes field driven coexistence below $p_{*}$ from a smooth crossover above it. Amplitude damping additionally reveals a strong dependence on channel placement: the post-strategy neutrality branch reaches $\Gamma=0$ at $p=1/3$ and then disappears. Resource negativity further shows that microscopic two-qubit entanglement and collective interaction strength are distinct quantities. The resulting extended lattice remains an ordinary classical Ising system.

The exact N-soliton solutions are derived for the coupled negative first order Klein-Gordon (CNKG) equation subject to vanishing boundary conditions by using the inverse scattering method via Gelfand-Levitan-Marchenko equation. Based on the zero-curvature representation, the conservation laws, integrals of motion, and Hamiltonian structure of the aforementioned coupled nonlinear equations are constructed. The Jost functions and their analyticity properties for the Manakov spectral problem are recalled. The integral equations for the eigenfunctions are then used to formulate the Gel'fand-Levitan-Marchenko equations. Solving these equations establishes a direct correspondence between the kernel functions and the potential, yielding the general N-soliton expressions.

The late-time ringdown dynamics of near-extremal black holes (BHs) are expected to be dominated by zero-damped modes (ZDMs), whose decay rates are parametrically suppressed relative to those of ordinary quasinormal modes. In this paper, we demonstrate that exact WKB methods provide an exceptionally powerful framework for analyzing the ZDM spectrum of near-extremal Reissner–Nordström (RN) BHs. Focusing on massless, neutral scalar modes propagating on an RN background, we present the full Stokes geometry derived from the radial eigenvalue problem and establish the corresponding exact quantization condition (EQC). Our analytic computation of the Voros symbols entering the EQC achieves higher-order accuracy for the ZDM spectrum compared to previous studies and is systematically improvable. Ergo, this work serves as a proof of concept for investigations of ZDM spectra of other systems using exact WKB methods.

A standard approach for studying magic angle twisted bilayer graphene (MATBG)'s correlated electronic phase diagram is to project the Coulomb interaction down to effective models only involving electrons in single-particle flat bands and some nearby remote bands. We provide a novel systematic derivation of a single-particle continuum model of MATBG's single-particle properties which incorporates structural relaxation while remaining in the Lagrangian frame. We project the Coulomb interactions down to electrons occupying the flat bands of this model and compute the Hartree-Fock many-body ground states at fillings $\nu = \pm 2$. We find that incorporating relaxation effects drives the model into a semi-metallic phase at $- 2$ because of particle-hole asymmetry in the relaxed model's single-particle dispersion and because the flat band wavefunctions become more concentrated leading to an enhanced Hartree potential. Our results corroborate recent ab initio density functional theory studies which also found semi-metallic phases at $-2$. We discuss potential explanations for why such phases have not been seen in experiments.

We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.

We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert–Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman–Fredholm determinant of the centered interaction operator. We show that the Hilbert–Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert–Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.

We investigate the power-law decay of the survival probability of a Brownian particle diffusing in an infinite planar wedge whose two sides are partially reactive and described by Robin boundary conditions. We employ matched asymptotic analysis to relate the long-time asymptotics to a stationary harmonic Robin problem near the apex. This approach determines the persistence exponent for arbitrary opening angles and yields the prefactor explicitly for the family of wedges with $\alpha=\pi/n$ ($n=1,2,\ldots$). A simple extension of the apex prefactor to arbitrary angles is conjectured and supported numerically. This asymptotic behavior describes the crossover to the well-known result for perfectly absorbing wedges. As immediate applications, we also deduce the long-time behavior of the probability density function of the boundary local time on wedge sides, as well as the probability density function of the associated first-crossing time.

We study the classical $XY$-model on random geometric graphs $\mathcal{G}_{n, \varepsilon}$, which are obtained by sampling $n \in \mathbb{N}$ independent points in a finite domain $\Omega \subset \mathbb{R}^d$, $d \geq 2$, and connecting two points by and edge if their distance is of order $\varepsilon > 0$. We refer to $\mathcal{G}_{n, \varepsilon}$ as the random environment. Letting $\varepsilon \to 0$ as $n \to \infty$ at a sufficiently slow rate, these graphs capture the geometry of $\Omega$. Denoting the inverse temperature by $\beta$, we show that in the limit $\beta \to \infty$ at a rate depending on $n$ and $\varepsilon$, the $XY$-model on $\mathcal{G}_{n, \varepsilon}$ exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as $n \to \infty$. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp–Lieb inequality.

Tipping points—abrupt, potentially irreversible reorganizations of a system's statistical state—are commonly anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectra redden. This paradigm is powerful near simple equilibrium bifurcations but is not a general theory for stochastic, multistable, or metastable systems. We develop such a theory from the Ruelle–Pollicott (RP) spectrum of Kolmogorov generators of hypoelliptic Itô diffusions. Resolving the sensitivity of invariant statistics on RP spectral blocks shows that each contribution factorizes into a spectral denominator and a residue coupling the block to both the observable and perturbation direction. Small denominators permit large responses; residues produce them. Thus a closing RP gap is not sufficient for an early warning, while growing residues can generate the classical signature with no gap closure. An early-warning signal is therefore a property of a triple: RP block, observable, and perturbation direction. We demonstrate this on a stochastic non-normal system whose RP spectrum is exactly frozen, yet classical indicators become more alarming than during genuine gap closure. The RP decomposition attributes this to residue growth and yields an index $\mathcal{N}(f)$ satisfying $\mathcal{N}(f)\leq 1$ for reversible dynamics; hence $\mathcal{N}(f)>1$ certifies residue-driven amplification. For metastable systems, one killed problem yields two complementary spectral objects: the Doob $Q$-process isolates in-well recovery, while escape clocks and committor-weighted destination probabilities govern interwell transitions. In a one-dimensional fold, these scale as $(\epsilon_c-\epsilon)^{1/2}$ and $(\epsilon_c-\epsilon)^{3/2}$, separating bifurcation-induced from noise-induced tipping. Doob drift also connects to optimal Girsanov sampling.

This paper presents the closed-form Eshelby's tensor for an arbitrarily oriented polyhedral inclusion in a bimaterial domain under general uniform eigenstrain. Existing bimaterial solutions are mainly restricted to special inclusion shapes or dilatational eigenstrains, because the bimaterial Green's function contains two Boussinesq's displacement potentials in addition to the harmonic and biharmonic potentials in Kelvin's solution. This paper derives the missing domain integrals of the two Boussinesq's potentials by reducing the volume integrals to surface and elementary line integrals. The formulae provide the complete elastic and thermoelastic bimaterial Eshelby's tensors, which are verified against analytical solutions for spherical and cuboidal inclusions parallel to the bimaterial interface, and finite element results of an inclined cuboid. Singularity analysis demonstrates that the interface-related contribution remains regular when the inclusion is separated from the bimaterial interface, while additional logarithmic singularities arise when an edge or vertex touches the interface without increasing the dominant singularity order.