Papers

There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools – spectral analysis and mean first-passage time (MFPT) analysis – and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.

We study the transverse field Ising model, defined by the Hamiltonian $H =\frac{1}{2}\sum_{i, j\in [n]} J_{ij} Z_i Z_j +\sum_{i=1}^n h_i^z Z_i + \eta\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $\eta$ is the transverse field strength. Let $\Delta(J)=\lambda_{\max}(J)-\lambda_{\min}(J)$ be the spectral width of $J.$ When the inverse temperature $\beta\geq0$ satisfies $\Delta(J)\cdot\frac{\tanh(\beta\eta)}{\eta}\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(\beta)=\operatorname{Tr}(e^{-\beta H})$ to a given relative error $\epsilon\in(0,1)$ in time polynomial in $n$, $\beta$, the model parameters, and $\epsilon^{-1}$. When $ \Delta(J) \cdot \frac{\tanh(\beta \eta)}{\eta} > 1 ,$ we show that approximating $ Z(\beta)$ within an $\exp(o(n))$-multiplicative factor is $\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $\Delta (J)\cdot \frac{\tanh(\beta \eta)}{\eta}\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ \rho_\beta = \frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \in \{I, X\}^{\otimes n}$, the algorithm further achieves arbitrarily small relative error.

The critical Ising model admits several natural scaling limits. For an Ising model on a lattice approximation $D^a$ of a planar domain $D$, the rescaled spin field converges, as $a\to0$, to the critical Ising magnetization field (IMF), a rough random distribution on $D$, while the spin interfaces converge to a nested $CLE_3$ and the FK–Ising representation converges to a colored $CLE_{16/3}$. In this work, we prove that the IMF can be directly reconstructed from the nested $CLE_3$. This complements the previously known construction of the IMF from the colored $CLE_{16/3}$ [CGN15]. Our result also implies that %We further show that if one constructs a colored $CLE_{16/3}$ from the nested $CLE_3$ through CLE percolation [MSW17], the resulting construction recovers the same magnetization field. A main difficulty in this present reconstruction comes from the fact that the renormalization exponent of the IMF, $15/8$, is smaller than the Hausdorff dimension of the $CLE_3$ carpet, $187/96$. Consequently, the field cannot be recovered by a direct Minkowski-content construction from the nested $CLE_3$ as is the case for $CLE_{16/3}$. We conjecture that the converse measurability also holds, namely that the nested $CLE_3$ can itself be reconstructed from the IMF.

A new method of analyzing grid, or aliasing, instabilities is described, in particle-in-cell (PIC) codes for plasma kinetic theory. It starts with a meshfree approach with $N_{p}$ macroparticles, with $N_{p}<\infty$. Linearization of these equations is done about a uniform density non-drifting equilibrium prescribed by macroparticles on a uniform lattice. Macroparticle positions perturbations are induced about this lattice. The resulting linear equations are analyzed for stability and dispersion relations. The linearized equations are then discretized on a grid of $N_{g}$ points with $N_{g} \le N_{p}$, in both momentum conserving (MCP) and energy conserving (ECP) formulations. For a cold stationary plasma, this leads a dynamical matrix. For ECP, the matrix is symmetric and positive definite (SPD) and immediately shows stability. For MCP, the dynamical matrix is neither symmetric nor positive definite, preventing immediate conclusions on stability. For both MCP and ECP and for a cold stationary plasma, the resulting matrix elements vary relative to the meshfree value, a result of aliasing. This deviation is periodic in the displacement of the particle lattice relative to the grid. In the MCP discretization, eigenvalues occur in complex conjugate pairs for general placement of the particle lattice relative to the grid. These conjugate pairs indicate nonnegative growth rates. The nature of the aliasing is studied and shown to be related to the trapezoidal rule error over the grid. The scaling of the linear growth rates with respect to $N_{g}$ and especially with respect to $N_{ppc}$ is studied. These analytical results are compared with PIC simulations and found to be in excellent agreement. The connection with a cold drifting beam is discussed briefly, specifically, grid-induced instabilities for both the MCP and ECP.

Let $G$ be a simply connected semisimple algebraic group. On the (undecorated) braid variety associated with a power of a Coxeter element, there is a finite-order autoequivalence, called the Zamolodchikov transformation, induced by cyclic rotation of words. In this paper, we show that the coarse moduli space of the fixed-point stack of this autoequivalence is isomorphic to the spectrum of the fusion ring of the corresponding affine Lie algebra. The above results also extend to twisted Coxeter elements, and the fusion rings of all affine Lie algebras are obtained in this way.

Knill-Laflamme coefficients characterize exact quantum error detection through scalar compressions of error operators to the code space. For prescribed Pauli observables and a fixed code dimension, attainable coefficient vectors form a joint higher-rank numerical range, whose Euclidean norm image is the signature spectrum. We study how operator representation, code dimension, and detection constraints govern this geometry. Our main theorem identifies representation multiplicity as sufficient to lift active expectation data to exact scalar compressions: a lift exists whenever each occupied block has multiplicity at least the code dimension times the rank of the corresponding active state. When multiplicity suffices to realize all active states, the coefficient range is the full active expectation body and is therefore convex and connected, with a closed interval as its signature spectrum. This framework unifies commuting Pauli families, where sufficient common-eigenspace degeneracy yields polytopes, and subsystem stabilizer codes, where protected logical subsystems supply multiplicity spaces for noncommuting gauge observables. Explicit examples show how increasing the code dimension can make a range shrink, collapse, or become empty, while increasing multiplicity can turn a Bloch sphere into a filled ball. In the subsystem setting, parent-Hamiltonian constructions realize exact detecting codes as degenerate ground spaces, selecting distinguished vectors and continuous paths within these ranges. A final three-qubit example shows additional noncommuting detection constraints reduce a tetrahedral range to its barycenter and four vertices, producing a disconnected signature spectrum. The results provide a structural framework for Knill-Laflamme coefficient geometry and motivate sharper criteria for connected coefficient ranges and interval signature spectra under structured error models.

Many quantum systems are believed to thermalize slowly at low temperatures due to the existence of metastable phases. However, there are cases where further cooling can restore polynomial-time thermalization: we demonstrate that the Davies dynamics of the mean-field interchange model with local dimension $d\geq 3$ and single-site generalized Pauli couplings exhibits this behavior. The dynamical phase diagram is beholden to two distinguished inverse temperatures, $\beta_{\mathrm{min}}(d)$ and $\beta=d$, with the static first-order phase transition lying between them. At low and high temperatures surrounding the interval $(\beta_{\mathrm{min}}(d),d)$, the spectral gap of the Davies generator is bounded below by an inverse polynomial in system size $n$. Meanwhile, in this interval, the gap vanishes exponentially in $n$, indicating metastability. The proof uses Schur-Weyl duality to decompose the Davies generator into a classical Markov generator on Young diagrams, and the restriction to the remaining symmetry sectors whose gap is $\Omega(n^{-1})$. Thus, exponential slowdown arises entirely from the Young diagram chain.

In this paper, we present an algorithm for constructing all Calabi-Yau orbifolds and their mirror pairs associated with an arbitrary Berglund-Hubsch-type model, and prove its correctness. In particular, the proposed method yields explicit generators for the groups defining the corresponding orbifolds. The method for classifying maximal admissible groups is proposed. We illustrate the application of the classification approach to an arbitrary Fermat-type Calabi-Yau manifold. In addition, as an illustrative example, we classify all maximal admissible groups of Fermat-type Calabi-Yau threefolds.

We prove a generalization of Grauert's higher coherence theorem for a class of curved differential graded (non-commutative) Fréchet algebras. This allows us to extend the Fourier-Mukai calculus to derived categories arising in many new contexts. We then apply it and prove equivalence of derived categories of dual generalized complex tori using a non-commutative version of the Poincaré line bundle. This lays the foundation for categories of generalized complex branes on generalized complex tori. Examples include complex tori, symplectic tori as well as their non-commutative and B-field deformations.

We propose a novel approach to state preparation by embracing the full power of open quantum systems. Instead of working with the whole unitary group in $n$ qubits, we identify a small Lie group $G$ in $(n+3)$ qubits. The dimension of its Lie algebra is $\text{poly}(n)$. Every state can be approximated starting from any fixed input state by repeatedly using partial trace and state preparation in the environment alongside group operations. Moreover, we also identify a single (open system magic) interaction Hamiltonian whose unitary group can be combined with open system quantum technology mentioned above to achieve transitivity on the space of all densities. Similarly, we show that all channels on $n$ qubits can be approximated by using a larger environment, achieving channel universality. The interesting small groups we identify are derived from Lie algebras of local Pauli strings and lead to a new landscape of cheap and expensive states, densities, and channels.

We study classical and quantum superintegrability of a resonant three-dimensional Hamiltonian with indefinite kinetic energy and three Poisson descriptions of the same flow. Besides three commuting quadratic Hamiltonians, the polynomial first-integral ring contains two primitive degree-four generators, $I_4^{\rm o}$ and $I_4^{\rm e}$, of odd and even momentum parity. A single octic relation leaves the polynomial invariant ring with transcendence degree four. Solving the complete first-integral equation shows that every rational first integral is a rational function of $H_1,H_2,H_3,I_4^{\rm o}$, so that the rational invariant field likewise has generic functional rank four. A branch-free transcendental invariant raises the generic smooth functional rank to five on the invariant open set $\Delta>0$. The system is therefore minimally superintegrable in the polynomial and rational classes and maximally superintegrable on this invariant domain when transcendental integrals are admitted. Weyl quantisation yields exact higher-order differential symmetries in all three Poisson realisations. The branch-free fifth classical invariant also admits a densely defined nonlocal quantum counterpart, providing a concrete quantum realisation of the transcendental completion.

Analogue and digital Hamiltonian simulation represent two distinct approaches to making one quantum system replicate the physical properties of another, with consequences ranging from practical applications to complexity theory and quantum gravity. They are rooted in two distinct regimes of quantum dynamics: time-independent Hamiltonians for the former; fully controllable time-dependent quantum circuits in the latter. Time-periodic Hamiltonians represent an intermediate regime. In this work, we establish a theory of Floquet simulation where we use periodically driven Hamiltonians to synthesise time-independent ones. We show that a set of interactions $S$ can Floquet-simulate every Hamiltonian in the Lie algebra $\mathrm{Lie}(S)$, and consequently we give a complete and constructive characterisation of Floquet-universal Hamiltonians that are able to produce any target Hamiltonian. Notably, our construction uses only $O(1)$ local interaction strengths and ratios thereof, avoiding the impractical multi-scale local interaction strengths often required in analogue simulation via time-independent Hamiltonians. Instead, the construction requires a range of driving frequencies which scales polynomially with system size for important classes such as $k$-local lattice Hamiltonians. This has immediate complexity-theoretic implications, including numerous BQP-completeness and QMA-hardness results for natural physical properties and quantities studied in Floquet physics.

We study the focusing one-dimensional cubic nonlinear Schrödinger / Gross–Pitaevskii equation (NLS-GP), where the potential of the underlying linear Schrödinger operator, $H_V=-\partial_x^2+V(x)$, is compactly supported. In \cite{turner2026resonance} the authors proved that purely imaginary zeros (transmission resonances) or purely imaginary poles in the lower half plane (scattering resonances) of a reflection coefficient of $H_V$ seed branches of resonance-induced nonlinear bound states, and that these states bifurcate at a strictly positive $L^2(\mathbb R)$ excitation threshold. This is in contrast to nonlinear bound states which bifurcate at zero $L^2$-norm from point spectra of $H_V$ (corresponding to poles in the upper half plane). In this paper we establish precise criteria for the nonlinear orbital stability and instability of the resonance-induced states near the bifurcation point. A corollary is that resonance-induced nonlinear states are orbitally stable if i) the underlying linear (scattering or transmission) resonance frequency is sufficiently small, and ii) the corresponding linear resonance eigenmode is strictly positive. Numerical simulations are presented to explore regimes not accessible to our theory, and to explore the large time dynamics for initial conditions near stable and unstable resonance-induced nonlinear bound states.

We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an $sl(2)$ algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra $U(sl(3))$. We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the $sl(2)$ algebraization after parabolic separation, and the $sl(3)$ algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common $sl(3)$ structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.

A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order \(D^{-1/2}\) for \(D\)-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a \(k\)-copy Gaussian state that is \(n\)-exchangeable under graded fermionic copy permutations is \(O(k/n)\)-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as \(O(r^{-1})\), and hence, by recoverability, the state admits an \(O(r^{-1/2})\) trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic \(h\)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

In an earlier paper of P. Rossi and the first author, a construction of a wide class of quantum hierarchies was presented, with the input data given by an arbitrary cohomological field theory. A crucial role in the construction is played by the double ramification (DR) cycles on the moduli spaces of stable curves. As a special case, one gets a quantization of the hierarchy of the Intermediate Long Wave (ILW) equation. The ILW equation has two famous limits: the Korteweg–de Vries equation and the Benjamin–Ono (BO) equation, and for the latter one a quantum hierarchy was constructed by Nazarov and Sklyanin. In our paper, we propose a conjecture describing a relation between a limit of the quantum ILW hierarchy and the Nazarov–Sklyanin hierarchy and present various checks to support it.

Closed bosonic lattice systems do not generally admit operator norm Lieb–Robinson bounds uniformly in the initial state. The reason is that the information propagation velocity can grow with the local boson occupancy, which can be macroscopically large. Here, we show that local dissipation restores an operator-norm Lieb–Robinson bound. We consider the dissipative Bose–Hubbard model described by a Lindbladian operator with on-site $\ell$-photon loss, $\ell>2$ and, for our main result, we establish an almost-ballistic Lieb-Robinson bound. Dissipation rapidly depletes highly occupied sites, thus regularizing the state on the Sobolev-type scale of local particle moments. The resulting moment bounds diverge as time approaches zero but remain integrable near zero at sufficiently low orders. We extend these ideas to treat cat-code dissipation and, for initial states in the code space, we prove a local adiabatic approximation uniform in the total volume. Further consequences include local channel approximation, a thermodynamic limit, and efficient digital quantum simulation. The point is that these applications are now available uniformly in the input state, as for quantum spin systems, but in contrast to closed Bose–Hubbard systems.

We construct the Buchholz algebra without a Fock representation, as the bounded inverse limit of intrinsic particle-cutoff quotients of the gauge-fixed resolvent algebra, with isometric particle-degree coordinates. Its finite matrix corners carry canonical traces, KMS states, and trace-norm product formulas, together with closed-path expansions that lay the groundwork for path-integral representations. The known Fock-space description and a proper quasilocal subalgebra arise as realizations.

We develop a conservative micro-continuum-cellular automaton method for simulating multispecies biofilm dynamics in complex flows. The proposed method couples the Darcy-Brinkman-Stokes equations, reactive transport, suspended bacteria, and biofilm dynamics with a two-stage cellular automaton algorithm for biofilm redistribution and interface evolution. While treating biofilms as evolving porous media, we ensure conservative redistribution of multispecies biomass across partially occupied cells. Donor and recipient cell volumes are explicitly accounted for to conserve biomass and preserve species composition on non-uniform meshes. The proposed method is assessed against diffusion-dominated benchmark cases, including single-species fingering and multispecies stratification, and is further evaluated through a mesh-convergence study for flow and growth over a rectangular bump. The framework is then applied to counter-diffusional biofilms in a membrane-aerated biofilm reactor as a canonical example. The results demonstrate that the framework captures the expected biofilm morphology and stratification in systems involving coupled flow, substrate transport, and biofilm dynamics. The proposed method provides a flexible computational approach for simulating multispecies biofilm dynamics in complex flows and geometries.

We prove a factorization theorem for ordinary topological recursion near a regular nodal degeneration of a spectral curve, allowing logarithmic spectral coordinates. The partition function for closed genera $g\ge2$ factors into the partition function of the normalization, a universal Gaussian vacuum, and the exponential of a connected graph sum. This graph sum has strictly positive order in the vanishing period, and thus the factorization recovers the conifold gap, identifies the term of degree zero in the vanishing period with the free energy of the normalization, and gives finite graph formulas for the coefficients of positive powers of the period. At fixed genus, the regular series converges jointly in the period and the parameters along the nodal locus. The Gaussian neck tensors can be expressed in terms of relative Gromov–Witten invariants of a parametrized $\mathbb P^1$. As examples, we use remodeling to obtain Gromov–Witten factorizations for local $\mathbb P^2$ and local $\mathbb F_0$ in the conifold frame, with backgrounds $\mathbb C^3$ and the resolved conifold, respectively. The analogous local $\mathbb F_1$ factorization has background $\mathcal O(1)\oplus\mathcal O(-3)$ over $\mathbb P^1$.

We consider the semiclassical magnetic Schrödinger operator in dimensions one and two. The first result is that an even one dimensional semiclassical potential is spectrally determined by the Schrödinger operator. The second result states that a two dimensional radial magnetic field is spectrally determined by a purely magnetic Schrödinger operator. This extends the earlier work of Guillemin-Wang and answers a conjecture of theirs therein.

The Sun's atmosphere is significantly influenced by events that often measure only tens of kilometres in the horizontal direction. They happen in both quiet and active regions and are determined by magneto-hydro-dynamic processes. Their origins often lie in the lowest layers of the solar atmosphere. To improve the understanding of chromospheric and coronal heating, the solar dynamo and its activity cycle, or to improve the prediction of space weather events we must therefore enhance our understanding of such small-scale events. In this work I first describe the process of calibrating spectropolarimetric observations with very high spatial resolution and put together the building blocks for the mathematical proof the MOMFBD image reconstruction algorithm. In the second part I describe the FISS-SP prototype which is designed to fully exploit the resolution power of a 1.6m telescope. This instrument is built to provide suitable data for image reconstruction of spectropolarimetric data. The resulting data can resolve small-scale features down to the diffraction limit. The simultaneous interpretation of a few lines is still the norm for spatially highly resolved solar spectra. Theoretical works have shown that the simultaneous interpretation of multiple solar lines can greatly improve the robustness against noise. Using observations from FISS-SP I advance the theoretical idea into a practical technique. With this I can retrieve a coherent model of the local solar atmosphere, resolving structures down to a horizontal size of about 49~km. Finally, I use those new techniques to interpret an observation of a solar pore. In this last part I describe and analyse small-scale structures, which could not be spectropolarimetrically resolved before. Typical pore-boundary features - namely, dark striations with bright grains moving into the pore - are identified as convective features.

We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in $\mathcal{O}\left(\log (n/\varepsilon)\right)$ time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order $1$. We further provide $\mathcal{O}(\text{polylog}(n/\varepsilon))$ depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.

We study the wetting transition for a two-dimensional Solid-On-Solid (SOS) interface above a hard wall in the presence of quenched disorder. The interface is represented by a nonnegative integer-valued height function $\phi$, with Hamiltonian $\mathcal H^\omega(\phi)=\beta\sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_x(h+\alpha\omega_x-\lambda(\alpha))\mathbf 1_{\{\phi(x)=0\}}$, where $(\omega_x)_{x\in\mathbb Z^2}$ is an i.i.d. centered field, $\alpha\ge0$ is the disorder strength, and $\lambda(\alpha)=\log\mathbb E[e^{\alpha\omega_0}]$. This normalization is chosen so that the annealed model coincides with the corresponding homogeneous wetting model. As the wall attraction $h$ increases, the interface undergoes a transition from a delocalized phase, in which contacts with the wall have vanishing density, to a localized phase with a positive density of contacts. For sufficiently large $\beta$, the homogeneous wetting point and its sharp near-critical free-energy behavior are known. We prove that, for every fixed $\alpha\ge0$, the quenched critical point coincides with the homogeneous wetting point, $h_c(\beta,\alpha)=h_w(\beta)=-\log(1-e^{-4\beta})$. Moreover, writing $u=h-h_w(\beta)$, we show that the quenched excess free energy satisfies $\overline F_\beta(\alpha,u)=\overline F_{\mathrm{homo}}(\beta,u)+o(u^3)$ as $u\downarrow0$, where $\overline F_{\mathrm{homo}}(\beta,u)$ denotes the leading homogeneous wetting asymptotic and satisfies $\overline F_{\mathrm{homo}}(\beta,u)\asymp u^3$. Thus, at the level of both the critical point and the leading near-critical free-energy asymptotics, the quenched model has the same behavior as the homogeneous SOS wetting model. This contrasts with the disordered SOS pinning model, in which quenched disorder leaves the critical point unchanged but modifies the leading critical behavior of the free energy.

We prove the Gromov-Hausdorff convergence of the state spaces, equipped with Connes' distance function, of the operator systems associated with spectral truncations for noncommutative $n$-tori, $n\ge 2$. The action of the $n$-torus on the noncommutative $n$-torus, together with an appropriate use of Bochner integrals, enables us to construct an $C^1$-approximate order isomorphism that establishes the convergence. The earlier convergence result for the classical case of tori [20] follows as a special case of our result. We conclude by studying the $C^*$-envelopes and propagation numbers of the truncated operator systems and show that the structural properties of the truncated operator systems for tori persist under deformation.

We study the spectral properties of the harmonic modification of the quantum mechanical Kronig-Penney model motivated by the theory of the dressing chain. In particular, we show that in the special cases the corresponding ground states can be given explicitly and all spectral gaps have the same length with the ends expressed as zeros of the corresponding Weber-Hermite functions.

We study endpoint mapping properties of low-energy wave operators for Schrödinger operators $H=-\Delta+V$ on $\mathbb R^n$, $n\ge3$. In dimension four, we prove that a zero-energy resonance prevents $L^1$ boundedness, whether or not zero is also an eigenvalue, under $|V(x)|\lesssim\langle x\rangle^{-\beta}$ with $\beta>10$. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy $L^p$ classification in every dimension $n\ge3$ under $\beta>n+4$. In particular, $L^\infty$ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of $V\psi$ for every zero-energy eigenfunction $\psi$. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.

A 35 input/single output quantum phase neural network (QPNN) is proposed to recognize the English letters' A 'and' B 'composed 7*5 of pixels. Each node of the proposed QPNN has the input with 0 or 1. Through normalization and quantum phase encoding, each set of 35 with 0 or 1 digital inputs is encoded into a quantum superposition state described by 35 quantum phases. Then, a quantum rotation gate is used to introduce adjustable phase, and a controlled NOT gate is used to entangle the relative phase between the two inputs. The relationship between the input and output of the network is derived. This paper also derived the phase weight learning training algorithm with adaptive learning rate. The solutions with the recognition probability of 1 is designed, and the analytical expressions for all adjustable phase solutions with zero errors are not unique. The experimental performance is verified on the Qiskit platform. This paper uses phase drive to convert the probability calculation into an analytical polynomial function with adjustable rotation angles in the network, avoiding the bottleneck of generating all 2^35 complex exponential amplitudes and providing an effective way to solve the "exponential wall" problem, which provides a new implementation solution for the practical application of multi-input recognition problems.

SIREN (Ship Inviscid Resistance and Elevation eNgine) is a Rankine-source panel solver for the steady wave resistance of a ship, in the tradition of Dawson's method. The free-surface condition is linearised about a base flow, and the linearisation is iterated. In the present formulation the base flow is the solution of a Neumann problem on the actual boundary: sources on the entire hull, wet and dry, and on the free surface panels, with impermeability imposed on both, so that the free surface acts as a rigid lid at the position it currently occupies. This replaces the classical double-body flow, whose image about z=0 is meaningful only for a flat surface and a hull cut at z=0, and it removes the need for desingularised free-surface panels. In the non-linear iteration the free-surface panels are moved rigidly in the vertical to the relaxed wave elevation at their centroid, the hull panels are marked wet or dry against the moved surface, and the base flow is recomputed on the moved geometry; impermeability applies to the whole hull regardless of this marking, which is used only to select which panels enter the force integral, and under-relaxation is confined to the passes where the non-linear correction accumulates without being reset. The radiation condition is imposed by a one-row upstream shift of the collocation points, and the wave resistance is obtained by pressure integration and, independently, by transverse wave-cut analysis.